Abstract group 46656.ir: $C_3:S_3^4.D_6$

• Label: 46656.ir
• Order: \( 2^{6} \cdot 3^{6} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $15$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3:S_3^4.D_6$
Order:	\(46656\)\(\medspace = 2^{6} \cdot 3^{6} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^5.A_4.C_8.C_2^3$, of order \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \)
Composition factors:	$C_2$ x 6, $C_3$ x 6
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	1359	890	8496	12654	1296	18072	1296	2592	46656
Conjugacy classes	1	11	14	20	46	3	34	1	2	132
Divisions	1	11	14	12	46	3	21	1	1	110
Autjugacy classes	1	11	9	12	33	2	18	1	1	88

Dimension	1	2	3	4	6	8	12	16	24	32	48	64
Irr. complex chars.	16	8	16	8	16	12	28	4	8	4	10	2	132
Irr. rational chars.	8	8	8	10	12	8	22	4	13	5	10	2	110

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$146764800$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{12}=c^{6}=d^{6}=e^{6}=f^{3}=g^{3}=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(2,3)(6,8)(7,9)(12,13)(14,15), (1,2,3,5,8,6)(4,7,9)(10,11,13,15)(12,14), (1,2,4,7)(3,6,9)(5,8)(10,12,13)(11,14,15)\rangle$
Transitive group:	36T16101				more information
Direct product:	$(C_3^2:C_4)$ $\, \times\, $ $(S_3\wr S_3)$
Semidirect product:	$C_3^5$ $\,\rtimes\,$ $(C_2^4.D_6)$	$(C_3\wr C_4:D_6)$ $\,\rtimes\,$ $D_6$	$(C_3^2\times S_3\wr S_3)$ $\,\rtimes\,$ $C_4$	$C_3^2$ $\,\rtimes\,$ $(C_4\times S_3\wr S_3)$	all 42
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3:S_3^4)$ . $D_6$	$(C_3:S_3)$ . $(S_3^3:D_6)$	$(C_3^4:(D_6\times S_4))$ . $C_2$	$(C_3^4:(S_3\times S_4))$ . $C_2^2$	all 10

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

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 860936 subgroups in 4970 conjugacy classes, 56 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3:S_3^4.D_6$
Commutator:	$G' \simeq$ $C_3^5:A_4$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3:S_3^4.D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^5$	$G/\operatorname{Fit} \simeq$ $C_2^4.D_6$
Radical:	$R \simeq$ $C_3:S_3^4.D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^5$	$G/\operatorname{soc} \simeq$ $C_2^4.D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5:C_3$

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 46656: $C_3:S_3^4.D_6$

Order 23328: $C_3^4:(D_6\times S_4)$, $C_3^5:(C_4\times S_4)$, $C_3^5:(C_4\times S_4)$, $C_3^5:(C_4\times S_4)$, $C_3^5:(C_4\times S_4)$, $S_3^3:C_3^3:C_4$, $(C_3^2\times S_3^3):C_{12}$

Order 15552: $C_3^4:C_4^2:D_6$

Order 11664: $C_3^4:(S_3\times S_4)$, $C_3^5:(C_2\times S_4)$, $C_3^4:(A_4\times D_6)$, $C_3^2\times S_3\wr S_3$, $C_3^4:(S_3\times S_4)$, $C_3^4:(S_3\times S_4)$, $C_3^4:(S_3\times S_4)$, $C_3^5:A_4:C_4$, $C_3^5:A_4:C_4$, $C_3^5:(C_4\times A_4)$, $C_3^5:(C_4\times A_4)$, $C_3^5:(C_4\times D_6)$

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

Order 5832: $C_3^5:S_4$, $C_3^3.S_3^3$, $C_3^5:S_4$, $C_3^4:(S_3\times A_4)$, $S_3^3:C_3^3$, $C_3^5:S_4$, $C_3^5:S_4$, $C_3^4:(S_3\times A_4)$, $C_3^5:(C_4\times S_3)$, $C_3^5:(C_4\times S_3)$, $C_3^5:(C_4\times S_3)$, $C_3^5:(C_4\times S_3)$, $C_3^2\wr C_2.S_3^2$, $C_3^5:(C_2\times C_{12})$

Order 5184: $S_3^2:C_6^2:C_4$, $C_4\times S_3\wr S_3$

Order 3888: $(C_3\times S_3^3):S_3$ x 4, $C_3\wr C_4:D_6$ x 3, $C_3^3:C_4\times S_3^2$ x 3, $C_3^2:C_{12}\times S_3^2$ x 3, $C_3\wr C_4:D_6$ x 3, $C_3^5:C_4^2$ x 2, $C_3^4:C_{12}:C_4$ x 2, $C_3\wr C_4:D_6$ x 2, $C_3^3:(C_4\times S_3^2)$ x 2, $S_3\times C_3^4:D_4$ 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, $C_3^3:C_6^2:C_4$ x 2, $C_3^2\wr C_2.D_{12}$ x 2, $(C_3^3\times S_3^2):C_4$ x 2, $(C_3^2\times S_3^2):C_{12}$ x 2, $S_3\times C_3^3:S_4$ x 2, $S_3\times C_3^3:S_4$ x 2, $C_3^3:(S_3\times S_4)$ x 2, $S_3^4:C_3$ x 2, $C_3\times S_3\wr S_3$ x 2, $C_3^3:(S_3\times S_4)$ x 2, $C_3^3:C_{12}:C_{12}$, $C_3^5.C_4^2$, $C_3^2\wr C_2.D_{12}$, $C_3^5:C_4^2$, $C_3^4:(C_6\times D_4)$, $C_3\times S_3^3:C_6$, $S_3\times C_3^3:D_{12}$, $C_3^3:(S_3\times D_{12})$, $C_3^4:(S_3\times D_4)$, $C_3^4:(S_3\times D_4)$, $C_3^4:(C_4\times D_6)$, $C_3^4:(C_4\times D_6)$

Order 2916: $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^4:(C_6\times S_3)$, $C_3^5:D_6$, $C_3^5:A_4$, $(C_3^2\times \He_3):C_{12}$, $C_3^5:C_3:C_4$, $C_3^5:C_{12}$, $C_3^5:C_{12}$

Order 2592: $C_3^4:(C_4\times D_4)$ x 8, $C_3\wr C_4:C_2^3$ x 3, $S_3^4:C_2$ x 2, $C_3:S_3^3:C_4$ x 2, $C_2\times C_3^4:C_4^2$, $C_3^2:C_6^2.D_4$, $(C_6\times S_3^2):D_6$, $C_4\times C_3^3:S_4$, $C_4\times C_3^3:S_4$, $C_3^3:(C_4\times S_4)$, $C_3^3:(C_4\times S_4)$, $S_3\wr C_3:C_4$, $S_3^3:C_{12}$, $S_3^3:D_6$

