Abstract group 69984.jj: $S_4\times C_3^4.S_3^2$

• Label: 69984.jj
• Order: \( 2^{5} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $31$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$S_4\times C_3^4.S_3^2$
Order:	\(69984\)\(\medspace = 2^{5} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$S_4\times C_3^4.S_3^2$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \)
Composition factors:	$C_2$ x 5, $C_3$ x 7
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	9	12	18	36
Elements	1	1359	3158	816	32418	3402	11496	12150	5184	69984
Conjugacy classes	1	11	27	4	85	16	27	28	11	210
Divisions	1	11	23	4	65	12	21	20	8	165
Autjugacy classes	1	11	27	4	85	16	27	28	11	210

Dimension	1	2	3	4	6	8	9	12	18	24	36	48	54	72	108
Irr. complex chars.	8	24	24	18	48	4	16	24	24	3	10	0	4	1	2	210
Irr. rational chars.	8	24	8	18	32	4	0	24	16	7	14	1	4	3	2	165

Minimal presentations

Permutation degree:	$31$
Transitive degree:	not computed
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	54	54	54
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{3}=d^{6}=e^{6}=f^{6}=g^{9}= \!\cdots\! \rangle}$
Permutation group:	Degree $31$ $\langle(3,8)(4,11)(6,14)(7,18)(9,21)(12,20)(13,22)(15,26)(17,27), (1,2,5)(3,6,12) \!\cdots\! \rangle$
Direct product:	$S_4$ $\, \times\, $ $(C_3^4.S_3^2)$
Semidirect product:	$(C_3^4.D_6^2)$ $\,\rtimes\,$ $S_3$	$(C_3^4:S_3)$ $\,\rtimes\,$ $(S_3\times S_4)$	$(C_3^4.S_3)$ $\,\rtimes\,$ $(S_3\times S_4)$	$(C_2\times C_3^4:D_6)$ $\,\rtimes\,$ $S_3^2$	all 31
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(S_4\times C_3^4)$ . $S_3^2$	$C_3^4$ . $(S_4\times S_3^2)$	$(S_4\times C_3^4:C_3)$ . $D_6$	$(C_3^2\times C_6^2)$ . $S_3^3$	all 40
Aut. group:	$\Aut(C_6:D_{18})$	$\Aut(C_6^2:D_9)$	$\Aut(A_4\times C_3:D_9)$	$\Aut(C_{18}:(C_6\times S_3))$	all 10

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 1087518 subgroups in 11724 conjugacy classes, 129 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_4\times C_3^4.S_3^2$
Commutator:	$G' \simeq$ $A_4\times C_3^3.C_3^3$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3^3$	$G/\Phi \simeq$ $C_3:S_3^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_3^3.C_3^3$	$G/\operatorname{Fit} \simeq$ $C_2\times D_6$
Radical:	$R \simeq$ $S_4\times C_3^4.S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3^3:S_3^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_3^3.C_3^3$

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 36 (order at least 1944) are shown, as well as all normal subgroups of any index.

Order 69984: $S_4\times C_3^4.S_3^2$

Order 34992: $C_6^2.C_3^3.C_3.D_6$, $C_6^2.C_3^4.D_6$, $C_6^2.C_3^4.D_6$, $S_4\times (C_3^3.C_3^3):C_2$, $C_3^4:C_3.D_6\times A_4$, $S_4\times (C_3^3.C_3^3):C_2$, $S_4\times (C_3^3.C_3^3):C_2$

Order 23328: $S_4\times (C_9.C_3^3):C_2^2$, $S_4\times (C_3\times C_9):C_3^2:C_2^2$, $S_4\times (C_3^4.C_3):C_2^2$, $S_4\times C_3^4:C_3:C_2^2$, $S_4\times \He_3:C_3^2:C_2^2$, $D_4\times C_3^4.S_3^2$

Order 17496: $(C_3^3.C_3^3):C_2\times A_4$, $S_4\times C_3^3.C_3^3$, $(C_3\times A_4).C_3^4:C_6$, $C_6^2.C_3^4.C_6$, $(C_3^3.C_3^3):C_2\times A_4$, $C_6^2.C_3^4.S_3$, $(C_3^3.C_3^3):C_2\times A_4$, $S_3\times C_3^4.C_3.D_6$

Order 11664: $S_4\times C_3\wr C_3:C_6$ x 2, $C_3^4.C_3.C_6.C_2^3$ x 2, $S_4\times (\He_3.C_3):C_6$ x 2, $S_4\times (C_9.C_3^3):C_2$ x 2, $C_3^4.D_6^2$ x 2, $C_6^2.C_3^4.C_2^2$, $(C_3\times C_9):C_3^2:C_2^2\times A_4$, $S_4\times (C_3\times C_9):(C_3\times C_6)$, $C_6^2.C_3^4.C_2^2$, $S_4\times \He_3:(C_3\times S_3)$, $S_4\times C_3^3:(C_3\times S_3)$, $S_4\times C_3:D_9:C_3^2$, $(C_3^4.C_3):C_2^2\times A_4$, $S_4\times C_3^2:(C_2\times C_9:C_3)$, $C_6^2.C_3^3.D_6$, $S_4\times C_3\times C_3\wr C_3:C_2$, $\He_3:C_3^2:C_2^2\times A_4$, $C_4\times C_3^4.C_3.D_6$, $C_6^2.C_3^3.D_6$, $S_4\times C_3.\He_3:S_3$, $C_3^4:C_3:C_2^2\times A_4$, $C_6^2.C_3^3.D_6$, $C_6^2.C_3^4.C_2^2$, $(C_9\times A_4).C_3^3.C_2^2$, $C_6^2.\He_3.D_6$, $S_4\times C_3^4:C_6$, $C_6^2.C_3^4.C_2^2$, $C_6^2.C_3^3.D_6$, $C_6^2.\He_3.D_6$, $C_3^4.C_3.D_{12}.C_2$, $C_6^2.C_3^4.C_2^2$, $C_6^2.C_3^4.C_2^2$, $C_6^2.C_3^4.C_2^2$, $S_4\times (C_3^4.C_3):C_2$, $C_3^4.C_3.C_6.C_2^3$, $C_6^2.C_3^4.C_2^2$, $C_6^2.C_3^4.C_2^2$, $D_4\times (C_3^3.C_3^3):C_2$, $C_9.C_3^3.D_6.C_2^2$, $C_3^3.C_3^3.D_4.C_2$, $S_4\times \He_3:C_3^2:C_2$, $C_6^2.C_3^3.D_6$, $C_6^2.C_3^4.C_2^2$, $D_4\times (C_3^3.C_3^3):C_2$, $S_4\times (\He_3.C_6):C_3$, $C_3^4.D_6:D_6$, $C_3^4.C_3.C_6.C_2^3$, $C_6^2.C_3^4.C_2^2$, $S_4\times (C_3\times C_9):C_3^2:C_2$, $(C_9.C_3^3):C_2^2\times A_4$, $D_4\times (C_3^3.C_3^3):C_2$, $C_9.C_3^3.D_{12}.C_2$, $S_4\times (\He_3.S_3):C_3$, $C_3^4.(S_3\times S_4)$

Order 8748: $C_3^3.C_3^3.D_6$, $C_3\times C_3^4:C_3.D_6$, $C_3^3.C_3^3.D_6$, $A_4\times C_3^3.C_3^3$, $S_3\times (C_3^3.C_3^3):C_2$, $S_3\times (C_3^3.C_3^3):C_2$, $C_3^4.C_3^2.D_6$, $S_3\times (C_3^3.C_3^3):C_2$

Order 7776: $C_3\times S_4\times C_3.S_3^2$ x 3, $S_4\times S_3\times D_9:C_3$ x 2, $D_4\times \He_3:C_3^2:C_2^2$, $S_4\times C_9.S_3^2$, $S_4\times (\He_3.S_3):C_2$, $S_4\times C_3:S_3:D_9$, $C_3\times S_4\times C_3^3:C_2^2$, $D_4\times (C_3\times C_9):C_3^2:C_2^2$, $D_4\times C_3^4:C_3:C_2^2$, $S_4\times C_3:D_9:S_3$, $D_4\times (C_9.C_3^3):C_2^2$, $D_4\times (C_3^4.C_3):C_2^2$, $S_4\times C_3^2:S_3\times S_3$

Order 5832: $C_2\times C_3^4.S_3^2$ x 7, $C_3\wr C_3:C_6\times A_4$ x 2, $C_2^2\times (C_3^3.C_3^3):C_2$ x 2, $C_6^2.C_3^4.C_2$ x 2, $(C_9.C_3^3):C_2\times A_4$ x 2, $C_6^2.C_3^4.C_2$ x 2, $C_2\times C_3^4.S_3^2$ x 2, $C_2\times C_3^4.C_3.D_6$ x 2, $S_4\times (C_3\times C_9):C_3^2$ x 2, $(\He_3.C_3):C_6\times A_4$ x 2, $C_3^4:C_3.D_{12}$ x 2, $S_4\times \He_3.C_3^2$ x 2, $C_2^2\times (C_3^3.C_3^3):C_2$ x 2, $(C_6\times C_9:C_3).C_6^2$ x 2, $C_2^2\times (C_3^3.C_3^3):C_2$ x 2, $C_3^4:C_3.C_6.C_2^2$ x 2, $(C_6\times C_{18}).\He_3.C_2$, $C_6^2.C_3^3.C_6$, $(C_6\times C_9:C_3).C_6^2$, $C_6^2.C_3^3.C_6$, $C_6^2.\He_3.S_3$, $S_3\times (C_9.C_3^3):C_2^2$, $C_3^3.C_6^2.C_6$, $C_3^4.C_3^2.C_4.C_2$, $C_6^2.\He_3.S_3$, $C_3:D_9:C_3^2\times A_4$, $\He_3:(C_3\times S_3)\times A_4$, $S_4\times C_3\times C_3\wr C_3$, $C_4\times (C_3^3.C_3^3):C_2$, $(A_4\times \He_3).C_3.C_6$, $S_4\times C_3^4:C_3$, $C_4\times (C_3^3.C_3^3):C_2$, $C_6^2.C_3^3.C_6$, $S_3\times \He_3:C_3^2:C_2^2$, $(C_9\times A_4).C_3^3.C_2$, $(C_3\times A_4).C_3^4.C_2$, $(C_6\times C_{18}).\He_3.C_2$, $(C_3\times A_4).\He_3.C_6$, $C_3^4:C_3.C_{12}.C_2$, $C_6^2.\He_3:S_3$, $C_3^4:C_6\times A_4$, $C_6^2.C_3^4.C_2$, $C_6^2.\He_3:S_3$, $C_6^2.\He_3.S_3$, $(A_4\times \He_3).C_3.C_6$, $(C_3\times A_4).\He_3.C_6$, $C_4\times (C_3^3.C_3^3):C_2$, $C_6^2.C_3^3.C_6$, $C_3^4.C_3.D_{12}$, $C_3^3.C_3^3.D_4$, $A_4.C_3^4.C_6$, $C_3^4.C_3.C_{12}.C_2$, $C_3^4:C_3.D_{12}$, $S_4\times \He_3:C_3^2$, $C_3.\He_3:S_3\times A_4$, $C_6^2.C_3^3.C_6$, $C_3^3:(C_3\times S_3)\times A_4$, $A_4.C_3^2:C_9.C_6$, $(C_3^4.C_3):C_2\times A_4$, $C_3^4.D_6:S_3$, $C_3^3.C_3^3.D_4$, $(C_3\times C_9):C_3^2:C_2\times A_4$, $D_4\times C_3^3.C_3^3$, $C_3^3.C_3^3.C_4.C_2$, $S_4\times C_3^4.C_3$, $C_6^2.\He_3.S_3$, $C_6^2.\He_3.S_3$, $C_3^4.C_6^2.C_2$, $C_3^4.C_3^2.D_4$, $C_3^2:(C_2\times C_9:C_3)\times A_4$, $(\He_3.C_6):C_3\times A_4$, $(C_3\times C_9):(C_3\times C_6)\times A_4$, $C_6^2.\He_3:S_3$, $(C_6\times C_{18}).\He_3.C_2$, $C_3\times C_3\wr C_3:C_2\times A_4$, $(\He_3.S_3):C_3\times A_4$, $C_6^2.C_3^4.C_2$, $C_6^2.C_3^4.C_2$, $S_4\times C_9.C_3^3$, $C_6^2.C_3^3.C_6$, $(C_3\times A_4).\He_3.C_6$, $C_6^2.\He_3:S_3$, $(C_6\times C_{18}).\He_3.C_2$, $(A_4\times \He_3).C_3.C_6$, $\He_3:C_3^2:C_2\times A_4$, $C_6^2.C_3^4.C_2$, $S_3\times (C_3\times C_9):C_3^2:C_2^2$, $C_3^3.S_3^3$, $C_3^3:S_3^3$

