Abstract group 6144.bbp: $C_4^4.S_4$

• Label: 6144.bbp
• Order: \( 2^{11} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_4^4.S_4$
Order:	\(6144\)\(\medspace = 2^{11} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_4^2:A_4.C_2^5.C_2^4$, of order \(98304\)\(\medspace = 2^{15} \cdot 3 \)
Composition factors:	$C_2$ x 11, $C_3$
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	207	128	1200	384	2688	1536	6144
Conjugacy classes	1	11	1	62	3	40	12	130
Divisions	1	11	1	39	3	24	6	85
Autjugacy classes	1	7	1	22	2	7	3	43

Dimension	1	2	3	4	6	8	12	16	24	48
Irr. complex chars.	16	8	16	16	48	8	16	0	2	0	130
Irr. rational chars.	8	8	8	2	16	8	26	4	4	1	85

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$645120$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{12}=d^{4}=e^{2}=f^{4}=g^{4}=[a,g]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,4,2,3)(5,7,6,8)(9,17,11,19,10,18,12,20)(13,24,15,22,14,23,16,21), (1,19,6,22,2,20,5,21) \!\cdots\! \rangle$
Transitive group:	24T9263				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4^3.S_4)$ $\,\rtimes\,$ $C_4$	$(C_4^3.S_4)$ $\,\rtimes\,$ $C_4$	$(C_4^4.A_4)$ $\,\rtimes\,$ $C_2$	$(C_4.Q_8^2)$ $\,\rtimes\,$ $(C_4\times S_3)$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^4$ . $S_4$	$C_4^3$ . $(C_4\times S_4)$ (2)	$C_4$ . $(C_4^3.S_4)$	$C_2^4$ . $(C_2^4:S_4)$	all 26

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

Homology

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

Subgroups

There are 72313 subgroups in 4984 conjugacy classes, 54 normal (32 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_4^3.S_4$
Commutator:	$G' \simeq$ $(C_2\times C_4^2).A_4$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4^2$	$G/\Phi \simeq$ $C_2^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^4.C_2^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_4^4.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^6.D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.C_2^4.C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 6144: $C_4^4.S_4$

Order 3072: $(C_2\times C_4^3).S_4$ x 2, $(C_2\times C_4^3).S_4$ x 2, $(C_2\times C_4^3).S_4$, $C_4^4.A_4$, $(C_2\times C_4^3).S_4$

Order 2048: $C_2^3.C_2^4.C_2^4$

Order 1536: $C_4^3.S_4$ x 2, $C_4^3.S_4$ x 2, $(C_2^2\times C_4^2).S_4$ x 2, $(C_2\times C_4^3).A_4$ x 2, $(C_2^2\times C_4^2).S_4$, $(C_2^2\times C_4^2).S_4$, $C_4^2.(C_4^2\times S_3)$, $(C_2\times C_4^3).A_4$

Order 1024: $C_2^3.C_2^4.C_2^3$ x 4, $C_2^3.C_2^4.C_2^3$ x 4, $C_2.(C_4\times Q_8).C_2^4$, $C_2^2.C_2^5.C_2^3$, $C_2^3.C_2^4.C_2^3$, $(C_2\times C_4^3).C_2^3$, $C_2^3.C_2^5.C_2^2$, $C_4^4.C_2^2$, $C_2^3.C_2^4.C_2^3$

Order 768: $C_4^3:D_6$ x 3, $C_4\times C_2^3.S_4$ x 3, $(C_2\times C_4^2).S_4$ x 2, $(C_2\times C_4^2).S_4$ x 2, $C_4^3:D_6$, $C_2^4.(C_2\times S_4)$, $(C_2^2\times C_4^2).A_4$, $C_4^3.A_4$, $C_4^3.A_4$, $C_4^3:C_{12}$

Order 512: $C_2^3.C_2^5.C_2$ x 4, $(C_2^2\times Q_8).C_2^4$ x 4, $C_2^4.C_2^4.C_2$ x 4, $(C_2^2\times D_4:C_2).D_4$ x 4, $(C_2^2\times D_4:C_2).D_4$ x 4, $C_2^3.C_2^4.C_2^2$ x 2, $C_2^3.C_2^4.C_2^2$ x 2, $C_2^3.C_2^4.C_2^2$ x 2, $C_2^3.C_2^4.C_2^2$ x 2, $(C_2^2\times Q_8).C_2^4$ x 2, $(C_2^2\times Q_8).C_2^4$ x 2, $(C_2^2\times Q_8).C_2^4$ x 2, $(C_2^2\times Q_8).C_2^4$ x 2, $C_2.C_2^6.C_2^2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $(C_2\times C_4).D_4^2$ x 2, $(C_2^3\times C_4).C_2^4$ x 2, $(C_2\times C_4).D_4^2$ x 2, $C_2^3.D_4^2$ x 2, $C_4^3.C_2^3$, $C_4^3.C_2^3$, $(C_2\times C_4^2).C_2^4$, $C_2\times C_4.D_4^2$, $C_2^3.C_2^5.C_2$, $(C_2^3\times C_4).C_2^4$, $C_2^3.C_2^5.C_2$, $(C_2^2\times Q_8).C_2^4$, $C_2^3.C_2^5.C_2$, $(C_2^2\times Q_8).C_2^4$, $(C_2^2\times Q_8).C_2^4$, $C_2^4.C_2^4.C_2$, $C_2^3.C_2^5.C_2$, $(C_2^3\times C_4).C_2^4$, $C_4^4.C_2$, $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$, $C_2^3.C_2^5.C_2$, $(C_2^3\times C_4).C_2^4$, $C_4^4.C_2$, $C_2^2.C_2^5.C_2^2$

Order 384: $C_4\wr S_3$ x 14, $C_4^3:C_6$ x 3, $C_2^4.S_4$ x 3, $C_2^4.S_4$ x 2, $C_4^2:(C_4\times S_3)$ x 2, $(C_2\times C_4^2).A_4$, $C_2^4.(C_2\times A_4)$, $C_4^3:C_6$, $C_2^4.S_4$, $C_2^4.S_4$, $C_4^2\times S_4$, $C_2^4.S_4$

