Abstract group 62208.ce: $C_6^3.(D_6\times S_4)$

• Label: 62208.ce
• Order: \( 2^{8} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_6^3.(D_6\times S_4)$
Order:	\(62208\)\(\medspace = 2^{8} \cdot 3^{5} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$D_5^3.C_2^2$, of order \(1492992\)\(\medspace = 2^{11} \cdot 3^{6} \)
Composition factors:	$C_2$ x 8, $C_3$ x 5
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
Elements	1	3583	944	4608	15776	1728	30384	5184	62208
Conjugacy classes	1	17	13	15	181	2	91	4	324
Divisions	1	17	13	15	178	2	75	4	305
Autjugacy classes	1	15	9	12	89	2	33	3	164

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid c^{12}=d^{6}=e^{6}=f^{6}=g^{2}=[d,e]=[d,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,2,5,9,3,6)(4,8)(10,11)(12,13)(14,15,18,17)(16,20,19,21)(22,23,24), (1,4,6,3,7,2,5,8,9) \!\cdots\! \rangle$
Transitive group:	36T17208				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_6^4.S_3)$ . $D_4$	$(C_6^3:S_4)$ . $D_6$	$(C_6^3:S_4)$ . $D_6$	$(C_6^3.S_4)$ . $D_6$	all 87

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 99 normal subgroups (97 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 36 (order at least 1728) are shown, as well as all normal subgroups of any index.

Order 62208: $C_6^3.(D_6\times S_4)$

Order 31104: $C_2\times C_6^4.D_6$, $C_2\times C_6^4:D_6$, $C_6^3.(S_3\times S_4)$, $C_6^4:(C_4\times S_3)$, $(S_3\times C_6^3):S_4$, $C_6^3.(A_4\times D_6)$, $C_6^4:D_{12}$

Order 20736: $C_6^4.C_2^4$, $C_3^3:C_2\wr D_6$, $C_3^3:C_2\wr D_6$

Order 15552: $C_6^4.D_6$ x 2, $C_6^4:D_6$ x 2, $C_2\times C_2^4:\He_3.C_3.C_6$, $C_2\times C_6^4.C_6$, $C_2\times C_6^4.S_3$, $(S_3\times C_6^3):D_6$, $C_6^4:D_6$, $(C_3\times C_6^3).S_4$, $C_2\times C_6^4:C_6$, $C_6^4.C_{12}$

Order 10368: $C_6^3.(C_2\times S_4)$ x 2, $C_6^3:(C_2\times S_4)$ x 2, $C_3^4.C_2^6.C_2$, $C_6^4.C_2^3$, $C_3^4.C_2^6.C_2$, $C_3^4.C_2^5.C_2^2$, $C_3^4.C_2^6.C_2$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_2\times C_2^4:\He_3.D_6$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^3.(C_2\times S_4)$, $(C_2\times C_6^3):D_{12}$, $(D_6\times C_6^2):S_4$, $C_2\times C_6^3:S_4$, $(C_3\times C_6^2):\GL(2,\mathbb{Z}/4)$, $(C_3\times C_6^2):\GL(2,\mathbb{Z}/4)$, $(C_2\times C_6^3):D_{12}$, $C_6^3.(C_2^2\times A_4)$, $C_6^2.(A_4\times D_{12})$, $C_6^2.(D_6\times S_4)$, $(D_6\times C_6^2):S_4$, $(C_2\times C_6^3):S_4$

Order 7776: $C_6^4.S_3$ x 2, $C_2\times C_6^2.C_3^3.C_2^2$ x 2, $C_3^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_6^4.C_6$, $C_2\times C_6^4.C_3$, $C_2^4:\He_3.C_3.C_6$, $C_6^3:S_3^2$, $C_6^3.S_3^2$, $C_3^4:\GL(2,\mathbb{Z}/4)$, $(S_3\times C_6^3):C_6$, $C_6^3.S_3^2$, $C_6^4:S_3$, $C_6^4:C_6$

Order 6912: $C_6^3.C_2^3.C_2^2$ x 4, $C_6^3.C_2^3.C_2^2$, $C_6^3.C_2^3.C_2^2$, $C_2^5.S_3^3$, $C_6^2.(D_4\times S_4)$

Order 5184: $C_3^4.C_2^5.C_2$ x 8, $(C_2\times C_6^3):D_6$ x 6, $C_6^3:S_4$ x 5, $C_3^4.C_2\wr C_2^2$ x 4, $C_6^4.C_2^2$ x 4, $C_6^2.(C_6\times S_4)$ x 4, $C_3^4.C_2^5.C_2$ x 3, $C_3^4.C_2^5.C_2$ x 3, $C_3^4.C_2^4.C_2^2$ x 2, $C_6^4.C_2^2$ x 2, $C_6^4.C_2^2$ x 2, $C_3^4.D_4^2$ x 2, $C_3^4.C_2^5.C_2$ x 2, $C_6^4.C_2^2$ x 2, $C_3^4.C_2^5.C_2$ x 2, $C_6^4.C_2^2$ x 2, $C_3^4.D_4^2$ x 2, $C_3^4.C_2^5.C_2$ x 2, $C_6^4.C_2^2$ x 2, $C_2^4:\He_3.D_6$ x 2, $C_6^2.(A_4\times D_6)$ x 2, $C_6^3:S_4$ x 2, $(S_3\times C_6^2):S_4$ x 2, $C_6^3.S_4$ x 2, $C_6^3:S_4$ x 2, $(C_2\times C_6^3):C_{12}$ x 2, $C_6^4.C_2^2$, $C_2\times (C_2\times C_6^3).C_6$, $C_2^5\times C_3:(C_3^3:C_2)$, $C_6^4.C_2^2$, $C_2\times C_2^4:\He_3.C_6$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_2\times (C_3\times C_2^4:C_9).C_6$, $C_6^4.C_2^2$, $C_3^4.C_2^5.C_2$, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^3.C_2^3$, $C_3^3.C_3:D_4:D_4$, $C_6^4.C_2^2$, $(C_3\times C_6^3).C_2^3$, $C_6^4.C_2^2$, $C_3^4.C_2^5.C_2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^2.C_6^2.C_2^2$, $(C_3\times C_6^3).C_2^3$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_2\times C_2^4.C_3^3:S_3$, $C_3^4.C_2^4.C_2^2$, $C_3^3:(D_4\times S_4)$, $C_3^3:(D_4\times S_4)$, $C_6^3.S_4$, $C_6^2.(A_4\times D_6)$, $C_6^3.S_4$, $C_6^3:S_4$, $(D_6\times C_6^2):A_4$, $(C_2\times C_6^3):C_{12}$, $C_2\times C_6^3:A_4$

