Abstract group 51840.o: $S_5\times C_3^2:\GL(2,3)$

• Label: 51840.o
• Order: \( 2^{7} \cdot 3^{4} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{4} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $14$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$S_5\times C_3^2:\GL(2,3)$
Order:	\(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$S_5\times C_3^2:\GL(2,3)$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_3$ x 3, $A_5$
Derived length:	$5$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	40
Elements	1	1195	1700	4404	24	14924	6048	1080	8880	1920	1296	4320	3456	2592	51840
Conjugacy classes	1	8	7	7	1	24	8	2	7	3	1	4	2	2	77
Divisions	1	8	7	7	1	24	4	2	7	3	1	2	2	1	70
Autjugacy classes	1	8	7	7	1	24	8	2	7	3	1	4	2	2	77

Dimension	1	2	3	4	5	6	8	10	12	15	16	18	20	24	32	40	48	64	80	96
Irr. complex chars.	4	6	4	6	4	2	10	6	7	4	4	2	2	1	4	4	2	2	2	1	77
Irr. rational chars.	4	2	4	8	4	2	6	2	5	4	6	2	4	2	4	4	2	2	2	1	70

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	$5472$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	32	32	32
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $14$ $\langle(1,2,4)(3,6,7)(5,8,9)(10,12)(11,14,13), (2,3,5,4,7,9,8,6)(10,11,13,12)\rangle$
Transitive group:	45T837				more information
Direct product:	$S_5$ $\, \times\, $ $(C_3^2:\GL(2,3))$
Semidirect product:	$(S_5\times \PU(3,2))$ $\,\rtimes\,$ $C_2$	$\PU(3,2)$ $\,\rtimes\,$ $(C_2\times S_5)$	$(S_5\times \PSU(3,2))$ $\,\rtimes\,$ $S_3$	$\PSU(3,2)$ $\,\rtimes\,$ $(S_3\times S_5)$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(\GL(2,4):D_6)$ . $S_4$	$(C_3:S_3)$ . $(S_4\times S_5)$	$(\GL(2,4):S_3)$ . $(C_2\times S_4)$		more information
Aut. group:	$\Aut(C_3^2:S_5)$	$\Aut(\GL(2,4):S_3)$	$\Aut(\He_3\times A_5)$	$\Aut(\GL(2,4):D_6)$	all 14

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 495958 subgroups in 2209 conjugacy classes, 20 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_5\times C_3^2:\GL(2,3)$
Commutator:	$G' \simeq$ $A_5\times \PU(3,2)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_5\times C_3^2:\GL(2,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2$	$G/\operatorname{Fit} \simeq$ $S_5\times \GL(2,3)$
Radical:	$R \simeq$ $C_3^2:\GL(2,3)$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_3\times \GL(2,4)$	$G/\operatorname{soc} \simeq$ $C_2\times \GL(2,3)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times \SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times \He_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 51840: $S_5\times C_3^2:\GL(2,3)$

Order 25920: $(C_3\times \GL(2,4)):\GL(2,3)$, $A_5\times C_3^2:\GL(2,3)$, $S_5\times \PU(3,2)$

Order 17280: $F_9:C_2\times S_5$

Order 12960: $A_5\times \PU(3,2)$, $C_3^2:D_6\times S_5$

Order 10368: $S_4\times C_3^2:\GL(2,3)$

Order 8640: $F_9\times S_5$, $S_5\times \PSU(3,2)$, $S_5\times \SOPlus(4,2)$, $F_5\times C_3^2:\GL(2,3)$, $\SOPlus(4,2).S_5$, $A_5\times F_9:C_2$, $\PSU(3,2):S_5$, $F_9:S_5$

Order 6480: $C_3^2:C_6\times S_5$ x 2, $\GL(2,4).S_3^2$ x 2, $C_3^2:S_3\times S_5$, $\GL(2,4).S_3^2$, $\GL(2,4).S_3^2$

Order 5760: $S_5\times \GL(2,3)$

Order 5184: $C_3^2:\GL(2,3)\times D_6$, $(C_3^2\times A_4):\GL(2,3)$, $A_4\times C_3^2:\GL(2,3)$, $S_4\times \PU(3,2)$

Order 4320: $S_3^2\times S_5$ x 2, $\PU(3,2):F_5$, $D_5\times C_3^2:\GL(2,3)$, $F_5\times \PU(3,2)$, $A_5:F_9$, $A_5\times F_9$, $A_5\times \PSU(3,2)$, $A_5:\PSU(3,2)$, $A_5:\SOPlus(4,2)$, $C_3^2:C_4\times S_5$, $S_3^2:S_5$, $A_5\times \SOPlus(4,2)$

Order 3456: $F_9:C_2\times S_4$, $C_2^4.\SL(3,3)$

Order 3240: $\He_3:S_5$ x 2, $C_3^2:C_6\times A_5$ x 2, $\He_3\times S_5$, $C_3^2:S_3\times A_5$, $\He_3:S_5$

Order 2880: $F_5\times F_9:C_2$, $A_5:\GL(2,3)$, $S_5\times \SL(2,3)$, $A_5\times \GL(2,3)$

Order 2592: $S_3\times C_3^2:\GL(2,3)$ x 4, $C_2\times C_3^3:\GL(2,3)$, $D_6\times \PU(3,2)$, $C_6\times C_3^2:\GL(2,3)$, $A_4\times \PU(3,2)$, $C_3^2:D_6\times S_4$

Order 2160: $\GL(2,4):D_6$ x 4, $A_5:S_3^2$ x 3, $\GL(2,4):D_6$ x 2, $S_3^2\times A_5$ x 2, $A_5:S_3^2$ x 2, $\PSU(3,2):D_{15}$, $C_5\times C_3^2:\GL(2,3)$, $D_5\times \PU(3,2)$, $C_3^2:D_6\times F_5$, $C_3^2:C_4\times A_5$, $(C_3\times \GL(2,4)):C_4$

Order 1920: $\SD_{16}\times S_5$

Order 1728: $C_6^2:\GL(2,3)$ x 2, $C_6^2:\GL(2,3)$ x 2, $C_2\times F_9:D_6$, $S_4\times \PSU(3,2)$, $D_6^2:D_6$, $C_6^2:\GL(2,3)$, $A_4\times F_9:C_2$, $\PSU(3,2):S_4$, $\SOPlus(4,2).S_4$, $F_9:S_4$, $S_4\times F_9$, $D_4\times \PU(3,2)$, $\PSU(3,2):D_{12}$, $C_4\times C_3^2:\GL(2,3)$