Order 256: $C_4^3:C_2^2$ x 16, $C_4^3:C_2^2$ x 10, $(C_2\times C_4^2).D_4$ x 8, $(C_2\times C_4^2).D_4$ x 8, $C_4^3.C_2^2$ x 8, $C_4^3:C_2^2$ x 8, $C_4^3.C_4$ x 6, $C_4^3:C_4$ x 6, $C_4^2.(C_2\times D_4)$ x 6, $(C_2\times C_4^2).D_4$ x 4, $(C_2\times C_4^2).D_4$ x 4, $(C_2^2\times D_4).D_4$ x 4, $(C_2\times C_4^2).D_4$ x 4, $(C_2\times C_4^2).D_4$ x 4, $(C_2\times C_4^2).D_4$ x 4, $(C_2\times C_4^3):C_2$ x 4, $(C_2\times C_4^2).D_4$ x 4, $(C_2\times C_4^2).D_4$ x 4, $C_4^3.C_2^2$ x 4, $C_4^3:C_2^2$ x 4, $D_8:C_4^2$ x 4, $D_4.(C_2\times C_4^2)$ x 4, $C_4^3:C_2^2$ x 4, $(C_2\times D_4):\OD_{16}$ x 3, $(C_2\times C_4^2).D_4$ x 2, $C_4^3.C_2^2$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2^2\times C_4^2).C_4$ x 2, $C_4^3.C_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_2^3.C_2^5$ x 2, $C_4^3.C_2^2$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_4^3.C_2^2$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times D_4).C_4^2$ x 2, $(C_2\times D_4).C_4^2$ x 2, $(C_2\times D_4):C_4^2$ x 2, $\OD_{16}.C_4^2$ x 2, $\OD_{16}.C_4^2$ x 2, $\OD_{16}:C_4^2$ x 2, $C_4^2.C_4^2$ x 2, $C_4^2.C_4^2$ x 2, $\OD_{16}.C_4^2$ x 2, $(C_2^2\times C_8).D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_4^3.C_2^2$ x 2, $C_2\times C_4^2.D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_2^4.(C_2\times D_4)$ x 2, $(C_4\times D_4).D_4$ x 2, $(C_4\times D_4).D_4$ x 2, $(C_2^2\times D_4):D_4$ x 2, $C_4^3:C_2^2$ x 2, $C_2\times C_4^2.D_4$ x 2, $C_2\times C_4^2.D_4$ x 2, $C_2\times \OD_{16}.D_4$ x 2, $D_8:C_4^2$ x 2, $D_8:C_4^2$ x 2, $\OD_{16}.C_4^2$ x 2, $C_4.(C_2^2\times C_4^2)$ x 2, $C_4^4$, $C_4^3.C_4$, $(C_2^2\times C_4^2).C_4$, $C_4^3.C_4$, $C_4^2.Q_{16}$, $C_4^2.D_8$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4.Q_8^2$, $C_4^3.C_2^2$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^4.(C_2\times D_4)$, $C_2^4.(C_2\times D_4)$, $C_4^2:\OD_{16}$, $C_4^2:\OD_{16}$, $C_4^2.\OD_{16}$, $C_4^3.C_4$, $C_4^3.C_4$, $C_4^3:C_4$, $(C_2\times D_4).C_4^2$, $\OD_{16}:C_4^2$, $C_4.D_4^2$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $(C_4\times D_4).D_4$, $(C_4\times D_4):D_4$, $C_4^3:C_2^2$, $C_4^2.(C_2^2\times C_4)$, $C_2\times \OD_{16}:D_4$, $(C_2\times C_4^2).D_4$, $(C_2\times C_4^2).D_4$, $C_2^4.(C_2\times D_4)$, $C_4^3:C_2^2$, $C_4^2.(C_2\times D_4)$, $(C_2^2\times C_4^2):C_2^2$, $\OD_{16}.C_4^2$, $C_4^2:\OD_{16}$, $C_4\times \OD_{16}.C_2^2$, $C_2^4.C_2^4$, $C_4^2\times \OD_{16}$, $C_4^2.C_2^4$, $C_4^2.C_4^2$, $(C_2\times D_4):\OD_{16}$

Order 192: $C_4^2:D_6$ x 8, $C_2^3.S_4$ x 8, $C_4\wr C_3$ x 7, $C_4^2:D_6$ x 4, $A_4:C_4^2$ x 3, $C_2^4.D_6$ x 3, $A_4\times C_4^2$, $C_2^4.A_4$, $C_2^3.(C_2\times A_4)$, $C_2^3:\SL(2,3)$, $C_4^2:C_{12}$, $Q_8.(C_2\times A_4)$, $C_2^4.A_4$