Order 3888: $C_6^2.C_3^3.C_2^2$ x 6, $C_2\times C_3^4.A_4.C_2$ x 3, $C_2\times C_6^2.C_3^3.C_2$ x 2, $C_2\times C_6^2.C_3^3.C_2$ x 2, $S_3\times C_3^3:S_4$ x 2, $C_3\times C_6^3:S_3$ x 2, $(C_3\times C_6^3).S_3$ x 2, $C_6^4.C_3$, $(D_6\times C_3^3):D_6$, $C_6\wr C_3\times S_3$, $C_3\times C_6^3.C_6$

Order 3456: $C_6^3.C_2^4$ x 9, $C_3^3.C_2^6.C_2$ x 9, $C_6^3.D_4.C_2$ x 8, $(C_2\times C_6^3).C_2^3$ x 8, $C_6^3.C_2^2.C_2^2$ x 4, $C_6^3.C_2^2.C_2^2$ x 4, $C_6^3.C_2^3.C_2$ x 4, $C_6^3.C_2^2.C_2^2$ x 4, $C_6^3.C_2^4$ x 4, $C_6^3.C_2^4$ x 4, $(C_2\times C_6^3).C_2^3$ x 4, $C_3^3.C_2^5.C_2^2$ x 4, $C_6^3.C_2^4$ x 3, $(C_2\times C_6^3).C_2^3$ x 2, $C_6^3.D_4.C_2$ x 2, $C_6^2.(C_4\times S_4)$ x 2, $C_6^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_6^3.C_2^2.C_2^2$, $(C_2\times C_6^3).C_2^3$, $C_6^3.D_4.C_2$, $C_6^3.C_2^4$, $C_6^3.C_2^3.C_2$, $C_6^3.C_2^2.C_2^2$, $C_6^3.C_2^2.C_2^2$, $C_2\times C_6^3.C_2^3$, $C_6^3.C_2^4$, $C_6^3.C_2^2.C_2^2$, $C_6^3.C_2^2.C_2^2$, $C_3^3.C_2^5.C_2^2$, $C_6^3.C_2^4$, $(C_2\times C_6^3).C_2^3$, $C_6^3.D_4.C_2$, $(C_3\times C_6^2).C_2^5$, $C_6^3.C_2^2.C_2^2$, $C_6^3.C_2^4$, $C_2\times (C_2^3\times C_6).S_3^2$, $(C_2\times C_6^3).C_2^3$, $C_3^3.C_2^6.C_2$, $C_6^3.D_4.C_2$, $C_6^3.C_2^3.C_2$, $C_6^2.(C_4\times S_4)$, $(C_2^3\times C_6^2):D_6$, $(C_2^2\times C_6^2):D_{12}$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $C_6^2:(C_2^2\times S_4)$, $C_2^4:(S_3\times D_{18})$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $(C_2^2\times C_6^2):D_{12}$, $C_6^2.(C_2^2\times S_4)$, $C_3^2:C_2\wr C_6$, $C_3^2:C_2\wr C_6$, $(C_2^2\times C_6^2):S_4$

Order 2592: $C_3^4.C_2^4.C_2$ x 26, $C_3^4.C_2^4.C_2$ x 17, $C_3^4.C_2^4.C_2$ x 15, $C_3^4.C_2^4.C_2$ x 14, $C_3^4.C_2^4.C_2$ x 12, $C_2^4\times C_3:(C_3^3:C_2)$ x 11, $(C_3\times C_6^2):S_4$ x 9, $C_3^4.C_2^4.C_2$ x 7, $C_6^4.C_2$ x 6, $C_6^4.C_2$ x 6, $C_6^4.C_2$ x 6, $C_3^4.C_2^4.C_2$ x 5, $C_6^3:D_6$ x 5, $C_6^4.C_2$ x 4, $(C_3\times C_6^3).C_2^2$ x 4, $(C_3\times C_6^3).C_2^2$ x 4, $C_6^3:A_4$ x 4, $C_6^4.C_2$ x 3, $C_6^4.C_2$ x 3, $C_3^4.C_2^4.C_2$ x 3, $(C_3\times C_6^3).C_2^2$ x 3, $(C_2\times C_6^3):C_6$ x 3, $(C_3\times C_6^2):S_4$ x 3, $C_6^4.C_2$ x 2, $C_3^4.(C_4\times D_4)$ x 2, $C_2\times C_6^2.S_3^2$ x 2, $(C_3^2\times C_6^2).C_2^3$ x 2, $C_6^2.C_6^2.C_2$ x 2, $C_6^2.C_6^2.C_2$ x 2, $C_6^4.C_2$ x 2, $C_3^3.C_2^3:D_6$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_6^4.C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.(C_4\times D_4)$ x 2, $C_3^4.C_4:D_4$ x 2, $C_3^4.C_2^2:D_4$ x 2, $C_3^4.C_2^2:D_4$ x 2, $C_3^3.D_6:Q_8$ x 2, $C_3^4.(C_4.D_4)$ x 2, $C_2^4.C_3^3:S_3$ x 2, $C_3^4.C_2^4.C_2$ x 2, $(C_3^3\times C_6).C_2^4$ x 2, $C_6^2.C_6^2.C_2$ x 2, $C_3^4.(C_4\times D_4)$ x 2, $(C_3\times C_6^3).C_2^2$ x 2, $C_3^4.C_2^2:D_4$ x 2, $(C_3\times C_6^3).C_2^2$ x 2, $C_3^3.D_6:C_4.C_2$ x 2, $C_6^2.C_6^2.C_2$ x 2, $(C_3\times C_2^4:C_9).C_6$ x 2, $C_3^4.(C_4\times D_4)$ x 2, $C_3^3.D_6:C_4.C_2$ x 2, $C_6^3.D_6$ x 2, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_6^2.(C_6\times A_4)$ x 2, $C_3\times C_6^2.S_4$ x 2, $C_2\times (C_2\times C_6^3).C_3$, $(C_3\times C_6^3).C_2^2$, $(C_3\times C_6^3).C_2^2$, $(C_3\times C_6^3).C_2^2$, $C_2^3\times S_3\times C_3^3:C_2$, $C_3^4.C_2^2:D_4$, $C_3^3.C_4:D_{12}$, $C_3^4.C_2^2\wr C_2$, $C_3.C_6^3.C_2^2$, $C_6.C_6^3.C_2$, $C_2\times C_6^4$, $C_3^4.C_2^2:D_4$, $C_3.C_6^3.C_2^2$, $(C_3\times C_6^3).C_2^2$, $C_3^3.D_6:C_4.C_2$, $C_3^4.D_4.C_2^2$, $C_3^4.C_2^3.C_2^2$, $C_3^4.C_2^2\wr C_2$, $C_3^3.C_2^3:D_6$, $(C_3\times C_6^3).C_2^2$, $C_3^4.D_4.C_2^2$, $C_3^4.C_2^3.C_2^2$, $(C_3\times C_6^3).C_2^2$, $C_2^4:\He_3.C_6$, $(C_3^2\times C_6^2).C_2^3$, $(C_3\times C_6^3).C_2^2$, $(C_3^2\times C_6^2).C_2^3$, $C_3.C_6^3.C_2^2$, $(C_3\times C_6^3).C_2^2$, $C_3^3.D_6:C_4.C_2$, $C_3^3.(D_6.D_4)$, $(C_2\times C_6^3).C_6$, $C_6^2.(C_6\times C_{12})$, $C_2\times (C_2\times C_6^3).C_3$, $(C_3\times C_6^3).C_2^2$, $C_6^3.D_6$, $C_6^3.D_6$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $C_6^3:D_6$, $C_6^3.D_6$, $C_6^2:(C_3\times D_{12})$, $C_6^3.D_6$, $C_6^3:D_6$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $(C_2\times C_6^3):C_6$, $C_3\times C_6^2:S_4$, $(C_2\times C_6^3):C_6$, $C_6^3:A_4$

