Abstract group 93312.hk: $C_6^3.C_6^2:D_6$

• Label: 93312.hk
• Order: \( 2^{7} \cdot 3^{6} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_6^3.C_6^2:D_6$
Order:	\(93312\)\(\medspace = 2^{7} \cdot 3^{6} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^3.C_2^4:\He_3.C_2^6$, of order \(746496\)\(\medspace = 2^{10} \cdot 3^{6} \)
Composition factors:	$C_2$ x 7, $C_3$ x 6
Derived length:	$4$

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	871	2834	6552	15086	5184	47232	5184	10368	93312
Conjugacy classes	1	9	15	14	117	4	66	4	4	234
Divisions	1	9	15	9	99	4	29	4	2	172
Autjugacy classes	1	8	15	7	96	2	22	2	1	154

Dimension	1	2	3	4	6	8	12	16	24	48	96
Irr. complex chars.	8	8	8	2	52	12	66	6	50	22	0	234
Irr. rational chars.	4	8	4	3	20	6	39	9	46	31	2	172

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid c^{6}=d^{6}=e^{6}=f^{6}=g^{6}=[a,f]=[a,g]= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,2,4)(3,6,7,9)(5,8)(14,15,17,19)(20,21)(23,24,26,27,29,30)(25,28), (1,3) \!\cdots\! \rangle$
Transitive group:	36T19651				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^5$ $\,\rtimes\,$ $(C_2^5.D_6)$	$(C_3:D_6^2)$ $\,\rtimes\,$ $(C_6.S_3^2)$	$((C_3^2\times C_6^3).S_4)$ $\,\rtimes\,$ $C_2$	$(C_6^2:C_4)$ $\,\rtimes\,$ $(C_3^3:S_4)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^3$ . $(C_6^2:D_6)$	$C_3^4$ . $(C_2^5.S_3^2)$	$(C_3\times C_6^3)$ . $(S_3\times S_4)$	$(C_3\times C_6)$ . $(C_6^3:S_4)$	all 27

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

Homology

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

Subgroups

There are 1205680 subgroups in 11668 conjugacy classes, 46 normal (42 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^3.C_2^4:\He_3.C_2^2$
Commutator:	$G' \simeq$ $C_3^3.C_2^4:\He_3$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2\times C_6$	$G/\Phi \simeq$ $S_3\times C_3^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_6^3$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $C_6^3.C_6^2:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^3$	$G/\operatorname{soc} \simeq$ $S_3\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5:C_3$

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 128 (order at least 729) are shown, as well as all normal subgroups of any index.

Order 93312: $C_6^3.C_6^2:D_6$

Order 46656: $C_3^5.C_2^5.C_6$, $C_2\times C_3^3.C_2^4:\He_3.C_2$, $(C_3^2\times C_6^3).S_4$

Order 31104: $(C_3\times C_6^2).A_4.C_{12}.C_2$, $C_3^5.C_2^4.C_2^3$

Order 23328: $C_3^3.C_2^4:\He_3.C_2$ x 2, $C_2\times C_3^3.C_2^4:\He_3$, $C_3^5:(C_4\times S_4)$

Order 15552: $C_3\times C_6^3.S_4$ x 2, $C_3^5.C_2^3.C_2^3$, $C_3^5.(C_4\times D_4).C_2$, $C_3^5.(C_2^5.C_2)$, $C_3\times C_3^4.C_2^3.D_4$, $C_6\times C_3^2.D_6\wr C_2$, $C_3^5.C_2^4.C_2^2$, $C_2\times (C_3\times C_6^2).A_4.S_3$, $C_3\times C_3^4.C_2^3.C_2^3$, $C_6^3.(S_3\times A_4)$

Order 11664: $C_3^3.C_2^4:\He_3$, $C_2\times C_6^2.C_3^3.C_6$, $(C_3^4\times C_6).S_4$, $C_2\times C_3^5:S_4$, $(C_3^4\times C_6).S_4$, $(C_3^3\times C_6^2):C_{12}$

Order 10368: $C_3^4.C_2^4.C_2^3$, $C_3^4.C_2^4.C_2^3$, $C_3^4.C_2^4.C_2^3$, $(C_6^2\times C_{12}):S_4$

Order 7776: $C_3\times C_3^2.D_6\wr C_2$ x 4, $(C_3\times C_6^2).C_6.C_6.C_2$ x 2, $C_6\times (C_3^3\times C_6).D_4$ x 2, $C_3\times C_3^3.D_6:C_4.C_2$ x 2, $C_6\times C_3^3:C_2^2.A_4$ x 2, $C_3^5.C_2^2.D_4$ x 2, $(C_3^2\times C_6^2):S_4$ x 2, $C_3^5.(C_4\times D_4)$, $C_2\times C_6^2\times C_3^3:C_2^2$, $C_3\times C_3^2.(S_3\times C_6.D_4)$, $C_6\times C_3^3.D_6:C_4$, $C_2\times C_6\times C_3^4:D_4$, $C_2\times (C_3^4\times C_6).D_4$, $C_3^2\times (C_6\times C_3:S_3).C_2^3$, $C_3\times C_6\times C_3^2:(C_6.D_4)$, $C_3^5.(C_4\times D_4)$, $C_6\times C_3^4.D_4.C_2$, $C_3^5.(C_4\times D_4)$, $(C_3^3\times C_6^2).C_2^3$, $C_3\times C_3^4.(C_4\times D_4)$, $C_6^2.C_3^3.C_4.C_2$, $C_2\times C_3^3.(C_3\times D_6:C_4)$, $C_3^4.C_4^2:S_3$, $C_6^3.S_3^2$

Order 5832: $C_6^2.C_3^3.C_6$ x 2, $C_2\times C_3^3.C_6^2:C_3$ x 2, $C_3^5:S_4$ x 2, $C_3^4.C_3^2.C_4.C_2$, $C_2\times C_6^2.C_3^3.C_3$

