Abstract group 6912.ia: $D_6.S_4^2$

• Label: 6912.ia
• Order: \( 2^{8} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$D_6.S_4^2$
Order:	\(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^3\times A_4^2.C_2^2\times S_3$, of order \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \)
Composition factors:	$C_2$ x 8, $C_3$ x 3
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	8	12	24
Elements	1	559	242	720	2438	768	1224	960	6912
Conjugacy classes	1	17	7	14	35	16	16	14	120
Divisions	1	17	7	14	35	8	16	7	105
Autjugacy classes	1	13	7	12	27	8	12	7	87

Dimension	1	2	3	4	6	8	9	12	16	18	24
Irr. complex chars.	8	20	16	18	24	7	8	12	1	4	2	120
Irr. rational chars.	8	12	16	14	16	9	8	12	2	4	4	105

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$2419200$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{2}=d^{6}=e^{2}=g^{12}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(5,7), (4,7)(5,6), (4,5,6)(9,12,13)(10,14,11), (4,6,7,5)(8,9,15,10)(11,14,13,12) \!\cdots\! \rangle$
Direct product:	$S_3$ $\, \times\, $ $S_4$ $\, \times\, $ $\GL(2,3)$
Semidirect product:	$(Q_8\times S_4)$ $\,\rtimes\,$ $S_3^2$	$Q_8$ $\,\rtimes\,$ $(S_4\times S_3^2)$	$(Q_8:D_6^2)$ $\,\rtimes\,$ $S_3$	$(C_6.S_4^2)$ $\,\rtimes\,$ $C_2$	all 53
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6$ . $S_4^2$	$(D_6\times S_4)$ . $S_4$	$C_2$ . $(S_3\times S_4^2)$	$C_6$ . $(C_2\times S_4^2)$	all 13
Aut. group:	$\Aut(S_3\times C_6^2)$	$\Aut(C_6^2:S_3^2)$

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

Homology

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

Subgroups

There are 150202 subgroups in 4918 conjugacy classes, 83 normal (67 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times S_4^2$
Commutator:	$G' \simeq$ $C_6.A_4^2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_3\times S_4^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_6.C_2^4$	$G/\operatorname{Fit} \simeq$ $S_3\times D_6$
Radical:	$R \simeq$ $D_6.S_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $D_6\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 6912: $D_6.S_4^2$

Order 3456: $C_6.S_4^2$, $S_3\times A_4:\GL(2,3)$, $S_3\times S_4\times \SL(2,3)$, $C_6.S_4^2$, $C_6.S_4^2$, $S_3\times A_4\times \GL(2,3)$, $C_6.S_4^2$

Order 2304: $S_3\times D_4\times \GL(2,3)$, $S_3\times \SD_{16}\times S_4$, $C_2^2.S_4^2$

Order 1728: $A_4:D_6^2$, $D_6.A_4^2$, $(C_3\times A_4):\GL(2,3)$, $C_3\times A_4:\GL(2,3)$, $A_4\times C_3:\GL(2,3)$, $C_3:S_4\times \SL(2,3)$, $C_3\times A_4\times \GL(2,3)$, $C_3\times S_4\times \SL(2,3)$, $S_3^2\times \GL(2,3)$

Order 1152: $S_4\times \GL(2,3)$ x 6, $Q_8:D_6^2$ x 2, $C_3:(D_4\times \GL(2,3))$ x 2, $C_3:D_4\times \GL(2,3)$ x 2, $D_6.\GL(2,\mathbb{Z}/4)$ x 2, $S_3\times Q_8:S_4$, $C_2^3.D_6^2$, $\GL(2,\mathbb{Z}/4):D_6$, $C_2\times A_4:\GL(2,3)$, $C_2\times A_4\times \GL(2,3)$, $C_2\times S_4\times \SL(2,3)$, $S_3\times A_4\times \SD_{16}$, $S_3\times D_4\times \SL(2,3)$, $C_{24}:(C_2\times S_4)$, $D_4\times C_3:\GL(2,3)$, $C_3\times \SD_{16}\times S_4$, $C_3\times D_4\times \GL(2,3)$, $Q_8:(S_3\times S_4)$, $Q_8:S_3\times S_4$, $D_6.\GL(2,\mathbb{Z}/4)$, $C_3:(\SD_{16}\times S_4)$, $C_3:\SD_{16}\times S_4$, $D_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:(C_2\times S_4)$, $C_{24}:C_2\times S_4$, $S_3\times C_8:S_4$, $S_3\times C_8\times S_4$, $D_{12}:\GL(2,3)$, $D_{12}\times \GL(2,3)$, $S_3\times C_4:\GL(2,3)$, $C_4\times S_3\times \GL(2,3)$

Order 864: $S_4\times S_3^2$ x 8, $S_3\times C_6:S_4$ x 2, $C_6\times S_3\times S_4$ x 2, $\SL(2,3):S_3^2$ x 2, $C_3\times S_3\times \GL(2,3)$ x 2, $D_6^2:C_6$, $C_2\times A_4:S_3^2$, $C_6:S_3\times S_4$, $C_6.A_4^2$, $S_3^2\times \SL(2,3)$, $\SL(2,3):S_3^2$, $C_3:S_3\times \GL(2,3)$

Order 768: $C_2\times D_4\times \GL(2,3)$, $C_2\times \SD_{16}\times S_4$, $D_6.D_4^2$

Order 576: $D_6\times \GL(2,3)$ x 12, $A_4:\GL(2,3)$ x 5, $S_4\times \SL(2,3)$ x 4, $A_4\times \GL(2,3)$ x 4, $C_2^2:D_6^2$ x 2, $C_2\times D_6\times \SL(2,3)$ x 2, $C_2\times C_6:\GL(2,3)$ x 2, $C_2\times C_6\times \GL(2,3)$ x 2, $C_3:D_4\times \SL(2,3)$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $D_6:\GL(2,3)$ x 2, $D_6:\GL(2,3)$ x 2, $(C_6\times \GL(2,3)):C_2$ x 2, $C_4:D_6^2$, $S_3\times Q_8:A_4$, $S_3\times Q_8\times A_4$, $S_3\times D_4\times A_4$, $(C_2\times C_6):\GL(2,3)$, $C_3\times Q_8:S_4$, $Q_8\times C_3:S_4$, $D_4\times C_3:S_4$, $C_3\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_6$, $C_2^2.A_4^2$, $\GL(2,\mathbb{Z}/4):S_3$, $C_2^4:S_3^2$, $S_3\times \GL(2,\mathbb{Z}/4)$, $D_{12}:S_4$, $S_4\times D_{12}$, $S_3\times C_4:S_4$, $C_4\times S_3\times S_4$, $S_3\times A_4:Q_8$, $C_3:(Q_8\times S_4)$, $C_3:Q_8\times S_4$, $C_3\times A_4\times \SD_{16}$, $C_3\times D_4\times \SL(2,3)$, $\SD_{16}\times S_3^2$, $A_4\times Q_8:S_3$, $A_4\times C_3:\SD_{16}$, $A_4\times C_{24}:C_2$, $S_3\times C_8\times A_4$, $D_{12}\times \SL(2,3)$, $C_4\times S_3\times \SL(2,3)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:S_4$, $C_{24}:S_4$, $C_{12}:\GL(2,3)$, $C_{12}:\GL(2,3)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:S_4$, $C_{24}\times S_4$, $C_{12}:\GL(2,3)$, $C_{12}\times \GL(2,3)$, $D_6.\GL(2,3)$, $C_3:(C_4\times \GL(2,3))$, $C_3:C_4\times \GL(2,3)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $S_3\times A_4:C_8$, $C_{12}.(C_2\times S_4)$, $C_{12}.(C_2\times S_4)$, $C_3:(C_8\times S_4)$, $C_3:C_8\times S_4$

