Abstract group 10368.qo: $\SOPlus(4,2)^2.C_2$

• Label: 10368.qo
• Order: \( 2^{7} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $4$

Group information

Description:	$\SOPlus(4,2)^2.C_2$
Order:	\(10368\)\(\medspace = 2^{7} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3^4.C_4^2.C_2^4$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \)
Composition factors:	$C_2$ x 7, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	483	80	2556	2352	1440	2304	1152	10368
Conjugacy classes	1	11	5	13	23	10	10	8	81
Divisions	1	11	5	13	23	5	10	4	72
Autjugacy classes	1	8	3	11	13	8	6	4	54

Dimension	1	2	4	8	16	32
Irr. complex chars.	16	16	19	20	2	8	81
Irr. rational chars.	16	8	21	13	6	8	72

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$48$
Rank:	$4$
Inequivalent generating quadruples:	$170951639040$

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

Presentation:	${\langle a, b, c, d, e, f \mid b^{8}=c^{6}=d^{12}=e^{3}=f^{3}=[a,e]=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(11,13)(14,15), (1,2,5)(3,7,6)(4,8,9), (1,3,9)(2,7,4)(5,6,8), (10,11,13) \!\cdots\! \rangle$
Direct product:	$\SOPlus(4,2)$ $\, \times\, $ $(F_9:C_2)$
Semidirect product:	$(S_3\times F_9:D_6)$ $\,\rtimes\,$ $C_2$	$(S_3^2\times F_9)$ $\,\rtimes\,$ $C_2^2$	$(S_3^2:F_9)$ $\,\rtimes\,$ $C_2^2$	$(F_9:S_3^2)$ $\,\rtimes\,$ $C_2^2$	all 47
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(S_3^4:C_2)$ . $C_2^2$	$\SOPlus(4,2)^2$ . $C_2$	$(C_3:S_3\times F_9)$ . $C_2^3$	$(C_3^2\wr C_2)$ . $D_4^2$	all 39
Aut. group:	$\Aut(C_3^3:(C_4\times S_3))$	$\Aut((C_3^2\times S_3^2):C_4)$	$\Aut(C_3:(S_3\times F_9))$	$\Aut(C_3^2:C_4\times S_3^2)$	all 23

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 105240 subgroups in 2197 conjugacy classes, 118 normal (64 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\SOPlus(4,2)^2.C_2$
Commutator:	$G' \simeq$ $C_3^3:(C_4\times S_3)$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\SOPlus(4,2)^2.C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^4$	$G/\operatorname{Fit} \simeq$ $D_4\times \SD_{16}$
Radical:	$R \simeq$ $\SOPlus(4,2)^2.C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $D_4\times \SD_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times \SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

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

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 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 1728: $C_2\times F_9:D_6$ x 2

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: $D_4\times F_9:C_2$, $\SD_{16}\times \SOPlus(4,2)$

Order 864: $F_9:D_6$ x 16, $S_3^3:C_2^2$ x 3, $D_6\times \PSU(3,2)$ 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 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 2, $C_3^2:D_4^2$ x 2, $C_6^2:\SD_{16}$ x 2, $C_6^2:\SD_{16}$ x 2, $F_9:D_4$ 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, $D_4\times \PSU(3,2)$, $Q_8\times \SOPlus(4,2)$, $D_4\times F_9$, $\PSU(3,2):D_4$, $(C_3\times C_{12}):\SD_{16}$, $F_9:D_4$, $C_4\times F_9:C_2$, $C_3^2:C_4\times \SD_{16}$, $Q_8:\SOPlus(4,2)$, $D_4.\SOPlus(4,2)$, $C_8:\SOPlus(4,2)$, $C_8\times \SOPlus(4,2)$

Order 432: $S_3^3:C_2$ x 20, $S_3\times \PSU(3,2)$ x 12, $F_9:S_3$ 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, $C_3^2:C_4\times D_6$ 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, $C_6\times \PSU(3,2)$ x 2, $C_6:F_9$ x 2, $C_6\times F_9$ x 2