Order 2304: $C_6^2.C_2^4.C_2^2$ x 5, $C_6^2.C_2^4.C_2^2$ x 4, $C_6^2.C_2^4.C_2^2$, $C_2^4:(S_3\times S_4)$, $(C_2^2\times D_{12}):S_4$

Order 1944: $C_3^3:(S_3\times A_4)$ x 4, $C_3^4:S_4$ x 4, $C_6^3:C_3^2$ x 3, $C_3^4:S_4$ x 3, $C_3^4:S_4$ x 2, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $S_3\times C_3^3:A_4$, $(C_3\times \He_3):D_{12}$, $C_3^4:(C_4\times S_3)$, $C_3^3:(C_3\times D_{12})$, $(D_6\times C_3^3):S_3$, $C_3^4:D_{12}$

Order 1728: $C_6^3.C_2^3$ x 105, $C_6^3:D_4$ x 56, $C_6^3.D_4$ x 48, $C_6^3:D_4$ x 36, $C_6^3.D_4$ x 24, $C_6^3:C_2^3$ x 22, $C_6^3.C_2^3$ x 22, $C_6^3:D_4$ x 18, $D_6^2:D_6$ x 16, $C_2^3.S_3^3$ x 16, $C_6^3.C_2^3$ x 16, $C_2^3.S_3^3$ x 16, $C_6^3:D_4$ x 12, $C_6^3:C_2^3$ x 11, $C_6^3.D_4$ x 9, $C_6^3:C_2^3$ x 8, $C_6^3:D_4$ x 8, $C_6^3.D_4$ x 8, $C_6^3.D_4$ x 8, $C_6^3:C_2^3$ x 4, $C_2\times C_6^2:S_4$ x 4, $C_2\times C_6^2.S_4$ x 4, $C_6^3:D_4$ x 4, $C_6^3.C_2^3$ x 4, $D_6^2:D_6$ x 4, $C_2^3.S_3^3$ x 4, $D_6^2:D_6$ x 4, $C_2^3.S_3^3$ x 4, $C_6^3:D_4$ x 4, $C_6^3.D_4$ x 4, $C_6^3.C_2^3$ x 4, $C_6^3:D_4$ x 4, $C_6^3.C_2^3$ x 4, $C_6^3.D_4$ x 4, $C_6^2.(C_2\times S_4)$ x 4, $C_2^4:(S_3\times D_9)$ x 4, $(C_2\times C_6^2).S_4$ x 3, $C_6^3.D_4$ x 3, $C_6^3.D_4$ x 3, $C_6^3.D_4$ x 3, $C_6^2:(C_2\times S_4)$ x 2, $C_6^3:C_2^3$ x 2, $C_6^2:(C_2^2\times A_4)$ x 2, $C_6^2:(C_2\times S_4)$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3.C_2^3$ x 2, $C_6^3:D_4$ x 2, $C_3^3.D_4^2$ x 2, $C_6^3.C_2^3$ x 2, $C_3^3:D_4^2$ x 2, $C_3^3:D_4^2$ x 2, $C_3^3:D_4^2$ x 2, $C_6^3.C_2^3$ x 2, $C_6^3.C_2^3$ x 2, $C_6^3.C_2^3$ x 2, $(C_6\times D_6):S_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $(C_2^2\times C_6^2):C_{12}$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $(C_2\times C_6^2):S_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^2:(C_2^2\times A_4)$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2:S_4$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $(C_2\times C_6^2).S_4$, $C_2^3.S_3^3$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2.S_4$, $C_3\times C_2^4:D_{18}$, $C_6^3:D_4$, $C_6^3:D_4$, $C_6^3:D_4$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3:C_2^3$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $(C_2^3\times C_6^2):C_6$, $C_6^2:(C_2\times S_4)$, $C_6^3.D_4$, $(C_2^2\times C_6^2):C_{12}$, $(C_2^2\times C_6^2):A_4$, $C_6^2.(C_2^2\times A_4)$, $C_2^4:(S_3\times C_{18})$, $C_6^3.D_4$, $C_6^3.D_4$

Order 1296: $C_6^4$, $C_6^3:S_3$, $S_3\times C_6^3$, $C_6^3.C_6$, $C_3\times C_6^2:A_4$, $(C_2\times C_6^3):C_3$

Order 864: $C_2^2\times C_6^3$ x 2, $C_2^5:\He_3$

Order 648: $C_3\times C_6^3$

Order 576: $C_6^2.C_2^4$

Order 432: $C_2\times C_6^3$ x 2, $D_6\times C_6^2$, $C_6^3.C_2$, $C_6^2:A_4$

Order 324: $C_3^2\times C_6^2$, $C_3^3:D_6$

Order 288: $C_2^3\times C_6^2$ x 2

Order 256: $C_2^5:D_4$

Order 243: $C_3^4:C_3$

Order 216: $C_6^3$ x 2

Order 192: $C_2^4:D_6$

Order 162: $C_3^3\times C_6$

Order 144: $C_2^2\times C_6^2$ x 2, $C_6^2:C_2^2$

Order 108: $C_3\times C_6^2$ x 2

Order 96: $C_2^4\times C_6$ x 2

Order 81: $C_3^4$

Order 72: $C_2\times C_6^2$ x 2

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

Order 48: $C_2^3\times C_6$ x 2, $C_2^2\times D_6$

Order 36: $C_6^2$ x 2

Order 32: $C_2^5$

Order 27: $C_3^3$ x 2

Order 24: $C_2^2\times C_6$ x 2

Order 18: $C_3\times C_6$ x 2

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 2

Order 9: $C_3^2$ x 2

Order 8: $C_2^3$

Order 6: $C_6$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 36 (order at least 1728) are shown, as well as all normal subgroups of any index.