Order 128: $D_4:C_4^2$ x 64, $C_4^2.C_2^3$ x 24, $C_4^2.C_2^3$ x 24, $C_2^4.D_4$ x 20, $C_4^2.D_4$ x 16, $C_4^2.D_4$ x 16, $\OD_{16}.D_4$ x 16, $(C_2\times C_4).C_4^2$ x 14, $\OD_{16}:(C_2\times C_4)$ x 12, $D_4.C_4^2$ x 12, $C_4^2:C_2^3$ x 12, $C_2\times C_4^2.C_4$ x 10, $C_4^2.D_4$ x 10, $C_2^3.C_4^2$ x 10, $C_4^2.D_4$ x 8, $C_4^3:C_2$ x 8, $\OD_{16}:D_4$ x 8, $C_4^2:(C_2\times C_4)$ x 8, $C_4^2:(C_2\times C_4)$ x 8, $C_4.(C_4\times D_4)$ x 8, $(C_2\times D_4).D_4$ x 8, $D_4.C_4^2$ x 8, $D_4.C_4^2$ x 8, $C_2^4.D_4$ x 8, $C_4^2.C_2^3$ x 8, $C_4^2.C_2^3$ x 8, $C_4^2:C_2^3$ x 8, $C_4^3:C_2$ x 8, $C_2^3.C_4^2$ x 7, $C_4^3:C_2$ x 7, $C_2\times C_4^3$ x 6, $C_4.(C_4\times D_4)$ x 6, $C_4^2.D_4$ x 6, $C_4^2.C_2^3$ x 6, $C_4^2.C_2^3$ x 6, $C_4^2.C_2^3$ x 6, $C_4^2.D_4$ x 6, $C_2^3.C_2^4$ x 6, $C_4^2:D_4$ x 6, $C_4^2.C_2^3$ x 5, $D_4.C_4^2$ x 5, $D_4.C_4^2$ x 5, $C_2^3.C_2^4$ x 5, $D_4\times C_4^2$ x 5, $C_2^3.Q_{16}$ x 4, $C_2^3.D_8$ x 4, $\OD_{16}:(C_2\times C_4)$ x 4, $C_2^4.D_4$ x 4, $C_4^2.D_4$ x 4, $C_2^4.D_4$ x 4, $C_8.C_4^2$ x 4, $C_8.C_4^2$ x 4, $C_2^3.C_4^2$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.C_2^3$ x 4, $C_4^2.C_2^3$ x 4, $C_2^3.C_2^4$ x 4, $C_4^3:C_2$ x 4, $C_4^2:D_4$ x 4, $C_2^4.D_4$ x 3, $C_4^2.D_4$ x 3, $C_8:C_4^2$ x 3, $(C_2\times Q_8):C_8$ x 3, $C_8.C_2^4$ x 3, $C_4^2.C_2^3$ x 3, $(C_2\times D_4):C_8$ x 3, $C_4^2.D_4$ x 3, $C_4^2:C_2^3$ x 3, $C_2^3.C_2^4$ x 3, $C_2.D_4^2$ x 3, $C_4^2:D_4$ x 3, $C_2^3.C_2^4$ x 3, $C_2^3.C_2^4$ x 3, $C_2^3.C_2^4$ x 3, $C_2\times C_4^2.C_4$ x 2, $C_4.(C_4\times D_4)$ x 2, $C_2^4.D_4$ x 2, $C_4^2.D_4$ x 2, $C_2^4.D_4$ x 2, $(C_2\times D_4).D_4$ x 2, $C_8.C_4^2$ x 2, $(C_2\times Q_8):C_8$ x 2, $C_8.C_4^2$ x 2, $(C_2\times D_4):C_8$ x 2, $C_2^3:C_4^2$ x 2, $C_2^4.C_2^3$ x 2, $C_2^4.Q_8$ x 2, $C_2^3.C_4^2$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_2^3:\OD_{16}$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_2^4.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.C_2^3$ x 2, $C_2^4.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.C_2^3$ x 2, $C_2^4.Q_8$ x 2, $C_2^4.D_4$ x 2, $\OD_{16}.C_2^3$ x 2, $C_2.Q_8^2$ x 2, $C_2.Q_8^2$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_4^2:Q_8$ x 2, $C_2^3.C_2^4$ x 2, $Q_8\times C_4^2$ x 2, $C_2^4.D_4$, $C_4^2:C_8$, $C_4^2:C_8$, $(C_2\times C_4^2).C_4$, $C_4^2.D_4$, $C_4^3.C_2$, $C_4^2:C_8$, $C_2^3.C_4^2$, $C_4^2\times C_8$, $C_4.C_2^5$, $D_8:C_2^3$, $C_4^2.C_2^3$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2:C_2^3$, $C_2^4.D_4$, $C_4^2.C_2^3$, $C_2^3.C_2^4$, $C_4^2:Q_8$, $C_4^2:D_4$, $C_2.Q_8^2$, $C_2^3.C_2^4$, $C_4^3:C_2$, $C_2^3.C_2^4$

Order 96: $C_4\times S_4$ x 12, $C_4^2:S_3$ x 8, $C_4^2:C_6$ x 4, $C_2^3:C_{12}$ x 3, $C_2^2.S_4$ x 3, $C_4^2:C_6$ x 2, $S_3\times C_4^2$, $C_2^2\times S_4$, $Q_8:A_4$

Order 64: $C_4^2:C_2^2$ x 124, $C_4^2.C_2^2$ x 67, $C_4^2:C_4$ x 64, $C_4^2:C_2^2$ x 56, $C_4\times \OD_{16}$ x 50, $C_4^2:C_2^2$ x 34, $D_8:C_4$ x 32, $D_8:C_4$ x 32, $C_8.C_2^3$ x 27, $C_4^3$ x 26, $C_4^2.C_2^2$ x 26, $\OD_{16}:C_4$ x 22, $C_2^2:C_4^2$ x 22, $D_4.D_4$ x 20, $C_4:C_4^2$ x 19, $C_4^2:C_2^2$ x 18, $C_2^3.C_2^3$ x 17, $C_2^3.C_2^3$ x 16, $D_4:D_4$ x 16, $C_4^2:C_2^2$ x 15, $C_2^2\times C_4^2$ x 15, $C_4^2.C_4$ x 14, $C_4^2.C_2^2$ x 14, $C_4^2.C_2^2$ x 13, $D_8:C_2^2$ x 12, $D_4.D_4$ x 12, $\SD_{16}:C_4$ x 12, $C_4\times \SD_{16}$ x 12, $C_2^3.Q_8$ x 12, $C_4.C_4^2$ x 11, $C_2^2\times \OD_{16}$ x 11, $C_2^3.D_4$ x 10, $C_2\times C_4\times C_8$ x 10, $C_2^3.D_4$ x 10, $C_2^3:D_4$ x 10, $\OD_{16}:C_4$ x 10, $C_8.C_2^3$ x 10, $\OD_{16}:C_4$ x 10, $C_4.C_2^4$ x 9, $\OD_{16}.C_2^2$ x 8, $D_4:C_2^3$ x 8, $\OD_{16}:C_4$ x 8, $C_4^2:C_2^2$ x 8, $C_2^3:D_4$ x 8, $C_4^2.C_2^2$ x 8, $C_4^2:C_2^2$ x 8, $D_8:C_4$ x 8, $C_4\times D_8$ x 8, $C_4:\OD_{16}$ x 8, $C_4^2.C_2^2$ x 8, $C_2^3.D_4$ x 7, $C_2^3.C_2^3$ x 7, $C_8:C_2^3$ x 7, $C_2^3.D_4$ x 6, $C_4^2:C_4$ x 6, $C_4^2.C_2^2$ x 6, $\OD_{16}.C_2^2$ x 5, $C_2^3.C_2^3$ x 5, $C_2^3.C_2^3$ x 5, $C_2^3:Q_8$ x 5, $C_4^2:C_4$ x 5, $C_4^2:C_2^2$ x 5, $C_4^2.C_4$ x 5, $C_2^3.D_4$ x 4, $\OD_{16}:C_2^2$ x 4, $C_2^3.C_2^3$ x 4, $C_4^2:C_4$ x 4, $C_4^2.C_4$ x 4, $C_4.C_4^2$ x 4, $C_2^3.D_4$ x 4, $Q_8:D_4$ x 4, $C_4:D_8$ x 4, $D_4:D_4$ x 4, $Q_{16}:C_4$ x 4, $C_4\times Q_{16}$ x 4, $C_4^2.C_4$ x 4, $C_2^3.Q_8$ x 4, $C_2^2:\OD_{16}$ x 3, $C_2^3.D_4$ x 3, $C_2^2.\OD_{16}$ x 3, $C_2^3:C_8$ x 3, $C_4.C_2^4$ x 3, $D_4:C_2^3$ x 3, $Q_{16}:C_2^2$ x 3, $C_4^2.C_2^2$ x 3, $C_2^3.D_4$ x 3, $D_8:C_2^2$ x 2, $C_2^2.C_4^2$ x 2, $C_2^2.C_4^2$ x 2, $C_2^3.D_4$ x 2, $C_2^2.D_8$ x 2, $Q_8:Q_8$ x 2, $D_4:Q_8$ x 2, $Q_8:Q_8$ x 2, $D_4:Q_8$ x 2, $Q_8:D_4$ x 2, $D_4:D_4$ x 2, $C_2^3.D_4$ x 2, $C_2^4:C_4$, $C_4^2.C_4$

