Abstract group 24576.bds: $(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$

• Label: 24576.bds
• Order: \( 2^{13} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $66$
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$
Order:	\(24576\)\(\medspace = 2^{13} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_{34}:F_{17}$, of order \(98304\)\(\medspace = 2^{15} \cdot 3 \)
Composition factors:	$C_2$ x 13, $C_3$
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
Elements	1	1119	128	9120	4992	6144	3072	24576
Conjugacy classes	1	37	1	68	21	22	6	156
Divisions	1	37	1	68	15	22	4	148
Autjugacy classes	1	30	1	45	16	11	3	107

Minimal presentations

Permutation degree:	$66$
Transitive degree:	not computed
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	48	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid b^{2}=c^{2}=d^{4}=e^{6}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $66$ $\langle(6,19)(8,17)(13,35)(14,33)(16,28)(20,29)(22,26)(24,47)(25,48)(27,51)(30,52) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $((C_2\times C_4^3).\GL(2,\mathbb{Z}/4))$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^7$ . $(C_2^3:S_4)$	$C_2^6$ . $(C_2^4:S_4)$ (2)	$C_2^5$ . $(C_2^5:S_4)$	$C_2^5$ . $(C_2\wr D_6)$ (4)	all 90
Aut. group:	$\Aut(C_4^2.(C_2\times D_6))$	$\Aut(C_2^6:A_4)$	$\Aut(C_4^3:A_4)$	$\Aut(C_4^3:A_4)$	all 9

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 214 normal subgroups (112 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $(C_2\times C_4^3).\GL(2,\mathbb{Z}/4)$
Commutator:	$G' \simeq$ $C_4^2:A_4.C_2^3$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_4^2:C_2^3$	$G/\Phi \simeq$ $C_2^3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_4^2.C_2^6.C_2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^5:\GL(2,\mathbb{Z}/4)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_2^2.C_2^6.C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 16 (order at least 1536) are shown, as well as all normal subgroups of any index.

Order 24576: $(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$

Order 12288: $(C_2\times C_4^3).\GL(2,\mathbb{Z}/4)$ x 8, $C_2\times C_4^2:A_4.C_2^2.D_4$ x 2, $C_2\times C_4^2:A_4.C_2^2\wr C_2$, $C_2\times C_4^2:A_4.C_2.C_2^4$, $C_2\times C_4^2:A_4.C_2^3.C_2^2$, $C_2\times C_4^2.C_2^4.C_6.C_2^2$, $C_2^7.\GL(2,\mathbb{Z}/4)$

Order 8192: $C_2\times C_2^2.C_2^6.C_2^4$

Order 6144: $C_4^2:A_4.C_2^2.D_4$ x 8, $C_4^2:A_4.C_2.C_2^4$ x 8, $C_4^2.C_2^4.C_6.C_2^2$ x 6, $C_2^6.\GL(2,\mathbb{Z}/4)$ x 6, $C_4^2:A_4.C_2^2\wr C_2$ x 4, $C_4^2:A_4.C_2^3.C_2^2$ x 4, $C_2\times C_4^2:A_4.D_4.C_2$ x 2, $C_4^2:A_4.C_2.D_4.C_2$ x 2, $C_4^2.(D_4\times S_4).C_2$ x 2, $C_2\times C_4^2:A_4.C_2.D_4$, $C_2\times C_4^2.(C_2^5.S_3)$, $C_2\times C_4^2.(C_2\times D_4\times A_4)$, $C_2\times C_4^2.(C_2^5.S_3)$, $C_2\times C_4^2:A_4.D_4.C_2$, $C_2\times C_4^2:A_4.D_4:C_2$, $C_2\times C_4^2:A_4.C_2^3.C_2$, $C_2\times C_4^2.C_2^4.C_6.C_2$, $C_2\times C_4^2:A_4.D_4.C_2$, $C_2\times C_4^2:C_3.C_2\wr C_2^2$, $C_2\times C_4^2.C_2^4.C_6.C_2$, $C_2\times C_4^2.(D_4\times A_4).C_2$, $C_2\times C_4^2:A_4.C_2.D_4$, $C_2\times C_4^2:A_4.D_4.C_2$, $C_2\times C_4^2:A_4.D_4.C_2$, $C_2\times C_4^2:A_4.C_2^4$, $C_4^3:(C_2^2\times S_4)$, $C_4^3.(C_2^2\times S_4)$

Order 4096: $C_2^2.C_2^6.C_2^4$ x 16, $C_2\times C_2^2.C_2^6.C_2^3$ x 4, $C_2\times C_2^2.C_2^6.C_2^3$ x 2, $C_2\times C_2^2.C_2^6.C_2^3$ x 2, $C_2\times C_2^2.C_2^6.C_2^3$, $C_2\times C_2^3.C_2^5.C_2^3$, $C_2\times C_2^2.C_2^6.C_2^3$, $C_2\times C_4^2.C_2^6.C_2$, $C_2\times C_2^2.C_2^6.C_2^3$, $C_2\times C_4^2.C_2^5.C_2^2$, $C_2\times C_2^2.C_2^6.C_2^3$