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 288: $F_9:C_2^2$ x 17, $C_6^2:Q_8$ x 4, $D_6\wr C_2$ x 4, $S_3^2:Q_8$ x 4, $D_4\times S_3^2$ x 3, $C_3^2:C_4\times D_4$ x 3, $C_6^2:D_4$ x 3, $S_3^2:D_4$ x 3, $C_4\times \SOPlus(4,2)$ x 3, $\PSU(3,2):C_4$ x 3, $\SOPlus(4,2):C_4$ x 3, $C_6^2:D_4$ x 3, $Q_8\times S_3^2$ x 2, $C_4:\PSU(3,2)$ x 2, $C_4:\SOPlus(4,2)$ x 2, $C_4.\SOPlus(4,2)$ x 2, $C_4.\SOPlus(4,2)$ x 2, $C_6^2:C_8$ 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, $C_6^2:Q_8$ x 2, $C_2^2\times F_9$ x 2, $C_3^2:C_4\times Q_8$, $C_4\times \PSU(3,2)$, $C_4:F_9$, $C_4\times F_9$, $F_9:C_4$, $C_{24}:D_6$, $(Q_8\times C_3^2):C_4$, $(D_4\times C_3^2):C_4$, $(C_3\times C_{24}):C_4$, $C_3^2:C_4\times C_8$, $C_4.\SOPlus(4,2)$

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, $C_3:F_9$ x 4, $C_3\times F_9$ x 4

Order 192: $D_6\times \SD_{16}$ x 2

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 144: $S_3^2:C_2^2$ x 21, $F_9:C_2$ x 18, $C_2\times \PSU(3,2)$ x 11, $S_3^2:C_4$ x 9, $C_2\times F_9$ x 8, $D_6^2$ x 6, $C_2.\SOPlus(4,2)$ x 6, $C_6^2:C_4$ x 5, $C_4\times S_3^2$ x 5, $C_6^2:C_4$ x 5, $(C_3\times C_{12}):C_4$ x 5, $C_2.\PSU(3,2)$ x 5, $D_6:D_6$ x 4, $D_6:D_6$ x 4, $C_{12}.D_6$ x 4, $C_{12}.D_6$ x 4, $C_3^2:C_4^2$ x 4, $C_{12}:D_6$ x 3, $C_{12}:D_6$ x 3, $S_3\times D_{12}$ x 3, $C_{12}.D_6$ 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, $C_{12}:D_6$ x 2, $C_{12}.D_6$ x 2, $C_3^2:\SD_{16}$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:S_3$, $C_{24}:S_3$, $C_{12}.D_6$, $C_3^2:(C_2\times C_8)$, $C_3^2\times \SD_{16}$

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

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

Order 96: $S_3\times \SD_{16}$ x 16, $D_4\times D_6$ x 3, $Q_8\times D_6$ 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

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 51, $\SOPlus(4,2)$ x 24, $C_2\times C_3^2:C_4$ x 22, $\PSU(3,2)$ x 11, $C_6\times D_6$ x 8, $C_6:D_6$ x 5, $F_9$ x 5, $S_3\times C_{12}$ x 5, $C_6.D_6$ x 5, $C_6.D_6$ x 5, $C_6\wr C_2$ x 4, $C_3^2:Q_8$ x 4, $C_3^2:Q_8$ x 4, $C_3:D_{12}$ x 4, $C_6^2:C_2$ x 3, $C_{12}:S_3$ x 3, $C_3\times D_{12}$ x 3, $D_6:S_3$ x 3, $D_4\times C_3^2$ x 2, $C_3:D_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_3:C_{24}$ x 2, $Q_8\times C_3^2$, $C_3^2:C_8$, $C_3\times C_{24}$, $C_3^2:C_8$

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 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 20, $S_3\times Q_8$ x 12, $C_{24}:C_2$ x 8, $S_3\times C_8$ x 8, $C_3\times \SD_{16}$ x 8, $Q_8:S_3$ x 8, $C_3:\SD_{16}$ x 8, $C_4\times D_6$ x 5, $C_2^2\times D_6$ x 4, $C_6:D_4$ x 4, $C_6:Q_8$ x 4, $C_6\times D_4$ x 3, $C_2\times D_{12}$ x 3, $C_6:C_8$ x 2, $C_6\times Q_8$ x 2, $C_2\times C_{24}$ x 2

Order 36: $S_3^2$ x 75, $C_6\times S_3$ x 47, $C_6:S_3$ x 27, $C_3^2:C_4$ x 17, $C_3:C_{12}$ x 5, $C_6^2$ x 5, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 3