Order 432: $A_4:S_3^2$ x 8, $C_6^2:D_6$ x 8, $A_4\times S_3^2$ x 4, $A_4:S_3^2$ x 4, $C_6^2:D_6$ x 4, $C_6\times S_3\times A_4$ x 2, $C_6^2:D_6$ x 2, $C_3\times S_3\times \SL(2,3)$ x 2, $C_3^2:\GL(2,3)$ x 2, $C_6:S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$, $C_2\times S_3^3$, $C_3:S_3\times \SL(2,3)$, $C_3^2:\GL(2,3)$, $C_3^2\times \GL(2,3)$

Order 384: $\SD_{16}\times S_4$ x 9, $D_4\times \GL(2,3)$ x 9, $C_2^3\times \GL(2,3)$ x 2, $C_{24}:C_2^4$ x 2, $C_2^3:\GL(2,3)$ x 2, $C_6.D_4^2$ x 2, $C_6.D_4^2$ x 2, $S_3\times C_8:D_4$ x 2, $C_6.D_4^2$ x 2, $S_3\times C_2^2:\SD_{16}$ x 2, $C_6.D_4^2$ x 2, $S_3\times Q_8:D_4$ x 2, $C_2^3:\GL(2,3)$, $C_2\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $S_3\times D_4\times Q_8$, $S_3\times D_4^2$, $C_2\times A_4\times \SD_{16}$, $C_2\times D_4\times \SL(2,3)$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_{12}$, $C_2\times C_8\times S_4$, $C_2\times C_4:\GL(2,3)$, $C_2\times C_4\times \GL(2,3)$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $D_{12}:\SD_{16}$, $S_3\times C_8:D_4$, $D_{12}:\SD_{16}$, $S_3\times C_4:\SD_{16}$, $D_{12}:\SD_{16}$, $S_3\times Q_8:D_4$, $\SD_{16}\times D_{12}$, $C_4\times S_3\times \SD_{16}$, $C_6.D_4^2$, $S_3\times C_8\times D_4$

Order 288: $D_6\times S_4$ x 23, $S_3\times \GL(2,3)$ x 19, $D_4\times S_3^2$ x 17, $D_6\times \SL(2,3)$ x 6, $C_6:\GL(2,3)$ x 6, $C_6\times \GL(2,3)$ x 6, $D_6^2:C_2$ x 4, $A_4\times \SL(2,3)$ x 4, $C_6^2:C_2^3$ x 2, $Q_8:C_6^2$ x 2, $D_6^2:C_2$ x 2, $D_6\times D_{12}$ x 2, $C_{24}:D_6$ x 2, $Q_8:S_3^2$ x 2, $D_{12}.D_6$ x 2, $C_{24}:D_6$ x 2, $C_2\times D_6^2$ x 2, $C_2\times A_4\times D_6$ x 2, $C_2\times C_6:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $C_3\times Q_8:A_4$, $C_2^3.C_6^2$, $C_2^3.C_6^2$, $Q_8\times S_3^2$, $D_{12}:D_6$, $C_2.D_6^2$, $A_4\times C_3:D_4$, $A_4\times D_{12}$, $C_4\times S_3\times A_4$, $A_4\times C_3:Q_8$, $C_3:\GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $C_{12}:S_4$, $C_{12}.S_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $C_{12}\times S_4$, $C_{12}.S_4$, $D_6:S_4$, $D_6:S_4$, $D_6.S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_{24}:D_6$, $A_4\times C_{24}$, $C_{12}\times \SL(2,3)$, $Q_8:S_3^2$, $D_4.S_3^2$, $C_{24}:D_6$, $C_8\times S_3^2$, $C_3:C_4\times \SL(2,3)$, $A_4\times C_3:C_8$, $C_6.\GL(2,3)$, $C_{12}.S_4$, $\SL(2,3):C_{12}$, $A_4:C_{24}$, $C_6^2:C_2^3$

Order 256: $C_4.D_4^2$

Order 216: $S_3^3$ x 8, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 4, $C_6:S_3^2$ x 3, $C_6\times S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_6^2:C_6$, $C_6:S_3^2$, $C_3^2\times \SL(2,3)$

Order 192: $D_6\times \SD_{16}$ x 26, $C_2^2\times \GL(2,3)$ x 17, $C_2^2:\GL(2,3)$ x 10, $D_4\times S_4$ x 7, $Q_8:S_4$ x 5, $A_4:\SD_{16}$ x 5, $C_8:S_4$ x 5, $C_8\times S_4$ x 5, $C_4:\GL(2,3)$ x 5, $C_4\times \GL(2,3)$ x 5, $Q_8\times S_4$ x 5, $A_4\times \SD_{16}$ x 5, $D_4\times \SL(2,3)$ x 5, $C_{24}:D_4$ x 4, $D_6:\SD_{16}$ x 4, $D_6:\SD_{16}$ x 4, $Q_8:D_{12}$ x 4, $D_6:\SD_{16}$ x 4, $C_{12}:C_2^4$ x 4, $D_6.Q_{16}$ x 3, $D_6.D_8$ x 3, $Q_8:S_4$ x 3, $C_2^4:D_6$ x 3, $D_{12}:D_4$ x 3, $S_3\times C_4:D_4$ x 3, $C_{24}:D_4$ x 2, $C_3\times C_2^2:\SD_{16}$ x 2, $C_3\times Q_8:D_4$ x 2, $(C_3\times Q_8):D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $C_{24}:D_4$ x 2, $(C_3\times Q_8):D_4$ x 2, $C_{12}.(C_2\times D_4)$ x 2, $C_{24}:D_4$ x 2, $C_{24}:D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $D_{12}.D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $C_8:D_{12}$ x 2, $D_{12}.D_4$ x 2, $D_6:\SD_{16}$ x 2, $D_6:\SD_{16}$ x 2, $C_{12}.(C_2\times D_4)$ x 2, $C_2^3.D_{12}$ x 2, $D_{12}.D_4$ x 2, $C_3:(C_8\times D_4)$ x 2, $D_6.\OD_{16}$ x 2, $C_{12}.C_2^4$ x 2, $C_2^3\times \SL(2,3)$ x 2, $C_{24}:C_2^3$ x 2, $C_3:D_4\times Q_8$ x 2, $C_{12}.C_2^4$ x 2, $C_{12}.C_2^4$ x 2, $C_{24}:C_2^3$ x 2, $C_{24}:C_2^3$ x 2, $C_3:(D_4\times Q_8)$ x 2, $S_3\times C_2^2:Q_8$ x 2, $C_2^4:D_6$ x 2, $C_2^4:D_6$ x 2, $D_4\times D_{12}$ x 2, $C_4^2:D_6$ x 2, $C_2\times C_4\times \SL(2,3)$, $C_2^3.S_4$, $C_2\times A_4:C_8$, $C_{24}:D_4$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}\times \SD_{16}$, $D_4\times C_{24}$, $C_3:C_4\times \SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $Q_8:D_{12}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_4^2.D_6$, $D_6.\SD_{16}$, $C_8:(C_4\times S_3)$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $D_{12}:C_8$, $C_4^2.D_6$, $Q_8:(C_4\times S_3)$, $C_3:(C_4\times \SD_{16})$, $C_8:D_{12}$, $C_{12}:\SD_{16}$, $C_8\times D_{12}$, $D_6.C_4^2$, $C_2^3\times S_4$, $C_2^3:\SL(2,3)$, $C_2\times Q_8\times A_4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^4.D_6$, $C_3\times D_4\times Q_8$, $C_3\times D_4^2$, $D_{12}:Q_8$, $C_4^2.D_6$, $C_3:D_4^2$, $C_4^2:D_6$, $Q_8\times D_{12}$, $C_4^2.D_6$, $C_4^2.D_6$, $C_2^3:C_{24}$

