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 ($PSU(3,3)$)	${}^{2}A_{2}(3^{2})$
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	$M_{11}$
projective special linear group:PSL(2,27)	9828	16	projective special linear group ($PSL(2,27)$)