Order 3072: $C_4^2:A_4.D_4.C_2$ x 9, $C_4^2:C_3.C_2\wr C_2^2$ x 8, $C_4^2:A_4.D_4.C_2$ x 8, $C_4^2:A_4.C_2^3.C_2$ x 8, $C_4^3:(C_2\times S_4)$ x 8, $C_4^2.(D_4\times S_4)$ x 8, $C_4^2:A_4.C_2^4$ x 7, $C_4^2:A_4.D_4.C_2$ x 6, $C_4^2:A_4.D_4:C_2$ x 4, $C_4^2:A_4.D_4.C_2$ x 4, $C_4^2:A_4.D_4:C_2$ x 4, $C_4^2:A_4.D_4.C_2$ x 4, $C_4^2.(C_2^5.S_3)$ x 4, $C_4^2.(C_2\times D_4\times A_4)$ x 4, $C_4^2.(D_4\times A_4).C_2$ x 4, $C_4^2:A_4.D_4.C_2$ x 4, $C_4^2.(C_2^5.S_3)$ x 4, $C_4^2.C_2^4.C_6.C_2$ x 4, $C_4^2:A_4.C_2^3.C_2$ x 3, $C_4^2:A_4.C_2.D_4$ x 3, $C_4^2.C_2^4.C_6.C_2$ x 3, $C_4^2:A_4.C_2.D_4$ x 3, $C_2\times C_4^2.C_2^4:S_3$ x 3, $C_2\times C_4^2:A_4.D_4$ x 2, $C_2\times C_4^2:A_4.C_2^3$ x 2, $C_2\times C_4^2:A_4.C_2^3$ x 2, $C_2\times C_4^2:C_3.C_2^3.C_2^2$ x 2, $C_2\times C_4^2:A_4.C_4.C_2$ x 2, $C_4^2.(D_4\times A_4).C_2$ x 2, $C_4^2.(C_2\times D_4\times A_4)$ x 2, $C_4^2:A_4.C_2^3.C_2$ x 2, $C_2\times C_4^2:A_4.C_2^3$ x 2, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times C_4^2:A_4.C_2^3$, $C_2\times C_4^2.C_2^3.C_6.C_2$, $C_2\times C_4^2.D_{12}:C_2^2$, $C_2\times C_4^2:A_4.D_4$, $C_2\times C_4^2:A_4.C_2^2.C_2$, $C_2\times C_4^2:A_4.C_4.C_2$, $C_2\times C_4^2.C_2^3.C_6.C_2$, $C_2\times C_4^2:A_4.Q_8$, $C_2\times C_2^4.C_2^4:S_3$, $C_2\times C_4^2.(C_6\times D_4).C_2$, $C_2\times C_4^2:A_4.C_4.C_2$, $C_2\times C_4^2.(D_4\times A_4)$, $C_2\times C_4^2.(D_4\times A_4)$, $C_2\times C_4^2:A_4.D_4$, $C_2\times C_4^2.(C_6\times D_4).C_2$, $C_2\times C_4^3:S_4$, $C_2\times C_4^3.S_4$, $C_2\times C_4^3:S_4$, $C_4^2.(C_2^3\times S_4)$, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$, $C_4^2.(C_2^3\times S_4)$, $C_4^3.(C_2^2\times A_4)$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $C_4^3:(C_2^2\times A_4)$, $C_2^6:(C_2\times S_4)$, $C_2^5:\GL(2,\mathbb{Z}/4)$

Order 2048: $C_2^2.C_2^6.C_2^3$ x 32, $C_2^2.C_2^6.C_2^3$ x 20, $C_2^2.C_2^6.C_2^3$ x 20, $C_2^2.C_2^6.C_2^3$ x 16, $C_2^2.C_2^6.C_2^3$ x 16, $C_4^2.C_2^4.C_2^3$ x 16, $C_4^2.C_2^6.C_2$ x 12, $C_2^2.C_2^5.C_2^4$ x 12, $C_2^3.C_2^4.C_2^4$ x 12, $C_2^2.C_2^6.C_2^3$ x 8, $C_2^2.C_2^6.C_2^3$ x 8, $C_2^3.C_2^5.C_2^3$ x 8, $C_2\times C_2^2.C_2^6.C_2^2$ x 4, $C_2\times C_2^2.C_2^6.C_2^2$ x 4, $C_4^2.C_2^5.C_2^2$ x 4, $C_2\times C_2^2.C_2^6.C_2^2$ x 4, $C_2\times C_2^2.C_2^6.C_2^2$ x 3, $C_2\times C_2^2.C_2^6.C_2^2$ x 3, $C_2\times C_2^2.C_2^6.C_2^2$ x 3, $C_2\times C_2^2.C_2^6.C_2^2$ x 3, $C_2\times C_2^3.C_2^4.C_2^3$ x 2, $C_2\times C_2^3.C_2^5.C_2^2$ x 2, $C_2\times C_4.D_4.Q_8:C_2^2$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_4^2.C_2^5.C_2$ x 2, $C_2\times C_4^2.C_2^5.C_2$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_4^2.C_2^5.C_2$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_2^3.C_2^4.C_2^3$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_2^3.C_2^4.C_2^3$ x 2, $C_2\times C_2^3.C_2^4.C_2^3$ x 2, $C_2\times C_4.D_4.Q_8:C_2^2$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_4.D_4.Q_8:C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4^2.C_2^5.C_2$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_4.D_4.Q_8:C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4^2.D_4^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_2^3.C_2^4.C_2^3$, $C_2\times C_2^3.C_2^4.C_2^3$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times C_2^3.C_2^4.C_2^3$, $C_2\times C_4^2.C_2^4.C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_2^3.C_2^4.C_2^3$, $C_2\times C_4.D_4.Q_8:C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4.D_4.Q_8:C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times C_4^2.C_2^5.C_2$, $C_2\times C_4^2.C_2^4.C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4^2.C_2^4.C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4^2.D_4^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times C_4^2.C_2^6$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_4^2.D_4^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_4^2.C_2^4.C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4^2.D_4^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times C_2^2.C_2^6.C_2^2$