Order 144: $S_3\times S_4$ x 38, $D_6:D_6$ x 34, $D_6^2$ x 33, $C_{12}:D_6$ x 18, $S_3\times D_{12}$ x 18, $D_6:D_6$ x 17, $C_6:S_4$ x 14, $A_4\times D_6$ x 13, $C_6\times S_4$ x 13, $C_3:\GL(2,3)$ x 10, $C_{12}:D_6$ x 9, $C_{12}:D_6$ x 9, $C_4\times S_3^2$ x 9, $S_3\times \SL(2,3)$ x 9, $C_3\times \GL(2,3)$ x 9, $C_6^2:C_2^2$ x 4, $C_6^2:C_2^2$ x 4, $C_6:D_{12}$ x 4, $C_6\times \SL(2,3)$ x 3, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{24}:C_6$ x 2, $S_3\times C_{24}$ x 2, $C_6.D_{12}$ x 2, $C_6.D_{12}$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_6^2:C_2^2$ x 2, $C_2^2:C_6^2$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}.D_6$ x 2, $C_6\times D_{12}$ x 2, $C_{12}\times D_6$ x 2, $D_6:D_6$ x 2, $D_6.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:S_3$, $C_{24}:S_3$, $C_{12}.D_6$, $C_4:C_6^2$, $C_{12}.D_6$, $C_6:D_{12}$, $C_{12}:D_6$, $C_{12}\times A_4$, $C_2^2.S_3^2$, $C_{12}.D_6$, $A_4\times C_3:C_4$, $C_6.S_4$, $A_4:C_{12}$, $C_3^2\times \SD_{16}$

Order 128: $D_4\times \SD_{16}$ x 16, $C_2^3\times \SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_4^2.C_2^3$, $C_2\times D_4^2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.C_2^3$

Order 108: $C_3:S_3^2$ x 12, $C_3\times S_3^2$ x 12, $C_3:S_3^2$ x 4, $C_3^2:D_6$ x 3, $C_3^2\times D_6$ x 3, $C_3^2:D_6$, $C_3^2\times A_4$

Order 96: $D_4\times D_6$ x 60, $S_3\times \SD_{16}$ x 56, $C_2\times \GL(2,3)$ x 28, $C_6\times \SD_{16}$ x 14, $Q_8:D_6$ x 14, $C_6:\SD_{16}$ x 14, $C_8:D_6$ x 14, $C_8\times D_6$ x 13, $C_2^2\times S_4$ x 12, $Q_8\times D_6$ x 12, $C_2^2\times \SL(2,3)$ x 9, $C_2^2:D_{12}$ x 6, $C_2^3:D_6$ x 6, $D_6:D_4$ x 6, $D_6.D_4$ x 5, $\GL(2,\mathbb{Z}/4)$ x 5, $C_4\times S_4$ x 5, $D_6.D_4$ x 4, $D_6:D_4$ x 4, $D_6:C_8$ x 4, $C_2^3\times D_6$ x 4, $C_{12}:C_2^3$ x 4, $C_2^2\times D_{12}$ x 4, $C_{12}:C_2^3$ x 4, $D_4\times A_4$ x 4, $C_4:S_4$ x 4, $A_4:Q_8$ x 4, $C_{12}:D_4$ x 4, $C_2^3.D_6$ x 4, $C_2^3.D_6$ x 3, $C_8\times A_4$ x 3, $C_4\times \SL(2,3)$ x 3, $\SL(2,3):C_4$ x 3, $A_4:C_8$ x 3, $Q_8:C_{12}$ x 3, $D_4:C_{12}$ x 3, $C_6.Q_{16}$ x 3, $C_6.D_8$ x 3, $D_{12}:C_4$ x 3, $C_6.Q_{16}$ x 3, $Q_8\times A_4$ x 3, $C_4.D_{12}$ x 3, $C_{12}:D_4$ x 3, $D_{12}:C_4$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_4:D_{12}$ x 3, $C_{12}:Q_8$ x 2, $C_2^3.D_6$ x 2, $C_4\times D_{12}$ x 2, $C_2^2:C_{24}$ x 2, $C_{12}.D_4$ x 2, $C_6.C_2^4$ x 2, $C_{12}.D_4$ x 2, $C_6.C_2^4$ x 2, $Q_8:A_4$ x 2, $C_{24}:C_4$ x 2, $C_2^2\times C_{24}$ x 2, $C_{12}.D_4$ x 2, $C_2^4:C_6$ x 2, $D_4\times C_{12}$ x 2, $C_2^4:S_3$ x 2, $D_6:Q_8$ x 2, $C_{12}.D_4$ x 2, $C_{12}.Q_8$ x 2, $C_2^3.D_6$ x 2, $C_{12}.D_4$ x 2, $C_{12}.C_2^3$ x 2, $D_6:Q_8$ x 2, $D_6:Q_8$ x 2, $D_{12}:C_4$ x 2, $C_3:(C_4\times Q_8)$, $C_{12}:C_8$, $C_4:D_{12}$, $S_3\times C_4^2$, $C_{12}:Q_8$, $C_{12}:Q_8$, $C_8:C_{12}$, $C_4:C_{24}$, $C_4\times C_{24}$, $C_{24}:C_4$, $C_2^3\times A_4$, $C_2^3:C_{12}$, $C_2^2.S_4$, $C_{12}:Q_8$, $C_{12}:D_4$, $Q_8\times C_{12}$, $C_3:C_4\times Q_8$, $C_{12}:C_8$

Order 72: $S_3\times D_6$ x 110, $C_6\times D_6$ x 34, $C_6\wr C_2$ x 18, $C_3:D_{12}$ x 18, $C_6:D_6$ x 17, $C_3:S_4$ x 15, $S_3\times A_4$ x 13, $C_3\times S_4$ x 13, $C_3\times D_{12}$ x 10, $S_3\times C_{12}$ x 10, $C_6.D_6$ x 10, $C_6\times A_4$ x 9, $C_6^2:C_2$ x 9, $D_6:S_3$ x 9, $C_3\times \SL(2,3)$ x 7, $D_4\times C_3^2$ x 5, $C_3:D_{12}$ x 5, $C_{12}:S_3$ x 5, $C_6.D_6$ x 5, $C_2\times C_6^2$ x 2, $C_6:C_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_3:C_{24}$ x 2, $Q_8\times C_3^2$, $C_6\times C_{12}$, $C_6.D_6$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_2^2\times \SD_{16}$ x 33, $C_8:D_4$ x 16, $C_2^2:\SD_{16}$ x 16, $Q_8:D_4$ x 16, $D_4^2$ x 12, $D_4\times Q_8$ x 8, $C_8:D_4$ x 8, $C_4:\SD_{16}$ x 8, $Q_8:D_4$ x 8, $C_4\times \SD_{16}$ x 8, $C_8\times D_4$ x 8, $C_2^2.Q_{16}$ x 3, $C_2^2.D_8$ x 3, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_2^3:C_8$ x 2, $Q_8\times C_2^3$ x 2, $C_2^3\times C_8$ x 2, $C_2^3:Q_8$ x 2, $C_2^3:D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_2\times C_4\times C_8$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2.C_2^2$, $C_2^3.D_4$, $C_4^2.C_4$

Order 54: $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 6, $C_3^2:S_3$ x 2, $C_3^2\times C_6$

