List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

Counts

(sequence A000001 in the OEIS)

Total number of nonisomorphic groups by order

+	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	0	1	1	1	2	1	2	1	5	2	2	1	5	1	2	1
16	14	1	5	1	5	2	2	1	15	2	2	5	4	1	4	1
32	51	1	2	1	14	1	2	2	14	1	6	1	4	2	2	1
48	52	2	5	1	5	1	15	2	13	2	2	1	13	1	2	4
64	267	1	4	1	5	1	4	1	50	1	2	3	4	1	6	1
80	52	15	2	1	15	1	2	1	12	1	10	1	4	2	2	1
96	231	1	5	2	16	1	4	1	14	2	2	1	45	1	6	2
112	43	1	6	1	5	4	2	1	47	2	2	1	4	5	16	1
128	2328	2	4	1	10	1	2	5	15	1	4	1	11	1	2	1

For labeled groups, see  A034383.

Glossary

Each group is named by their Small Groups library index as G_oⁱ, where o is the order of the group, and i is the index of the group within that order.

Common group names:

• Z_n: the cyclic group of order n (the notation C_n is also used; it is isomorphic to the additive group of Z/nZ).
• Dih_n: the dihedral group of order 2n (often the notation D_n or D_2n is used )
  • K₄: the Klein four-group of order 4, same as Z₂ × Z₂ or Dih₂.
• S_n: the symmetric group of degree n, containing the n! permutations of n elements.
• A_n: the alternating group of degree n, containing the even permutations of n elements, of order 1 for for n = 1, and order n!/2 otherwise.
• Dic_n or Q_4n: the dicyclic group of order 4n.
  • Q₈: the quaternion group of order 8, also Dic₂.

The notations Z_n and Dih_n have the advantage that point groups in three dimensions C_n and D_n do not have the same notation. There are more isometry groups than these two, of the same abstract group type.

The notation G × H denotes the direct product of the two groups; Gⁿ denotes the direct product of a group with itself n times. G ⋊ H denotes a semidirect product where H acts on G; this may also depend on the choice of action of H on G

Abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups Z_n, for prime n.) The equality sign ("=") denotes isomorphism.

The identity element in the cycle graphs is represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16.

In the lists of subgroups, the trivial group and the group itself are not listed. Where there are multiple isomorphic subgroups, their number is indicated in parentheses.

List of small abelian groups

The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups.

(sequence A000688 in the OEIS)

Number of nonisomorphic abelian groups by order

+	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	0	1	1	1	2	1	1	1	3	2	1	1	2	1	1	1
16	5	1	2	1	2	1	1	1	3	2	1	3	2	1	1	1
32	7	1	1	1	4	1	1	1	3	1	1	1	2	2	1	1
48	5	2	2	1	2	1	3	1	3	1	1	1	2	1	1	2
64	11	1	1	1	2	1	1	1	6	1	1	2	2	1	1	1
80	5	5	1	1	2	1	1	1	3	1	2	1	2	1	1	1
96	7	1	2	2	4	1	1	1	3	1	1	1	6	1	1	1
112	5	1	1	1	2	2	1	1	3	2	1	1	2	3	2	1
128	15	1	1	1	2	1	1	3	3	1	1	1	2	1	1	1

For labeled Abelian groups, see  A034382.

