Abstract group 13824.hc: $(C_2^2\times C_6):S_4^2$

• Label: 13824.hc
• Order: \( 2^{9} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $2$

Group information

Description:	$(C_2^2\times C_6):S_4^2$
Order:	\(13824\)\(\medspace = 2^{9} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^2\times C_2^6.C_3^4.C_2^4$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \)
Composition factors:	$C_2$ x 9, $C_3$ 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	12
Elements	1	871	890	2712	4070	5280	13824
Conjugacy classes	1	20	8	23	32	18	102
Divisions	1	20	7	23	31	18	100
Autjugacy classes	1	10	7	11	20	9	58

Dimension	1	2	3	4	6	8	9	12	16	18	24	32	36	48
Irr. complex chars.	4	6	16	9	18	6	12	12	5	8	4	0	1	1	102
Irr. rational chars.	4	6	16	7	18	7	12	12	3	8	4	1	1	1	100

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{2}=b^{6}=c^{2}=d^{6}=e^{6}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2)(4,6)(8,14)(9,11)(10,12)(13,15), (1,2,3)(4,7)(8,14)(9,11)(10,15)(12,13) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(Q_8:A_4^2)$ $\,\rtimes\,$ $D_6$	$(C_2^5:A_4)$ $\,\rtimes\,$ $S_3^2$	$C_2^5$ $\,\rtimes\,$ $(A_4:S_3^2)$	$(C_2^2\times C_6)$ $\,\rtimes\,$ $S_4^2$	all 32
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_2^2:S_4^2)$	$(C_6:S_4)$ . $(C_2^2:S_4)$	$(C_6\times A_4)$ . $(C_2^3:S_4)$	$C_2$ . $((C_2\times C_6):S_4^2)$	all 7

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

Homology

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

Subgroups

There are 558666 subgroups in 11272 conjugacy classes, 56 normal (33 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $(C_2\times C_6):S_4^2$
Commutator:	$G' \simeq$ $C_3\times Q_8:A_4^2$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $(C_2\times C_6):S_4^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}.C_2^5$	$G/\operatorname{Fit} \simeq$ $S_3^2$
Radical:	$R \simeq$ $(C_2^2\times C_6):S_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^4:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:D_4^2$
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 13824: $(C_2^2\times C_6):S_4^2$

Order 6912: $Q_8:A_4^2:S_3$, $Q_8:A_4^2:C_6$, $Q_8:A_4^2:S_3$

Order 4608: $C_2^5:(S_3\times S_4)$, $C_2^5.D_6^2$, $C_2^3:S_4^2$

Order 3456: $C_6:S_4^2$ x 3, $C_3\times Q_8:A_4^2$, $Q_8:A_4:S_3^2$

Order 2304: $(C_2^3\times C_{12}):D_{12}$ x 3, $C_2^4.D_6^2$ x 3, $C_2^3.(D_6\times S_4)$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $A_4\times C_2^3:S_4$ x 2, $Q_8:A_4\times S_4$ x 2, $(C_2^4\times C_6):S_4$, $C_3\times C_2^5:S_4$, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_2^3\times S_4)$, $C_2^4.(S_3\times S_4)$, $C_2^4.(S_3\times S_4)$, $C_2^5.(C_6\times D_6)$, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^5:(S_3\times A_4)$, $(C_2^3\times A_4):S_4$, $S_3\times C_2^4:S_4$

Order 1728: $C_3:S_4^2$ x 12, $A_4^2:D_6$ x 6, $C_6\times \PSOPlus(4,3)$ x 3, $(C_2\times C_6^2):S_4$ x 2, $D_4:C_6^2:C_6$, $C_3:S_4\times \SL(2,3)$

Order 1536: $D_4\times C_2^3:S_4$, $C_2\wr C_2^2\times S_4$, $(C_2\times C_{12}):D_4^2$

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ x 6, $C_2^5.S_3^2$ x 6, $Q_8:A_4^2$ x 4, $C_2\times S_4^2$ x 3, $(C_6\times D_4):S_4$ x 3, $C_6:S_4\times D_4$ x 3, $S_3\times C_2^3:S_4$ x 3, $A_4\times D_6:D_4$ x 3, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$ x 3, $D_6:\GL(2,\mathbb{Z}/4)$ x 3, $D_6:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^5.S_3^2$ x 3, $C_2^5.S_3^2$ x 3, $(C_2\times S_4):D_{12}$ x 3, $D_6:(C_4\times S_4)$ x 3, $(C_2^3\times C_{12}):C_{12}$ x 3, $C_3\times C_2^2.\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^3\times C_6):S_4$ x 2, $C_2^3.D_6^2$, $C_2^3:D_6^2$, $Q_8:A_4\times D_6$, $C_6\times C_2^3:S_4$, $(D_4\times A_4):D_6$, $(D_4\times A_4):D_6$, $C_2^4.(S_3\times A_4)$, $C_3\times C_2^4.S_4$, $C_3\times C_2^5:A_4$, $C_6\times C_2^4:A_4$, $C_2^5:C_6^2$

Order 864: $A_4^2:S_3$ x 12, $A_4^2:C_6$ x 6, $C_2\times A_4:S_3^2$ x 4, $C_6\times A_4^2$ x 3, $C_3^2\times Q_8:A_4$, $C_6.A_4^2$

Order 768: $D_6:D_4^2$ x 3, $(C_2\times C_6).D_4^2$ x 3, $(C_2^2\times D_{12}):D_4$ x 3, $(C_2^3\times C_{12}):D_4$ x 3, $(C_2\times C_6).D_4^2$ x 3, $C_2^5:D_{12}$ x 3, $(C_2^2\times D_{12}):D_4$ x 3, $C_2^5:(C_4\times S_3)$ x 3, $C_2^5:S_4$ x 3, $C_2^6:D_6$ x 3, $C_2^5:D_{12}$ x 3, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^3:C_4\times S_4$ x 3, $\GL(2,\mathbb{Z}/4):C_2^3$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.(D_4\times D_6)$, $(C_2^2\times D_{12}):D_4$, $C_3\times C_2^5:D_4$, $C_2^3.(D_4\times D_6)$, $C_2^5:S_4$, $(C_2^3\times C_4):S_4$, $C_4\times C_2^3:S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $D_4\times Q_8:A_4$, $A_4\times C_2\wr C_2^2$, $(D_4\times C_2^3):D_6$, $(C_2^3\times C_{12}):D_4$, $C_2\wr C_2^2\times D_6$