Order 4374: $C_3\times (C_3^3.C_3^3):C_2$, $C_3^5.C_3.S_3$, $S_3\times C_3^3.C_3^3$, $C_3\times (C_3^3.C_3^3):C_2$, $C_3\times (C_3^3.C_3^3):C_2$, $C_3^4.\He_3.C_2$, $C_3^3.C_3^3.C_6$

Order 3888: $C_3\times S_4\times \He_3:C_2$ x 6, $C_3\times S_4\times C_3^2:S_3$ x 5, $C_3\times C_6^2:S_3^2$ x 5, $C_3.C_3^3.(C_6\times D_4)$ x 4, $C_6^2.C_3^3.C_2^2$ x 3, $S_4\times S_3\times C_9:C_3$ x 3, $C_3\times C_3.S_3^2\times A_4$ x 3, $C_3^2\times S_4\times C_3:S_3$ x 3, $S_4\times (C_3.C_3^3):C_2$ x 3, $C_3\times S_4\times D_9:C_3$ x 3, $C_3^3:S_3\times S_4$ x 3, $C_2^2\times (C_3\times C_9):C_3^2:C_2^2$ x 2, $C_2^2\times (C_9.C_3^3):C_2^2$ x 2, $C_9.(C_3\times S_3\times D_{12})$ x 2, $S_4\times \He_3.C_6$ x 2, $D_4\times C_3\wr C_3:C_6$ x 2, $\He_3.C_3.(C_6\times D_4)$ x 2, $S_4\times C_3^2:S_3:C_3$ x 2, $S_4\times C_9:C_3:S_3$ x 2, $C_3\times C_6^2.S_3^2$ x 2, $D_4\times (C_9.C_3^3):C_2$ x 2, $\He_3.C_3.(C_6\times D_4)$ x 2, $C_2^2\times (C_3^4.C_3):C_2^2$ x 2, $D_4\times (\He_3.C_3):C_6$ x 2, $C_3^4.(S_3\times D_4)$ x 2, $C_3.C_3^3.(C_6\times D_4)$ x 2, $S_4\times (\He_3.C_3):C_2$ x 2, $S_4\times C_9:(C_3\times S_3)$ x 2, $S_3\times D_9:C_3\times A_4$ x 2, $S_4\times C_3:S_3:C_9$ x 2, $C_2^2\times C_3^4:C_3:C_2^2$ x 2, $\He_3:C_3.(C_6\times D_4)$ x 2, $C_3\times (C_3^2\times A_4).D_6$ x 2, $S_4\times C_3\times D_9:C_3$ x 2, $C_2^2\times \He_3:C_3^2:C_2^2$ x 2, $C_3^3.(S_3\times S_4)$ x 2, $(C_3\times C_6^2):S_3^2$ x 2, $C_6^2.(C_3\times S_3^2)$ x 2, $C_3^3.C_6^2.C_2^2$, $C_3^2:C_9.(C_6\times D_4)$, $S_4\times (C_3\times C_9):S_3$, $C_6^2.C_3^3.C_2^2$, $D_4\times (C_3\times C_9):C_3^2:C_2$, $(A_4\times \He_3).C_2\times S_3$, $S_4\times \He_3:S_3$, $(A_4\times \He_3).D_6$, $C_4\times C_3^4:C_3:C_2^2$, $\He_3.C_3.(C_6\times D_4)$, $C_6^2.C_3^3.C_2^2$, $C_3\times (C_3^2\times A_4).D_6$, $(C_3^3\times A_4).D_6$, $\He_3.(C_6\times A_4).C_2$, $C_6^2.C_3^3.C_2^2$, $D_4\times C_3^4:C_6$, $\He_3.C_3.(C_6\times D_4)$, $C_9.S_3^2\times A_4$, $C_3:D_9:S_3\times A_4$, $C_3^2:S_3\times S_3\times A_4$, $\He_3.C_3.(C_6\times D_4)$, $C_4\times (C_3\times C_9):C_3^2:C_2^2$, $\He_3.A_4.C_6.C_2$, $\He_3:C_3.(C_6\times D_4)$, $D_4\times (\He_3.C_6):C_3$, $C_3:S_4\times D_9:C_3$, $(C_9\times A_4).(C_6\times S_3)$, $(A_4\times \He_3).D_6$, $D_4\times \He_3:(C_3\times S_3)$, $C_3^3.(C_6\times D_{12})$, $C_3\times (C_3^2\times A_4).D_6$, $C_6^2.C_3^3.C_2^2$, $S_4\times S_3\times \He_3$, $D_4\times C_3\times C_3\wr C_3:C_2$, $(C_3^3\times A_4).D_6$, $C_3\times C_3^3:C_2^2\times A_4$, $C_3:S_4\times C_3^2:S_3$, $(C_3\times C_9\times A_4).D_6$, $S_4\times C_3:D_9:C_3$, $C_3^4.(S_3\times D_4)$, $D_4\times C_3.\He_3:S_3$, $\He_3:C_3.(C_6\times D_4)$, $(A_4\times \He_3).D_6$, $S_4\times \He_3.S_3$, $D_4\times (\He_3.S_3):C_3$, $C_4\times (C_3^4.C_3):C_2^2$, $D_4\times \He_3:C_3^2:C_2$, $(C_9\times A_4).S_3^2$, $D_4\times (C_3^4.C_3):C_2$, $\He_3.C_6^2.C_2^2$, $\He_3.C_3.(C_6\times D_4)$, $C_3^3.C_6^2.C_2^2$, $C_3^2:C_9.(C_6\times D_4)$, $D_4\times C_3:D_9:C_3^2$, $\He_3.(C_6\times D_{12})$, $(C_9\times A_4).(C_6\times S_3)$, $(C_3\times C_9\times A_4).C_6.C_2$, $C_3^2.(A_4\times S_3^2)$, $D_4\times C_3^3:(C_3\times S_3)$, $C_3^4.D_{12}.C_2$, $(C_3\times C_9\times A_4).D_6$, $\He_3:C_3.(C_6\times D_4)$, $\He_3.C_6^2.C_2^2$, $S_4\times \He_3:C_3:C_2$, $C_3^2:C_9.(C_6\times D_4)$, $(C_3^3\times A_4).D_6$, $\He_3:C_3.(C_6\times D_4)$, $\He_3.C_6^2.C_2^2$, $C_3:S_3:D_9\times A_4$, $C_3^2:C_9.(C_6\times D_4)$, $S_4\times (C_3\times C_9):C_6$, $C_3^4.D_{12}.C_2$, $C_6^2.C_3^3.C_2^2$, $(C_3\times C_9\times A_4).D_6$, $C_6^2.C_3^3.C_2^2$, $C_9.(A_4\times S_3^2)$, $C_4\times (C_9.C_3^3):C_2^2$, $C_6^2.C_3^3.C_2^2$, $(C_9\times A_4).C_6\times S_3$, $C_3^4:C_3.D_4.C_2$, $C_3^4.D_6.C_2^2$, $D_4\times C_3^2:(C_2\times C_9:C_3)$, $C_3\times C_6^2.S_3^2$, $D_4\times (C_3\times C_9):(C_3\times C_6)$, $\He_3.C_3.A_4.C_2^2$, $(\He_3.S_3):C_2\times A_4$, $S_4\times C_3\times \He_3:C_2$, $C_3.C_3^3.(C_6\times D_4)$, $(C_3^3\times A_4).D_6$, $C_4\times \He_3:C_3^2:C_2^2$, $(A_4\times \He_3).D_6$, $(A_4\times \He_3).D_6$, $C_3^4:(C_2\times S_4)$, $C_3^3.(C_6\times S_4)$

Order 2916: $C_3^4.S_3^2$ x 15, $C_2\times (C_3^3.C_3^3):C_2$ x 5, $C_2\times (C_3^3.C_3^3):C_2$ x 5, $C_2\times (C_3^3.C_3^3):C_2$ x 5, $A_4\times \He_3.C_3^2$ x 2, $C_2^2\times C_3^3.C_3^3$ x 2, $S_3\times (\He_3.C_3):C_6$ x 2, $S_3\times C_3\wr C_3:C_6$ x 2, $C_3.C_3^5.C_2^2$ x 2, $A_4\times (C_3\times C_9):C_3^2$ x 2, $S_3\times (C_9.C_3^3):C_2$ x 2, $C_3^4.(C_6\times S_3)$, $S_3\times \He_3:(C_3\times S_3)$, $C_6^2.C_3^3.C_3$, $S_3\times (\He_3.S_3):C_3$, $S_3\times (C_3\times C_9):C_3^2:C_2$, $C_6^2.\He_3.C_3$, $C_4\times C_3^3.C_3^3$, $\He_3.C_3^3.C_2^2$, $C_3^4.C_3^2.C_4$, $\He_3.C_3^3.C_2^2$, $C_3^3.C_3^3.C_2^2$, $C_3\times C_3^4:C_3:C_2^2$, $C_3^4.(C_6\times S_3)$, $C_6^2.C_3^3.C_3$, $C_6^2.\He_3.C_3$, $C_9.C_3^3.D_6$, $A_4\times C_3^4.C_3$, $A_4\times \He_3:C_3^2$, $C_6^2.\He_3.C_3$, $C_3^5.D_6$, $S_3\times C_3.\He_3:S_3$, $C_3.C_3^5.C_2^2$, $S_3\times (\He_3.C_6):C_3$, $C_3^2.C_3^4.C_2^2$, $C_3^4.S_3^2$, $A_4\times C_3\times C_3\wr C_3$, $C_6^2.\He_3.C_3$, $C_6^2.\He_3.C_3$, $S_3\times (C_3\times C_9):(C_3\times C_6)$, $C_3^3.C_3^3.C_2^2$, $S_3\times \He_3:C_3^2:C_2$, $S_3\times C_3^4:C_6$, $\He_3.C_3^3.C_2^2$, $(C_2\times C_3^4.C_3).C_6$, $S_3\times C_3:D_9:C_3^2$, $A_4\times C_3^4:C_3$, $A_4\times C_9.C_3^3$, $C_6^2.C_3^3.C_3$, $C_6^2.C_3^3.C_3$, $C_3^4:C_3.C_{12}$, $C_3^4.S_3^2$, $C_3^4.S_3^2$, $C_3^4.S_3^2$, $C_3^5.D_6$, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^5:D_6$