List of all abelian groups up to order 30

Order	ID	Go^i	Group	Nontrivial proper Subgroups	Cycle graph	Properties
1 [1]	1	G1^1	Z1[2] = S1 = A2	–		Trivial. Cyclic. Alternating. Symmetric. Elementary.
2 [3]	2	G2^1	Z2[4] = S2 = Dih1	–		Simple. Symmetric. Cyclic. Elementary. (Smallest non-trivial group.)
3 [5]	3	G3^1	Z3[6] = A3	–		Simple. Alternating. Cyclic. Elementary.
4[7]	4	G4^1	Z4[8] = Dic1	Z2		Cyclic.
	5	G4^2	Z2^2 = K4[9] = Dih2	Z2 (3)		Elementary. Product. (Klein four-group. The smallest non-cyclic group.)
5[10]	6	G5^1	Z5[11]	–		Simple. Cyclic. Elementary.
6[12]	8	G6^2	Z6[13] = Z3 × Z2[14]	Z3, Z2		Cyclic. Product.
7[15]	9	G7^1	Z7[16]	–		Simple. Cyclic. Elementary.
8[17]	10	G8^1	Z8[18]	Z4, Z2		Cyclic.
	11	G8^2	Z4 × Z2[19]	Z2^2, Z4 (2), Z2 (3)		Product.
	14	G8^5	Z2^3[20]	Z2^2 (7), Z2 (7)		Product. Elementary. (The non-identity elements correspond to the points in the Fano plane, the Z2 × Z2 subgroups to the lines.)
9[21]	15	G9^1	Z9[22]	Z3		Cyclic.
	16	G9^2	Z3^2[23]	Z3 (4)		Elementary. Product.
10[24]	18	G10^2	Z10[25] = Z5 × Z2	Z5, Z2		Cyclic. Product.
11	19	G11^1	Z11[26]	–		Simple. Cyclic. Elementary.
12[27]	21	G12^2	Z12[28] = Z4 × Z3	Z6, Z4, Z3, Z2		Cyclic. Product.
	24	G12^5	Z6 × Z2[29] = Z3 × Z2^2	Z6 (3), Z3, Z2 (3), Z2^2		Product.
13	25	G13^1	Z13[30]	–		Simple. Cyclic. Elementary.
14[31]	27	G14^2	Z14[32] = Z7 × Z2	Z7, Z2		Cyclic. Product.
15[33]	28	G15^1	Z15[34] = Z5 × Z3	Z5, Z3		Cyclic. Product.
16[35]	29	G16^1	Z16[36]	Z8, Z4, Z2		Cyclic.
	30	G16^2	Z4^2[37]	Z2 (3), Z4 (6), Z2^2, Z4 × Z2 (3)		Product.
	33	G16^5	Z8 × Z2[38]	Z2 (3), Z4 (2), Z2^2, Z8 (2), Z4 × Z2		Product.
	38	G16^10	Z4 × Z2^2[39]	Z2 (7), Z4 (4), Z2^2 (7), Z2^3, Z4 × Z2 (6)		Product.
	42	G16^14	Z2^4[19] = K4^2	Z2 (15), Z2^2 (35), Z2^3 (15)		Product. Elementary.
17	43	G17^1	Z17[40]	–		Simple. Cyclic. Elementary.
18[41]	45	G18^2	Z18[42] = Z9 × Z2	Z9, Z6, Z3, Z2		Cyclic. Product.
	48	G18^5	Z6 × Z3[43] = Z3^2 × Z2	Z6, Z3, Z2		Product.
19	49	G19^1	Z19[44]	–		Simple. Cyclic. Elementary.
20[45]	51	G20^2	Z20[46] = Z5 × Z4	Z20, Z10, Z5, Z4, Z2		Cyclic. Product.
	54	G20^5	Z10 × Z2[47] = Z5 × Z2^2	Z5, Z2		Product.
21	56	G21^2	Z21[48] = Z7 × Z3	Z7, Z3		Cyclic. Product.
22	58	G22^2	Z22[49] = Z11 × Z2	Z11, Z2		Cyclic. Product.
23	59	G23^1	Z23[50]	–		Simple. Cyclic. Elementary.
24[51]	61	G24^2	Z24[52] = Z8 × Z3	Z12, Z8, Z6, Z4, Z3, Z2		Cyclic. Product.
	68	G24^9	Z12 × Z2[53] = Z6 × Z4 = Z4 × Z3 × Z2	Z12, Z6, Z4, Z3, Z2		Product.
	74	G24^15	Z6 × Z2^2[39] = Z3 × Z2^3	Z6, Z3, Z2		Product.
25	75	G25^1	Z25	Z5		Cyclic.
	76	G25^2	Z5^2	Z5		Product. Elementary.
26	78	G26^2	Z26 = Z13 × Z2	Z13, Z2		Cyclic. Product.
27[54]	79	G27^1	Z27	Z9, Z3		Cyclic.
	80	G27^2	Z9 × Z3	Z9, Z3		Product.
	83	G27^5	Z3^3	Z3		Product. Elementary.
28	85	G28^2	Z28 = Z7 × Z4	Z14, Z7, Z4, Z2		Cyclic. Product.
	87	G28^4	Z14 × Z2 = Z7 × Z2^2	Z14, Z7, Z4, Z2		Product.
29	88	G29^1	Z29	–		Simple. Cyclic. Elementary.
30[55]	92	G30^4	Z30 = Z15 × Z2 = Z10 × Z3 = Z6 × Z5 = Z5 × Z3 × Z2	Z15, Z10, Z6, Z5, Z3, Z2		Cyclic. Product.