Order 5184: $C_3^4.C_2^3.C_2^3$ x 3, $C_3^4.C_2^4.C_2^2$ x 3, $(C_3\times C_6^3).C_2^3$ x 2, $C_3\times C_3^3.C_2^2:C_4:C_4$ x 2, $C_3\times C_3^3.(C_2^5.C_2)$, $C_6\times C_3^3.C_2^2\wr C_2$, $C_3^4.C_2^4.C_2^2$, $C_2\times C_3^2.D_6\wr C_2$, $C_6\times C_6.S_3\wr C_2.C_2$, $(C_3\times C_6^3).C_2^3$, $C_3^4.(C_4\times D_4).C_2$, $C_3^4.C_2^3.C_2^3$, $C_3^4.C_2^3.D_4$, $C_3^4.(C_4\times D_4).C_2$, $C_2\times C_3^3:C_2^2.S_4$, $C_3^4.(C_4\times D_4).C_2$, $C_3^4.C_2^3.C_2^3$, $C_3^4.(C_4\times D_4).C_2$, $C_3^3:C_2^2.A_4.C_4$, $(C_3\times C_6^2).A_4.C_4$, $C_3\times C_3^3.(C_2^5.C_2)$, $(C_6^2\times C_{12}):A_4$

Order 3888: $C_6\times C_3^4:D_4$ x 8, $C_6^2\times C_3^3:C_2^2$ x 6, $C_3\times C_3^4.D_4.C_2$ x 6, $C_3^2\times C_3^2:(C_6.D_4)$ x 5, $C_3\times C_3^3.D_6:C_4$ x 4, $C_6\times C_3^4:(C_2\times C_4)$ x 3, $C_6^2.(C_3\times C_6).C_6$ x 2, $C_3\times C_3^4.C_2^2:C_4$ x 2, $C_3\times (C_3^3\times C_6).D_4$ x 2, $C_3^4.D_6:C_4$ x 2, $C_3\times C_3^3.A_4:C_4$ x 2, $C_2\times C_6^2.C_3^3.C_2$ x 2, $C_3\times C_3^3:C_2^2.A_4$ x 2, $C_3^2.A_4.C_6.C_6$ x 2, $C_3\times C_6^3.S_3$ x 2, $(C_3\times C_6^3).C_6$ x 2, $C_2\times C_6^2\times C_3\times C_3:S_3$, $C_3^5.C_4:C_4$, $C_2\times C_6\times C_3^3:C_2:S_3$, $C_6^3.C_3.C_6$, $C_3^4.D_6:C_4$, $C_3^2\times C_2^2:(C_3^3:C_4)$, $C_3\times C_6^2\times C_3^2:C_4$, $C_3\times C_3^4.C_4^2$, $C_3\times C_3^4.C_4:C_4$, $C_6\times C_3^4:D_4$, $C_3^5.C_4^2$, $C_3\times C_3^3.D_6:C_4$, $C_2^2\times C_3^5:C_4$, $C_2\times C_6^2.C_3^3.C_2$, $C_2\times C_3^5.C_4.C_2$, $C_6^2\times C_3^2:(C_3:C_4)$, $C_3\times C_6\times C_2\times C_3^2:(C_3:C_4)$, $C_3\times C_6\times (C_3\times C_6^2):C_2$, $C_2\times C_6^2.C_3^3.C_2$, $C_6\times C_3^4:(C_2\times C_4)$, $C_3^3.(C_3\times D_6:C_4)$, $C_3\times C_3^3.D_6:C_4$, $C_3\times (C_3^3\times C_6).D_4$, $C_6^2:C_3.C_6.C_6$, $C_3\wr A_4:C_4$, $(C_3\times C_6^3).S_3$, $(C_3^2\times C_6^2):C_{12}$, $C_2\times C_3^4:S_4$

Order 3456: $(C_3\times C_6^2).C_2^4.C_2$, $C_6^3.C_2^3.C_2$, $(C_3\times C_6^2).C_2^4.C_2$, $(C_3\times C_6^2).C_2^4.C_2$, $C_2^4:\He_3.C_4.C_2$

Order 2916: $C_3^5:A_4$ x 2, $C_6^2.C_3^3.C_3$, $C_3^4.(C_3\times C_{12})$, $C_2\times (C_3^4.C_3^2):C_2$, $(C_3^4\times C_6).S_3$

Order 2592: $C_3^4.C_2^3.C_2^2$ x 8, $C_3\times C_3^3.C_2^2\wr C_2$ x 6, $C_3^2.D_6\wr C_2$ x 6, $C_2\times C_3^3.D_6:C_4$ x 5, $(C_6\times S_3)\wr C_2$ x 4, $C_6\times C_6.S_3\wr C_2$ x 3, $C_3\times (C_3\times C_6^2).C_2^3$ x 3, $C_6\times C_3^2:(C_6.D_4)$ x 3, $C_3\times (C_6\times C_3:S_3).C_2^3$ x 3, $C_6\times (C_2\times C_6):S_3^2$ x 3, $C_6\times C_3:(S_3^2:C_4)$ x 3, $C_3^4.(C_4\times D_4)$ x 3, $C_3\times C_3^3.(C_4\times D_4)$ x 3, $C_3^4.(C_4\times D_4)$ x 3, $C_2\times (C_3^3\times C_6).D_4$ x 2, $C_2\times (C_3^3\times C_6).D_4$ x 2, $C_3^4.C_2^3.C_2^2$ x 2, $C_6^2.C_3^2:C_4.C_2$ x 2, $C_3\times C_6.S_3\wr C_2.C_2$ x 2, $C_2\times C_3^3:C_2^2.A_4$ x 2, $C_6\times D_6:(C_3^2:C_4)$ x 2, $C_2^2\times C_6\times C_3^3:C_2^2$ x 2, $C_2\times (C_3\times C_6^2).C_6.C_2$ x 2, $C_6^3.D_6$ x 2, $(C_3\times C_6^2):S_4$ x 2, $C_3^4.(C_4\times D_4)$, $C_3^4.(C_4\times D_4)$, $C_3^4.(C_4\times D_4)$, $C_2\times C_3^3.D_6:C_4$, $C_3^4.(C_4\times D_4)$, $C_3^4.(C_4\times D_4)$, $C_2\times C_6\times C_3:S_3\wr C_2$, $C_3^3.(C_2\times D_6:C_4)$, $C_3^3.(C_2\times D_6:C_4)$, $C_2\times C_6^2\times S_3^2$, $C_2\times C_3^4.D_4.C_2$, $C_3^4.(C_4\times D_4)$, $C_2\times C_3^4.C_2^2:C_4$, $C_6^2.C_3^2:C_4.C_2$, $C_2\times (C_3^3\times C_6).D_4$, $C_3\times C_3^3.(C_4\times D_4)$, $(C_3^2\times C_6^2).C_2^3$, $C_2\times C_6\times S_3^2:S_3$, $C_3^2.(S_3\times D_6:C_4)$, $C_3^4.C_4:C_4.C_2$, $C_3^4.(C_4\times D_4)$, $C_3^4.C_4:C_4.C_2$, $C_3^4.(C_4\times D_4)$, $C_3^4.(C_4\times D_4)$, $C_2^2\times C_3^4:D_4$, $C_3^4.C_4:C_4.C_2$, $C_3^2\times C_2\times C_3:S_3.D_4$, $C_3\times C_6\times C_3:S_3.D_4$, $C_3^3.C_4^2:S_3$, $C_3\times C_3^3.(C_4\times D_4)$, $C_3^3.C_4^2:S_3$, $C_3^4.(C_4\times D_4)$, $C_6^2.(C_{12}\times S_3)$, $C_6\times S_3\times C_6^2:C_2$, $C_3^4.(C_4\times D_4)$, $C_3^3.(C_2\times D_6:C_4)$, $C_3^4.(C_4\times D_4)$, $C_3^2.(S_3\times C_6.D_4)$, $C_6\times C_3:((C_2\times C_{12}):S_3)$, $C_3\times C_3:(C_6.D_4)\times S_3$, $C_6^3.D_6$