Order 576: $\GL(2,\mathbb{Z}/4):S_3$ x 36, $C_2^2:D_6^2$ x 13, $A_4^2:C_2^2$ x 12, $S_4^2$ x 12, $D_4\times C_3:S_4$ x 9, $C_2^2\times C_6:S_4$ x 7, $C_3\times C_2^3:S_4$ x 7, $C_6:D_4\times A_4$ x 6, $C_6:\GL(2,\mathbb{Z}/4)$ x 6, $C_6\times \GL(2,\mathbb{Z}/4)$ x 6, $C_6:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4.S_3^2$ x 6, $C_2^4.S_3^2$ x 6, $(C_2\times C_{12}):D_{12}$ x 6, $C_2^2:S_4\times C_6$ x 3, $A_4^2:C_2^2$ x 3, $C_2^4:C_6^2$ x 3, $S_3\times Q_8:A_4$ x 3, $(C_2^2\times C_6):S_4$ x 3, $(C_3\times D_4):S_4$ x 3, $(C_2\times C_{12}):S_4$ x 3, $C_2\times C_{12}:S_4$ x 3, $C_2\times C_{12}:S_4$ x 3, $S_4\times \SL(2,3)$ x 3, $D_6^2:C_2^2$ x 3, $A_4\times C_6.D_4$ x 3, $A_4\times D_6:C_4$ x 3, $C_6.\GL(2,\mathbb{Z}/4)$ x 3, $C_6.\GL(2,\mathbb{Z}/4)$ x 3, $C_6.\GL(2,\mathbb{Z}/4)$ x 3, $C_6.\GL(2,\mathbb{Z}/4)$ x 3, $C_6.\GL(2,\mathbb{Z}/4)$ x 3, $(C_6\times D_4).D_6$ x 3, $(C_6\times C_{12}):D_4$ x 2, $C_2^2\times A_4\times D_6$, $C_2^2\times C_6\times S_4$, $C_6^2.C_2^4$, $C_6\times Q_8:A_4$, $C_2^4.C_6^2$, $(C_3\times Q_8):S_4$, $Q_8\times C_3:S_4$, $C_3:D_4\times \SL(2,3)$

Order 512: $C_2^3:D_4^2$

Order 432: $A_4:S_3^2$ x 16, $C_6^2:D_6$ x 8, $C_6:S_3\times A_4$ x 4, $C_3\times A_4^2$ x 3, $C_3:S_3\times \SL(2,3)$

Order 384: $C_2^4:D_{12}$ x 12, $\GL(2,\mathbb{Z}/4):C_2^2$ x 12, $C_2\wr C_2^2\times S_3$ x 9, $(C_6\times D_4):D_4$ x 8, $C_2^4:S_4$ x 7, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^5.D_6$ x 6, $C_2^4:(C_4\times S_3)$ x 6, $C_2^3.(C_4\times D_6)$ x 6, $C_2^4.D_{12}$ x 6, $(C_2^2\times C_{12}).D_4$ x 6, $C_2^4.D_{12}$ x 6, $\GL(2,\mathbb{Z}/4):C_2^2$ x 6, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4:D_{12}$ x 6, $C_2^5.D_6$ x 6, $C_2^4:D_{12}$ x 6, $(C_2^3\times C_{12}):C_2^2$ x 4, $C_2\wr C_2^2\times C_6$ x 4, $C_6.D_4^2$ x 3, $C_6.D_4^2$ x 3, $C_6.D_4^2$ x 3, $C_6.D_4^2$ x 3, $(C_4\times D_6):D_4$ x 3, $C_6.D_4^2$ x 3, $C_6.D_4^2$ x 3, $C_6.D_4^2$ x 3, $C_6.D_4^2$ x 3, $D_6:(C_4\times D_4)$ x 3, $C_2^4:D_{12}$ x 3, $C_6.D_4^2$ x 3, $C_2^5:C_{12}$ x 3, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^5:C_{12}$ x 3, $C_2^5.D_6$ x 3, $(C_2^2\times D_{12}):C_4$ x 3, $(C_6\times D_4).D_4$ x 3, $(C_2\times C_{12}):C_4^2$ x 3, $(C_2^3\times C_4):D_6$ x 3, $(C_6\times D_4).Q_8$ x 3, $C_2^5:A_4$ x 3, $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $(C_6\times D_4):D_4$ x 3, $(C_6\times D_4):D_4$ x 3, $C_2^5:D_6$ x 3, $C_6:D_4^2$ x 3, $C_2^5:D_6$ x 3, $C_6:D_4^2$ x 3, $C_2^5:D_6$ x 3, $C_2^6:C_6$ x 3, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 3, $C_6.D_4^2$ x 3, $C_6.D_4^2$ x 3, $C_2^5.D_6$ x 3, $C_2^5.D_6$ x 3, $C_2^5:D_6$ x 3, $C_2^5:D_6$ x 3, $C_6.D_4^2$ x 3, $C_6.D_4^2$ x 3, $C_2^3:C_4\times D_6$ x 3, $C_2^5:A_4$ x 2, $A_4\times D_4:C_2^2$ x 2, $C_2^4\times S_4$, $C_{12}.C_2^5$, $D_{12}:C_2^4$, $C_{12}:C_2^5$, $(C_2^2\times D_4):D_6$, $(C_2^2\times D_4):D_6$, $C_{12}.(C_2^3\times C_4)$, $(C_2^2\times D_4):D_6$, $(C_2^2\times D_4):D_6$, $C_4\times Q_8:A_4$, $C_2^4.S_4$

Order 288: $D_6\times S_4$ x 62, $C_2\times C_6:S_4$ x 35, $A_4\times S_4$ x 24, $A_4\times C_3:D_4$ x 18, $C_3\times \GL(2,\mathbb{Z}/4)$ x 18, $D_6:S_4$ x 18, $C_2^3.S_3^2$ x 18, $C_2^3.S_3^2$ x 18, $C_2\times A_4^2$ x 12, $C_3\times Q_8:A_4$ x 10, $C_2\times A_4\times D_6$ x 8, $C_2\times C_6\times S_4$ x 8, $C_2^3:C_6^2$ x 7, $C_6:C_4\times A_4$ x 6, $C_3:\GL(2,\mathbb{Z}/4)$ x 6, $C_2\times A_4:C_{12}$ x 6, $A_4\times \SL(2,3)$ x 6, $C_6^2:D_4$ x 6, $C_6^2:D_4$ x 6, $C_6^2.D_4$ x 6, $C_6^2.D_4$ x 6, $C_3\times C_2^2:S_4$ x 6, $\PSOPlus(4,3)$ x 6, $D_6^2:C_2$ x 4, $C_{12}:S_4$ x 4, $C_2^3.C_6^2$ x 3, $C_2^3.C_6^2$ x 3, $C_2\times C_6.S_4$ x 3, $C_{12}:S_4$ x 3, $D_6^2:C_2$ x 3, $C_2^3.S_3^2$ x 3, $D_6:D_{12}$ x 3, $D_6^2.C_2$ x 3, $C_2^5:C_3^2$ x 3, $C_6^2.C_2^3$ x 3, $C_6^2:C_2^3$ x 3, $C_3\times Q_8:A_4$, $C_2^3.C_6^2$, $Q_8:C_6^2$, $D_6\times \SL(2,3)$, $C_{12}.S_4$, $C_3:C_4\times \SL(2,3)$, $C_2\times D_6^2$, $D_4:C_6^2$, $C_6^2.C_2^3$, $C_6^2.C_2^3$