Order 48: $C_2\times S_4$ x 6, $C_4\times A_4$ x 6, $A_4:C_4$ x 6, $C_4\times D_6$ x 3, $C_{12}:C_4$ x 3, $C_4^2:C_3$ x 2, $C_2^2\times A_4$, $\SL(2,3):C_2$, $C_4\times C_{12}$

Order 32: $C_4\times D_4$ x 140, $C_4\wr C_2$ x 136, $C_2\times C_4^2$ x 117, $C_2\times \OD_{16}$ x 91, $D_4:C_2^2$ x 81, $C_4^2:C_2$ x 77, $\OD_{16}:C_2$ x 64, $C_4\times Q_8$ x 54, $D_8:C_2$ x 36, $C_2^3:C_4$ x 36, $C_8:C_4$ x 34, $C_2^2.D_4$ x 34, $D_4:C_2^2$ x 31, $C_4\times C_8$ x 31, $C_4.C_2^3$ x 24, $C_4^2:C_2$ x 24, $C_4:D_4$ x 22, $D_4:C_4$ x 20, $Q_{16}:C_2$ x 20, $C_2^2\times D_4$ x 19, $C_2^3\times C_4$ x 19, $C_2.C_4^2$ x 18, $C_4.D_4$ x 16, $D_8:C_2$ x 16, $C_8.C_4$ x 16, $C_2^2\times C_8$ x 12, $Q_8:C_4$ x 12, $C_4:D_4$ x 11, $C_4:C_8$ x 11, $\OD_{16}:C_2$ x 9, $C_4:Q_8$ x 9, $C_2^2\times Q_8$ x 8, $C_2\times \SD_{16}$ x 8, $C_8:C_4$ x 8, $C_8:C_4$ x 8, $C_2^2:C_8$ x 7, $C_2\times D_8$ x 6, $C_2^3:C_4$ x 4, $C_2\times Q_{16}$ x 2

Order 24: $C_4\times S_3$ x 12, $S_4$ x 4, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_2\times A_4$ x 3, $\SL(2,3)$, $C_2\times D_6$

Order 16: $D_4:C_2$ x 138, $C_2^2\times C_4$ x 128, $C_4^2$ x 118, $C_2\times D_4$ x 108, $\OD_{16}$ x 82, $C_2^2:C_4$ x 62, $C_2\times C_8$ x 60, $C_4:C_4$ x 55, $C_2\times Q_8$ x 40, $\SD_{16}$ x 24, $D_8$ x 16, $Q_{16}$ x 8, $C_2^4$ x 5

Order 12: $D_6$ x 6, $C_{12}$ x 6, $C_3:C_4$ x 6, $C_2\times C_6$, $A_4$

Order 8: $C_2\times C_4$ x 166, $D_4$ x 68, $C_2^3$ x 29, $Q_8$ x 24, $C_8$ x 24

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_4$ x 39, $C_2^2$ x 33

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6144: $C_4^4.S_4$

Order 3072: $(C_2\times C_4^3).S_4$, $(C_2\times C_4^3).S_4$, $(C_2\times C_4^3).S_4$, $C_4^4.A_4$, $(C_2\times C_4^3).S_4$

Order 2048: $C_2^3.C_2^4.C_2^4$

Order 1536: $(C_2^2\times C_4^2).S_4$, $(C_2^2\times C_4^2).S_4$, $C_4^3.S_4$, $C_4^3.S_4$, $(C_2^2\times C_4^2).S_4$, $C_4^2.(C_4^2\times S_3)$, $(C_2\times C_4^3).A_4$, $(C_2\times C_4^3).A_4$

Order 1024: $C_2^3.C_2^4.C_2^3$ x 2, $C_2^3.C_2^4.C_2^3$ x 2, $C_2.(C_4\times Q_8).C_2^4$, $C_2^2.C_2^5.C_2^3$, $C_2^3.C_2^4.C_2^3$, $(C_2\times C_4^3).C_2^3$, $C_2^3.C_2^5.C_2^2$, $C_4^4.C_2^2$, $C_2^3.C_2^4.C_2^3$

Order 768: $C_4^3:D_6$ x 2, $C_4\times C_2^3.S_4$ x 2, $(C_2\times C_4^2).S_4$, $(C_2\times C_4^2).S_4$, $C_4^3:D_6$, $C_2^4.(C_2\times S_4)$, $(C_2^2\times C_4^2).A_4$, $C_4^3.A_4$, $C_4^3.A_4$, $C_4^3:C_{12}$