Order 1620: $\He_3\times A_5$

Order 1440: $D_6\times S_5$ x 2, $F_9:D_{10}$, $F_5\times \PSU(3,2)$, $F_5\times \SOPlus(4,2)$, $F_9:F_5$, $\PSU(3,2):F_5$, $\SOPlus(4,2):F_5$, $F_5\times F_9$, $A_5\times \SL(2,3)$

Order 1296: $S_3\times \PU(3,2)$ x 2, $C_3^3:\GL(2,3)$ x 2, $C_3^3:\GL(2,3)$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6\times \PU(3,2)$, $C_3^2:D_6^2$, $A_4\times C_3^2:D_6$, $C_6^2:S_3^2$, $C_6^2:S_3^2$

Order 1152: $S_4\times \GL(2,3)$, $D_4\times F_9:C_2$

Order 1080: $S_3\times \GL(2,4)$ x 4, $C_3^2:S_5$ x 4, $C_3^2\times S_5$ x 3, $\GL(2,4):S_3$ x 2, $C_3^2:S_5$ x 2, $C_3^2:(S_3\times F_5)$ x 2, $C_3^2:C_6\times F_5$ x 2, $C_5\times \PU(3,2)$, $D_5\times C_3^2:D_6$, $C_3^2:(S_3\times F_5)$, $C_3^2:S_3\times F_5$

Order 960: $\SD_{16}\times A_5$, $Q_8:S_5$, $A_5:\SD_{16}$, $C_8:S_5$, $C_8\times S_5$, $F_5\times \GL(2,3)$, $Q_8\times S_5$, $D_4\times S_5$

Order 864: $F_9:D_6$ x 8, $C_2\times C_3^2:\GL(2,3)$ x 6, $S_4\times S_3^2$ x 2, $C_6^2:\SL(2,3)$ x 2, $D_6\times \PSU(3,2)$, $S_3^3:C_2^2$, $F_9:D_6$, $\PSU(3,2):D_6$, $C_2\times C_3^3:\SD_{16}$, $C_2\times F_9:C_6$, $D_6\times F_9$, $A_4\times \PSU(3,2)$, $D_6^2:C_6$, $S_3^2:S_4$, $C_6^2:D_{12}$, $C_3^2:C_4\times S_4$, $A_4:\PSU(3,2)$, $C_6^2:\SL(2,3)$, $A_4\times F_9$, $A_4:F_9$, $C_4\times \PU(3,2)$, $(C_3\times C_6).\GL(2,3)$, $C_6.D_6^2$

Order 720: $S_3\times S_5$ x 9, $F_5\times S_3^2$ x 2, $D_6\times A_5$ x 2, $C_6:S_5$ x 2, $C_6\times S_5$ x 2, $D_5\times \PSU(3,2)$, $S_3^2:D_{10}$, $F_9:D_5$, $\PSU(3,2):D_5$, $(C_3\times C_{15}):\SD_{16}$, $F_9:C_{10}$, $D_5\times F_9$, $D_5:F_9$, $D_5.\PSU(3,2)$, $D_5.\SOPlus(4,2)$, $S_3^2:F_5$, $C_3^2:C_4\times F_5$

Order 648: $C_3.S_3^3$ x 8, $C_2\times C_3^2:S_3^2$ x 2, $S_3\times C_3^2:D_6$ x 2, $A_4\times C_3^2:C_6$ x 2, $\He_3:S_4$ x 2, $C_3^3:\SL(2,3)$, $C_2\times C_3^2:S_3^2$, $S_3\times C_3^2:D_6$, $C_6\times C_3^2:D_6$, $A_4\times C_3^2:S_3$, $\He_3:S_4$, $S_4\times \He_3$

Order 576: $F_9:C_2^3$ x 2, $C_6^2:\SD_{16}$ x 2, $C_6^2:\SD_{16}$ x 2, $F_9:D_4$ x 2, $D_4\times \PSU(3,2)$, $C_3^2:D_4^2$, $D_4\times F_9$, $D_6\times \GL(2,3)$, $\PSU(3,2):D_4$, $(C_3\times C_{12}):\SD_{16}$, $F_9:D_4$, $C_4\times F_9:C_2$, $A_4:\GL(2,3)$, $S_4\times \SL(2,3)$, $A_4\times \GL(2,3)$

Order 540: $C_3\times \GL(2,4)$ x 3, $C_3^2:D_{30}$ x 2, $\He_3:D_{10}$ x 2, $\He_3:F_5$ x 2, $\He_3:D_{10}$, $C_{15}.S_3^2$, $\He_3:D_{10}$, $\He_3:F_5$, $F_5\times \He_3$

Order 480: $D_5\times \GL(2,3)$, $F_5\times \SL(2,3)$, $\SL(2,3):F_5$, $Q_8\times A_5$, $D_4\times A_5$, $C_2^2:S_5$, $A_5:Q_8$, $C_4:S_5$, $C_4\times S_5$, $C_8\times A_5$, $A_5:C_8$, $C_2^2\times S_5$

Order 432: $S_3^3:C_2$ x 6, $C_3^2:\GL(2,3)$ x 6, $C_6^2:D_6$ x 4, $S_3\times \PSU(3,2)$ x 4, $F_9:S_3$ x 4, $C_3^3:\SD_{16}$ x 4, $C_3^3:\SD_{16}$ x 4, $F_9:C_6$ x 4, $S_3\times F_9$ x 4, $C_6^2:D_6$ x 4, $A_4:S_3^2$ x 3, $C_2\times \PU(3,2)$ x 3, $C_2\times S_3^3$ x 2, $A_4\times S_3^2$ x 2, $A_4:S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_{12}.S_3^2$ x 2, $C_6:\PSU(3,2)$, $C_6\times \PSU(3,2)$, $C_2\times C_3^2:D_{12}$, $S_3^2:D_6$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$, $C_6:F_9$, $C_6\times F_9$, $C_6^2:C_{12}$, $(C_3^2\times A_4):C_4$, $C_6^2:D_6$, $C_{12}.S_3^2$, $C_{12}.S_3^2$

Order 384: $\SD_{16}\times S_4$, $D_4\times \GL(2,3)$

Order 360: $S_3\times A_5$ x 5, $C_3:S_5$ x 5, $C_3\times S_5$ x 5, $D_{15}:C_{12}$ x 4, $D_5.S_3^2$ x 3, $D_5\times S_3^2$ x 2, $D_5.S_3^2$ x 2, $C_3:S_3\times F_5$ x 2, $C_6\times A_5$ x 2, $C_5:\PSU(3,2)$, $C_5\times \PSU(3,2)$, $C_3^2:D_{20}$, $S_3^2:D_5$, $S_3^2:C_{10}$, $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$, $C_5:F_9$, $C_5:F_9$, $C_5\times F_9$