Order 48: $S_3\times D_4$ x 146, $C_2^2\times D_6$ x 63, $C_6:D_4$ x 56, $C_4\times D_6$ x 34, $C_6\times D_4$ x 31, $C_2\times D_{12}$ x 31, $C_2\times S_4$ x 29, $S_3\times Q_8$ x 21, $C_{24}:C_2$ x 18, $S_3\times C_8$ x 18, $C_3\times \SD_{16}$ x 18, $Q_8:S_3$ x 18, $C_3:\SD_{16}$ x 18, $\GL(2,3)$ x 15, $D_6:C_4$ x 10, $C_6:Q_8$ x 9, $C_2\times \SL(2,3)$ x 9, $C_6:C_8$ x 7, $C_2^2\times A_4$ x 7, $C_2\times C_{24}$ x 7, $C_6\times Q_8$ x 6, $C_6.D_4$ x 6, $C_2^2:C_{12}$ x 5, $C_6.D_4$ x 5, $C_2^3\times C_6$ x 4, $C_2^2\times C_{12}$ x 4, $C_6.C_2^3$ x 4, $C_4:C_{12}$ x 4, $C_{12}:C_4$ x 4, $C_4\times A_4$ x 3, $A_4:C_4$ x 3, $C_{12}:C_4$ x 3, $C_4\times C_{12}$

Order 36: $S_3^2$ x 82, $C_6\times S_3$ x 68, $C_6:S_3$ x 35, $C_6^2$ x 9, $C_3:C_{12}$ x 6, $C_3\times A_4$ x 6, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 3

Order 32: $C_2\times \SD_{16}$ x 99, $C_2^2\times D_4$ x 44, $C_4:D_4$ x 20, $C_2^2\times C_8$ x 16, $C_2^2\wr C_2$ x 16, $C_2^2\times Q_8$ x 15, $C_4\times D_4$ x 14, $D_4:C_4$ x 12, $C_2^2:Q_8$ x 12, $Q_8:C_4$ x 12, $C_2^2:C_8$ x 8, $C_4:Q_8$ x 6, $C_4:D_4$ x 6, $C_2^3:C_4$ x 5, $C_4\times C_8$ x 4, $C_4\times Q_8$ x 4, $C_2^2.D_4$ x 4, $C_8:C_4$ x 4, $C_4:C_8$ x 4, $C_2^3\times C_4$ x 3, $C_2^5$ x 2, $C_2\times C_4^2$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 220, $C_3:D_4$ x 78, $C_4\times S_3$ x 48, $D_{12}$ x 43, $C_3\times D_4$ x 43, $C_2^2\times C_6$ x 32, $C_2\times C_{12}$ x 18, $C_6:C_4$ x 18, $S_4$ x 15, $C_3:Q_8$ x 12, $C_2\times A_4$ x 11, $C_3\times Q_8$ x 8, $C_{24}$ x 7, $C_3:C_8$ x 7, $\SL(2,3)$ x 6

Order 18: $C_3\times S_3$ x 38, $C_3:S_3$ x 21, $C_3\times C_6$ x 14

Order 16: $C_2\times D_4$ x 130, $\SD_{16}$ x 52, $C_2\times Q_8$ x 29, $C_2^4$ x 27, $C_2^2\times C_4$ x 26, $C_2\times C_8$ x 25, $C_2^2:C_4$ x 20, $C_4:C_4$ x 14, $C_4^2$ x 4

Order 12: $D_6$ x 192, $C_2\times C_6$ x 62, $C_{12}$ x 16, $C_3:C_4$ x 16, $A_4$ x 4

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 88, $D_4$ x 71, $C_2\times C_4$ x 43, $Q_8$ x 13, $C_8$ x 8

Order 6: $S_3$ x 42, $C_6$ x 35

Order 4: $C_2^2$ x 76, $C_4$ x 14

Order 3: $C_3$ x 7

Order 2: $C_2$ x 17

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6912: $D_6.S_4^2$

Order 3456: $C_6.S_4^2$, $S_3\times A_4:\GL(2,3)$, $S_3\times S_4\times \SL(2,3)$, $C_6.S_4^2$, $C_6.S_4^2$, $S_3\times A_4\times \GL(2,3)$, $C_6.S_4^2$

Order 2304: $S_3\times D_4\times \GL(2,3)$, $S_3\times \SD_{16}\times S_4$, $C_2^2.S_4^2$

Order 1728: $A_4:D_6^2$, $D_6.A_4^2$, $(C_3\times A_4):\GL(2,3)$, $C_3\times A_4:\GL(2,3)$, $A_4\times C_3:\GL(2,3)$, $C_3:S_4\times \SL(2,3)$, $C_3\times A_4\times \GL(2,3)$, $C_3\times S_4\times \SL(2,3)$, $S_3^2\times \GL(2,3)$

Order 1152: $S_4\times \GL(2,3)$ x 6, $Q_8:D_6^2$ x 2, $C_3:(D_4\times \GL(2,3))$ x 2, $C_3:D_4\times \GL(2,3)$ x 2, $D_6.\GL(2,\mathbb{Z}/4)$ x 2, $S_3\times Q_8:S_4$, $C_2^3.D_6^2$, $\GL(2,\mathbb{Z}/4):D_6$, $C_2\times A_4:\GL(2,3)$, $C_2\times A_4\times \GL(2,3)$, $C_2\times S_4\times \SL(2,3)$, $S_3\times A_4\times \SD_{16}$, $S_3\times D_4\times \SL(2,3)$, $C_{24}:(C_2\times S_4)$, $D_4\times C_3:\GL(2,3)$, $C_3\times \SD_{16}\times S_4$, $C_3\times D_4\times \GL(2,3)$, $Q_8:(S_3\times S_4)$, $Q_8:S_3\times S_4$, $D_6.\GL(2,\mathbb{Z}/4)$, $C_3:(\SD_{16}\times S_4)$, $C_3:\SD_{16}\times S_4$, $D_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:(C_2\times S_4)$, $C_{24}:C_2\times S_4$, $S_3\times C_8:S_4$, $S_3\times C_8\times S_4$, $D_{12}:\GL(2,3)$, $D_{12}\times \GL(2,3)$, $S_3\times C_4:\GL(2,3)$, $C_4\times S_3\times \GL(2,3)$

Order 864: $S_3\times C_6:S_4$ x 2, $C_6\times S_3\times S_4$ x 2, $\SL(2,3):S_3^2$ x 2, $C_3\times S_3\times \GL(2,3)$ x 2, $D_6^2:C_6$, $C_2\times A_4:S_3^2$, $C_6:S_3\times S_4$, $S_4\times S_3^2$, $C_6.A_4^2$, $S_3^2\times \SL(2,3)$, $\SL(2,3):S_3^2$, $C_3:S_3\times \GL(2,3)$

Order 768: $C_2\times D_4\times \GL(2,3)$, $C_2\times \SD_{16}\times S_4$, $D_6.D_4^2$