Order 512: $C_2^3.C_2^5.C_2$ x 2, $(C_2^2\times Q_8).C_2^4$ x 2, $C_2.C_2^6.C_2^2$ x 2, $(C_2^2\times Q_8).C_2^4$ x 2, $C_2^4.C_2^4.C_2$ x 2, $(C_2^2\times D_4:C_2).D_4$ x 2, $(C_2^2\times D_4:C_2).D_4$ x 2, $C_4^3.C_2^3$, $C_4^3.C_2^3$, $(C_2\times C_4^2).C_2^4$, $C_2\times C_4.D_4^2$, $C_2^3.C_2^5.C_2$, $(C_2^3\times C_4).C_2^4$, $C_2^3.C_2^5.C_2$, $C_2^3.C_2^4.C_2^2$, $C_2^3.C_2^4.C_2^2$, $C_2^3.C_2^4.C_2^2$, $C_2^3.C_2^4.C_2^2$, $(C_2^2\times Q_8).C_2^4$, $(C_2^2\times Q_8).C_2^4$, $(C_2^2\times Q_8).C_2^4$, $(C_2^2\times Q_8).C_2^4$, $C_2^3.C_2^5.C_2$, $C_2^3.C_2^5.C_2$, $(C_2^2\times Q_8).C_2^4$, $(C_2^2\times Q_8).C_2^4$, $C_2^4.C_2^4.C_2$, $C_2^3.C_2^5.C_2$, $C_2^4.C_2^4.C_2$, $(C_2^3\times C_4).C_2^4$, $(C_2\times C_4).D_4^2$, $(C_2^3\times C_4).C_2^4$, $(C_2\times C_4).D_4^2$, $C_2^3.D_4^2$, $C_4^4.C_2$, $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$, $C_2^3.C_2^5.C_2$, $(C_2^3\times C_4).C_2^4$, $C_4^4.C_2$, $C_2^2.C_2^5.C_2^2$

Order 384: $C_4\wr S_3$ x 4, $C_4^3:C_6$ x 2, $C_2^4.S_4$ x 2, $(C_2\times C_4^2).A_4$, $C_2^4.S_4$, $C_2^4.(C_2\times A_4)$, $C_4^3:C_6$, $C_2^4.S_4$, $C_4^2:(C_4\times S_3)$, $C_2^4.S_4$, $C_4^2\times S_4$, $C_2^4.S_4$

Order 256: $C_4^3:C_2^2$ x 7, $(C_2\times C_4^2).D_4$ x 4, $(C_2\times C_4^2).D_4$ x 4, $(C_2^2\times D_4).D_4$ x 4, $C_4^3:C_4$ x 4, $C_4^2.(C_2\times D_4)$ x 4, $C_4^3.C_2^2$ x 4, $C_4^3:C_2^2$ x 4, $C_4^3.C_4$ x 3, $C_4^3:C_2^2$ x 3, $C_4^3:C_2^2$ x 3, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_4^3.C_2^2$ x 2, $(C_2\times D_4):\OD_{16}$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2^2\times C_4^2).C_4$ x 2, $C_4^2.C_2^4$ x 2, $C_2^3.C_2^5$ x 2, $C_4^3.C_2^2$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_4^3.C_2^2$ x 2, $(C_2\times C_4^3):C_2$ x 2, $C_4^3.C_2^2$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_2^4.(C_2\times D_4)$ x 2, $(C_2^2\times D_4):D_4$ x 2, $C_4^3.C_2^2$ x 2, $C_4^3:C_2^2$ x 2, $D_4.(C_2\times C_4^2)$ x 2, $C_4^4$, $(C_2\times C_4^2).D_4$, $(C_2\times C_4^2).D_4$, $(C_2\times C_4^2).D_4$, $(C_2\times C_4^2).D_4$, $C_4^3.C_4$, $C_4^3.C_4$, $(C_2\times C_4^2).D_4$, $(C_2^2\times C_4^2).C_4$, $C_4^3.C_4$, $C_4^2.Q_{16}$, $C_4^2.D_8$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4.Q_8^2$, $C_4^3.C_2^2$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^4.(C_2\times D_4)$, $C_2^4.(C_2\times D_4)$, $C_4^2:\OD_{16}$, $C_4^2:\OD_{16}$, $C_4^2.\OD_{16}$, $(C_2\times C_4^2).D_4$, $C_4^3.C_4$, $C_4^3.C_4$, $(C_2\times D_4).C_4^2$, $(C_2\times D_4).C_4^2$, $(C_2\times D_4):C_4^2$, $\OD_{16}.C_4^2$, $\OD_{16}.C_4^2$, $\OD_{16}:C_4^2$, $C_4^2.C_4^2$, $C_4^2.C_4^2$, $C_4^3:C_4$, $\OD_{16}.C_4^2$, $(C_2^2\times C_8).D_4$, $(C_2\times C_4^2).D_4$, $(C_2\times D_4).C_4^2$, $\OD_{16}:C_4^2$, $(C_2\times C_4^2).D_4$, $(C_2\times C_4^2).D_4$, $C_4.D_4^2$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $C_2\times C_4^2.D_4$, $(C_4\times D_4).D_4$, $(C_4\times D_4).D_4$, $(C_4\times D_4).D_4$, $(C_4\times D_4):D_4$, $C_4^3:C_2^2$, $C_4^3:C_2^2$, $C_2\times C_4^2.D_4$, $C_2\times C_4^2.D_4$, $C_4^2.(C_2^2\times C_4)$, $C_2\times \OD_{16}:D_4$, $(C_2\times C_4^2).D_4$, $C_2\times \OD_{16}.D_4$, $(C_2\times C_4^2).D_4$, $C_2^4.(C_2\times D_4)$, $C_4^3:C_2^2$, $C_4^2.(C_2\times D_4)$, $(C_2^2\times C_4^2):C_2^2$, $D_8:C_4^2$, $D_8:C_4^2$, $D_8:C_4^2$, $\OD_{16}.C_4^2$, $\OD_{16}.C_4^2$, $C_4^2:\OD_{16}$, $C_4\times \OD_{16}.C_2^2$, $C_2^4.C_2^4$, $C_4.(C_2^2\times C_4^2)$, $C_4^2\times \OD_{16}$, $C_4^2.C_2^4$, $C_4^2.C_4^2$, $(C_2\times D_4):\OD_{16}$