Order 2592: $C_3\times S_4\times S_3^2$ x 5, $C_3\times D_4\times C_3.S_3^2$ x 3, $C_3^2:D_6\times S_4$ x 3, $S_4\times S_3\times D_9$ x 2, $D_4\times S_3\times D_9:C_3$ x 2, $C_2\times S_4\times C_3^2:S_3$, $D_4\times C_3^2:S_3\times S_3$, $D_4\times C_9.S_3^2$, $D_4\times (\He_3.S_3):C_2$, $D_4\times C_3:D_9:S_3$, $D_4\times C_3:S_3:D_9$, $C_3\times D_4\times C_3^3:C_2^2$, $C_3\times S_4\times S_3^2$, $C_{18}:C_6\times S_4$, $S_4\times C_3:S_3^2$

Order 2187: $C_3\times C_3^3.C_3^3$

Order 1944: $C_2\times C_3^3.S_3^2$ x 11, $C_2\times C_3^3:S_3^2$ x 11, $C_2\times C_3^3.S_3^2$ x 11, $C_2\times C_3^3.S_3^2$ x 11, $C_2\times C_3^3.S_3^2$ x 11, $A_4\times C_3^3:C_6$ x 7, $C_3\times \He_3:S_4$ x 7, $C_9:C_3^2\times S_4$ x 6, $C_3\times S_4\times \He_3$ x 6, $C_3\wr C_3\times S_4$ x 6, $A_4\times C_3^3:S_3$ x 5, $A_4\times D_9:C_3^2$ x 5, $C_3\times \He_3:S_4$ x 5, $C_9:(C_3^2\times S_4)$ x 5, $(A_4\times C_3^3):S_3$ x 5, $(C_3\times C_{18}):C_6^2$ x 4, $C_9.C_3^2\times S_4$ x 4, $(C_3\times C_{18}).C_6^2$ x 4, $C_{18}.(C_3\times S_3^2)$ x 4, $\He_3.(C_6\times D_6)$ x 4, $C_3^3:(C_6\times D_6)$ x 4, $A_4\times C_3^3:S_3$ x 4, $C_9:C_3^2:S_4$ x 4, $S_4\times C_3.\He_3$ x 4, $(C_6\times \He_3).D_6$ x 4, $C_3^3:C_6\wr C_2$ x 4, $C_3^2\wr C_2\times A_4$ x 3, $C_3^4:S_4$ x 3, $C_6^2.(S_3\times C_3^2)$ x 3, $A_4\times \He_3.S_3$ x 3, $(D_6\times C_{18}):C_3^2$ x 3, $(A_4\times \He_3).S_3$ x 3, $C_9.(C_3^2\times S_4)$ x 3, $C_3^3.(C_3\times S_4)$ x 3, $C_3^3.(C_3\times S_4)$ x 3, $C_3^2.S_3^3$ x 3, $C_3^3:(S_3\times A_4)$ x 3, $C_3\wr C_3:S_4$ x 3, $C_9:C_3^2:S_4$ x 3, $C_3^3:(C_3\times S_4)$ x 3, $C_2\times \He_3.C_6^2$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $A_4\times \He_3.C_6$ x 2, $C_3^3.(S_3\times A_4)$ x 2, $A_4\times C_3^3.S_3$ x 2, $\He_3.(C_3\times S_4)$ x 2, $(A_4\times C_3^3).S_3$ x 2, $C_3^3.(C_3\times S_4)$ x 2, $C_3^2.S_3^3$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $C_{36}.(S_3\times C_3^2)$ x 2, $C_{36}.(S_3\times C_3^2)$ x 2, $C_{18}.(C_3\times S_3^2)$ x 2, $C_{18}.(C_3\times S_3^2)$ x 2, $C_2\times C_3^4:D_6$ x 2, $\He_3:(C_6\times D_6)$ x 2, $C_6\times C_3^3:D_6$ x 2, $C_2\times C_3^4.D_6$ x 2, $C_2\times C_3^4:D_6$ x 2, $(C_3\times C_{18}):C_6^2$ x 2, $C_2\times \He_3.C_6^2$ x 2, $(C_3\times C_{18}):C_6^2$ x 2, $C_2\times \He_3.C_6^2$ x 2, $(C_3\times C_{18}):C_6^2$ x 2, $C_2\times C_3^3.C_6^2$ x 2, $C_2^2\times C_3^4:C_6$ x 2, $\He_3.(S_3\times A_4)$ x 2, $A_4\times \He_3.S_3$ x 2, $C_3^3:(S_3\times A_4)$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $A_4\times \He_3.C_6$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $A_4\times \He_3.C_6$ x 2, $C_3^3.(C_6\times A_4)$ x 2, $A_4\times C_3^2:C_{18}$ x 2, $(A_4\times \He_3).S_3$ x 2, $(A_4\times C_3^3).S_3$ x 2, $C_9:C_3^2:S_4$ x 2, $(A_4\times C_3^3):S_3$ x 2, $(C_3\times \He_3):S_4$ x 2, $\He_3.(C_3\times S_4)$ x 2, $C_3^3:(C_3\times S_4)$ x 2, $\He_3:(C_3\times S_4)$ x 2, $\He_3.(C_3\times S_4)$ x 2, $\He_3.(C_3\times S_4)$ x 2, $C_3^2:(C_9\times S_4)$ x 2, $S_4\times \He_3:C_3$ x 2, $C_3^2:C_9\times S_4$ x 2, $D_4\times C_3^2.C_3^3$ x 2, $D_4\times C_3^3:C_3^2$ x 2, $(C_3^2\times C_6^2):S_3$ x 2, $\He_3:C_6\wr C_2$ x 2, $(C_3^2\times C_6^2):S_3$ x 2, $(C_3^2\times C_6^2).S_3$ x 2, $(C_3^2\times C_6^2):S_3$ x 2, $(C_{12}\times \He_3).S_3$ x 2, $\He_3.(S_3\times C_{12})$ x 2, $C_3^3:(C_3\times D_{12})$ x 2, $C_3^3:(S_3\times C_{12})$ x 2, $(C_3\times C_{18}).C_6^2$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $(C_3\times C_{18}).C_6^2$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $(C_3\times C_{18}).C_6^2$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $(C_3^2\times C_6^2):C_6$ x 2, $(C_3\times \He_3):D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $(C_3^3\times C_6).D_6$ x 2, $(C_3^2\times C_6).S_3^2$ x 2, $\He_3.(C_3\times D_{12})$ x 2, $(C_3^2\times C_6).S_3^2$ x 2, $(C_3^2\times C_6).S_3^2$ x 2, $\He_3.(C_3\times D_{12})$ x 2, $(C_3^2\times C_6).S_3^2$ x 2, $(C_3^3\times C_6).D_6$ x 2, $C_3^4.D_{12}$ x 2, $(C_3^3\times C_6).D_6$ x 2, $S_4\times C_3^4$, $C_3^2:S_3^3$, $A_4\times C_3^3:S_3$, $C_3^3.(S_3\times A_4)$, $S_3\times A_4\times \He_3$, $(A_4\times C_3^3):S_3$, $C_3:S_4\times \He_3$, $C_9.S_3^3$, $C_3^2:S_3^3$, $D_4\times C_9.C_3^3$, $C_3^3.(C_3\times D_{12})$, $C_3^3.(S_3\times C_{12})$, $C_{18}.(C_3\times S_3^2)$, $C_3^3:(S_3\times A_4)$, $\He_3:(S_3\times A_4)$, $A_4\times \He_3:S_3$, $C_3^3.(S_3\times A_4)$, $A_4\times C_3^2:D_9$, $(C_3^2\times C_6^2):C_6$, $C_3^3.(C_6\times A_4)$, $C_3^3.(S_3\times A_4)$, $C_3^3.(S_3\times A_4)$, $C_3^3.(S_3\times A_4)$, $C_3^3.(S_3\times A_4)$, $C_3^3.(S_3\times A_4)$, $C_6^2.(S_3\times C_3^2)$, $A_4\times \He_3.S_3$, $C_6^2.(S_3\times C_3^2)$, $A_4\times \He_3.S_3$, $C_3^3.(S_3\times A_4)$, $A_4\times C_3^2:D_9$, $C_3^4:S_4$, $(A_4\times \He_3):S_3$, $(C_3^2\times A_4):D_9$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4.S_4$, $C_9:C_3^2:S_4$, $C_3\times \He_3:S_4$, $(A_4\times \He_3).S_3$, $(A_4\times \He_3).S_3$, $(C_3^2\times A_4):D_9$, $C_3^4.S_4$, $C_3^3.(C_3\times S_4)$, $C_3^3.(C_3\times S_4)$, $C_3^3.(C_3\times S_4)$, $C_3^3:(C_3\times S_4)$, $C_3^3.(C_3\times S_4)$, $C_3^3.(C_3\times S_4)$, $C_3^2.S_3^3$, $C_3^2.S_3^3$, $C_3^2.S_3^3$, $D_4\times \He_3:C_3^2$, $C_3^4:C_3\times D_4$, $C_3^4.C_3\times D_4$, $C_3^4:C_3\times D_4$, $C_3^4:D_{12}$, $C_4\times C_3^4:S_3$, $(C_{12}\times \He_3):S_3$, $\He_3:(S_3\times C_{12})$, $C_3^4:D_{12}$, $C_3^3:S_3\times C_{12}$, $C_3^4.D_{12}$, $C_4\times C_3^4.S_3$, $C_3^4:D_{12}$, $C_4\times C_3^4:S_3$, $(C_3\times C_{18}).C_6^2$, $(C_3\times C_{18}).C_6^2$, $C_3^3.(C_3\times D_{12})$, $C_3^3.(S_3\times C_{12})$, $(C_3\times C_{18}).C_6^2$, $(C_3\times C_{18}).C_6^2$, $C_3^3.(C_3\times D_{12})$, $C_3^3.(S_3\times C_{12})$, $C_3^4.D_{12}$, $C_4\times C_3^4.S_3$, $C_3^3.(C_3\times D_{12})$, $C_4\times C_3^4.C_6$, $C_3^3:(C_3\times D_{12})$, $C_4\times C_3^4:C_6$, $C_3^4:(C_4\times S_3)$, $C_3^4:(C_4\times S_3)$, $(C_3^3\times C_6).D_6$, $\He_3.(S_3\times C_{12})$, $(C_3^2\times C_6).S_3^2$, $\He_3.(S_3\times C_{12})$, $\He_3.(S_3\times C_{12})$, $(C_3^2\times C_6).S_3^2$, $\He_3.(S_3\times C_{12})$, $C_3^3.(S_3\times C_{12})$, $(C_3^3\times C_6).D_6$, $C_3^3.(S_3\times C_{12})$