Order 1944: $C_3^5:C_2^3$ x 16, $C_3^2:C_6^3$ x 8, $C_3^4:D_{12}$ x 8, $C_3\wr C_4\times C_6$ x 7, $C_3^5:C_2^3$ x 5, $C_2\times C_3\wr A_4$ x 5, $C_3^3:(S_3\times C_{12})$ x 5, $C_6^3:C_3^2$ x 5, $C_3^5:D_4$ x 4, $C_3^4:D_{12}$ x 4, $C_3^4.S_4$ x 4, $C_6\times C_3^3:C_{12}$ x 3, $C_3^4:S_4$ x 2, $C_2\times C_3^4:C_{12}$ x 2, $C_6^3.C_3^2$ x 2, $C_3^4:S_4$ x 2, $C_3^4:S_4$ x 2, $C_3^3.(S_3\times C_{12})$ x 2, $C_3^2\times C_6^3$, $C_3^5:C_2^3$, $C_6\times C_3^3:C_{12}$, $C_3^4:(C_4\times S_3)$, $C_3^3:(S_3\times C_{12})$, $C_6^3.C_3^2$, $C_3^4:(C_4\times S_3)$, $C_3^4:(C_4\times S_3)$

Order 1728: $C_6^3.D_4$ x 3, $C_2^3.S_3^3$ x 3, $C_6^3.D_4$ x 3, $C_6^3.(C_2\times C_4)$ x 2, $C_6^3:D_4$, $C_6^3:D_4$, $C_6^3:D_4$, $C_2\times C_6^2:S_4$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3:D_4$, $C_2^3.S_3^3$, $C_2^3.S_3^3$, $D_6^2.D_6$, $D_6^2.D_6$, $D_6^2.D_6$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$, $D_6^2:C_{12}$, $(C_2\times C_6^2).S_4$, $(C_2^2\times C_6^2):C_{12}$, $C_6^3.C_2^3$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$, $(C_6^2\times C_{12}):C_4$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$

Order 1458: $C_3^5:S_3$ x 2, $C_3^5:C_6$

Order 1296: $C_6^2.S_3^2$ x 22, $C_6^2:S_3^2$ x 20, $C_6^2:S_3^2$ x 18, $(C_3\times C_6^2):C_{12}$ x 16, $C_6^3.C_6$ x 12, $C_6^2.S_3^2$ x 12, $C_6^2\times S_3^2$ x 11, $C_2\times C_3^3:D_{12}$ x 8, $C_6\times C_3^2:D_{12}$ x 8, $C_2\times C_3\wr D_4$ x 8, $C_6^2:S_3^2$ x 8, $C_3\wr C_2^2:C_4$ x 8, $C_3\wr C_2^2:C_4$ x 8, $(C_3\times S_3^2):C_{12}$ x 8, $C_6^2.S_3^2$ x 8, $C_6^3:C_6$ x 7, $C_6^3:C_6$ x 7, $C_6^3.C_6$ x 7, $C_6^2.S_3^2$ x 7, $C_6^2.S_3^2$ x 7, $C_3\wr C_4\times C_2^2$ x 6, $C_6^2:S_3^2$ x 6, $C_6^2.S_3^2$ x 6, $C_6^2.S_3^2$ x 6, $C_6^3.S_3$ x 5, $C_3\times C_6^2:C_{12}$ x 5, $C_6^2.S_3^2$ x 4, $C_6^2.S_3^2$ x 4, $C_6^2.S_3^2$ x 4, $C_6^3.S_3$ x 4, $C_6^2:D_{18}$ x 4, $C_6^3:C_6$ x 3, $C_6^3.C_6$ x 3, $C_6^2.S_3^2$ x 3, $C_6^2.S_3^2$ x 3, $(C_3^2\times C_6).S_4$ x 3, $C_6^3:S_3$ x 3, $C_6^3.C_6$ x 3, $C_6^2.S_3^2$ x 3, $C_6^2.S_3^2$ x 3, $C_6^2.S_3^2$ x 3, $C_6^2.S_3^2$ x 3, $C_6^2:C_{36}$ x 2, $C_6^3.S_3$ x 2, $S_3\times C_6^3$ x 2, $C_3\times C_6^2:C_{12}$ x 2, $(C_3\times C_6^2):A_4$ x 2, $C_6.C_6^3$ x 2, $C_6^3.S_3$ x 2, $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3.C_6$ x 2, $C_6^3.S_3$ x 2, $C_6^3:S_3$ x 2, $C_6^3.C_6$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_2\times C_3^3:S_4$, $C_6^3.S_3$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^3.C_6$, $C_6^3.C_6$, $C_6^3.S_3$