Order 192: $C_4\wr C_3$ x 4, $C_4^2:D_6$ x 3, $C_2^3.S_4$ x 3, $A_4:C_4^2$ x 2, $C_2^4.D_6$ x 2, $A_4\times C_4^2$, $C_2^4.A_4$, $C_4^2:D_6$, $C_2^3.(C_2\times A_4)$, $C_2^3:\SL(2,3)$, $C_4^2:C_{12}$, $Q_8.(C_2\times A_4)$, $C_2^4.A_4$

Order 128: $D_4:C_4^2$ x 21, $C_2^4.D_4$ x 10, $(C_2\times C_4).C_4^2$ x 8, $C_2^3.C_4^2$ x 7, $C_4^3:C_2$ x 7, $\OD_{16}:(C_2\times C_4)$ x 6, $C_4^2.C_2^3$ x 6, $C_4^2.C_2^3$ x 6, $C_4^2.D_4$ x 6, $C_4^2:C_2^3$ x 6, $C_2^3.C_2^4$ x 6, $C_4^2:D_4$ x 6, $C_4^3:C_2$ x 6, $C_2\times C_4^3$ x 5, $C_2\times C_4^2.C_4$ x 5, $C_2^3.C_2^4$ x 5, $D_4\times C_4^2$ x 5, $C_4^2:(C_2\times C_4)$ x 4, $C_4^2:(C_2\times C_4)$ x 4, $C_4.(C_4\times D_4)$ x 4, $C_2^4.D_4$ x 4, $(C_2\times D_4).D_4$ x 4, $C_2^4.D_4$ x 4, $D_4.C_4^2$ x 4, $C_4^2.C_2^3$ x 4, $C_4^2.C_2^3$ x 4, $C_4^2.D_4$ x 4, $C_2^3.C_4^2$ x 4, $C_2^3.C_2^4$ x 4, $C_4^3:C_2$ x 4, $C_4^2:D_4$ x 4, $C_4.(C_4\times D_4)$ x 3, $C_2^4.D_4$ x 3, $C_4^2.D_4$ x 3, $C_4^2.D_4$ x 3, $C_2^4.D_4$ x 3, $C_4^2.C_2^3$ x 3, $C_4^2.C_2^3$ x 3, $C_4^2:C_2^3$ x 3, $C_4^2:C_2^3$ x 3, $D_4.C_4^2$ x 3, $D_4.C_4^2$ x 3, $C_2^3.C_2^4$ x 3, $C_2.D_4^2$ x 3, $C_4^2:D_4$ x 3, $C_2^3.C_2^4$ x 3, $C_2^3.C_2^4$ x 3, $C_2^3.C_2^4$ x 3, $C_2^3.Q_{16}$ x 2, $C_2^3.D_8$ x 2, $\OD_{16}:(C_2\times C_4)$ x 2, $C_2^4.D_4$ x 2, $C_4^2.D_4$ x 2, $C_2^4.D_4$ x 2, $(C_2\times D_4).D_4$ x 2, $D_4.C_4^2$ x 2, $D_4.C_4^2$ x 2, $C_2^3:C_4^2$ x 2, $C_2^3.C_4^2$ x 2, $C_8:C_4^2$ x 2, $(C_2\times Q_8):C_8$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_8.C_2^4$ x 2, $(C_2\times D_4):C_8$ x 2, $C_2^3:\OD_{16}$ x 2, $C_4^2.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2.C_2^3$ x 2, $C_2^4.D_4$ x 2, $\OD_{16}.C_2^3$ x 2, $C_2.Q_8^2$ x 2, $C_2.Q_8^2$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_4^2:Q_8$ x 2, $C_2^3.C_2^4$ x 2, $Q_8\times C_4^2$ x 2, $C_4^2.D_4$, $C_2\times C_4^2.C_4$, $C_4^3:C_2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4.(C_4\times D_4)$, $C_2^4.D_4$, $\OD_{16}:D_4$, $\OD_{16}.D_4$, $C_4^2:C_8$, $C_4^2:C_8$, $(C_2\times C_4^2).C_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_8.C_4^2$, $(C_2\times Q_8):C_8$, $C_8.C_4^2$, $C_8.C_4^2$, $C_8.C_4^2$, $(C_2\times D_4):C_8$, $C_4^3.C_2$, $C_4^2:C_8$, $C_2^3.C_4^2$, $C_2^4.C_2^3$, $C_2^4.Q_8$, $C_2^3.C_4^2$, $C_4^2\times C_8$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4.C_2^5$, $D_8:C_2^3$, $C_4^2.C_2^3$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_4^2:C_2^3$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_4^2.D_4$, $C_4^2.D_4$, $C_2^4.Q_8$, $C_2^4.D_4$, $C_4^2.C_2^3$, $C_2^3.C_2^4$, $C_4^2:Q_8$, $C_4^2:D_4$, $C_2.Q_8^2$, $C_2^3.C_2^4$, $C_4^3:C_2$, $C_2^3.C_2^4$

Order 96: $C_4^2:C_6$ x 3, $C_4\times S_4$ x 3, $C_4^2:S_3$ x 2, $C_2^3:C_{12}$ x 2, $C_2^2.S_4$ x 2, $S_3\times C_4^2$, $C_4^2:C_6$, $C_2^2\times S_4$, $Q_8:A_4$