Order 1458: $C_3^4:C_3:S_3$, $C_3^4.(C_3\times S_3)$, $C_3^4.(C_3\times S_3)$

Order 1296: $C_6^2:S_3^2$ x 7, $C_6^2.S_3^2$

Order 972: $C_6^2:C_3^3$ x 3, $A_4\times C_3^4$, $C_3^3.C_6^2$, $C_6^2.C_3^3$, $\He_3:C_6^2$, $C_3^3.C_6^2$, $C_3^3.C_6^2$, $C_3^3:C_6^2$

Order 729: $C_3^3.C_3^3$

Order 648: $S_4\times C_3^3$ x 2, $C_{18}:(C_6\times S_3)$, $S_3\times C_3^2:D_6$, $A_4\times C_3:D_9$, $(C_3\times C_9):S_4$, $C_3\times C_9\times S_4$

Order 486: $C_9.(S_3\times C_3^2)$, $C_3^4:S_3$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^4.S_3$

Order 324: $C_3^2:S_3^2$ x 7, $C_3^2:C_6^2$ x 3, $A_4\times C_3^3$ x 2, $C_3^2\times C_6^2$, $C_9:C_6^2$, $C_6^2.C_3^2$

Order 288: $D_6\times S_4$ x 4

Order 243: $C_9.C_3^3$, $\He_3:C_3^2$, $C_3^2.C_3^3$, $C_3^4.C_3$, $C_3^4:C_3$

Order 216: $C_6^2:S_3$ x 4, $C_3^2\times S_4$ x 2, $C_6:D_{18}$

Order 162: $C_3^3.S_3$

Order 144: $S_3\times S_4$ x 24

Order 108: $C_3^2:D_6$ x 23, $C_3\times C_6^2$ x 2, $C_3^2\times A_4$ x 2, $C_6\times C_{18}$

Order 96: $C_2^2\times S_4$

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

Order 72: $S_3\times D_6$ x 4, $C_3\times S_4$

Order 54: $C_3^2:S_3$ x 28, $C_3:D_9$

Order 48: $C_2\times S_4$ x 12

Order 36: $S_3^2$ x 24, $C_6^2$ x 2, $C_3\times A_4$

Order 32: $C_2^2\times D_4$

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

Order 24: $S_4$ x 19, $C_2\times D_6$ x 17

Order 12: $D_6$ x 78, $C_2\times C_6$, $A_4$

Order 9: $C_3^2$ x 2

Order 8: $C_2^3$ x 4

Order 6: $S_3$ x 52

Order 4: $C_2^2$ x 18

Order 3: $C_3$

Order 2: $C_2$ x 10

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 36 (order at least 1944) are shown, as well as all normal subgroups of any index.

Order 69984: $S_4\times C_3^4.S_3^2$

Order 34992: $C_6^2.C_3^3.C_3.D_6$, $C_6^2.C_3^4.D_6$, $C_6^2.C_3^4.D_6$, $S_4\times (C_3^3.C_3^3):C_2$, $C_3^4:C_3.D_6\times A_4$, $S_4\times (C_3^3.C_3^3):C_2$, $S_4\times (C_3^3.C_3^3):C_2$

Order 23328: $S_4\times (C_9.C_3^3):C_2^2$, $S_4\times (C_3\times C_9):C_3^2:C_2^2$, $S_4\times (C_3^4.C_3):C_2^2$, $S_4\times C_3^4:C_3:C_2^2$, $S_4\times \He_3:C_3^2:C_2^2$, $D_4\times C_3^4.S_3^2$

Order 17496: $(C_3^3.C_3^3):C_2\times A_4$, $S_4\times C_3^3.C_3^3$, $(C_3\times A_4).C_3^4:C_6$, $C_6^2.C_3^4.C_6$, $(C_3^3.C_3^3):C_2\times A_4$, $C_6^2.C_3^4.S_3$, $(C_3^3.C_3^3):C_2\times A_4$, $S_3\times C_3^4.C_3.D_6$

Order 11664: $S_4\times C_3\wr C_3:C_6$ x 2, $C_3^4.C_3.C_6.C_2^3$ x 2, $S_4\times (\He_3.C_3):C_6$ x 2, $S_4\times (C_9.C_3^3):C_2$ x 2, $C_3^4.D_6^2$ x 2, $C_6^2.C_3^4.C_2^2$, $(C_3\times C_9):C_3^2:C_2^2\times A_4$, $S_4\times (C_3\times C_9):(C_3\times C_6)$, $C_6^2.C_3^4.C_2^2$, $S_4\times \He_3:(C_3\times S_3)$, $S_4\times C_3^3:(C_3\times S_3)$, $S_4\times C_3:D_9:C_3^2$, $(C_3^4.C_3):C_2^2\times A_4$, $S_4\times C_3^2:(C_2\times C_9:C_3)$, $C_6^2.C_3^3.D_6$, $S_4\times C_3\times C_3\wr C_3:C_2$, $\He_3:C_3^2:C_2^2\times A_4$, $C_4\times C_3^4.C_3.D_6$, $C_6^2.C_3^3.D_6$, $S_4\times C_3.\He_3:S_3$, $C_3^4:C_3:C_2^2\times A_4$, $C_6^2.C_3^3.D_6$, $C_6^2.C_3^4.C_2^2$, $(C_9\times A_4).C_3^3.C_2^2$, $C_6^2.\He_3.D_6$, $S_4\times C_3^4:C_6$, $C_6^2.C_3^4.C_2^2$, $C_6^2.C_3^3.D_6$, $C_6^2.\He_3.D_6$, $C_3^4.C_3.D_{12}.C_2$, $C_6^2.C_3^4.C_2^2$, $C_6^2.C_3^4.C_2^2$, $C_6^2.C_3^4.C_2^2$, $S_4\times (C_3^4.C_3):C_2$, $C_3^4.C_3.C_6.C_2^3$, $C_6^2.C_3^4.C_2^2$, $C_6^2.C_3^4.C_2^2$, $D_4\times (C_3^3.C_3^3):C_2$, $C_9.C_3^3.D_6.C_2^2$, $C_3^3.C_3^3.D_4.C_2$, $S_4\times \He_3:C_3^2:C_2$, $C_6^2.C_3^3.D_6$, $C_6^2.C_3^4.C_2^2$, $D_4\times (C_3^3.C_3^3):C_2$, $S_4\times (\He_3.C_6):C_3$, $C_3^4.D_6:D_6$, $C_3^4.C_3.C_6.C_2^3$, $C_6^2.C_3^4.C_2^2$, $S_4\times (C_3\times C_9):C_3^2:C_2$, $(C_9.C_3^3):C_2^2\times A_4$, $D_4\times (C_3^3.C_3^3):C_2$, $C_9.C_3^3.D_{12}.C_2$, $S_4\times (\He_3.S_3):C_3$, $C_3^4.(S_3\times S_4)$

Order 8748: $C_3^3.C_3^3.D_6$, $C_3\times C_3^4:C_3.D_6$, $C_3^3.C_3^3.D_6$, $A_4\times C_3^3.C_3^3$, $S_3\times (C_3^3.C_3^3):C_2$, $S_3\times (C_3^3.C_3^3):C_2$, $C_3^4.C_3^2.D_6$, $S_3\times (C_3^3.C_3^3):C_2$

Order 7776: $C_3\times S_4\times C_3.S_3^2$ x 3, $S_4\times S_3\times D_9:C_3$ x 2, $D_4\times \He_3:C_3^2:C_2^2$, $S_4\times C_9.S_3^2$, $S_4\times (\He_3.S_3):C_2$, $S_4\times C_3:S_3:D_9$, $C_3\times S_4\times C_3^3:C_2^2$, $D_4\times (C_3\times C_9):C_3^2:C_2^2$, $D_4\times C_3^4:C_3:C_2^2$, $S_4\times C_3:D_9:S_3$, $D_4\times (C_9.C_3^3):C_2^2$, $D_4\times (C_3^4.C_3):C_2^2$, $S_4\times C_3^2:S_3\times S_3$

Order 5832: $C_2\times C_3^4.S_3^2$ x 7, $C_3\wr C_3:C_6\times A_4$ x 2, $C_2^2\times (C_3^3.C_3^3):C_2$ x 2, $C_6^2.C_3^4.C_2$ x 2, $(C_9.C_3^3):C_2\times A_4$ x 2, $C_6^2.C_3^4.C_2$ x 2, $C_2\times C_3^4.S_3^2$ x 2, $C_2\times C_3^4.C_3.D_6$ x 2, $S_4\times (C_3\times C_9):C_3^2$ x 2, $(\He_3.C_3):C_6\times A_4$ x 2, $C_3^4:C_3.D_{12}$ x 2, $S_4\times \He_3.C_3^2$ x 2, $C_2^2\times (C_3^3.C_3^3):C_2$ x 2, $(C_6\times C_9:C_3).C_6^2$ x 2, $C_2^2\times (C_3^3.C_3^3):C_2$ x 2, $C_3^4:C_3.C_6.C_2^2$ x 2, $(C_6\times C_{18}).\He_3.C_2$, $C_6^2.C_3^3.C_6$, $(C_6\times C_9:C_3).C_6^2$, $C_6^2.C_3^3.C_6$, $C_6^2.\He_3.S_3$, $S_3\times (C_9.C_3^3):C_2^2$, $C_3^3.C_6^2.C_6$, $C_3^4.C_3^2.C_4.C_2$, $C_6^2.\He_3.S_3$, $C_3:D_9:C_3^2\times A_4$, $\He_3:(C_3\times S_3)\times A_4$, $S_4\times C_3\times C_3\wr C_3$, $C_4\times (C_3^3.C_3^3):C_2$, $(A_4\times \He_3).C_3.C_6$, $S_4\times C_3^4:C_3$, $C_4\times (C_3^3.C_3^3):C_2$, $C_6^2.C_3^3.C_6$, $S_3\times \He_3:C_3^2:C_2^2$, $(C_9\times A_4).C_3^3.C_2$, $(C_3\times A_4).C_3^4.C_2$, $(C_6\times C_{18}).\He_3.C_2$, $(C_3\times A_4).\He_3.C_6$, $C_3^4:C_3.C_{12}.C_2$, $C_6^2.\He_3:S_3$, $C_3^4:C_6\times A_4$, $C_6^2.C_3^4.C_2$, $C_6^2.\He_3:S_3$, $C_6^2.\He_3.S_3$, $(A_4\times \He_3).C_3.C_6$, $(C_3\times A_4).\He_3.C_6$, $C_4\times (C_3^3.C_3^3):C_2$, $C_6^2.C_3^3.C_6$, $C_3^4.C_3.D_{12}$, $C_3^3.C_3^3.D_4$, $A_4.C_3^4.C_6$, $C_3^4.C_3.C_{12}.C_2$, $C_3^4:C_3.D_{12}$, $S_4\times \He_3:C_3^2$, $C_3.\He_3:S_3\times A_4$, $C_6^2.C_3^3.C_6$, $C_3^3:(C_3\times S_3)\times A_4$, $A_4.C_3^2:C_9.C_6$, $(C_3^4.C_3):C_2\times A_4$, $C_3^4.D_6:S_3$, $C_3^3.C_3^3.D_4$, $(C_3\times C_9):C_3^2:C_2\times A_4$, $D_4\times C_3^3.C_3^3$, $C_3^3.C_3^3.C_4.C_2$, $S_4\times C_3^4.C_3$, $C_6^2.\He_3.S_3$, $C_6^2.\He_3.S_3$, $C_3^4.C_6^2.C_2$, $C_3^4.C_3^2.D_4$, $C_3^2:(C_2\times C_9:C_3)\times A_4$, $(\He_3.C_6):C_3\times A_4$, $(C_3\times C_9):(C_3\times C_6)\times A_4$, $C_6^2.\He_3:S_3$, $(C_6\times C_{18}).\He_3.C_2$, $C_3\times C_3\wr C_3:C_2\times A_4$, $(\He_3.S_3):C_3\times A_4$, $C_6^2.C_3^4.C_2$, $C_6^2.C_3^4.C_2$, $S_4\times C_9.C_3^3$, $C_6^2.C_3^3.C_6$, $(C_3\times A_4).\He_3.C_6$, $C_6^2.\He_3:S_3$, $(C_6\times C_{18}).\He_3.C_2$, $(A_4\times \He_3).C_3.C_6$, $\He_3:C_3^2:C_2\times A_4$, $C_6^2.C_3^4.C_2$, $S_3\times (C_3\times C_9):C_3^2:C_2^2$, $C_3^3.S_3^3$, $C_3^3:S_3^3$