Order 324: $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 4, $C_3^2:S_3^2$ x 4, $C_3^3:D_6$ x 4, $C_3^3:D_6$ x 2, $C_3^2:C_6^2$ x 2, $C_3^3:D_6$, $C_3^3:D_6$, $D_6\times \He_3$, $A_4\times \He_3$

Order 320: $\SD_{16}\times F_5$

Order 288: $F_9:C_2^2$ x 13, $S_3\times \GL(2,3)$ x 4, $\PSU(3,2):C_4$ x 3, $\SOPlus(4,2):C_4$ x 3, $D_4\times S_3^2$ x 2, $C_6^2:Q_8$ x 2, $C_6^2:D_4$ x 2, $D_6\wr C_2$ x 2, $C_6^2:C_8$ x 2, $C_6^2:Q_8$ x 2, $C_6^2:D_4$ x 2, $C_2^2\times F_9$ x 2, $D_6\times S_4$ x 2, $C_3^2:C_4\times D_4$, $D_6\times \SL(2,3)$, $C_6:\GL(2,3)$, $C_6\times \GL(2,3)$, $C_4:\PSU(3,2)$, $C_4\times \PSU(3,2)$, $C_4:\SOPlus(4,2)$, $S_3^2:D_4$, $C_4\times \SOPlus(4,2)$, $C_4:F_9$, $C_4\times F_9$, $A_4\times \SL(2,3)$, $F_9:C_4$

Order 270: $C_3^2:D_{15}$ x 2, $C_3^2:C_{30}$ x 2, $D_5\times \He_3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

Order 240: $C_2\times S_5$ x 4, $D_6\times F_5$ x 2, $C_4\times A_5$, $A_5:C_4$, $C_2^2\times A_5$, $D_5\times \SL(2,3)$, $C_5:\GL(2,3)$, $C_5\times \GL(2,3)$

Order 216: $S_3^3$ x 14, $C_6.S_3^2$ x 13, $C_6\times S_3^2$ x 6, $C_6:S_3^2$ x 5, $C_6^2:C_6$ x 4, $C_3^2:D_{12}$ x 4, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 4, $S_3^2:S_3$ x 4, $C_3^2:C_4\times S_3$ x 4, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 3, $C_3^2\times S_4$ x 3, $C_3^3:Q_8$ x 3, $C_3^2:D_{12}$ x 3, $S_3^2:C_6$ x 3, $\PU(3,2)$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_6^2:S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_4\times C_3^2:C_6$ x 2, $\He_3:D_4$ x 2, $C_6.S_3^2$ x 2, $C_6:S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^3:Q_8$ x 2, $C_3:F_9$ x 2, $C_3\times F_9$ x 2, $C_6^2:S_3$ x 2, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $C_6.S_3^2$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$

Order 192: $C_2^2:\GL(2,3)$ x 2, $C_2^2\times \GL(2,3)$ x 2, $Q_8:S_4$, $A_4:\SD_{16}$, $C_8:S_4$, $C_8\times S_4$, $C_4:\GL(2,3)$, $C_4\times \GL(2,3)$, $Q_8:S_4$, $Q_8\times S_4$, $D_4\times S_4$, $D_6\times \SD_{16}$, $A_4\times \SD_{16}$, $D_4\times \SL(2,3)$

Order 180: $C_{15}:D_6$ x 4, $C_{15}:C_{12}$ x 4, $S_3\times D_{15}$ x 3, $C_3^2\times F_5$ x 3, $\GL(2,4)$ x 3, $C_{15}:D_6$ x 2, $C_5\times S_3^2$ x 2, $C_{15}:D_6$ x 2, $C_3^2:F_5$ x 2, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$

Order 162: $C_3^3:C_6$ x 4, $C_3^3:C_6$ x 4, $C_3^3:S_3$ x 2, $C_3^3:S_3$ x 2, $S_3\times \He_3$ x 2, $C_6\times \He_3$

Order 160: $Q_8:F_5$, $D_{20}:C_4$, $C_{40}:C_4$, $C_8\times F_5$, $Q_8\times F_5$, $D_4\times F_5$, $D_5\times \SD_{16}$

Order 144: $F_9:C_2$ x 13, $S_3\times S_4$ x 12, $S_3^2:C_2^2$ x 9, $D_6^2$ x 6, $C_2\times F_9$ x 6, $D_6:D_6$ x 6, $C_2\times \PSU(3,2)$ x 5, $C_{12}:D_6$ x 4, $D_6:D_6$ x 4, $S_3\times D_{12}$ x 3, $S_3^2:C_4$ x 3, $C_6^2:C_4$ x 2, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $C_{12}:D_6$ x 2, $C_{12}:D_6$ x 2, $C_4\times S_3^2$ x 2, $C_6^2:C_4$ x 2, $S_3\times \SL(2,3)$ x 2, $C_3:\GL(2,3)$ x 2, $C_3\times \GL(2,3)$ x 2, $C_2.\PSU(3,2)$ x 2, $C_6\times \SL(2,3)$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$, $C_2.\SOPlus(4,2)$

Order 135: $C_5\times \He_3$

Order 128: $D_4\times \SD_{16}$

Order 120: $S_3\times F_5$ x 9, $S_5$ x 3, $S_3\times D_{10}$ x 2, $C_{30}:C_4$ x 2, $C_6\times F_5$ x 2, $C_2\times A_5$ x 2, $C_5\times \SL(2,3)$

Order 108: $C_3\times S_3^2$ x 23, $C_3:S_3^2$ x 20, $C_3^2:D_6$ x 19, $C_3^2:D_6$ x 14, $C_3^2:D_6$ x 8, $C_3^2\times D_6$ x 7, $C_3:S_3^2$ x 7, $C_3^2:D_6$ x 6, $C_3^2\times A_4$ x 3, $C_3^2:A_4$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2, $C_2^2\times \He_3$ x 2, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $S_3\times \SD_{16}$ x 8, $C_2\times \GL(2,3)$ x 6, $D_4\times D_6$ x 3, $C_2^2\times \SL(2,3)$ x 2, $C_8\times A_4$, $C_4\times \SL(2,3)$, $\SL(2,3):C_4$, $A_4:C_8$, $C_2^2\times S_4$, $Q_8\times D_6$, $Q_8:A_4$, $Q_8\times A_4$, $D_4\times A_4$, $\GL(2,\mathbb{Z}/4)$, $C_4:S_4$, $C_4\times S_4$, $A_4:Q_8$, $C_6\times \SD_{16}$, $Q_8:D_6$, $C_6:\SD_{16}$, $C_8:D_6$, $C_8\times D_6$

Order 90: $C_3\times D_{15}$ x 4, $S_3\times C_{15}$ x 4, $C_3^2\times D_5$ x 3, $C_3:D_{15}$ x 2, $C_{15}:S_3$ x 2