Order 1536: $C_2^4.\GL(2,\mathbb{Z}/4)$ x 20, $C_4^2:A_4.C_2^3$ x 14, $C_4^2:A_4.C_2^3$ x 14, $C_4^2:A_4.C_2^3$ x 11, $(C_2\times C_4^3):D_6$ x 10, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 8, $C_4^2:A_4.D_4$ x 8, $C_2^4.C_2^4:S_3$ x 8, $C_2^5.C_2^3.S_3$ x 8, $C_2^3.C_2^5.S_3$ x 6, $C_2^4.(C_2^3\times A_4)$ x 6, $C_4^2.(C_2^2\times S_4)$ x 5, $C_4^2.(C_2^2\times S_4)$ x 4, $C_2^5:(C_2\times S_4)$ x 4, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_4^2:A_4.D_4$ x 4, $C_4^2:A_4.D_4$ x 4, $C_4^2:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^2\times C_4^2:C_3.D_4$ x 4, $C_2\times C_4^2:C_3.C_2^3.C_2$ x 4, $C_2\times C_4^2.(C_6.D_4)$ x 4, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 4, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_4^2.C_2^4.S_3$ x 4, $C_4^3:S_4$ x 4, $C_4^3.S_4$ x 4, $C_4^3:S_4$ x 4, $C_4^2.A_4:Q_8$ x 4, $C_4^2:A_4.C_2^3$ x 4, $C_4^2.(D_4\times A_4)$ x 4, $C_4^2.(D_4\times A_4)$ x 4, $C_4^2.(D_4\times A_4)$ x 4, $C_4^3:(C_2\times A_4)$ x 4, $C_2^3.C_2^5.S_3$ x 3, $C_2^3.C_2^5.S_3$ x 3, $C_2\times C_4^2.(C_6.D_4)$ x 3, $C_4^2:A_4.C_2^3$ x 3, $C_4^2.C_2^4.C_6$ x 3, $C_2^3.C_2^5.C_6$ x 3, $C_4^2.(C_2^2\times S_4)$ x 2, $(C_2^2\times C_4^2).S_4$ x 2, $C_2^6:S_4$ x 2, $C_2\times C_4^2.D_4:S_3$ x 2, $C_2\times C_4^3:D_6$ x 2, $C_2\times C_2^4.(C_6.D_4)$ x 2, $C_4^2:A_4.C_2^3$ x 2, $C_4^2:A_4.C_2^3$ x 2, $C_2^6:(C_2\times A_4)$ x 2, $C_4^2:A_4.C_2^3$ x 2, $C_4^2:C_3.C_2^5$ x 2, $(C_2^2\times C_4^2):S_4$, $C_2^6:S_4$, $(C_2^2\times C_4^2).S_4$, $C_2^6.S_4$, $C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^4:\GL(2,\mathbb{Z}/4)$, $(C_2\times C_4^3):D_6$, $C_2\times C_2^4.(C_6.D_4)$, $C_4^2.(C_2^2\times S_4)$, $C_2^4.C_2^5.C_3$, $C_4^2.(C_2^3\times A_4)$, $C_2^7:A_4$, $C_2^7.A_4$, $C_2^2\times C_4^2.(C_3\times D_4)$, $C_2^5.C_2^4.C_3$, $C_4^2.C_2^5.C_3$, $C_4^2.(C_2^3\times A_4)$, $C_2\times (C_2\times C_4^2).C_{12}.C_2$, $C_2^5.C_2^4.C_3$, $C_2\times C_4^3:A_4$, $C_4^2.C_2^4.C_6$

Order 1024: $C_4^2.C_2^5.C_2$ x 4, $C_4^2.C_2^6$ x 2, $C_4^2.C_2^5.C_2$ x 2, $C_2\times C_4^2.C_2^5$, $C_2\times C_4^2.C_2^4.C_2$, $C_2\times C_4^2.C_2^5$, $C_2\times C_2^4.C_2^5$

Order 768: $C_4^2.(C_2^2\times A_4)$ x 2, $C_2^5:(C_2\times A_4)$ x 2, $C_4^2:A_4.C_2^2$ x 2, $(C_2^2\times C_4^2):A_4$

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

Order 384: $(C_2\times C_4^2):A_4$ x 2, $C_2^5:A_4$

Order 256: $C_4^3:C_2^2$ x 4, $C_2^4.C_2^4$ x 2, $D_4^2:C_2^2$ x 2, $D_4^2:C_2^2$ x 2, $D_4^2:C_2^2$ x 2, $C_4^2:C_2^4$ x 2, $C_4^2:C_2^4$ x 2, $D_4^2:C_2^2$, $C_2^2\times C_4^3$

Order 192: $C_4^2:A_4$

Order 128: $C_4^2:C_2^3$ x 8, $C_2\times C_4^3$ x 2, $D_4^2:C_2$ x 2, $C_2^7$, $C_4^2:C_2^3$, $C_2^3\times C_4^2$

Order 64: $C_4^2:C_2^2$ x 4, $C_2^6$ x 3, $C_2^2\times C_4^2$ x 3, $C_4^2:C_2^2$

Order 32: $C_2^5$ x 7, $C_2\times C_4^2$ x 3

Order 16: $C_2^4$ x 7, $C_4^2$

Order 8: $C_2^3$ x 3

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 16 (order at least 1536) are shown, as well as all normal subgroups of any index.

Order 24576: $(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$

Order 12288: $C_2\times C_4^2:A_4.C_2^2.D_4$ x 2, $C_2\times C_4^2:A_4.C_2^2\wr C_2$, $C_2\times C_4^2:A_4.C_2.C_2^4$, $C_2\times C_4^2:A_4.C_2^3.C_2^2$, $C_2\times C_4^2.C_2^4.C_6.C_2^2$, $C_2^7.\GL(2,\mathbb{Z}/4)$, $(C_2\times C_4^3).\GL(2,\mathbb{Z}/4)$