Order 576: $D_6\times \GL(2,3)$ x 12, $A_4:\GL(2,3)$ x 5, $S_4\times \SL(2,3)$ x 4, $A_4\times \GL(2,3)$ x 4, $C_2^2:D_6^2$ x 2, $C_2\times D_6\times \SL(2,3)$ x 2, $C_2\times C_6:\GL(2,3)$ x 2, $C_2\times C_6\times \GL(2,3)$ x 2, $C_3:D_4\times \SL(2,3)$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $D_6:\GL(2,3)$ x 2, $D_6:\GL(2,3)$ x 2, $(C_6\times \GL(2,3)):C_2$ x 2, $C_4:D_6^2$, $S_3\times Q_8:A_4$, $S_3\times Q_8\times A_4$, $S_3\times D_4\times A_4$, $(C_2\times C_6):\GL(2,3)$, $C_3\times Q_8:S_4$, $Q_8\times C_3:S_4$, $D_4\times C_3:S_4$, $C_3\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_6$, $C_2^2.A_4^2$, $\GL(2,\mathbb{Z}/4):S_3$, $C_2^4:S_3^2$, $S_3\times \GL(2,\mathbb{Z}/4)$, $D_{12}:S_4$, $S_4\times D_{12}$, $S_3\times C_4:S_4$, $C_4\times S_3\times S_4$, $S_3\times A_4:Q_8$, $C_3:(Q_8\times S_4)$, $C_3:Q_8\times S_4$, $C_3\times A_4\times \SD_{16}$, $C_3\times D_4\times \SL(2,3)$, $\SD_{16}\times S_3^2$, $A_4\times Q_8:S_3$, $A_4\times C_3:\SD_{16}$, $A_4\times C_{24}:C_2$, $S_3\times C_8\times A_4$, $D_{12}\times \SL(2,3)$, $C_4\times S_3\times \SL(2,3)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:S_4$, $C_{24}:S_4$, $C_{12}:\GL(2,3)$, $C_{12}:\GL(2,3)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:S_4$, $C_{24}\times S_4$, $C_{12}:\GL(2,3)$, $C_{12}\times \GL(2,3)$, $D_6.\GL(2,3)$, $C_3:(C_4\times \GL(2,3))$, $C_3:C_4\times \GL(2,3)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $S_3\times A_4:C_8$, $C_{12}.(C_2\times S_4)$, $C_{12}.(C_2\times S_4)$, $C_3:(C_8\times S_4)$, $C_3:C_8\times S_4$

Order 432: $C_6\times S_3\times A_4$ x 2, $C_6^2:D_6$ x 2, $A_4:S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_3\times S_3\times \SL(2,3)$ x 2, $C_3^2:\GL(2,3)$ x 2, $C_6:S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$, $C_2\times S_3^3$, $A_4\times S_3^2$, $A_4:S_3^2$, $C_6^2:D_6$, $C_3:S_3\times \SL(2,3)$, $C_3^2:\GL(2,3)$, $C_3^2\times \GL(2,3)$

Order 384: $\SD_{16}\times S_4$ x 9, $D_4\times \GL(2,3)$ x 9, $C_2^3\times \GL(2,3)$ x 2, $C_{24}:C_2^4$ x 2, $C_2^3:\GL(2,3)$ x 2, $C_6.D_4^2$ x 2, $C_6.D_4^2$ x 2, $S_3\times C_8:D_4$ x 2, $C_6.D_4^2$ x 2, $S_3\times C_2^2:\SD_{16}$ x 2, $C_6.D_4^2$ x 2, $S_3\times Q_8:D_4$ x 2, $C_2^3:\GL(2,3)$, $C_2\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $S_3\times D_4\times Q_8$, $S_3\times D_4^2$, $C_2\times A_4\times \SD_{16}$, $C_2\times D_4\times \SL(2,3)$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_{12}$, $C_2\times C_8\times S_4$, $C_2\times C_4:\GL(2,3)$, $C_2\times C_4\times \GL(2,3)$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $D_{12}:\SD_{16}$, $S_3\times C_8:D_4$, $D_{12}:\SD_{16}$, $S_3\times C_4:\SD_{16}$, $D_{12}:\SD_{16}$, $S_3\times Q_8:D_4$, $\SD_{16}\times D_{12}$, $C_4\times S_3\times \SD_{16}$, $C_6.D_4^2$, $S_3\times C_8\times D_4$

Order 288: $S_3\times \GL(2,3)$ x 19, $D_6\times S_4$ x 13, $D_6\times \SL(2,3)$ x 6, $C_6:\GL(2,3)$ x 6, $C_6\times \GL(2,3)$ x 6, $D_6^2:C_2$ x 4, $A_4\times \SL(2,3)$ x 4, $D_4\times S_3^2$ x 3, $C_6^2:C_2^3$ x 2, $Q_8:C_6^2$ x 2, $D_6^2:C_2$ x 2, $D_6\times D_{12}$ x 2, $C_{24}:D_6$ x 2, $Q_8:S_3^2$ x 2, $D_{12}.D_6$ x 2, $C_{24}:D_6$ x 2, $C_2\times D_6^2$ x 2, $C_2\times A_4\times D_6$ x 2, $C_2\times C_6:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $C_3\times Q_8:A_4$, $C_2^3.C_6^2$, $C_2^3.C_6^2$, $Q_8\times S_3^2$, $D_{12}:D_6$, $C_2.D_6^2$, $A_4\times C_3:D_4$, $A_4\times D_{12}$, $C_4\times S_3\times A_4$, $A_4\times C_3:Q_8$, $C_3:\GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $C_{12}:S_4$, $C_{12}.S_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $C_{12}\times S_4$, $C_{12}.S_4$, $D_6:S_4$, $D_6:S_4$, $D_6.S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_{24}:D_6$, $A_4\times C_{24}$, $C_{12}\times \SL(2,3)$, $Q_8:S_3^2$, $D_4.S_3^2$, $C_{24}:D_6$, $C_8\times S_3^2$, $C_3:C_4\times \SL(2,3)$, $A_4\times C_3:C_8$, $C_6.\GL(2,3)$, $C_{12}.S_4$, $\SL(2,3):C_{12}$, $A_4:C_{24}$, $C_6^2:C_2^3$

Order 256: $C_4.D_4^2$

Order 216: $C_6:S_3^2$ x 3, $C_6\times S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_6^2:C_6$, $C_6:S_3^2$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2\times S_4$, $S_3^3$, $C_3^2\times \SL(2,3)$