Order 64: $C_4^2.C_2^2$ x 36, $C_4^2:C_2^2$ x 28, $C_4^2:C_2^2$ x 26, $C_4\times \OD_{16}$ x 24, $C_4^2:C_4$ x 20, $C_2^2:C_4^2$ x 19, $C_4^3$ x 18, $C_2^3.C_2^3$ x 17, $C_2^3.C_2^3$ x 16, $C_4^2.C_2^2$ x 16, $C_4:C_4^2$ x 15, $C_4^2:C_2^2$ x 14, $C_2^2\times C_4^2$ x 13, $C_4^2:C_2^2$ x 11, $C_4^2.C_2^2$ x 11, $C_2^3.D_4$ x 10, $C_2^3:D_4$ x 10, $C_4^2.C_2^2$ x 10, $C_4^2:C_2^2$ x 8, $C_2^3:D_4$ x 8, $C_4.C_4^2$ x 7, $C_2\times C_4\times C_8$ x 7, $C_2^3.C_2^3$ x 7, $D_4:C_2^3$ x 7, $C_8.C_2^3$ x 7, $C_2^3.D_4$ x 6, $\OD_{16}:C_4$ x 6, $C_2^3.D_4$ x 6, $C_2^2\times \OD_{16}$ x 6, $D_4.D_4$ x 6, $\SD_{16}:C_4$ x 6, $C_4\times \SD_{16}$ x 6, $\OD_{16}:C_4$ x 6, $C_4^2:C_2^2$ x 6, $C_2^3.C_2^3$ x 5, $C_2^3:Q_8$ x 5, $C_4^2:C_4$ x 5, $C_4^2:C_4$ x 5, $C_4^2.C_4$ x 5, $C_4^2:C_2^2$ x 5, $C_4^2.C_2^2$ x 5, $C_2^3.D_4$ x 4, $\OD_{16}.C_2^2$ x 4, $\OD_{16}:C_2^2$ x 4, $C_2^3.C_2^3$ x 4, $C_4^2:C_4$ x 4, $C_8:C_2^3$ x 4, $\OD_{16}:C_4$ x 4, $D_8:C_4$ x 4, $C_4\times D_8$ x 4, $C_4^2.C_4$ x 4, $C_4^2.C_4$ x 4, $C_4:\OD_{16}$ x 4, $C_2^2:\OD_{16}$ x 3, $C_2^3.D_4$ x 3, $C_4.C_2^4$ x 3, $D_4:C_2^3$ x 3, $C_8.C_2^3$ x 3, $C_4^2.C_2^2$ x 3, $D_4:D_4$ x 3, $C_2^3.D_4$ x 2, $\OD_{16}.C_2^2$ x 2, $C_2^3.C_2^3$ x 2, $C_2^2.\OD_{16}$ x 2, $C_2^3:C_8$ x 2, $C_4.C_2^4$ x 2, $D_8:C_2^2$ x 2, $Q_{16}:C_2^2$ x 2, $\OD_{16}:C_4$ x 2, $C_4^2.C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_2^3.D_4$ x 2, $C_2^2.D_8$ x 2, $Q_8:D_4$ x 2, $C_4:D_8$ x 2, $D_4.D_4$ x 2, $D_4:D_4$ x 2, $Q_8:D_4$ x 2, $D_4:D_4$ x 2, $Q_{16}:C_4$ x 2, $C_4\times Q_{16}$ x 2, $C_2^3.Q_8$ x 2, $C_2^3.D_4$ x 2, $C_4^2.C_2^2$ x 2, $C_2^4:C_4$, $C_4^2.C_4$, $D_8:C_2^2$, $C_2^2.C_4^2$, $C_4.C_4^2$, $C_2^2.C_4^2$, $Q_8:Q_8$, $D_4:Q_8$, $Q_8:Q_8$, $D_4:Q_8$, $C_4^2:C_2^2$, $D_8:C_4$, $D_8:C_4$, $C_2^3.Q_8$, $C_2^3.D_4$, $C_4^2.C_4$

Order 48: $C_4\times A_4$ x 3, $C_2\times S_4$ x 2, $C_4\times D_6$ x 2, $A_4:C_4$ x 2, $C_4^2:C_3$ x 2, $C_{12}:C_4$ x 2, $C_2^2\times A_4$, $\SL(2,3):C_2$, $C_4\times C_{12}$

Order 32: $C_4\times D_4$ x 75, $C_2\times C_4^2$ x 74, $C_4^2:C_2$ x 37, $D_4:C_2^2$ x 35, $C_2^3:C_4$ x 31, $C_4\times Q_8$ x 30, $C_2\times \OD_{16}$ x 29, $C_2^2.D_4$ x 28, $C_2^2\times D_4$ x 17, $C_2^3\times C_4$ x 17, $C_4\wr C_2$ x 17, $C_4^2:C_2$ x 16, $C_2.C_4^2$ x 16, $C_4:D_4$ x 13, $C_8:C_4$ x 12, $C_4\times C_8$ x 12, $D_4:C_4$ x 10, $D_4:C_2^2$ x 8, $C_4:D_4$ x 8, $C_4.D_4$ x 7, $C_2^2\times Q_8$ x 7, $D_8:C_2$ x 7, $C_4.C_2^3$ x 6, $C_2^2:C_8$ x 6, $\OD_{16}:C_2$ x 6, $C_2^2\times C_8$ x 6, $C_4:Q_8$ x 6, $Q_8:C_4$ x 6, $C_4:C_8$ x 5, $\OD_{16}:C_2$ x 4, $C_2\times \SD_{16}$ x 4, $C_8:C_4$ x 4, $C_8:C_4$ x 4, $Q_{16}:C_2$ x 3, $C_2\times D_8$ x 3, $C_2^3:C_4$, $D_8:C_2$, $C_2\times Q_{16}$, $C_8.C_4$

Order 24: $C_4\times S_3$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $C_2\times A_4$ x 2, $\SL(2,3)$, $C_2\times D_6$, $S_4$

Order 16: $C_2^2\times C_4$ x 77, $C_4^2$ x 65, $C_2\times D_4$ x 51, $D_4:C_2$ x 37, $C_2^2:C_4$ x 36, $C_4:C_4$ x 30, $\OD_{16}$ x 21, $C_2\times Q_8$ x 20, $C_2\times C_8$ x 15, $C_2^4$ x 5, $\SD_{16}$ x 4, $D_8$ x 3, $Q_{16}$

Order 12: $C_{12}$ x 3, $D_6$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$, $A_4$

Order 8: $C_2\times C_4$ x 84, $D_4$ x 23, $C_2^3$ x 21, $Q_8$ x 11, $C_8$ x 7

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

Order 4: $C_4$ x 22, $C_2^2$ x 21

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6144: $C_4^4.S_4$ ($C_1$)