Order 256: $C_2^5:D_4$ x 6, $C_2^5:D_4$ x 6, $C_2^2:D_4^2$ x 3, $C_2^2.D_4^2$ x 3, $C_2^2.D_4^2$ x 3, $C_2^2.D_4^2$ x 3, $C_2^5:D_4$ x 2, $C_2^5:D_4$ x 2, $D_4^2:C_2^2$, $C_2^4.C_2^4$, $C_2^4.C_2^4$

Order 216: $C_3^2:S_4$ x 16, $C_6^2:C_6$ x 8, $C_6^2:C_6$ x 4, $C_6:S_3^2$, $C_3^2\times \SL(2,3)$

Order 192: $D_4\times S_4$ x 63, $C_2^4:D_6$ x 33, $C_2^3\times S_4$ x 28, $C_2^4:D_6$ x 27, $C_{12}:C_2^4$ x 26, $C_2\times \GL(2,\mathbb{Z}/4)$ x 24, $C_2^4:D_6$ x 24, $C_2^3.D_{12}$ x 21, $C_2^3:D_{12}$ x 21, $C_2^4:D_6$ x 21, $C_2^3:D_{12}$ x 18, $(C_2\times C_{12}):D_4$ x 16, $C_2^4:D_6$ x 15, $C_2^3:C_4\times S_3$ x 15, $C_2^3.D_{12}$ x 12, $C_2^4:C_{12}$ x 12, $C_2^4.D_6$ x 12, $C_2^3.D_{12}$ x 12, $C_2^5:C_6$ x 12, $C_2^3:D_{12}$ x 12, $C_2^4.D_6$ x 12, $C_2^4.D_6$ x 12, $C_3\times C_2\wr C_2^2$ x 11, $C_2.\GL(2,\mathbb{Z}/4)$ x 9, $(C_2\times C_{12}):D_4$ x 9, $C_2^4.D_6$ x 9, $C_2^3:D_{12}$ x 9, $C_2^4.D_6$ x 9, $C_2^4.D_6$ x 9, $(C_2\times C_4):D_{12}$ x 9, $C_2^3:S_4$ x 9, $D_4:S_4$ x 9, $\GL(2,\mathbb{Z}/4):C_2$ x 9, $C_2^5:C_6$ x 9, $C_2^4:D_6$ x 9, $(C_2\times D_4):D_6$ x 9, $C_2^4.D_6$ x 9, $C_2^4:D_6$ x 9, $D_6:C_2^4$ x 8, $D_{12}:C_2^3$ x 7, $C_2^4:C_{12}$ x 6, $C_2.\GL(2,\mathbb{Z}/4)$ x 6, $C_3\times C_2^3:D_4$ x 6, $C_2^4.D_6$ x 6, $C_2^4.D_6$ x 6, $C_2^4.D_6$ x 6, $C_2^4.D_6$ x 6, $C_{12}.C_2^4$ x 5, $C_{12}:C_2^4$ x 4, $C_2^4:C_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^3.D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^3.D_{12}$ x 3, $C_2^3.D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $(C_2\times C_4).D_{12}$ x 3, $(C_2\times C_4):D_{12}$ x 3, $(C_2\times C_4).D_{12}$ x 3, $(C_4\times D_6):C_4$ x 3, $D_6:C_4^2$ x 3, $C_2^3:S_4$ x 3, $D_{12}:C_2^3$ x 3, $C_2^4:A_4$ x 3, $A_4\times D_4:C_2$ x 3, $Q_8:S_4$ x 3, $Q_8\times S_4$ x 3, $C_2^4.D_6$ x 3, $(C_3\times Q_8):D_4$ x 3, $(C_2\times C_{12}):D_4$ x 3, $(C_2\times C_{12}):D_4$ x 3, $(C_2\times C_{12}):D_4$ x 3, $(C_2\times C_{12}):D_4$ x 3, $(C_3\times D_4):D_4$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^3.D_{12}$ x 3, $D_6\times C_2^4$ x 2, $A_4\times C_2^4$, $C_{12}.C_2^4$, $D_{12}:C_2^3$, $D_{12}:C_2^3$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$, $(C_3\times Q_8):D_4$, $C_{12}.(C_2^2\times C_4)$, $C_{12}.(C_2^2\times C_4)$, $(C_3\times Q_8):D_4$, $C_3:D_4\times Q_8$, $D_4\times \SL(2,3)$

Order 144: $S_3\times S_4$ x 56, $C_6:S_4$ x 47, $C_6\times S_4$ x 37, $A_4\times D_6$ x 30, $D_6:D_6$ x 20, $C_2^2:C_6^2$ x 16, $A_4^2$ x 12, $D_6^2$ x 11, $C_{12}:D_6$ x 9, $A_4\times C_3:C_4$ x 9, $A_4:C_{12}$ x 9, $C_6^2:C_2^2$ x 8, $C_6:D_{12}$ x 8, $C_6^2:C_4$ x 6, $D_6:C_{12}$ x 6, $C_6.D_{12}$ x 6, $C_6^2:C_2^2$ x 6, $C_6^2:C_2^2$ x 5, $C_2^2.S_3^2$ x 4, $S_3\times \SL(2,3)$ x 4, $C_6.D_{12}$ x 3, $C_2^4:C_3^2$ x 3, $C_4:C_6^2$ x 3, $C_{12}.D_6$ x 3, $C_{12}.D_6$ x 3, $C_6:D_{12}$ x 3, $C_{12}:D_6$ x 3, $C_6^2:C_2^2$ x 2, $C_4.C_6^2$, $C_{12}.D_6$, $C_{12}.D_6$, $C_6\times \SL(2,3)$, $C_{12}\times A_4$, $C_6.S_4$

Order 128: $C_2^4:D_4$ x 29, $C_2^4:D_4$ x 18, $C_2^4.D_4$ x 12, $C_2^4:D_4$ x 12, $C_2^4:D_4$ x 12, $C_2^4.D_4$ x 9, $C_2^4.D_4$ x 7, $C_2^4.D_4$ x 6, $C_2^4.C_2^3$ x 6, $C_2^5:C_4$ x 6, $D_4^2:C_2$ x 6, $C_2\times D_4^2$ x 6, $C_2^4:D_4$ x 6, $C_2^5:C_4$ x 6, $C_2^3\wr C_2$ x 6, $C_2.D_4^2$ x 6, $C_2.D_4^2$ x 6, $C_2.D_4^2$ x 6, $C_2^4.D_4$ x 3, $C_2^4.C_2^3$ x 3, $C_2^3:C_4^2$ x 3, $D_4^2:C_2$ x 3, $C_2^4:D_4$ x 3, $C_2^3.C_2^4$ x 3, $D_4:C_2^4$ x 2, $D_4^2:C_2$ x 2, $D_4\times C_2^4$, $D_4^2:C_2$, $D_4^2:C_2$, $C_4^2.C_2^3$