Order 192: $D_6\times \SD_{16}$ x 26, $C_2^2\times \GL(2,3)$ x 17, $C_2^2:\GL(2,3)$ x 10, $D_4\times S_4$ x 7, $Q_8:S_4$ x 5, $A_4:\SD_{16}$ x 5, $C_8:S_4$ x 5, $C_8\times S_4$ x 5, $C_4:\GL(2,3)$ x 5, $C_4\times \GL(2,3)$ x 5, $Q_8\times S_4$ x 5, $A_4\times \SD_{16}$ x 5, $D_4\times \SL(2,3)$ x 5, $C_{24}:D_4$ x 4, $D_6:\SD_{16}$ x 4, $D_6:\SD_{16}$ x 4, $Q_8:D_{12}$ x 4, $D_6:\SD_{16}$ x 4, $C_{12}:C_2^4$ x 4, $D_6.Q_{16}$ x 3, $D_6.D_8$ x 3, $Q_8:S_4$ x 3, $C_2^4:D_6$ x 3, $D_{12}:D_4$ x 3, $S_3\times C_4:D_4$ x 3, $C_{24}:D_4$ x 2, $C_3\times C_2^2:\SD_{16}$ x 2, $C_3\times Q_8:D_4$ x 2, $(C_3\times Q_8):D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $C_{24}:D_4$ x 2, $(C_3\times Q_8):D_4$ x 2, $C_{12}.(C_2\times D_4)$ x 2, $C_{24}:D_4$ x 2, $C_{24}:D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $D_{12}.D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $C_8:D_{12}$ x 2, $D_{12}.D_4$ x 2, $D_6:\SD_{16}$ x 2, $D_6:\SD_{16}$ x 2, $C_{12}.(C_2\times D_4)$ x 2, $C_2^3.D_{12}$ x 2, $D_{12}.D_4$ x 2, $C_3:(C_8\times D_4)$ x 2, $D_6.\OD_{16}$ x 2, $C_{12}.C_2^4$ x 2, $C_2^3\times \SL(2,3)$ x 2, $C_{24}:C_2^3$ x 2, $C_3:D_4\times Q_8$ x 2, $C_{12}.C_2^4$ x 2, $C_{12}.C_2^4$ x 2, $C_{24}:C_2^3$ x 2, $C_{24}:C_2^3$ x 2, $C_3:(D_4\times Q_8)$ x 2, $S_3\times C_2^2:Q_8$ x 2, $C_2^4:D_6$ x 2, $C_2^4:D_6$ x 2, $D_4\times D_{12}$ x 2, $C_4^2:D_6$ x 2, $C_2\times C_4\times \SL(2,3)$, $C_2^3.S_4$, $C_2\times A_4:C_8$, $C_{24}:D_4$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}\times \SD_{16}$, $D_4\times C_{24}$, $C_3:C_4\times \SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $Q_8:D_{12}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_4^2.D_6$, $D_6.\SD_{16}$, $C_8:(C_4\times S_3)$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $D_{12}:C_8$, $C_4^2.D_6$, $Q_8:(C_4\times S_3)$, $C_3:(C_4\times \SD_{16})$, $C_8:D_{12}$, $C_{12}:\SD_{16}$, $C_8\times D_{12}$, $D_6.C_4^2$, $C_2^3\times S_4$, $C_2^3:\SL(2,3)$, $C_2\times Q_8\times A_4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^4.D_6$, $C_3\times D_4\times Q_8$, $C_3\times D_4^2$, $D_{12}:Q_8$, $C_4^2.D_6$, $C_3:D_4^2$, $C_4^2:D_6$, $Q_8\times D_{12}$, $C_4^2.D_6$, $C_4^2.D_6$, $C_2^3:C_{24}$

Order 144: $D_6^2$ x 17, $S_3\times S_4$ x 11, $C_6:S_4$ x 10, $C_3:\GL(2,3)$ x 10, $A_4\times D_6$ x 9, $C_6\times S_4$ x 9, $S_3\times \SL(2,3)$ x 9, $C_3\times \GL(2,3)$ x 9, $D_6:D_6$ x 8, $C_{12}:D_6$ x 6, $C_6^2:C_2^2$ x 4, $C_6^2:C_2^2$ x 4, $D_6:D_6$ x 4, $C_6:D_{12}$ x 4, $S_3\times D_{12}$ x 4, $C_{12}:D_6$ x 3, $C_6\times \SL(2,3)$ x 3, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{24}:C_6$ x 2, $S_3\times C_{24}$ x 2, $C_6.D_{12}$ x 2, $C_6.D_{12}$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_6^2:C_2^2$ x 2, $C_2^2:C_6^2$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}.D_6$ x 2, $C_6\times D_{12}$ x 2, $C_{12}\times D_6$ x 2, $D_6:D_6$ x 2, $D_6.D_6$ x 2, $C_{12}:D_6$ x 2, $C_4\times S_3^2$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:S_3$, $C_{24}:S_3$, $C_{12}.D_6$, $C_4:C_6^2$, $C_{12}.D_6$, $C_6:D_{12}$, $C_{12}:D_6$, $C_{12}\times A_4$, $C_2^2.S_3^2$, $C_{12}.D_6$, $A_4\times C_3:C_4$, $C_6.S_4$, $A_4:C_{12}$, $C_3^2\times \SD_{16}$

Order 128: $D_4\times \SD_{16}$ x 16, $C_2^3\times \SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_4^2.C_2^3$, $C_2\times D_4^2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.C_2^3$

Order 108: $C_3^2:D_6$ x 3, $C_3^2\times D_6$ x 3, $C_3:S_3^2$ x 3, $C_3\times S_3^2$ x 3, $C_3^2:D_6$, $C_3^2\times A_4$, $C_3:S_3^2$

Order 96: $S_3\times \SD_{16}$ x 56, $D_4\times D_6$ x 40, $C_2\times \GL(2,3)$ x 28, $C_6\times \SD_{16}$ x 14, $Q_8:D_6$ x 14, $C_6:\SD_{16}$ x 14, $C_8:D_6$ x 14, $C_8\times D_6$ x 13, $Q_8\times D_6$ x 12, $C_2^2\times S_4$ x 9, $C_2^2\times \SL(2,3)$ x 9, $C_2^2:D_{12}$ x 6, $C_2^3:D_6$ x 6, $D_6:D_4$ x 6, $D_6.D_4$ x 5, $\GL(2,\mathbb{Z}/4)$ x 5, $C_4\times S_4$ x 5, $D_6.D_4$ x 4, $D_6:D_4$ x 4, $D_6:C_8$ x 4, $C_2^3\times D_6$ x 4, $C_{12}:C_2^3$ x 4, $C_2^2\times D_{12}$ x 4, $C_{12}:C_2^3$ x 4, $D_4\times A_4$ x 4, $C_4:S_4$ x 4, $A_4:Q_8$ x 4, $C_{12}:D_4$ x 4, $C_2^3.D_6$ x 4, $C_2^3.D_6$ x 3, $C_8\times A_4$ x 3, $C_4\times \SL(2,3)$ x 3, $\SL(2,3):C_4$ x 3, $A_4:C_8$ x 3, $Q_8:C_{12}$ x 3, $D_4:C_{12}$ x 3, $C_6.Q_{16}$ x 3, $C_6.D_8$ x 3, $D_{12}:C_4$ x 3, $C_6.Q_{16}$ x 3, $Q_8\times A_4$ x 3, $C_4.D_{12}$ x 3, $C_{12}:D_4$ x 3, $D_{12}:C_4$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_4:D_{12}$ x 3, $C_{12}:Q_8$ x 2, $C_2^3.D_6$ x 2, $C_4\times D_{12}$ x 2, $C_2^2:C_{24}$ x 2, $C_{12}.D_4$ x 2, $C_6.C_2^4$ x 2, $C_{12}.D_4$ x 2, $C_6.C_2^4$ x 2, $Q_8:A_4$ x 2, $C_{24}:C_4$ x 2, $C_2^2\times C_{24}$ x 2, $C_{12}.D_4$ x 2, $C_2^4:C_6$ x 2, $D_4\times C_{12}$ x 2, $C_2^4:S_3$ x 2, $D_6:Q_8$ x 2, $C_{12}.D_4$ x 2, $C_{12}.Q_8$ x 2, $C_2^3.D_6$ x 2, $C_{12}.D_4$ x 2, $C_{12}.C_2^3$ x 2, $D_6:Q_8$ x 2, $D_6:Q_8$ x 2, $D_{12}:C_4$ x 2, $C_3:(C_4\times Q_8)$, $C_{12}:C_8$, $C_4:D_{12}$, $S_3\times C_4^2$, $C_{12}:Q_8$, $C_{12}:Q_8$, $C_8:C_{12}$, $C_4:C_{24}$, $C_4\times C_{24}$, $C_{24}:C_4$, $C_2^3\times A_4$, $C_2^3:C_{12}$, $C_2^2.S_4$, $C_{12}:Q_8$, $C_{12}:D_4$, $Q_8\times C_{12}$, $C_3:C_4\times Q_8$, $C_{12}:C_8$