Order 8192: $C_2\times C_2^2.C_2^6.C_2^4$

Order 6144: $C_4^2:A_4.C_2.C_2^4$ x 3, $C_2\times C_4^2:A_4.D_4.C_2$ x 2, $C_4^2.C_2^4.C_6.C_2^2$ x 2, $C_4^2:A_4.C_2^2.D_4$ x 2, $C_2^6.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times C_4^2:A_4.C_2.D_4$, $C_2\times C_4^2.(C_2^5.S_3)$, $C_2\times C_4^2.(C_2\times D_4\times A_4)$, $C_2\times C_4^2.(C_2^5.S_3)$, $C_2\times C_4^2:A_4.D_4.C_2$, $C_2\times C_4^2:A_4.D_4:C_2$, $C_2\times C_4^2:A_4.C_2^3.C_2$, $C_2\times C_4^2.C_2^4.C_6.C_2$, $C_2\times C_4^2:A_4.D_4.C_2$, $C_4^2:A_4.C_2^2\wr C_2$, $C_4^2:A_4.C_2.D_4.C_2$, $C_2\times C_4^2:C_3.C_2\wr C_2^2$, $C_2\times C_4^2.C_2^4.C_6.C_2$, $C_4^2.(D_4\times S_4).C_2$, $C_2\times C_4^2.(D_4\times A_4).C_2$, $C_2\times C_4^2:A_4.C_2.D_4$, $C_2\times C_4^2:A_4.D_4.C_2$, $C_2\times C_4^2:A_4.D_4.C_2$, $C_2\times C_4^2:A_4.C_2^4$, $C_4^2:A_4.C_2^3.C_2^2$, $C_4^3:(C_2^2\times S_4)$, $C_4^3.(C_2^2\times S_4)$

Order 4096: $C_2\times C_2^2.C_2^6.C_2^3$ x 4, $C_2\times C_2^2.C_2^6.C_2^3$ x 2, $C_2\times C_2^2.C_2^6.C_2^3$ x 2, $C_2^2.C_2^6.C_2^4$ x 2, $C_2\times C_2^2.C_2^6.C_2^3$, $C_2\times C_2^3.C_2^5.C_2^3$, $C_2\times C_2^2.C_2^6.C_2^3$, $C_2\times C_4^2.C_2^6.C_2$, $C_2\times C_2^2.C_2^6.C_2^3$, $C_2\times C_4^2.C_2^5.C_2^2$, $C_2\times C_2^2.C_2^6.C_2^3$

Order 3072: $C_4^2:A_4.C_2^4$ x 5, $C_4^2:A_4.D_4.C_2$ x 4, $C_4^2:A_4.D_4.C_2$ x 4, $C_4^2.(C_2\times D_4\times A_4)$ x 3, $C_2\times C_4^2.C_2^4:S_3$ x 3, $C_2\times C_4^2:A_4.D_4$ x 2, $C_4^2:A_4.C_2^3.C_2$ x 2, $C_2\times C_4^2:A_4.C_2^3$ x 2, $C_4^2:A_4.C_2.D_4$ x 2, $C_2\times C_4^2:A_4.C_2^3$ x 2, $C_2\times C_4^2:C_3.C_2^3.C_2^2$ x 2, $C_4^2.C_2^4.C_6.C_2$ x 2, $C_4^2:A_4.C_2.D_4$ x 2, $C_2\times C_4^2:A_4.C_4.C_2$ x 2, $C_4^2:A_4.D_4.C_2$ x 2, $C_4^2:A_4.C_2^3.C_2$ x 2, $C_2\times C_4^2:A_4.C_2^3$ x 2, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_4^3:(C_2\times S_4)$ x 2, $C_4^2.(D_4\times S_4)$ x 2, $C_2\times C_4^2:A_4.C_2^3$, $C_4^2:A_4.D_4:C_2$, $C_4^2:A_4.D_4.C_2$, $C_2\times C_4^2.C_2^3.C_6.C_2$, $C_4^2:A_4.D_4:C_2$, $C_4^2:A_4.D_4.C_2$, $C_2\times C_4^2.D_{12}:C_2^2$, $C_2\times C_4^2:A_4.D_4$, $C_2\times C_4^2:A_4.C_2^2.C_2$, $C_2\times C_4^2:A_4.C_4.C_2$, $C_2\times C_4^2.C_2^3.C_6.C_2$, $C_4^2:C_3.C_2\wr C_2^2$, $C_2\times C_4^2:A_4.Q_8$, $C_2\times C_2^4.C_2^4:S_3$, $C_2\times C_4^2.(C_6\times D_4).C_2$, $C_4^2.(C_2^5.S_3)$, $C_2\times C_4^2:A_4.C_4.C_2$, $C_4^2.(D_4\times A_4).C_2$, $C_2\times C_4^2.(D_4\times A_4)$, $C_2\times C_4^2.(D_4\times A_4)$, $C_4^2.(D_4\times A_4).C_2$, $C_4^2.(C_2\times D_4\times A_4)$, $C_4^2:A_4.D_4.C_2$, $C_4^2:A_4.C_2^3.C_2$, $C_2\times C_4^2:A_4.D_4$, $C_2\times C_4^2.(C_6\times D_4).C_2$, $C_4^2.(C_2^5.S_3)$, $C_4^2.C_2^4.C_6.C_2$, $C_2\times C_4^3:S_4$, $C_2\times C_4^3.S_4$, $C_2\times C_4^3:S_4$, $C_4^2.(C_2^3\times S_4)$, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$, $C_4^2.(C_2^3\times S_4)$, $C_4^3.(C_2^2\times A_4)$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $C_4^3:(C_2^2\times A_4)$, $C_2^6:(C_2\times S_4)$, $C_2^5:\GL(2,\mathbb{Z}/4)$