Order 1152: $C_6^2.(C_4\times D_4)$, $C_6^2.(C_4\times D_4)$, $C_6^2.(C_4\times D_4)$, $C_6^2.(C_4\times D_4)$, $C_2^5.S_3^2$

Order 972: $C_3^3:C_6^2$ x 12, $C_3^3:C_6^2$ x 10, $C_3\wr A_4$ x 5, $C_3^4:A_4$ x 5, $C_3\wr C_3:C_{12}$ x 5, $C_3^3:C_6^2$ x 4, $C_3^4:C_{12}$ x 4, $C_3^3\times C_6^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^4:C_{12}$ x 3, $C_3^4.D_6$ x 2, $C_3^4.A_4$ x 2, $(C_3^3\times C_6).S_3$ x 2, $C_3^4.C_{12}$ x 2, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^4:D_6$, $C_3^3:C_6^2$, $C_3^4:A_4$, $(C_3\times \He_3):C_{12}$, $C_3^4:C_{12}$, $C_3^4:C_{12}$

Order 864: $C_6^3.C_2^2$ x 15, $C_6^2.D_{12}$ x 11, $C_6^2.D_{12}$ x 10, $C_6^2.D_{12}$ x 8, $C_6^2.D_{12}$ x 7, $C_6^3:C_4$ x 6, $C_6^2:D_{12}$ x 6, $D_6^2:S_3$ x 6, $D_6^2:C_6$ x 6, $C_6^2:D_{12}$ x 6, $C_6^3.C_2^2$ x 4, $C_6\times D_6^2$ x 3, $C_6^3:C_2^2$ x 3, $D_6^2:C_6$ x 3, $C_6^2.D_{12}$ x 3, $D_6^2.S_3$ x 3, $D_6^2.C_6$ x 3, $C_6^3.C_2^2$ x 3, $C_6^3.C_2^2$ x 3, $C_6^2.D_{12}$ x 3, $C_6^3.C_2^2$ x 3, $C_6^3.C_2^2$ x 3, $C_6^3.C_2^2$ x 3, $D_6^2.S_3$ x 3, $C_6^3:C_4$ x 2, $C_6^2:S_4$ x 2, $C_6^3:C_4$ x 2, $C_6^3:C_4$ x 2, $C_2^2.S_3^3$ x 2, $C_2^2.S_3^3$ x 2, $C_2^2.S_3^3$ x 2, $C_2^2.S_3^3$ x 2, $C_2^2.S_3^3$ x 2, $D_6^2.S_3$ x 2, $(C_6\times S_3^2):C_4$ x 2, $C_4\times C_3^2.S_4$ x 2, $C_6.(S_3\times S_4)$ x 2, $C_6^2.D_{12}$ x 2, $C_6^3:C_2^2$, $C_6:D_6^2$, $C_6^2:D_{12}$, $D_6^2:S_3$, $D_6^2:C_6$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $D_6^2:C_6$, $C_6^2:D_{12}$, $C_3^2:(C_4\times S_4)$, $C_2^5:\He_3$, $C_6^3:C_4$, $C_6^3.C_2^2$, $C_6^2.D_{12}$, $C_6^3.C_2^2$, $(C_6\times C_{12}):D_6$, $C_3^2:(C_4\times D_{12})$, $C_3^3:(C_4\times D_4)$, $(C_6\times C_{12}).D_6$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_4\times C_3^2:S_4$, $C_3^2:(C_4\times S_4)$

Order 729: $C_3^5:C_3$

Order 648: $C_3^3:S_4$ x 2, $C_3\times C_6^3$

Order 486: $C_3^4\times C_6$

Order 432: $C_3:D_6^2$

Order 384: $C_2^5.D_6$

Order 324: $C_3^2\times C_6^2$

Order 243: $C_3^5$

Order 216: $C_6.S_3^2$, $C_6^3$

Order 162: $C_3^3\times C_6$

Order 144: $C_6^2:C_4$

Order 128: $C_2^4.D_4$

Order 108: $C_3\times C_6^2$

Order 81: $C_3^4$

Order 72: $C_2\times C_6^2$

Order 54: $C_3^2\times C_6$

Order 36: $C_6^2$

Order 27: $C_3^3$

Order 24: $C_6:C_4$ x 4, $S_4$ x 2, $C_4\times S_3$, $C_2^2\times C_6$

Order 18: $C_3\times C_6$

Order 12: $C_3:C_4$ x 26, $C_2\times C_6$

Order 9: $C_3^2$

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

Order 6: $S_3$ x 2, $C_6$

Order 4: $C_4$ x 5, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 128 (order at least 729) are shown, as well as all normal subgroups of any index.

Order 93312: $C_6^3.C_6^2:D_6$

Order 46656: $C_3^5.C_2^5.C_6$, $C_2\times C_3^3.C_2^4:\He_3.C_2$, $(C_3^2\times C_6^3).S_4$

Order 31104: $(C_3\times C_6^2).A_4.C_{12}.C_2$, $C_3^5.C_2^4.C_2^3$

Order 23328: $C_3^3.C_2^4:\He_3.C_2$, $C_2\times C_3^3.C_2^4:\He_3$, $C_3^5:(C_4\times S_4)$

Order 15552: $C_3\times C_6^3.S_4$ x 2, $C_3^5.C_2^3.C_2^3$, $C_3^5.(C_4\times D_4).C_2$, $C_3^5.(C_2^5.C_2)$, $C_3\times C_3^4.C_2^3.D_4$, $C_6\times C_3^2.D_6\wr C_2$, $C_3^5.C_2^4.C_2^2$, $C_2\times (C_3\times C_6^2).A_4.S_3$, $C_3\times C_3^4.C_2^3.C_2^3$, $C_6^3.(S_3\times A_4)$

