Abstract group 31104.mn: $S_3^5:C_2^2$

• Label: 31104.mn
• Order: \( 2^{7} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $5$

Group information

Description:	$S_3^5:C_2^2$
Order:	\(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_3^3\times C_{54}):C_3$, of order \(248832\)\(\medspace = 2^{10} \cdot 3^{5} \)
Composition factors:	$C_2$ x 7, $C_3$ x 5
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	1935	242	4464	15318	9144	31104
Conjugacy classes	1	31	17	18	143	33	243
Divisions	1	31	17	18	143	33	243
Autjugacy classes	1	11	5	8	28	10	63

Dimension	1	2	4	8	16	32
Irr. complex chars.	32	32	74	49	40	16	243
Irr. rational chars.	32	32	74	49	40	16	243

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$48$
Rank:	$5$
Inequivalent generating 5-tuples:	not computed

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:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{2}=c^{12}=d^{6}=e^{12}=f^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2), (4,5,6), (13,15), (1,2,3)(4,9,5,7,6,8)(10,14,11,15)(12,13), (4,5) \!\cdots\! \rangle$
Direct product:	$S_3$ $\, \times\, $ $\SOPlus(4,2)$ ${}^2$
Semidirect product:	$S_3^5$ $\,\rtimes\,$ $C_2^2$	$(S_3^4:D_6)$ $\,\rtimes\,$ $C_2$	$(S_3^4:D_6)$ $\,\rtimes\,$ $C_2$	$(S_3^4:C_2)$ $\,\rtimes\,$ $D_6$	all 152
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3:S_3^4)$ . $C_2^3$	$(C_3:S_3^4)$ . $C_2^3$	$(C_3^2:S_3^3)$ . $C_2^4$	$(C_3^2:S_3^3)$ . $C_2^4$	all 39
Aut. group:	$\Aut((C_3\times S_3^2):S_3^2)$	$\Aut(C_3^4:(S_3\times D_4))$	$\Aut(C_3^3:(S_3\times D_{12}))$	$\Aut(C_3^2\wr C_2.D_{12})$	all 11

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

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 2216888 subgroups in 30158 conjugacy classes, 628 normal (30 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_3^5:C_2^2$
Commutator:	$G' \simeq$ $C_3^3:S_3^2$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3^5:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^5$	$G/\operatorname{Fit} \simeq$ $C_2\times D_4^2$
Radical:	$R \simeq$ $S_3^5:C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^5$	$G/\operatorname{soc} \simeq$ $C_2\times D_4^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5$

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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Classes of subgroups up to conjugation

All subgroups of index up to 64 (order at least 486) are shown, as well as all normal subgroups of any index.

Order 31104: $S_3^5:C_2^2$

Order 15552: $C_3^5:D_4^2$ x 4, $S_3^5:C_2$ x 4, $S_3^3:\SOPlus(4,2)$ x 4, $S_3^4:D_6$ x 4, $S_3^4:D_6$ x 4, $S_3^5:C_2$ x 4, $C_3^4:C_4^2:D_6$ x 2, $C_3^5:D_4^2$ x 2, $C_3^5:D_4^2$, $C_3^4:C_4^2:D_6$, $C_3^5:D_4^2$

Order 10368: $S_3^4:C_2^3$

Order 7776: $S_3^4:S_3$ x 16, $S_3^2:S_3^3$ x 16, $C_3^4:(D_4\times D_6)$ x 8, $C_3^2:D_{12}\times S_3^2$ x 8, $S_3^4:C_6$ x 8, $S_3^2:S_3^3$ x 8, $C_3^4:C_{12}:D_4$ x 8, $(C_3^3\times S_3^2):D_4$ x 8, $(C_3^2\times S_3^2):D_{12}$ x 8, $(C_3^3\times S_3^2):D_4$ x 8, $(C_3\times S_3^3):D_6$ x 6, $C_3^4:C_{12}:D_4$ x 4, $(C_3^3\times S_3^2):D_4$ x 4, $C_3^4:C_{12}:D_4$ x 4, $C_3^4:(D_4\times D_6)$ x 4, $C_3^5:(C_4\times D_4)$ x 4, $C_3^3:C_{12}:D_{12}$ x 4, $C_3^4:(D_4\times D_6)$ x 4, $C_3^2:C_4\times S_3^3$ x 4, $S_3^5$ x 4, $C_3^5:(C_4\times D_4)$ x 4, $(C_3^2\times S_3^2):D_{12}$ x 4, $C_3^2:S_3^3:C_4$ x 4, $(C_3^2\times S_3^3):C_4$ x 4, $C_3:S_3^2\wr C_2$ x 4, $C_3:S_3^2\wr C_2$ x 4, $S_3^4:S_3$ x 4, $S_3^4:C_6$ x 4, $C_3^4:(D_4\times C_{12})$ x 2, $C_3^4:(C_4\times D_{12})$ x 2, $C_3^5:(C_4\times D_4)$ x 2, $C_3^4:C_{12}:D_4$ x 2, $C_3^3:C_6^2.D_4$ x 2, $C_3^5:(C_4\times D_4)$ x 2, $C_3^4:(D_4\times D_6)$ x 2, $S_3\times C_3^4:C_4^2$, $C_3^4:C_{12}:D_4$, $C_3^3:C_{12}:D_{12}$

Order 5184: $\SOPlus(4,2)^2$ x 16, $S_3^2:D_6^2$ x 8, $C_3:S_3^3:D_4$ x 4, $S_3^4:C_2^2$ x 4, $S_3^2:C_6^2:C_4$ x 2, $C_3^4:C_4^2:C_2^2$

Order 3888: $S_3\times C_3^3:D_{12}$ x 24, $C_3\wr D_4\times S_3$ x 24, $C_3:S_3^4$ x 20, $C_3\wr C_4:D_6$ x 16, $C_3:S_3^4$ x 16, $C_3^3:(S_3\times D_{12})$ x 16, $C_3^4:(S_3\times D_4)$ x 16, $(C_3\times S_3^2):S_3^2$ x 16, $C_3\times S_3^2:S_3^2$ x 16, $S_3\times C_3^4:D_4$ x 12, $C_3\wr C_4:D_6$ x 12, $S_3\times C_3^4:D_4$ x 12, $S_3\times C_3^3:D_{12}$ x 12, $C_3\times S_3^4$ x 12, $C_3\wr C_4:D_6$ x 12, $C_3\wr C_4:D_6$ x 12, $C_3:S_3^4$ x 12, $C_3^4:(C_6\times D_4)$ x 8, $C_3^3:(S_3\times D_{12})$ x 8, $C_3^3:C_4\times S_3^2$ x 8, $C_3^2:C_{12}\times S_3^2$ x 8, $C_3\wr C_4:D_6$ x 8, $C_3^2\wr C_2.D_{12}$ x 8, $C_3^4:(C_6\times D_4)$ x 6, $C_3\times S_3^3:C_6$ x 6, $S_3\times C_3^3:D_{12}$ x 6, $C_3^3:(S_3\times D_{12})$ x 6, $C_3^4:(S_3\times D_4)$ x 6, $C_3\wr C_4:D_6$ x 6, $C_3^4:C_{12}:C_4$ x 4, $C_3\wr C_4:D_6$ x 4, $C_3^3:(C_4\times S_3^2)$ x 4, $C_3:S_3^4$ x 4, $C_3^3:S_3^2:C_4$ x 4, $\GL(2,3).C_2^6$ x 4, $C_3^3:C_6^2:C_4$ x 4, $C_3^2\wr C_2.D_{12}$ x 4, $(C_3^3\times S_3^2):C_4$ x 4, $(C_3^2\times S_3^2):C_{12}$ x 4, $C_3^5:(C_2\times D_4)$ x 4, $C_3^3:C_{12}:C_{12}$ x 2, $C_3^5:C_4^2$ x 2, $C_3^2\wr C_2.D_{12}$ x 2, $C_3^4:(S_3\times D_4)$ x 2, $S_3\times C_3^4:D_4$ x 2, $C_3^4:(C_2\times D_{12})$ x 2, $C_3^4:(S_3\times D_4)$ x 2, $C_3^4:(C_6\times D_4)$ x 2, $C_3^5.C_4^2$, $C_3^5:C_4^2$, $C_3^4:(C_4\times D_6)$