Order 62208: $C_6^3.(D_6\times S_4)$

Order 31104: $C_2\times C_6^4.D_6$, $C_2\times C_6^4:D_6$, $C_6^3.(S_3\times S_4)$, $C_6^4:(C_4\times S_3)$, $(S_3\times C_6^3):S_4$, $C_6^3.(A_4\times D_6)$, $C_6^4:D_{12}$

Order 20736: $C_6^4.C_2^4$, $C_3^3:C_2\wr D_6$, $C_3^3:C_2\wr D_6$

Order 15552: $C_6^4.D_6$, $C_2\times C_2^4:\He_3.C_3.C_6$, $C_2\times C_6^4.C_6$, $C_2\times C_6^4.S_3$, $(S_3\times C_6^3):D_6$, $C_6^4:D_6$, $C_6^4:D_6$, $(C_3\times C_6^3).S_4$, $C_2\times C_6^4:C_6$, $C_6^4.C_{12}$

Order 10368: $C_6^3.(C_2\times S_4)$ x 2, $C_6^3:(C_2\times S_4)$ x 2, $C_3^4.C_2^6.C_2$, $C_6^4.C_2^3$, $C_3^4.C_2^6.C_2$, $C_3^4.C_2^5.C_2^2$, $C_3^4.C_2^6.C_2$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_2\times C_2^4:\He_3.D_6$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^4.C_2^3$, $C_6^3.(C_2\times S_4)$, $(C_2\times C_6^3):D_{12}$, $(D_6\times C_6^2):S_4$, $C_2\times C_6^3:S_4$, $(C_3\times C_6^2):\GL(2,\mathbb{Z}/4)$, $(C_3\times C_6^2):\GL(2,\mathbb{Z}/4)$, $(C_2\times C_6^3):D_{12}$, $C_6^3.(C_2^2\times A_4)$, $C_6^2.(A_4\times D_{12})$, $C_6^2.(D_6\times S_4)$, $(D_6\times C_6^2):S_4$, $(C_2\times C_6^3):S_4$

Order 7776: $C_2\times C_6^2.C_3^3.C_2^2$ x 2, $C_3^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_6^4.S_3$, $C_6^4.C_6$, $C_2\times C_6^4.C_3$, $C_2^4:\He_3.C_3.C_6$, $C_6^3:S_3^2$, $C_6^3.S_3^2$, $C_3^4:\GL(2,\mathbb{Z}/4)$, $(S_3\times C_6^3):C_6$, $C_6^3.S_3^2$, $C_6^4:S_3$, $C_6^4:C_6$

Order 6912: $C_6^3.C_2^3.C_2^2$ x 2, $C_6^3.C_2^3.C_2^2$, $C_6^3.C_2^3.C_2^2$, $C_2^5.S_3^3$, $C_6^2.(D_4\times S_4)$

Order 5184: $C_6^3:S_4$ x 5, $C_3^4.C_2^5.C_2$ x 3, $C_3^4.C_2^5.C_2$ x 3, $C_3^4.C_2^5.C_2$ x 3, $C_3^4.C_2^4.C_2^2$ x 2, $C_6^4.C_2^2$ x 2, $C_6^4.C_2^2$ x 2, $C_3^4.D_4^2$ x 2, $C_3^4.C_2^5.C_2$ x 2, $C_6^4.C_2^2$ x 2, $C_3^4.C_2^5.C_2$ x 2, $C_6^4.C_2^2$ x 2, $C_6^4.C_2^2$ x 2, $C_3^4.D_4^2$ x 2, $C_3^4.C_2^5.C_2$ x 2, $C_6^4.C_2^2$ x 2, $C_6^2.(A_4\times D_6)$ x 2, $(C_2\times C_6^3):D_6$ x 2, $C_6^3.S_4$ x 2, $C_6^3:S_4$ x 2, $(C_2\times C_6^3):C_{12}$ x 2, $C_6^4.C_2^2$, $C_2\times (C_2\times C_6^3).C_6$, $C_2^5\times C_3:(C_3^3:C_2)$, $C_6^4.C_2^2$, $C_2\times C_2^4:\He_3.C_6$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_2\times (C_3\times C_2^4:C_9).C_6$, $C_6^4.C_2^2$, $C_3^4.C_2^5.C_2$, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^3.C_2^3$, $C_3^3.C_3:D_4:D_4$, $C_6^4.C_2^2$, $C_3^4.C_2\wr C_2^2$, $(C_3\times C_6^3).C_2^3$, $C_6^4.C_2^2$, $C_3^4.C_2^5.C_2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^2.C_6^2.C_2^2$, $(C_3\times C_6^3).C_2^3$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_6^4.C_2^2$, $C_2\times C_2^4.C_3^3:S_3$, $C_2^4:\He_3.D_6$, $C_3^4.C_2^4.C_2^2$, $C_3^3:(D_4\times S_4)$, $C_3^3:(D_4\times S_4)$, $C_6^3:S_4$, $C_6^2.(C_6\times S_4)$, $(S_3\times C_6^2):S_4$, $C_6^3.S_4$, $C_6^2.(A_4\times D_6)$, $C_6^3.S_4$, $C_6^3:S_4$, $(D_6\times C_6^2):A_4$, $(C_2\times C_6^3):C_{12}$, $C_2\times C_6^3:A_4$

Order 3888: $C_2\times C_3^4.A_4.C_2$ x 3, $C_2\times C_6^2.C_3^3.C_2$ x 2, $C_6^2.C_3^3.C_2^2$ x 2, $C_2\times C_6^2.C_3^3.C_2$ x 2, $C_3\times C_6^3:S_3$ x 2, $(C_3\times C_6^3).S_3$ x 2, $C_6^4.C_3$, $(D_6\times C_3^3):D_6$, $S_3\times C_3^3:S_4$, $C_6\wr C_3\times S_3$, $C_3\times C_6^3.C_6$

