List of simple non-abelian groups of small order

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 $M_{11}$