Order 4374: $C_3\times (C_3^3.C_3^3):C_2$, $C_3^5.C_3.S_3$, $S_3\times C_3^3.C_3^3$, $C_3\times (C_3^3.C_3^3):C_2$, $C_3\times (C_3^3.C_3^3):C_2$, $C_3^4.\He_3.C_2$, $C_3^3.C_3^3.C_6$

Order 3888: $C_3\times S_4\times \He_3:C_2$ x 6, $C_3\times S_4\times C_3^2:S_3$ x 5, $C_3\times C_6^2:S_3^2$ x 5, $C_3.C_3^3.(C_6\times D_4)$ x 4, $C_6^2.C_3^3.C_2^2$ x 3, $S_4\times S_3\times C_9:C_3$ x 3, $C_3\times C_3.S_3^2\times A_4$ x 3, $C_3^2\times S_4\times C_3:S_3$ x 3, $S_4\times (C_3.C_3^3):C_2$ x 3, $C_3\times S_4\times D_9:C_3$ x 3, $C_3^3:S_3\times S_4$ x 3, $C_2^2\times (C_3\times C_9):C_3^2:C_2^2$ x 2, $C_2^2\times (C_9.C_3^3):C_2^2$ x 2, $C_9.(C_3\times S_3\times D_{12})$ x 2, $S_4\times \He_3.C_6$ x 2, $D_4\times C_3\wr C_3:C_6$ x 2, $\He_3.C_3.(C_6\times D_4)$ x 2, $S_4\times C_3^2:S_3:C_3$ x 2, $S_4\times C_9:C_3:S_3$ x 2, $C_3\times C_6^2.S_3^2$ x 2, $D_4\times (C_9.C_3^3):C_2$ x 2, $\He_3.C_3.(C_6\times D_4)$ x 2, $C_2^2\times (C_3^4.C_3):C_2^2$ x 2, $D_4\times (\He_3.C_3):C_6$ x 2, $C_3^4.(S_3\times D_4)$ x 2, $C_3.C_3^3.(C_6\times D_4)$ x 2, $S_4\times (\He_3.C_3):C_2$ x 2, $S_4\times C_9:(C_3\times S_3)$ x 2, $S_3\times D_9:C_3\times A_4$ x 2, $S_4\times C_3:S_3:C_9$ x 2, $C_2^2\times C_3^4:C_3:C_2^2$ x 2, $\He_3:C_3.(C_6\times D_4)$ x 2, $C_3\times (C_3^2\times A_4).D_6$ x 2, $S_4\times C_3\times D_9:C_3$ x 2, $C_2^2\times \He_3:C_3^2:C_2^2$ x 2, $C_3^3.(S_3\times S_4)$ x 2, $(C_3\times C_6^2):S_3^2$ x 2, $C_6^2.(C_3\times S_3^2)$ x 2, $C_3^3.C_6^2.C_2^2$, $C_3^2:C_9.(C_6\times D_4)$, $S_4\times (C_3\times C_9):S_3$, $C_6^2.C_3^3.C_2^2$, $D_4\times (C_3\times C_9):C_3^2:C_2$, $(A_4\times \He_3).C_2\times S_3$, $S_4\times \He_3:S_3$, $(A_4\times \He_3).D_6$, $C_4\times C_3^4:C_3:C_2^2$, $\He_3.C_3.(C_6\times D_4)$, $C_6^2.C_3^3.C_2^2$, $C_3\times (C_3^2\times A_4).D_6$, $(C_3^3\times A_4).D_6$, $\He_3.(C_6\times A_4).C_2$, $C_6^2.C_3^3.C_2^2$, $D_4\times C_3^4:C_6$, $\He_3.C_3.(C_6\times D_4)$, $C_9.S_3^2\times A_4$, $C_3:D_9:S_3\times A_4$, $C_3^2:S_3\times S_3\times A_4$, $\He_3.C_3.(C_6\times D_4)$, $C_4\times (C_3\times C_9):C_3^2:C_2^2$, $\He_3.A_4.C_6.C_2$, $\He_3:C_3.(C_6\times D_4)$, $D_4\times (\He_3.C_6):C_3$, $C_3:S_4\times D_9:C_3$, $(C_9\times A_4).(C_6\times S_3)$, $(A_4\times \He_3).D_6$, $D_4\times \He_3:(C_3\times S_3)$, $C_3^3.(C_6\times D_{12})$, $C_3\times (C_3^2\times A_4).D_6$, $C_6^2.C_3^3.C_2^2$, $S_4\times S_3\times \He_3$, $D_4\times C_3\times C_3\wr C_3:C_2$, $(C_3^3\times A_4).D_6$, $C_3\times C_3^3:C_2^2\times A_4$, $C_3:S_4\times C_3^2:S_3$, $(C_3\times C_9\times A_4).D_6$, $S_4\times C_3:D_9:C_3$, $C_3^4.(S_3\times D_4)$, $D_4\times C_3.\He_3:S_3$, $\He_3:C_3.(C_6\times D_4)$, $(A_4\times \He_3).D_6$, $S_4\times \He_3.S_3$, $D_4\times (\He_3.S_3):C_3$, $C_4\times (C_3^4.C_3):C_2^2$, $D_4\times \He_3:C_3^2:C_2$, $(C_9\times A_4).S_3^2$, $D_4\times (C_3^4.C_3):C_2$, $\He_3.C_6^2.C_2^2$, $\He_3.C_3.(C_6\times D_4)$, $C_3^3.C_6^2.C_2^2$, $C_3^2:C_9.(C_6\times D_4)$, $D_4\times C_3:D_9:C_3^2$, $\He_3.(C_6\times D_{12})$, $(C_9\times A_4).(C_6\times S_3)$, $(C_3\times C_9\times A_4).C_6.C_2$, $C_3^2.(A_4\times S_3^2)$, $D_4\times C_3^3:(C_3\times S_3)$, $C_3^4.D_{12}.C_2$, $(C_3\times C_9\times A_4).D_6$, $\He_3:C_3.(C_6\times D_4)$, $\He_3.C_6^2.C_2^2$, $S_4\times \He_3:C_3:C_2$, $C_3^2:C_9.(C_6\times D_4)$, $(C_3^3\times A_4).D_6$, $\He_3:C_3.(C_6\times D_4)$, $\He_3.C_6^2.C_2^2$, $C_3:S_3:D_9\times A_4$, $C_3^2:C_9.(C_6\times D_4)$, $S_4\times (C_3\times C_9):C_6$, $C_3^4.D_{12}.C_2$, $C_6^2.C_3^3.C_2^2$, $(C_3\times C_9\times A_4).D_6$, $C_6^2.C_3^3.C_2^2$, $C_9.(A_4\times S_3^2)$, $C_4\times (C_9.C_3^3):C_2^2$, $C_6^2.C_3^3.C_2^2$, $(C_9\times A_4).C_6\times S_3$, $C_3^4:C_3.D_4.C_2$, $C_3^4.D_6.C_2^2$, $D_4\times C_3^2:(C_2\times C_9:C_3)$, $C_3\times C_6^2.S_3^2$, $D_4\times (C_3\times C_9):(C_3\times C_6)$, $\He_3.C_3.A_4.C_2^2$, $(\He_3.S_3):C_2\times A_4$, $S_4\times C_3\times \He_3:C_2$, $C_3.C_3^3.(C_6\times D_4)$, $(C_3^3\times A_4).D_6$, $C_4\times \He_3:C_3^2:C_2^2$, $(A_4\times \He_3).D_6$, $(A_4\times \He_3).D_6$, $C_3^4:(C_2\times S_4)$, $C_3^3.(C_6\times S_4)$

Order 2916: $C_3^4.S_3^2$ x 15, $C_2\times (C_3^3.C_3^3):C_2$ x 5, $C_2\times (C_3^3.C_3^3):C_2$ x 5, $C_2\times (C_3^3.C_3^3):C_2$ x 5, $A_4\times \He_3.C_3^2$ x 2, $C_2^2\times C_3^3.C_3^3$ x 2, $S_3\times (\He_3.C_3):C_6$ x 2, $S_3\times C_3\wr C_3:C_6$ x 2, $C_3.C_3^5.C_2^2$ x 2, $A_4\times (C_3\times C_9):C_3^2$ x 2, $S_3\times (C_9.C_3^3):C_2$ x 2, $C_3^4.(C_6\times S_3)$, $S_3\times \He_3:(C_3\times S_3)$, $C_6^2.C_3^3.C_3$, $S_3\times (\He_3.S_3):C_3$, $S_3\times (C_3\times C_9):C_3^2:C_2$, $C_6^2.\He_3.C_3$, $C_4\times C_3^3.C_3^3$, $\He_3.C_3^3.C_2^2$, $C_3^4.C_3^2.C_4$, $\He_3.C_3^3.C_2^2$, $C_3^3.C_3^3.C_2^2$, $C_3\times C_3^4:C_3:C_2^2$, $C_3^4.(C_6\times S_3)$, $C_6^2.C_3^3.C_3$, $C_6^2.\He_3.C_3$, $C_9.C_3^3.D_6$, $A_4\times C_3^4.C_3$, $A_4\times \He_3:C_3^2$, $C_6^2.\He_3.C_3$, $C_3^5.D_6$, $S_3\times C_3.\He_3:S_3$, $C_3.C_3^5.C_2^2$, $S_3\times (\He_3.C_6):C_3$, $C_3^2.C_3^4.C_2^2$, $C_3^4.S_3^2$, $A_4\times C_3\times C_3\wr C_3$, $C_6^2.\He_3.C_3$, $C_6^2.\He_3.C_3$, $S_3\times (C_3\times C_9):(C_3\times C_6)$, $C_3^3.C_3^3.C_2^2$, $S_3\times \He_3:C_3^2:C_2$, $S_3\times C_3^4:C_6$, $\He_3.C_3^3.C_2^2$, $(C_2\times C_3^4.C_3).C_6$, $S_3\times C_3:D_9:C_3^2$, $A_4\times C_3^4:C_3$, $A_4\times C_9.C_3^3$, $C_6^2.C_3^3.C_3$, $C_6^2.C_3^3.C_3$, $C_3^4:C_3.C_{12}$, $C_3^4.S_3^2$, $C_3^4.S_3^2$, $C_3^4.S_3^2$, $C_3^5.D_6$, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^5:D_6$