Order 3456: $C_6^3.C_2^4$ x 5, $C_3^3.C_2^6.C_2$ x 5, $C_6^3.D_4.C_2$ x 4, $(C_2\times C_6^3).C_2^3$ x 4, $C_6^3.C_2^4$ x 3, $C_6^3.C_2^2.C_2^2$ x 2, $C_6^3.C_2^2.C_2^2$ x 2, $C_6^3.C_2^3.C_2$ x 2, $C_6^3.C_2^2.C_2^2$ x 2, $C_6^3.C_2^4$ x 2, $C_6^3.C_2^4$ x 2, $(C_2\times C_6^3).C_2^3$ x 2, $(C_2\times C_6^3).C_2^3$ x 2, $C_6^3.D_4.C_2$ x 2, $C_3^3.C_2^5.C_2^2$ x 2, $C_6^2.(C_4\times S_4)$ x 2, $C_6^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_6^3.C_2^2.C_2^2$, $(C_2\times C_6^3).C_2^3$, $C_6^3.D_4.C_2$, $C_6^3.C_2^4$, $C_6^3.C_2^3.C_2$, $C_6^3.C_2^2.C_2^2$, $C_6^3.C_2^2.C_2^2$, $C_2\times C_6^3.C_2^3$, $C_6^3.C_2^4$, $C_6^3.C_2^2.C_2^2$, $C_6^3.C_2^2.C_2^2$, $C_3^3.C_2^5.C_2^2$, $C_6^3.C_2^4$, $(C_2\times C_6^3).C_2^3$, $C_6^3.D_4.C_2$, $(C_3\times C_6^2).C_2^5$, $C_6^3.C_2^2.C_2^2$, $C_6^3.C_2^4$, $C_2\times (C_2^3\times C_6).S_3^2$, $(C_2\times C_6^3).C_2^3$, $C_3^3.C_2^6.C_2$, $C_6^3.D_4.C_2$, $C_6^3.C_2^3.C_2$, $C_6^2.(C_4\times S_4)$, $(C_2^3\times C_6^2):D_6$, $(C_2^2\times C_6^2):D_{12}$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $C_6^2:(C_2^2\times S_4)$, $C_2^4:(S_3\times D_{18})$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $(C_2^2\times C_6^2):D_{12}$, $C_6^2.(C_2^2\times S_4)$, $C_3^2:C_2\wr C_6$, $C_3^2:C_2\wr C_6$, $(C_2^2\times C_6^2):S_4$

Order 2592: $C_3^4.C_2^4.C_2$ x 16, $C_3^4.C_2^4.C_2$ x 12, $C_2^4\times C_3:(C_3^3:C_2)$ x 10, $C_3^4.C_2^4.C_2$ x 8, $C_3^4.C_2^4.C_2$ x 7, $C_6^4.C_2$ x 6, $C_3^4.C_2^4.C_2$ x 5, $C_6^4.C_2$ x 5, $(C_3\times C_6^2):S_4$ x 5, $C_6^4.C_2$ x 4, $C_3^4.C_2^4.C_2$ x 4, $C_3^4.C_2^4.C_2$ x 4, $C_6^3:A_4$ x 4, $C_6^3:D_6$ x 4, $C_6^4.C_2$ x 3, $C_6^4.C_2$ x 3, $C_6^4.C_2$ x 3, $C_3^4.C_2^4.C_2$ x 3, $(C_3\times C_6^3).C_2^2$ x 3, $C_6^4.C_2$ x 2, $C_3^4.(C_4\times D_4)$ x 2, $C_2\times C_6^2.S_3^2$ x 2, $(C_3^2\times C_6^2).C_2^3$ x 2, $C_6^2.C_6^2.C_2$ x 2, $C_6^2.C_6^2.C_2$ x 2, $C_6^4.C_2$ x 2, $C_3^3.C_2^3:D_6$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_6^4.C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $(C_3\times C_6^3).C_2^2$ x 2, $C_3^4.(C_4\times D_4)$ x 2, $C_3^4.C_4:D_4$ x 2, $C_3^4.C_2^2:D_4$ x 2, $C_3^4.C_2^2:D_4$ x 2, $C_3^3.D_6:Q_8$ x 2, $C_3^4.(C_4.D_4)$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_6^2.C_6^2.C_2$ x 2, $C_3^4.(C_4\times D_4)$ x 2, $C_3^4.C_2^2:D_4$ x 2, $C_3^3.D_6:C_4.C_2$ x 2, $(C_3\times C_6^3).C_2^2$ x 2, $C_3^4.(C_4\times D_4)$ x 2, $C_3^3.D_6:C_4.C_2$ x 2, $(C_2\times C_6^3):C_6$ x 2, $C_6^3.D_6$ x 2, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 2, $(C_3\times C_6^2):S_4$ x 2, $C_2\times (C_2\times C_6^3).C_3$, $(C_3\times C_6^3).C_2^2$, $(C_3\times C_6^3).C_2^2$, $(C_3\times C_6^3).C_2^2$, $C_2^3\times S_3\times C_3^3:C_2$, $C_3^4.C_2^2:D_4$, $C_3^3.C_4:D_{12}$, $C_3^4.C_2^2\wr C_2$, $C_3.C_6^3.C_2^2$, $C_6.C_6^3.C_2$, $C_2\times C_6^4$, $C_3^4.C_2^2:D_4$, $C_3.C_6^3.C_2^2$, $(C_3\times C_6^3).C_2^2$, $C_3^3.D_6:C_4.C_2$, $C_3^4.D_4.C_2^2$, $C_3^4.C_2^3.C_2^2$, $C_3^4.C_2^2\wr C_2$, $C_3^3.C_2^3:D_6$, $(C_3\times C_6^3).C_2^2$, $C_3^4.D_4.C_2^2$, $C_3^4.C_2^3.C_2^2$, $(C_3\times C_6^3).C_2^2$, $C_2^4:\He_3.C_6$, $(C_3^2\times C_6^2).C_2^3$, $(C_3\times C_6^3).C_2^2$, $C_2^4.C_3^3:S_3$, $(C_3^2\times C_6^2).C_2^3$, $(C_3^3\times C_6).C_2^4$, $C_3.C_6^3.C_2^2$, $(C_3\times C_6^3).C_2^2$, $(C_3\times C_6^3).C_2^2$, $C_3^3.D_6:C_4.C_2$, $C_3^3.(D_6.D_4)$, $(C_3\times C_6^3).C_2^2$, $(C_2\times C_6^3).C_6$, $C_6^2.C_6^2.C_2$, $C_6^2.(C_6\times C_{12})$, $(C_3\times C_2^4:C_9).C_6$, $C_2\times (C_2\times C_6^3).C_3$, $(C_3\times C_6^3).C_2^2$, $C_6^3.D_6$, $C_6^3.D_6$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $C_6^3:D_6$, $C_6^3.D_6$, $C_6^2:(C_3\times D_{12})$, $C_6^3.D_6$, $C_6^3:D_6$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $(C_2\times C_6^3):C_6$, $C_6^2.(C_6\times A_4)$, $C_3\times C_6^2.S_4$, $C_3\times C_6^2:S_4$, $(C_2\times C_6^3):C_6$, $C_6^3:A_4$