Order 1944: $C_3\wr C_4\times S_3$ x 4, $S_3^3:C_3^2$ x 4, $C_3^4:S_4$ x 4, $C_3^4:S_4$ 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, $C_3\wr C_4:S_3$ x 3, $C_3^4:(C_4\times S_3)$ x 3, $C_3\wr C_4:S_3$ x 3, $C_3^4:C_4\times S_3$ x 3, $C_3^3:(S_3\times C_{12})$ x 3, $C_3^3:(S_3\times C_{12})$ x 3, $S_3\times C_3^3:C_{12}$ x 3, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^2\times S_3^3$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^5:D_4$ x 2, $C_3^5:D_4$ x 2, $C_3^5:D_4$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $S_3\times C_3^3:A_4$ x 2, $C_3^3:(S_3\times A_4)$ x 2, $C_3^4:S_4$ x 2, $C_3\wr S_4$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3^4:(C_2\times C_{12})$ x 2, $C_3^2.S_3^3$ x 2, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3^4:(C_4\times S_3)$, $C_3^5:(C_2\times C_4)$, $C_3^4:C_4\times S_3$, $C_3^4:(C_2\times C_{12})$, $C_3^2.S_3^3$, $C_3^4:(C_4\times S_3)$, $C_3^4:(C_4\times S_3)$, $(C_3^2\times D_9):C_{12}$

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

Order 1458: $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $C_3^2:C_4\times S_3^2$ x 24, $(C_3\times S_3^2):D_6$ x 12, $S_3^3:S_3$ x 8, $(C_3^2\times S_3^2):C_4$ x 8, $S_3\wr S_3$ x 6, $C_3^3:C_4\times D_6$ x 4, $C_3^2:C_{12}\times D_6$ x 4, $S_3^4$ x 4, $S_3\times C_3^2:D_{12}$ x 4, $S_3^2:S_3^2$ x 4, $S_3^3:C_6$ x 4, $C_3^2\wr C_2.D_4$ x 4, $C_3^4:C_4^2$ x 4, $C_6:S_3^3$ x 3, $C_3^3:(C_4\times D_6)$ x 3, $C_3^3:(C_4\times D_6)$ x 3, $C_2\times C_3^4:D_4$ x 2, $C_3^3:(S_3\times D_4)$ x 2, $C_4\times C_3^3:D_6$, $C_2\times C_3^3:D_{12}$, $S_3^2:C_6^2$, $C_3^4:(C_2^2\times C_4)$, $S_3^3:C_6$, $C_2\times C_3^3:S_4$, $C_2\times C_3^3:S_4$, $C_4\times C_3^3:A_4$, $C_3^3:(C_4\times A_4)$, $(C_3^2\times C_6).S_4$, $(C_3^2\times C_6).S_4$

Order 972: $C_3^3:S_3^2$ x 4, $C_3^3:C_6^2$ x 4, $C_3\wr A_4$ x 4, $C_3^3:S_3^2$ x 4, $C_3^4:D_6$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3\times S_3^2$ x 3, $C_3^4:C_{12}$ x 3, $C_3^4:C_{12}$ x 3, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^5:C_4$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $\He_3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^5:C_4$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^4:D_6$, $(C_3^2\times \He_3):C_4$, $(C_3^2\times C_9):C_{12}$, $C_3^4.C_{12}$, $C_3^4:C_{12}$

Order 864: $C_3^2:C_4\times S_4$ x 4, $C_6^2:C_4\times S_3$ x 3, $S_3^3:C_2^2$ x 3, $C_4\times S_3^3$ x 2, $C_4\times S_3^2:S_3$ x 2, $C_6^2:(C_4\times S_3)$ x 2, $C_6^2:(C_4\times S_3)$ x 2, $C_6^2:C_4\times S_3$ x 2, $S_3\times C_3^2:C_4^2$ x 2, $S_3^2:(C_4\times S_3)$ x 2, $S_3^3:C_4$ x 2, $C_6:S_3\times S_4$, $C_2\times C_6^2:C_{12}$, $C_6^2:C_6:C_4$, $C_6.D_6^2$, $C_3^3:C_4\times D_4$, $D_4\times C_3^2:C_{12}$, $C_4\times C_3^2:D_{12}$, $C_{12}\times \SOPlus(4,2)$, $(C_3^2\times D_{12}):C_4$, $C_3^2:C_4\times D_{12}$, $C_{12}:S_3.D_6$, $C_3^3:(C_4\times D_4)$, $C_3^3:(C_4\times D_4)$, $D_6.\SOPlus(4,2)$

Order 729: $C_3^5:C_3$

Order 648: $C_3:S_3^3$ x 38, $C_3^3:C_4\times S_3$ x 24, $S_3\times C_3^2:C_{12}$ x 24, $C_3^3:(C_4\times S_3)$ x 18, $C_3^4:D_4$ x 16, $C_3^3:(C_4\times S_3)$ x 14, $C_3:S_3^3$ x 10, $C_3\times S_3^3$ x 10, $C_3^3:D_{12}$ x 8, $C_3^3:D_{12}$ x 8, $C_3\wr D_4$ x 8, $C_3^4:D_4$ x 8, $C_3^4:C_2^3$ x 4, $C_3^4:C_2^3$ x 4, $C_2\times C_3\wr C_4$ x 4, $C_3:S_3^3$ x 4, $C_3^4:D_4$ x 4, $C_3^3:D_{12}$ x 4, $S_3\wr C_3$ x 4, $C_3^3:S_4$ x 4, $C_3^3:S_4$ x 4, $C_3^4:C_2^3$ x 3, $C_3^4:C_2^3$ x 3, $C_2\times C_3^4:C_4$ x 3, $C_6\times C_3^2:C_{12}$ x 3, $C_3^4:(C_2\times C_4)$ x 3, $C_3^4:C_2^3$ x 2, $C_3.S_3^3$ x 2, $C_2\times C_3^4:C_4$, $C_2\times C_3^3:A_4$, $\He_3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_3^2:C_4\times D_9$, $C_2\times C_3^3:D_6$, $C_4\times C_3^3:S_3$, $C_4\times C_3^3:C_6$, $C_4\times C_3\wr S_3$

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

Order 486: $C_3^4:S_3$ x 8, $C_3^4:C_6$ x 4, $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 4, $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 2, $S_3\times C_3^4$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:S_3$, $D_9:C_3^3$