Order 11664: $C_3^3.C_2^4:\He_3$, $C_2\times C_6^2.C_3^3.C_6$, $(C_3^4\times C_6).S_4$, $C_2\times C_3^5:S_4$, $(C_3^4\times C_6).S_4$, $(C_3^3\times C_6^2):C_{12}$

Order 10368: $C_3^4.C_2^4.C_2^3$, $C_3^4.C_2^4.C_2^3$, $C_3^4.C_2^4.C_2^3$, $(C_6^2\times C_{12}):S_4$

Order 7776: $C_6\times (C_3^3\times C_6).D_4$ x 2, $C_3\times C_3^2.D_6\wr C_2$ x 2, $C_6\times C_3^3:C_2^2.A_4$ x 2, $C_3^5.(C_4\times D_4)$, $C_2\times C_6^2\times C_3^3:C_2^2$, $C_3\times C_3^2.(S_3\times C_6.D_4)$, $(C_3\times C_6^2).C_6.C_6.C_2$, $C_6\times C_3^3.D_6:C_4$, $C_2\times C_6\times C_3^4:D_4$, $C_2\times (C_3^4\times C_6).D_4$, $C_3^2\times (C_6\times C_3:S_3).C_2^3$, $C_3\times C_6\times C_3^2:(C_6.D_4)$, $C_3^5.(C_4\times D_4)$, $C_6\times C_3^4.D_4.C_2$, $C_3\times C_3^3.D_6:C_4.C_2$, $C_3^5.(C_4\times D_4)$, $(C_3^3\times C_6^2).C_2^3$, $C_3\times C_3^4.(C_4\times D_4)$, $C_6^2.C_3^3.C_4.C_2$, $C_2\times C_3^3.(C_3\times D_6:C_4)$, $C_3^4.C_4^2:S_3$, $C_3^5.C_2^2.D_4$, $(C_3^2\times C_6^2):S_4$, $C_6^3.S_3^2$

Order 5832: $C_2\times C_3^3.C_6^2:C_3$ x 2, $C_6^2.C_3^3.C_6$, $C_3^4.C_3^2.C_4.C_2$, $C_2\times C_6^2.C_3^3.C_3$, $C_3^5:S_4$

Order 5184: $C_3^4.C_2^3.C_2^3$ x 3, $C_3^4.C_2^4.C_2^2$ x 3, $(C_3\times C_6^3).C_2^3$ x 2, $C_3\times C_3^3.C_2^2:C_4:C_4$ x 2, $C_3\times C_3^3.(C_2^5.C_2)$, $C_6\times C_3^3.C_2^2\wr C_2$, $C_3^4.C_2^4.C_2^2$, $C_2\times C_3^2.D_6\wr C_2$, $C_6\times C_6.S_3\wr C_2.C_2$, $(C_3\times C_6^3).C_2^3$, $C_3^4.(C_4\times D_4).C_2$, $C_3^4.C_2^3.C_2^3$, $C_3^4.C_2^3.D_4$, $C_3^4.(C_4\times D_4).C_2$, $C_2\times C_3^3:C_2^2.S_4$, $C_3^4.(C_4\times D_4).C_2$, $C_3^4.C_2^3.C_2^3$, $C_3^4.(C_4\times D_4).C_2$, $C_3^3:C_2^2.A_4.C_4$, $(C_3\times C_6^2).A_4.C_4$, $C_3\times C_3^3.(C_2^5.C_2)$, $(C_6^2\times C_{12}):A_4$

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

Order 3456: $(C_3\times C_6^2).C_2^4.C_2$, $C_6^3.C_2^3.C_2$, $(C_3\times C_6^2).C_2^4.C_2$, $(C_3\times C_6^2).C_2^4.C_2$, $C_2^4:\He_3.C_4.C_2$

Order 2916: $C_3^5:A_4$ x 2, $C_6^2.C_3^3.C_3$, $C_3^4.(C_3\times C_{12})$, $C_2\times (C_3^4.C_3^2):C_2$, $(C_3^4\times C_6).S_3$

Order 2592: $C_2\times C_3^3.D_6:C_4$ x 5, $C_6\times C_6.S_3\wr C_2$ x 3, $C_3\times C_3^3.C_2^2\wr C_2$ x 3, $C_3\times (C_3\times C_6^2).C_2^3$ x 3, $C_3^2.D_6\wr C_2$ x 3, $C_6\times C_3^2:(C_6.D_4)$ x 3, $C_3\times (C_6\times C_3:S_3).C_2^3$ x 3, $C_6\times (C_2\times C_6):S_3^2$ x 3, $C_6\times C_3:(S_3^2:C_4)$ x 3, $C_3^4.(C_4\times D_4)$ x 3, $C_3^4.C_2^3.C_2^2$ x 3, $C_3\times C_3^3.(C_4\times D_4)$ x 3, $C_3^4.(C_4\times D_4)$ x 3, $C_2\times (C_3^3\times C_6).D_4$ x 2, $C_2\times (C_3^3\times C_6).D_4$ x 2, $C_2\times C_3^3:C_2^2.A_4$ x 2, $C_6\times D_6:(C_3^2:C_4)$ x 2, $C_2^2\times C_6\times C_3^3:C_2^2$ x 2, $C_2\times (C_3\times C_6^2).C_6.C_2$ x 2, $(C_6\times S_3)\wr C_2$ x 2, $C_3^4.(C_4\times D_4)$, $C_3^4.(C_4\times D_4)$, $C_3^4.(C_4\times D_4)$, $C_2\times C_3^3.D_6:C_4$, $C_3^4.(C_4\times D_4)$, $C_3^4.(C_4\times D_4)$, $C_2\times C_6\times C_3:S_3\wr C_2$, $C_3^3.(C_2\times D_6:C_4)$, $C_3^3.(C_2\times D_6:C_4)$, $C_2\times C_6^2\times S_3^2$, $C_2\times C_3^4.D_4.C_2$, $C_3^4.(C_4\times D_4)$, $C_2\times C_3^4.C_2^2:C_4$, $C_6^2.C_3^2:C_4.C_2$, $C_2\times (C_3^3\times C_6).D_4$, $C_3\times C_3^3.(C_4\times D_4)$, $C_3^4.C_2^3.C_2^2$, $(C_3^2\times C_6^2).C_2^3$, $C_2\times C_6\times S_3^2:S_3$, $C_3^2.(S_3\times D_6:C_4)$, $C_3^4.C_4:C_4.C_2$, $C_6^2.C_3^2:C_4.C_2$, $C_3^4.(C_4\times D_4)$, $C_3\times C_6.S_3\wr C_2.C_2$, $C_3^4.C_4:C_4.C_2$, $C_3^4.(C_4\times D_4)$, $C_3^4.(C_4\times D_4)$, $C_2^2\times C_3^4:D_4$, $C_3^4.C_4:C_4.C_2$, $C_3^2\times C_2\times C_3:S_3.D_4$, $C_3\times C_6\times C_3:S_3.D_4$, $C_3^3.C_4^2:S_3$, $C_3\times C_3^3.(C_4\times D_4)$, $C_3^3.C_4^2:S_3$, $C_3^4.(C_4\times D_4)$, $C_6^2.(C_{12}\times S_3)$, $C_6\times S_3\times C_6^2:C_2$, $C_3^4.(C_4\times D_4)$, $C_3^3.(C_2\times D_6:C_4)$, $C_3^4.(C_4\times D_4)$, $C_3^2.(S_3\times C_6.D_4)$, $C_6\times C_3:((C_2\times C_{12}):S_3)$, $C_3\times C_3:(C_6.D_4)\times S_3$, $C_6^3.D_6$, $C_6^3.D_6$, $(C_3\times C_6^2):S_4$

