Abstract group 103680.g: $F_9:C_2\times S_6$

• Label: 103680.g
• Order: \( 2^{8} \cdot 3^{4} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$F_9:C_2\times S_6$
Order:	\(103680\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$\PSU(3,2):C_2.A_6.C_2^2$, of order \(207360\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2, $A_6$
Derived length:	$3$

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	1671	728	17784	144	18984	9216	3024	23040	1152	7776	11520	3456	5184	103680
Conjugacy classes	1	11	5	18	1	23	12	2	12	1	2	8	1	2	99
Divisions	1	11	5	18	1	23	6	2	12	1	2	4	1	1	88
Autjugacy classes	1	8	3	16	1	13	10	2	8	1	2	4	1	2	72

Dimension	1	2	4	5	8	9	10	16	18	20	32	36	40	64	72	80	128
Irr. complex chars.	8	6	0	16	4	8	20	4	6	6	3	0	8	0	4	4	2	99
Irr. rational chars.	8	2	2	16	4	8	12	4	2	6	1	2	10	1	4	4	2	88

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$1619896320$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	40	40	40
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(7,12)(8,11)(9,10)(13,15), (7,8,9)(10,11,12)(13,14,15), (7,12)(9,13)(10,15) \!\cdots\! \rangle$
Direct product:	$(F_9:C_2)$ $\, \times\, $ $S_6$
Semidirect product:	$(F_9\times S_6)$ $\,\rtimes\,$ $C_2$	$(F_9:S_6)$ $\,\rtimes\,$ $C_2$	$F_9$ $\,\rtimes\,$ $(C_2\times S_6)$	$(F_9\times A_6)$ $\,\rtimes\,$ $C_2^2$	all 17
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3:S_3\times S_6)$ . $D_4$	$(C_3:S_3)$ . $(D_4\times S_6)$	$(C_3^2:C_4\times S_6)$ . $C_2^2$	$((C_3^2\times A_6):C_4)$ . $D_4$	all 11

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1712050 subgroups in 5351 conjugacy classes, 32 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $F_9:C_2\times S_6$
Commutator:	$G' \simeq$ $C_3^2:C_4\times A_6$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $F_9:C_2\times S_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2$	$G/\operatorname{Fit} \simeq$ $\SD_{16}\times S_6$
Radical:	$R \simeq$ $F_9:C_2$	$G/R \simeq$ $S_6$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times A_6$	$G/\operatorname{soc} \simeq$ $C_2\times \SD_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$
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 103680: $F_9:C_2\times S_6$

Order 51840: $S_6\times \SOPlus(4,2)$, $S_6\times \PSU(3,2)$, $\PSU(3,2):S_6$, $F_9:C_2\times A_6$, $\SOPlus(4,2).S_6$, $F_9:S_6$, $F_9\times S_6$

Order 25920: $C_3^2:C_4\times S_6$ x 2, $A_6:\PSU(3,2)$ x 2, $A_6:\SOPlus(4,2)$, $S_3^2\times S_6$, $S_3^2:S_6$, $A_6\times \SOPlus(4,2)$, $A_6:F_9$, $F_9\times A_6$, $A_6\times \PSU(3,2)$

Order 17280: $F_9:C_2\times S_5$ x 2

Order 12960: $(C_3^2\times A_6):C_4$ x 2, $C_3^2:C_4\times A_6$ x 2, $C_3:S_3\times S_6$, $A_6:S_3^2$, $C_3\times S_3\times S_6$, $A_6:S_3^2$, $S_3^2\times A_6$

Order 11520: $\SD_{16}\times S_6$

Order 10368: $\SOPlus(4,2)^2.C_2$

Order 8640: $F_9\times S_5$ x 2, $S_5\times \PSU(3,2)$ x 2, $S_5\times \SOPlus(4,2)$ x 2, $\SOPlus(4,2).S_5$ x 2, $A_5\times F_9:C_2$ x 2, $\PSU(3,2):S_5$ x 2, $F_9:S_5$ x 2, $D_6\times S_6$

Order 6912: $F_9:C_2^2\times S_4$ x 2

Order 6480: $C_3:S_3\times A_6$, $C_3^2:S_6$, $C_3^2\times S_6$, $C_3^2:S_6$, $C_3\times S_3\times A_6$

Order 5760: $C_8\times S_6$, $C_8:S_6$, $\SD_{16}\times A_6$, $A_6:\SD_{16}$, $Q_8\times S_6$, $Q_8:S_6$, $D_4\times S_6$

Order 5184: $F_9:\SOPlus(4,2)$ x 2, $(C_3^2\times S_3^2):\SD_{16}$ x 2, $S_3\times F_9:D_6$ x 2, $(C_3^2\times S_3^2):\SD_{16}$ x 2, $F_9\times \SOPlus(4,2)$, $F_9:\SOPlus(4,2)$, $C_3^3:C_{12}:\SD_{16}$, $C_3^4:(C_4\times \SD_{16})$, $C_3^4:(D_4\times Q_8)$, $C_3^3:C_{12}:\SD_{16}$, $\SOPlus(4,2)^2$

Order 4320: $A_5:\PSU(3,2)$ x 4, $C_3^2:C_4\times S_5$ x 4, $S_3\times S_6$ x 4, $A_5:F_9$ x 2, $A_5\times F_9$ x 2, $A_5\times \PSU(3,2)$ x 2, $A_5:\SOPlus(4,2)$ x 2, $S_3^2\times S_5$ x 2, $S_3^2:S_5$ x 2, $A_5\times \SOPlus(4,2)$ x 2, $C_6:S_6$, $D_6\times A_6$, $C_6\times S_6$

Order 3456: $F_9:C_2\times S_4$ x 16, $A_4\times F_9:C_2^2$ x 2, $C_2\times S_4\times F_9$ x 2, $C_2\times F_9:S_4$ x 2, $A_4:F_9:C_2^2$ x 2, $A_4:F_9:C_2^2$ x 2, $C_6^2:(Q_8\times D_6)$ x 2, $C_2\times D_6^2:D_6$ x 2

Order 3240: $C_3^2\times A_6$

Order 2880: $A_6:Q_8$ x 2, $C_4\times S_6$ x 2, $C_8\times A_6$, $A_6:C_8$, $Q_8\times A_6$, $F_5\times F_9:C_2$, $C_4:S_6$, $C_2^2:S_6$, $D_4\times A_6$, $C_2^2\times S_6$

Order 2592: $S_3^2:\PSU(3,2)$ x 4, $S_3^2:\SOPlus(4,2)$ x 3, $C_3^4:(C_4\times D_4)$ x 3, $S_3^4:C_2$ x 3, $C_3^3:C_{12}:Q_8$ x 2, $C_3^2\wr C_2.\SD_{16}$ x 2, $F_9:S_3^2$ x 2, $F_9:S_3^2$ x 2, $S_3^2\times F_9$ x 2, $\PSU(3,2):S_3^2$ x 2, $S_3^2\times \PSU(3,2)$ x 2, $\PSU(3,2):S_3^2$ x 2, $S_3\times C_3^3:\SD_{16}$ x 2, $F_9:C_6\times S_3$ x 2, $C_3^2\wr C_2.\SD_{16}$ x 2, $C_3^4:(C_2\times \SD_{16})$ x 2, $S_3^2:F_9$ x 2, $\SOPlus(4,2).S_3^2$ x 2, $S_3^2\wr C_2$ x 2, $C_3^3:C_{12}:C_8$, $(C_3^2\times F_9):C_4$, $C_3^2:C_4\times F_9$, $(C_3\times F_9):D_6$, $C_3^2\wr C_2.\SD_{16}$, $C_3^4:(C_4\times Q_8)$, $C_3^3:C_{12}:D_4$, $C_3^2\wr C_2.\SD_{16}$

Order 2304: $D_4\times F_9:C_2^2$

Order 2160: $C_3^2:C_4\times A_5$ x 4, $(C_3\times \GL(2,4)):C_4$ x 4, $\GL(2,4):D_6$ x 2, $S_3^2\times A_5$ x 2, $A_5:S_3^2$ x 2, $A_5:S_3^2$ x 2, $\GL(2,4):D_6$ x 2, $C_3:S_6$ x 2, $S_3\times A_6$ x 2, $C_3\times S_6$ x 2, $C_6\times A_6$