Order 2048: $C_4^2.C_2^4.C_2^3$ x 7, $C_2^2.C_2^6.C_2^3$ x 6, $C_2^2.C_2^6.C_2^3$ x 6, $C_2^2.C_2^6.C_2^3$ x 6, $C_2\times C_2^2.C_2^6.C_2^2$ x 4, $C_2\times C_2^2.C_2^6.C_2^2$ x 4, $C_4^2.C_2^6.C_2$ x 4, $C_2\times C_2^2.C_2^6.C_2^2$ x 4, $C_2^2.C_2^6.C_2^3$ x 3, $C_2\times C_2^2.C_2^6.C_2^2$ x 3, $C_2\times C_2^2.C_2^6.C_2^2$ x 3, $C_2\times C_2^2.C_2^6.C_2^2$ x 3, $C_2\times C_2^2.C_2^6.C_2^2$ x 3, $C_2^2.C_2^5.C_2^4$ x 3, $C_2^3.C_2^4.C_2^4$ x 3, $C_2\times C_2^3.C_2^4.C_2^3$ x 2, $C_2\times C_2^3.C_2^5.C_2^2$ x 2, $C_2\times C_4.D_4.Q_8:C_2^2$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_4^2.C_2^5.C_2$ x 2, $C_2\times C_4^2.C_2^5.C_2$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_4^2.C_2^5.C_2$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_2^3.C_2^4.C_2^3$ x 2, $C_2\times C_2^2.C_2^6.C_2^2$ x 2, $C_2\times C_2^3.C_2^4.C_2^3$ x 2, $C_2\times C_2^3.C_2^4.C_2^3$ x 2, $C_2\times C_4.D_4.Q_8:C_2^2$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_4.D_4.Q_8:C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4^2.C_2^5.C_2$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_4.D_4.Q_8:C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4^2.D_4^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_2^3.C_2^4.C_2^3$, $C_2\times C_2^3.C_2^4.C_2^3$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times C_2^3.C_2^4.C_2^3$, $C_2\times C_4^2.C_2^4.C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_2^3.C_2^4.C_2^3$, $C_2\times C_4.D_4.Q_8:C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4.D_4.Q_8:C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times C_4^2.C_2^5.C_2$, $C_2\times C_4^2.C_2^4.C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4^2.C_2^4.C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2\times C_4^2.D_4^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times C_4^2.C_2^6$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_4^2.D_4^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_4^2.C_2^5.C_2^2$, $C_2^2.C_2^6.C_2^3$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_4^2.C_2^4.C_2^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2\times C_4^2.D_4^2$, $C_2\times C_2^2.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2\times C_2^2.C_2^6.C_2^2$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times C_2^2.C_2^6.C_2^2$

Order 1536: $C_4^2:A_4.C_2^3$ x 9, $C_4^2:A_4.C_2^3$ x 8, $C_4^2:A_4.C_2^3$ x 8, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^3.C_2^5.S_3$ x 4, $C_2^2\times C_4^2:C_3.D_4$ x 4, $C_2\times C_4^2:C_3.C_2^3.C_2$ x 4, $(C_2\times C_4^3):D_6$ x 4, $C_2\times C_4^2.(C_6.D_4)$ x 4, $C_4^2.(C_2^2\times S_4)$ x 3, $C_2^5:(C_2\times S_4)$ x 3, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 3, $C_4^2:A_4.D_4$ x 3, $C_2\times C_4^2.(C_6.D_4)$ x 3, $C_4^2:A_4.C_2^3$ x 3, $C_4^2:A_4.C_2^3$ x 3, $C_4^2.(C_2^2\times S_4)$ x 2, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.C_2^5.S_3$ x 2, $C_2^3.C_2^5.S_3$ x 2, $C_4^2:A_4.D_4$ x 2, $(C_2^2\times C_4^2).S_4$ x 2, $C_2^6:S_4$ x 2, $C_2\times C_4^2.D_4:S_3$ x 2, $C_2\times C_4^3:D_6$ x 2, $C_2^4.C_2^4:S_3$ x 2, $C_2\times C_2^4.(C_6.D_4)$ x 2, $C_4^2.(C_2^2\times S_4)$ x 2, $C_2^5.C_2^3.S_3$ x 2, $C_4^2:A_4.C_2^3$ x 2, $C_4^2:A_4.C_2^3$ x 2, $C_2^6:(C_2\times A_4)$ x 2, $C_4^2:A_4.C_2^3$ x 2, $C_4^2:C_3.C_2^5$ x 2, $C_2^4.(C_2^3\times A_4)$ x 2, $C_4^2.(D_4\times A_4)$ x 2, $C_4^3:(C_2\times A_4)$ x 2, $C_4^2.C_2^4.C_6$ x 2, $C_2^3.C_2^5.C_6$ x 2, $(C_2^2\times C_4^2):S_4$, $C_2^6:S_4$, $C_4^2:A_4.D_4$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times C_4^2).S_4$, $C_2^6.S_4$, $C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^4:\GL(2,\mathbb{Z}/4)$, $(C_2\times C_4^3):D_6$, $C_2\times C_2^4.(C_6.D_4)$, $C_4^2.(C_2^2\times S_4)$, $C_2^4.\GL(2,\mathbb{Z}/4)$, $C_4^2.\GL(2,\mathbb{Z}/4)$, $C_4^2.\GL(2,\mathbb{Z}/4)$, $C_4^2.C_2^4.S_3$, $C_4^3:S_4$, $C_4^3.S_4$, $C_4^3:S_4$, $C_4^2.A_4:Q_8$, $C_2^4.C_2^5.C_3$, $C_4^2.(C_2^3\times A_4)$, $C_2^7:A_4$, $C_2^7.A_4$, $C_2^2\times C_4^2.(C_3\times D_4)$, $C_2^5.C_2^4.C_3$, $C_4^2.C_2^5.C_3$, $C_4^2.(C_2^3\times A_4)$, $C_2\times (C_2\times C_4^2).C_{12}.C_2$, $C_2^5.C_2^4.C_3$, $C_4^2.(D_4\times A_4)$, $C_4^2.(D_4\times A_4)$, $C_2\times C_4^3:A_4$, $C_4^2.C_2^4.C_6$

Order 1024: $C_4^2.C_2^5.C_2$ x 2, $C_4^2.C_2^6$, $C_4^2.C_2^5.C_2$, $C_2\times C_4^2.C_2^5$, $C_2\times C_4^2.C_2^4.C_2$, $C_2\times C_4^2.C_2^5$, $C_2\times C_2^4.C_2^5$

Order 768: $C_4^2.(C_2^2\times A_4)$ x 2, $C_2^5:(C_2\times A_4)$ x 2, $C_4^2:A_4.C_2^2$ x 2, $(C_2^2\times C_4^2):A_4$