Order 2304: $C_6^2.C_2^4.C_2^2$ x 3, $C_6^2.C_2^4.C_2^2$ x 2, $C_6^2.C_2^4.C_2^2$, $C_2^4:(S_3\times S_4)$, $(C_2^2\times D_{12}):S_4$

Order 1944: $C_6^3:C_3^2$ x 3, $C_3^3:(S_3\times A_4)$ x 3, $C_3^4:S_4$ x 2, $C_3^4:S_4$ x 2, $C_3^4:S_4$ x 2, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $S_3\times C_3^3:A_4$, $(C_3\times \He_3):D_{12}$, $C_3^4:(C_4\times S_3)$, $C_3^3:(C_3\times D_{12})$, $(D_6\times C_3^3):S_3$, $C_3^4:D_{12}$

Order 1728: $C_6^3.D_4$ x 30, $C_6^3:D_4$ x 28, $C_6^3:D_4$ x 22, $C_6^3.C_2^3$ x 16, $C_6^3.D_4$ x 12, $C_6^3:C_2^3$ x 10, $C_6^3:D_4$ x 10, $C_6^3.C_2^3$ x 10, $C_6^3:D_4$ x 8, $D_6^2:D_6$ x 8, $C_2^3.S_3^3$ x 8, $C_6^3:C_2^3$ x 7, $C_6^3.D_4$ x 5, $C_2\times C_6^2:S_4$ x 4, $C_2\times C_6^2.S_4$ x 4, $C_6^3:D_4$ x 4, $C_6^3:D_4$ x 4, $C_6^3.C_2^3$ x 4, $C_6^3.D_4$ x 4, $C_6^3.D_4$ x 4, $C_6^3:C_2^3$ x 3, $(C_2\times C_6^2).S_4$ x 3, $C_6^3.D_4$ x 3, $C_6^3.D_4$ x 3, $C_6^3.D_4$ x 3, $C_6^3:C_2^3$ x 2, $C_6^3:C_2^3$ x 2, $C_6^2:(C_2^2\times A_4)$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3.C_2^3$ x 2, $C_6^3:D_4$ x 2, $C_6^3.C_2^3$ x 2, $C_3^3.D_4^2$ x 2, $C_6^3.C_2^3$ x 2, $C_3^3:D_4^2$ x 2, $C_3^3:D_4^2$ x 2, $C_3^3:D_4^2$ x 2, $C_6^3.C_2^3$ x 2, $C_6^3.C_2^3$ x 2, $C_6^3.C_2^3$ x 2, $D_6^2:D_6$ x 2, $C_2^3.S_3^3$ x 2, $D_6^2:D_6$ x 2, $C_2^3.S_3^3$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $(C_2^2\times C_6^2):C_{12}$ x 2, $C_6^3:D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.C_2^3$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_2^3.S_3^3$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^2:(C_2^2\times A_4)$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2:S_4$, $C_6^2:(C_2\times S_4)$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $(C_2\times C_6^2).S_4$, $C_2^3.S_3^3$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2.S_4$, $C_3\times C_2^4:D_{18}$, $C_6^2:(C_2\times S_4)$, $C_6^3:D_4$, $C_6^3:D_4$, $C_6^3:D_4$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3:C_2^3$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $C_6^3.C_2^3$, $(C_2^3\times C_6^2):C_6$, $(C_6\times D_6):S_4$, $C_6^2:(C_2\times S_4)$, $C_6^3.D_4$, $C_6^3:D_4$, $C_6^3.C_2^3$, $(C_2^2\times C_6^2):C_{12}$, $(C_2^2\times C_6^2):A_4$, $C_6^2.(C_2^2\times A_4)$, $C_2^4:(S_3\times C_{18})$, $(C_2\times C_6^2):S_4$, $C_6^2.(C_2\times S_4)$, $C_2^4:(S_3\times D_9)$, $C_6^3.D_4$, $C_6^3.D_4$

Order 1296: $C_6^4$, $C_6^3:S_3$, $S_3\times C_6^3$, $C_6^3.C_6$, $C_3\times C_6^2:A_4$, $(C_2\times C_6^3):C_3$

Order 864: $C_2^2\times C_6^3$ x 2, $C_2^5:\He_3$

Order 648: $C_3\times C_6^3$

Order 576: $C_6^2.C_2^4$

Order 432: $C_2\times C_6^3$ x 2, $D_6\times C_6^2$, $C_6^3.C_2$, $C_6^2:A_4$

Order 324: $C_3^2\times C_6^2$, $C_3^3:D_6$

Order 288: $C_2^3\times C_6^2$ x 2

Order 256: $C_2^5:D_4$

Order 243: $C_3^4:C_3$

Order 216: $C_6^3$ x 2

Order 192: $C_2^4:D_6$

Order 162: $C_3^3\times C_6$

Order 144: $C_2^2\times C_6^2$ x 2, $C_6^2:C_2^2$

Order 108: $C_3\times C_6^2$ x 2

Order 96: $C_2^4\times C_6$ x 2

Order 81: $C_3^4$

Order 72: $C_2\times C_6^2$ x 2

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

Order 48: $C_2^3\times C_6$ x 2, $C_2^2\times D_6$

Order 36: $C_6^2$ x 2

Order 32: $C_2^5$

Order 27: $C_3^3$ x 2

Order 24: $C_2^2\times C_6$ x 2

Order 18: $C_3\times C_6$ x 2

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 2

Order 9: $C_3^2$ x 2

Order 8: $C_2^3$

Order 6: $C_6$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 62208: $C_6^3.(D_6\times S_4)$ ($C_1$)

Order 31104: $C_2\times C_6^4.D_6$ ($C_2$), $C_2\times C_6^4:D_6$ ($C_2$), $C_6^3.(S_3\times S_4)$ ($C_2$), $C_6^4:(C_4\times S_3)$ ($C_2$), $(S_3\times C_6^3):S_4$ ($C_2$), $C_6^3.(A_4\times D_6)$ ($C_2$), $C_6^4:D_{12}$ ($C_2$)

Order 15552: $C_2\times C_2^4:\He_3.C_3.C_6$ ($C_2^2$), $C_2\times C_6^4.C_6$ ($C_2^2$), $C_2\times C_6^4.S_3$ ($C_2^2$), $C_6^4:D_6$ ($C_2^2$), $(C_3\times C_6^3).S_4$ ($C_2^2$), $C_2\times C_6^4:C_6$ ($C_2^2$), $C_6^4.C_{12}$ ($C_2^2$)