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

Order 96: $D_4\times D_6$ x 191, $C_2^2\times S_4$ x 136, $D_6:D_4$ x 87, $C_2^3:D_6$ x 86, $C_2^2:D_{12}$ x 63, $\GL(2,\mathbb{Z}/4)$ x 54, $C_2^2.D_{12}$ x 51, $C_2^3.D_6$ x 48, $C_{12}:C_2^3$ x 37, $C_2^3\times D_6$ x 34, $C_2^4:C_6$ x 33, $C_2^3.D_6$ x 33, $C_4\times S_4$ x 30, $C_2^4:S_3$ x 30, $D_6.D_4$ x 27, $D_4\times A_4$ x 27, $C_4:S_4$ x 27, $D_6:D_4$ x 24, $C_2^3.D_6$ x 24, $D_4:D_6$ x 24, $C_2^3:C_{12}$ x 24, $C_2^2.D_{12}$ x 24, $C_2^3\times A_4$ x 22, $C_2^3:C_{12}$ x 21, $C_{12}:D_4$ x 21, $C_2^2\times D_{12}$ x 19, $C_2^3.D_6$ x 18, $C_6.C_2^4$ x 18, $C_{12}:C_2^3$ x 16, $C_{12}:D_4$ x 16, $Q_8:A_4$ x 13, $C_6.C_4^2$ x 12, $C_2^3:C_{12}$ x 12, $C_2^2.S_4$ x 12, $C_{12}:D_4$ x 12, $C_2^3.D_6$ x 12, $C_{12}:D_4$ x 12, $D_4:D_6$ x 11, $D_4:D_6$ x 11, $C_2^3.D_6$ x 9, $C_6.C_2^4$ x 8, $C_6.C_2^4$ x 7, $C_6.C_4^2$ x 6, $C_2^3:A_4$ x 6, $C_2^2:S_4$ x 6, $C_2^3.D_6$ x 6, $C_2^3.D_6$ x 6, $C_6:C_4^2$ x 6, $C_2^4\times C_6$ x 5, $C_2^3\times C_{12}$ x 4, $D_6:Q_8$ x 4, $C_{12}.D_4$ x 4, $D_4:D_6$ x 3, $D_{12}:C_2^2$ x 3, $Q_8:A_4$ x 3, $Q_8\times A_4$ x 3, $C_2^2\times \SL(2,3)$ x 3, $A_4:Q_8$ x 3, $C_{12}.D_4$ x 3, $C_{12}.D_4$ x 3, $C_2^3.D_6$ x 3, $C_2^2.D_{12}$ x 3, $C_3:C_4\times Q_8$ x 2, $C_4\times \SL(2,3)$, $C_6.C_2^4$, $Q_8:D_6$, $Q_8\times D_6$, $C_{12}.D_4$

Order 72: $S_3\times D_6$ x 43, $C_3\times S_4$ x 38, $C_6\times A_4$ x 36, $C_3:S_4$ x 25, $S_3\times A_4$ x 24, $C_6:D_6$ x 23, $C_6\wr C_2$ x 20, $C_3:D_{12}$ x 20, $C_6\times D_6$ x 12, $C_6.D_6$ x 10, $C_3\times \SL(2,3)$ x 9, $C_6:C_{12}$ x 8, $C_6^2:C_2$ x 6, $C_2\times C_6^2$ x 5, $C_{12}:S_3$ x 4, $D_4\times C_3^2$ x 3, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3, $C_3:D_{12}$ x 3, $Q_8\times C_3^2$, $C_3^2:Q_8$

Order 64: $C_2^3:D_4$ x 87, $C_2\wr C_2^2$ x 55, $D_4^2$ x 51, $C_2^4:C_4$ x 48, $D_4\times C_2^3$ x 39, $C_2^3.D_4$ x 36, $C_2^3:D_4$ x 27, $C_2^4:C_4$ x 27, $C_2^3:D_4$ x 24, $D_4:D_4$ x 18, $D_4:C_2^3$ x 14, $C_2^2.C_4^2$ x 12, $C_4^2:C_2^2$ x 12, $C_2^4:C_4$ x 12, $C_2^3.D_4$ x 9, $C_2^3.C_2^3$ x 9, $C_2^3:D_4$ x 9, $C_2^3.D_4$ x 6, $C_2^3.D_4$ x 6, $C_2^2:C_4^2$ x 6, $C_4^2:C_2^2$ x 6, $C_4^2:C_2^2$ x 6, $C_4^2:C_2^2$ x 6, $Q_8:D_4$ x 3, $D_4:D_4$ x 3, $C_2^2.C_2^4$ x 3, $C_4^2.C_2^2$ x 3, $C_4^2:C_2^2$ x 3, $C_2^6$ x 2, $D_4:C_2^3$ x 2, $Q_8:D_4$ x 2, $C_2^4\times C_4$, $D_4\times Q_8$, $C_4^2.C_2^2$, $C_4^2.C_2^2$

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

Order 48: $S_3\times D_4$ x 259, $C_6:D_4$ x 232, $C_2^2\times D_6$ x 220, $C_2\times S_4$ x 158, $C_6\times D_4$ x 116, $C_2\times D_{12}$ x 86, $D_6:C_4$ x 85, $C_4\times D_6$ x 72, $C_2^2\times A_4$ x 54, $C_6.D_4$ x 49, $C_2^2:C_{12}$ x 48, $C_6.C_2^3$ x 35, $C_2^3\times C_6$ x 32, $C_2^2\times C_{12}$ x 21, $D_4:S_3$ x 21, $D_{12}:C_2$ x 21, $D_4:C_6$ x 18, $C_6.D_4$ x 18, $C_4\times A_4$ x 12, $A_4:C_4$ x 12, $C_{12}:C_4$ x 10, $D_{12}:C_2$ x 7, $C_2^2:A_4$ x 6, $S_3\times Q_8$ x 5, $C_{12}:C_4$ x 4, $C_2\times \SL(2,3)$ x 3, $C_6\times Q_8$ x 2, $C_6:Q_8$

Order 36: $C_6\times S_3$ x 33, $S_3^2$ x 28, $C_3\times A_4$ x 27, $C_6:S_3$ x 26, $C_3:C_{12}$ x 10, $C_6^2$ x 10, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$ x 323, $C_2^2\wr C_2$ x 195, $C_2^3:C_4$ x 129, $C_4:D_4$ x 96, $C_4\times D_4$ x 52, $C_2^3:C_4$ x 51, $D_4:C_2^2$ x 42, $C_2^5$ x 38, $C_2^3\times C_4$ x 27, $D_4:C_2^2$ x 23, $C_4:D_4$ x 21, $C_2.C_4^2$ x 18, $C_2^2.D_4$ x 12, $C_2^2.D_4$ x 12, $C_2^2:Q_8$ x 8, $C_4^2:C_2$ x 6, $C_2\times C_4^2$ x 6, $C_4^2:C_2$ x 3, $C_2^2\times Q_8$ x 2, $C_4\times Q_8$ x 2, $C_4:Q_8$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 396, $C_3:D_4$ x 145, $C_2^2\times C_6$ x 93, $C_3\times D_4$ x 68, $D_{12}$ x 59, $C_4\times S_3$ x 58, $S_4$ x 56, $C_6:C_4$ x 55, $C_2\times C_{12}$ x 52, $C_2\times A_4$ x 48, $\SL(2,3)$ x 8, $C_3\times Q_8$ x 6, $C_3:Q_8$ x 5