Order 432: $C_3^2:C_4\times D_6$ x 34, $S_3^3:C_2$ x 26, $C_2\times S_3^3$ x 8, $S_3^2:D_6$ x 6, $C_6^2:D_6$ x 4, $(C_3^2\times D_6):C_4$ x 4, $C_3^3:C_4^2$ x 4, $C_3:D_6^2$ x 3, $C_2^2\times C_3^3:C_4$ x 3, $C_6^2:C_{12}$ x 3, $C_2\times C_3^2:D_{12}$ x 3, $C_6\times \SOPlus(4,2)$ x 3, $C_{12}:S_3^2$ x 3, $C_{12}\times S_3^2$ x 3, $C_2.S_3^3$ x 3, $C_2.S_3^3$ x 3, $C_6^2:C_{12}$ x 2, $(C_3^2\times A_4):C_4$ x 2, $C_{12}:S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $(C_3\times C_6^2):C_4$ x 2, $C_3^3:C_4^2$ x 2, $C_6^2:C_{12}$ x 2, $C_3^3:C_4^2$ x 2, $C_2.S_3^3$ x 2, $C_6.\SOPlus(4,2)$ x 2, $C_6.\SOPlus(4,2)$ x 2, $(C_3\times S_3^2):C_4$ x 2, $C_6.\SOPlus(4,2)$ x 2, $S_3^2:C_{12}$ x 2, $C_3^2:C_{12}:C_4$ x 2, $C_6:S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$, $C_6^2:D_6$, $C_6^2:D_6$, $C_{12}:C_6^2$, $C_{12}:S_3^2$, $C_{12}:S_3^2$, $C_{12}:S_3^2$, $(C_3^2\times C_{12}):C_4$, $(C_3\times C_{12}):C_{12}$, $C_6.\SOPlus(4,2)$, $C_6.\SOPlus(4,2)$, $C_{12}.S_3^2$

Order 324: $C_3^2:S_3^2$ x 44, $C_3^2:S_3^2$ x 36, $C_3^2:S_3^2$ x 28, $C_3^2\times S_3^2$ x 28, $C_3^2:S_3^2$ x 18, $C_3\wr C_4$ x 16, $C_3^4:C_4$ x 12, $C_3^3:C_{12}$ x 12, $C_3\wr C_2^2$ x 10, $C_3^3:D_6$ x 8, $C_3^2:S_3^2$ x 8, $C_3^3:D_6$ x 4, $C_3^2:C_6^2$ x 4, $D_6\times C_3^3$ x 4, $C_3^2:S_3^2$ x 4, $C_3^3:A_4$ x 3, $C_3^3:D_6$ x 2, $C_3^4:C_4$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2.S_3^2$ x 2, $C_3^3:D_6$ x 2, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^3:C_{12}$, $C_3\wr C_3:C_4$, $C_3^3:C_{12}$, $\He_3:C_{12}$, $C_3^2:D_{18}$, $(C_3^2\times C_9):C_4$, $C_3^2:C_{36}$

Order 288: $C_3^2:C_4\times D_4$ x 8, $C_4\times \SOPlus(4,2)$ x 8, $C_2.D_6^2$ x 3, $D_4\times S_3^2$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_2\times C_3^2:C_4^2$ x 2, $C_6^2.D_4$ x 2, $C_2\times C_6^2:C_4$ x 2, $D_6\times S_4$ x 2, $(C_6\times C_{12}):C_4$, $C_6^2.D_4$, $C_6^2:D_4$, $C_6^2:C_2^3$

Order 243: $C_3^4:C_3$ x 8, $C_3^5$, $C_9:C_3^3$, $C_3^2\times \He_3$

Order 216: $S_3^3$ x 60, $C_3^2:C_4\times S_3$ x 60, $C_6:S_3^2$ x 51, $S_3^2:S_3$ x 28, $C_2\times C_3^3:C_4$ x 20, $C_2\times C_3^2:C_{12}$ x 20, $C_6\times S_3^2$ x 17, $C_3^2:D_{12}$ x 14, $S_3^2:C_6$ x 14, $C_6:S_3^2$ x 8, $C_6.S_3^2$ x 4, $C_6^2:S_3$ x 3, $C_6^2:C_6$ x 3, $S_3\times C_6^2$ x 3, $C_3:S_3\times C_{12}$ x 3, $C_6.C_6^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_4\times C_3^2:C_6$ x 2, $C_6.S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_6^2:S_3$ x 2, $C_4\times C_3^2:S_3$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_3^3:D_4$ x 2, $C_4\times C_3^2:S_3$, $C_{36}:C_6$, $C_6.S_3^2$, $C_6^2:C_6$, $D_4\times C_3^3$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_3^2\times D_{12}$, $C_6.S_3^2$

Order 192: $C_2^4.D_6$, $C_4^2:D_6$

Order 162: $C_3^3:C_6$ x 28, $C_3^2\wr C_2$ x 28, $S_3\times C_3^3$ x 28, $C_3^3:S_3$ x 14, $C_3^3:S_3$ x 14, $C_3^3:C_6$ x 8, $C_3^3:C_6$ x 8, $C_3^3:S_3$ x 6, $C_3^3:C_6$ x 6, $C_3\wr S_3$ x 6, $C_3^3.S_3$ x 4, $D_9:C_3^2$ x 4, $C_3^3\times C_6$ x 2, $C_3^3:S_3$ x 2, $C_3^3.C_6$ x 2, $S_3\times \He_3$ x 2, $C_3^2:D_9$, $C_3^2:C_{18}$, $C_3^2\times D_9$, $C_3^3:C_6$

Order 144: $C_4\times S_3^2$ x 24, $C_6^2:C_4$ x 18, $S_3^2:C_2^2$ x 12, $C_{12}:D_6$ x 12, $D_6^2$ x 9, $S_3\times S_4$ x 8, $D_6:D_6$ x 8, $C_6^2:C_4$ x 8, $C_3^2:C_4^2$ x 8, $S_3^2:C_4$ x 8, $C_6:S_4$ x 4, $C_{12}:D_6$ x 4, $C_{12}\times D_6$ x 4, $D_6:D_6$ x 4, $D_6.D_6$ x 4, $S_3\times D_{12}$ x 4, $(C_3\times C_{12}):C_4$ x 4, $C_2.\SOPlus(4,2)$ x 4, $C_{12}:D_6$ x 3, $C_2^2.S_3^2$ x 3, $C_6^2:C_2^2$ x 2, $A_4\times D_6$ x 2, $C_6\times S_4$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $C_4:C_6^2$, $C_6:D_{12}$

Order 108: $C_3:S_3^2$ x 172, $C_3\times S_3^2$ x 88, $C_3^2:D_6$ x 37, $C_3^2\times D_6$ x 37, $C_3:S_3^2$ x 35, $C_3^2:D_6$ x 30, $C_3^3:C_4$ x 24, $C_3^2:C_{12}$ x 24, $C_3^2:D_6$ x 8, $C_3\times C_6^2$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2\times C_{12}$ x 2, $C_3^3:C_4$ x 2, $C_3^2:D_6$ x 2, $S_3\times D_9$ x 2, $C_9:C_{12}$, $C_3^2\times A_4$, $C_3^2:D_6$, $C_{18}:C_6$, $C_9:C_{12}$, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $C_4\times S_4$ x 4, $D_4\times D_6$ x 3, $C_{12}:C_2^3$ x 3, $C_2^3.D_6$ x 2, $D_6.D_4$ x 2, $C_{12}:D_4$ x 2, $D_6.D_4$, $C_4\times D_{12}$, $S_3\times C_4^2$, $C_2^2\times S_4$, $C_2^3:C_{12}$, $C_2^2.S_4$, $D_4\times C_{12}$, $C_2^3.D_6$, $D_{12}:C_4$