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

Order 384: $(C_2\times C_4^2):A_4$ x 2, $C_2^5:A_4$

Order 256: $D_4^2:C_2^2$ x 2, $D_4^2:C_2^2$ x 2, $D_4^2:C_2^2$ x 2, $C_4^2:C_2^4$ x 2, $C_4^3:C_2^2$ x 2, $C_2^4.C_2^4$, $D_4^2:C_2^2$, $C_4^2:C_2^4$, $C_2^2\times C_4^3$

Order 192: $C_4^2:A_4$

Order 128: $C_4^2:C_2^3$ x 7, $D_4^2:C_2$ x 2, $C_2\times C_4^3$, $C_2^7$, $C_4^2:C_2^3$, $C_2^3\times C_4^2$

Order 64: $C_4^2:C_2^2$ x 4, $C_2^2\times C_4^2$ x 3, $C_2^6$ x 2, $C_4^2:C_2^2$

Order 32: $C_2^5$ x 7, $C_2\times C_4^2$ x 3

Order 16: $C_2^4$ x 7, $C_4^2$

Order 8: $C_2^3$ x 3

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 24576: $(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 12288: $(C_2\times C_4^3).\GL(2,\mathbb{Z}/4)$ ($C_2$) x 8, $C_2\times C_4^2:A_4.C_2^2.D_4$ ($C_2$) x 2, $C_2\times C_4^2:A_4.C_2^2\wr C_2$ ($C_2$), $C_2\times C_4^2:A_4.C_2.C_2^4$ ($C_2$), $C_2\times C_4^2:A_4.C_2^3.C_2^2$ ($C_2$), $C_2\times C_4^2.C_2^4.C_6.C_2^2$ ($C_2$), $C_2^7.\GL(2,\mathbb{Z}/4)$ ($C_2$)

Order 6144: $C_4^2:A_4.C_2^2.D_4$ ($C_2^2$) x 8, $C_4^2.C_2^4.C_6.C_2^2$ ($C_2^2$) x 4, $C_4^2:A_4.C_2^2\wr C_2$ ($C_2^2$) x 4, $C_4^2:A_4.C_2.C_2^4$ ($C_2^2$) x 4, $C_4^2:A_4.C_2^3.C_2^2$ ($C_2^2$) x 4, $C_2^6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 4, $C_2\times C_4^2:A_4.C_2.D_4$ ($C_2^2$), $C_2\times C_4^2.(C_2\times D_4\times A_4)$ ($C_2^2$), $C_2\times C_4^2:A_4.D_4.C_2$ ($C_2^2$), $C_2\times C_4^2.C_2^4.C_6.C_2$ ($C_2^2$), $C_2\times C_4^2:A_4.C_2.D_4$ ($C_2^2$), $C_2\times C_4^2:A_4.D_4.C_2$ ($C_2^2$), $C_2\times C_4^2:A_4.C_2^4$ ($C_2^2$)

Order 4096: $C_2\times C_4^2.C_2^6.C_2$ ($S_3$)

Order 3072: $C_4^2:A_4.D_4.C_2$ ($C_2^3$) x 2, $C_4^2:A_4.D_4.C_2$ ($D_4$) x 2, $C_4^2:A_4.C_2.D_4$ ($C_2^3$) x 2, $C_4^2.C_2^4.C_6.C_2$ ($C_2^3$) x 2, $C_4^2:A_4.C_2^4$ ($C_2^3$) x 2, $C_4^2:A_4.C_2.D_4$ ($C_2^3$) x 2, $C_4^2:A_4.D_4.C_2$ ($C_2^3$) x 2, $C_4^2:A_4.D_4.C_2$ ($D_4$) x 2, $C_4^2.(D_4\times A_4).C_2$ ($D_4$) x 2, $C_4^2.(C_2\times D_4\times A_4)$ ($C_2^3$) x 2, $C_2\times C_4^2:A_4.C_2^3$ ($D_4$), $C_2\times C_4^2:A_4.C_2^3$ ($C_2^3$), $C_2\times C_4^2.C_2^3.C_6.C_2$ ($D_4$), $C_2\times C_4^2:A_4.C_4.C_2$ ($D_4$), $C_2\times C_4^2:A_4.C_4.C_2$ ($D_4$), $C_4^2.(C_2^3\times S_4)$ ($D_4$), $C_2^6:(C_2\times S_4)$ ($D_4$)

Order 2048: $C_4^2.C_2^6.C_2$ ($D_6$) x 4, $C_2\times C_4^2.C_2^5.C_2$ ($D_6$), $C_2\times C_4^2.C_2^5.C_2$ ($D_6$), $C_2\times C_4^2.C_2^6$ ($D_6$)

Order 1536: $C_4^2:A_4.C_2^3$ ($C_2\times D_4$) x 3, $C_4^2.(C_2^2\times S_4)$ ($C_2\times D_4$) x 2, $C_2^5:(C_2\times S_4)$ ($C_2\times D_4$) x 2, $C_2^3.C_2^5.S_3$ ($C_2\times D_4$) x 2, $C_2^3.C_2^5.S_3$ ($C_2\times D_4$) x 2, $C_4^2:A_4.C_2^3$ ($C_2\times D_4$) x 2, $C_4^2.C_2^4.C_6$ ($C_2\times D_4$) x 2, $C_4^2:A_4.C_2^3$ ($C_2^4$), $C_4^2.(C_2^3\times A_4)$ ($C_2\times D_4$), $C_2^6:(C_2\times A_4)$ ($C_2\times D_4$), $C_4^2:A_4.C_2^3$ ($C_2\times D_4$)