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

Order 16: $C_2\times D_4$ x 606, $C_2^4$ x 229, $C_2^2:C_4$ x 161, $C_2^2\times C_4$ x 118, $D_4:C_2$ x 43, $C_4:C_4$ x 22, $C_4^2$ x 10, $C_2\times Q_8$ x 5

Order 12: $D_6$ x 207, $C_2\times C_6$ x 91, $A_4$ x 22, $C_3:C_4$ x 21, $C_{12}$ x 18

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 379, $D_4$ x 196, $C_2\times C_4$ x 123, $Q_8$ x 5

Order 6: $S_3$ x 33, $C_6$ x 31

Order 4: $C_2^2$ x 174, $C_4$ x 23

Order 3: $C_3$ x 7

Order 2: $C_2$ x 20

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 13824: $(C_2^2\times C_6):S_4^2$

Order 6912: $Q_8:A_4^2:S_3$, $Q_8:A_4^2:C_6$, $Q_8:A_4^2:S_3$

Order 4608: $C_2^5:(S_3\times S_4)$, $C_2^5.D_6^2$, $C_2^3:S_4^2$

Order 3456: $C_3\times Q_8:A_4^2$, $Q_8:A_4:S_3^2$, $C_6:S_4^2$

Order 2304: $A_4\times C_2^3:S_4$ x 2, $Q_8:A_4\times S_4$ x 2, $(C_2^4\times C_6):S_4$, $(C_2^3\times C_{12}):D_{12}$, $C_3\times C_2^5:S_4$, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_2^3\times S_4)$, $C_2^4.D_6^2$, $C_2^4.(S_3\times S_4)$, $C_2^4.(S_3\times S_4)$, $C_2^5.(C_6\times D_6)$, $C_2^3.(D_6\times S_4)$, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^5:(S_3\times A_4)$, $(C_2^3\times A_4):S_4$, $S_3\times C_2^4:S_4$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$

Order 1728: $A_4^2:D_6$ x 2, $(C_2\times C_6^2):S_4$ x 2, $C_6\times \PSOPlus(4,3)$, $C_3:S_4^2$, $D_4:C_6^2:C_6$, $C_3:S_4\times \SL(2,3)$

Order 1536: $D_4\times C_2^3:S_4$, $C_2\wr C_2^2\times S_4$, $(C_2\times C_{12}):D_4^2$

Order 1152: $Q_8:A_4^2$ x 4, $S_3\times C_2^3:S_4$ x 3, $(C_2^3\times C_6):S_4$ x 2, $\GL(2,\mathbb{Z}/4):D_6$ x 2, $C_2^3.D_6^2$, $C_2^3:D_6^2$, $C_2\times S_4^2$, $Q_8:A_4\times D_6$, $C_6\times C_2^3:S_4$, $(D_4\times A_4):D_6$, $(D_4\times A_4):D_6$, $(C_6\times D_4):S_4$, $C_6:S_4\times D_4$, $C_2^4.(S_3\times A_4)$, $A_4\times D_6:D_4$, $C_3\times C_2^4.S_4$, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$, $D_6:\GL(2,\mathbb{Z}/4)$, $D_6:\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $(C_2\times S_4):D_{12}$, $D_6:(C_4\times S_4)$, $(C_2^3\times C_{12}):C_{12}$, $C_3\times C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_3\times C_2^5:A_4$, $C_6\times C_2^4:A_4$, $C_2^5:C_6^2$

Order 864: $C_2\times A_4:S_3^2$ x 2, $A_4^2:S_3$ x 2, $C_6\times A_4^2$, $A_4^2:C_6$, $C_3^2\times Q_8:A_4$, $C_6.A_4^2$

Order 768: $C_2^5:S_4$ x 3, $\GL(2,\mathbb{Z}/4):C_2^3$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $D_6:D_4^2$, $C_2^3.(D_4\times D_6)$, $(C_2^2\times D_{12}):D_4$, $C_3\times C_2^5:D_4$, $(C_2\times C_6).D_4^2$, $(C_2^2\times D_{12}):D_4$, $(C_2^3\times C_{12}):D_4$, $(C_2\times C_6).D_4^2$, $C_2^5:D_{12}$, $(C_2^2\times D_{12}):D_4$, $C_2^5:(C_4\times S_3)$, $C_2^3.(D_4\times D_6)$, $C_2^5:S_4$, $(C_2^3\times C_4):S_4$, $C_4\times C_2^3:S_4$, $C_2^6:D_6$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^5:D_{12}$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^3:C_4\times S_4$, $D_4\times Q_8:A_4$, $A_4\times C_2\wr C_2^2$, $(D_4\times C_2^3):D_6$, $(C_2^3\times C_{12}):D_4$, $C_2\wr C_2^2\times D_6$

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

Order 512: $C_2^3:D_4^2$

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

Order 384: $C_2^4:S_4$ x 7, $C_2\wr C_2^2\times S_3$ x 5, $\GL(2,\mathbb{Z}/4):C_2^2$ x 4, $(C_6\times D_4):D_4$ x 4, $C_2^5:A_4$ x 3, $(C_2^3\times C_{12}):C_2^2$ x 2, $C_2^4:D_{12}$ x 2, $C_2^4.D_{12}$ x 2, $C_2^5:A_4$ x 2, $A_4\times D_4:C_2^2$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:D_{12}$ x 2, $C_2^5.D_6$ x 2, $C_2\wr C_2^2\times C_6$ x 2, $C_2^4:D_{12}$ x 2, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $(C_4\times D_6):D_4$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $D_6:(C_4\times D_4)$, $C_2^4:D_{12}$, $C_6.D_4^2$, $C_2^5:C_{12}$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^5:C_{12}$, $C_2^5.D_6$, $C_2^5.D_6$, $C_2^4:(C_4\times S_3)$, $(C_2^2\times D_{12}):C_4$, $C_2^3.(C_4\times D_6)$, $(C_6\times D_4).D_4$, $(C_2\times C_{12}):C_4^2$, $(C_2^2\times C_{12}).D_4$, $(C_2^3\times C_4):D_6$, $(C_6\times D_4).Q_8$, $C_2^4.D_{12}$, $C_2^4\times S_4$, $C_{12}.C_2^5$, $D_{12}:C_2^4$, $C_{12}:C_2^5$, $(C_2^2\times D_4):D_6$, $(C_6\times D_4):D_4$, $(C_2^2\times D_4):D_6$, $(C_6\times D_4):D_4$, $C_{12}.(C_2^3\times C_4)$, $(C_2^2\times D_4):D_6$, $(C_2^2\times D_4):D_6$, $C_2^5:D_6$, $C_6:D_4^2$, $C_2^5:D_6$, $C_6:D_4^2$, $C_2^5:D_6$, $C_4\times Q_8:A_4$, $C_2^6:C_6$, $C_2^4.S_4$, $C_2^2:\GL(2,\mathbb{Z}/4)$, $C_6.D_4^2$, $C_6.D_4^2$, $C_2^5.D_6$, $C_2^5.D_6$, $C_2^5:D_6$, $C_2^5:D_6$, $C_6.D_4^2$, $C_6.D_4^2$, $C_2^3:C_4\times D_6$