List of small non-abelian groups

(sequence A060689 in the OEIS)

Number of nonisomorphic nonabelian groups by order

+	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0	0	0	0	0	0	0	1	0	2	0	1	0	3	0	1	0
16	9	0	3	0	3	1	1	0	12	0	1	2	2	0	3	0
32	44	0	1	0	10	0	1	1	11	0	5	0	2	0	1	0
48	47	0	3	0	3	0	12	1	10	1	1	0	11	0	1	2
64	256	0	3	0	3	0	3	0	44	0	1	1	2	0	5	0
80	47	10	1	0	13	0	1	0	9	0	8	0	2	1	1	0
96	224	0	3	0	12	0	3	0	11	1	1	0	39	0	5	1
112	38	0	5	0	3	2	1	0	44	0	1	0	2	2	14	0
128	2313	1	3	0	8	0	1	2	12	0	3	0	9	0	1	0

Order of non-abelian groups are

6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120, 122, 124, 125, 126, 128, 129, 130, 132, 134, 135, 136, 138, 140, 142, ... (sequence A060652 in the OEIS)

List of all nonabelian groups up to order 30

Order	ID	Go^i	Group	Nontrivial proper Subgroups	Cycle graph	Properties
6[12]	7	G6^1	Dih3 = S3[56] = D6	Z3, Z2 (3)		Dihedral group, the smallest non-abelian group, symmetric group, Frobenius group
8[17]	12	G8^3	Dih4 = D8[57]	Z4, Z2^2 (2), Z2 (5)		Dihedral group. Extraspecial group. Nilpotent.
	13	G8^4	Q8[58] = Dic2 = <2,2,2>	Z4 (3), Z2		Quaternion group, Hamiltonian group. all subgroups are normal without the group being abelian. The smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G. Extraspecial group Binary dihedral group. Nilpotent.
10[24]	17	G10^1	Dih5 = D10[59]	Z5, Z2 (5)		Dihedral group, Frobenius group
12[27]	20	G12^1	Q12[60] = Dic3 = <3,2,2> = Z3 ⋊ Z4	Z2, Z3, Z4 (3), Z6		Binary dihedral group
	22	G12^3	A4[61]	Z2^2, Z3 (4), Z2 (3)		Alternating group. No subgroups of order 6, although 6 divides its order. Frobenius group
	23	G12^4	Dih6 = D12[62] = Dih3 × Z2	Z6, Dih3 (2), Z2^2 (3), Z3, Z2 (7)		Dihedral group, product
14[31]	26	G14^1	Dih7	Z7, Z2 (7)		Dihedral group, Frobenius group
16[35][63]	31	G16^3	G4,4 = K4 ⋊ Z4 (Z4×Z2) ⋊ Z2	E8, Z4 × Z2 (2), Z4 (4), K4 (6), Z2 (6)		Has the same number of elements of every order as the Pauli group. Nilpotent.
	32	G16^4	Z4 ⋊ Z4			The squares of elements do not form a subgroup. Has the same number of elements of every order as Q8 × Z2. Nilpotent.
	34	G16^6	Z8 ⋊ Z2			Sometimes called the modular group of order 16, though this is misleading as abelian groups and Q8 × Z2 are also modular. Nilpotent.
	35	G16^7	Dih8	Z8, Dih4 (2), Z2^2 (4), Z4, Z2 (9)		Dihedral group. Nilpotent.
	36	G16^8	QD16			The order 16 quasidihedral group. Nilpotent.
	37	G16^9	Q16 = Dic4 = <4,2,2>			generalized quaternion group, binary dihedral group. Nilpotent.
	39	G16^11	Dih4 × Z2	Dih4 (2), Z4 × Z2, Z2^3 (2), Z2^2 (11), Z4 (2), Z2 (11)		Product. Nilpotent.
	40	G16^12	Q8 × Z2			Hamiltonian, product. Nilpotent.
	41	G16^13	(Z4 × Z2) ⋊ Z2			The Pauli group generated by the Pauli matrices. Nilpotent.
18[41]	44	G18^1	Dih9			Dihedral group, Frobenius group
	46	G18^3	S3×Z3			Product
	47	G18^4	(Z3 × Z3)⋊ Z2			Frobenius group
20[45]	50	G20^1	Q20 = Dic5 = <5,2,2>			Binary dihedral group
	52	G20^3	Z5 ⋊ Z4			Frobenius group
	53	G20^4	Dih10 = Dih5 × Z2			Dihedral group, product
21	55	G21^1	Z7 ⋊ Z3			Smallest non-abelian group of odd order. Frobenius group
22	57	G22^1	Dih11			Dihedral group, Frobenius group
24[51]	60	G24^1	Z3 ⋊ Z8			Central extension of S3
	62	G24^3	SL(2,3) = 2T = Q8 ⋊ Z3			Binary tetrahedral group
	63	G24^4	Q24 = Dic6 = <6,2,2> = Z3 ⋊ Q8			Binary dihedral
	64	G24^5	Z4 × S3			Product
	65	G24^6	Dih12			Dihedral group
	66	G24^7	Dic3 × Z2 = Z2 × (Z3 × Z4)			Product
	67	G24^8	(Z6 × Z2)⋊ Z2 = Z3 ⋊ Dih4			Double cover of dihedral group
	69	G24^10	Dih4 × Z3			Product. Nilpotent.
	70	G24^11	Q8 × Z3			Product. Nilpotent.
	71	G24^12	S4			Symmetric group. Has no normal Sylow subgroups.
	72	G24^13	A4 × Z2			Product
	73	G24^14	D12× Z2			Product
26	77	G26^1	Dih13			Dihedral group, Frobenius group
27[54]	81	G27^3	Z3^2 ⋊ Z3			All non-trivial elements have order 3. Extraspecial group. Nilpotent.
	82	G27^4	Z9 ⋊ Z3			Extraspecial group. Nilpotent.
28	84	G28^1	Z7 ⋊ Z4			Binary dihedral group
	86	G28^3	Dih14			Dihedral group, product
30[55]	89	G30^1	Z5 × S3			Product
	90	G30^2	Z3 × Dih5			Product
	91	G30^3	Dih15			Dihedral group, Frobenius group