Order 3456: $D_4\times S_3^3:C_2$ x 2

Order 2592: $S_3^4:C_2$ x 116, $S_3^2:\SOPlus(4,2)$ x 32, $S_3^2\wr C_2$ x 32, $S_3^3:D_6$ x 24, $C_3^4:(C_4\times D_4)$ x 16, $C_2\times S_3^2:S_3^2$ x 16, $C_2\times S_3^3:C_6$ x 12, $C_2\times S_3^4$ x 12, $C_3^2:D_{12}\times D_6$ x 12, $C_3^3:(D_4\times D_6)$ x 8, $C_3\wr C_4:C_2^3$ x 8, $C_3^3:C_{12}:D_4$ x 8, $(C_6\times S_3^2):D_6$ x 6, $C_3:S_3^3:C_4$ x 4, $C_3:S_3^3:C_4$ x 4, $C_3^2:C_6^2.D_4$ x 2, $C_3^4:(C_2^2\times D_4)$ x 2, $C_2\times C_3^4:C_4^2$

Order 1944: $C_3^2:S_3^3$ x 36, $C_3^2:S_3^3$ x 36, $C_3^2:S_3^3$ x 32, $C_3^2:S_3^3$ x 24, $C_3^4:D_{12}$ x 24, $C_3^2:S_3^3$ x 17, $C_3^2:S_3^3$ x 16, $C_3^2:S_3^3$ x 16, $C_3^2\times S_3^3$ x 16, $C_3^2:S_3^3$ x 12, $C_3^2:S_3^3$ x 12, $C_3^4:D_{12}$ x 12, $C_3^4:D_{12}$ x 12, $C_3^4:D_{12}$ x 12, $C_3^5:D_4$ x 12, $C_3^5:D_4$ x 12, $C_3^5:D_4$ x 12, $C_3^5:D_4$ x 12, $C_3\wr C_4\times S_3$ x 12, $C_3^4:(C_4\times S_3)$ x 8, $C_3^3:(S_3\times C_{12})$ x 8, $C_3^4:D_{12}$ x 6, $C_3^4:D_{12}$ x 6, $C_3\wr C_4:S_3$ x 6, $C_3\wr C_4:S_3$ x 6, $C_3^4:C_4\times S_3$ x 6, $C_3^3:(S_3\times C_{12})$ x 6, $S_3\times C_3^3:C_{12}$ x 6, $C_3^5:D_4$ x 4, $C_3^5:D_4$ x 4, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3^4:(C_4\times S_3)$, $C_3^5:(C_2\times C_4)$, $C_3^4:C_4\times S_3$, $C_3^4:(C_2\times C_{12})$

Order 1728: $D_6^2:D_6$ x 8, $D_6^2:D_6$ x 8, $D_6^2:D_6$ x 8, $D_4\times S_3^3$ x 4, $D_6^2:D_6$ x 4, $C_3^3:D_4^2$ x 4, $S_3\times C_6^2:D_4$ x 4, $D_6^2:D_6$ x 4, $C_3^3:D_4^2$ x 4, $S_3^3:D_4$ x 4, $C_3^3:D_4^2$ x 2, $C_3^3:D_4^2$ x 2, $C_6^2:(C_4\times D_6)$ x 2, $C_3^3:D_4^2$ x 2, $(C_2\times S_3^3).C_2^2$ x 2, $C_3^3:D_4^2$ x 2, $C_4\times S_3^3:C_2$ x 2

Order 1296: $S_3^3:S_3$ x 208, $S_3^4$ x 144, $S_3^2:S_3^2$ x 120, $S_3\times C_3^2:D_{12}$ x 104, $S_3^3:C_6$ x 104, $C_3^3:(S_3\times D_4)$ x 60, $C_3^2:C_4\times S_3^2$ x 60, $(C_3\times S_3^2):D_6$ x 52, $C_6:S_3^3$ x 36, $C_6:S_3^3$ x 32, $C_6\times S_3^3$ x 32, $C_2\times C_3^3:D_{12}$ x 24, $C_2\times C_3\wr D_4$ x 24, $(C_3^2\times S_3^2):C_4$ x 16, $C_3^2\wr C_2.D_4$ x 16, $C_6:S_3^3$ x 12, $C_2\times C_3^4:D_4$ x 12, $C_2\times C_3^4:D_4$ x 12, $C_6\times C_3^2:D_{12}$ x 12, $C_3^3:C_4\times D_6$ x 12, $C_3^2:C_{12}\times D_6$ x 12, $C_3^4:(C_2\times D_4)$ x 10, $C_3^3:(C_4\times D_6)$ x 8, $C_3^2\wr C_2.D_4$ x 8, $C_2\times C_3^3:D_{12}$ x 6, $S_3^2:C_6^2$ x 6, $C_3^3:(C_4\times D_6)$ x 6, $C_3^4:C_4^2$ x 4, $C_2\times C_3^4:D_4$ x 2, $C_3^4:(C_2^2\times C_4)$

Order 1152: $D_6^2:C_2^3$ x 2

Order 972: $C_3^3:S_3^2$ x 36, $C_3^3:C_6^2$ x 30, $C_3^3:S_3^2$ x 20, $C_3^3:S_3^2$ x 17, $C_3^3:S_3^2$ x 17, $C_3^3:C_6^2$ x 16, $C_3^3:S_3^2$ x 12, $C_3^3\times S_3^2$ x 12, $C_3^3:S_3^2$ x 10, $C_3^4:C_{12}$ x 6, $C_3^4:C_{12}$ x 6, $C_3^3:S_3^2$ x 5, $C_3^5:C_4$ x 2, $C_3^4:C_{12}$ x 2, $C_3^5:C_4$, $C_3^4:C_{12}$

Order 864: $S_3^3:C_2^2$ x 166, $S_3\times D_6^2$ x 20, $D_6^2:S_3$ x 16, $C_2^2:S_3^3$ x 16, $D_6^2:S_3$ x 16, $S_3^2:D_{12}$ x 16, $D_6:\SOPlus(4,2)$ x 16, $S_3^3:C_4$ x 12, $C_6^2:D_{12}$ x 8, $D_6^2:C_6$ x 8, $C_6^2:C_4\times S_3$ x 8, $D_6^2:C_6$ x 8, $C_2^2:S_3^3$ x 8, $C_4:S_3^3$ x 8, $D_{12}\times S_3^2$ x 8, $C_6^2:D_{12}$ x 8, $(C_3\times C_6^2):D_4$ x 8, $D_6^2:S_3$ x 8, $(S_3\times D_6):D_6$ x 8, $D_6^2:C_6$ x 8, $(S_3\times D_6).D_6$ x 8, $(S_3\times D_6).D_6$ x 8, $(C_3\times S_3^2):D_4$ x 8, $D_6:\SOPlus(4,2)$ x 8, $C_6.D_6^2$ x 6, $C_6.D_6^2$ x 4, $C_4:S_3^3$ x 4, $C_4\times S_3^3$ x 4, $C_6^2:D_{12}$ x 4, $C_{12}:\SOPlus(4,2)$ x 4, $C_{12}:\SOPlus(4,2)$ x 4, $S_3^2:D_{12}$ x 4, $C_{12}:\SOPlus(4,2)$ x 4, $C_4\times S_3^2:S_3$ x 4, $C_3\times C_6^2:D_4$ x 4, $C_{12}:\SOPlus(4,2)$ x 4, $C_6^2:(C_4\times S_3)$ x 4, $C_6^2:(C_4\times S_3)$ x 4, $C_6^2:C_4\times S_3$ x 4, $D_6:\SOPlus(4,2)$ x 4, $D_6:\SOPlus(4,2)$ x 4, $S_3^2:(C_4\times S_3)$ x 4, $C_3^3:C_4\times D_4$ x 2, $D_4\times C_3^2:C_{12}$ x 2, $C_{12}:\SOPlus(4,2)$ x 2, $C_4\times C_3^2:D_{12}$ x 2, $C_{12}:\SOPlus(4,2)$ x 2, $C_{12}\times \SOPlus(4,2)$ x 2, $(C_3^2\times D_{12}):C_4$ x 2, $C_3^2:C_4\times D_{12}$ x 2, $C_{12}:S_3.D_6$ x 2, $S_3\times C_3^2:C_4^2$ x 2, $C_3^3:(C_4\times D_4)$ x 2, $C_3^3:(C_4\times D_4)$ x 2, $D_6.\SOPlus(4,2)$ x 2