Order 288: $D_6\times S_4$ x 12, $C_2\times C_6:S_4$ x 9, $C_3\times Q_8:A_4$ x 8, $A_4\times \SL(2,3)$ x 4, $C_2\times A_4^2$ x 4, $A_4\times S_4$ x 4, $A_4\times C_3:D_4$ x 3, $C_3\times \GL(2,\mathbb{Z}/4)$ x 3, $C_2^3:C_6^2$ x 3, $C_2\times A_4\times D_6$ x 3, $C_2\times C_6\times S_4$ x 3, $D_6^2:C_2$ x 2, $C_6:C_4\times A_4$ x 2, $C_{12}:S_4$ x 2, $C_2\times A_4:C_{12}$ x 2, $D_6:S_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.D_4$ x 2, $C_3\times Q_8:A_4$, $C_2^3.C_6^2$, $Q_8:C_6^2$, $C_2^3.C_6^2$, $C_2^3.C_6^2$, $D_6\times \SL(2,3)$, $C_3:\GL(2,\mathbb{Z}/4)$, $C_2\times C_6.S_4$, $C_{12}:S_4$, $C_{12}.S_4$, $D_6^2:C_2$, $C_2^3.S_3^2$, $D_6:D_{12}$, $D_6^2.C_2$, $C_3:C_4\times \SL(2,3)$, $C_6^2.D_4$, $C_2^5:C_3^2$, $C_2\times D_6^2$, $C_3\times C_2^2:S_4$, $\PSOPlus(4,3)$, $D_4:C_6^2$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2:C_2^3$

Order 256: $C_2^5:D_4$ x 2, $C_2^5:D_4$ x 2, $C_2^5:D_4$ x 2, $C_2^5:D_4$ x 2, $D_4^2:C_2^2$, $C_2^2:D_4^2$, $C_2^4.C_2^4$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^4.C_2^4$

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

Order 192: $C_2^3\times S_4$ x 11, $C_2^4:D_6$ x 11, $(C_2\times C_{12}):D_4$ x 8, $C_{12}:C_2^4$ x 8, $C_2\times \GL(2,\mathbb{Z}/4)$ x 8, $C_2^3.D_{12}$ x 7, $C_2^3:D_{12}$ x 7, $C_2^3:S_4$ x 7, $D_4\times S_4$ x 7, $C_2^4:D_6$ x 7, $C_2^3:D_{12}$ x 6, $C_3\times C_2\wr C_2^2$ x 5, $D_{12}:C_2^3$ x 5, $C_2^4:D_6$ x 5, $C_2^4:D_6$ x 4, $C_2^3:C_4\times S_3$ x 4, $D_6:C_2^4$ x 4, $C_2^5:C_6$ x 4, $C_2^3:D_{12}$ x 4, $C_2^4.D_6$ x 4, $C_2^4.D_6$ x 4, $C_2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4:C_{12}$ x 3, $(C_2\times C_{12}):D_4$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $(C_2\times C_4):D_{12}$ x 3, $C_{12}.C_2^4$ x 3, $C_2^4:A_4$ x 3, $C_2^5:C_6$ x 3, $C_2^4:D_6$ x 3, $C_2^4:C_{12}$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.D_{12}$ x 2, $C_3\times C_2^3:D_4$ x 2, $C_2^3.D_{12}$ x 2, $D_6\times C_2^4$ x 2, $C_{12}:C_2^4$ x 2, $A_4\times D_4:C_2$ x 2, $Q_8:S_4$ x 2, $Q_8\times S_4$ x 2, $D_4:S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_2$ x 2, $(C_3\times Q_8):D_4$ x 2, $(C_2\times D_4):D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4:D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4:D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4:C_{12}$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^3.D_{12}$, $C_2^4.D_6$, $C_2^3.D_{12}$, $C_2^3.D_{12}$, $C_2^4.D_6$, $C_2^4.D_6$, $(C_2\times C_4).D_{12}$, $(C_2\times C_4):D_{12}$, $(C_2\times C_4).D_{12}$, $(C_4\times D_6):C_4$, $D_6:C_4^2$, $A_4\times C_2^4$, $C_2^3:S_4$, $C_{12}.C_2^4$, $D_{12}:C_2^3$, $D_{12}:C_2^3$, $D_{12}:C_2^3$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$, $C_2^4.D_6$, $(C_2\times C_{12}):D_4$, $(C_2\times C_{12}):D_4$, $(C_3\times Q_8):D_4$, $(C_2\times C_{12}):D_4$, $(C_2\times C_{12}):D_4$, $C_{12}.(C_2^2\times C_4)$, $C_{12}.(C_2^2\times C_4)$, $(C_3\times D_4):D_4$, $(C_3\times Q_8):D_4$, $C_3:D_4\times Q_8$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^3.D_{12}$, $D_4\times \SL(2,3)$

Order 144: $C_6:S_4$ x 13, $C_6\times S_4$ x 13, $A_4\times D_6$ x 10, $C_2^2:C_6^2$ x 6, $S_3\times S_4$ x 6, $D_6^2$ x 4, $A_4^2$ x 4, $C_6^2:C_2^2$ x 4, $C_6:D_{12}$ x 4, $C_6^2:C_2^2$ x 3, $D_6:D_6$ x 3, $S_3\times \SL(2,3)$ x 3, $C_6^2:C_4$ x 2, $D_6:C_{12}$ x 2, $C_6.D_{12}$ x 2, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $C_2^2.S_3^2$ x 2, $A_4\times C_3:C_4$ x 2, $A_4:C_{12}$ x 2, $C_6.D_{12}$, $C_2^4:C_3^2$, $C_4.C_6^2$, $C_4:C_6^2$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $C_6:D_{12}$, $C_{12}:D_6$, $C_6\times \SL(2,3)$, $C_{12}\times A_4$, $C_6.S_4$

Order 128: $C_2^4:D_4$ x 13, $C_2^4:D_4$ x 4, $C_2^4:D_4$ x 4, $C_2^4:D_4$ x 3, $C_2^4.D_4$ x 3, $C_2^4.D_4$ x 3, $C_2^4.D_4$ x 2, $C_2^5:C_4$ x 2, $D_4:C_2^4$ x 2, $D_4^2:C_2$ x 2, $D_4^2:C_2$ x 2, $C_2\times D_4^2$ x 2, $C_2^4:D_4$ x 2, $C_2^5:C_4$ x 2, $C_2^3\wr C_2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^4.D_4$, $C_2^4.C_2^3$, $C_2^4.D_4$, $C_2^4.C_2^3$, $C_2^3:C_4^2$, $D_4\times C_2^4$, $D_4^2:C_2$, $D_4^2:C_2$, $D_4^2:C_2$, $C_4^2.C_2^3$, $C_2^4:D_4$, $C_2^3.C_2^4$