Classifying groups of small order

Small groups of prime power order pⁿ are given as follows:

• Order p: The only group is cyclic.
• Order p²: There are just two groups, both abelian.
• Order p³: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p² by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2.
• Order p⁴: The classification is complicated, and gets much harder as the exponent of p increases.

Most groups of small order have a Sylow p subgroup P with a normal p-complement N for some prime p dividing the order, so can be classified in terms of the possible primes p, p-groups P, groups N, and actions of P on N. In some sense this reduces the classification of these groups to the classification of p-groups. Some of the small groups that do not have a normal p complement include:

• Order 24: The symmetric group S₄
• Order 48: The binary octahedral group and the product S₄ × Z₂
• Order 60: The alternating group A₅.

Small groups library

The group theoretical computer algebra system GAP contains the "Small Groups library" which provides access to descriptions of small order groups. The groups are listed up to isomorphism. At present, the library contains the following groups:^[64]

• those of order at most 2000, except for order 1024 (423164062 groups in the library; the ones of order 1024 had to be skipped, as there are an additional 49487365422 nonisomorphic 2-groups of order 1024.);
• those of cubefree order at most 50000 (395 703 groups);
• those of squarefree order;
• those of order pⁿ for n at most 6 and p prime;
• those of order p⁷ for p = 3, 5, 7, 11 (907 489 groups);
• those of order pqⁿ where qⁿ divides 2⁸, 3⁶, 5⁵ or 7⁴ and p is an arbitrary prime which differs from q;
• those whose orders factorise into at most 3 primes (count with multiplicity).

It contains explicit descriptions of the available groups in computer readable format.

The smallest 10 orders for which the SmallGroups library does not have information are 1024, 2016, 2024, 2025, 2040, 2048, 2052, 2058, 2064, 2072.

See also

• Classification of finite simple groups
• Composition series
• List of finite simple groups
• Number of groups of a given order
• Small Latin squares and quasigroups

Notes

1. Groups of order 1
2. Z1
3. Groups of order 2
4. Z2
5. Groups of order 3
6. Z3
7. Groups of order 4
8. Z4
9. Klein group
10. Groups of order 5
11. Z5
12. Groups of order 6
13. Z6
14. See a worked example showing the isomorphism Z₆ = Z₃ × Z₂.
15. Groups of order 7
16. Z7
17. Groups of order 8
18. Z8
19. Z4×Z2
20. Elementary abelian group:E8
21. Groups of order 9
22. Z9
23. Z3×Z3
24. Groups of order 10
25. Z10
26. Z11
27. Groups of order 12
28. Z12
29. Z6×Z2
30. Z13
31. Groups of order 14
32. Z14
33. Groups of order 15
34. Z15
35. Groups of order 16
36. Z16
37. Z4×Z4
38. Z8×Z2
39. Z4×Z2×Z2
40. Z17
41. Groups of order 18
42. Z18
43. Z6×Z3
44. Z19
45. Groups of order 20
46. Z20
47. Z10×Z2
48. Z21
49. Z22
50. Z23
51. Groups of order 24
52. Z24
53. Z12×Z2
54. Groups of order 27
55. Groups of order 30
56. S3
57. D8
58. Q8
59. D10
60. Q12
61. A4
62. D12
63. Wild, Marcel. "The Groups of Order Sixteen Made Easy", American Mathematical Monthly, Jan 2005
64. Hans Ulrich Besche The Small Groups library

References

• Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9., Table 1, Nonabelian groups order<32.
• Hall, Jr., Marshall; Senior, James K. (1964). "The Groups of Order 2ⁿ (n ≤ 6)". Macmillan. MR 168631. A catalog of the 340 groups of order dividing 64 with tables of defining relations, constants, and lattice of subgroups of each group.

External links

• Particular groups in the Group Properties Wiki
• Number of groups of given order
• Groups of given order
• Besche, H. U.; Eick, B.; O'Brien, E. "small group library".
• GroupNames database