Order 1944: $C_3^5:C_2^3$ x 16, $C_3^2:C_6^3$ x 8, $C_3\wr C_4\times C_6$ x 6, $C_3^5:C_2^3$ x 4, $C_3^4:D_{12}$ x 4, $C_2\times C_3\wr A_4$ x 4, $C_3^3:(S_3\times C_{12})$ x 4, $C_6^3:C_3^2$ x 4, $C_6\times C_3^3:C_{12}$ x 3, $C_2\times C_3^4:C_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^4:D_{12}$ x 2, $C_3^2\times C_6^3$, $C_3^5:C_2^3$, $C_3^4:S_4$, $C_6\times C_3^3:C_{12}$, $C_3^4:(C_4\times S_3)$, $C_3^3:(S_3\times C_{12})$, $C_6^3.C_3^2$, $C_6^3.C_3^2$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4.S_4$, $C_3^4:(C_4\times S_3)$, $C_3^3.(S_3\times C_{12})$, $C_3^4:(C_4\times S_3)$

Order 1728: $C_6^3.D_4$ x 3, $C_2^3.S_3^3$ x 3, $C_6^3.D_4$ x 3, $C_6^3.(C_2\times C_4)$ x 2, $C_6^3:D_4$, $C_6^3:D_4$, $C_6^3:D_4$, $C_2\times C_6^2:S_4$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3:D_4$, $C_2^3.S_3^3$, $C_2^3.S_3^3$, $D_6^2.D_6$, $D_6^2.D_6$, $D_6^2.D_6$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$, $D_6^2:C_{12}$, $(C_2\times C_6^2).S_4$, $(C_2^2\times C_6^2):C_{12}$, $C_6^3.C_2^3$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$, $(C_6^2\times C_{12}):C_4$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$

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

Order 1296: $C_6^2:S_3^2$ x 20, $C_6^2.S_3^2$ x 18, $C_6^3.C_6$ x 12, $C_6^2:S_3^2$ x 12, $C_6^2.S_3^2$ x 12, $(C_3\times C_6^2):C_{12}$ x 12, $C_6^2\times S_3^2$ x 11, $C_6^2.S_3^2$ x 8, $C_6^3:C_6$ x 7, $C_6^3:C_6$ x 7, $C_6^3.C_6$ x 7, $C_6^2.S_3^2$ x 7, $C_6^2.S_3^2$ x 7, $C_3\wr C_4\times C_2^2$ x 6, $C_2\times C_3^3:D_{12}$ x 6, $C_6\times C_3^2:D_{12}$ x 6, $C_2\times C_3\wr D_4$ x 6, $C_6^2.S_3^2$ x 6, $C_3\wr C_2^2:C_4$ x 6, $C_3\wr C_2^2:C_4$ x 6, $(C_3\times S_3^2):C_{12}$ x 6, $C_6^2.S_3^2$ x 6, $C_6^3.S_3$ x 4, $C_6^2:S_3^2$ x 4, $C_6^2.S_3^2$ x 4, $C_6^2:S_3^2$ x 4, $C_3\times C_6^2:C_{12}$ x 4, $C_6^2.S_3^2$ x 4, $C_6^2.S_3^2$ x 4, $C_6^3:C_6$ x 3, $C_6^3.C_6$ x 3, $C_6^2.S_3^2$ x 3, $C_6^2.S_3^2$ x 3, $C_6^3:S_3$ x 3, $C_6^3.C_6$ x 3, $C_6^2.S_3^2$ x 3, $C_6^2.S_3^2$ x 3, $C_6^2.S_3^2$ x 3, $C_6^2.S_3^2$ x 3, $S_3\times C_6^3$ x 2, $C_3\times C_6^2:C_{12}$ x 2, $(C_3\times C_6^2):A_4$ x 2, $C_6.C_6^3$ x 2, $(C_3^2\times C_6).S_4$ x 2, $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3.S_3$ x 2, $C_6^3.S_3$ x 2, $C_6^3:S_3$ x 2, $C_6^2:D_{18}$ x 2, $C_6^3.C_6$, $C_6^2:C_{36}$, $C_6^3.S_3$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_2\times C_3^3:S_4$, $C_6^3.S_3$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^3.S_3$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^3.C_6$, $C_6^3.C_6$, $C_6^3.C_6$, $C_6^3.S_3$

Order 1152: $C_6^2.(C_4\times D_4)$, $C_6^2.(C_4\times D_4)$, $C_6^2.(C_4\times D_4)$, $C_6^2.(C_4\times D_4)$, $C_2^5.S_3^2$