Order 32: $C_2\times \SD_{16}$ x 17, $C_2^2:Q_8$ x 4, $D_4:C_4$ x 3, $C_2^2\times D_4$ x 3, $C_4:D_4$ x 3, $C_4\times D_4$ 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, $C_4:Q_8$ x 2, $C_2^2\wr C_2$ x 2, $C_4:D_4$, $C_4\times C_8$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$

Order 27: $C_3^3$ x 5

Order 24: $C_2\times D_6$ x 41, $C_4\times S_3$ x 20, $C_3:D_4$ x 16, $C_3:Q_8$ x 14, $D_{12}$ x 10, $C_3\times D_4$ x 10, $C_3\times Q_8$ x 6, $C_2\times C_{12}$ x 5, $C_6:C_4$ x 5, $C_{24}$ x 4, $C_2^2\times C_6$ x 4, $C_3:C_8$ x 4

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 20, $\SD_{16}$ x 18, $C_2\times Q_8$ x 11, $C_2\times C_8$ x 8, $C_4:C_4$ x 8, $C_2^2:C_4$ x 7, $C_2^2\times C_4$ x 5, $C_4^2$ x 2, $C_2^4$ x 2

Order 12: $D_6$ x 78, $C_2\times C_6$ x 21, $C_{12}$ x 10, $C_3:C_4$ x 10

Order 9: $C_3^2$ x 10

Order 8: $D_4$ x 22, $C_2\times C_4$ x 20, $C_2^3$ x 15, $Q_8$ x 11, $C_8$ x 5

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

Order 4: $C_2^2$ x 27, $C_4$ x 13

Order 3: $C_3$ x 5

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

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

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 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 1728: $C_2\times F_9:D_6$

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: $D_4\times F_9:C_2$, $\SD_{16}\times \SOPlus(4,2)$

Order 864: $F_9:D_6$ x 8, $S_3^3:C_2^2$ x 2, $D_6\times \PSU(3,2)$, $F_9:D_6$, $\PSU(3,2):D_6$, $C_2\times C_3^3:\SD_{16}$, $C_2\times F_9:C_6$, $D_6\times F_9$

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

Order 432: $S_3^3:C_2$ x 12, $S_3\times \PSU(3,2)$ x 6, $F_9:S_3$ 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, $C_2\times S_3^3$ 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, $C_6\times \PSU(3,2)$, $C_6:F_9$, $C_6\times F_9$

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 288: $F_9:C_2^2$ x 9, $C_3^2:C_4\times D_4$ x 3, $C_4\times \SOPlus(4,2)$ x 3, $D_4\times S_3^2$ x 2, $C_6^2:Q_8$ x 2, $C_4:\PSU(3,2)$ x 2, $C_6^2:D_4$ x 2, $D_6\wr C_2$ x 2, $C_4:\SOPlus(4,2)$ x 2, $S_3^2:D_4$ x 2, $C_4.\SOPlus(4,2)$ x 2, $S_3^2:Q_8$ x 2, $\PSU(3,2):C_4$ x 2, $\SOPlus(4,2):C_4$ x 2, $C_6^2:D_4$ x 2, $Q_8\times S_3^2$, $C_3^2:C_4\times Q_8$, $C_4\times \PSU(3,2)$, $C_4.\SOPlus(4,2)$, $C_6^2:C_8$, $C_4:F_9$, $C_4\times F_9$, $F_9:C_4$, $C_{24}:D_6$, $C_{24}:D_6$, $Q_8:S_3^2$, $Q_8:S_3^2$, $D_4.S_3^2$, $D_{12}.D_6$, $C_{24}:D_6$, $C_{24}:D_6$, $C_8\times S_3^2$, $(Q_8\times C_3^2):C_4$, $(D_4\times C_3^2):C_4$, $(C_3\times C_{24}):C_4$, $C_3^2:C_4\times C_8$, $C_4.\SOPlus(4,2)$, $C_4.\SOPlus(4,2)$, $C_4.\SOPlus(4,2)$, $S_3^2:C_8$, $C_6^2:Q_8$, $C_2^2\times F_9$

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, $C_3:F_9$ x 2, $C_3\times F_9$ x 2