Order 81: $C_3\times \He_3$

Order 80: $C_{20}:C_4$ x 2, $C_{40}:C_2$, $C_2^2\times F_5$, $Q_8\times D_5$, $C_8\times D_5$, $D_4\times D_5$, $D_{10}:C_4$, $C_4\times F_5$, $D_5:C_8$, $C_5\times \SD_{16}$, $Q_8:D_5$, $C_5:\SD_{16}$

Order 72: $S_3\times D_6$ x 47, $C_6\times D_6$ x 12, $S_3\times A_4$ x 9, $C_3\times S_4$ x 9, $\SOPlus(4,2)$ x 9, $C_3:S_4$ x 8, $C_6\wr C_2$ x 8, $C_6:D_6$ x 6, $C_2\times C_3^2:C_4$ x 6, $C_3:D_{12}$ x 6, $\PSU(3,2)$ x 5, $F_9$ x 4, $C_6^2:C_2$ x 4, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $D_4\times C_3^2$ x 3, $D_6:S_3$ x 3, $C_6.D_6$ x 3, $C_6\times A_4$ x 2, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2, $C_6.D_6$ x 2, $C_3\times \SL(2,3)$

Order 64: $C_2^2\times \SD_{16}$ x 2, $C_8:D_4$ x 2, $C_2^2:\SD_{16}$ x 2, $Q_8:D_4$ x 2, $D_4\times Q_8$, $D_4^2$, $C_8:D_4$, $C_4:\SD_{16}$, $Q_8:D_4$, $C_4\times \SD_{16}$, $C_8\times D_4$

Order 60: $S_3\times D_5$ x 9, $C_{15}:C_4$ x 5, $C_3\times F_5$ x 5, $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_3\times D_{10}$ x 2, $A_5$

Order 54: $C_3^2:C_6$ x 14, $S_3\times C_3^2$ x 14, $C_3^2:C_6$ x 12, $C_3^2:S_3$ x 9, $C_2\times \He_3$ x 5, $C_3^2:S_3$ x 4, $C_3^2\times C_6$ x 3

Order 48: $S_3\times D_4$ x 23, $C_2\times S_4$ x 6, $\GL(2,3)$ x 6, $C_2^2\times D_6$ x 5, $C_6:D_4$ x 5, $C_{24}:C_2$ x 4, $S_3\times Q_8$ x 4, $S_3\times C_8$ x 4, $C_3\times \SD_{16}$ x 4, $Q_8:S_3$ x 4, $C_3:\SD_{16}$ x 4, $C_6\times D_4$ x 3, $C_2\times D_{12}$ x 3, $C_4\times D_6$ x 3, $C_2\times \SL(2,3)$ x 3, $C_6:C_8$, $C_2^2\times A_4$, $C_6\times Q_8$, $C_6:Q_8$, $C_4\times A_4$, $A_4:C_4$, $C_2\times C_{24}$

Order 45: $C_3\times C_{15}$ x 3

Order 40: $C_2\times F_5$ x 5, $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$, $C_5:C_8$, $C_{40}$, $C_2\times D_{10}$, $C_5\times Q_8$, $C_5\times D_4$, $C_5:C_8$

Order 36: $S_3^2$ x 67, $C_6\times S_3$ x 54, $C_6:S_3$ x 26, $C_6^2$ x 8, $C_3\times A_4$ x 8, $C_3^2:C_4$ x 4, $C_3:C_{12}$ x 4, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 2

Order 32: $C_2\times \SD_{16}$ x 13, $D_4:C_4$ x 3, $C_2^2\times D_4$ x 3, $C_4:D_4$ x 3, $Q_8:C_4$ x 3, $C_2^2:C_8$ x 2, $C_2^2\times Q_8$ x 2, $C_2^2\times C_8$ x 2, $C_2^2:Q_8$ x 2, $C_2^2\wr C_2$ x 2, $C_4\times D_4$ x 2, $C_4:Q_8$, $C_4:D_4$, $C_4\times C_8$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$

Order 30: $D_{15}$ x 5, $C_3\times D_5$ x 5, $C_5\times S_3$ x 5, $C_{30}$ x 2

Order 27: $\He_3$ x 4, $C_3^3$ x 3

Order 24: $C_2\times D_6$ x 44, $C_3:D_4$ x 22, $C_4\times S_3$ x 13, $D_{12}$ x 12, $C_3\times D_4$ x 12, $S_4$ x 8, $C_2^2\times C_6$ x 5, $C_2\times A_4$ x 4, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_3:Q_8$ x 3, $\SL(2,3)$ x 3, $C_{24}$ x 2, $C_3\times Q_8$ x 2, $C_3:C_8$ x 2

Order 20: $D_{10}$ x 4, $F_5$ x 4, $C_2\times C_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 48, $C_3:S_3$ x 22, $C_3\times C_6$ x 18

Order 16: $C_2\times D_4$ x 17, $\SD_{16}$ x 13, $C_2\times C_8$ x 6, $C_2^2:C_4$ x 5, $C_2\times Q_8$ x 5, $C_4:C_4$ x 4, $C_2^2\times C_4$ x 3, $C_2^4$ x 2, $C_4^2$

Order 15: $C_{15}$ x 3

Order 12: $D_6$ x 78, $C_2\times C_6$ x 23, $C_{12}$ x 7, $C_3:C_4$ x 7, $A_4$ x 4

Order 10: $D_5$ x 3, $C_{10}$ x 2

Order 9: $C_3^2$ x 11

Order 8: $D_4$ x 19, $C_2^3$ x 12, $C_2\times C_4$ x 10, $Q_8$ x 5, $C_8$ x 4

Order 6: $S_3$ x 31, $C_6$ x 24

Order 5: $C_5$

Order 4: $C_2^2$ x 20, $C_4$ x 7

Order 3: $C_3$ x 7

Order 2: $C_2$ x 8

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 51840: $S_5\times C_3^2:\GL(2,3)$

Order 25920: $(C_3\times \GL(2,4)):\GL(2,3)$, $A_5\times C_3^2:\GL(2,3)$, $S_5\times \PU(3,2)$

Order 17280: $F_9:C_2\times S_5$

Order 12960: $A_5\times \PU(3,2)$, $C_3^2:D_6\times S_5$

Order 10368: $S_4\times C_3^2:\GL(2,3)$

Order 8640: $F_9\times S_5$, $S_5\times \PSU(3,2)$, $S_5\times \SOPlus(4,2)$, $F_5\times C_3^2:\GL(2,3)$, $\SOPlus(4,2).S_5$, $A_5\times F_9:C_2$, $\PSU(3,2):S_5$, $F_9:S_5$