Order 3072: $(C_2\times C_4^3).S_4$ ($C_2$) x 2, $(C_2\times C_4^3).S_4$ ($C_2$) x 2, $(C_2\times C_4^3).S_4$ ($C_2$), $C_4^4.A_4$ ($C_2$), $(C_2\times C_4^3).S_4$ ($C_2$)

Order 1536: $C_4^3.S_4$ ($C_4$) x 2, $C_4^3.S_4$ ($C_4$) x 2, $(C_2^2\times C_4^2).S_4$ ($C_2^2$) x 2, $(C_2\times C_4^3).A_4$ ($C_2^2$) x 2, $(C_2^2\times C_4^2).S_4$ ($C_2^2$), $(C_2^2\times C_4^2).S_4$ ($C_2^2$), $(C_2\times C_4^3).A_4$ ($C_2^2$)

Order 1024: $C_4^4.C_2^2$ ($S_3$)

Order 768: $(C_2\times C_4^2).S_4$ ($C_2\times C_4$) x 2, $(C_2\times C_4^2).S_4$ ($C_2\times C_4$) x 2, $(C_2^2\times C_4^2).A_4$ ($C_2^3$), $C_4^3.A_4$ ($C_2\times C_4$), $C_4^3.A_4$ ($C_2\times C_4$)

Order 512: $C_2^3.C_2^5.C_2$ ($D_6$) x 2, $C_4^3.C_2^3$ ($D_6$)

Order 384: $(C_2\times C_4^2).A_4$ ($C_2^2\times C_4$)

Order 256: $C_4^4$ ($S_4$), $C_2^3.C_2^5$ ($C_2\times D_6$), $C_4.Q_8^2$ ($C_4\times S_3$), $C_4^3.C_2^2$ ($C_4\times S_3$)

Order 128: $C_2\times C_4^3$ ($C_2\times S_4$) x 3, $C_2.Q_8^2$ ($C_4\times D_6$)

Order 64: $C_4^3$ ($C_4\times S_4$) x 2, $C_2^2\times C_4^2$ ($C_2^2\times S_4$)

Order 32: $C_2^3\times C_4$ ($C_2^3:S_4$), $C_2\times C_4^2$ ($C_2^4.D_6$)

Order 16: $C_2^2\times C_4$ ($C_2^5.D_6$) x 2, $C_2^4$ ($C_2^4:S_4$)

Order 8: $C_2^3$ ($(C_2^3\times C_4):S_4$) x 2, $C_2^3$ ($C_2^6.D_6$)

Order 4: $C_2^2$ ($C_2^4.C_2^4.S_3$), $C_4$ ($C_4^3.S_4$)

Order 2: $C_2$ ($C_2\times C_4^3.S_4$)

Order 1: $C_1$ ($C_4^4.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6144: $C_4^4.S_4$ ($C_1$)

Order 3072: $(C_2\times C_4^3).S_4$ ($C_2$), $(C_2\times C_4^3).S_4$ ($C_2$), $(C_2\times C_4^3).S_4$ ($C_2$), $C_4^4.A_4$ ($C_2$), $(C_2\times C_4^3).S_4$ ($C_2$)

Order 1536: $(C_2^2\times C_4^2).S_4$ ($C_2^2$), $(C_2^2\times C_4^2).S_4$ ($C_2^2$), $C_4^3.S_4$ ($C_4$), $C_4^3.S_4$ ($C_4$), $(C_2^2\times C_4^2).S_4$ ($C_2^2$), $(C_2\times C_4^3).A_4$ ($C_2^2$), $(C_2\times C_4^3).A_4$ ($C_2^2$)

Order 1024: $C_4^4.C_2^2$ ($S_3$)

Order 768: $(C_2\times C_4^2).S_4$ ($C_2\times C_4$), $(C_2\times C_4^2).S_4$ ($C_2\times C_4$), $(C_2^2\times C_4^2).A_4$ ($C_2^3$), $C_4^3.A_4$ ($C_2\times C_4$), $C_4^3.A_4$ ($C_2\times C_4$)

Order 512: $C_4^3.C_2^3$ ($D_6$), $C_2^3.C_2^5.C_2$ ($D_6$)

Order 384: $(C_2\times C_4^2).A_4$ ($C_2^2\times C_4$)

Order 256: $C_4^4$ ($S_4$), $C_2^3.C_2^5$ ($C_2\times D_6$), $C_4.Q_8^2$ ($C_4\times S_3$), $C_4^3.C_2^2$ ($C_4\times S_3$)

Order 128: $C_2\times C_4^3$ ($C_2\times S_4$) x 2, $C_2.Q_8^2$ ($C_4\times D_6$)

Order 64: $C_4^3$ ($C_4\times S_4$) x 2, $C_2^2\times C_4^2$ ($C_2^2\times S_4$)

Order 32: $C_2^3\times C_4$ ($C_2^3:S_4$), $C_2\times C_4^2$ ($C_2^4.D_6$)

Order 16: $C_2^2\times C_4$ ($C_2^5.D_6$) x 2, $C_2^4$ ($C_2^4:S_4$)

Order 8: $C_2^3$ ($(C_2^3\times C_4):S_4$), $C_2^3$ ($C_2^6.D_6$)

Order 4: $C_2^2$ ($C_2^4.C_2^4.S_3$), $C_4$ ($C_4^3.S_4$)

Order 2: $C_2$ ($C_2\times C_4^3.S_4$)

Order 1: $C_1$ ($C_4^4.S_4$)

Series

Derived series

$C_4^4.S_4$

$\rhd$

$(C_2\times C_4^2).A_4$

$\rhd$

$C_2.Q_8^2$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_4^4.S_4$

$\rhd$

$C_4^4.A_4$

$\rhd$

$(C_2\times C_4^3).A_4$

$\rhd$

$(C_2^2\times C_4^2).A_4$

$\rhd$

$(C_2\times C_4^2).A_4$

$\rhd$

$C_2.Q_8^2$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4^4.S_4$

$\rhd$

$(C_2\times C_4^2).A_4$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

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

Character theory

Complex character table

See the $130 \times 130$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $85 \times 85$ rational character table.