Order 192: $D_6\times \SD_{16}$

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 144: $S_3^2:C_2^2$ x 14, $F_9:C_2$ x 11, $C_2\times \PSU(3,2)$ x 6, $C_2\times F_9$ x 5, $(C_3\times C_{12}):C_4$ x 5, $C_2.\SOPlus(4,2)$ x 5, $S_3^2:C_4$ x 5, $C_3^2:C_4^2$ x 4, $C_2.\PSU(3,2)$ x 4, $D_6^2$ x 3, $C_6^2:C_4$ x 3, $C_4\times S_3^2$ x 3, $C_6^2:C_4$ x 3, $C_{12}:D_6$ x 2, $C_{12}:D_6$ x 2, $D_6:D_6$ x 2, $D_6:D_6$ x 2, $C_{12}:D_6$ x 2, $S_3\times D_{12}$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:S_3$, $C_{24}:S_3$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:C_6$, $S_3\times C_{24}$, $C_6.D_{12}$, $C_6.D_{12}$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $C_3^2:(C_2\times C_8)$, $C_3^2:\SD_{16}$, $C_3^2\times \SD_{16}$

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

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

Order 96: $S_3\times \SD_{16}$ x 8, $D_4\times D_6$ x 2, $Q_8\times D_6$, $C_6\times \SD_{16}$, $Q_8:D_6$, $C_6:\SD_{16}$, $C_8:D_6$, $C_8\times D_6$

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 26, $\SOPlus(4,2)$ x 17, $C_2\times C_3^2:C_4$ x 16, $\PSU(3,2)$ x 7, $C_6\times D_6$ x 4, $F_9$ x 4, $C_6:D_6$ x 3, $C_{12}:S_3$ x 3, $S_3\times C_{12}$ x 3, $C_6.D_6$ x 3, $C_6.D_6$ x 3, $D_4\times C_3^2$ x 2, $C_6^2:C_2$ x 2, $C_3:D_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_6\wr C_2$ x 2, $C_3\times D_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_3^2:Q_8$ x 2, $C_3:D_{12}$ x 2, $D_6:S_3$ x 2, $Q_8\times C_3^2$, $C_3^2:C_8$, $C_3\times C_{24}$, $C_3^2:C_8$, $C_3:C_{24}$

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

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 12, $S_3\times Q_8$ x 6, $C_{24}:C_2$ 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, $C_4\times D_6$ x 3, $C_2^2\times D_6$ x 2, $C_6\times D_4$ x 2, $C_6:D_4$ x 2, $C_2\times D_{12}$ x 2, $C_6:Q_8$ x 2, $C_6:C_8$, $C_6\times Q_8$, $C_2\times C_{24}$

Order 36: $S_3^2$ x 39, $C_6\times S_3$ x 24, $C_6:S_3$ x 16, $C_3^2:C_4$ x 15, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 3, $C_3:C_{12}$ x 3, $C_6^2$ x 3

Order 32: $C_2\times \SD_{16}$ x 9, $C_4\times D_4$ x 3, $D_4:C_4$ x 2, $C_2^2\times D_4$ x 2, $C_4:Q_8$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $Q_8:C_4$ x 2, $C_2^2:C_8$, $C_2^2\times Q_8$, $C_2^2\times C_8$, $C_4:D_4$, $C_4\times C_8$, $C_2^2\wr C_2$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$

Order 27: $C_3^3$ x 3

Order 24: $C_2\times D_6$ x 21, $C_4\times S_3$ x 12, $C_3:D_4$ x 8, $C_3:Q_8$ x 7, $D_{12}$ x 6, $C_3\times D_4$ x 6, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_3\times Q_8$ x 3, $C_{24}$ x 2, $C_2^2\times C_6$ x 2, $C_3:C_8$ x 2

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 13, $\SD_{16}$ x 11, $C_4:C_4$ x 7, $C_2\times Q_8$ x 6, $C_2\times C_8$ x 5, $C_2^2:C_4$ x 4, $C_2^2\times C_4$ x 3, $C_4^2$ x 2, $C_2^4$

Order 12: $D_6$ x 42, $C_2\times C_6$ x 11, $C_{12}$ x 6, $C_3:C_4$ x 6

Order 9: $C_3^2$ x 7

Order 8: $D_4$ x 15, $C_2\times C_4$ x 14, $C_2^3$ x 8, $Q_8$ x 7, $C_8$ x 4

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

Order 4: $C_2^2$ x 16, $C_4$ x 11

Order 3: $C_3$ x 3

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 10368: $\SOPlus(4,2)^2.C_2$ ($C_1$)