Order 1920: $\SD_{16}\times S_5$ x 2

Order 1728: $S_4\times \PSU(3,2)$ x 12, $D_6^2:D_6$ x 12, $A_4\times F_9:C_2$ x 8, $\PSU(3,2):S_4$ x 8, $\SOPlus(4,2).S_4$ x 8, $F_9:S_4$ x 8, $S_4\times F_9$ x 8, $C_6^2:(C_4\times D_6)$ x 4, $C_6^2:C_6:Q_8$ x 4, $A_4:D_6^2$ x 2, $C_6^2:(C_6\times Q_8)$ x 2, $C_2\times D_6^2:C_6$ x 2, $D_6^2:D_6$ x 2, $C_2\times C_6^2:D_{12}$ x 2, $C_2\times A_4\times F_9$ x 2, $C_2\times A_4:F_9$ x 2, $C_2\times F_9:D_6$ x 2

Order 1440: $C_2\times S_6$ x 4, $D_6\times S_5$ x 2, $C_4\times A_6$ x 2, $A_6:C_4$ x 2, $F_9:D_{10}$, $F_5\times \PSU(3,2)$, $F_5\times \SOPlus(4,2)$, $F_9:F_5$, $\PSU(3,2):F_5$, $\SOPlus(4,2):F_5$, $F_5\times F_9$, $C_2^2\times A_6$

Order 1296: $C_3^2:C_4\times S_3^2$ x 5, $(C_3^2\times S_3^2):C_4$ x 5, $C_3^3:(S_3\times Q_8)$ x 4, $S_3\times C_3^3:Q_8$ x 4, $S_3^2:S_3^2$ x 4, $S_3^3:S_3$ x 4, $C_3^3:(S_3\times D_4)$ x 3, $S_3\times C_3^2:D_{12}$ x 3, $S_3^3:C_6$ x 3, $C_3^2\wr C_2.D_4$ x 3, $C_3^2\wr C_2.D_4$ x 3, $S_3^4$ x 2, $S_3\times C_3^3:Q_8$ x 2, $(C_3\times S_3^2):D_6$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3:(S_3\times F_9)$ x 2, $S_3\times C_3:F_9$ x 2, $C_3\times S_3\times F_9$ x 2, $C_3^2\wr C_2.Q_8$ x 2, $C_3^2\wr C_2.D_4$ x 2, $C_3^4:C_4^2$ x 2, $C_3^3:(S_3\times Q_8)$, $C_3^4:(C_2\times D_4)$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^2:(C_2\times F_9)$, $C_3:S_3\times F_9$

Order 1152: $D_4\times F_9:C_2$ x 16, $C_2\times C_6^2:\SD_{16}$ x 2, $C_2\times C_6^2:\SD_{16}$ x 2, $F_9:C_2^4$ x 2, $C_2\times F_9:D_4$ x 2, $(C_6\times C_{12}):\SD_{16}$, $(C_6\times C_{12}):\SD_{16}$, $C_4\times F_9:C_2^2$, $C_2\times D_4\times F_9$, $C_6^2:(C_2^2\times Q_8)$, $D_6^2:C_2^3$, $\SD_{16}\times \SOPlus(4,2)$, $C_2\times F_9:D_4$

Order 1080: $S_3\times \GL(2,4)$ x 2, $\GL(2,4):S_3$ x 2, $C_3^2\times S_5$ x 2, $C_3^2:S_5$ x 2, $C_3^2:S_5$ x 2, $C_3\times A_6$

Order 960: $\SD_{16}\times A_5$ x 2, $Q_8:S_5$ x 2, $A_5:\SD_{16}$ x 2, $C_8:S_5$ x 2, $C_8\times S_5$ x 2, $Q_8\times S_5$ x 2, $D_4\times S_5$ x 2

Order 864: $C_3^2:C_4\times S_4$ x 16, $F_9:D_6$ x 16, $A_4:\PSU(3,2)$ x 14, $S_4\times S_3^2$ x 12, $S_3^2:S_4$ x 8, $A_4\times \PSU(3,2)$ x 6, $D_6^2:C_6$ x 6, $C_6^2:D_{12}$ x 6, $C_2\times C_6^2:C_{12}$ x 4, $C_6^2:C_6:C_4$ x 4, $A_4\times F_9$ x 4, $A_4:F_9$ x 4, $S_3^3:C_2^2$ x 3, $D_6^2:C_6$ x 2, $C_2\times A_4:S_3^2$ x 2, $S_3\times C_6:S_4$ x 2, $C_6:S_3\times S_4$ x 2, $C_6\times S_3\times S_4$ x 2, $D_6\times \PSU(3,2)$ x 2, $F_9:D_6$ x 2, $\PSU(3,2):D_6$ x 2, $C_2\times C_3^3:\SD_{16}$ x 2, $C_2\times F_9:C_6$ x 2, $D_6\times F_9$ x 2

Order 768: $C_2\times \SD_{16}\times S_4$ x 2

Order 720: $S_3\times S_5$ x 8, $S_6$ x 3, $D_5.\PSU(3,2)$ x 2, $C_3^2:C_4\times F_5$ x 2, $D_6\times A_5$ x 2, $C_6:S_5$ x 2, $C_6\times S_5$ x 2, $C_2\times A_6$ x 2, $D_5\times \PSU(3,2)$, $S_3^2:D_{10}$, $F_5\times S_3^2$, $F_9:D_5$, $\PSU(3,2):D_5$, $(C_3\times C_{15}):\SD_{16}$, $F_9:C_{10}$, $D_5\times F_9$, $D_5:F_9$, $D_5.\SOPlus(4,2)$, $S_3^2:F_5$

Order 648: $C_3:S_3^3$ x 5, $C_3^3:(C_4\times S_3)$ x 5, $C_3^3:C_4\times S_3$ x 5, $S_3\times C_3^2:C_{12}$ x 5, $C_3:S_3^3$ x 4, $C_3\times S_3^3$ x 4, $C_3^4:Q_8$ x 4, $C_3^4:Q_8$ x 4, $C_3^3:D_{12}$ x 4, $C_3\wr D_4$ x 4, $C_3^4:D_4$ x 3, $C_3^4:D_4$ x 3, $C_3^3:D_{12}$ x 3, $C_3^3:(C_4\times S_3)$ x 3, $C_3:S_3^3$ x 2, $C_3^4:Q_8$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3^4:(C_2\times C_4)$ x 2, $C_3^3:C_{24}$ x 2, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^2:F_9$, $C_3^2:F_9$, $C_3^2\times F_9$

Order 576: $F_9:C_2^3$ x 25, $C_6^2:\SD_{16}$ x 16, $C_6^2:\SD_{16}$ x 16, $F_9:D_4$ x 16, $C_3^2:D_4^2$ x 13, $D_4\times \PSU(3,2)$ x 12, $D_4\times F_9$ x 8, $\PSU(3,2):D_4$ x 8, $(C_3\times C_{12}):\SD_{16}$ x 8, $F_9:D_4$ x 8, $C_4\times F_9:C_2$ x 8, $C_2\times C_6^2:Q_8$ x 4, $C_6^2.\SD_{16}$ x 3, $C_6^2.\SD_{16}$ x 3, $C_2^3\times \PSU(3,2)$ x 2, $D_6^2:C_2^2$ x 2, $C_2^3\times F_9$ x 2, $C_2^2:D_6^2$ x 2, $C_6^2:(C_2^2\times C_4)$ x 2, $(C_6\times C_{12}):Q_8$ x 2, $D_6^2.C_2^2$ x 2, $D_6^2:C_2^2$ x 2, $C_2^3:F_9$ x 2, $\SD_{16}\times S_3^2$ x 2, $S_3^2:\SD_{16}$ x 2, $S_3^2:\SD_{16}$ x 2, $S_3^2:\SD_{16}$ x 2, $C_4:D_6^2$, $(C_6\times C_{12}):Q_8$, $Q_8\times \SOPlus(4,2)$, $D_6^2.C_2^2$, $D_6^2.C_2^2$, $D_6^2.C_2^2$, $C_2\times C_4:F_9$, $C_2\times C_4\times F_9$, $C_6^2.\SD_{16}$, $C_3^2:C_4\times \SD_{16}$, $Q_8:\SOPlus(4,2)$, $D_4.\SOPlus(4,2)$, $C_8:\SOPlus(4,2)$, $C_8\times \SOPlus(4,2)$

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

Order 480: $A_5:Q_8$ x 4, $C_4\times S_5$ x 4, $Q_8\times A_5$ x 2, $D_4\times A_5$ x 2, $C_2^2:S_5$ x 2, $C_4:S_5$ x 2, $C_8\times A_5$ x 2, $A_5:C_8$ x 2, $C_2^2\times S_5$ x 2