Order 72: $S_3\times D_6$ x 42, $C_6\times D_6$ x 22, $C_6:D_6$ x 11, $C_6\times A_4$ x 8, $C_3:S_4$ x 8, $C_6\wr C_2$ x 8, $S_3\times A_4$ x 7, $C_3\times S_4$ x 7, $C_3\times \SL(2,3)$ x 7, $C_3:D_{12}$ x 6, $C_6^2:C_2$ x 4, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $C_6.D_6$ x 4, $D_4\times C_3^2$ x 3, $D_6:S_3$ x 3, $C_2\times C_6^2$ x 2, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2, $C_6:C_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_6.D_6$ x 2, $C_3:C_{24}$ x 2, $Q_8\times C_3^2$, $C_6\times C_{12}$, $C_6.D_6$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_2^2\times \SD_{16}$ x 33, $C_8:D_4$ x 16, $C_2^2:\SD_{16}$ x 16, $Q_8:D_4$ x 16, $D_4^2$ x 12, $D_4\times Q_8$ x 8, $C_8:D_4$ x 8, $C_4:\SD_{16}$ x 8, $Q_8:D_4$ x 8, $C_4\times \SD_{16}$ x 8, $C_8\times D_4$ x 8, $C_2^2.Q_{16}$ x 3, $C_2^2.D_8$ x 3, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_2^3:C_8$ x 2, $Q_8\times C_2^3$ x 2, $C_2^3\times C_8$ x 2, $C_2^3:Q_8$ x 2, $C_2^3:D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_2\times C_4\times C_8$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2.C_2^2$, $C_2^3.D_4$, $C_4^2.C_4$

Order 54: $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3, $C_3^2\times C_6$, $C_3^2:S_3$

Order 48: $S_3\times D_4$ x 63, $C_2^2\times D_6$ x 41, $C_6:D_4$ x 38, $C_4\times D_6$ x 24, $C_6\times D_4$ x 23, $S_3\times Q_8$ x 21, $C_2\times D_{12}$ x 21, $C_{24}:C_2$ x 18, $S_3\times C_8$ x 18, $C_3\times \SD_{16}$ x 18, $Q_8:S_3$ x 18, $C_3:\SD_{16}$ x 18, $C_2\times S_4$ x 17, $\GL(2,3)$ x 15, $D_6:C_4$ x 10, $C_6:Q_8$ x 9, $C_2\times \SL(2,3)$ x 9, $C_6:C_8$ x 7, $C_2\times C_{24}$ x 7, $C_2^2\times A_4$ x 6, $C_6\times Q_8$ x 6, $C_6.D_4$ x 6, $C_2^2:C_{12}$ x 5, $C_6.D_4$ x 5, $C_2^3\times C_6$ x 4, $C_2^2\times C_{12}$ x 4, $C_6.C_2^3$ x 4, $C_4:C_{12}$ x 4, $C_{12}:C_4$ x 4, $C_4\times A_4$ x 3, $A_4:C_4$ x 3, $C_{12}:C_4$ x 3, $C_4\times C_{12}$

Order 36: $C_6\times S_3$ x 40, $S_3^2$ x 22, $C_6:S_3$ x 21, $C_6^2$ x 8, $C_3\times A_4$ x 6, $C_3:C_{12}$ x 4, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2

Order 32: $C_2\times \SD_{16}$ x 99, $C_2^2\times D_4$ x 38, $C_4:D_4$ x 20, $C_2^2\times C_8$ x 16, $C_2^2\wr C_2$ x 16, $C_2^2\times Q_8$ x 15, $C_4\times D_4$ x 14, $D_4:C_4$ x 12, $C_2^2:Q_8$ x 12, $Q_8:C_4$ x 12, $C_2^2:C_8$ x 8, $C_4:Q_8$ x 6, $C_4:D_4$ x 6, $C_2^3:C_4$ x 5, $C_4\times C_8$ x 4, $C_4\times Q_8$ x 4, $C_2^2.D_4$ x 4, $C_8:C_4$ x 4, $C_4:C_8$ x 4, $C_2^3\times C_4$ x 3, $C_2^5$ x 2, $C_2\times C_4^2$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 112, $C_3:D_4$ x 41, $C_3\times D_4$ x 27, $C_4\times S_3$ x 26, $C_2^2\times C_6$ x 25, $D_{12}$ x 22, $C_2\times C_{12}$ x 14, $C_6:C_4$ x 14, $C_3:Q_8$ x 12, $C_2\times A_4$ x 9, $S_4$ x 9, $C_3\times Q_8$ x 8, $C_{24}$ x 7, $C_3:C_8$ x 7, $\SL(2,3)$ x 6

Order 18: $C_3\times S_3$ x 20, $C_3\times C_6$ x 12, $C_3:S_3$ x 11

Order 16: $C_2\times D_4$ x 95, $\SD_{16}$ x 52, $C_2\times Q_8$ x 29, $C_2\times C_8$ x 25, $C_2^2\times C_4$ x 23, $C_2^4$ x 21, $C_2^2:C_4$ x 20, $C_4:C_4$ x 14, $C_4^2$ x 4

Order 12: $D_6$ x 94, $C_2\times C_6$ x 46, $C_{12}$ x 12, $C_3:C_4$ x 12, $A_4$ x 4

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 58, $D_4$ x 51, $C_2\times C_4$ x 34, $Q_8$ x 13, $C_8$ x 8

Order 6: $C_6$ x 27, $S_3$ x 24

Order 4: $C_2^2$ x 50, $C_4$ x 12

Order 3: $C_3$ x 7

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6912: $D_6.S_4^2$ ($C_1$)

Order 3456: $C_6.S_4^2$ ($C_2$), $S_3\times A_4:\GL(2,3)$ ($C_2$), $S_3\times S_4\times \SL(2,3)$ ($C_2$), $C_6.S_4^2$ ($C_2$), $C_6.S_4^2$ ($C_2$), $S_3\times A_4\times \GL(2,3)$ ($C_2$), $C_6.S_4^2$ ($C_2$)

Order 1728: $D_6.A_4^2$ ($C_2^2$), $(C_3\times A_4):\GL(2,3)$ ($C_2^2$), $C_3\times A_4:\GL(2,3)$ ($C_2^2$), $A_4\times C_3:\GL(2,3)$ ($C_2^2$), $C_3:S_4\times \SL(2,3)$ ($C_2^2$), $C_3\times A_4\times \GL(2,3)$ ($C_2^2$), $C_3\times S_4\times \SL(2,3)$ ($C_2^2$)

Order 1152: $C_2^3.D_6^2$ ($S_3$), $Q_8:D_6^2$ ($S_3$), $S_4\times \GL(2,3)$ ($S_3$)

Order 864: $C_6.A_4^2$ ($C_2^3$)

Order 576: $S_3\times Q_8\times A_4$ ($D_6$), $C_2\times D_6\times \SL(2,3)$ ($D_6$), $Q_8\times C_3:S_4$ ($D_6$), $C_2\times C_6:\GL(2,3)$ ($D_6$), $C_3\times Q_8\times S_4$ ($D_6$), $C_2\times C_6\times \GL(2,3)$ ($D_6$), $A_4:\GL(2,3)$ ($D_6$), $S_4\times \SL(2,3)$ ($D_6$), $A_4\times \GL(2,3)$ ($D_6$)

Order 288: $C_2^3.C_6^2$ ($C_2\times D_6$), $Q_8:C_6^2$ ($C_2\times D_6$), $A_4\times \SL(2,3)$ ($C_2\times D_6$), $S_3\times \GL(2,3)$ ($S_4$), $D_6\times S_4$ ($S_4$)

Order 192: $C_{12}.C_2^4$ ($S_3^2$), $Q_8\times S_4$ ($S_3^2$), $C_2^2\times \GL(2,3)$ ($S_3^2$)