Order 5184: $F_9:\SOPlus(4,2)$ ($C_2$) x 2, $(C_3^2\times S_3^2):\SD_{16}$ ($C_2$) x 2, $S_3\times F_9:D_6$ ($C_2$) x 2, $(C_3^2\times S_3^2):\SD_{16}$ ($C_2$) x 2, $F_9\times \SOPlus(4,2)$ ($C_2$), $F_9:\SOPlus(4,2)$ ($C_2$), $C_3^3:C_{12}:\SD_{16}$ ($C_2$), $C_3^4:(C_4\times \SD_{16})$ ($C_2$), $C_3^4:(D_4\times Q_8)$ ($C_2$), $C_3^3:C_{12}:\SD_{16}$ ($C_2$), $\SOPlus(4,2)^2$ ($C_2$)

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

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

Order 648: $C_3:S_3^3$ ($C_2\times D_4$) x 2, $C_3^4:D_4$ ($\SD_{16}$) x 2, $C_3^3:(C_4\times S_3)$ ($C_2\times D_4$) x 2, $C_3^4:Q_8$ ($C_2\times D_4$), $C_3^4:Q_8$ ($C_2\times D_4$), $C_3^3:D_{12}$ ($\SD_{16}$), $C_3^3:D_{12}$ ($C_2\times D_4$), $C_3^4:D_4$ ($\SD_{16}$), $C_3^4:D_4$ ($C_2\times D_4$), $C_3^4:(C_2\times C_4)$ ($C_2\times D_4$), $C_3^3:(C_4\times S_3)$ ($C_2^4$), $C_3^3:(C_4\times S_3)$ ($C_2\times D_4$), $C_3^2:F_9$ ($C_2\times D_4$), $C_3^2\times F_9$ ($C_2\times D_4$)

Order 324: $C_3^2:S_3^2$ ($C_2\times \SD_{16}$) x 2, $C_3^2\times S_3^2$ ($C_2\times \SD_{16}$) x 2, $C_3^2:S_3^2$ ($C_2^2\times D_4$), $C_3^4:C_4$ ($D_4:C_2^2$), $C_3^4:C_4$ ($C_2\times \SD_{16}$), $C_3^3:C_{12}$ ($C_2^2\times D_4$), $C_3^3:C_{12}$ ($C_2\times \SD_{16}$)

Order 162: $C_3^3:S_3$ ($D_8:C_2^2$), $C_3^2\wr C_2$ ($C_2^2\times \SD_{16}$), $C_3^2\wr C_2$ ($D_4^2$)

Order 144: $F_9:C_2$ ($\SOPlus(4,2)$)

Order 81: $C_3^4$ ($D_4\times \SD_{16}$)

Order 72: $\PSU(3,2)$ ($S_3^2:C_2^2$), $\SOPlus(4,2)$ ($S_3^2:C_2^2$), $\SOPlus(4,2)$ ($F_9:C_2$), $F_9$ ($S_3^2:C_2^2$)

Order 36: $S_3^2$ ($F_9:C_2^2$) x 2, $C_3^2:C_4$ ($C_6^2:D_4$), $C_3^2:C_4$ ($F_9:C_2^2$)

Order 18: $C_3:S_3$ ($F_9:C_2^3$), $C_3:S_3$ ($C_3^2:D_4^2$)

Order 9: $C_3^2$ ($D_4\times F_9:C_2$), $C_3^2$ ($\SD_{16}\times \SOPlus(4,2)$)

Order 1: $C_1$ ($\SOPlus(4,2)^2.C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 10368: $\SOPlus(4,2)^2.C_2$ ($C_1$)

Order 5184: $F_9\times \SOPlus(4,2)$ ($C_2$), $F_9:\SOPlus(4,2)$ ($C_2$), $F_9:\SOPlus(4,2)$ ($C_2$), $C_3^3:C_{12}:\SD_{16}$ ($C_2$), $C_3^4:(C_4\times \SD_{16})$ ($C_2$), $C_3^4:(D_4\times Q_8)$ ($C_2$), $C_3^3:C_{12}:\SD_{16}$ ($C_2$), $(C_3^2\times S_3^2):\SD_{16}$ ($C_2$), $S_3\times F_9:D_6$ ($C_2$), $\SOPlus(4,2)^2$ ($C_2$), $(C_3^2\times S_3^2):\SD_{16}$ ($C_2$)