Order 432: $S_3^3:C_2$ x 20, $S_3\times \PSU(3,2)$ x 12, $A_4:S_3^2$ x 8, $C_6^2:D_6$ x 8, $C_6^2:D_6$ x 8, $C_6^2:C_{12}$ x 8, $(C_3^2\times A_4):C_4$ x 8, $F_9:S_3$ x 8, $C_3^3:\SD_{16}$ x 8, $C_3^3:\SD_{16}$ x 8, $F_9:C_6$ x 8, $S_3\times F_9$ x 8, $A_4\times S_3^2$ x 6, $A_4:S_3^2$ x 6, $C_3^2:C_4\times D_6$ x 5, $C_2\times S_3^3$ x 4, $C_6:\PSU(3,2)$ x 4, $S_3^2:D_6$ x 4, $C_2\times C_3^2:D_{12}$ x 3, $C_6\times \SOPlus(4,2)$ x 3, $C_6:S_3\times A_4$ x 2, $C_6\times S_3\times A_4$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $A_4:C_6^2$ x 2, $C_6\times \PSU(3,2)$ x 2, $C_6:F_9$ x 2, $C_6\times F_9$ x 2

Order 384: $\SD_{16}\times S_4$ x 16, $C_2\times Q_8\times S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2\times A_4\times \SD_{16}$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.D_{12}$ x 2, $C_2\times C_8\times S_4$ x 2

Order 360: $S_3\times A_5$ x 4, $C_3:S_5$ x 4, $C_3\times S_5$ x 4, $C_5:\PSU(3,2)$ x 2, $(C_3^2\times D_5):C_4$ x 2, $C_3^2:C_4\times D_5$ x 2, $C_6\times A_5$ x 2, $D_5\times S_3^2$, $C_5\times \PSU(3,2)$, $C_3^2:D_{20}$, $S_3^2:D_5$, $S_3^2:C_{10}$, $D_5.S_3^2$, $D_5.S_3^2$, $C_3:S_3\times F_5$, $D_{15}:C_{12}$, $C_5:F_9$, $C_5:F_9$, $C_5\times F_9$, $A_6$

Order 324: $C_3^2:S_3^2$ x 7, $C_3^2:S_3^2$ x 5, $C_3^2\times S_3^2$ x 5, $C_3\wr C_4$ x 5, $C_3\wr C_2^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$ x 3, $C_3^4:C_4$ x 3, $C_3^3:C_{12}$ x 3, $C_3^4:C_4$ x 2

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

Order 288: $F_9:C_2^2$ x 62, $C_6^2:Q_8$ x 28, $D_6\times S_4$ x 26, $C_6^2:D_4$ x 19, $D_6\wr C_2$ x 18, $C_3^2:C_4\times D_4$ x 17, $C_6^2:Q_8$ x 17, $D_4\times S_3^2$ x 14, $C_4:\PSU(3,2)$ x 14, $C_6^2:D_4$ x 13, $\PSU(3,2):C_4$ x 12, $\SOPlus(4,2):C_4$ x 12, $C_2^2\times F_9$ x 12, $S_3^2:D_4$ x 10, $C_4\times \SOPlus(4,2)$ x 8, $C_6^2:C_8$ x 8, $C_4:\SOPlus(4,2)$ x 7, $C_4\times \PSU(3,2)$ x 6, $C_6^2.Q_8$ x 5, $C_2\times C_6^2:C_4$ x 4, $S_3^2:Q_8$ x 4, $C_4:F_9$ x 4, $C_4\times F_9$ x 4, $F_9:C_4$ x 4, $C_2\times C_6^2:C_4$ x 4, $C_6^2.D_4$ x 3, $D_6^2:C_2$ x 2, $D_6^2:C_2$ x 2, $Q_8\times S_3^2$ x 2, $(C_6\times C_{12}):C_4$ x 2, $C_2\times C_3^2:C_4^2$ x 2, $C_4.\SOPlus(4,2)$ x 2, $C_4.\SOPlus(4,2)$ x 2, $C_{24}:D_6$ x 2, $Q_8:S_3^2$ x 2, $Q_8:S_3^2$ x 2, $D_4.S_3^2$ x 2, $D_{12}.D_6$ x 2, $C_{24}:D_6$ x 2, $C_{24}:D_6$ x 2, $C_8\times S_3^2$ x 2, $C_4.\SOPlus(4,2)$ x 2, $C_4.\SOPlus(4,2)$ x 2, $S_3^2:C_8$ x 2, $C_2\times D_6^2$ x 2, $C_2\times A_4\times D_6$ x 2, $C_2\times C_6:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $C_6^2:C_2^3$, $D_{12}:D_6$, $D_6\times D_{12}$, $C_2.D_6^2$, $C_3^2:C_4\times Q_8$, $C_6^2.D_4$, $C_{24}:D_6$, $(Q_8\times C_3^2):C_4$, $(D_4\times C_3^2):C_4$, $(C_3\times C_{24}):C_4$, $C_3^2:C_4\times C_8$, $C_4.\SOPlus(4,2)$, $C_6^2:C_2^3$

Order 256: $C_4.D_4^2$

Order 240: $C_2\times S_5$ x 8, $C_4\times A_5$ x 4, $A_5:C_4$ x 4, $C_2^2\times A_5$ x 2, $D_6\times F_5$

Order 216: $S_3^3$ x 26, $C_3^2:C_4\times S_3$ x 20, $S_3^2:S_3$ x 16, $C_3^3:Q_8$ x 14, $C_3^2:D_{12}$ x 10, $S_3^2:C_6$ x 10, $C_6\times S_3^2$ x 8, $C_6:S_3^2$ x 7, $C_3^3:Q_8$ x 6, $C_2\times C_3^3:C_4$ x 5, $C_2\times C_3^2:C_{12}$ x 5, $C_6:S_3^2$ x 4, $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 4, $C_3^2:S_4$ x 4, $C_3^2\times S_4$ x 4, $C_3:F_9$ x 4, $C_3\times F_9$ x 4, $C_6^2:C_6$ x 2

Order 192: $Q_8\times S_4$ x 12, $D_4\times S_4$ x 12, $Q_8:S_4$ x 8, $A_4:\SD_{16}$ x 8, $C_8:S_4$ x 8, $C_8\times S_4$ x 8, $A_4\times \SD_{16}$ x 8, $C_2^4.D_6$ x 4, $C_2^4.D_6$ x 4, $C_2\times A_4:C_8$ x 2, $C_2^3\times S_4$ x 2, $C_2\times Q_8\times A_4$ x 2, $C_2^5:C_6$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:D_{12}$ x 2, $D_6\times \SD_{16}$ x 2, $C_2^3:C_{24}$ x 2, $C_{12}:C_2^4$

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

Order 162: $C_3^2\wr C_2$ x 4, $C_3^3:C_6$ x 3, $S_3\times C_3^3$ x 3, $C_3^3:S_3$

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: $S_3^2:C_2^2$ x 48, $S_3\times S_4$ x 40, $C_2\times \PSU(3,2)$ x 33, $F_9:C_2$ x 32, $C_6^2:C_4$ x 25, $D_6^2$ x 21, $C_2.\PSU(3,2)$ x 19, $D_6:D_6$ x 18, $S_3^2:C_4$ x 18, $C_6^2:C_4$ x 17, $C_2\times F_9$ x 16, $A_4\times D_6$ x 14, $C_6:S_4$ x 14, $C_6\times S_4$ x 14, $D_6:D_6$ x 14, $(C_3\times C_{12}):C_4$ x 11, $C_{12}:D_6$ x 10, $S_3\times D_{12}$ x 10, $C_4\times S_3^2$ x 10, $C_3^2:C_4^2$ x 10, $C_{12}:D_6$ x 9, $C_{12}:D_6$ x 8, $C_2.\SOPlus(4,2)$ x 8, $C_{12}.D_6$ x 4, $C_{12}.D_6$ x 4, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{24}:C_6$ x 2, $S_3\times C_{24}$ x 2, $C_6.D_{12}$ x 2, $C_6.D_{12}$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$ x 2, $C_2^2:C_6^2$ x 2, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}.D_6$ x 2, $C_6:D_{12}$ x 2, $C_3^2:\SD_{16}$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:S_3$, $C_{24}:S_3$, $C_4:C_6^2$, $C_{12}.D_6$, $C_6:D_{12}$, $C_{12}:D_6$, $C_6\times D_{12}$, $C_{12}\times D_6$, $D_6:D_6$, $C_2^2.S_3^2$, $D_6.D_6$, $C_3^2:(C_2\times C_8)$, $C_3^2\times \SD_{16}$