Order 648: $C_3:S_3^3$ x 320, $C_3:S_3^3$ x 248, $C_3\times S_3^3$ x 248, $C_3^3:D_{12}$ x 112, $C_3\wr D_4$ x 112, $C_3:S_3^3$ x 80, $C_3^4:C_2^3$ x 60, $C_3^4:D_4$ x 56, $C_3^4:D_4$ x 56, $C_3^3:D_{12}$ x 56, $C_3^3:C_4\times S_3$ x 56, $S_3\times C_3^2:C_{12}$ x 56, $C_3^4:C_2^3$ x 36, $C_3^4:C_2^3$ x 36, $C_3^4:C_2^3$ x 32, $C_3^3:(C_4\times S_3)$ x 32, $C_3^3:D_{12}$ x 28, $C_3^4:D_4$ x 28, $C_3^3:(C_4\times S_3)$ x 28, $C_3^4:C_2^3$ x 20, $C_3^4:C_2^3$ x 17, $C_2\times C_3\wr C_4$ x 12, $C_3^4:D_4$ x 8, $C_2\times C_3^4:C_4$ x 6, $C_6\times C_3^2:C_{12}$ x 6, $C_3^4:(C_2\times C_4)$ x 3, $C_2\times C_3^4:C_4$

Order 576: $C_3^2:D_4^2$ x 32, $C_4:D_6^2$ x 8, $D_6^2:C_2^2$ x 8, $D_6^2:C_2^2$ x 4, $D_6^2.C_2^2$ x 4, $D_6^2.C_2^2$ x 4, $C_6^2:(C_2^2\times C_4)$ x 2, $D_6^2.C_2^2$ x 2, $D_6^2.C_2^2$ x 2

Order 486: $C_3^4:C_6$ x 10, $C_3^4:C_6$ x 10, $C_3^4:C_6$ x 5, $S_3\times C_3^4$ x 5, $C_3^4:S_3$

Order 432: $S_3^3:C_2$ x 170

Order 384: $S_3\times D_4^2$

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

Order 288: $C_6^2:D_4$ x 32

Order 243: $C_3^5$

Order 216: $S_3^3$ x 4, $S_3^2:S_3$ x 4, $C_3^2:D_{12}$ x 2, $S_3^2:C_6$ x 2, $C_3^2:C_4\times S_3$ x 2

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

Order 144: $S_3^2:C_2^2$ x 202

Order 128: $C_2\times D_4^2$

Order 108: $C_3:S_3^2$ x 4, $C_3\times S_3^2$ x 4, $C_3:S_3^2$ x 2, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2

Order 96: $D_4\times D_6$ x 36

Order 81: $C_3^4$

Order 72: $\SOPlus(4,2)$ x 152

Order 64: $D_4^2$ x 16

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

Order 48: $S_3\times D_4$ x 164

Order 36: $S_3^2$ x 4, $C_3^2:C_4$ x 2

Order 32: $C_2^2\times D_4$ x 32

Order 27: $C_3^3$ x 2

Order 24: $C_2\times D_6$ x 100

Order 18: $C_3:S_3$ x 2

Order 16: $C_2\times D_4$ x 196

Order 12: $D_6$ x 216

Order 9: $C_3^2$ x 2

Order 8: $D_4$ x 136, $C_2^3$ x 64

Order 6: $S_3$ x 64

Order 4: $C_2^2$ x 112

Order 3: $C_3$

Order 2: $C_2$ x 28

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 64 (order at least 486) are shown, as well as all normal subgroups of any index.

Order 31104: $S_3^5:C_2^2$

Order 15552: $C_3^4:C_4^2:D_6$, $C_3^5:D_4^2$, $C_3^5:D_4^2$, $C_3^5:D_4^2$, $S_3^5:C_2$, $S_3^3:\SOPlus(4,2)$, $C_3^4:C_4^2:D_6$, $S_3^4:D_6$, $C_3^5:D_4^2$, $S_3^4:D_6$, $S_3^5:C_2$

Order 10368: $S_3^4:C_2^3$

Order 7776: $C_3^4:(D_4\times D_6)$ x 2, $C_3^2:D_{12}\times S_3^2$ x 2, $S_3^4:C_6$ x 2, $(C_3\times S_3^3):D_6$ x 2, $S_3^4:S_3$ x 2, $S_3^2:S_3^3$ x 2, $S_3\times C_3^4:C_4^2$, $C_3^4:C_{12}:D_4$, $C_3^4:C_{12}:D_4$, $(C_3^3\times S_3^2):D_4$, $C_3^4:C_{12}:D_4$, $C_3^4:(D_4\times C_{12})$, $C_3^4:(C_4\times D_{12})$, $C_3^5:(C_4\times D_4)$, $C_3^4:C_{12}:D_4$, $C_3^3:C_{12}:D_{12}$, $C_3^4:(D_4\times D_6)$, $C_3^5:(C_4\times D_4)$, $C_3^3:C_{12}:D_{12}$, $C_3^4:(D_4\times D_6)$, $C_3^2:C_4\times S_3^3$, $C_3^3:C_6^2.D_4$, $S_3^5$, $C_3^5:(C_4\times D_4)$, $C_3^5:(C_4\times D_4)$, $S_3^2:S_3^3$, $C_3^4:C_{12}:D_4$, $(C_3^2\times S_3^2):D_{12}$, $(C_3^3\times S_3^2):D_4$, $C_3^2:S_3^3:C_4$, $(C_3^2\times S_3^3):C_4$, $C_3^4:(D_4\times D_6)$, $C_3:S_3^2\wr C_2$, $C_3:S_3^2\wr C_2$, $(C_3^2\times S_3^2):D_{12}$, $S_3^4:S_3$, $(C_3^3\times S_3^2):D_4$, $S_3^4:C_6$

Order 5184: $\SOPlus(4,2)^2$ x 6, $S_3^2:D_6^2$ x 2, $S_3^2:C_6^2:C_4$, $C_3^4:C_4^2:C_2^2$, $C_3:S_3^3:D_4$, $S_3^4:C_2^2$