Order 6480: $C_3^2:C_6\times S_5$ x 2, $\GL(2,4).S_3^2$ x 2, $C_3^2:S_3\times S_5$, $\GL(2,4).S_3^2$, $\GL(2,4).S_3^2$

Order 5760: $S_5\times \GL(2,3)$

Order 5184: $C_3^2:\GL(2,3)\times D_6$, $(C_3^2\times A_4):\GL(2,3)$, $A_4\times C_3^2:\GL(2,3)$, $S_4\times \PU(3,2)$

Order 4320: $S_3^2\times S_5$ x 2, $\PU(3,2):F_5$, $D_5\times C_3^2:\GL(2,3)$, $F_5\times \PU(3,2)$, $A_5:F_9$, $A_5\times F_9$, $A_5\times \PSU(3,2)$, $A_5:\PSU(3,2)$, $A_5:\SOPlus(4,2)$, $C_3^2:C_4\times S_5$, $S_3^2:S_5$, $A_5\times \SOPlus(4,2)$

Order 3456: $F_9:C_2\times S_4$, $C_2^4.\SL(3,3)$

Order 3240: $\He_3:S_5$ x 2, $C_3^2:C_6\times A_5$ x 2, $\He_3\times S_5$, $C_3^2:S_3\times A_5$, $\He_3:S_5$

Order 2880: $F_5\times F_9:C_2$, $A_5:\GL(2,3)$, $S_5\times \SL(2,3)$, $A_5\times \GL(2,3)$

Order 2592: $S_3\times C_3^2:\GL(2,3)$ x 4, $C_2\times C_3^3:\GL(2,3)$, $D_6\times \PU(3,2)$, $C_6\times C_3^2:\GL(2,3)$, $A_4\times \PU(3,2)$, $C_3^2:D_6\times S_4$

Order 2160: $\GL(2,4):D_6$ x 4, $A_5:S_3^2$ x 3, $\GL(2,4):D_6$ x 2, $S_3^2\times A_5$ x 2, $A_5:S_3^2$ x 2, $\PSU(3,2):D_{15}$, $C_5\times C_3^2:\GL(2,3)$, $D_5\times \PU(3,2)$, $C_3^2:D_6\times F_5$, $C_3^2:C_4\times A_5$, $(C_3\times \GL(2,4)):C_4$

Order 1920: $\SD_{16}\times S_5$

Order 1728: $C_6^2:\GL(2,3)$ x 2, $C_6^2:\GL(2,3)$ x 2, $C_2\times F_9:D_6$, $S_4\times \PSU(3,2)$, $D_6^2:D_6$, $C_6^2:\GL(2,3)$, $A_4\times F_9:C_2$, $\PSU(3,2):S_4$, $\SOPlus(4,2).S_4$, $F_9:S_4$, $S_4\times F_9$, $D_4\times \PU(3,2)$, $\PSU(3,2):D_{12}$, $C_4\times C_3^2:\GL(2,3)$

Order 1620: $\He_3\times A_5$

Order 1440: $D_6\times S_5$ x 2, $F_9:D_{10}$, $F_5\times \PSU(3,2)$, $F_5\times \SOPlus(4,2)$, $F_9:F_5$, $\PSU(3,2):F_5$, $\SOPlus(4,2):F_5$, $F_5\times F_9$, $A_5\times \SL(2,3)$

Order 1296: $S_3\times \PU(3,2)$ x 2, $C_3^3:\GL(2,3)$ x 2, $C_3^3:\GL(2,3)$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6\times \PU(3,2)$, $C_3^2:D_6^2$, $A_4\times C_3^2:D_6$, $C_6^2:S_3^2$, $C_6^2:S_3^2$

Order 1152: $S_4\times \GL(2,3)$, $D_4\times F_9:C_2$

Order 1080: $S_3\times \GL(2,4)$ x 4, $C_3^2:S_5$ x 4, $C_3^2\times S_5$ x 3, $\GL(2,4):S_3$ x 2, $C_3^2:S_5$ x 2, $C_3^2:(S_3\times F_5)$ x 2, $C_3^2:C_6\times F_5$ x 2, $C_5\times \PU(3,2)$, $D_5\times C_3^2:D_6$, $C_3^2:(S_3\times F_5)$, $C_3^2:S_3\times F_5$

Order 960: $\SD_{16}\times A_5$, $Q_8:S_5$, $A_5:\SD_{16}$, $C_8:S_5$, $C_8\times S_5$, $F_5\times \GL(2,3)$, $Q_8\times S_5$, $D_4\times S_5$

Order 864: $F_9:D_6$ x 8, $C_2\times C_3^2:\GL(2,3)$ x 6, $S_4\times S_3^2$ x 2, $C_6^2:\SL(2,3)$ x 2, $D_6\times \PSU(3,2)$, $S_3^3:C_2^2$, $F_9:D_6$, $\PSU(3,2):D_6$, $C_2\times C_3^3:\SD_{16}$, $C_2\times F_9:C_6$, $D_6\times F_9$, $A_4\times \PSU(3,2)$, $D_6^2:C_6$, $S_3^2:S_4$, $C_6^2:D_{12}$, $C_3^2:C_4\times S_4$, $A_4:\PSU(3,2)$, $C_6^2:\SL(2,3)$, $A_4\times F_9$, $A_4:F_9$, $C_4\times \PU(3,2)$, $(C_3\times C_6).\GL(2,3)$, $C_6.D_6^2$

Order 720: $S_3\times S_5$ x 9, $F_5\times S_3^2$ x 2, $D_6\times A_5$ x 2, $C_6:S_5$ x 2, $C_6\times S_5$ x 2, $D_5\times \PSU(3,2)$, $S_3^2:D_{10}$, $F_9:D_5$, $\PSU(3,2):D_5$, $(C_3\times C_{15}):\SD_{16}$, $F_9:C_{10}$, $D_5\times F_9$, $D_5:F_9$, $D_5.\PSU(3,2)$, $D_5.\SOPlus(4,2)$, $S_3^2:F_5$, $C_3^2:C_4\times F_5$

Order 648: $C_3.S_3^3$ x 8, $C_2\times C_3^2:S_3^2$ x 2, $S_3\times C_3^2:D_6$ x 2, $A_4\times C_3^2:C_6$ x 2, $\He_3:S_4$ x 2, $C_3^3:\SL(2,3)$, $C_2\times C_3^2:S_3^2$, $S_3\times C_3^2:D_6$, $C_6\times C_3^2:D_6$, $A_4\times C_3^2:S_3$, $\He_3:S_4$, $S_4\times \He_3$