Order 128: $D_4\times \SD_{16}$ x 16, $C_2^3\times \SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_4^2.C_2^3$, $C_2\times D_4^2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.C_2^3$

Order 120: $S_5$ x 6, $S_3\times F_5$ x 4, $C_2\times A_5$ x 4, $S_3\times D_{10}$, $C_{30}:C_4$, $C_6\times F_5$

Order 108: $C_3:S_3^2$ x 34, $C_3\times S_3^2$ x 34, $C_3:S_3^2$ x 14, $C_3^3:C_4$ x 10, $C_3^2:C_{12}$ x 10, $C_3^2:D_6$ x 7, $C_3^2\times D_6$ x 7, $C_3^2:D_6$ x 3, $C_3^2\times A_4$ x 2

Order 96: $D_4\times D_6$ x 26, $C_2^2\times S_4$ x 22, $C_4\times S_4$ x 16, $S_3\times \SD_{16}$ x 16, $A_4:Q_8$ x 14, $\GL(2,\mathbb{Z}/4)$ x 8, $Q_8\times A_4$ x 6, $D_4\times A_4$ x 6, $C_4:S_4$ x 6, $C_8\times A_4$ x 4, $A_4:C_8$ x 4, $C_2^3:C_{12}$ x 4, $C_2^2.S_4$ x 4, $C_2^3\times D_6$ x 2, $C_2^3\times A_4$ x 2, $C_2^3:D_6$ x 2, $Q_8\times D_6$ x 2, $C_6\times \SD_{16}$ x 2, $Q_8:D_6$ x 2, $C_6:\SD_{16}$ x 2, $C_8:D_6$ x 2, $C_8\times D_6$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

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

Order 81: $C_3^4$

Order 80: $C_{20}:C_4$ x 3, $C_4\times F_5$ x 2, $C_{40}:C_2$, $C_2^2\times F_5$, $Q_8\times D_5$, $C_8\times D_5$, $D_4\times D_5$, $D_{10}:C_4$, $D_5:C_8$, $C_5\times \SD_{16}$, $Q_8:D_5$, $C_5:\SD_{16}$

Order 72: $S_3\times D_6$ x 73, $C_2\times C_3^2:C_4$ x 38, $\SOPlus(4,2)$ x 32, $C_6\times D_6$ x 17, $\PSU(3,2)$ x 16, $C_6:D_6$ x 14, $S_3\times A_4$ x 12, $C_3:S_4$ x 12, $C_3\times S_4$ x 12, $C_6\wr C_2$ x 10, $C_3:D_{12}$ x 10, $C_6^2:C_2$ x 9, $C_6\times A_4$ x 8, $S_3\times C_{12}$ x 8, $C_6.D_6$ x 8, $C_6.D_6$ x 7, $F_9$ x 6, $C_{12}:S_3$ x 6, $C_3\times D_{12}$ x 6, $D_6:S_3$ x 6, $D_4\times C_3^2$ x 5, $C_3:D_{12}$ x 5, $C_3^2:Q_8$ x 4, $C_3^2:Q_8$ x 4, $C_2\times C_6^2$ x 2, $C_3^2:Q_8$ x 2, $C_3:C_{24}$ x 2, $Q_8\times C_3^2$, $C_6\times C_{12}$, $C_6.D_6$, $C_6:C_{12}$, $C_3^2:C_8$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_2^2\times \SD_{16}$ x 25, $C_8:D_4$ x 16, $C_2^2:\SD_{16}$ x 16, $Q_8:D_4$ x 16, $D_4\times Q_8$ x 12, $D_4^2$ x 12, $C_8:D_4$ x 8, $C_4:\SD_{16}$ x 8, $Q_8:D_4$ x 8, $C_4\times \SD_{16}$ x 8, $C_8\times D_4$ x 8, $C_2^3:Q_8$ x 4, $C_2^2.Q_{16}$ x 3, $C_2^2.D_8$ x 3, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_4^2:C_2^2$ x 3, $C_2^3:C_8$ x 2, $Q_8\times C_2^3$ x 2, $C_2^3\times C_8$ x 2, $C_4^2.C_2^2$ x 2, $C_2^3:D_4$ x 2, $C_2\times C_4\times C_8$, $C_4^2:C_2^2$, $C_4^2.C_2^2$, $C_2^3.D_4$, $C_4^2.C_4$

Order 60: $S_3\times D_5$ x 4, $C_{15}:C_4$ x 2, $C_3\times F_5$ x 2, $A_5$ x 2, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$

Order 54: $C_3^2:C_6$ x 20, $S_3\times C_3^2$ x 20, $C_3^2:S_3$ x 8, $C_3^2\times C_6$ x 3

Order 48: $S_3\times D_4$ x 76, $C_2\times S_4$ x 48, $C_2^2\times D_6$ x 29, $C_6:D_4$ x 26, $C_4\times D_6$ x 16, $C_6\times D_4$ x 14, $C_2\times D_{12}$ x 14, $C_2^2\times A_4$ x 12, $S_3\times Q_8$ x 12, $C_{24}:C_2$ x 8, $S_3\times C_8$ x 8, $C_4\times A_4$ x 8, $A_4:C_4$ x 8, $C_3\times \SD_{16}$ x 8, $Q_8:S_3$ x 8, $C_3:\SD_{16}$ x 8, $C_6:Q_8$ x 4, $C_6:C_8$ x 2, $C_2^3\times C_6$ x 2, $C_6\times Q_8$ x 2, $C_2\times C_{24}$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$

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

Order 40: $C_2\times F_5$ x 6, $C_4\times D_5$ x 2, $C_5:Q_8$ x 2, $C_5:D_4$, $D_{20}$, $C_5:C_8$, $C_{40}$, $C_2\times D_{10}$, $C_5\times Q_8$, $C_5\times D_4$, $C_5:C_8$

Order 36: $S_3^2$ x 80, $C_6\times S_3$ x 52, $C_6:S_3$ x 32, $C_3^2:C_4$ x 19, $C_6^2$ x 8, $C_3:C_{12}$ x 6, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4, $C_3\times A_4$ x 4

Order 32: $C_2\times \SD_{16}$ x 62, $C_2^2\times D_4$ x 38, $C_2^2:Q_8$ x 28, $C_4\times D_4$ x 22, $C_4:D_4$ x 20, $C_2^2\times Q_8$ x 17, $C_2^2\wr C_2$ x 16, $C_4:Q_8$ x 14, $D_4:C_4$ x 12, $C_2^2\times C_8$ x 12, $Q_8:C_4$ x 12, $C_2^2:C_8$ x 8, $C_2^2.D_4$ x 8, $C_2^3:C_4$ x 7, $C_4:D_4$ x 6, $C_4\times Q_8$ x 6, $C_2^3\times C_4$ x 5, $C_4\times C_8$ x 4, $C_8:C_4$ x 4, $C_4:C_8$ x 4, $C_2^5$ x 2, $C_2\times C_4^2$ x 2

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

Order 27: $C_3^3$ x 5

Order 24: $C_2\times D_6$ x 94, $C_3:D_4$ x 40, $C_4\times S_3$ x 32, $D_{12}$ x 22, $C_3\times D_4$ x 22, $S_4$ x 20, $C_2\times A_4$ x 16, $C_2^2\times C_6$ x 15, $C_3:Q_8$ x 14, $C_2\times C_{12}$ x 10, $C_6:C_4$ x 10, $C_3\times Q_8$ x 6, $C_{24}$ x 4, $C_3:C_8$ x 4

Order 20: $F_5$ x 5, $D_{10}$ x 4, $C_{20}$ x 2, $C_5:C_4$ x 2, $C_2\times C_{10}$

Order 18: $C_3\times S_3$ x 44, $C_3:S_3$ x 25, $C_3\times C_6$ x 15

Order 16: $C_2\times D_4$ x 107, $C_2^2\times C_4$ x 34, $C_2\times Q_8$ x 33, $\SD_{16}$ x 32, $C_4:C_4$ x 30, $C_2^2:C_4$ x 28, $C_2^4$ x 21, $C_2\times C_8$ x 16, $C_4^2$ x 8

Order 15: $C_{15}$

Order 12: $D_6$ x 96, $C_2\times C_6$ x 29, $C_{12}$ x 12, $C_3:C_4$ x 12, $A_4$ x 4