Order 1024: $C_4^2.C_2^6$ ($C_2\times D_6$) x 2, $C_4^2.C_2^5.C_2$ ($C_3:D_4$) x 2, $C_4^2.C_2^5.C_2$ ($C_2\times D_6$) x 2, $C_4^2.C_2^5.C_2$ ($C_2\times D_6$) x 2, $C_2\times C_4^2.C_2^5$ ($C_2\times D_6$), $C_2\times C_4^2.C_2^4.C_2$ ($C_3:D_4$), $C_2\times C_4^2.C_2^5$ ($C_3:D_4$), $C_2\times C_2^4.C_2^5$ ($S_4$)

Order 768: $C_4^2.(C_2^2\times A_4)$ ($C_2^2\times D_4$), $C_4^2.(C_2^2\times A_4)$ ($C_2^2\wr C_2$), $C_2^5:(C_2\times A_4)$ ($C_2^2\times D_4$), $C_2^5:(C_2\times A_4)$ ($C_2^2\wr C_2$), $C_4^2:A_4.C_2^2$ ($C_2^2\times D_4$), $C_4^2:A_4.C_2^2$ ($C_2^2\wr C_2$), $(C_2^2\times C_4^2):A_4$ ($C_2^2\wr C_2$)

Order 512: $C_2^4.C_2^5$ ($C_2\times S_4$) x 4, $C_4^2.C_2^4.C_2$ ($C_6:D_4$) x 2, $C_4^2.C_2^5$ ($C_6:D_4$) x 2, $C_4^2.C_2^5$ ($S_3\times D_4$) x 2, $C_2^2\times C_4^3.C_2$ ($C_2\times S_4$), $C_2\times C_2^4.C_2^4$ ($C_2\times S_4$), $C_4^2.C_2^5$ ($C_2^2\times D_6$), $C_4^2.C_2^5$ ($C_6:D_4$), $C_2\times C_2^2.C_2^6$ ($S_3\times D_4$), $D_4^2:C_2^3$ ($S_3\times D_4$), $C_2\times C_4^2.C_2^4$ ($C_6:D_4$), $C_2^3\times (C_2\times C_4).C_2^3$ ($C_2\times S_4$)

Order 384: $(C_2\times C_4^2):A_4$ ($C_2^3:D_4$), $(C_2\times C_4^2):A_4$ ($C_2\wr C_2^2$), $C_2^5:A_4$ ($C_2\wr C_2^2$)

Order 256: $C_2^4.C_2^4$ ($C_2^2\times S_4$) x 2, $C_4^2:C_2^4$ ($C_2^2\times S_4$) x 2, $C_4^3:C_2^2$ ($C_2^2\times S_4$) x 2, $C_4^3:C_2^2$ ($\GL(2,\mathbb{Z}/4)$) x 2, $D_4^2:C_2^2$ ($D_4\times D_6$), $D_4^2:C_2^2$ ($D_6:D_4$), $D_4^2:C_2^2$ ($D_4\times D_6$), $D_4^2:C_2^2$ ($D_6:D_4$), $D_4^2:C_2^2$ ($C_2^3:D_6$), $D_4^2:C_2^2$ ($D_6:D_4$), $D_4^2:C_2^2$ ($D_6:D_4$), $C_4^2:C_2^4$ ($C_2^2\times S_4$), $C_4^2:C_2^4$ ($\GL(2,\mathbb{Z}/4)$), $C_2^2\times C_4^3$ ($\GL(2,\mathbb{Z}/4)$)

Order 192: $C_4^2:A_4$ ($C_2^4:D_4$)

Order 128: $C_4^2:C_2^3$ ($D_4\times S_4$) x 4, $C_4^2:C_2^3$ ($C_2\times \GL(2,\mathbb{Z}/4)$) x 3, $C_2\times C_4^3$ ($C_2\times \GL(2,\mathbb{Z}/4)$) x 2, $C_2^7$ ($C_2^3:S_4$), $D_4^2:C_2$ ($C_2^4:D_6$), $D_4^2:C_2$ ($C_2^4:D_6$), $C_4^2:C_2^3$ ($C_2^4:D_6$), $C_4^2:C_2^3$ ($C_2^3\times S_4$), $C_2^3\times C_4^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 64: $C_2^6$ ($C_2^4:S_4$) x 3, $C_4^2:C_2^2$ ($\GL(2,\mathbb{Z}/4):C_2^2$) x 2, $C_4^2:C_2^2$ ($C_2^2:\GL(2,\mathbb{Z}/4)$) x 2, $C_2^2\times C_4^2$ ($C_2^2:\GL(2,\mathbb{Z}/4)$) x 2, $C_4^2:C_2^2$ ($C_2^5:D_6$), $C_2^2\times C_4^2$ ($C_2^2\times \GL(2,\mathbb{Z}/4)$)

Order 32: $C_2^5$ ($C_2\wr D_6$) x 4, $C_2^5$ ($C_2^3:\GL(2,\mathbb{Z}/4)$) x 2, $C_2\times C_4^2$ ($C_2^3:\GL(2,\mathbb{Z}/4)$) x 2, $C_2^5$ ($C_2^5:S_4$), $C_2\times C_4^2$ ($C_2^3:\GL(2,\mathbb{Z}/4)$)

Order 16: $C_2^4$ ($C_2^4:\GL(2,\mathbb{Z}/4)$) x 4, $C_2^4$ ($C_2^7:D_6$) x 2, $C_4^2$ ($C_2^4:\GL(2,\mathbb{Z}/4)$), $C_2^4$ ($C_2^4:\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_2^5:\GL(2,\mathbb{Z}/4)$) x 2, $C_2^3$ ($C_2\times C_2^4.C_2^3:D_6$)

Order 4: $C_2^2$ ($C_2\times C_2^4:C_3.C_2\wr C_2^2$)