Order 576: $F_9:C_2^3$ x 2, $C_6^2:\SD_{16}$ x 2, $C_6^2:\SD_{16}$ x 2, $F_9:D_4$ x 2, $D_4\times \PSU(3,2)$, $C_3^2:D_4^2$, $D_4\times F_9$, $D_6\times \GL(2,3)$, $\PSU(3,2):D_4$, $(C_3\times C_{12}):\SD_{16}$, $F_9:D_4$, $C_4\times F_9:C_2$, $A_4:\GL(2,3)$, $S_4\times \SL(2,3)$, $A_4\times \GL(2,3)$

Order 540: $C_3\times \GL(2,4)$ x 3, $C_3^2:D_{30}$ x 2, $\He_3:D_{10}$ x 2, $\He_3:F_5$ x 2, $\He_3:D_{10}$, $C_{15}.S_3^2$, $\He_3:D_{10}$, $\He_3:F_5$, $F_5\times \He_3$

Order 480: $D_5\times \GL(2,3)$, $F_5\times \SL(2,3)$, $\SL(2,3):F_5$, $Q_8\times A_5$, $D_4\times A_5$, $C_2^2:S_5$, $A_5:Q_8$, $C_4:S_5$, $C_4\times S_5$, $C_8\times A_5$, $A_5:C_8$, $C_2^2\times S_5$

Order 432: $S_3^3:C_2$ x 6, $C_3^2:\GL(2,3)$ x 6, $C_6^2:D_6$ x 4, $S_3\times \PSU(3,2)$ x 4, $F_9:S_3$ x 4, $C_3^3:\SD_{16}$ x 4, $C_3^3:\SD_{16}$ x 4, $F_9:C_6$ x 4, $S_3\times F_9$ x 4, $C_6^2:D_6$ x 4, $A_4:S_3^2$ x 3, $C_2\times \PU(3,2)$ x 3, $C_2\times S_3^3$ x 2, $A_4\times S_3^2$ x 2, $A_4:S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_{12}.S_3^2$ x 2, $C_6:\PSU(3,2)$, $C_6\times \PSU(3,2)$, $C_2\times C_3^2:D_{12}$, $S_3^2:D_6$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$, $C_6:F_9$, $C_6\times F_9$, $C_6^2:C_{12}$, $(C_3^2\times A_4):C_4$, $C_6^2:D_6$, $C_{12}.S_3^2$, $C_{12}.S_3^2$

Order 384: $\SD_{16}\times S_4$, $D_4\times \GL(2,3)$

Order 360: $S_3\times A_5$ x 5, $C_3:S_5$ x 5, $C_3\times S_5$ x 5, $D_{15}:C_{12}$ x 4, $D_5.S_3^2$ x 3, $D_5\times S_3^2$ x 2, $D_5.S_3^2$ x 2, $C_3:S_3\times F_5$ x 2, $C_6\times A_5$ x 2, $C_5:\PSU(3,2)$, $C_5\times \PSU(3,2)$, $C_3^2:D_{20}$, $S_3^2:D_5$, $S_3^2:C_{10}$, $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$, $C_5:F_9$, $C_5:F_9$, $C_5\times F_9$

Order 324: $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 4, $C_3^2:S_3^2$ x 4, $C_3^3:D_6$ x 4, $C_3^3:D_6$ x 2, $C_3^2:C_6^2$ x 2, $C_3^3:D_6$, $C_3^3:D_6$, $D_6\times \He_3$, $A_4\times \He_3$

Order 320: $\SD_{16}\times F_5$

Order 288: $F_9:C_2^2$ x 13, $S_3\times \GL(2,3)$ x 4, $\PSU(3,2):C_4$ x 3, $\SOPlus(4,2):C_4$ x 3, $D_4\times S_3^2$ x 2, $C_6^2:Q_8$ x 2, $C_6^2:D_4$ x 2, $D_6\wr C_2$ x 2, $C_6^2:C_8$ x 2, $C_6^2:Q_8$ x 2, $C_6^2:D_4$ x 2, $C_2^2\times F_9$ x 2, $D_6\times S_4$ x 2, $C_3^2:C_4\times D_4$, $D_6\times \SL(2,3)$, $C_6:\GL(2,3)$, $C_6\times \GL(2,3)$, $C_4:\PSU(3,2)$, $C_4\times \PSU(3,2)$, $C_4:\SOPlus(4,2)$, $S_3^2:D_4$, $C_4\times \SOPlus(4,2)$, $C_4:F_9$, $C_4\times F_9$, $A_4\times \SL(2,3)$, $F_9:C_4$

Order 270: $C_3^2:D_{15}$ x 2, $C_3^2:C_{30}$ x 2, $D_5\times \He_3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

Order 240: $C_2\times S_5$ x 4, $D_6\times F_5$ x 2, $C_4\times A_5$, $A_5:C_4$, $C_2^2\times A_5$, $D_5\times \SL(2,3)$, $C_5:\GL(2,3)$, $C_5\times \GL(2,3)$

Order 216: $S_3^3$ x 14, $C_6.S_3^2$ x 13, $C_6\times S_3^2$ x 6, $C_6:S_3^2$ x 5, $C_6^2:C_6$ x 4, $C_3^2:D_{12}$ x 4, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 4, $S_3^2:S_3$ x 4, $C_3^2:C_4\times S_3$ x 4, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 3, $C_3^2\times S_4$ x 3, $C_3^3:Q_8$ x 3, $C_3^2:D_{12}$ x 3, $S_3^2:C_6$ x 3, $\PU(3,2)$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_6^2:S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_4\times C_3^2:C_6$ x 2, $\He_3:D_4$ x 2, $C_6.S_3^2$ x 2, $C_6:S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^3:Q_8$ x 2, $C_3:F_9$ x 2, $C_3\times F_9$ x 2, $C_6^2:S_3$ x 2, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $C_6.S_3^2$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$

Order 192: $C_2^2:\GL(2,3)$ x 2, $C_2^2\times \GL(2,3)$ x 2, $Q_8:S_4$, $A_4:\SD_{16}$, $C_8:S_4$, $C_8\times S_4$, $C_4:\GL(2,3)$, $C_4\times \GL(2,3)$, $Q_8:S_4$, $Q_8\times S_4$, $D_4\times S_4$, $D_6\times \SD_{16}$, $A_4\times \SD_{16}$, $D_4\times \SL(2,3)$

Order 180: $C_{15}:D_6$ x 4, $C_{15}:C_{12}$ x 4, $S_3\times D_{15}$ x 3, $C_3^2\times F_5$ x 3, $\GL(2,4)$ x 3, $C_{15}:D_6$ x 2, $C_5\times S_3^2$ x 2, $C_{15}:D_6$ x 2, $C_3^2:F_5$ x 2, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$

Order 162: $C_3^3:C_6$ x 4, $C_3^3:C_6$ x 4, $C_3^3:S_3$ x 2, $C_3^3:S_3$ x 2, $S_3\times \He_3$ x 2, $C_6\times \He_3$