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

Order 9: $C_3^2$ x 10

Order 8: $D_4$ x 60, $C_2^3$ x 57, $C_2\times C_4$ x 50, $Q_8$ x 16, $C_8$ x 6

Order 6: $S_3$ x 28, $C_6$ x 23

Order 5: $C_5$

Order 4: $C_2^2$ x 45, $C_4$ x 18

Order 3: $C_3$ x 5

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 103680: $F_9:C_2\times S_6$

Order 51840: $S_6\times \SOPlus(4,2)$, $S_6\times \PSU(3,2)$, $\PSU(3,2):S_6$, $F_9:C_2\times A_6$, $\SOPlus(4,2).S_6$, $F_9:S_6$, $F_9\times S_6$

Order 25920: $C_3^2:C_4\times S_6$ x 2, $A_6:\PSU(3,2)$ x 2, $A_6:\SOPlus(4,2)$, $S_3^2\times S_6$, $S_3^2:S_6$, $A_6\times \SOPlus(4,2)$, $A_6:F_9$, $F_9\times A_6$, $A_6\times \PSU(3,2)$

Order 17280: $F_9:C_2\times S_5$

Order 12960: $(C_3^2\times A_6):C_4$ x 2, $C_3^2:C_4\times A_6$ x 2, $C_3:S_3\times S_6$, $A_6:S_3^2$, $C_3\times S_3\times S_6$, $A_6:S_3^2$, $S_3^2\times A_6$

Order 11520: $\SD_{16}\times S_6$

Order 10368: $\SOPlus(4,2)^2.C_2$

Order 8640: $F_9\times S_5$, $S_5\times \PSU(3,2)$, $D_6\times S_6$, $S_5\times \SOPlus(4,2)$, $\SOPlus(4,2).S_5$, $A_5\times F_9:C_2$, $\PSU(3,2):S_5$, $F_9:S_5$

Order 6912: $F_9:C_2^2\times S_4$

Order 6480: $C_3:S_3\times A_6$, $C_3^2:S_6$, $C_3^2\times S_6$, $C_3^2:S_6$, $C_3\times S_3\times A_6$

Order 5760: $C_8\times S_6$, $C_8:S_6$, $\SD_{16}\times A_6$, $A_6:\SD_{16}$, $Q_8\times S_6$, $Q_8:S_6$, $D_4\times S_6$

Order 5184: $F_9\times \SOPlus(4,2)$, $F_9:\SOPlus(4,2)$, $F_9:\SOPlus(4,2)$, $C_3^3:C_{12}:\SD_{16}$, $C_3^4:(C_4\times \SD_{16})$, $C_3^4:(D_4\times Q_8)$, $C_3^3:C_{12}:\SD_{16}$, $(C_3^2\times S_3^2):\SD_{16}$, $S_3\times F_9:D_6$, $\SOPlus(4,2)^2$, $(C_3^2\times S_3^2):\SD_{16}$

Order 4320: $S_3\times S_6$ x 4, $A_5:\PSU(3,2)$ x 2, $C_3^2:C_4\times S_5$ x 2, $A_5:F_9$, $A_5\times F_9$, $A_5\times \PSU(3,2)$, $C_6:S_6$, $D_6\times A_6$, $C_6\times S_6$, $A_5:\SOPlus(4,2)$, $S_3^2\times S_5$, $S_3^2:S_5$, $A_5\times \SOPlus(4,2)$

Order 3456: $F_9:C_2\times S_4$ x 8, $A_4\times F_9:C_2^2$, $C_2\times S_4\times F_9$, $C_2\times F_9:S_4$, $A_4:F_9:C_2^2$, $A_4:F_9:C_2^2$, $C_6^2:(Q_8\times D_6)$, $C_2\times D_6^2:D_6$

Order 3240: $C_3^2\times A_6$

Order 2880: $A_6:Q_8$ x 2, $C_4\times S_6$ x 2, $C_8\times A_6$, $A_6:C_8$, $Q_8\times A_6$, $F_5\times F_9:C_2$, $C_4:S_6$, $C_2^2:S_6$, $D_4\times A_6$, $C_2^2\times S_6$

Order 2592: $C_3^4:(C_4\times D_4)$ x 3, $C_3^3:C_{12}:Q_8$ x 2, $S_3^2:\SOPlus(4,2)$ x 2, $S_3^4:C_2$ x 2, $S_3^2:\PSU(3,2)$ x 2, $C_3^3:C_{12}:C_8$, $C_3^2\wr C_2.\SD_{16}$, $(C_3^2\times F_9):C_4$, $C_3^2:C_4\times F_9$, $(C_3\times F_9):D_6$, $C_3^2\wr C_2.\SD_{16}$, $C_3^4:(C_4\times Q_8)$, $F_9:S_3^2$, $F_9:S_3^2$, $S_3^2\times F_9$, $\PSU(3,2):S_3^2$, $S_3^2\times \PSU(3,2)$, $\PSU(3,2):S_3^2$, $S_3\times C_3^3:\SD_{16}$, $F_9:C_6\times S_3$, $C_3^3:C_{12}:D_4$, $C_3^2\wr C_2.\SD_{16}$, $C_3^4:(C_2\times \SD_{16})$, $S_3^2:F_9$, $\SOPlus(4,2).S_3^2$, $C_3^2\wr C_2.\SD_{16}$, $S_3^2\wr C_2$

Order 2304: $D_4\times F_9:C_2^2$

Order 2160: $C_3^2:C_4\times A_5$ x 2, $(C_3\times \GL(2,4)):C_4$ x 2, $C_3:S_6$ x 2, $S_3\times A_6$ x 2, $C_3\times S_6$ x 2, $\GL(2,4):D_6$, $C_6\times A_6$, $S_3^2\times A_5$, $A_5:S_3^2$, $A_5:S_3^2$, $\GL(2,4):D_6$

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

Order 1728: $S_4\times \PSU(3,2)$ x 6, $D_6^2:D_6$ x 6, $A_4\times F_9:C_2$ x 4, $\PSU(3,2):S_4$ x 4, $\SOPlus(4,2).S_4$ x 4, $F_9:S_4$ x 4, $S_4\times F_9$ x 4, $C_6^2:(C_4\times D_6)$ x 2, $C_6^2:C_6:Q_8$ x 2, $A_4:D_6^2$, $C_6^2:(C_6\times Q_8)$, $C_2\times D_6^2:C_6$, $D_6^2:D_6$, $C_2\times C_6^2:D_{12}$, $C_2\times A_4\times F_9$, $C_2\times A_4:F_9$, $C_2\times F_9:D_6$

Order 1440: $C_2\times S_6$ x 4, $C_4\times A_6$ x 2, $A_6:C_4$ x 2, $F_9:D_{10}$, $F_5\times \PSU(3,2)$, $F_5\times \SOPlus(4,2)$, $F_9:F_5$, $\PSU(3,2):F_5$, $\SOPlus(4,2):F_5$, $F_5\times F_9$, $D_6\times S_5$, $C_2^2\times A_6$

Order 1296: $C_3^2:C_4\times S_3^2$ x 3, $C_3^2\wr C_2.D_4$ x 3, $(C_3^2\times S_3^2):C_4$ x 3, $C_3^3:(S_3\times Q_8)$ x 2, $S_3\times C_3^3:Q_8$ x 2, $C_3^3:(S_3\times D_4)$ x 2, $S_3\times C_3^2:D_{12}$ x 2, $S_3^2:S_3^2$ x 2, $(C_3\times S_3^2):D_6$ x 2, $S_3^3:S_3$ x 2, $S_3^3:C_6$ x 2, $C_3^2\wr C_2.Q_8$ x 2, $C_3^2\wr C_2.D_4$ x 2, $C_3^4:C_4^2$ x 2, $S_3^4$, $C_3^3:(S_3\times Q_8)$, $S_3\times C_3^3:Q_8$, $C_3^4:(C_2\times D_4)$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^2:(C_2\times F_9)$, $C_3:(S_3\times F_9)$, $C_3:S_3\times F_9$, $S_3\times C_3:F_9$, $C_3\times S_3\times F_9$, $C_3^2\wr C_2.D_4$

Order 1152: $D_4\times F_9:C_2$ x 12, $(C_6\times C_{12}):\SD_{16}$, $(C_6\times C_{12}):\SD_{16}$, $C_2\times C_6^2:\SD_{16}$, $C_2\times C_6^2:\SD_{16}$, $C_4\times F_9:C_2^2$, $C_2\times D_4\times F_9$, $F_9:C_2^4$, $C_6^2:(C_2^2\times Q_8)$, $D_6^2:C_2^3$, $\SD_{16}\times \SOPlus(4,2)$, $C_2\times F_9:D_4$, $C_2\times F_9:D_4$