Order 3888: $C_3:S_3^4$ x 3, $S_3\times C_3^3:D_{12}$ x 3, $C_3:S_3^4$ x 3, $S_3\times C_3^3:D_{12}$ x 3, $C_3\wr D_4\times S_3$ x 3, $C_3^4:(C_6\times D_4)$ x 2, $S_3\times C_3^4:D_4$ x 2, $C_3^4:(C_6\times D_4)$ x 2, $C_3\wr C_4:D_6$ x 2, $C_3\times S_3^3:C_6$ x 2, $S_3\times C_3^3:D_{12}$ x 2, $C_3^3:(S_3\times D_{12})$ x 2, $C_3^4:(S_3\times D_4)$ x 2, $C_3\wr C_4:D_6$ x 2, $C_3\wr C_4:D_6$ x 2, $S_3\times C_3^4:D_4$ x 2, $C_3^3:(S_3\times D_{12})$ x 2, $C_3^3:C_4\times S_3^2$ x 2, $C_3^2:C_{12}\times S_3^2$ x 2, $C_3\times S_3^4$ x 2, $C_3\wr C_4:D_6$ x 2, $C_3\wr C_4:D_6$ x 2, $C_3:S_3^4$ x 2, $C_3\wr C_4:D_6$ x 2, $C_3^3:(S_3\times D_{12})$ x 2, $C_3^4:(S_3\times D_4)$ x 2, $(C_3\times S_3^2):S_3^2$ x 2, $C_3\times S_3^2:S_3^2$ x 2, $C_3^3:C_{12}:C_{12}$, $C_3^5:C_4^2$, $C_3^5.C_4^2$, $C_3^4:C_{12}:C_4$, $C_3^2\wr C_2.D_{12}$, $C_3^5:C_4^2$, $C_3\wr C_4:D_6$, $C_3^3:(C_4\times S_3^2)$, $C_3^4:(S_3\times D_4)$, $C_3:S_3^4$, $S_3\times C_3^4:D_4$, $C_3^4:(C_4\times D_6)$, $C_3^3:S_3^2:C_4$, $C_3^2\wr C_2.D_{12}$, $\GL(2,3).C_2^6$, $C_3^3:C_6^2:C_4$, $C_3^2\wr C_2.D_{12}$, $(C_3^3\times S_3^2):C_4$, $(C_3^2\times S_3^2):C_{12}$, $C_3^4:(C_2\times D_{12})$, $C_3^4:(S_3\times D_4)$, $C_3^5:(C_2\times D_4)$, $C_3^4:(C_6\times D_4)$

Order 3456: $D_4\times S_3^3:C_2$

Order 2592: $S_3^4:C_2$ x 22, $S_3^2:\SOPlus(4,2)$ x 6, $C_3^4:(C_4\times D_4)$ x 6, $S_3^2\wr C_2$ x 6, $C_3^3:C_{12}:D_4$ x 4, $C_2\times S_3^3:C_6$ x 3, $C_3^2:D_{12}\times D_6$ x 3, $S_3^3:D_6$ x 3, $C_3^3:(D_4\times D_6)$ x 2, $(C_6\times S_3^2):D_6$ x 2, $C_2\times S_3^4$ x 2, $C_3\wr C_4:C_2^3$ x 2, $C_2\times S_3^2:S_3^2$ x 2, $C_2\times C_3^4:C_4^2$, $C_3^2:C_6^2.D_4$, $C_3:S_3^3:C_4$, $C_3^4:(C_2^2\times D_4)$, $C_3:S_3^3:C_4$

Order 1944: $C_3^2:S_3^3$ x 6, $C_3^2:S_3^3$ x 6, $C_3^2:S_3^3$ x 4, $C_3^2:S_3^3$ x 4, $C_3^2:S_3^3$ x 4, $C_3^2:S_3^3$ x 3, $C_3^2:S_3^3$ x 3, $C_3^2:S_3^3$ x 3, $C_3^2\times S_3^3$ x 3, $C_3^4:D_{12}$ x 3, $C_3\wr C_4\times S_3$ x 3, $C_3^2:S_3^3$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^5:D_4$ x 2, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3\wr C_4:S_3$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3\wr C_4:S_3$ x 2, $C_3^4:C_4\times S_3$ x 2, $C_3^3:(S_3\times C_{12})$ x 2, $C_3^3:(S_3\times C_{12})$ x 2, $S_3\times C_3^3:C_{12}$ x 2, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3^5:D_4$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3^5:D_4$, $C_3^4:(C_4\times S_3)$, $C_3^5:(C_2\times C_4)$, $C_3^4:C_4\times S_3$, $C_3^4:(C_2\times C_{12})$, $C_3^4:(C_4\times S_3)$

Order 1728: $D_6^2:D_6$ x 2, $D_4\times S_3^3$, $C_3^3:D_4^2$, $D_6^2:D_6$, $C_3^3:D_4^2$, $C_6^2:(C_4\times D_6)$, $C_3^3:D_4^2$, $S_3\times C_6^2:D_4$, $D_6^2:D_6$, $D_6^2:D_6$, $D_6^2:D_6$, $C_3^3:D_4^2$, $(C_2\times S_3^3).C_2^2$, $C_3^3:D_4^2$, $S_3^3:D_4$, $C_3^3:D_4^2$, $C_4\times S_3^3:C_2$

Order 1296: $S_3^3:S_3$ x 26, $S_3^4$ x 20, $S_3\times C_3^2:D_{12}$ x 20, $S_3^3:C_6$ x 20, $S_3^2:S_3^2$ x 15, $C_3^2:C_4\times S_3^2$ x 15, $(C_3\times S_3^2):D_6$ x 13, $C_3^3:(S_3\times D_4)$ x 12, $C_6:S_3^3$ x 6, $C_6:S_3^3$ x 4, $C_6\times S_3^3$ x 4, $C_3^4:(C_2\times D_4)$ x 4, $(C_3^2\times S_3^2):C_4$ x 4, $C_2\times C_3^3:D_{12}$ x 3, $C_6\times C_3^2:D_{12}$ x 3, $C_2\times C_3\wr D_4$ x 3, $C_3^3:C_4\times D_6$ x 3, $C_3^2:C_{12}\times D_6$ x 3, $C_3^2\wr C_2.D_4$ x 3, $C_3^2\wr C_2.D_4$ x 3, $C_3^4:C_4^2$ x 3, $C_6:S_3^3$ x 2, $C_2\times C_3^3:D_{12}$ x 2, $C_2\times C_3^4:D_4$ x 2, $C_2\times C_3^4:D_4$ x 2, $S_3^2:C_6^2$ x 2, $C_3^3:(C_4\times D_6)$ x 2, $C_3^3:(C_4\times D_6)$ x 2, $C_2\times C_3^4:D_4$, $C_3^4:(C_2^2\times C_4)$

Order 1152: $D_6^2:C_2^3$

Order 972: $C_3^3:S_3^2$ x 6, $C_3^3:C_6^2$ x 6, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:C_6^2$ x 3, $C_3^3\times S_3^2$ x 3, $C_3^3:S_3^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4:C_{12}$ x 2, $C_3^5:C_4$, $C_3^4:C_{12}$, $C_3^5:C_4$, $C_3^4:C_{12}$

Order 864: $S_3^3:C_2^2$ x 32, $S_3\times D_6^2$ x 3, $C_6^2:D_{12}$ x 2, $D_6^2:S_3$ x 2, $D_6^2:C_6$ x 2, $C_6^2:C_4\times S_3$ x 2, $C_6.D_6^2$ x 2, $D_6^2:C_6$ x 2, $C_2^2:S_3^3$ x 2, $D_6^2:S_3$ x 2, $C_4:S_3^3$ x 2, $D_{12}\times S_3^2$ x 2, $(S_3\times D_6).D_6$ x 2, $S_3^2:D_{12}$ x 2, $D_6:\SOPlus(4,2)$ x 2, $S_3^3:C_4$ x 2, $C_6.D_6^2$, $C_3^3:C_4\times D_4$, $D_4\times C_3^2:C_{12}$, $C_2^2:S_3^3$, $C_4:S_3^3$, $C_4\times S_3^3$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_{12}:\SOPlus(4,2)$, $C_{12}:\SOPlus(4,2)$, $C_4\times C_3^2:D_{12}$, $(C_3\times C_6^2):D_4$, $D_6^2:S_3$, $(S_3\times D_6):D_6$, $C_{12}:\SOPlus(4,2)$, $S_3^2:D_{12}$, $C_{12}:\SOPlus(4,2)$, $C_4\times S_3^2:S_3$, $C_3\times C_6^2:D_4$, $D_6^2:C_6$, $C_{12}:\SOPlus(4,2)$, $C_{12}:\SOPlus(4,2)$, $C_{12}\times \SOPlus(4,2)$, $C_6^2:(C_4\times S_3)$, $C_6^2:(C_4\times S_3)$, $C_6^2:C_4\times S_3$, $(C_3^2\times D_{12}):C_4$, $C_3^2:C_4\times D_{12}$, $C_{12}:S_3.D_6$, $S_3\times C_3^2:C_4^2$, $(S_3\times D_6).D_6$, $(C_3\times S_3^2):D_4$, $D_6:\SOPlus(4,2)$, $D_6:\SOPlus(4,2)$, $D_6:\SOPlus(4,2)$, $C_3^3:(C_4\times D_4)$, $S_3^2:(C_4\times S_3)$, $C_3^3:(C_4\times D_4)$, $D_6.\SOPlus(4,2)$