Order 160: $Q_8:F_5$, $D_{20}:C_4$, $C_{40}:C_4$, $C_8\times F_5$, $Q_8\times F_5$, $D_4\times F_5$, $D_5\times \SD_{16}$

Order 144: $F_9:C_2$ x 13, $S_3\times S_4$ x 12, $S_3^2:C_2^2$ x 9, $D_6^2$ x 6, $C_2\times F_9$ x 6, $D_6:D_6$ x 6, $C_2\times \PSU(3,2)$ x 5, $C_{12}:D_6$ x 4, $D_6:D_6$ x 4, $S_3\times D_{12}$ x 3, $S_3^2:C_4$ x 3, $C_6^2:C_4$ x 2, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $C_{12}:D_6$ x 2, $C_{12}:D_6$ x 2, $C_4\times S_3^2$ x 2, $C_6^2:C_4$ x 2, $S_3\times \SL(2,3)$ x 2, $C_3:\GL(2,3)$ x 2, $C_3\times \GL(2,3)$ x 2, $C_2.\PSU(3,2)$ x 2, $C_6\times \SL(2,3)$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$, $C_2.\SOPlus(4,2)$

Order 135: $C_5\times \He_3$

Order 128: $D_4\times \SD_{16}$

Order 120: $S_3\times F_5$ x 9, $S_5$ x 3, $S_3\times D_{10}$ x 2, $C_{30}:C_4$ x 2, $C_6\times F_5$ x 2, $C_2\times A_5$ x 2, $C_5\times \SL(2,3)$

Order 108: $C_3\times S_3^2$ x 23, $C_3:S_3^2$ x 20, $C_3^2:D_6$ x 19, $C_3^2:D_6$ x 14, $C_3^2:D_6$ x 8, $C_3^2\times D_6$ x 7, $C_3:S_3^2$ x 7, $C_3^2:D_6$ x 6, $C_3^2\times A_4$ x 3, $C_3^2:A_4$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2, $C_2^2\times \He_3$ x 2, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $S_3\times \SD_{16}$ x 8, $C_2\times \GL(2,3)$ x 6, $D_4\times D_6$ x 3, $C_2^2\times \SL(2,3)$ x 2, $C_8\times A_4$, $C_4\times \SL(2,3)$, $\SL(2,3):C_4$, $A_4:C_8$, $C_2^2\times S_4$, $Q_8\times D_6$, $Q_8:A_4$, $Q_8\times A_4$, $D_4\times A_4$, $\GL(2,\mathbb{Z}/4)$, $C_4:S_4$, $C_4\times S_4$, $A_4:Q_8$, $C_6\times \SD_{16}$, $Q_8:D_6$, $C_6:\SD_{16}$, $C_8:D_6$, $C_8\times D_6$

Order 90: $C_3\times D_{15}$ x 4, $S_3\times C_{15}$ x 4, $C_3^2\times D_5$ x 3, $C_3:D_{15}$ x 2, $C_{15}:S_3$ x 2

Order 81: $C_3\times \He_3$

Order 80: $C_{20}:C_4$ x 2, $C_{40}:C_2$, $C_2^2\times F_5$, $Q_8\times D_5$, $C_8\times D_5$, $D_4\times D_5$, $D_{10}:C_4$, $C_4\times F_5$, $D_5:C_8$, $C_5\times \SD_{16}$, $Q_8:D_5$, $C_5:\SD_{16}$

Order 72: $S_3\times D_6$ x 47, $C_6\times D_6$ x 12, $S_3\times A_4$ x 9, $C_3\times S_4$ x 9, $\SOPlus(4,2)$ x 9, $C_3:S_4$ x 8, $C_6\wr C_2$ x 8, $C_6:D_6$ x 6, $C_2\times C_3^2:C_4$ x 6, $C_3:D_{12}$ x 6, $\PSU(3,2)$ x 5, $F_9$ x 4, $C_6^2:C_2$ x 4, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $D_4\times C_3^2$ x 3, $D_6:S_3$ x 3, $C_6.D_6$ x 3, $C_6\times A_4$ x 2, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2, $C_6.D_6$ x 2, $C_3\times \SL(2,3)$

Order 64: $C_2^2\times \SD_{16}$ x 2, $C_8:D_4$ x 2, $C_2^2:\SD_{16}$ x 2, $Q_8:D_4$ x 2, $D_4\times Q_8$, $D_4^2$, $C_8:D_4$, $C_4:\SD_{16}$, $Q_8:D_4$, $C_4\times \SD_{16}$, $C_8\times D_4$

Order 60: $S_3\times D_5$ x 9, $C_{15}:C_4$ x 5, $C_3\times F_5$ x 5, $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_3\times D_{10}$ x 2, $A_5$

Order 54: $C_3^2:C_6$ x 14, $S_3\times C_3^2$ x 14, $C_3^2:C_6$ x 12, $C_3^2:S_3$ x 9, $C_2\times \He_3$ x 5, $C_3^2:S_3$ x 4, $C_3^2\times C_6$ x 3

Order 48: $S_3\times D_4$ x 23, $C_2\times S_4$ x 6, $\GL(2,3)$ x 6, $C_2^2\times D_6$ x 5, $C_6:D_4$ x 5, $C_{24}:C_2$ x 4, $S_3\times Q_8$ x 4, $S_3\times C_8$ x 4, $C_3\times \SD_{16}$ x 4, $Q_8:S_3$ x 4, $C_3:\SD_{16}$ x 4, $C_6\times D_4$ x 3, $C_2\times D_{12}$ x 3, $C_4\times D_6$ x 3, $C_2\times \SL(2,3)$ x 3, $C_6:C_8$, $C_2^2\times A_4$, $C_6\times Q_8$, $C_6:Q_8$, $C_4\times A_4$, $A_4:C_4$, $C_2\times C_{24}$

Order 45: $C_3\times C_{15}$ x 3

Order 40: $C_2\times F_5$ x 5, $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$, $C_5:C_8$, $C_{40}$, $C_2\times D_{10}$, $C_5\times Q_8$, $C_5\times D_4$, $C_5:C_8$

Order 36: $S_3^2$ x 67, $C_6\times S_3$ x 54, $C_6:S_3$ x 26, $C_6^2$ x 8, $C_3\times A_4$ x 8, $C_3^2:C_4$ x 4, $C_3:C_{12}$ x 4, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 2

Order 32: $C_2\times \SD_{16}$ x 13, $D_4:C_4$ x 3, $C_2^2\times D_4$ x 3, $C_4:D_4$ x 3, $Q_8:C_4$ x 3, $C_2^2:C_8$ x 2, $C_2^2\times Q_8$ x 2, $C_2^2\times C_8$ x 2, $C_2^2:Q_8$ x 2, $C_2^2\wr C_2$ x 2, $C_4\times D_4$ x 2, $C_4:Q_8$, $C_4:D_4$, $C_4\times C_8$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$