Order 1080: $S_3\times \GL(2,4)$, $\GL(2,4):S_3$, $C_3^2\times S_5$, $C_3^2:S_5$, $C_3^2:S_5$, $C_3\times A_6$

Order 960: $\SD_{16}\times A_5$, $Q_8:S_5$, $A_5:\SD_{16}$, $C_8:S_5$, $C_8\times S_5$, $Q_8\times S_5$, $D_4\times S_5$

Order 864: $C_3^2:C_4\times S_4$ x 8, $F_9:D_6$ x 8, $A_4:\PSU(3,2)$ x 7, $S_4\times S_3^2$ x 6, $S_3^2:S_4$ x 4, $A_4\times \PSU(3,2)$ x 3, $D_6^2:C_6$ x 3, $C_6^2:D_{12}$ x 3, $C_2\times C_6^2:C_{12}$ x 2, $C_6^2:C_6:C_4$ x 2, $S_3^3:C_2^2$ x 2, $A_4\times F_9$ x 2, $A_4:F_9$ x 2, $D_6^2:C_6$, $C_2\times A_4:S_3^2$, $S_3\times C_6:S_4$, $C_6:S_3\times S_4$, $C_6\times S_3\times S_4$, $D_6\times \PSU(3,2)$, $F_9:D_6$, $\PSU(3,2):D_6$, $C_2\times C_3^3:\SD_{16}$, $C_2\times F_9:C_6$, $D_6\times F_9$

Order 768: $C_2\times \SD_{16}\times S_4$

Order 720: $S_3\times S_5$ x 4, $S_6$ x 3, $D_5.\PSU(3,2)$ x 2, $C_3^2:C_4\times F_5$ x 2, $C_2\times A_6$ x 2, $D_5\times \PSU(3,2)$, $S_3^2:D_{10}$, $F_5\times S_3^2$, $F_9:D_5$, $\PSU(3,2):D_5$, $(C_3\times C_{15}):\SD_{16}$, $F_9:C_{10}$, $D_5\times F_9$, $D_5:F_9$, $D_5.\SOPlus(4,2)$, $S_3^2:F_5$, $D_6\times A_5$, $C_6:S_5$, $C_6\times S_5$

Order 648: $C_3:S_3^3$ x 3, $C_3^3:(C_4\times S_3)$ x 3, $C_3^3:(C_4\times S_3)$ x 3, $C_3^3:C_4\times S_3$ x 3, $S_3\times C_3^2:C_{12}$ x 3, $C_3:S_3^3$ x 2, $C_3\times S_3^3$ x 2, $C_3^4:Q_8$ x 2, $C_3^4:Q_8$ x 2, $C_3^4:Q_8$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3^4:D_4$ x 2, $C_3^3:D_{12}$ x 2, $C_3\wr D_4$ x 2, $C_3^4:D_4$ x 2, $C_3^4:(C_2\times C_4)$ x 2, $C_3:S_3^3$, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^2:F_9$, $C_3^2:F_9$, $C_3^3:C_{24}$, $C_3^2\times F_9$

Order 576: $F_9:C_2^3$ x 13, $C_3^2:D_4^2$ x 10, $D_4\times \PSU(3,2)$ x 9, $C_6^2:\SD_{16}$ x 8, $C_6^2:\SD_{16}$ x 8, $F_9:D_4$ x 8, $C_4\times F_9:C_2$ x 8, $D_4\times F_9$ x 6, $\PSU(3,2):D_4$ x 6, $(C_3\times C_{12}):\SD_{16}$ x 6, $F_9:D_4$ x 6, $C_6^2:(C_2^2\times C_4)$ x 2, $C_2\times C_6^2:Q_8$ x 2, $(C_6\times C_{12}):Q_8$ x 2, $C_6^2.\SD_{16}$ x 2, $C_6^2.\SD_{16}$ x 2, $C_2^3\times \PSU(3,2)$, $D_6^2:C_2^2$, $C_2^3\times F_9$, $C_2^2:D_6^2$, $C_4:D_6^2$, $(C_6\times C_{12}):Q_8$, $D_6^2.C_2^2$, $D_6^2:C_2^2$, $Q_8\times \SOPlus(4,2)$, $D_6^2.C_2^2$, $D_6^2.C_2^2$, $D_6^2.C_2^2$, $C_2^3:F_9$, $C_2\times C_4:F_9$, $C_2\times C_4\times F_9$, $C_6^2.\SD_{16}$, $\SD_{16}\times S_3^2$, $C_3^2:C_4\times \SD_{16}$, $Q_8:\SOPlus(4,2)$, $S_3^2:\SD_{16}$, $D_4.\SOPlus(4,2)$, $S_3^2:\SD_{16}$, $C_8:\SOPlus(4,2)$, $S_3^2:\SD_{16}$, $C_8\times \SOPlus(4,2)$

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

Order 480: $A_5:Q_8$ x 2, $C_4\times S_5$ x 2, $Q_8\times A_5$, $D_4\times A_5$, $C_2^2:S_5$, $C_4:S_5$, $C_8\times A_5$, $A_5:C_8$, $C_2^2\times S_5$

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

Order 384: $\SD_{16}\times S_4$ x 8, $C_2\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2\times A_4\times \SD_{16}$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_{12}$, $C_2\times C_8\times S_4$

Order 360: $C_5:\PSU(3,2)$ x 2, $(C_3^2\times D_5):C_4$ x 2, $C_3^2:C_4\times D_5$ x 2, $S_3\times A_5$ x 2, $C_3:S_5$ x 2, $C_3\times S_5$ x 2, $D_5\times S_3^2$, $C_5\times \PSU(3,2)$, $C_3^2:D_{20}$, $S_3^2:D_5$, $S_3^2:C_{10}$, $D_5.S_3^2$, $D_5.S_3^2$, $C_3:S_3\times F_5$, $D_{15}:C_{12}$, $C_5:F_9$, $C_5:F_9$, $C_5\times F_9$, $C_6\times A_5$, $A_6$

Order 324: $C_3^2:S_3^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2\times S_3^2$ x 3, $C_3^4:C_4$ x 3, $C_3\wr C_4$ x 3, $C_3^3:C_{12}$ x 3, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3\wr C_2^2$ x 2, $C_3^4:C_4$ x 2

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

Order 288: $F_9:C_2^2$ x 33, $C_6^2:Q_8$ x 14, $C_3^2:C_4\times D_4$ x 13, $D_6\times S_4$ x 13, $C_4:\PSU(3,2)$ x 11, $C_6^2:D_4$ x 11, $D_4\times S_3^2$ x 10, $D_6\wr C_2$ x 9, $C_6^2:Q_8$ x 9, $C_4\times \SOPlus(4,2)$ x 8, $\PSU(3,2):C_4$ x 8, $\SOPlus(4,2):C_4$ x 8, $C_6^2:D_4$ x 7, $S_3^2:D_4$ x 7, $C_2^2\times F_9$ x 7, $C_4\times \PSU(3,2)$ x 6, $C_4:\SOPlus(4,2)$ x 6, $C_6^2.Q_8$ x 4, $C_6^2:C_8$ x 4, $C_4\times F_9$ x 4, $F_9:C_4$ x 4, $C_4:F_9$ x 3, $C_2\times C_6^2:C_4$ x 2, $(C_6\times C_{12}):C_4$ x 2, $C_2\times C_3^2:C_4^2$ x 2, $C_6^2.D_4$ x 2, $C_4.\SOPlus(4,2)$ x 2, $S_3^2:Q_8$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_6^2:C_2^3$, $D_6^2:C_2$, $D_6^2:C_2$, $Q_8\times S_3^2$, $D_{12}:D_6$, $D_6\times D_{12}$, $C_2.D_6^2$, $C_3^2:C_4\times Q_8$, $C_6^2.D_4$, $C_4.\SOPlus(4,2)$, $C_{24}:D_6$, $C_{24}:D_6$, $Q_8:S_3^2$, $Q_8:S_3^2$, $D_4.S_3^2$, $D_{12}.D_6$, $C_{24}:D_6$, $C_{24}:D_6$, $C_8\times S_3^2$, $(Q_8\times C_3^2):C_4$, $(D_4\times C_3^2):C_4$, $(C_3\times C_{24}):C_4$, $C_3^2:C_4\times C_8$, $C_4.\SOPlus(4,2)$, $C_4.\SOPlus(4,2)$, $C_4.\SOPlus(4,2)$, $S_3^2:C_8$, $C_2\times D_6^2$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$, $C_6^2:C_2^3$