Order 972: $C_3^3:C_6^2$ x 12, $C_3^3:C_6^2$ x 10, $C_3\wr A_4$ x 4, $C_3^4:A_4$ x 4, $C_3\wr C_3:C_{12}$ x 4, $C_3^3\times C_6^2$ x 3, $C_3^3:C_6^2$ x 3, $C_3^4:C_{12}$ x 3, $C_3^3:S_3^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^4:D_6$, $C_3^4.D_6$, $C_3^3:C_6^2$, $C_3^4:A_4$, $C_3^4.A_4$, $(C_3^3\times C_6).S_3$, $(C_3\times \He_3):C_{12}$, $C_3^4:C_{12}$, $C_3^4.C_{12}$, $C_3^4:C_{12}$

Order 864: $C_6^3.C_2^2$ x 15, $C_6^2.D_{12}$ x 11, $C_6^2.D_{12}$ x 10, $C_6^2.D_{12}$ x 7, $C_6^3:C_4$ x 6, $C_6^3.C_2^2$ x 4, $C_6\times D_6^2$ x 3, $C_6^3:C_2^2$ x 3, $D_6^2:C_6$ x 3, $C_6^2:D_{12}$ x 3, $C_6^2.D_{12}$ x 3, $D_6^2:S_3$ x 3, $D_6^2.S_3$ x 3, $D_6^2:C_6$ x 3, $D_6^2.C_6$ x 3, $C_6^3.C_2^2$ x 3, $C_6^3.C_2^2$ x 3, $C_6^2.D_{12}$ x 3, $C_6^2:D_{12}$ x 3, $C_6^3.C_2^2$ x 3, $C_6^3.C_2^2$ x 3, $C_6^3.C_2^2$ x 3, $D_6^2.S_3$ x 3, $C_6^2.D_{12}$ x 3, $C_6^3:C_4$ x 2, $C_6^3:C_4$ x 2, $C_6^3:C_4$ x 2, $C_2^2.S_3^3$ x 2, $C_2^2.S_3^3$ x 2, $C_2^2.S_3^3$ x 2, $C_2^2.S_3^3$ x 2, $C_2^2.S_3^3$ x 2, $D_6^2.S_3$ x 2, $C_6^3:C_2^2$, $C_6:D_6^2$, $C_6^2:D_{12}$, $D_6^2:S_3$, $D_6^2:C_6$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $D_6^2:C_6$, $C_6^2:D_{12}$, $C_3^2:(C_4\times S_4)$, $C_2^5:\He_3$, $C_6^2:S_4$, $C_6^3:C_4$, $C_6^3.C_2^2$, $C_6^2.D_{12}$, $C_6^3.C_2^2$, $(C_6\times C_{12}):D_6$, $C_3^2:(C_4\times D_{12})$, $C_3^3:(C_4\times D_4)$, $(C_6\times C_{12}).D_6$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $C_2^2.S_3^3$, $(C_6\times S_3^2):C_4$, $C_4\times C_3^2:S_4$, $C_4\times C_3^2.S_4$, $C_3^2:(C_4\times S_4)$, $C_6.(S_3\times S_4)$, $C_6^2.D_{12}$

Order 729: $C_3^5:C_3$

Order 648: $C_3\times C_6^3$, $C_3^3:S_4$

Order 486: $C_3^4\times C_6$

Order 432: $C_3:D_6^2$

Order 384: $C_2^5.D_6$

Order 324: $C_3^2\times C_6^2$

Order 243: $C_3^5$

Order 216: $C_6.S_3^2$, $C_6^3$

Order 162: $C_3^3\times C_6$

Order 144: $C_6^2:C_4$

Order 128: $C_2^4.D_4$

Order 108: $C_3\times C_6^2$

Order 81: $C_3^4$

Order 72: $C_2\times C_6^2$

Order 54: $C_3^2\times C_6$

Order 36: $C_6^2$

Order 27: $C_3^3$

Order 24: $C_6:C_4$ x 3, $C_4\times S_3$, $C_2^2\times C_6$, $S_4$

Order 18: $C_3\times C_6$

Order 12: $C_3:C_4$ x 20, $C_2\times C_6$

Order 9: $C_3^2$

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

Order 6: $C_6$, $S_3$

Order 4: $C_4$ x 5, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 93312: $C_6^3.C_6^2:D_6$ ($C_1$)

Order 46656: $C_3^5.C_2^5.C_6$ ($C_2$), $C_2\times C_3^3.C_2^4:\He_3.C_2$ ($C_2$), $(C_3^2\times C_6^3).S_4$ ($C_2$)

Order 23328: $C_3^3.C_2^4:\He_3.C_2$ ($C_4$) x 2, $C_2\times C_3^3.C_2^4:\He_3$ ($C_2^2$)

Order 15552: $C_3^5.(C_2^5.C_2)$ ($S_3$), $C_2\times (C_3\times C_6^2).A_4.S_3$ ($S_3$)

Order 11664: $C_3^3.C_2^4:\He_3$ ($C_2\times C_4$)

Order 7776: $(C_3^2\times C_6^2):S_4$ ($C_3:C_4$) x 2, $C_2\times C_6^2\times C_3^3:C_2^2$ ($D_6$), $C_6\times C_3^3:C_2^2.A_4$ ($D_6$)

Order 3888: $C_3\times C_6^2\times C_3^2:C_4$ ($S_4$), $C_6^2\times C_3^3:C_2^2$ ($C_4\times S_3$), $C_3\times C_3^3:C_2^2.A_4$ ($C_6:C_4$)

Order 2592: $C_2^2\times C_6\times C_3^3:C_2^2$ ($S_3^2$)

Order 1944: $C_3^2\times C_6^3$ ($C_2\times S_4$)

Order 1296: $C_6^2:S_3^2$ ($C_6.D_6$)

Order 972: $C_3^3\times C_6^2$ ($C_4\times S_4$)

Order 864: $C_6:D_6^2$ ($C_3^2:D_6$)

Order 648: $C_3\times C_6^3$ ($S_3\times S_4$)

Order 486: $C_3^4\times C_6$ ($C_2^3:S_4$)

Order 432: $C_3:D_6^2$ ($C_6.S_3^2$)