Order 2592: $C_3\times S_4\times S_3^2$ x 5, $C_3\times D_4\times C_3.S_3^2$ x 3, $C_3^2:D_6\times S_4$ x 3, $S_4\times S_3\times D_9$ x 2, $D_4\times S_3\times D_9:C_3$ x 2, $C_2\times S_4\times C_3^2:S_3$, $D_4\times C_3^2:S_3\times S_3$, $D_4\times C_9.S_3^2$, $D_4\times (\He_3.S_3):C_2$, $D_4\times C_3:D_9:S_3$, $D_4\times C_3:S_3:D_9$, $C_3\times D_4\times C_3^3:C_2^2$, $C_3\times S_4\times S_3^2$, $C_{18}:C_6\times S_4$, $S_4\times C_3:S_3^2$

Order 2187: $C_3\times C_3^3.C_3^3$

Order 1944: $C_2\times C_3^3.S_3^2$ x 11, $C_2\times C_3^3:S_3^2$ x 11, $C_2\times C_3^3.S_3^2$ x 11, $C_2\times C_3^3.S_3^2$ x 11, $C_2\times C_3^3.S_3^2$ x 11, $A_4\times C_3^3:C_6$ x 7, $C_3\times \He_3:S_4$ x 7, $C_9:C_3^2\times S_4$ x 6, $C_3\times S_4\times \He_3$ x 6, $C_3\wr C_3\times S_4$ x 6, $A_4\times C_3^3:S_3$ x 5, $A_4\times D_9:C_3^2$ x 5, $C_3\times \He_3:S_4$ x 5, $C_9:(C_3^2\times S_4)$ x 5, $(A_4\times C_3^3):S_3$ x 5, $(C_3\times C_{18}):C_6^2$ x 4, $C_9.C_3^2\times S_4$ x 4, $(C_3\times C_{18}).C_6^2$ x 4, $C_{18}.(C_3\times S_3^2)$ x 4, $\He_3.(C_6\times D_6)$ x 4, $C_3^3:(C_6\times D_6)$ x 4, $A_4\times C_3^3:S_3$ x 4, $C_9:C_3^2:S_4$ x 4, $S_4\times C_3.\He_3$ x 4, $(C_6\times \He_3).D_6$ x 4, $C_3^3:C_6\wr C_2$ x 4, $C_3^2\wr C_2\times A_4$ x 3, $C_3^4:S_4$ x 3, $C_6^2.(S_3\times C_3^2)$ x 3, $A_4\times \He_3.S_3$ x 3, $(D_6\times C_{18}):C_3^2$ x 3, $(A_4\times \He_3).S_3$ x 3, $C_9.(C_3^2\times S_4)$ x 3, $C_3^3.(C_3\times S_4)$ x 3, $C_3^3.(C_3\times S_4)$ x 3, $C_3^2.S_3^3$ x 3, $C_3^3:(S_3\times A_4)$ x 3, $C_3\wr C_3:S_4$ x 3, $C_9:C_3^2:S_4$ x 3, $C_3^3:(C_3\times S_4)$ x 3, $C_2\times \He_3.C_6^2$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $A_4\times \He_3.C_6$ x 2, $C_3^3.(S_3\times A_4)$ x 2, $A_4\times C_3^3.S_3$ x 2, $\He_3.(C_3\times S_4)$ x 2, $(A_4\times C_3^3).S_3$ x 2, $C_3^3.(C_3\times S_4)$ x 2, $C_3^2.S_3^3$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $C_{36}.(S_3\times C_3^2)$ x 2, $C_{36}.(S_3\times C_3^2)$ x 2, $C_{18}.(C_3\times S_3^2)$ x 2, $C_{18}.(C_3\times S_3^2)$ x 2, $C_2\times C_3^4:D_6$ x 2, $\He_3:(C_6\times D_6)$ x 2, $C_6\times C_3^3:D_6$ x 2, $C_2\times C_3^4.D_6$ x 2, $C_2\times C_3^4:D_6$ x 2, $(C_3\times C_{18}):C_6^2$ x 2, $C_2\times \He_3.C_6^2$ x 2, $(C_3\times C_{18}):C_6^2$ x 2, $C_2\times \He_3.C_6^2$ x 2, $(C_3\times C_{18}):C_6^2$ x 2, $C_2\times C_3^3.C_6^2$ x 2, $C_2^2\times C_3^4:C_6$ x 2, $\He_3.(S_3\times A_4)$ x 2, $A_4\times \He_3.S_3$ x 2, $C_3^3:(S_3\times A_4)$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $A_4\times \He_3.C_6$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $A_4\times \He_3.C_6$ x 2, $C_3^3.(C_6\times A_4)$ x 2, $A_4\times C_3^2:C_{18}$ x 2, $(A_4\times \He_3).S_3$ x 2, $(A_4\times C_3^3).S_3$ x 2, $C_9:C_3^2:S_4$ x 2, $(A_4\times C_3^3):S_3$ x 2, $(C_3\times \He_3):S_4$ x 2, $\He_3.(C_3\times S_4)$ x 2, $C_3^3:(C_3\times S_4)$ x 2, $\He_3:(C_3\times S_4)$ x 2, $\He_3.(C_3\times S_4)$ x 2, $\He_3.(C_3\times S_4)$ x 2, $C_3^2:(C_9\times S_4)$ x 2, $S_4\times \He_3:C_3$ x 2, $C_3^2:C_9\times S_4$ x 2, $D_4\times C_3^2.C_3^3$ x 2, $D_4\times C_3^3:C_3^2$ x 2, $(C_3^2\times C_6^2):S_3$ x 2, $\He_3:C_6\wr C_2$ x 2, $(C_3^2\times C_6^2):S_3$ x 2, $(C_3^2\times C_6^2).S_3$ x 2, $(C_3^2\times C_6^2):S_3$ x 2, $(C_{12}\times \He_3).S_3$ x 2, $\He_3.(S_3\times C_{12})$ x 2, $C_3^3:(C_3\times D_{12})$ x 2, $C_3^3:(S_3\times C_{12})$ x 2, $(C_3\times C_{18}).C_6^2$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $(C_3\times C_{18}).C_6^2$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $(C_3\times C_{18}).C_6^2$ x 2, $C_6^2.(S_3\times C_3^2)$ x 2, $(C_3^2\times C_6^2):C_6$ x 2, $(C_3\times \He_3):D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $(C_3^3\times C_6).D_6$ x 2, $(C_3^2\times C_6).S_3^2$ x 2, $\He_3.(C_3\times D_{12})$ x 2, $(C_3^2\times C_6).S_3^2$ x 2, $(C_3^2\times C_6).S_3^2$ x 2, $\He_3.(C_3\times D_{12})$ x 2, $(C_3^2\times C_6).S_3^2$ x 2, $(C_3^3\times C_6).D_6$ x 2, $C_3^4.D_{12}$ x 2, $(C_3^3\times C_6).D_6$ x 2, $S_4\times C_3^4$, $C_3^2:S_3^3$, $A_4\times C_3^3:S_3$, $C_3^3.(S_3\times A_4)$, $S_3\times A_4\times \He_3$, $(A_4\times C_3^3):S_3$, $C_3:S_4\times \He_3$, $C_9.S_3^3$, $C_3^2:S_3^3$, $D_4\times C_9.C_3^3$, $C_3^3.(C_3\times D_{12})$, $C_3^3.(S_3\times C_{12})$, $C_{18}.(C_3\times S_3^2)$, $C_3^3:(S_3\times A_4)$, $\He_3:(S_3\times A_4)$, $A_4\times \He_3:S_3$, $C_3^3.(S_3\times A_4)$, $A_4\times C_3^2:D_9$, $(C_3^2\times C_6^2):C_6$, $C_3^3.(C_6\times A_4)$, $C_3^3.(S_3\times A_4)$, $C_3^3.(S_3\times A_4)$, $C_3^3.(S_3\times A_4)$, $C_3^3.(S_3\times A_4)$, $C_3^3.(S_3\times A_4)$, $C_6^2.(S_3\times C_3^2)$, $A_4\times \He_3.S_3$, $C_6^2.(S_3\times C_3^2)$, $A_4\times \He_3.S_3$, $C_3^3.(S_3\times A_4)$, $A_4\times C_3^2:D_9$, $C_3^4:S_4$, $(A_4\times \He_3):S_3$, $(C_3^2\times A_4):D_9$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4.S_4$, $C_9:C_3^2:S_4$, $C_3\times \He_3:S_4$, $(A_4\times \He_3).S_3$, $(A_4\times \He_3).S_3$, $(C_3^2\times A_4):D_9$, $C_3^4.S_4$, $C_3^3.(C_3\times S_4)$, $C_3^3.(C_3\times S_4)$, $C_3^3.(C_3\times S_4)$, $C_3^3:(C_3\times S_4)$, $C_3^3.(C_3\times S_4)$, $C_3^3.(C_3\times S_4)$, $C_3^2.S_3^3$, $C_3^2.S_3^3$, $C_3^2.S_3^3$, $D_4\times \He_3:C_3^2$, $C_3^4:C_3\times D_4$, $C_3^4.C_3\times D_4$, $C_3^4:C_3\times D_4$, $C_3^4:D_{12}$, $C_4\times C_3^4:S_3$, $(C_{12}\times \He_3):S_3$, $\He_3:(S_3\times C_{12})$, $C_3^4:D_{12}$, $C_3^3:S_3\times C_{12}$, $C_3^4.D_{12}$, $C_4\times C_3^4.S_3$, $C_3^4:D_{12}$, $C_4\times C_3^4:S_3$, $(C_3\times C_{18}).C_6^2$, $(C_3\times C_{18}).C_6^2$, $C_3^3.(C_3\times D_{12})$, $C_3^3.(S_3\times C_{12})$, $(C_3\times C_{18}).C_6^2$, $(C_3\times C_{18}).C_6^2$, $C_3^3.(C_3\times D_{12})$, $C_3^3.(S_3\times C_{12})$, $C_3^4.D_{12}$, $C_4\times C_3^4.S_3$, $C_3^3.(C_3\times D_{12})$, $C_4\times C_3^4.C_6$, $C_3^3:(C_3\times D_{12})$, $C_4\times C_3^4:C_6$, $C_3^4:(C_4\times S_3)$, $C_3^4:(C_4\times S_3)$, $(C_3^3\times C_6).D_6$, $\He_3.(S_3\times C_{12})$, $(C_3^2\times C_6).S_3^2$, $\He_3.(S_3\times C_{12})$, $\He_3.(S_3\times C_{12})$, $(C_3^2\times C_6).S_3^2$, $\He_3.(S_3\times C_{12})$, $C_3^3.(S_3\times C_{12})$, $(C_3^3\times C_6).D_6$, $C_3^3.(S_3\times C_{12})$

