List of simple non-abelian groups of small order

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Listing by order

The simple abelian groups are precisely the groups of prime order, and there is one such group for each prime number.

The first few simple non-abelian groups are listed below:

Group Order Number of conjugacy classes Families of simple non-abelian groups that it is a member of Shorthand notations
alternating group:A5 60 5 alternating group (parameter n = 5), projective special linear group (PSL(2,4), also PSL(2,5)), projective symplectic group (PSp(2,4), PSp(2,5)), Chevalley group of type B (B_1(4),B_1(5)) A_5, A_1(4),A_1(5),B_1(4),B_1(5),C_1(4),C_1(5)
projective special linear group:PSL(3,2) 168 6 projective special linear group (PSL(3,2), also PSL(2,7)), projective symplectic group (PSp(2,7)), Chevalley group of type B (B_1(7)) A_2(2), A_1(7), B_1(7), C_1(7).
alternating group:A6 360 7 alternating group (parameter n = 6), projective special linear group (PSL(2,9)), projective symplectic group (PSp(2,9)), Chevalley group of type B (B_1(9)) A_6, A_1(9), B_1(9), C_1(9). Also, B_2(2)'
projective special linear group:PSL(2,8) 504 9 projective special linear group (PSL(2,8)), Chevalley group of type B (B_1(8)), projective symplectic group (PSp(2,8)) A_1(8), B_1(8), C_1(8)
projective special linear group:PSL(2,11) 660 8 projective special linear group (PSL(2,11)),Chevalley group of type B (B_1(11)), projective symplectic group (PSp(2,11)) A_1(11), B_1(11), C_1(11)
projective special linear group:PSL(2,13) 1092 9 projective special linear group (PSL(2,13)),Chevalley group of type B (B_1(13)), projective symplectic group (PSp(2,13)) A_1(13), B_1(13), C_1(13)
projective special linear group:PSL(2,17) 2448 11 projective special linear group (PSL(2,17)),Chevalley group of type B (B_1(17)), projective symplectic group (PSp(2,17)) A_1(17), B_1(17), C_1(17)
alternating group:A7 2520 9 alternating group (A_7) A_7
projective special linear group:PSL(2,19) 3420 12 projective special linear group (PSL(2,19)), Chevalley group of type B (B_1(19)), projective symplectic group (PSp(2,19)) A_1(19), B_1(19), C_1(19)
projective special linear group:PSL(3,3) 5616 12 projective special linear group (PSL(3,3)) A_2(3)
projective special unitary group:PSU(3,3) 6048 14 projective special unitary group (PSU(3,3)) {}^2A_2(3^2)
projective special linear group:PSL(2,23) 6072 14 projective special linear group (PSL(2,23)),Chevalley group of type B (B_1(23)), projective symplectic group (PSp(2,23)) A_1(23), B_1(23), C_1(23)
projective special linear group:PSL(2,25) 7800 15 projective special linear group (PSL(2,25)),Chevalley group of type B (B_1(25)), projective symplectic group (PSp(2,25)) A_1(25), B_1(25), C_1(25)
Mathieu group:M11 7920 10 sporadic simple group, among the Mathieu groups M_{11}