Order 324: $C_3^2\times C_6^2$ ($C_2^3.S_3^2$)

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

Order 216: $C_6^3$ ($C_6^2:D_6$)

Order 162: $C_3^3\times C_6$ ($C_2^4:S_3^2$)

Order 144: $C_6^2:C_4$ ($C_3^3:S_4$)

Order 108: $C_3\times C_6^2$ ($C_3^2:(C_4\times S_4)$)

Order 81: $C_3^4$ ($C_2^5.S_3^2$)

Order 72: $C_2\times C_6^2$ ($C_2\times C_3^3:S_4$)

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

Order 36: $C_6^2$ ($C_3^3.(C_4\times S_4)$)

Order 27: $C_3^3$ ($C_2^4:\He_3.C_4.C_2$)

Order 24: $C_2^2\times C_6$ ($S_3\times C_3^3:S_4$)

Order 18: $C_3\times C_6$ ($C_6^3:S_4$)

Order 12: $C_2\times C_6$ ($C_3^3.(C_3:C_4\times S_4)$)

Order 9: $C_3^2$ ($(C_6^2\times C_{12}):S_4$)

Order 8: $C_2^3$ ($C_3^3.C_3:S_3:S_4$)

Order 6: $C_6$ ($(C_3\times C_6^2).A_4.D_6$)

Order 4: $C_2^2$ ($C_3^3.C_6^2:C_3.C_4.C_2$)

Order 3: $C_3$ ($(C_3\times C_6^2).A_4.C_6.C_2^2$)

Order 2: $C_2$ ($C_3^3.C_2^4:\He_3.C_2^2$)

Order 1: $C_1$ ($C_6^3.C_6^2:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 93312: $C_6^3.C_6^2:D_6$ ($C_1$)

Order 46656: $C_3^5.C_2^5.C_6$ ($C_2$), $C_2\times C_3^3.C_2^4:\He_3.C_2$ ($C_2$), $(C_3^2\times C_6^3).S_4$ ($C_2$)

Order 23328: $C_3^3.C_2^4:\He_3.C_2$ ($C_4$), $C_2\times C_3^3.C_2^4:\He_3$ ($C_2^2$)

Order 15552: $C_3^5.(C_2^5.C_2)$ ($S_3$), $C_2\times (C_3\times C_6^2).A_4.S_3$ ($S_3$)

Order 11664: $C_3^3.C_2^4:\He_3$ ($C_2\times C_4$)

Order 7776: $C_2\times C_6^2\times C_3^3:C_2^2$ ($D_6$), $C_6\times C_3^3:C_2^2.A_4$ ($D_6$), $(C_3^2\times C_6^2):S_4$ ($C_3:C_4$)

Order 3888: $C_3\times C_6^2\times C_3^2:C_4$ ($S_4$), $C_6^2\times C_3^3:C_2^2$ ($C_4\times S_3$), $C_3\times C_3^3:C_2^2.A_4$ ($C_6:C_4$)

Order 2592: $C_2^2\times C_6\times C_3^3:C_2^2$ ($S_3^2$)

Order 1944: $C_3^2\times C_6^3$ ($C_2\times S_4$)

Order 1296: $C_6^2:S_3^2$ ($C_6.D_6$)

Order 972: $C_3^3\times C_6^2$ ($C_4\times S_4$)

Order 864: $C_6:D_6^2$ ($C_3^2:D_6$)

Order 648: $C_3\times C_6^3$ ($S_3\times S_4$)

Order 486: $C_3^4\times C_6$ ($C_2^3:S_4$)

Order 432: $C_3:D_6^2$ ($C_6.S_3^2$)

Order 324: $C_3^2\times C_6^2$ ($C_2^3.S_3^2$)

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

Order 216: $C_6^3$ ($C_6^2:D_6$)

Order 162: $C_3^3\times C_6$ ($C_2^4:S_3^2$)

Order 144: $C_6^2:C_4$ ($C_3^3:S_4$)

Order 108: $C_3\times C_6^2$ ($C_3^2:(C_4\times S_4)$)

Order 81: $C_3^4$ ($C_2^5.S_3^2$)

Order 72: $C_2\times C_6^2$ ($C_2\times C_3^3:S_4$)

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

Order 36: $C_6^2$ ($C_3^3.(C_4\times S_4)$)

Order 27: $C_3^3$ ($C_2^4:\He_3.C_4.C_2$)

Order 24: $C_2^2\times C_6$ ($S_3\times C_3^3:S_4$)

Order 18: $C_3\times C_6$ ($C_6^3:S_4$)

Order 12: $C_2\times C_6$ ($C_3^3.(C_3:C_4\times S_4)$)

Order 9: $C_3^2$ ($(C_6^2\times C_{12}):S_4$)

Order 8: $C_2^3$ ($C_3^3.C_3:S_3:S_4$)

Order 6: $C_6$ ($(C_3\times C_6^2).A_4.D_6$)

Order 4: $C_2^2$ ($C_3^3.C_6^2:C_3.C_4.C_2$)

Order 3: $C_3$ ($(C_3\times C_6^2).A_4.C_6.C_2^2$)

Order 2: $C_2$ ($C_3^3.C_2^4:\He_3.C_2^2$)

Order 1: $C_1$ ($C_6^3.C_6^2:D_6$)

Series

Derived series

$C_6^3.C_6^2:D_6$

$\rhd$

$C_3^3.C_2^4:\He_3$

$\rhd$

$C_6^2:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_6^3.C_6^2:D_6$

$\rhd$

$(C_3^2\times C_6^3).S_4$

$\rhd$

$C_2\times C_3^3.C_2^4:\He_3$

$\rhd$

$C_3^3.C_2^4:\He_3$

$\rhd$

$C_6^2\times C_3^3:C_2^2$

$\rhd$

$C_3\times C_3^3:C_2^2.A_4$

$\rhd$

$C_6^2:S_3^2$

$\rhd$

$C_3:D_6^2$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Lower central series

$C_6^3.C_6^2:D_6$

$\rhd$

$C_3^3.C_2^4:\He_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 6 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 $234 \times 234$ 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 $172 \times 172$ rational character table (warning: may be slow to load).