Order 1458: $C_3^4:C_3:S_3$, $C_3^4.(C_3\times S_3)$, $C_3^4.(C_3\times S_3)$

Order 1296: $C_6^2:S_3^2$ x 7, $C_6^2.S_3^2$

Order 972: $C_6^2:C_3^3$ x 3, $A_4\times C_3^4$, $C_3^3.C_6^2$, $C_6^2.C_3^3$, $\He_3:C_6^2$, $C_3^3.C_6^2$, $C_3^3.C_6^2$, $C_3^3:C_6^2$

Order 729: $C_3^3.C_3^3$

Order 648: $S_4\times C_3^3$ x 2, $C_{18}:(C_6\times S_3)$, $S_3\times C_3^2:D_6$, $A_4\times C_3:D_9$, $(C_3\times C_9):S_4$, $C_3\times C_9\times S_4$

Order 486: $C_9.(S_3\times C_3^2)$, $C_3^4:S_3$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^4.S_3$

Order 324: $C_3^2:S_3^2$ x 7, $C_3^2:C_6^2$ x 3, $A_4\times C_3^3$ x 2, $C_3^2\times C_6^2$, $C_9:C_6^2$, $C_6^2.C_3^2$

Order 288: $D_6\times S_4$ x 4

Order 243: $C_9.C_3^3$, $\He_3:C_3^2$, $C_3^2.C_3^3$, $C_3^4.C_3$, $C_3^4:C_3$

Order 216: $C_6^2:S_3$ x 4, $C_3^2\times S_4$ x 2, $C_6:D_{18}$

Order 162: $C_3^3.S_3$

Order 144: $S_3\times S_4$ x 24

Order 108: $C_3^2:D_6$ x 23, $C_3\times C_6^2$ x 2, $C_3^2\times A_4$ x 2, $C_6\times C_{18}$

Order 96: $C_2^2\times S_4$

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

Order 72: $S_3\times D_6$ x 4, $C_3\times S_4$

Order 54: $C_3^2:S_3$ x 28, $C_3:D_9$

Order 48: $C_2\times S_4$ x 12

Order 36: $S_3^2$ x 24, $C_6^2$ x 2, $C_3\times A_4$

Order 32: $C_2^2\times D_4$

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

Order 24: $S_4$ x 19, $C_2\times D_6$ x 17

Order 12: $D_6$ x 78, $C_2\times C_6$, $A_4$

Order 9: $C_3^2$ x 2

Order 8: $C_2^3$ x 4

Order 6: $S_3$ x 52

Order 4: $C_2^2$ x 18

Order 3: $C_3$

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 69984: $S_4\times C_3^4.S_3^2$ ($C_1$)

Order 34992: $C_6^2.C_3^3.C_3.D_6$ ($C_2$), $C_6^2.C_3^4.D_6$ ($C_2$), $C_6^2.C_3^4.D_6$ ($C_2$), $S_4\times (C_3^3.C_3^3):C_2$ ($C_2$), $C_3^4:C_3.D_6\times A_4$ ($C_2$), $S_4\times (C_3^3.C_3^3):C_2$ ($C_2$), $S_4\times (C_3^3.C_3^3):C_2$ ($C_2$)

Order 17496: $(C_3^3.C_3^3):C_2\times A_4$ ($C_2^2$), $S_4\times C_3^3.C_3^3$ ($C_2^2$), $(C_3\times A_4).C_3^4:C_6$ ($C_2^2$), $C_6^2.C_3^4.C_6$ ($C_2^2$), $(C_3^3.C_3^3):C_2\times A_4$ ($C_2^2$), $C_6^2.C_3^4.S_3$ ($C_2^2$), $(C_3^3.C_3^3):C_2\times A_4$ ($C_2^2$)

Order 11664: $S_4\times (C_3\times C_9):(C_3\times C_6)$ ($S_3$), $S_4\times C_3:D_9:C_3^2$ ($S_3$), $S_4\times C_3.\He_3:S_3$ ($S_3$), $S_4\times (C_9.C_3^3):C_2$ ($S_3$), $S_4\times (\He_3.S_3):C_3$ ($S_3$), $C_3^4.D_6^2$ ($S_3$)

Order 8748: $A_4\times C_3^3.C_3^3$ ($C_2^3$)

Order 5832: $C_2^2\times (C_3^3.C_3^3):C_2$ ($D_6$), $C_3:D_9:C_3^2\times A_4$ ($D_6$), $(C_9.C_3^3):C_2\times A_4$ ($D_6$), $S_4\times C_3^4:C_3$ ($D_6$), $(C_9\times A_4).C_3^3.C_2$ ($D_6$), $(C_3\times A_4).\He_3.C_6$ ($D_6$), $S_4\times (C_3\times C_9):C_3^2$ ($D_6$), $(C_3\times A_4).\He_3.C_6$ ($D_6$), $S_4\times \He_3:C_3^2$ ($D_6$), $C_3.\He_3:S_3\times A_4$ ($D_6$), $C_2^2\times (C_3^3.C_3^3):C_2$ ($D_6$), $S_4\times C_3^4.C_3$ ($D_6$), $(C_3\times C_9):(C_3\times C_6)\times A_4$ ($D_6$), $(\He_3.S_3):C_3\times A_4$ ($D_6$), $C_6^2.C_3^4.C_2$ ($D_6$), $C_2^2\times (C_3^3.C_3^3):C_2$ ($D_6$), $S_4\times C_9.C_3^3$ ($D_6$), $(C_3\times A_4).\He_3.C_6$ ($D_6$)

Order 3888: $S_4\times C_9:(C_3\times S_3)$ ($C_3:S_3$)

Order 2916: $C_2^2\times C_3^3.C_3^3$ ($C_2\times D_6$), $A_4\times C_3^4.C_3$ ($C_2\times D_6$), $A_4\times \He_3:C_3^2$ ($C_2\times D_6$), $A_4\times (C_3\times C_9):C_3^2$ ($C_2\times D_6$), $A_4\times C_3^4:C_3$ ($C_2\times D_6$), $A_4\times C_9.C_3^3$ ($C_2\times D_6$), $C_3^4.S_3^2$ ($S_4$)

Order 1944: $C_3\times S_4\times \He_3$ ($S_3^2$) x 3, $S_4\times C_3^4$ ($S_3^2$), $(C_3\times C_{18}):C_6^2$ ($S_3^2$), $A_4\times C_3^3.S_3$ ($C_6:S_3$), $(A_4\times C_3^3).S_3$ ($C_6:S_3$), $C_9:C_3^2\times S_4$ ($C_6:S_3$), $C_2\times C_3^4:D_6$ ($S_3^2$), $(C_3\times C_{18}):C_6^2$ ($S_3^2$), $(C_3\times C_{18}):C_6^2$ ($S_3^2$), $(C_3\times C_{18}):C_6^2$ ($S_3^2$)

Order 1458: $C_3^4:C_3:S_3$ ($C_2\times S_4$), $C_3^4.(C_3\times S_3)$ ($C_2\times S_4$), $C_3^4.(C_3\times S_3)$ ($C_2\times S_4$)

Order 1296: $C_6^2.S_3^2$ ($C_3^2:S_3$)

Order 972: $C_6^2:C_3^3$ ($S_3\times D_6$) x 3, $A_4\times C_3^4$ ($S_3\times D_6$), $C_3^3.C_6^2$ ($S_3\times D_6$), $C_6^2.C_3^3$ ($C_6:D_6$), $\He_3:C_6^2$ ($S_3\times D_6$), $C_3^3.C_6^2$ ($S_3\times D_6$), $C_3^3.C_6^2$ ($S_3\times D_6$), $C_3^3:C_6^2$ ($S_3\times D_6$)

Order 729: $C_3^3.C_3^3$ ($C_2^2\times S_4$)

Order 648: $S_4\times C_3^3$ ($C_3:S_3^2$), $S_4\times C_3^3$ ($C_3^2:D_6$), $C_{18}:(C_6\times S_3)$ ($C_3:S_3^2$), $A_4\times C_3:D_9$ ($C_3^2:D_6$), $(C_3\times C_9):S_4$ ($C_3^2:D_6$), $C_3\times C_9\times S_4$ ($C_3^2:D_6$)

Order 486: $C_9.(S_3\times C_3^2)$ ($S_3\times S_4$), $C_3^4:S_3$ ($S_3\times S_4$), $C_3^3.(C_3\times S_3)$ ($S_3\times S_4$), $C_3^3.(C_3\times S_3)$ ($S_3\times S_4$), $C_3^4.S_3$ ($S_3\times S_4$)

Order 324: $C_3^2:C_6^2$ ($S_3^3$) x 3, $C_3^2\times C_6^2$ ($S_3^3$), $A_4\times C_3^3$ ($C_6:S_3^2$), $A_4\times C_3^3$ ($C_6.S_3^2$), $C_9:C_6^2$ ($C_6:S_3^2$), $C_6^2.C_3^2$ ($C_6^2:S_3$)

Order 243: $C_9.C_3^3$ ($D_6\times S_4$), $\He_3:C_3^2$ ($D_6\times S_4$), $C_3^2.C_3^3$ ($D_6\times S_4$), $C_3^4.C_3$ ($D_6\times S_4$), $C_3^4:C_3$ ($D_6\times S_4$)

Order 216: $C_3^2\times S_4$ ($C_3^2:S_3^2$), $C_3^2\times S_4$ ($C_3^2:S_3^2$), $C_6:D_{18}$ ($C_3^2:S_3^2$)

Order 162: $C_3^3.S_3$ ($C_6^2:D_6$)

Order 108: $C_3\times C_6^2$ ($C_3:S_3^3$), $C_3\times C_6^2$ ($C_3.S_3^3$), $C_3^2\times A_4$ ($S_3\times C_3^2:D_6$), $C_3^2\times A_4$ ($C_2\times C_3^2:S_3^2$), $C_6\times C_{18}$ ($S_3\times C_3^2:D_6$)

Order 81: $C_3\times \He_3$ ($S_4\times S_3^2$) x 3, $C_3^4$ ($S_4\times S_3^2$), $C_9:C_3^2$ ($C_6:S_3\times S_4$)