Order 81: $C_3\wr C_3$ x 13, $C_3^4$ x 12, $C_9:C_3^2$ x 6, $C_3\times \He_3$ x 6, $C_3^2\times C_9$

Order 72: $S_3\times D_6$ x 98, $C_2\times C_3^2:C_4$ x 37, $C_6:D_6$ x 29, $S_3\times C_{12}$ x 24, $C_6.D_6$ x 24, $C_{12}:S_3$ x 18, $C_6\times D_6$ x 16, $\SOPlus(4,2)$ x 16, $C_6^2:C_2$ x 16, $C_6.D_6$ x 14, $C_3:S_4$ x 8, $D_4\times C_3^2$ x 8, $C_3:D_{12}$ x 8, $C_6\wr C_2$ x 8, $C_3:D_{12}$ x 8, $C_6\times A_4$ x 4, $S_3\times A_4$ x 4, $C_3\times S_4$ x 4, $C_6:C_{12}$ x 4, $C_3\times D_{12}$ x 4, $D_6:S_3$ x 4, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3, $C_2\times C_6^2$ x 2, $C_4\times D_9$

Order 64: $C_4^2:C_2^2$

Order 54: $C_3^2:C_6$ x 100, $S_3\times C_3^2$ x 100, $C_3^2:S_3$ x 72, $C_3^2\times C_6$ x 23, $C_3^2:C_6$ x 12, $C_3^2:S_3$ x 8, $C_9:C_6$ x 6, $C_3:D_9$ x 4, $S_3\times C_9$ x 2, $C_3\times D_9$ x 2, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_4\times D_6$ x 33, $S_3\times D_4$ x 26, $C_2\times S_4$ x 8, $C_2^2\times D_6$ x 7, $C_6:D_4$ x 6, $D_6:C_4$ x 4, $C_6\times D_4$ x 3, $C_2^2\times C_{12}$ x 3, $C_6.C_2^3$ x 3, $C_2\times D_{12}$ x 3, $C_{12}:C_4$ x 3, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_2^2:C_{12}$ x 2, $C_6.D_4$ x 2, $C_6.D_4$ x 2, $C_2^2\times A_4$, $C_4:C_{12}$, $C_4\times C_{12}$, $C_{12}:C_4$

Order 36: $S_3^2$ x 164, $C_6:S_3$ x 115, $C_6\times S_3$ x 104, $C_6^2$ x 21, $C_3^2:C_4$ x 18, $C_3:C_{12}$ x 16, $C_3\times C_{12}$ x 12, $C_3^2:C_4$ x 12, $C_3\times A_4$ x 4, $D_{18}$, $C_{36}$, $C_9:C_4$

Order 32: $C_4\times D_4$ x 8, $C_2^3\times C_4$ x 2, $C_2^3:C_4$ x 2, $C_2^2\times D_4$, $C_2^2.D_4$, $C_2\times C_4^2$

Order 27: $C_3^3$ x 49, $C_9:C_3$ x 7, $\He_3$ x 7, $C_3\times C_9$ x 4

Order 24: $C_2\times D_6$ x 71, $C_4\times S_3$ x 56, $C_3:D_4$ x 28, $C_2\times C_{12}$ x 19, $C_6:C_4$ x 19, $D_{12}$ x 14, $C_3\times D_4$ x 14, $S_4$ x 8, $C_2^2\times C_6$ x 7, $C_2\times A_4$ x 5

Order 18: $C_3\times S_3$ x 120, $C_3:S_3$ x 118, $C_3\times C_6$ x 65, $D_9$ x 4, $C_{18}$

Order 16: $C_2^2\times C_4$ x 17, $C_2\times D_4$ x 12, $C_2^2:C_4$ x 8, $C_4:C_4$ x 4, $C_4^2$ x 4, $C_2^4$ x 2

Order 12: $D_6$ x 142, $C_2\times C_6$ x 38, $C_{12}$ x 21, $C_3:C_4$ x 21, $A_4$ x 3

Order 9: $C_3^2$ x 53, $C_9$ x 3

Order 8: $C_2\times C_4$ x 31, $C_2^3$ x 17, $D_4$ x 16

Order 6: $S_3$ x 60, $C_6$ x 46

Order 4: $C_2^2$ x 28, $C_4$ x 12

Order 3: $C_3$ x 14

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 46656: $C_3:S_3^4.D_6$

Order 23328: $C_3^4:(D_6\times S_4)$, $C_3^5:(C_4\times S_4)$, $C_3^5:(C_4\times S_4)$, $C_3^5:(C_4\times S_4)$, $C_3^5:(C_4\times S_4)$, $S_3^3:C_3^3:C_4$, $(C_3^2\times S_3^3):C_{12}$

Order 15552: $C_3^4:C_4^2:D_6$

Order 11664: $C_3^4:(S_3\times S_4)$, $C_3^5:(C_2\times S_4)$, $C_3^4:(A_4\times D_6)$, $C_3^2\times S_3\wr S_3$, $C_3^4:(S_3\times S_4)$, $C_3^4:(S_3\times S_4)$, $C_3^4:(S_3\times S_4)$, $C_3^5:A_4:C_4$, $C_3^5:A_4:C_4$, $C_3^5:(C_4\times A_4)$, $C_3^5:(C_4\times A_4)$, $C_3^5:(C_4\times D_6)$

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

Order 5832: $C_3^5:S_4$, $C_3^3.S_3^3$, $C_3^5:S_4$, $C_3^4:(S_3\times A_4)$, $S_3^3:C_3^3$, $C_3^5:S_4$, $C_3^5:S_4$, $C_3^4:(S_3\times A_4)$, $C_3^5:(C_4\times S_3)$, $C_3^5:(C_4\times S_3)$, $C_3^5:(C_4\times S_3)$, $C_3^5:(C_4\times S_3)$, $C_3^2\wr C_2.S_3^2$, $C_3^5:(C_2\times C_{12})$

Order 5184: $S_3^2:C_6^2:C_4$, $C_4\times S_3\wr S_3$

Order 3888: $C_3\wr C_4:D_6$ x 3, $C_3^3:C_4\times S_3^2$ x 3, $C_3^2:C_{12}\times S_3^2$ x 3, $C_3\wr C_4:D_6$ x 3, $C_3^5:C_4^2$ x 2, $C_3^4:C_{12}:C_4$ x 2, $C_3\wr C_4:D_6$ x 2, $C_3^3:(C_4\times S_3^2)$ x 2, $S_3\times C_3^4:D_4$ 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, $C_3^3:C_6^2:C_4$ x 2, $C_3^2\wr C_2.D_{12}$ x 2, $(C_3^3\times S_3^2):C_4$ x 2, $(C_3^2\times S_3^2):C_{12}$ x 2, $(C_3\times S_3^3):S_3$ x 2, $C_3^3:C_{12}:C_{12}$, $C_3^5.C_4^2$, $C_3^2\wr C_2.D_{12}$, $C_3^5:C_4^2$, $C_3^4:(C_6\times D_4)$, $C_3\times S_3^3:C_6$, $S_3\times C_3^3:D_{12}$, $C_3^3:(S_3\times D_{12})$, $C_3^4:(S_3\times D_4)$, $C_3^4:(S_3\times D_4)$, $C_3^4:(C_4\times D_6)$, $C_3^4:(C_4\times D_6)$, $S_3\times C_3^3:S_4$, $S_3\times C_3^3:S_4$, $C_3^3:(S_3\times S_4)$, $S_3^4:C_3$, $C_3\times S_3\wr S_3$, $C_3^3:(S_3\times S_4)$