Order 648: $C_3:S_3^3$ x 51, $C_3:S_3^3$ x 32, $C_3\times S_3^3$ x 32, $C_3^3:D_{12}$ x 14, $C_3\wr D_4$ x 14, $C_3^3:C_4\times S_3$ x 14, $S_3\times C_3^2:C_{12}$ x 14, $C_3:S_3^3$ x 12, $C_3^3:D_{12}$ x 11, $C_3^4:C_2^3$ x 9, $C_3^4:D_4$ x 9, $C_3^3:(C_4\times S_3)$ x 9, $C_3^4:D_4$ x 8, $C_3^3:(C_4\times S_3)$ x 8, $C_3^3:D_{12}$ x 7, $C_3^4:D_4$ x 7, $C_3^4:C_2^3$ x 6, $C_3^4:C_2^3$ x 6, $C_3^4:C_2^3$ x 4, $C_3^4:C_2^3$ x 4, $C_3^4:C_2^3$ x 4, $C_2\times C_3\wr C_4$ x 3, $C_3^4:D_4$ x 3, $C_3^4:(C_2\times C_4)$ x 3, $C_2\times C_3^4:C_4$ x 2, $C_6\times C_3^2:C_{12}$ x 2, $C_2\times C_3^4:C_4$

Order 576: $C_3^2:D_4^2$ x 9, $C_4:D_6^2$ x 2, $D_6^2:C_2^2$, $C_6^2:(C_2^2\times C_4)$, $D_6^2.C_2^2$, $D_6^2:C_2^2$, $D_6^2.C_2^2$, $D_6^2.C_2^2$, $D_6^2.C_2^2$

Order 486: $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 2, $S_3\times C_3^4$ x 2, $C_3^4:S_3$

Order 432: $S_3^3:C_2$ x 33

Order 384: $S_3\times D_4^2$

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

Order 288: $C_6^2:D_4$ x 6

Order 243: $C_3^5$

Order 216: $S_3^3$, $C_3^2:D_{12}$, $S_3^2:S_3$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$

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

Order 144: $S_3^2:C_2^2$ x 37

Order 128: $C_2\times D_4^2$

Order 108: $C_3:S_3^2$, $C_3:S_3^2$, $C_3\times S_3^2$, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 96: $D_4\times D_6$ x 7

Order 81: $C_3^4$

Order 72: $\SOPlus(4,2)$ x 30

Order 64: $D_4^2$ x 6

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

Order 48: $S_3\times D_4$ x 32

Order 36: $C_3^2:C_4$, $S_3^2$

Order 32: $C_2^2\times D_4$ x 6

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 15

Order 18: $C_3:S_3$

Order 16: $C_2\times D_4$ x 36

Order 12: $D_6$ x 36

Order 9: $C_3^2$

Order 8: $D_4$ x 27, $C_2^3$ x 10

Order 6: $S_3$ x 15

Order 4: $C_2^2$ x 21

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 31104: $S_3^5:C_2^2$ ($C_1$)

Order 15552: $C_3^5:D_4^2$ ($C_2$) x 4, $S_3^5:C_2$ ($C_2$) x 4, $S_3^3:\SOPlus(4,2)$ ($C_2$) x 4, $S_3^4:D_6$ ($C_2$) x 4, $S_3^4:D_6$ ($C_2$) x 4, $S_3^5:C_2$ ($C_2$) x 4, $C_3^4:C_4^2:D_6$ ($C_2$) x 2, $C_3^5:D_4^2$ ($C_2$) x 2, $C_3^5:D_4^2$ ($C_2$), $C_3^4:C_4^2:D_6$ ($C_2$), $C_3^5:D_4^2$ ($C_2$)

Order 7776: $S_3^4:S_3$ ($C_2^2$) x 8, $S_3^2:S_3^3$ ($C_2^2$) x 8, $C_3^4:C_{12}:D_4$ ($C_2^2$) x 8, $(C_3^3\times S_3^2):D_4$ ($C_2^2$) x 8, $S_3^2:S_3^3$ ($C_2^2$) x 8, $(C_3^2\times S_3^2):D_{12}$ ($C_2^2$) x 8, $(C_3^3\times S_3^2):D_4$ ($C_2^2$) x 8, $C_3^4:C_{12}:D_4$ ($C_2^2$) x 4, $(C_3^3\times S_3^2):D_4$ ($C_2^2$) x 4, $C_3^4:C_{12}:D_4$ ($C_2^2$) x 4, $C_3^4:(D_4\times D_6)$ ($C_2^2$) x 4, $C_3^5:(C_4\times D_4)$ ($C_2^2$) x 4, $C_3^4:(D_4\times D_6)$ ($C_2^2$) x 4, $C_3^2:D_{12}\times S_3^2$ ($C_2^2$) x 4, $C_3^3:C_{12}:D_{12}$ ($C_2^2$) x 4, $C_3^4:(D_4\times D_6)$ ($C_2^2$) x 4, $C_3^2:C_4\times S_3^3$ ($C_2^2$) x 4, $S_3^4:C_6$ ($C_2^2$) x 4, $S_3^5$ ($C_2^2$) x 4, $C_3^5:(C_4\times D_4)$ ($C_2^2$) x 4, $(C_3^2\times S_3^2):D_{12}$ ($C_2^2$) x 4, $C_3^2:S_3^3:C_4$ ($C_2^2$) x 4, $(C_3^2\times S_3^3):C_4$ ($C_2^2$) x 4, $C_3:S_3^2\wr C_2$ ($C_2^2$) x 4, $C_3:S_3^2\wr C_2$ ($C_2^2$) x 4, $S_3^4:S_3$ ($C_2^2$) x 4, $S_3^4:C_6$ ($C_2^2$) x 4, $C_3^4:(D_4\times C_{12})$ ($C_2^2$) x 2, $C_3^4:(C_4\times D_{12})$ ($C_2^2$) x 2, $C_3^5:(C_4\times D_4)$ ($C_2^2$) x 2, $C_3^4:C_{12}:D_4$ ($C_2^2$) x 2, $C_3^3:C_6^2.D_4$ ($C_2^2$) x 2, $C_3^5:(C_4\times D_4)$ ($C_2^2$) x 2, $(C_3\times S_3^3):D_6$ ($C_2^2$) x 2, $C_3^4:(D_4\times D_6)$ ($C_2^2$) x 2, $S_3\times C_3^4:C_4^2$ ($C_2^2$), $C_3^4:C_{12}:D_4$ ($C_2^2$), $C_3^3:C_{12}:D_{12}$ ($C_2^2$)

Order 5184: $\SOPlus(4,2)^2$ ($S_3$)