Order 256: $C_4.D_4^2$

Order 240: $C_2\times S_5$ x 4, $C_4\times A_5$ x 2, $A_5:C_4$ x 2, $D_6\times F_5$, $C_2^2\times A_5$

Order 216: $S_3^3$ x 13, $C_3^2:C_4\times S_3$ x 12, $S_3^2:S_3$ x 8, $C_3^3:Q_8$ x 7, $C_3^2:D_{12}$ x 6, $S_3^2:C_6$ x 6, $C_6:S_3^2$ x 4, $C_6\times S_3^2$ x 4, $C_2\times C_3^3:C_4$ x 3, $C_2\times C_3^2:C_{12}$ x 3, $C_3^3:Q_8$ x 3, $C_6:S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_3:F_9$ x 2, $C_3\times F_9$ x 2, $C_6^2:C_6$

Order 192: $Q_8\times S_4$ x 6, $D_4\times S_4$ x 6, $Q_8:S_4$ x 4, $A_4:\SD_{16}$ x 4, $C_8:S_4$ x 4, $C_8\times S_4$ x 4, $A_4\times \SD_{16}$ x 4, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2\times A_4:C_8$, $C_2^3\times S_4$, $C_{12}:C_2^4$, $C_2\times Q_8\times A_4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3:D_{12}$, $D_6\times \SD_{16}$, $C_2^3:C_{24}$

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

Order 162: $C_3^2\wr C_2$ x 3, $C_3^3:C_6$ x 2, $S_3\times C_3^3$ x 2, $C_3^3:S_3$

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: $S_3^2:C_2^2$ x 29, $S_3\times S_4$ x 20, $C_2\times \PSU(3,2)$ x 18, $F_9:C_2$ x 18, $C_6^2:C_4$ x 15, $C_2.\PSU(3,2)$ x 14, $D_6^2$ x 11, $S_3^2:C_4$ x 11, $C_2\times F_9$ x 10, $C_3^2:C_4^2$ x 10, $D_6:D_6$ x 9, $C_6^2:C_4$ x 9, $(C_3\times C_{12}):C_4$ x 9, $C_4\times S_3^2$ x 8, $A_4\times D_6$ x 7, $C_6:S_4$ x 7, $C_6\times S_4$ x 7, $C_{12}:D_6$ x 7, $C_{12}:D_6$ x 7, $D_6:D_6$ x 7, $S_3\times D_{12}$ x 7, $C_2.\SOPlus(4,2)$ x 7, $C_{12}:D_6$ x 6, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:S_3$, $C_{24}:S_3$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:C_6$, $S_3\times C_{24}$, $C_6.D_{12}$, $C_6.D_{12}$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_2^2:C_6^2$, $C_4:C_6^2$, $C_6^2:C_2^2$, $C_{12}.D_6$, $C_6:D_{12}$, $C_{12}:D_6$, $C_6^2:C_2^2$, $C_{12}.D_6$, $C_6\times D_{12}$, $C_{12}\times D_6$, $C_6:D_{12}$, $D_6:D_6$, $C_2^2.S_3^2$, $D_6.D_6$, $C_3^2:(C_2\times C_8)$, $C_3^2:\SD_{16}$, $C_3^2\times \SD_{16}$

Order 128: $D_4\times \SD_{16}$ x 12, $C_2^3\times \SD_{16}$, $C_4^2.C_2^3$, $C_2\times D_4^2$, $C_4^2.D_4$, $C_2^3:\SD_{16}$, $C_4^2.D_4$, $C_4^2.D_4$, $C_2^3:\SD_{16}$, $C_2^3:\SD_{16}$, $C_4^2.D_4$, $C_4^2.C_2^3$

Order 120: $S_3\times F_5$ x 4, $S_5$ x 3, $C_2\times A_5$ x 2, $S_3\times D_{10}$, $C_{30}:C_4$, $C_6\times F_5$

Order 108: $C_3:S_3^2$ x 19, $C_3\times S_3^2$ x 17, $C_3:S_3^2$ x 7, $C_3^3:C_4$ x 6, $C_3^2:C_{12}$ x 6, $C_3^2:D_6$ x 4, $C_3^2\times D_6$ x 4, $C_3^2:D_6$ x 2, $C_3^2\times A_4$

Order 96: $D_4\times D_6$ x 19, $C_2^2\times S_4$ x 11, $C_4\times S_4$ x 8, $S_3\times \SD_{16}$ x 8, $A_4:Q_8$ x 7, $\GL(2,\mathbb{Z}/4)$ x 4, $Q_8\times A_4$ x 3, $D_4\times A_4$ x 3, $C_4:S_4$ x 3, $C_8\times A_4$ x 2, $A_4:C_8$ x 2, $C_2^3:C_{12}$ x 2, $C_2^2.S_4$ x 2, $C_2^3\times D_6$, $C_2^3\times A_4$, $C_{12}:C_2^3$, $C_2^3:D_6$, $Q_8\times D_6$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_6\times \SD_{16}$, $Q_8:D_6$, $C_6:\SD_{16}$, $C_8:D_6$, $C_8\times D_6$

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

Order 81: $C_3^4$

Order 80: $C_{20}:C_4$ x 3, $C_4\times F_5$ x 2, $C_{40}:C_2$, $C_2^2\times F_5$, $Q_8\times D_5$, $C_8\times D_5$, $D_4\times D_5$, $D_{10}:C_4$, $D_5:C_8$, $C_5\times \SD_{16}$, $Q_8:D_5$, $C_5:\SD_{16}$

Order 72: $S_3\times D_6$ x 38, $C_2\times C_3^2:C_4$ x 26, $\SOPlus(4,2)$ x 22, $\PSU(3,2)$ x 10, $C_6\times D_6$ x 9, $C_6:D_6$ x 8, $S_3\times A_4$ x 6, $C_3:S_4$ x 6, $C_3\times S_4$ x 6, $C_{12}:S_3$ x 6, $S_3\times C_{12}$ x 6, $C_6.D_6$ x 6, $F_9$ x 5, $C_6^2:C_2$ x 5, $C_6\wr C_2$ x 5, $C_3:D_{12}$ x 5, $C_6.D_6$ x 5, $C_6\times A_4$ x 4, $D_4\times C_3^2$ x 4, $C_3:D_{12}$ x 4, $C_3\times D_{12}$ x 4, $D_6:S_3$ x 4, $C_3^2:Q_8$ x 2, $C_3^2:Q_8$ x 2, $C_3^2:Q_8$ x 2, $C_2\times C_6^2$, $Q_8\times C_3^2$, $C_6\times C_{12}$, $C_6.D_6$, $C_6:C_{12}$, $C_3^2:C_8$, $C_3\times C_{24}$, $C_3^2:C_8$, $C_3:C_{24}$

Order 64: $C_2^2\times \SD_{16}$ x 13, $D_4\times Q_8$ x 9, $D_4^2$ x 9, $C_8:D_4$ x 8, $C_2^2:\SD_{16}$ x 8, $Q_8:D_4$ x 8, $C_4\times \SD_{16}$ x 8, $C_8:D_4$ x 6, $C_4:\SD_{16}$ x 6, $Q_8:D_4$ x 6, $C_8\times D_4$ x 6, $C_4^2:C_2^2$ x 3, $C_2^2.Q_{16}$ x 2, $C_2^2.D_8$ x 2, $D_4\times C_2^3$ x 2, $C_4^2.C_2^2$ x 2, $C_2^3:Q_8$ x 2, $C_2^3:D_4$ x 2, $C_2^3:C_8$, $C_2\times C_4\times C_8$, $Q_8\times C_2^3$, $C_2^3\times C_8$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2.C_2^2$, $C_2^3.D_4$, $C_4^2.C_4$

Order 60: $S_3\times D_5$ x 4, $C_{15}:C_4$ x 2, $C_3\times F_5$ x 2, $A_5$, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$

Order 54: $C_3^2:C_6$ x 11, $S_3\times C_3^2$ x 11, $C_3^2:S_3$ x 5, $C_3^2\times C_6$ x 2