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

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

Order 648: $C_3:S_3^3$ ($C_2\times D_4$), $C_3^4:Q_8$ ($C_2\times D_4$), $C_3^4:Q_8$ ($C_2\times D_4$), $C_3^3:D_{12}$ ($\SD_{16}$), $C_3^3:D_{12}$ ($C_2\times D_4$), $C_3^4:D_4$ ($\SD_{16}$), $C_3^4:D_4$ ($\SD_{16}$), $C_3^4:D_4$ ($C_2\times D_4$), $C_3^4:(C_2\times C_4)$ ($C_2\times D_4$), $C_3^3:(C_4\times S_3)$ ($C_2\times D_4$), $C_3^3:(C_4\times S_3)$ ($C_2^4$), $C_3^3:(C_4\times S_3)$ ($C_2\times D_4$), $C_3^2:F_9$ ($C_2\times D_4$), $C_3^2\times F_9$ ($C_2\times D_4$)

Order 324: $C_3^2:S_3^2$ ($C_2\times \SD_{16}$), $C_3^2:S_3^2$ ($C_2^2\times D_4$), $C_3^2\times S_3^2$ ($C_2\times \SD_{16}$), $C_3^4:C_4$ ($D_4:C_2^2$), $C_3^4:C_4$ ($C_2\times \SD_{16}$), $C_3^3:C_{12}$ ($C_2^2\times D_4$), $C_3^3:C_{12}$ ($C_2\times \SD_{16}$)

Order 162: $C_3^3:S_3$ ($D_8:C_2^2$), $C_3^2\wr C_2$ ($C_2^2\times \SD_{16}$), $C_3^2\wr C_2$ ($D_4^2$)

Order 144: $F_9:C_2$ ($\SOPlus(4,2)$)

Order 81: $C_3^4$ ($D_4\times \SD_{16}$)

Order 72: $\PSU(3,2)$ ($S_3^2:C_2^2$), $\SOPlus(4,2)$ ($S_3^2:C_2^2$), $\SOPlus(4,2)$ ($F_9:C_2$), $F_9$ ($S_3^2:C_2^2$)

Order 36: $C_3^2:C_4$ ($C_6^2:D_4$), $C_3^2:C_4$ ($F_9:C_2^2$), $S_3^2$ ($F_9:C_2^2$)

Order 18: $C_3:S_3$ ($F_9:C_2^3$), $C_3:S_3$ ($C_3^2:D_4^2$)

Order 9: $C_3^2$ ($D_4\times F_9:C_2$), $C_3^2$ ($\SD_{16}\times \SOPlus(4,2)$)

Order 1: $C_1$ ($\SOPlus(4,2)^2.C_2$)

Series

Derived series

$\SOPlus(4,2)^2.C_2$

$\rhd$

$\SOPlus(4,2)^2.C_2$

$\rhd$

$C_3^3:(C_4\times S_3)$

$\rhd$

$C_3^3:(C_4\times S_3)$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$\SOPlus(4,2)^2.C_2$

$\rhd$

$\SOPlus(4,2)^2.C_2$

$\rhd$

$S_3\times F_9:D_6$

$\rhd$

$S_3\times F_9:D_6$

$\rhd$

$(C_3\times F_9):D_6$

$\rhd$

$(C_3\times F_9):D_6$

$\rhd$

$(C_3\times S_3^2):D_6$

$\rhd$

$(C_3\times S_3^2):D_6$

$\rhd$

$C_3^3:(C_4\times S_3)$

$\rhd$

$C_3^3:(C_4\times S_3)$

$\rhd$

$C_3^3:C_{12}$

$\rhd$

$C_3^3:C_{12}$

$\rhd$

$C_3^2\wr C_2$

$\rhd$

$C_3^2\wr C_2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$\SOPlus(4,2)^2.C_2$

$\rhd$

$\SOPlus(4,2)^2.C_2$

$\rhd$

$C_3^3:(C_4\times S_3)$

$\rhd$

$C_3^3:(C_4\times S_3)$

$\rhd$

$C_3^2\wr C_2$

$\rhd$

$C_3^2\wr C_2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 2 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 $81 \times 81$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $72 \times 72$ rational character table.