Order 3888: $C_3\wr C_4:D_6$ ($C_2^3$) x 8, $C_3:S_3^4$ ($C_2^3$) x 8, $C_3^2\wr C_2.D_{12}$ ($C_2^3$) x 8, $C_3^3:(S_3\times D_{12})$ ($C_2^3$) x 8, $C_3^4:(S_3\times D_4)$ ($C_2^3$) x 8, $(C_3\times S_3^2):S_3^2$ ($C_2^3$) x 8, $C_3\times S_3^2:S_3^2$ ($C_2^3$) x 8, $C_3^4:C_{12}:C_4$ ($C_2^3$) x 4, $C_3^4:(C_6\times D_4)$ ($C_2^3$) x 4, $C_3\wr C_4:D_6$ ($C_2^3$) x 4, $C_3^3:(C_4\times S_3^2)$ ($C_2^3$) x 4, $S_3\times C_3^4:D_4$ ($D_4$) x 4, $C_3\wr C_4:D_6$ ($C_2^3$) x 4, $C_3^3:(S_3\times D_{12})$ ($C_2^3$) x 4, $C_3:S_3^4$ ($C_2^3$) x 4, $C_3^3:C_4\times S_3^2$ ($C_2^3$) x 4, $C_3^2:C_{12}\times S_3^2$ ($C_2^3$) x 4, $C_3:S_3^4$ ($C_2^3$) x 4, $C_3\times S_3^4$ ($C_2^3$) x 4, $C_3\wr C_4:D_6$ ($D_4$) x 4, $C_3:S_3^4$ ($C_2^3$) x 4, $C_3\wr C_4:D_6$ ($C_2^3$) x 4, $C_3^3:S_3^2:C_4$ ($C_2^3$) x 4, $\GL(2,3).C_2^6$ ($C_2^3$) x 4, $C_3^3:C_6^2:C_4$ ($C_2^3$) x 4, $C_3^2\wr C_2.D_{12}$ ($C_2^3$) x 4, $(C_3^3\times S_3^2):C_4$ ($C_2^3$) x 4, $(C_3^2\times S_3^2):C_{12}$ ($C_2^3$) x 4, $C_3^5:(C_2\times D_4)$ ($C_2^3$) x 4, $C_3^3:C_{12}:C_{12}$ ($C_2^3$) x 2, $C_3^5:C_4^2$ ($C_2^3$) x 2, $C_3^2\wr C_2.D_{12}$ ($C_2^3$) x 2, $C_3^4:(C_6\times D_4)$ ($C_2^3$) x 2, $C_3\times S_3^3:C_6$ ($D_4$) x 2, $S_3\times C_3^3:D_{12}$ ($D_4$) x 2, $C_3^3:(S_3\times D_{12})$ ($C_2^3$) x 2, $C_3^4:(S_3\times D_4)$ ($D_4$) x 2, $C_3^4:(S_3\times D_4)$ ($D_4$) x 2, $C_3\wr C_4:D_6$ ($C_2^3$) x 2, $C_3^4:(C_2\times D_{12})$ ($C_2^3$) x 2, $C_3^4:(C_6\times D_4)$ ($C_2^3$) x 2, $C_3^5.C_4^2$ ($C_2^3$), $C_3^5:C_4^2$ ($C_2^3$), $C_3^4:(C_4\times D_6)$ ($C_2^3$)

Order 2592: $S_3^2:\SOPlus(4,2)$ ($D_6$) x 4, $S_3^4:C_2$ ($D_6$) x 4, $S_3^2\wr C_2$ ($D_6$) x 4, $C_3^4:(C_4\times D_4)$ ($D_6$) x 2, $C_3^3:C_{12}:D_4$ ($D_6$)

Order 1944: $C_3^2:S_3^3$ ($C_2^4$) x 4, $C_3^2:S_3^3$ ($C_2\times D_4$) x 4, $C_3^2:S_3^3$ ($C_2^4$) x 4, $C_3^2:S_3^3$ ($C_2\times D_4$) x 4, $C_3^2:S_3^3$ ($C_2\times D_4$) x 4, $C_3^2:S_3^3$ ($C_2^4$) x 4, $C_3^2:S_3^3$ ($C_2^4$) x 4, $C_3^2\times S_3^3$ ($C_2\times D_4$) x 4, $C_3^4:D_{12}$ ($C_2\times D_4$) x 4, $C_3^4:D_{12}$ ($C_2\times D_4$) x 4, $C_3^5:D_4$ ($C_2\times D_4$) x 4, $C_3^5:D_4$ ($C_2\times D_4$) x 4, $C_3^5:D_4$ ($C_2\times D_4$) x 4, $C_3^5:D_4$ ($C_2\times D_4$) x 4, $C_3^4:(C_4\times S_3)$ ($C_2^4$) x 4, $C_3^3:(S_3\times C_{12})$ ($C_2^4$) x 4, $C_3^4:D_{12}$ ($C_2\times D_4$) x 2, $C_3^4:D_{12}$ ($C_2\times D_4$) x 2, $C_3^4:D_{12}$ ($C_2\times D_4$) x 2, $C_3^5:D_4$ ($C_2\times D_4$) x 2, $C_3\wr C_4:S_3$ ($C_2\times D_4$) x 2, $C_3^4:(C_4\times S_3)$ ($C_2\times D_4$) x 2, $C_3\wr C_4:S_3$ ($C_2^4$) x 2, $C_3^4:C_4\times S_3$ ($C_2\times D_4$) x 2, $C_3^3:(S_3\times C_{12})$ ($C_2^4$) x 2, $S_3\times C_3^3:C_{12}$ ($C_2\times D_4$) x 2, $C_3^2:S_3^3$ ($C_2^4$), $C_3^5:(C_2\times C_4)$ ($C_2^4$), $C_3^4:(C_2\times C_{12})$ ($C_2^4$)

Order 1296: $S_3^2:S_3^2$ ($C_2\times D_6$) x 8, $S_3^4$ ($C_2\times D_6$) x 4, $C_3^3:(S_3\times D_4)$ ($C_2\times D_6$) x 4, $C_3^2:C_4\times S_3^2$ ($C_2\times D_6$) x 4, $(C_3^2\times S_3^2):C_4$ ($C_2\times D_6$) x 4, $C_3^2\wr C_2.D_4$ ($C_2\times D_6$) x 4, $C_3^4:(C_2\times D_4)$ ($C_2\times D_6$) x 2, $(C_3\times S_3^2):D_6$ ($C_2\times D_6$) x 2, $C_3^2\wr C_2.D_4$ ($C_2\times D_6$) x 2, $C_3^4:C_4^2$ ($C_2\times D_6$)

Order 972: $C_3^3:S_3^2$ ($C_2^2\times D_4$) x 4, $C_3^3:S_3^2$ ($C_2^2\times D_4$) x 4, $C_3^3:C_6^2$ ($C_2^2\times D_4$) x 4, $C_3^3\times S_3^2$ ($C_2^2\times D_4$) x 4, $C_3^3:S_3^2$ ($C_2^2\times D_4$) x 2, $C_3^3:C_6^2$ ($C_2^2\times D_4$) x 2, $C_3^5:C_4$ ($C_2^2\times D_4$) x 2, $C_3^4:C_{12}$ ($C_2^2\times D_4$) x 2, $C_3^4:C_{12}$ ($C_2^2\times D_4$) x 2, $C_3^4:C_{12}$ ($C_2^2\times D_4$) x 2, $C_3^3:S_3^2$ ($D_4:C_2^2$), $C_3^3:S_3^2$ ($D_4:C_2^2$), $C_3^3:S_3^2$ ($C_2^5$)

Order 648: $C_3:S_3^3$ ($C_2^2\times D_6$) x 4, $C_3:S_3^3$ ($C_2^2\times D_6$) x 4, $C_3^4:D_4$ ($S_3\times D_4$) x 4, $C_3^3:(C_4\times S_3)$ ($C_2^2\times D_6$) x 4, $C_3^3:D_{12}$ ($S_3\times D_4$) x 2, $C_3^4:D_4$ ($S_3\times D_4$) x 2, $C_3^3:(C_4\times S_3)$ ($C_2^2\times D_6$) x 2, $C_3^4:(C_2\times C_4)$ ($C_2^2\times D_6$)

Order 486: $C_3^4:C_6$ ($D_4^2$) x 2, $C_3^4:C_6$ ($D_4\times C_2^3$) x 2, $C_3^4:S_3$ ($D_4^2$), $C_3^4:C_6$ ($D_4:C_2^3$), $S_3\times C_3^4$ ($D_4^2$)