Order 72: $C_3\times S_4$ ($C_3^3:S_3^2$)

Order 54: $C_3:D_9$ ($C_6^2:S_3^2$)

Order 36: $C_6^2$ ($C_3^2:S_3^3$), $C_6^2$ ($C_3^2:S_3^3$), $C_3\times A_4$ ($C_2\times C_3^3:S_3^2$)

Order 27: $C_3^3$ ($C_3:S_3^2\times S_4$), $C_3^3$ ($C_3^2:D_6\times S_4$), $C_3\times C_9$ ($C_2\times S_4\times C_3^2:S_3$)

Order 24: $S_4$ ($C_3^4.S_3^2$)

Order 12: $C_2\times C_6$ ($C_3^3:S_3^3$), $A_4$ ($C_2\times C_9.C_3^3.D_6$)

Order 9: $C_3^2$ ($S_4\times \He_3:S_3:C_2$), $C_3^2$ ($S_4\times C_3^2:S_3\times S_3$)

Order 4: $C_2^2$ ($S_3\times C_3^4.C_3.D_6$)

Order 3: $C_3$ ($S_4\times C_3^4:C_3:C_2^2$)

Order 1: $C_1$ ($S_4\times C_3^4.S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 69984: $S_4\times C_3^4.S_3^2$ ($C_1$)

Order 34992: $C_6^2.C_3^3.C_3.D_6$ ($C_2$), $C_6^2.C_3^4.D_6$ ($C_2$), $C_6^2.C_3^4.D_6$ ($C_2$), $S_4\times (C_3^3.C_3^3):C_2$ ($C_2$), $C_3^4:C_3.D_6\times A_4$ ($C_2$), $S_4\times (C_3^3.C_3^3):C_2$ ($C_2$), $S_4\times (C_3^3.C_3^3):C_2$ ($C_2$)

Order 17496: $(C_3^3.C_3^3):C_2\times A_4$ ($C_2^2$), $S_4\times C_3^3.C_3^3$ ($C_2^2$), $(C_3\times A_4).C_3^4:C_6$ ($C_2^2$), $C_6^2.C_3^4.C_6$ ($C_2^2$), $(C_3^3.C_3^3):C_2\times A_4$ ($C_2^2$), $C_6^2.C_3^4.S_3$ ($C_2^2$), $(C_3^3.C_3^3):C_2\times A_4$ ($C_2^2$)

Order 11664: $S_4\times (C_3\times C_9):(C_3\times C_6)$ ($S_3$), $S_4\times C_3:D_9:C_3^2$ ($S_3$), $S_4\times C_3.\He_3:S_3$ ($S_3$), $S_4\times (C_9.C_3^3):C_2$ ($S_3$), $S_4\times (\He_3.S_3):C_3$ ($S_3$), $C_3^4.D_6^2$ ($S_3$)

Order 8748: $A_4\times C_3^3.C_3^3$ ($C_2^3$)

Order 5832: $C_2^2\times (C_3^3.C_3^3):C_2$ ($D_6$), $C_3:D_9:C_3^2\times A_4$ ($D_6$), $(C_9.C_3^3):C_2\times A_4$ ($D_6$), $S_4\times C_3^4:C_3$ ($D_6$), $(C_9\times A_4).C_3^3.C_2$ ($D_6$), $(C_3\times A_4).\He_3.C_6$ ($D_6$), $S_4\times (C_3\times C_9):C_3^2$ ($D_6$), $(C_3\times A_4).\He_3.C_6$ ($D_6$), $S_4\times \He_3:C_3^2$ ($D_6$), $C_3.\He_3:S_3\times A_4$ ($D_6$), $C_2^2\times (C_3^3.C_3^3):C_2$ ($D_6$), $S_4\times C_3^4.C_3$ ($D_6$), $(C_3\times C_9):(C_3\times C_6)\times A_4$ ($D_6$), $(\He_3.S_3):C_3\times A_4$ ($D_6$), $C_6^2.C_3^4.C_2$ ($D_6$), $C_2^2\times (C_3^3.C_3^3):C_2$ ($D_6$), $S_4\times C_9.C_3^3$ ($D_6$), $(C_3\times A_4).\He_3.C_6$ ($D_6$)

Order 3888: $S_4\times C_9:(C_3\times S_3)$ ($C_3:S_3$)

Order 2916: $C_2^2\times C_3^3.C_3^3$ ($C_2\times D_6$), $A_4\times C_3^4.C_3$ ($C_2\times D_6$), $A_4\times \He_3:C_3^2$ ($C_2\times D_6$), $A_4\times (C_3\times C_9):C_3^2$ ($C_2\times D_6$), $A_4\times C_3^4:C_3$ ($C_2\times D_6$), $A_4\times C_9.C_3^3$ ($C_2\times D_6$), $C_3^4.S_3^2$ ($S_4$)

Order 1944: $C_3\times S_4\times \He_3$ ($S_3^2$) x 3, $S_4\times C_3^4$ ($S_3^2$), $(C_3\times C_{18}):C_6^2$ ($S_3^2$), $A_4\times C_3^3.S_3$ ($C_6:S_3$), $(A_4\times C_3^3).S_3$ ($C_6:S_3$), $C_9:C_3^2\times S_4$ ($C_6:S_3$), $C_2\times C_3^4:D_6$ ($S_3^2$), $(C_3\times C_{18}):C_6^2$ ($S_3^2$), $(C_3\times C_{18}):C_6^2$ ($S_3^2$), $(C_3\times C_{18}):C_6^2$ ($S_3^2$)

Order 1458: $C_3^4:C_3:S_3$ ($C_2\times S_4$), $C_3^4.(C_3\times S_3)$ ($C_2\times S_4$), $C_3^4.(C_3\times S_3)$ ($C_2\times S_4$)

Order 1296: $C_6^2.S_3^2$ ($C_3^2:S_3$)

Order 972: $C_6^2:C_3^3$ ($S_3\times D_6$) x 3, $A_4\times C_3^4$ ($S_3\times D_6$), $C_3^3.C_6^2$ ($S_3\times D_6$), $C_6^2.C_3^3$ ($C_6:D_6$), $\He_3:C_6^2$ ($S_3\times D_6$), $C_3^3.C_6^2$ ($S_3\times D_6$), $C_3^3.C_6^2$ ($S_3\times D_6$), $C_3^3:C_6^2$ ($S_3\times D_6$)

Order 729: $C_3^3.C_3^3$ ($C_2^2\times S_4$)

Order 648: $S_4\times C_3^3$ ($C_3:S_3^2$), $S_4\times C_3^3$ ($C_3^2:D_6$), $C_{18}:(C_6\times S_3)$ ($C_3:S_3^2$), $A_4\times C_3:D_9$ ($C_3^2:D_6$), $(C_3\times C_9):S_4$ ($C_3^2:D_6$), $C_3\times C_9\times S_4$ ($C_3^2:D_6$)

Order 486: $C_9.(S_3\times C_3^2)$ ($S_3\times S_4$), $C_3^4:S_3$ ($S_3\times S_4$), $C_3^3.(C_3\times S_3)$ ($S_3\times S_4$), $C_3^3.(C_3\times S_3)$ ($S_3\times S_4$), $C_3^4.S_3$ ($S_3\times S_4$)

Order 324: $C_3^2:C_6^2$ ($S_3^3$) x 3, $C_3^2\times C_6^2$ ($S_3^3$), $A_4\times C_3^3$ ($C_6:S_3^2$), $A_4\times C_3^3$ ($C_6.S_3^2$), $C_9:C_6^2$ ($C_6:S_3^2$), $C_6^2.C_3^2$ ($C_6^2:S_3$)

Order 243: $C_9.C_3^3$ ($D_6\times S_4$), $\He_3:C_3^2$ ($D_6\times S_4$), $C_3^2.C_3^3$ ($D_6\times S_4$), $C_3^4.C_3$ ($D_6\times S_4$), $C_3^4:C_3$ ($D_6\times S_4$)

Order 216: $C_3^2\times S_4$ ($C_3^2:S_3^2$), $C_3^2\times S_4$ ($C_3^2:S_3^2$), $C_6:D_{18}$ ($C_3^2:S_3^2$)

Order 162: $C_3^3.S_3$ ($C_6^2:D_6$)

Order 108: $C_3\times C_6^2$ ($C_3:S_3^3$), $C_3\times C_6^2$ ($C_3.S_3^3$), $C_3^2\times A_4$ ($S_3\times C_3^2:D_6$), $C_3^2\times A_4$ ($C_2\times C_3^2:S_3^2$), $C_6\times C_{18}$ ($S_3\times C_3^2:D_6$)

Order 81: $C_3\times \He_3$ ($S_4\times S_3^2$) x 3, $C_3^4$ ($S_4\times S_3^2$), $C_9:C_3^2$ ($C_6:S_3\times S_4$)

Order 72: $C_3\times S_4$ ($C_3^3:S_3^2$)

Order 54: $C_3:D_9$ ($C_6^2:S_3^2$)

Order 36: $C_6^2$ ($C_3^2:S_3^3$), $C_6^2$ ($C_3^2:S_3^3$), $C_3\times A_4$ ($C_2\times C_3^3:S_3^2$)

Order 27: $C_3^3$ ($C_3:S_3^2\times S_4$), $C_3^3$ ($C_3^2:D_6\times S_4$), $C_3\times C_9$ ($C_2\times S_4\times C_3^2:S_3$)

Order 24: $S_4$ ($C_3^4.S_3^2$)

Order 12: $C_2\times C_6$ ($C_3^3:S_3^3$), $A_4$ ($C_2\times C_9.C_3^3.D_6$)

Order 9: $C_3^2$ ($S_4\times \He_3:S_3:C_2$), $C_3^2$ ($S_4\times C_3^2:S_3\times S_3$)

Order 4: $C_2^2$ ($S_3\times C_3^4.C_3.D_6$)

Order 3: $C_3$ ($S_4\times C_3^4:C_3:C_2^2$)

Order 1: $C_1$ ($S_4\times C_3^4.S_3^2$)

Series

Derived series

$S_4\times C_3^4.S_3^2$

$\rhd$

$A_4\times C_3^3.C_3^3$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_1$

Chief series

$S_4\times C_3^4.S_3^2$

$\rhd$

$S_4\times (C_3^3.C_3^3):C_2$

$\rhd$

$S_4\times C_3^3.C_3^3$

$\rhd$

$A_4\times C_3^3.C_3^3$

$\rhd$

$A_4\times C_3^4.C_3$

$\rhd$

$A_4\times C_9.C_3^3$

$\rhd$

$C_2^2\times C_3^3.C_3^3$

$\rhd$

$C_6^2:C_3^3$

$\rhd$

$C_6^2.C_3^3$

$\rhd$

$C_3^3.C_6^2$

$\rhd$

$A_4\times C_3^3$

$\rhd$

$C_3^2\times C_6^2$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$S_4\times C_3^4.S_3^2$

$\rhd$

$A_4\times C_3^3.C_3^3$

Upper central series

$C_1$

Character theory

Complex character table

See the $210 \times 210$ 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 $165 \times 165$ rational character table (warning: may be slow to load).