Order 2916: $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^4:(C_6\times S_3)$, $C_3^5:D_6$, $C_3^5:A_4$, $(C_3^2\times \He_3):C_{12}$, $C_3^5:C_3:C_4$, $C_3^5:C_{12}$, $C_3^5:C_{12}$

Order 2592: $C_3^4:(C_4\times D_4)$ x 8, $C_3\wr C_4:C_2^3$ x 3, $C_3:S_3^3:C_4$ x 2, $C_2\times C_3^4:C_4^2$, $C_3^2:C_6^2.D_4$, $(C_6\times S_3^2):D_6$, $C_4\times C_3^3:S_4$, $C_4\times C_3^3:S_4$, $C_3^3:(C_4\times S_4)$, $C_3^3:(C_4\times S_4)$, $S_3\wr C_3:C_4$, $S_3^3:C_{12}$, $S_3^4:C_2$, $S_3^3:D_6$

Order 1944: $C_3\wr C_4\times S_3$ 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, $C_3\wr C_4:S_3$ x 3, $C_3^4:(C_4\times S_3)$ x 3, $C_3\wr C_4:S_3$ x 3, $C_3^4:C_4\times S_3$ x 3, $C_3^3:(S_3\times C_{12})$ x 3, $C_3^3:(S_3\times C_{12})$ x 3, $S_3\times C_3^3:C_{12}$ x 3, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^2\times S_3^3$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^5:D_4$ x 2, $C_3^5:D_4$ x 2, $C_3^5:D_4$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $S_3^3:C_3^2$ x 2, $C_3^4:S_4$ x 2, $C_3^4:S_4$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3^4:(C_2\times C_{12})$ x 2, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3^4:(C_4\times S_3)$, $C_3^5:(C_2\times C_4)$, $C_3^4:C_4\times S_3$, $C_3^4:(C_2\times C_{12})$, $S_3\times C_3^3:A_4$, $C_3^3:(S_3\times A_4)$, $C_3^4:S_4$, $C_3\wr S_4$, $C_3^2.S_3^3$, $C_3^4:(C_4\times S_3)$, $C_3^4:(C_4\times S_3)$, $(C_3^2\times D_9):C_{12}$, $C_3^2.S_3^3$

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

Order 1458: $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $C_3^2:C_4\times S_3^2$ x 23, $(C_3\times S_3^2):D_6$ x 10, $(C_3^2\times S_3^2):C_4$ x 8, $S_3\wr S_3$ x 5, $C_3^3:C_4\times D_6$ x 4, $C_3^2:C_{12}\times D_6$ x 4, $S_3^3:S_3$ x 4, $C_3^2\wr C_2.D_4$ x 4, $C_3^4:C_4^2$ x 4, $C_6:S_3^3$ x 3, $C_3^3:(C_4\times D_6)$ x 3, $C_3^3:(C_4\times D_6)$ x 3, $C_2\times C_3^4:D_4$ x 2, $S_3^4$ x 2, $S_3\times C_3^2:D_{12}$ x 2, $S_3^2:S_3^2$ x 2, $S_3^3:C_6$ x 2, $C_4\times C_3^3:D_6$, $C_2\times C_3^3:D_{12}$, $S_3^2:C_6^2$, $C_3^4:(C_2^2\times C_4)$, $C_3^3:(S_3\times D_4)$, $S_3^3:C_6$, $C_2\times C_3^3:S_4$, $C_2\times C_3^3:S_4$, $C_4\times C_3^3:A_4$, $C_3^3:(C_4\times A_4)$, $(C_3^2\times C_6).S_4$, $(C_3^2\times C_6).S_4$

Order 972: $C_3^3:S_3^2$ x 4, $C_3^3:C_6^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3\times S_3^2$ x 3, $C_3^4:C_{12}$ x 3, $C_3^4:C_{12}$ x 3, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^5:C_4$ x 2, $C_3^4:C_{12}$ x 2, $C_3\wr A_4$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^5:C_4$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^4:D_6$, $(C_3^2\times \He_3):C_4$, $(C_3^2\times C_9):C_{12}$, $C_3^4.C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$

Order 864: $C_3^2:C_4\times S_4$ x 4, $C_6^2:C_4\times S_3$ x 3, $S_3^3:C_2^2$ x 2, $C_4\times S_3^3$ x 2, $C_4\times S_3^2:S_3$ x 2, $C_6^2:(C_4\times S_3)$ x 2, $C_6^2:(C_4\times S_3)$ x 2, $C_6^2:C_4\times S_3$ x 2, $S_3\times C_3^2:C_4^2$ x 2, $S_3^2:(C_4\times S_3)$ x 2, $S_3^3:C_4$ x 2, $C_6:S_3\times S_4$, $C_2\times C_6^2:C_{12}$, $C_6^2:C_6:C_4$, $C_6.D_6^2$, $C_3^3:C_4\times D_4$, $D_4\times C_3^2:C_{12}$, $C_4\times C_3^2:D_{12}$, $C_{12}\times \SOPlus(4,2)$, $(C_3^2\times D_{12}):C_4$, $C_3^2:C_4\times D_{12}$, $C_{12}:S_3.D_6$, $C_3^3:(C_4\times D_4)$, $C_3^3:(C_4\times D_4)$, $D_6.\SOPlus(4,2)$

Order 729: $C_3^5:C_3$

Order 648: $C_3:S_3^3$ x 30, $C_3^3:C_4\times S_3$ x 22, $S_3\times C_3^2:C_{12}$ x 22, $C_3^3:(C_4\times S_3)$ x 16, $C_3^3:(C_4\times S_3)$ x 13, $C_3^4:D_4$ x 12, $C_3^3:D_{12}$ x 6, $C_3^4:D_4$ x 6, $C_3:S_3^3$ x 5, $C_3\times S_3^3$ x 5, $C_3^4:C_2^3$ x 4, $C_3^4:C_2^3$ x 4, $C_2\times C_3\wr C_4$ x 4, $C_3^3:D_{12}$ x 4, $C_3\wr D_4$ x 4, $C_3^4:C_2^3$ x 3, $C_3^4:C_2^3$ x 3, $C_2\times C_3^4:C_4$ x 3, $C_6\times C_3^2:C_{12}$ x 3, $C_3^4:(C_2\times C_4)$ x 3, $S_3\wr C_3$ x 3, $C_3^3:S_4$ x 3, $C_3^3:S_4$ x 3, $C_3^4:C_2^3$ x 2, $C_3:S_3^3$ x 2, $C_3^4:D_4$ x 2, $C_3^3:D_{12}$ x 2, $C_2\times C_3^4:C_4$, $C_2\times C_3^3:A_4$, $\He_3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_3.S_3^3$, $C_3^2:C_4\times D_9$, $C_2\times C_3^3:D_6$, $C_4\times C_3^3:S_3$, $C_4\times C_3^3:C_6$, $C_4\times C_3\wr S_3$