Order 30: $D_{15}$ x 5, $C_3\times D_5$ x 5, $C_5\times S_3$ x 5, $C_{30}$ x 2

Order 27: $\He_3$ x 4, $C_3^3$ x 3

Order 24: $C_2\times D_6$ x 44, $C_3:D_4$ x 22, $C_4\times S_3$ x 13, $D_{12}$ x 12, $C_3\times D_4$ x 12, $S_4$ x 8, $C_2^2\times C_6$ x 5, $C_2\times A_4$ x 4, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_3:Q_8$ x 3, $\SL(2,3)$ x 3, $C_{24}$ x 2, $C_3\times Q_8$ x 2, $C_3:C_8$ x 2

Order 20: $D_{10}$ x 4, $F_5$ x 4, $C_2\times C_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 48, $C_3:S_3$ x 22, $C_3\times C_6$ x 18

Order 16: $C_2\times D_4$ x 17, $\SD_{16}$ x 13, $C_2\times C_8$ x 6, $C_2^2:C_4$ x 5, $C_2\times Q_8$ x 5, $C_4:C_4$ x 4, $C_2^2\times C_4$ x 3, $C_2^4$ x 2, $C_4^2$

Order 15: $C_{15}$ x 3

Order 12: $D_6$ x 78, $C_2\times C_6$ x 23, $C_{12}$ x 7, $C_3:C_4$ x 7, $A_4$ x 4

Order 10: $D_5$ x 3, $C_{10}$ x 2

Order 9: $C_3^2$ x 11

Order 8: $D_4$ x 19, $C_2^3$ x 12, $C_2\times C_4$ x 10, $Q_8$ x 5, $C_8$ x 4

Order 6: $S_3$ x 31, $C_6$ x 24

Order 5: $C_5$

Order 4: $C_2^2$ x 20, $C_4$ x 7

Order 3: $C_3$ x 7

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 51840: $S_5\times C_3^2:\GL(2,3)$ ($C_1$)

Order 25920: $(C_3\times \GL(2,4)):\GL(2,3)$ ($C_2$), $A_5\times C_3^2:\GL(2,3)$ ($C_2$), $S_5\times \PU(3,2)$ ($C_2$)

Order 12960: $A_5\times \PU(3,2)$ ($C_2^2$)

Order 8640: $S_5\times \PSU(3,2)$ ($S_3$)

Order 4320: $A_5\times \PSU(3,2)$ ($D_6$)

Order 2160: $\GL(2,4):D_6$ ($S_4$)

Order 1080: $\GL(2,4):S_3$ ($C_2\times S_4$), $C_3^2\times S_5$ ($\GL(2,3)$), $C_3^2:S_5$ ($\GL(2,3)$)

Order 540: $C_3\times \GL(2,4)$ ($C_2\times \GL(2,3)$)

Order 432: $C_3^2:\GL(2,3)$ ($S_5$)

Order 216: $\PU(3,2)$ ($C_2\times S_5$)

Order 120: $S_5$ ($C_3^2:\GL(2,3)$)

Order 72: $\PSU(3,2)$ ($S_3\times S_5$)

Order 60: $A_5$ ($C_2\times C_3^2:\GL(2,3)$)

Order 18: $C_3:S_3$ ($S_4\times S_5$)

Order 9: $C_3^2$ ($S_5\times \GL(2,3)$)

Order 1: $C_1$ ($S_5\times C_3^2:\GL(2,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 51840: $S_5\times C_3^2:\GL(2,3)$ ($C_1$)

Order 25920: $(C_3\times \GL(2,4)):\GL(2,3)$ ($C_2$), $A_5\times C_3^2:\GL(2,3)$ ($C_2$), $S_5\times \PU(3,2)$ ($C_2$)

Order 12960: $A_5\times \PU(3,2)$ ($C_2^2$)

Order 8640: $S_5\times \PSU(3,2)$ ($S_3$)

Order 4320: $A_5\times \PSU(3,2)$ ($D_6$)

Order 2160: $\GL(2,4):D_6$ ($S_4$)

Order 1080: $\GL(2,4):S_3$ ($C_2\times S_4$), $C_3^2\times S_5$ ($\GL(2,3)$), $C_3^2:S_5$ ($\GL(2,3)$)

Order 540: $C_3\times \GL(2,4)$ ($C_2\times \GL(2,3)$)

Order 432: $C_3^2:\GL(2,3)$ ($S_5$)

Order 216: $\PU(3,2)$ ($C_2\times S_5$)

Order 120: $S_5$ ($C_3^2:\GL(2,3)$)

Order 72: $\PSU(3,2)$ ($S_3\times S_5$)

Order 60: $A_5$ ($C_2\times C_3^2:\GL(2,3)$)

Order 18: $C_3:S_3$ ($S_4\times S_5$)

Order 9: $C_3^2$ ($S_5\times \GL(2,3)$)

Order 1: $C_1$ ($S_5\times C_3^2:\GL(2,3)$)

Series

Derived series

$S_5\times C_3^2:\GL(2,3)$

$\rhd$

$S_5\times C_3^2:\GL(2,3)$

$\rhd$

$A_5\times \PU(3,2)$

$\rhd$

$A_5\times \PU(3,2)$

$\rhd$

$A_5\times \PSU(3,2)$

$\rhd$

$A_5\times \PSU(3,2)$

$\rhd$

$\GL(2,4):S_3$

$\rhd$

$\GL(2,4):S_3$

$\rhd$

$C_3\times \GL(2,4)$

$\rhd$

$C_3\times \GL(2,4)$

$\rhd$

$A_5$

$\rhd$

$A_5$

Chief series

$S_5\times C_3^2:\GL(2,3)$

$\rhd$

$S_5\times C_3^2:\GL(2,3)$

$\rhd$

$A_5\times C_3^2:\GL(2,3)$

$\rhd$

$A_5\times C_3^2:\GL(2,3)$

$\rhd$

$C_3^2:\GL(2,3)$

$\rhd$

$C_3^2:\GL(2,3)$

$\rhd$

$\PU(3,2)$

$\rhd$

$\PU(3,2)$

$\rhd$

$\PSU(3,2)$

$\rhd$

$\PSU(3,2)$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$S_5\times C_3^2:\GL(2,3)$

$\rhd$

$S_5\times C_3^2:\GL(2,3)$

$\rhd$

$A_5\times \PU(3,2)$

$\rhd$

$A_5\times \PU(3,2)$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $77 \times 77$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $70 \times 70$ rational character table.