Order 144: $S_3\times S_4$ ($\GL(2,3)$) x 4, $A_4\times D_6$ ($C_2\times S_4$), $C_6:S_4$ ($C_2\times S_4$), $C_6\times S_4$ ($C_2\times S_4$), $S_3\times \SL(2,3)$ ($C_2\times S_4$), $C_3:\GL(2,3)$ ($C_2\times S_4$), $C_3\times \GL(2,3)$ ($C_2\times S_4$)

Order 96: $C_6.C_2^4$ ($S_3\times D_6$), $Q_8\times A_4$ ($S_3\times D_6$), $C_2^2\times \SL(2,3)$ ($S_3\times D_6$)

Order 72: $S_3\times A_4$ ($C_2\times \GL(2,3)$) x 2, $C_3:S_4$ ($C_2\times \GL(2,3)$) x 2, $C_3\times S_4$ ($C_2\times \GL(2,3)$) x 2, $C_6\times A_4$ ($C_2^2\times S_4$), $C_3\times \SL(2,3)$ ($C_2^2\times S_4$)

Order 48: $C_2^2\times D_6$ ($S_3\times S_4$), $C_2\times S_4$ ($S_3\times S_4$), $S_3\times Q_8$ ($S_3\times S_4$), $\GL(2,3)$ ($S_3\times S_4$)

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

Order 32: $C_2^2\times Q_8$ ($S_3^3$)

Order 24: $C_2\times D_6$ ($S_3\times \GL(2,3)$) x 2, $S_4$ ($S_3\times \GL(2,3)$) x 2, $\SL(2,3)$ ($D_6\times S_4$), $C_2^2\times C_6$ ($D_6\times S_4$), $C_2\times A_4$ ($D_6\times S_4$), $C_3\times Q_8$ ($D_6\times S_4$)

Order 12: $C_2\times C_6$ ($D_6\times \GL(2,3)$), $D_6$ ($S_4^2$), $A_4$ ($D_6\times \GL(2,3)$)

Order 8: $C_2^3$ ($S_4\times S_3^2$), $Q_8$ ($S_4\times S_3^2$)

Order 6: $S_3$ ($S_4\times \GL(2,3)$) x 2, $C_6$ ($C_2\times S_4^2$)

Order 4: $C_2^2$ ($S_3^2\times \GL(2,3)$)

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

Order 2: $C_2$ ($S_3\times S_4^2$)

Order 1: $C_1$ ($D_6.S_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6912: $D_6.S_4^2$ ($C_1$)

Order 3456: $C_6.S_4^2$ ($C_2$), $S_3\times A_4:\GL(2,3)$ ($C_2$), $S_3\times S_4\times \SL(2,3)$ ($C_2$), $C_6.S_4^2$ ($C_2$), $C_6.S_4^2$ ($C_2$), $S_3\times A_4\times \GL(2,3)$ ($C_2$), $C_6.S_4^2$ ($C_2$)

Order 1728: $D_6.A_4^2$ ($C_2^2$), $(C_3\times A_4):\GL(2,3)$ ($C_2^2$), $C_3\times A_4:\GL(2,3)$ ($C_2^2$), $A_4\times C_3:\GL(2,3)$ ($C_2^2$), $C_3:S_4\times \SL(2,3)$ ($C_2^2$), $C_3\times A_4\times \GL(2,3)$ ($C_2^2$), $C_3\times S_4\times \SL(2,3)$ ($C_2^2$)

Order 1152: $C_2^3.D_6^2$ ($S_3$), $Q_8:D_6^2$ ($S_3$), $S_4\times \GL(2,3)$ ($S_3$)

Order 864: $C_6.A_4^2$ ($C_2^3$)

Order 576: $S_3\times Q_8\times A_4$ ($D_6$), $C_2\times D_6\times \SL(2,3)$ ($D_6$), $Q_8\times C_3:S_4$ ($D_6$), $C_2\times C_6:\GL(2,3)$ ($D_6$), $C_3\times Q_8\times S_4$ ($D_6$), $C_2\times C_6\times \GL(2,3)$ ($D_6$), $A_4:\GL(2,3)$ ($D_6$), $S_4\times \SL(2,3)$ ($D_6$), $A_4\times \GL(2,3)$ ($D_6$)

Order 288: $C_2^3.C_6^2$ ($C_2\times D_6$), $Q_8:C_6^2$ ($C_2\times D_6$), $A_4\times \SL(2,3)$ ($C_2\times D_6$), $S_3\times \GL(2,3)$ ($S_4$), $D_6\times S_4$ ($S_4$)

Order 192: $C_{12}.C_2^4$ ($S_3^2$), $Q_8\times S_4$ ($S_3^2$), $C_2^2\times \GL(2,3)$ ($S_3^2$)

Order 144: $A_4\times D_6$ ($C_2\times S_4$), $C_6:S_4$ ($C_2\times S_4$), $C_6\times S_4$ ($C_2\times S_4$), $S_3\times S_4$ ($\GL(2,3)$), $S_3\times \SL(2,3)$ ($C_2\times S_4$), $C_3:\GL(2,3)$ ($C_2\times S_4$), $C_3\times \GL(2,3)$ ($C_2\times S_4$)

Order 96: $C_6.C_2^4$ ($S_3\times D_6$), $Q_8\times A_4$ ($S_3\times D_6$), $C_2^2\times \SL(2,3)$ ($S_3\times D_6$)

Order 72: $C_6\times A_4$ ($C_2^2\times S_4$), $S_3\times A_4$ ($C_2\times \GL(2,3)$), $C_3:S_4$ ($C_2\times \GL(2,3)$), $C_3\times S_4$ ($C_2\times \GL(2,3)$), $C_3\times \SL(2,3)$ ($C_2^2\times S_4$)

Order 48: $C_2^2\times D_6$ ($S_3\times S_4$), $C_2\times S_4$ ($S_3\times S_4$), $S_3\times Q_8$ ($S_3\times S_4$), $\GL(2,3)$ ($S_3\times S_4$)

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

Order 32: $C_2^2\times Q_8$ ($S_3^3$)

Order 24: $\SL(2,3)$ ($D_6\times S_4$), $C_2^2\times C_6$ ($D_6\times S_4$), $C_2\times D_6$ ($S_3\times \GL(2,3)$), $C_2\times A_4$ ($D_6\times S_4$), $S_4$ ($S_3\times \GL(2,3)$), $C_3\times Q_8$ ($D_6\times S_4$)

Order 12: $C_2\times C_6$ ($D_6\times \GL(2,3)$), $D_6$ ($S_4^2$), $A_4$ ($D_6\times \GL(2,3)$)

Order 8: $C_2^3$ ($S_4\times S_3^2$), $Q_8$ ($S_4\times S_3^2$)

Order 6: $C_6$ ($C_2\times S_4^2$), $S_3$ ($S_4\times \GL(2,3)$)

Order 4: $C_2^2$ ($S_3^2\times \GL(2,3)$)

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

Order 2: $C_2$ ($S_3\times S_4^2$)

Order 1: $C_1$ ($D_6.S_4^2$)

Series

Derived series

$D_6.S_4^2$

$\rhd$

$C_6.A_4^2$

$\rhd$

$C_2^2\times Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$D_6.S_4^2$

$\rhd$

$C_6.S_4^2$

$\rhd$

$C_3\times S_4\times \SL(2,3)$

$\rhd$

$C_6.A_4^2$

$\rhd$

$A_4\times \SL(2,3)$

$\rhd$

$C_2^2\times \SL(2,3)$

$\rhd$

$Q_8\times A_4$

$\rhd$

$C_2^2\times Q_8$

$\rhd$

$C_2^3$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_6.S_4^2$

$\rhd$

$C_6.A_4^2$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $105 \times 105$ rational character table.