Order 48: $S_3\times D_4$ x 46, $C_2\times S_4$ x 24, $C_2^2\times D_6$ x 15, $C_4\times D_6$ x 14, $C_6:D_4$ x 13, $C_6\times D_4$ x 10, $C_2\times D_{12}$ x 10, $C_2^2\times A_4$ x 6, $S_3\times Q_8$ x 6, $C_{24}:C_2$ x 4, $S_3\times C_8$ x 4, $C_4\times A_4$ x 4, $A_4:C_4$ x 4, $C_3\times \SD_{16}$ x 4, $Q_8:S_3$ x 4, $C_3:\SD_{16}$ x 4, $C_6:Q_8$ x 2, $C_6:C_8$, $C_2^3\times C_6$, $C_6\times Q_8$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_2\times C_{24}$

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

Order 40: $C_2\times F_5$ x 6, $C_4\times D_5$ x 2, $C_5:Q_8$ x 2, $C_5:D_4$, $D_{20}$, $C_5:C_8$, $C_{40}$, $C_2\times D_{10}$, $C_5\times Q_8$, $C_5\times D_4$, $C_5:C_8$

Order 36: $S_3^2$ x 42, $C_6\times S_3$ x 27, $C_6:S_3$ x 19, $C_3^2:C_4$ x 17, $C_6^2$ x 5, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4, $C_3:C_{12}$ x 4, $C_3\times A_4$ x 2

Order 32: $C_2\times \SD_{16}$ x 33, $C_2^2\times D_4$ x 25, $C_4\times D_4$ x 18, $C_2^2:Q_8$ x 14, $C_4:D_4$ x 12, $C_4:Q_8$ x 11, $C_2^2\times Q_8$ x 9, $D_4:C_4$ x 8, $C_2^2\wr C_2$ x 8, $Q_8:C_4$ x 8, $C_2^2\times C_8$ x 7, $C_2^2.D_4$ x 7, $C_4\times Q_8$ x 6, $C_4:D_4$ x 5, $C_2^2:C_8$ x 4, $C_4\times C_8$ x 4, $C_2^3:C_4$ x 4, $C_8:C_4$ x 4, $C_2^3\times C_4$ x 3, $C_4:C_8$ x 3, $C_2\times C_4^2$ x 2, $C_2^5$

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

Order 27: $C_3^3$ x 3

Order 24: $C_2\times D_6$ x 50, $C_4\times S_3$ x 24, $C_3:D_4$ x 20, $D_{12}$ x 13, $C_3\times D_4$ x 13, $S_4$ x 10, $C_2\times C_{12}$ x 8, $C_6:C_4$ x 8, $C_2^2\times C_6$ x 8, $C_2\times A_4$ x 8, $C_3:Q_8$ x 7, $C_3\times Q_8$ x 3, $C_{24}$ x 2, $C_3:C_8$ x 2

Order 20: $F_5$ x 5, $D_{10}$ x 4, $C_{20}$ x 2, $C_5:C_4$ x 2, $C_2\times C_{10}$

Order 18: $C_3\times S_3$ x 23, $C_3:S_3$ x 16, $C_3\times C_6$ x 9

Order 16: $C_2\times D_4$ x 66, $C_2^2\times C_4$ x 24, $C_4:C_4$ x 23, $\SD_{16}$ x 18, $C_2\times Q_8$ x 18, $C_2^2:C_4$ x 16, $C_2^4$ x 11, $C_2\times C_8$ x 10, $C_4^2$ x 8

Order 15: $C_{15}$

Order 12: $D_6$ x 52, $C_2\times C_6$ x 16, $C_{12}$ x 8, $C_3:C_4$ x 8, $A_4$ x 2

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

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 38, $D_4$ x 37, $C_2^3$ x 31, $Q_8$ x 10, $C_8$ x 5

Order 6: $S_3$ x 16, $C_6$ x 13

Order 5: $C_5$

Order 4: $C_2^2$ x 27, $C_4$ x 16

Order 3: $C_3$ x 3

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 103680: $F_9:C_2\times S_6$ ($C_1$)

Order 51840: $S_6\times \SOPlus(4,2)$ ($C_2$), $S_6\times \PSU(3,2)$ ($C_2$), $\PSU(3,2):S_6$ ($C_2$), $F_9:C_2\times A_6$ ($C_2$), $\SOPlus(4,2).S_6$ ($C_2$), $F_9:S_6$ ($C_2$), $F_9\times S_6$ ($C_2$)

Order 25920: $A_6:\SOPlus(4,2)$ ($C_2^2$), $C_3^2:C_4\times S_6$ ($C_2^2$), $A_6\times \SOPlus(4,2)$ ($C_2^2$), $A_6:F_9$ ($C_2^2$), $F_9\times A_6$ ($C_2^2$), $A_6\times \PSU(3,2)$ ($C_2^2$), $A_6:\PSU(3,2)$ ($C_2^2$)

Order 12960: $C_3:S_3\times S_6$ ($D_4$), $(C_3^2\times A_6):C_4$ ($D_4$), $C_3^2:C_4\times A_6$ ($C_2^3$)

Order 6480: $C_3:S_3\times A_6$ ($C_2\times D_4$), $C_3^2:S_6$ ($\SD_{16}$), $C_3^2\times S_6$ ($\SD_{16}$)

Order 3240: $C_3^2\times A_6$ ($C_2\times \SD_{16}$)

Order 720: $S_6$ ($F_9:C_2$)

Order 360: $A_6$ ($F_9:C_2^2$)

Order 144: $F_9:C_2$ ($S_6$)

Order 72: $\PSU(3,2)$ ($C_2\times S_6$), $\SOPlus(4,2)$ ($C_2\times S_6$), $F_9$ ($C_2\times S_6$)

Order 36: $C_3^2:C_4$ ($C_2^2\times S_6$)

Order 18: $C_3:S_3$ ($D_4\times S_6$)

Order 9: $C_3^2$ ($\SD_{16}\times S_6$)

Order 1: $C_1$ ($F_9:C_2\times S_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 103680: $F_9:C_2\times S_6$ ($C_1$)

Order 51840: $S_6\times \SOPlus(4,2)$ ($C_2$), $S_6\times \PSU(3,2)$ ($C_2$), $\PSU(3,2):S_6$ ($C_2$), $F_9:C_2\times A_6$ ($C_2$), $\SOPlus(4,2).S_6$ ($C_2$), $F_9:S_6$ ($C_2$), $F_9\times S_6$ ($C_2$)

Order 25920: $A_6:\SOPlus(4,2)$ ($C_2^2$), $C_3^2:C_4\times S_6$ ($C_2^2$), $A_6\times \SOPlus(4,2)$ ($C_2^2$), $A_6:F_9$ ($C_2^2$), $F_9\times A_6$ ($C_2^2$), $A_6\times \PSU(3,2)$ ($C_2^2$), $A_6:\PSU(3,2)$ ($C_2^2$)

Order 12960: $C_3:S_3\times S_6$ ($D_4$), $(C_3^2\times A_6):C_4$ ($D_4$), $C_3^2:C_4\times A_6$ ($C_2^3$)

Order 6480: $C_3:S_3\times A_6$ ($C_2\times D_4$), $C_3^2:S_6$ ($\SD_{16}$), $C_3^2\times S_6$ ($\SD_{16}$)

Order 3240: $C_3^2\times A_6$ ($C_2\times \SD_{16}$)

Order 720: $S_6$ ($F_9:C_2$)

Order 360: $A_6$ ($F_9:C_2^2$)

Order 144: $F_9:C_2$ ($S_6$)

Order 72: $\PSU(3,2)$ ($C_2\times S_6$), $\SOPlus(4,2)$ ($C_2\times S_6$), $F_9$ ($C_2\times S_6$)

Order 36: $C_3^2:C_4$ ($C_2^2\times S_6$)

Order 18: $C_3:S_3$ ($D_4\times S_6$)

Order 9: $C_3^2$ ($\SD_{16}\times S_6$)

Order 1: $C_1$ ($F_9:C_2\times S_6$)

Series

Derived series

$F_9:C_2\times S_6$

$\rhd$

$C_3^2:C_4\times A_6$

$\rhd$

$C_3^2\times A_6$

$\rhd$

$A_6$

Chief series

$F_9:C_2\times S_6$

$\rhd$

$F_9:C_2\times A_6$

$\rhd$

$F_9:C_2$

$\rhd$

$\SOPlus(4,2)$

$\rhd$

$C_3^2:C_4$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$F_9:C_2\times S_6$

$\rhd$

$C_3^2:C_4\times A_6$

$\rhd$

$C_3:S_3\times A_6$

$\rhd$

$C_3^2\times A_6$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $99 \times 99$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $88 \times 88$ rational character table.