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

Order 486: $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 2, $S_3\times C_3^4$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:S_3$, $D_9:C_3^3$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $C_3^2:C_4\times D_6$ x 33, $S_3^3:C_2$ x 17, $C_2\times S_3^3$ x 5, $S_3^2:D_6$ x 4, $C_6^2:D_6$ x 4, $(C_3^2\times D_6):C_4$ x 4, $C_3^3:C_4^2$ x 4, $C_3:D_6^2$ x 3, $C_2^2\times C_3^3:C_4$ x 3, $C_6^2:C_{12}$ x 3, $C_{12}:S_3^2$ x 3, $C_{12}\times S_3^2$ x 3, $C_2.S_3^3$ x 3, $C_2.S_3^3$ x 3, $C_2\times C_3^2:D_{12}$ x 2, $C_6\times \SOPlus(4,2)$ x 2, $C_6^2:C_{12}$ x 2, $(C_3^2\times A_4):C_4$ x 2, $C_{12}:S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $(C_3\times C_6^2):C_4$ x 2, $C_3^3:C_4^2$ x 2, $C_6^2:C_{12}$ x 2, $C_3^3:C_4^2$ x 2, $C_2.S_3^3$ x 2, $C_6.\SOPlus(4,2)$ x 2, $C_6.\SOPlus(4,2)$ x 2, $(C_3\times S_3^2):C_4$ x 2, $C_6.\SOPlus(4,2)$ x 2, $S_3^2:C_{12}$ x 2, $C_3^2:C_{12}:C_4$ x 2, $C_6:S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$, $C_6^2:D_6$, $C_6^2:D_6$, $C_{12}:C_6^2$, $C_{12}:S_3^2$, $C_{12}:S_3^2$, $C_{12}:S_3^2$, $(C_3^2\times C_{12}):C_4$, $(C_3\times C_{12}):C_{12}$, $C_6.\SOPlus(4,2)$, $C_6.\SOPlus(4,2)$, $C_{12}.S_3^2$

Order 324: $C_3^2:S_3^2$ x 32, $C_3^2:S_3^2$ x 28, $C_3^2:S_3^2$ x 20, $C_3^2\times S_3^2$ x 20, $C_3^2:S_3^2$ x 14, $C_3\wr C_4$ x 14, $C_3^4:C_4$ x 10, $C_3^3:C_{12}$ x 10, $C_3^3:D_6$ x 6, $C_3\wr C_2^2$ x 5, $C_3^3:D_6$ x 4, $C_3^2:C_6^2$ x 4, $D_6\times C_3^3$ x 4, $C_3^2:S_3^2$ x 4, $C_3^3:D_6$ x 2, $C_3^4:C_4$ x 2, $C_3^3:A_4$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^3:C_{12}$, $C_3\wr C_3:C_4$, $C_3^3:C_{12}$, $\He_3:C_{12}$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:D_{18}$, $C_3^2.S_3^2$, $C_3^3:D_6$, $(C_3^2\times C_9):C_4$, $C_3^2:C_{36}$

Order 288: $C_3^2:C_4\times D_4$ x 8, $C_4\times \SOPlus(4,2)$ x 8, $C_2.D_6^2$ x 3, $C_2\times C_6^2:C_4$ x 2, $C_2\times C_3^2:C_4^2$ x 2, $C_6^2.D_4$ x 2, $C_2\times C_6^2:C_4$ x 2, $D_4\times S_3^2$, $(C_6\times C_{12}):C_4$, $C_6^2.D_4$, $C_6^2:D_4$, $D_6\times S_4$, $C_6^2:C_2^3$

Order 243: $C_3^4:C_3$ x 4, $C_3^5$, $C_9:C_3^3$, $C_3^2\times \He_3$

Order 216: $C_3^2:C_4\times S_3$ x 54, $C_6:S_3^2$ x 41, $S_3^3$ x 35, $C_2\times C_3^3:C_4$ x 19, $C_2\times C_3^2:C_{12}$ x 19, $S_3^2:S_3$ x 18, $C_6\times S_3^2$ x 10, $C_3^2:D_{12}$ x 9, $S_3^2:C_6$ x 9, $C_6:S_3^2$ x 5, $C_6.S_3^2$ x 4, $C_6^2:S_3$ x 3, $C_6^2:C_6$ x 3, $S_3\times C_6^2$ x 3, $C_3:S_3\times C_{12}$ x 3, $C_6.C_6^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_4\times C_3^2:C_6$ x 2, $C_6.S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_6^2:S_3$ x 2, $C_4\times C_3^2:S_3$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_3^3:D_4$ x 2, $C_4\times C_3^2:S_3$, $C_{36}:C_6$, $C_6.S_3^2$, $C_6^2:C_6$, $D_4\times C_3^3$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_3^2\times D_{12}$, $C_6.S_3^2$

Order 192: $C_2^4.D_6$, $C_4^2:D_6$

Order 162: $C_3^3:C_6$ x 20, $C_3^2\wr C_2$ x 20, $S_3\times C_3^3$ x 20, $C_3^3:S_3$ x 10, $C_3^3:S_3$ x 7, $C_3^3:C_6$ x 4, $C_3^3:C_6$ x 4, $C_3^3:C_6$ x 4, $C_3\wr S_3$ x 4, $C_3^3:S_3$ x 3, $C_3^3\times C_6$ x 2, $C_3^3.S_3$ x 2, $D_9:C_3^2$ x 2, $C_3^2:D_9$, $C_3^3:S_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $S_3\times \He_3$, $C_3^2\times D_9$, $C_3^3:C_6$

Order 144: $C_4\times S_3^2$ x 23, $C_6^2:C_4$ x 18, $S_3^2:C_2^2$ x 12, $C_{12}:D_6$ x 10, $C_6^2:C_4$ x 8, $C_3^2:C_4^2$ x 8, $S_3^2:C_4$ x 8, $D_6^2$ x 6, $S_3\times S_4$ x 4, $C_{12}\times D_6$ x 4, $D_6:D_6$ x 4, $D_6.D_6$ x 4, $(C_3\times C_{12}):C_4$ x 4, $C_2.\SOPlus(4,2)$ x 4, $C_{12}:D_6$ x 3, $C_2^2.S_3^2$ x 3, $C_6^2:C_2^2$ x 2, $C_6:S_4$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $D_6:D_6$ x 2, $S_3\times D_{12}$ x 2, $A_4\times D_6$, $C_6\times S_4$, $C_4:C_6^2$, $C_6:D_{12}$, $C_{12}:D_6$