Order 432: $S_3^3:C_2$ ($\SOPlus(4,2)$) x 2

Order 324: $C_3^2:S_3^2$ ($D_4\times D_6$) x 4, $C_3^2\times S_3^2$ ($D_4\times D_6$) x 4, $C_3^4:C_4$ ($D_4\times D_6$) x 2, $C_3^3:C_{12}$ ($D_4\times D_6$) x 2, $C_3^2:S_3^2$ ($C_2^3\times D_6$)

Order 243: $C_3^5$ ($C_2\times D_4^2$)

Order 216: $S_3^3$ ($S_3^2:C_2^2$) x 4, $S_3^2:S_3$ ($S_3^2:C_2^2$) x 4, $C_3^2:D_{12}$ ($S_3^2:C_2^2$) x 2, $S_3^2:C_6$ ($S_3^2:C_2^2$) x 2, $C_3^2:C_4\times S_3$ ($S_3^2:C_2^2$) x 2

Order 162: $C_3^2\wr C_2$ ($C_{12}:C_2^4$) x 2, $C_3^3:S_3$ ($D_{12}:C_2^3$)

Order 108: $C_3:S_3^2$ ($C_6^2:D_4$) x 4, $C_3\times S_3^2$ ($C_6^2:D_4$) x 4, $C_3:S_3^2$ ($C_6^2:D_4$) x 2, $C_3^3:C_4$ ($C_6^2:D_4$) x 2, $C_3^2:C_{12}$ ($C_6^2:D_4$) x 2

Order 81: $C_3^4$ ($S_3\times D_4^2$)

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

Order 54: $C_3^2:S_3$ ($C_3^2:D_4^2$) x 2, $C_3^2:C_6$ ($D_6^2:C_2^2$) x 2, $S_3\times C_3^2$ ($C_3^2:D_4^2$) x 2

Order 36: $S_3^2$ ($S_3^3:C_2^2$) x 4, $C_3^2:C_4$ ($S_3^3:C_2^2$) x 2

Order 27: $C_3^3$ ($D_6^2:C_2^3$) x 2

Order 18: $C_3:S_3$ ($D_6^2:D_6$) x 2

Order 9: $C_3^2$ ($D_4\times S_3^3:C_2$) x 2

Order 6: $S_3$ ($C_3^4.D_4^2$)

Order 3: $C_3$ ($S_3^4:C_2^3$)

Order 1: $C_1$ ($S_3^5:C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 31104: $S_3^5:C_2^2$ ($C_1$)

Order 15552: $C_3^4:C_4^2:D_6$ ($C_2$), $C_3^5:D_4^2$ ($C_2$), $C_3^5:D_4^2$ ($C_2$), $C_3^5:D_4^2$ ($C_2$), $S_3^5:C_2$ ($C_2$), $S_3^3:\SOPlus(4,2)$ ($C_2$), $C_3^4:C_4^2:D_6$ ($C_2$), $S_3^4:D_6$ ($C_2$), $C_3^5:D_4^2$ ($C_2$), $S_3^4:D_6$ ($C_2$), $S_3^5:C_2$ ($C_2$)

Order 7776: $S_3\times C_3^4:C_4^2$ ($C_2^2$), $C_3^4:C_{12}:D_4$ ($C_2^2$), $C_3^4:C_{12}:D_4$ ($C_2^2$), $(C_3^3\times S_3^2):D_4$ ($C_2^2$), $C_3^4:C_{12}:D_4$ ($C_2^2$), $C_3^4:(D_4\times C_{12})$ ($C_2^2$), $C_3^4:(C_4\times D_{12})$ ($C_2^2$), $C_3^5:(C_4\times D_4)$ ($C_2^2$), $C_3^4:C_{12}:D_4$ ($C_2^2$), $C_3^3:C_{12}:D_{12}$ ($C_2^2$), $C_3^4:(D_4\times D_6)$ ($C_2^2$), $C_3^5:(C_4\times D_4)$ ($C_2^2$), $C_3^4:(D_4\times D_6)$ ($C_2^2$), $C_3^2:D_{12}\times S_3^2$ ($C_2^2$), $C_3^3:C_{12}:D_{12}$ ($C_2^2$), $C_3^4:(D_4\times D_6)$ ($C_2^2$), $C_3^2:C_4\times S_3^3$ ($C_2^2$), $S_3^4:C_6$ ($C_2^2$), $C_3^3:C_6^2.D_4$ ($C_2^2$), $S_3^5$ ($C_2^2$), $C_3^5:(C_4\times D_4)$ ($C_2^2$), $(C_3\times S_3^3):D_6$ ($C_2^2$), $S_3^4:S_3$ ($C_2^2$), $C_3^5:(C_4\times D_4)$ ($C_2^2$), $S_3^2:S_3^3$ ($C_2^2$), $C_3^4:C_{12}:D_4$ ($C_2^2$), $(C_3^2\times S_3^2):D_{12}$ ($C_2^2$), $(C_3^3\times S_3^2):D_4$ ($C_2^2$), $C_3^2:S_3^3:C_4$ ($C_2^2$), $(C_3^2\times S_3^3):C_4$ ($C_2^2$), $C_3^4:(D_4\times D_6)$ ($C_2^2$), $S_3^2:S_3^3$ ($C_2^2$), $C_3:S_3^2\wr C_2$ ($C_2^2$), $C_3:S_3^2\wr C_2$ ($C_2^2$), $(C_3^2\times S_3^2):D_{12}$ ($C_2^2$), $S_3^4:S_3$ ($C_2^2$), $(C_3^3\times S_3^2):D_4$ ($C_2^2$), $S_3^4:C_6$ ($C_2^2$)

Order 5184: $\SOPlus(4,2)^2$ ($S_3$)

Order 3888: $C_3^3:C_{12}:C_{12}$ ($C_2^3$), $C_3^5:C_4^2$ ($C_2^3$), $C_3^5.C_4^2$ ($C_2^3$), $C_3^4:C_{12}:C_4$ ($C_2^3$), $C_3^2\wr C_2.D_{12}$ ($C_2^3$), $C_3^4:(C_6\times D_4)$ ($C_2^3$), $C_3^5:C_4^2$ ($C_2^3$), $C_3\wr C_4:D_6$ ($C_2^3$), $C_3^3:(C_4\times S_3^2)$ ($C_2^3$), $S_3\times C_3^4:D_4$ ($D_4$), $C_3^4:(C_6\times D_4)$ ($C_2^3$), $C_3\wr C_4:D_6$ ($C_2^3$), $C_3\times S_3^3:C_6$ ($D_4$), $S_3\times C_3^3:D_{12}$ ($D_4$), $C_3^3:(S_3\times D_{12})$ ($C_2^3$), $C_3^4:(S_3\times D_4)$ ($D_4$), $C_3^4:(S_3\times D_4)$ ($D_4$), $C_3\wr C_4:D_6$ ($C_2^3$), $C_3\wr C_4:D_6$ ($C_2^3$), $C_3^3:(S_3\times D_{12})$ ($C_2^3$), $C_3:S_3^4$ ($C_2^3$), $C_3^3:C_4\times S_3^2$ ($C_2^3$), $C_3:S_3^4$ ($C_2^3$), $C_3^2:C_{12}\times S_3^2$ ($C_2^3$), $C_3:S_3^4$ ($C_2^3$), $C_3\times S_3^4$ ($C_2^3$), $C_3\wr C_4:D_6$ ($D_4$), $C_3:S_3^4$ ($C_2^3$), $C_3\wr C_4:D_6$ ($C_2^3$), $C_3^4:(C_4\times D_6)$ ($C_2^3$), $C_3^3:S_3^2:C_4$ ($C_2^3$), $C_3^2\wr C_2.D_{12}$ ($C_2^3$), $\GL(2,3).C_2^6$ ($C_2^3$), $C_3^3:C_6^2:C_4$ ($C_2^3$), $C_3^2\wr C_2.D_{12}$ ($C_2^3$), $(C_3^3\times S_3^2):C_4$ ($C_2^3$), $(C_3^2\times S_3^2):C_{12}$ ($C_2^3$), $C_3^4:(C_2\times D_{12})$ ($C_2^3$), $C_3^5:(C_2\times D_4)$ ($C_2^3$), $C_3^4:(C_6\times D_4)$ ($C_2^3$), $C_3^3:(S_3\times D_{12})$ ($C_2^3$), $C_3^4:(S_3\times D_4)$ ($C_2^3$), $(C_3\times S_3^2):S_3^2$ ($C_2^3$), $C_3\times S_3^2:S_3^2$ ($C_2^3$)