Order 10368: $C_3^4.C_2^6.C_2$ ($S_3$), $(C_2\times C_6^3):D_{12}$ ($S_3$), $(D_6\times C_6^2):S_4$ ($S_3$)

Order 7776: $C_6^4.S_3$ ($D_4$) x 2, $C_2\times C_6^4.C_3$ ($C_2^3$)

Order 5184: $C_2^5\times C_3:(C_3^3:C_2)$ ($D_6$), $C_2\times C_2^4:\He_3.C_6$ ($D_6$), $C_6^4.C_2^2$ ($D_6$), $C_6^4.C_2^2$ ($D_6$), $C_6^3.S_4$ ($D_6$), $C_6^3:S_4$ ($D_6$), $C_6^3:S_4$ ($D_6$), $(C_2\times C_6^3):C_{12}$ ($D_6$), $(D_6\times C_6^2):A_4$ ($D_6$)

Order 3888: $C_6^4.C_3$ ($C_2\times D_4$)

Order 2592: $C_2\times (C_2\times C_6^3).C_3$ ($C_2\times D_6$), $C_2\times C_6^4$ ($C_2\times D_6$), $C_3^4.D_4.C_2^2$ ($S_4$), $C_6^3:A_4$ ($C_2\times D_6$)

Order 1728: $C_2\times C_6^2:S_4$ ($S_3^2$), $C_6^3:D_4$ ($S_3^2$), $C_6^3.D_4$ ($S_3^2$)

Order 1296: $C_6^4$ ($S_3\times D_4$), $C_6^3:S_3$ ($C_2\times S_4$), $S_3\times C_6^3$ ($C_2\times S_4$), $C_6^3.C_6$ ($C_2\times S_4$), $C_3\times C_6^2:A_4$ ($S_3\times D_4$), $(C_2\times C_6^3):C_3$ ($S_3\times D_4$)

Order 864: $C_2^2\times C_6^3$ ($S_3\times D_6$) x 2, $C_2^5:\He_3$ ($S_3\times D_6$)

Order 648: $C_3\times C_6^3$ ($C_2^2\times S_4$)

Order 576: $C_6^2.C_2^4$ ($C_3^2:D_6$)

Order 432: $C_2\times C_6^3$ ($D_6:D_6$), $C_2\times C_6^3$ ($C_{12}:D_6$), $D_6\times C_6^2$ ($S_3\times S_4$), $C_6^3.C_2$ ($S_3\times S_4$), $C_6^2:A_4$ ($D_6:D_6$)

Order 324: $C_3^2\times C_6^2$ ($D_4\times S_4$), $C_3^3:D_6$ ($C_2^3:S_4$)

Order 288: $C_2^3\times C_6^2$ ($S_3^3$), $C_2^3\times C_6^2$ ($C_6.S_3^2$)

Order 216: $C_6^3$ ($D_6\times S_4$) x 2

Order 192: $C_2^4:D_6$ ($C_3^3:D_6$)

Order 162: $C_3^3\times C_6$ ($C_2^4:S_4$)

Order 144: $C_2^2\times C_6^2$ ($D_6:S_3^2$), $C_2^2\times C_6^2$ ($C_6^2:D_6$), $C_6^2:C_2^2$ ($C_6^2:D_6$)

Order 108: $C_3\times C_6^2$ ($\GL(2,\mathbb{Z}/4):S_3$), $C_3\times C_6^2$ ($D_{12}:S_4$)

Order 96: $C_2^4\times C_6$ ($C_3.S_3^3$), $C_2^4\times C_6$ ($C_2\times C_3^3:D_6$)

Order 81: $C_3^4$ ($C_2\wr D_6$)

Order 72: $C_2\times C_6^2$ ($S_4\times S_3^2$), $C_2\times C_6^2$ ($C_2\times C_6^2:D_6$)

Order 54: $C_3^2\times C_6$ ($S_3\times C_2^3:S_4$) x 2

Order 48: $C_2^3\times C_6$ ($C_6^2.S_3^2$), $C_2^3\times C_6$ ($C_6.S_3^3$), $C_2^2\times D_6$ ($C_6^2.S_3^2$)

Order 36: $C_6^2$ ($C_2^3.S_3^3$), $C_6^2$ ($C_6^2:(C_2\times S_4)$)

Order 32: $C_2^5$ ($C_3^2.S_3^3$)

Order 27: $C_3^3$ ($C_2^4:(S_3\times S_4)$), $C_3^3$ ($(C_2^2\times D_{12}):S_4$)

Order 24: $C_2^2\times C_6$ ($C_6^2.(C_2\times S_3^2)$), $C_2^2\times C_6$ ($S_3\times C_6^2:D_6$)

Order 18: $C_3\times C_6$ ($C_6^2.(C_2^2\times S_4)$), $C_3\times C_6$ ($D_6^2:S_4$)

Order 16: $C_2^4$ ($(D_6\times C_3^3):D_6$)

Order 12: $C_2\times C_6$ ($C_6^2.C_4:S_3^2$), $C_2\times C_6$ ($C_3^3:(D_4\times S_4)$)

Order 9: $C_3^2$ ($C_2^5.S_3^3$), $C_3^2$ ($C_6^2.(D_4\times S_4)$)

Order 8: $C_2^3$ ($C_6^2.S_3^3$)

Order 6: $C_6$ ($C_6^3:(C_2\times S_4)$), $C_6$ ($C_6^2.(D_6\times S_4)$)

Order 4: $C_2^2$ ($(S_3\times C_6^3):D_6$)

Order 3: $C_3$ ($C_2^4:\He_3.D_6.C_2^2$), $C_3$ ($C_3^3:C_2\wr D_6$)

Order 2: $C_2$ ($C_6^4:(C_2\times D_6)$)

Order 1: $C_1$ ($C_6^3.(D_6\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 62208: $C_6^3.(D_6\times S_4)$ ($C_1$)

Order 31104: $C_2\times C_6^4.D_6$ ($C_2$), $C_2\times C_6^4:D_6$ ($C_2$), $C_6^3.(S_3\times S_4)$ ($C_2$), $C_6^4:(C_4\times S_3)$ ($C_2$), $(S_3\times C_6^3):S_4$ ($C_2$), $C_6^3.(A_4\times D_6)$ ($C_2$), $C_6^4:D_{12}$ ($C_2$)

Order 15552: $C_2\times C_2^4:\He_3.C_3.C_6$ ($C_2^2$), $C_2\times C_6^4.C_6$ ($C_2^2$), $C_2\times C_6^4.S_3$ ($C_2^2$), $C_6^4:D_6$ ($C_2^2$), $(C_3\times C_6^3).S_4$ ($C_2^2$), $C_2\times C_6^4:C_6$ ($C_2^2$), $C_6^4.C_{12}$ ($C_2^2$)