Order 108: $C_3:S_3^2$ x 109, $C_3\times S_3^2$ x 49, $C_3^2:D_6$ x 27, $C_3^2\times D_6$ x 27, $C_3^2:D_6$ x 23, $C_3:S_3^2$ x 20, $C_3^3:C_4$ x 20, $C_3^2:C_{12}$ x 20, $C_3^2:D_6$ x 6, $C_3\times C_6^2$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2\times C_{12}$ x 2, $C_3^3:C_4$ x 2, $C_3^2:D_6$ x 2, $C_9:C_{12}$, $C_3^2\times A_4$, $C_3^2:D_6$, $C_{18}:C_6$, $S_3\times D_9$, $C_9:C_{12}$, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $C_4\times S_4$ x 4, $C_{12}:C_2^3$ x 3, $C_2^3.D_6$ x 2, $D_6.D_4$ x 2, $D_4\times D_6$ x 2, $C_{12}:D_4$ x 2, $D_6.D_4$, $C_4\times D_{12}$, $S_3\times C_4^2$, $C_2^2\times S_4$, $C_2^3:C_{12}$, $C_2^2.S_4$, $D_4\times C_{12}$, $C_2^3.D_6$, $D_{12}:C_4$

Order 81: $C_3^4$ x 8, $C_3\wr C_3$ x 6, $C_9:C_3^2$ x 3, $C_3\times \He_3$ x 3, $C_3^2\times C_9$

Order 72: $S_3\times D_6$ x 65, $C_2\times C_3^2:C_4$ x 36, $C_6:D_6$ x 23, $S_3\times C_{12}$ x 22, $C_6.D_6$ x 22, $\SOPlus(4,2)$ x 16, $C_{12}:S_3$ x 16, $C_6.D_6$ x 13, $C_6^2:C_2$ x 12, $C_6\times D_6$ x 10, $D_4\times C_3^2$ x 6, $C_3:D_{12}$ x 6, $C_3:S_4$ x 4, $C_6\wr C_2$ x 4, $C_6:C_{12}$ x 4, $C_3:D_{12}$ x 4, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3, $C_2\times C_6^2$ x 2, $C_6\times A_4$ x 2, $S_3\times A_4$ x 2, $C_3\times S_4$ x 2, $C_3\times D_{12}$ x 2, $D_6:S_3$ x 2, $C_4\times D_9$

Order 64: $C_4^2:C_2^2$

Order 54: $C_3^2:C_6$ x 57, $S_3\times C_3^2$ x 57, $C_3^2:S_3$ x 42, $C_3^2\times C_6$ x 16, $C_3^2:C_6$ x 8, $C_3^2:S_3$ x 5, $C_9:C_6$ x 4, $C_3:D_9$ x 2, $S_3\times C_9$, $C_3\times D_9$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_4\times D_6$ x 32, $S_3\times D_4$ x 17, $C_2\times S_4$ x 7, $C_2^2\times D_6$ x 5, $C_6:D_4$ x 4, $D_6:C_4$ x 4, $C_2^2\times C_{12}$ x 3, $C_6.C_2^3$ x 3, $C_{12}:C_4$ x 3, $C_6\times D_4$ x 2, $C_2\times D_{12}$ x 2, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_2^2:C_{12}$ x 2, $C_6.D_4$ x 2, $C_6.D_4$ x 2, $C_2^2\times A_4$, $C_4:C_{12}$, $C_4\times C_{12}$, $C_{12}:C_4$

Order 36: $S_3^2$ x 104, $C_6:S_3$ x 74, $C_6\times S_3$ x 66, $C_3^2:C_4$ x 16, $C_6^2$ x 15, $C_3:C_{12}$ x 14, $C_3\times C_{12}$ x 10, $C_3^2:C_4$ x 10, $C_3\times A_4$ x 2, $D_{18}$, $C_{36}$, $C_9:C_4$

Order 32: $C_4\times D_4$ x 8, $C_2^3\times C_4$ x 2, $C_2^3:C_4$ x 2, $C_2^2\times D_4$, $C_2^2.D_4$, $C_2\times C_4^2$

Order 27: $C_3^3$ x 26, $C_9:C_3$ x 4, $\He_3$ x 4, $C_3\times C_9$ x 2

Order 24: $C_2\times D_6$ x 53, $C_4\times S_3$ x 51, $C_2\times C_{12}$ x 18, $C_3:D_4$ x 18, $C_6:C_4$ x 18, $D_{12}$ x 9, $C_3\times D_4$ x 9, $S_4$ x 6, $C_2^2\times C_6$ x 5, $C_2\times A_4$ x 4

Order 18: $C_3\times S_3$ x 72, $C_3:S_3$ x 66, $C_3\times C_6$ x 38, $D_9$ x 3, $C_{18}$

Order 16: $C_2^2\times C_4$ x 17, $C_2\times D_4$ x 12, $C_2^2:C_4$ x 8, $C_4:C_4$ x 4, $C_4^2$ x 4, $C_2^4$ x 2

Order 12: $D_6$ x 106, $C_2\times C_6$ x 28, $C_{12}$ x 18, $C_3:C_4$ x 18, $A_4$ x 2

Order 9: $C_3^2$ x 28, $C_9$ x 2

Order 8: $C_2\times C_4$ x 31, $C_2^3$ x 17, $D_4$ x 16

Order 6: $S_3$ x 42, $C_6$ x 33

Order 4: $C_2^2$ x 28, $C_4$ x 12

Order 3: $C_3$ x 9

Order 2: $C_2$ x 11

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 46656: $C_3:S_3^4.D_6$ ($C_1$)

Order 23328: $C_3^4:(D_6\times S_4)$ ($C_2$), $C_3^5:(C_4\times S_4)$ ($C_2$), $C_3^5:(C_4\times S_4)$ ($C_2$), $C_3^5:(C_4\times S_4)$ ($C_2$), $C_3^5:(C_4\times S_4)$ ($C_2$), $S_3^3:C_3^3:C_4$ ($C_2$), $(C_3^2\times S_3^3):C_{12}$ ($C_2$)

Order 11664: $C_3^4:(S_3\times S_4)$ ($C_4$), $C_3^5:(C_2\times S_4)$ ($C_4$), $C_3^4:(A_4\times D_6)$ ($C_2^2$), $C_3^2\times S_3\wr S_3$ ($C_4$), $C_3^4:(S_3\times S_4)$ ($C_2^2$), $C_3^4:(S_3\times S_4)$ ($C_2^2$), $C_3^4:(S_3\times S_4)$ ($C_4$), $C_3^5:A_4:C_4$ ($C_2^2$), $C_3^5:A_4:C_4$ ($C_2^2$), $C_3^5:(C_4\times A_4)$ ($C_2^2$), $C_3^5:(C_4\times A_4)$ ($C_2^2$)

Order 7776: $C_3^2:C_4\times S_3^3$ ($S_3$)

Order 5832: $C_3^5:S_4$ ($C_2\times C_4$), $C_3^5:S_4$ ($C_2\times C_4$), $C_3^4:(S_3\times A_4)$ ($C_2^3$), $S_3^3:C_3^3$ ($C_2\times C_4$), $C_3^5:S_4$ ($C_2\times C_4$), $C_3^5:S_4$ ($C_2\times C_4$), $C_3^4:(S_3\times A_4)$ ($C_2\times C_4$)