Order 2592: $S_3^2:\SOPlus(4,2)$ ($D_6$), $C_3^4:(C_4\times D_4)$ ($D_6$), $S_3^4:C_2$ ($D_6$), $C_3^3:C_{12}:D_4$ ($D_6$), $S_3^2\wr C_2$ ($D_6$)

Order 1944: $C_3^2:S_3^3$ ($C_2^4$), $C_3^2:S_3^3$ ($C_2\times D_4$), $C_3^2:S_3^3$ ($C_2^4$), $C_3^2:S_3^3$ ($C_2\times D_4$), $C_3^2:S_3^3$ ($C_2^4$), $C_3^2:S_3^3$ ($C_2\times D_4$), $C_3^2:S_3^3$ ($C_2^4$), $C_3^2:S_3^3$ ($C_2^4$), $C_3^2\times S_3^3$ ($C_2\times D_4$), $C_3^4:D_{12}$ ($C_2\times D_4$), $C_3^4:D_{12}$ ($C_2\times D_4$), $C_3^4:D_{12}$ ($C_2\times D_4$), $C_3^5:D_4$ ($C_2\times D_4$), $C_3^5:D_4$ ($C_2\times D_4$), $C_3^4:D_{12}$ ($C_2\times D_4$), $C_3^5:D_4$ ($C_2\times D_4$), $C_3^4:D_{12}$ ($C_2\times D_4$), $C_3^5:D_4$ ($C_2\times D_4$), $C_3^5:D_4$ ($C_2\times D_4$), $C_3^5:(C_2\times C_4)$ ($C_2^4$), $C_3^4:(C_2\times C_{12})$ ($C_2^4$), $C_3\wr C_4:S_3$ ($C_2\times D_4$), $C_3^4:(C_4\times S_3)$ ($C_2\times D_4$), $C_3^4:(C_4\times S_3)$ ($C_2^4$), $C_3\wr C_4:S_3$ ($C_2^4$), $C_3^4:C_4\times S_3$ ($C_2\times D_4$), $C_3^3:(S_3\times C_{12})$ ($C_2^4$), $C_3^3:(S_3\times C_{12})$ ($C_2^4$), $S_3\times C_3^3:C_{12}$ ($C_2\times D_4$)

Order 1296: $S_3^4$ ($C_2\times D_6$), $C_3^4:(C_2\times D_4)$ ($C_2\times D_6$), $C_3^3:(S_3\times D_4)$ ($C_2\times D_6$), $S_3^2:S_3^2$ ($C_2\times D_6$), $(C_3\times S_3^2):D_6$ ($C_2\times D_6$), $C_3^2:C_4\times S_3^2$ ($C_2\times D_6$), $C_3^2\wr C_2.D_4$ ($C_2\times D_6$), $(C_3^2\times S_3^2):C_4$ ($C_2\times D_6$), $C_3^2\wr C_2.D_4$ ($C_2\times D_6$), $C_3^4:C_4^2$ ($C_2\times D_6$)

Order 972: $C_3^3:S_3^2$ ($D_4:C_2^2$), $C_3^3:S_3^2$ ($C_2^2\times D_4$), $C_3^3:S_3^2$ ($C_2^2\times D_4$), $C_3^3:S_3^2$ ($D_4:C_2^2$), $C_3^3:S_3^2$ ($C_2^2\times D_4$), $C_3^3:S_3^2$ ($C_2^5$), $C_3^3:C_6^2$ ($C_2^2\times D_4$), $C_3^3:C_6^2$ ($C_2^2\times D_4$), $C_3^3\times S_3^2$ ($C_2^2\times D_4$), $C_3^5:C_4$ ($C_2^2\times D_4$), $C_3^4:C_{12}$ ($C_2^2\times D_4$), $C_3^4:C_{12}$ ($C_2^2\times D_4$), $C_3^4:C_{12}$ ($C_2^2\times D_4$)

Order 648: $C_3:S_3^3$ ($C_2^2\times D_6$), $C_3:S_3^3$ ($C_2^2\times D_6$), $C_3^3:D_{12}$ ($S_3\times D_4$), $C_3^4:D_4$ ($S_3\times D_4$), $C_3^4:D_4$ ($S_3\times D_4$), $C_3^4:(C_2\times C_4)$ ($C_2^2\times D_6$), $C_3^3:(C_4\times S_3)$ ($C_2^2\times D_6$), $C_3^3:(C_4\times S_3)$ ($C_2^2\times D_6$)

Order 486: $C_3^4:S_3$ ($D_4^2$), $C_3^4:C_6$ ($D_4:C_2^3$), $C_3^4:C_6$ ($D_4^2$), $C_3^4:C_6$ ($D_4\times C_2^3$), $S_3\times C_3^4$ ($D_4^2$)

Order 432: $S_3^3:C_2$ ($\SOPlus(4,2)$)

Order 324: $C_3^2:S_3^2$ ($D_4\times D_6$), $C_3^2:S_3^2$ ($C_2^3\times D_6$), $C_3^2\times S_3^2$ ($D_4\times D_6$), $C_3^4:C_4$ ($D_4\times D_6$), $C_3^3:C_{12}$ ($D_4\times D_6$)

Order 243: $C_3^5$ ($C_2\times D_4^2$)

Order 216: $S_3^3$ ($S_3^2:C_2^2$), $C_3^2:D_{12}$ ($S_3^2:C_2^2$), $S_3^2:S_3$ ($S_3^2:C_2^2$), $S_3^2:C_6$ ($S_3^2:C_2^2$), $C_3^2:C_4\times S_3$ ($S_3^2:C_2^2$)

Order 162: $C_3^3:S_3$ ($D_{12}:C_2^3$), $C_3^2\wr C_2$ ($C_{12}:C_2^4$)

Order 108: $C_3:S_3^2$ ($C_6^2:D_4$), $C_3:S_3^2$ ($C_6^2:D_4$), $C_3\times S_3^2$ ($C_6^2:D_4$), $C_3^3:C_4$ ($C_6^2:D_4$), $C_3^2:C_{12}$ ($C_6^2:D_4$)

Order 81: $C_3^4$ ($S_3\times D_4^2$)

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

Order 54: $C_3^2:S_3$ ($C_3^2:D_4^2$), $C_3^2:C_6$ ($D_6^2:C_2^2$), $S_3\times C_3^2$ ($C_3^2:D_4^2$)

Order 36: $C_3^2:C_4$ ($S_3^3:C_2^2$), $S_3^2$ ($S_3^3:C_2^2$)

Order 27: $C_3^3$ ($D_6^2:C_2^3$)

Order 18: $C_3:S_3$ ($D_6^2:D_6$)

Order 9: $C_3^2$ ($D_4\times S_3^3:C_2$)

Order 6: $S_3$ ($C_3^4.D_4^2$)

Order 3: $C_3$ ($S_3^4:C_2^3$)

Order 1: $C_1$ ($S_3^5:C_2^2$)

Series

Derived series

$S_3^5:C_2^2$

$\rhd$

$C_3^3:S_3^2$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$S_3^5:C_2^2$

$\rhd$

$S_3^5:C_2$

$\rhd$

$S_3^4:C_6$

$\rhd$

$C_3^4:(C_6\times D_4)$

$\rhd$

$C_3^2:S_3^3$

$\rhd$

$C_3^3:S_3^2$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$S_3^5:C_2^2$

$\rhd$

$C_3^3:S_3^2$

$\rhd$

$C_3^5$

Upper central series

$C_1$

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

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $243 \times 243$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.