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

Order 96: $D_4\times D_6$ x 34, $C_2^2\times S_4$ x 31, $C_2^3:D_6$ x 25, $D_6:D_4$ x 22, $C_2^2.D_{12}$ x 17, $C_2^3\times D_6$ x 16, $C_2^2:D_{12}$ x 14, $C_{12}:C_2^3$ x 12, $C_2^3.D_6$ x 11, $C_6.C_2^4$ x 10, $D_6.D_4$ x 9, $C_2^3\times A_4$ x 9, $Q_8:A_4$ x 8, $C_2^4:C_6$ x 8, $C_2^3:C_{12}$ x 8, $D_4:D_6$ x 7, $\GL(2,\mathbb{Z}/4)$ x 7, $C_2^4:S_3$ x 7, $C_2^3.D_6$ x 7, $C_2^3:C_{12}$ x 6, $D_4:D_6$ x 6, $D_4:D_6$ x 6, $C_2^2\times D_{12}$ x 6, $C_4\times S_4$ x 6, $C_2^3.D_6$ x 5, $C_{12}:C_2^3$ x 5, $D_4\times A_4$ x 5, $C_{12}:D_4$ x 5, $C_6.C_4^2$ x 4, $C_6.C_2^4$ x 4, $C_2^3:C_{12}$ x 4, $C_2^2.S_4$ x 4, $C_4:S_4$ x 4, $C_2^2.D_{12}$ x 4, $C_2^3.D_6$ x 3, $C_2^4\times C_6$ x 3, $C_6.C_2^4$ x 3, $C_2^2\times \SL(2,3)$ x 3, $C_{12}:D_4$ x 3, $C_{12}:D_4$ x 3, $C_2^3.D_6$ x 3, $C_{12}:D_4$ x 3, $C_2^3.D_6$ x 3, $D_6:D_4$ x 2, $C_6.C_4^2$ x 2, $C_2^3:A_4$ x 2, $C_2^3\times C_{12}$ x 2, $Q_8:A_4$ x 2, $Q_8\times A_4$ x 2, $A_4:Q_8$ x 2, $D_6:Q_8$ x 2, $C_3:C_4\times Q_8$ x 2, $C_{12}.D_4$ x 2, $C_6:C_4^2$ x 2, $C_4\times \SL(2,3)$, $C_2^2:S_4$, $C_6.C_2^4$, $Q_8:D_6$, $Q_8\times D_6$, $D_4:D_6$, $D_{12}:C_2^2$, $C_{12}.D_4$, $C_{12}.D_4$, $C_{12}.D_4$, $C_2^3.D_6$, $C_2^3.D_6$, $C_2^3.D_6$, $C_2^2.D_{12}$

Order 72: $C_6\times A_4$ x 16, $S_3\times D_6$ x 11, $C_6:D_6$ x 9, $C_3\times S_4$ x 9, $C_3\times \SL(2,3)$ x 7, $C_6\times D_6$ x 6, $S_3\times A_4$ x 6, $C_3:S_4$ x 6, $C_6\wr C_2$ x 6, $C_6:C_{12}$ x 4, $C_3:D_{12}$ x 4, $C_2\times C_6^2$ x 3, $C_{12}:S_3$ x 2, $C_6.D_6$ x 2, $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_6\times C_{12}$, $C_6^2:C_2$, $C_6.D_6$, $C_3:D_{12}$, $C_3^2:Q_8$

Order 64: $C_2^3:D_4$ x 29, $C_2\wr C_2^2$ x 20, $D_4\times C_2^3$ x 15, $C_2^3.D_4$ x 12, $C_2^4:C_4$ x 11, $C_2^3:D_4$ x 9, $C_2^4:C_4$ x 9, $D_4:C_2^3$ x 8, $C_2^3:D_4$ x 8, $D_4^2$ x 7, $D_4:D_4$ x 4, $C_4^2:C_2^2$ x 4, $C_2^4:C_4$ x 4, $C_2^3.D_4$ x 3, $C_2^3.C_2^3$ x 3, $C_2^3.D_4$ x 2, $C_2^3.D_4$ x 2, $C_2^2:C_4^2$ x 2, $C_2^6$ x 2, $D_4:C_2^3$ x 2, $Q_8:D_4$ x 2, $C_2^2.C_4^2$ x 2, $Q_8:D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3:D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^4\times C_4$, $D_4\times Q_8$, $D_4:D_4$, $C_2^2.C_2^4$, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2.C_2^2$

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

Order 48: $C_2^2\times D_6$ x 63, $C_6:D_4$ x 57, $C_2\times S_4$ x 35, $C_6\times D_4$ x 31, $S_3\times D_4$ x 28, $D_6:C_4$ x 23, $C_2\times D_{12}$ x 21, $C_4\times D_6$ x 19, $C_2^2\times A_4$ x 18, $C_2^3\times C_6$ x 15, $C_6.D_4$ x 14, $C_2^2:C_{12}$ x 13, $C_6.C_2^3$ x 11, $D_4:C_6$ x 9, $C_2^2\times C_{12}$ x 8, $D_{12}:C_2$ x 6, $D_4:S_3$ x 5, $C_6.D_4$ x 5, $D_{12}:C_2$ x 4, $S_3\times Q_8$ x 4, $C_4\times A_4$ x 4, $A_4:C_4$ x 4, $C_2\times \SL(2,3)$ x 3, $C_{12}:C_4$ x 3, $C_2^2:A_4$ x 2, $C_6\times Q_8$ x 2, $C_{12}:C_4$ x 2, $C_6:Q_8$

Order 36: $C_6\times S_3$ x 17, $C_3\times A_4$ x 13, $C_6:S_3$ x 12, $C_6^2$ x 6, $S_3^2$ x 5, $C_3:C_{12}$ x 4, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$ x 73, $C_2^2\wr C_2$ x 45, $C_2^3:C_4$ x 43, $C_2^5$ x 18, $D_4:C_2^2$ x 17, $C_4:D_4$ x 15, $C_2^3:C_4$ x 12, $D_4:C_2^2$ x 11, $C_4\times D_4$ x 11, $C_2^3\times C_4$ x 10, $C_2.C_4^2$ x 6, $C_2^2:Q_8$ x 4, $C_2^2.D_4$ x 4, $C_4:D_4$ x 3, $C_2^2\times Q_8$ x 2, $C_4^2:C_2$ x 2, $C_2^2.D_4$ x 2, $C_4\times Q_8$ x 2, $C_2\times C_4^2$ x 2, $C_4:Q_8$, $C_4^2:C_2$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 95, $C_2^2\times C_6$ x 37, $C_3:D_4$ x 27, $C_6:C_4$ x 20, $C_2\times C_{12}$ x 19, $C_2\times A_4$ x 18, $C_3\times D_4$ x 16, $C_4\times S_3$ x 14, $S_4$ x 13, $D_{12}$ x 11, $\SL(2,3)$ x 6, $C_3\times Q_8$ x 5, $C_3:Q_8$ x 4