Order 3888: $C_3\wr C_4:D_6$ ($D_6$), $C_3^3:(C_4\times S_3^2)$ ($D_6$), $C_3:S_3^4$ ($D_6$)

Order 2916: $C_3^5:A_4$ ($C_2^2\times C_4$)

Order 1944: $C_3^2:S_3^3$ ($C_4\times S_3$), $C_3^2:S_3^3$ ($C_2\times D_6$), $C_3^2\times S_3^3$ ($C_4\times S_3$), $C_3^4:(C_4\times S_3)$ ($S_4$)

Order 1296: $S_3\wr S_3$ ($C_3^2:C_4$)

Order 972: $C_3^3:S_3^2$ ($C_2\times S_4$), $C_3^3:C_6^2$ ($C_4\times D_6$), $C_3^5:C_4$ ($C_2\times S_4$), $C_3^4:C_{12}$ ($C_2\times S_4$)

Order 648: $S_3\wr C_3$ ($C_2\times C_3^2:C_4$), $C_3^3:S_4$ ($C_2\times C_3^2:C_4$), $C_3^3:S_4$ ($C_2\times C_3^2:C_4$)

Order 486: $C_3^4:S_3$ ($C_4\times S_4$), $C_3^4:C_6$ ($C_4\times S_4$), $C_3^4:C_6$ ($C_2^2\times S_4$)

Order 324: $C_3^3:A_4$ ($C_6^2:C_4$)

Order 243: $C_3^5$ ($C_2^4.D_6$)

Order 216: $S_3^3$ ($C_3^2:C_4\times S_3$)

Order 108: $C_3:S_3^2$ ($C_3^2:C_4\times D_6$)

Order 54: $C_3^2:S_3$ ($C_3^2:C_4\times S_4$)

Order 36: $C_3^2:C_4$ ($S_3\wr S_3$)

Order 27: $C_3^3$ ($C_6^2:(C_4\times D_6)$)

Order 18: $C_3:S_3$ ($S_3^3:D_6$)

Order 9: $C_3^2$ ($C_4\times S_3\wr S_3$)

Order 1: $C_1$ ($C_3:S_3^4.D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 46656: $C_3:S_3^4.D_6$ ($C_1$)

Order 23328: $C_3^4:(D_6\times S_4)$ ($C_2$), $C_3^5:(C_4\times S_4)$ ($C_2$), $C_3^5:(C_4\times S_4)$ ($C_2$), $C_3^5:(C_4\times S_4)$ ($C_2$), $C_3^5:(C_4\times S_4)$ ($C_2$), $S_3^3:C_3^3:C_4$ ($C_2$), $(C_3^2\times S_3^3):C_{12}$ ($C_2$)

Order 11664: $C_3^4:(S_3\times S_4)$ ($C_4$), $C_3^5:(C_2\times S_4)$ ($C_4$), $C_3^4:(A_4\times D_6)$ ($C_2^2$), $C_3^2\times S_3\wr S_3$ ($C_4$), $C_3^4:(S_3\times S_4)$ ($C_2^2$), $C_3^4:(S_3\times S_4)$ ($C_2^2$), $C_3^4:(S_3\times S_4)$ ($C_4$), $C_3^5:A_4:C_4$ ($C_2^2$), $C_3^5:A_4:C_4$ ($C_2^2$), $C_3^5:(C_4\times A_4)$ ($C_2^2$), $C_3^5:(C_4\times A_4)$ ($C_2^2$)

Order 7776: $C_3^2:C_4\times S_3^3$ ($S_3$)

Order 5832: $C_3^5:S_4$ ($C_2\times C_4$), $C_3^5:S_4$ ($C_2\times C_4$), $C_3^4:(S_3\times A_4)$ ($C_2^3$), $S_3^3:C_3^3$ ($C_2\times C_4$), $C_3^5:S_4$ ($C_2\times C_4$), $C_3^5:S_4$ ($C_2\times C_4$), $C_3^4:(S_3\times A_4)$ ($C_2\times C_4$)

Order 3888: $C_3\wr C_4:D_6$ ($D_6$), $C_3^3:(C_4\times S_3^2)$ ($D_6$), $C_3:S_3^4$ ($D_6$)

Order 2916: $C_3^5:A_4$ ($C_2^2\times C_4$)

Order 1944: $C_3^2:S_3^3$ ($C_4\times S_3$), $C_3^2:S_3^3$ ($C_2\times D_6$), $C_3^2\times S_3^3$ ($C_4\times S_3$), $C_3^4:(C_4\times S_3)$ ($S_4$)

Order 1296: $S_3\wr S_3$ ($C_3^2:C_4$)

Order 972: $C_3^3:S_3^2$ ($C_2\times S_4$), $C_3^3:C_6^2$ ($C_4\times D_6$), $C_3^5:C_4$ ($C_2\times S_4$), $C_3^4:C_{12}$ ($C_2\times S_4$)

Order 648: $S_3\wr C_3$ ($C_2\times C_3^2:C_4$), $C_3^3:S_4$ ($C_2\times C_3^2:C_4$), $C_3^3:S_4$ ($C_2\times C_3^2:C_4$)

Order 486: $C_3^4:S_3$ ($C_4\times S_4$), $C_3^4:C_6$ ($C_4\times S_4$), $C_3^4:C_6$ ($C_2^2\times S_4$)

Order 324: $C_3^3:A_4$ ($C_6^2:C_4$)

Order 243: $C_3^5$ ($C_2^4.D_6$)

Order 216: $S_3^3$ ($C_3^2:C_4\times S_3$)

Order 108: $C_3:S_3^2$ ($C_3^2:C_4\times D_6$)

Order 54: $C_3^2:S_3$ ($C_3^2:C_4\times S_4$)

Order 36: $C_3^2:C_4$ ($S_3\wr S_3$)

Order 27: $C_3^3$ ($C_6^2:(C_4\times D_6)$)

Order 18: $C_3:S_3$ ($S_3^3:D_6$)

Order 9: $C_3^2$ ($C_4\times S_3\wr S_3$)

Order 1: $C_1$ ($C_3:S_3^4.D_6$)

Series

Derived series

$C_3:S_3^4.D_6$

$\rhd$

$C_3:S_3^4.D_6$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3:S_3^4.D_6$

$\rhd$

$C_3:S_3^4.D_6$

$\rhd$

$(C_3^2\times S_3^3):C_{12}$

$\rhd$

$(C_3^2\times S_3^3):C_{12}$

$\rhd$

$C_3^4:(A_4\times D_6)$

$\rhd$

$C_3^4:(A_4\times D_6)$

$\rhd$

$C_3^4:(S_3\times A_4)$

$\rhd$

$C_3^4:(S_3\times A_4)$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3:S_3^4.D_6$

$\rhd$

$C_3:S_3^4.D_6$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^5:A_4$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $132 \times 132$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $110 \times 110$ rational character table.