Order 10368: $C_3^4.C_2^6.C_2$ ($S_3$), $(C_2\times C_6^3):D_{12}$ ($S_3$), $(D_6\times C_6^2):S_4$ ($S_3$)

Order 7776: $C_6^4.S_3$ ($D_4$), $C_2\times C_6^4.C_3$ ($C_2^3$)

Order 5184: $C_2^5\times C_3:(C_3^3:C_2)$ ($D_6$), $C_2\times C_2^4:\He_3.C_6$ ($D_6$), $C_6^4.C_2^2$ ($D_6$), $C_6^4.C_2^2$ ($D_6$), $C_6^3.S_4$ ($D_6$), $C_6^3:S_4$ ($D_6$), $C_6^3:S_4$ ($D_6$), $(C_2\times C_6^3):C_{12}$ ($D_6$), $(D_6\times C_6^2):A_4$ ($D_6$)

Order 3888: $C_6^4.C_3$ ($C_2\times D_4$)

Order 2592: $C_2\times (C_2\times C_6^3).C_3$ ($C_2\times D_6$), $C_2\times C_6^4$ ($C_2\times D_6$), $C_3^4.D_4.C_2^2$ ($S_4$), $C_6^3:A_4$ ($C_2\times D_6$)

Order 1728: $C_2\times C_6^2:S_4$ ($S_3^2$), $C_6^3:D_4$ ($S_3^2$), $C_6^3.D_4$ ($S_3^2$)

Order 1296: $C_6^4$ ($S_3\times D_4$), $C_6^3:S_3$ ($C_2\times S_4$), $S_3\times C_6^3$ ($C_2\times S_4$), $C_6^3.C_6$ ($C_2\times S_4$), $C_3\times C_6^2:A_4$ ($S_3\times D_4$), $(C_2\times C_6^3):C_3$ ($S_3\times D_4$)

Order 864: $C_2^2\times C_6^3$ ($S_3\times D_6$) x 2, $C_2^5:\He_3$ ($S_3\times D_6$)

Order 648: $C_3\times C_6^3$ ($C_2^2\times S_4$)

Order 576: $C_6^2.C_2^4$ ($C_3^2:D_6$)

Order 432: $C_2\times C_6^3$ ($D_6:D_6$), $C_2\times C_6^3$ ($C_{12}:D_6$), $D_6\times C_6^2$ ($S_3\times S_4$), $C_6^3.C_2$ ($S_3\times S_4$), $C_6^2:A_4$ ($D_6:D_6$)

Order 324: $C_3^2\times C_6^2$ ($D_4\times S_4$), $C_3^3:D_6$ ($C_2^3:S_4$)

Order 288: $C_2^3\times C_6^2$ ($S_3^3$), $C_2^3\times C_6^2$ ($C_6.S_3^2$)

Order 216: $C_6^3$ ($D_6\times S_4$) x 2

Order 192: $C_2^4:D_6$ ($C_3^3:D_6$)

Order 162: $C_3^3\times C_6$ ($C_2^4:S_4$)

Order 144: $C_2^2\times C_6^2$ ($D_6:S_3^2$), $C_2^2\times C_6^2$ ($C_6^2:D_6$), $C_6^2:C_2^2$ ($C_6^2:D_6$)

Order 108: $C_3\times C_6^2$ ($\GL(2,\mathbb{Z}/4):S_3$), $C_3\times C_6^2$ ($D_{12}:S_4$)

Order 96: $C_2^4\times C_6$ ($C_3.S_3^3$), $C_2^4\times C_6$ ($C_2\times C_3^3:D_6$)

Order 81: $C_3^4$ ($C_2\wr D_6$)

Order 72: $C_2\times C_6^2$ ($S_4\times S_3^2$), $C_2\times C_6^2$ ($C_2\times C_6^2:D_6$)

Order 54: $C_3^2\times C_6$ ($S_3\times C_2^3:S_4$) x 2

Order 48: $C_2^3\times C_6$ ($C_6^2.S_3^2$), $C_2^3\times C_6$ ($C_6.S_3^3$), $C_2^2\times D_6$ ($C_6^2.S_3^2$)

Order 36: $C_6^2$ ($C_2^3.S_3^3$), $C_6^2$ ($C_6^2:(C_2\times S_4)$)

Order 32: $C_2^5$ ($C_3^2.S_3^3$)

Order 27: $C_3^3$ ($C_2^4:(S_3\times S_4)$), $C_3^3$ ($(C_2^2\times D_{12}):S_4$)

Order 24: $C_2^2\times C_6$ ($C_6^2.(C_2\times S_3^2)$), $C_2^2\times C_6$ ($S_3\times C_6^2:D_6$)

Order 18: $C_3\times C_6$ ($C_6^2.(C_2^2\times S_4)$), $C_3\times C_6$ ($D_6^2:S_4$)

Order 16: $C_2^4$ ($(D_6\times C_3^3):D_6$)

Order 12: $C_2\times C_6$ ($C_6^2.C_4:S_3^2$), $C_2\times C_6$ ($C_3^3:(D_4\times S_4)$)

Order 9: $C_3^2$ ($C_2^5.S_3^3$), $C_3^2$ ($C_6^2.(D_4\times S_4)$)

Order 8: $C_2^3$ ($C_6^2.S_3^3$)

Order 6: $C_6$ ($C_6^3:(C_2\times S_4)$), $C_6$ ($C_6^2.(D_6\times S_4)$)

Order 4: $C_2^2$ ($(S_3\times C_6^3):D_6$)

Order 3: $C_3$ ($C_2^4:\He_3.D_6.C_2^2$), $C_3$ ($C_3^3:C_2\wr D_6$)

Order 2: $C_2$ ($C_6^4:(C_2\times D_6)$)

Order 1: $C_1$ ($C_6^3.(D_6\times S_4)$)

Series

Derived series

$C_6^3.(D_6\times S_4)$

$\rhd$

$C_2\times C_6^4.C_3$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_1$

Chief series

$C_6^3.(D_6\times S_4)$

$\rhd$

$C_6^3.(A_4\times D_6)$

$\rhd$

$C_2\times C_6^4.C_6$

$\rhd$

$C_2\times C_6^4.C_3$

$\rhd$

$C_6^4.C_3$

$\rhd$

$C_3\times C_6^2:A_4$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^3.(D_6\times S_4)$

$\rhd$

$C_2\times C_6^4.C_3$

$\rhd$

$C_6^4.C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

The $324 \times 324$ character table is not available for this group.

Rational character table

The $305 \times 305$ rational character table is not available for this group.