Order 18: $C_3\times C_6$ x 10, $C_3\times S_3$ x 9, $C_3:S_3$ x 5

Order 16: $C_2\times D_4$ x 114, $C_2^4$ x 66, $C_2^2:C_4$ x 43, $C_2^2\times C_4$ x 34, $D_4:C_2$ x 16, $C_4:C_4$ x 7, $C_2\times Q_8$ x 5, $C_4^2$ x 3

Order 12: $D_6$ x 63, $C_2\times C_6$ x 40, $A_4$ x 10, $C_3:C_4$ x 10, $C_{12}$ x 9

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 92, $C_2\times C_4$ x 39, $D_4$ x 38, $Q_8$ x 5

Order 6: $C_6$ x 20, $S_3$ x 15

Order 4: $C_2^2$ x 52, $C_4$ x 11

Order 3: $C_3$ x 7

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 13824: $(C_2^2\times C_6):S_4^2$ ($C_1$)

Order 6912: $Q_8:A_4^2:S_3$ ($C_2$), $Q_8:A_4^2:C_6$ ($C_2$), $Q_8:A_4^2:S_3$ ($C_2$)

Order 3456: $C_3\times Q_8:A_4^2$ ($C_2^2$)

Order 2304: $(C_2^4\times C_6):S_4$ ($S_3$), $C_{12}.(C_2^3\times S_4)$ ($S_3$), $(C_2^3\times A_4):S_4$ ($S_3$)

Order 1152: $Q_8:A_4^2$ ($D_6$), $C_6\times C_2^4:A_4$ ($D_6$), $C_2^5:C_6^2$ ($D_6$)

Order 576: $C_2^2\times C_6:S_4$ ($S_4$) x 3, $(C_2^2\times C_6):S_4$ ($S_4$)

Order 384: $C_{12}.C_2^5$ ($S_3^2$), $C_2^5:A_4$ ($S_3^2$), $A_4\times D_4:C_2^2$ ($S_3^2$)

Order 288: $C_2^3:C_6^2$ ($C_2\times S_4$) x 3, $C_3\times Q_8:A_4$ ($C_2\times S_4$)

Order 144: $C_6:S_4$ ($C_2^2:S_4$)

Order 128: $D_4:C_2^4$ ($C_3:S_3^2$)

Order 96: $C_2^4\times C_6$ ($S_3\times S_4$) x 3, $C_2^3\times A_4$ ($S_3\times S_4$) x 3, $C_6.C_2^4$ ($S_3\times S_4$), $Q_8:A_4$ ($S_3\times S_4$)

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

Order 36: $C_3\times A_4$ ($C_2^4:S_4$)

Order 32: $C_2^5$ ($A_4:S_3^2$) x 3, $D_4:C_2^2$ ($A_4:S_3^2$)

Order 24: $C_2^2\times C_6$ ($S_4^2$) x 3, $C_2^2\times C_6$ ($C_2^4:S_3^2$), $C_2\times A_4$ ($C_2^4:S_3^2$)

Order 12: $C_2\times C_6$ ($S_3\times C_2^3:S_4$), $A_4$ ($S_3\times C_2^3:S_4$)

Order 8: $C_2^3$ ($C_3:S_4^2$) x 3, $C_2^3$ ($C_6^2:(C_2\times S_4)$)

Order 6: $C_6$ ($C_2^2:S_4^2$)

Order 4: $C_2^2$ ($Q_8:A_4:S_3^2$)

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

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

Order 1: $C_1$ ($(C_2^2\times C_6):S_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 13824: $(C_2^2\times C_6):S_4^2$ ($C_1$)

Order 6912: $Q_8:A_4^2:S_3$ ($C_2$), $Q_8:A_4^2:C_6$ ($C_2$), $Q_8:A_4^2:S_3$ ($C_2$)

Order 3456: $C_3\times Q_8:A_4^2$ ($C_2^2$)

Order 2304: $(C_2^4\times C_6):S_4$ ($S_3$), $C_{12}.(C_2^3\times S_4)$ ($S_3$), $(C_2^3\times A_4):S_4$ ($S_3$)

Order 1152: $Q_8:A_4^2$ ($D_6$), $C_6\times C_2^4:A_4$ ($D_6$), $C_2^5:C_6^2$ ($D_6$)

Order 576: $C_2^2\times C_6:S_4$ ($S_4$), $(C_2^2\times C_6):S_4$ ($S_4$)

Order 384: $C_{12}.C_2^5$ ($S_3^2$), $C_2^5:A_4$ ($S_3^2$), $A_4\times D_4:C_2^2$ ($S_3^2$)

Order 288: $C_3\times Q_8:A_4$ ($C_2\times S_4$), $C_2^3:C_6^2$ ($C_2\times S_4$)

Order 144: $C_6:S_4$ ($C_2^2:S_4$)

Order 128: $D_4:C_2^4$ ($C_3:S_3^2$)

Order 96: $C_2^4\times C_6$ ($S_3\times S_4$), $C_2^3\times A_4$ ($S_3\times S_4$), $C_6.C_2^4$ ($S_3\times S_4$), $Q_8:A_4$ ($S_3\times S_4$)

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

Order 36: $C_3\times A_4$ ($C_2^4:S_4$)

Order 32: $C_2^5$ ($A_4:S_3^2$), $D_4:C_2^2$ ($A_4:S_3^2$)

Order 24: $C_2^2\times C_6$ ($C_2^4:S_3^2$), $C_2^2\times C_6$ ($S_4^2$), $C_2\times A_4$ ($C_2^4:S_3^2$)

Order 12: $C_2\times C_6$ ($S_3\times C_2^3:S_4$), $A_4$ ($S_3\times C_2^3:S_4$)

Order 8: $C_2^3$ ($C_6^2:(C_2\times S_4)$), $C_2^3$ ($C_3:S_4^2$)

Order 6: $C_6$ ($C_2^2:S_4^2$)

Order 4: $C_2^2$ ($Q_8:A_4:S_3^2$)

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

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

Order 1: $C_1$ ($(C_2^2\times C_6):S_4^2$)

Series

Derived series

$(C_2^2\times C_6):S_4^2$

$\rhd$

$C_3\times Q_8:A_4^2$

$\rhd$

$D_4:C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$(C_2^2\times C_6):S_4^2$

$\rhd$

$Q_8:A_4^2:C_6$

$\rhd$

$C_3\times Q_8:A_4^2$

$\rhd$

$C_2^5:C_6^2$

$\rhd$

$Q_8:A_4^2$

$\rhd$

$C_{12}.C_2^5$

$\rhd$

$A_4\times D_4:C_2^2$

$\rhd$

$C_2^5:A_4$

$\rhd$

$D_4:C_2^4$

$\rhd$

$C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times C_6):S_4^2$

$\rhd$

$C_3\times Q_8:A_4^2$

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 0 larger groups in the database.

Character theory

Complex character table

See the $102 \times 102$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $100 \times 100$ rational character table.