Order 2: $C_2$ ($(C_2\times C_4^3).\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 24576: $(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 12288: $C_2\times C_4^2:A_4.C_2^2.D_4$ ($C_2$) x 2, $C_2\times C_4^2:A_4.C_2^2\wr C_2$ ($C_2$), $C_2\times C_4^2:A_4.C_2.C_2^4$ ($C_2$), $C_2\times C_4^2:A_4.C_2^3.C_2^2$ ($C_2$), $C_2\times C_4^2.C_2^4.C_6.C_2^2$ ($C_2$), $C_2^7.\GL(2,\mathbb{Z}/4)$ ($C_2$), $(C_2\times C_4^3).\GL(2,\mathbb{Z}/4)$ ($C_2$)

Order 6144: $C_4^2:A_4.C_2^2.D_4$ ($C_2^2$) x 2, $C_2\times C_4^2:A_4.C_2.D_4$ ($C_2^2$), $C_2\times C_4^2.(C_2\times D_4\times A_4)$ ($C_2^2$), $C_2\times C_4^2:A_4.D_4.C_2$ ($C_2^2$), $C_4^2.C_2^4.C_6.C_2^2$ ($C_2^2$), $C_4^2:A_4.C_2^2\wr C_2$ ($C_2^2$), $C_2\times C_4^2.C_2^4.C_6.C_2$ ($C_2^2$), $C_4^2:A_4.C_2.C_2^4$ ($C_2^2$), $C_2\times C_4^2:A_4.C_2.D_4$ ($C_2^2$), $C_2\times C_4^2:A_4.D_4.C_2$ ($C_2^2$), $C_2\times C_4^2:A_4.C_2^4$ ($C_2^2$), $C_4^2:A_4.C_2^3.C_2^2$ ($C_2^2$), $C_2^6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$)

Order 4096: $C_2\times C_4^2.C_2^6.C_2$ ($S_3$)

Order 3072: $C_2\times C_4^2:A_4.C_2^3$ ($D_4$), $C_4^2:A_4.D_4.C_2$ ($C_2^3$), $C_4^2:A_4.D_4.C_2$ ($D_4$), $C_2\times C_4^2:A_4.C_2^3$ ($C_2^3$), $C_4^2:A_4.C_2.D_4$ ($C_2^3$), $C_2\times C_4^2.C_2^3.C_6.C_2$ ($D_4$), $C_4^2.C_2^4.C_6.C_2$ ($C_2^3$), $C_4^2:A_4.C_2^4$ ($C_2^3$), $C_4^2:A_4.C_2.D_4$ ($C_2^3$), $C_2\times C_4^2:A_4.C_4.C_2$ ($D_4$), $C_2\times C_4^2:A_4.C_4.C_2$ ($D_4$), $C_4^2:A_4.D_4.C_2$ ($C_2^3$), $C_4^2:A_4.D_4.C_2$ ($D_4$), $C_4^2.(D_4\times A_4).C_2$ ($D_4$), $C_4^2.(C_2\times D_4\times A_4)$ ($C_2^3$), $C_4^2.(C_2^3\times S_4)$ ($D_4$), $C_2^6:(C_2\times S_4)$ ($D_4$)

Order 2048: $C_2\times C_4^2.C_2^5.C_2$ ($D_6$), $C_2\times C_4^2.C_2^5.C_2$ ($D_6$), $C_2\times C_4^2.C_2^6$ ($D_6$), $C_4^2.C_2^6.C_2$ ($D_6$)

Order 1536: $C_4^2:A_4.C_2^3$ ($C_2\times D_4$) x 3, $C_4^2.(C_2^2\times S_4)$ ($C_2\times D_4$), $C_2^5:(C_2\times S_4)$ ($C_2\times D_4$), $C_2^3.C_2^5.S_3$ ($C_2\times D_4$), $C_2^3.C_2^5.S_3$ ($C_2\times D_4$), $C_4^2:A_4.C_2^3$ ($C_2^4$), $C_4^2.(C_2^3\times A_4)$ ($C_2\times D_4$), $C_2^6:(C_2\times A_4)$ ($C_2\times D_4$), $C_4^2:A_4.C_2^3$ ($C_2\times D_4$), $C_4^2:A_4.C_2^3$ ($C_2\times D_4$), $C_4^2.C_2^4.C_6$ ($C_2\times D_4$)

Order 1024: $C_4^2.C_2^6$ ($C_2\times D_6$), $C_4^2.C_2^5.C_2$ ($C_3:D_4$), $C_4^2.C_2^5.C_2$ ($C_2\times D_6$), $C_4^2.C_2^5.C_2$ ($C_2\times D_6$), $C_2\times C_4^2.C_2^5$ ($C_2\times D_6$), $C_2\times C_4^2.C_2^4.C_2$ ($C_3:D_4$), $C_2\times C_4^2.C_2^5$ ($C_3:D_4$), $C_2\times C_2^4.C_2^5$ ($S_4$)

Order 768: $C_4^2.(C_2^2\times A_4)$ ($C_2^2\times D_4$), $C_4^2.(C_2^2\times A_4)$ ($C_2^2\wr C_2$), $C_2^5:(C_2\times A_4)$ ($C_2^2\times D_4$), $C_2^5:(C_2\times A_4)$ ($C_2^2\wr C_2$), $C_4^2:A_4.C_2^2$ ($C_2^2\times D_4$), $C_4^2:A_4.C_2^2$ ($C_2^2\wr C_2$), $(C_2^2\times C_4^2):A_4$ ($C_2^2\wr C_2$)

Order 512: $C_4^2.C_2^5$ ($S_3\times D_4$) x 2, $C_4^2.C_2^4.C_2$ ($C_6:D_4$), $C_2^2\times C_4^3.C_2$ ($C_2\times S_4$), $C_4^2.C_2^5$ ($C_6:D_4$), $C_2^4.C_2^5$ ($C_2\times S_4$), $C_2\times C_2^4.C_2^4$ ($C_2\times S_4$), $C_4^2.C_2^5$ ($C_2^2\times D_6$), $C_4^2.C_2^5$ ($C_6:D_4$), $C_2\times C_2^2.C_2^6$ ($S_3\times D_4$), $D_4^2:C_2^3$ ($S_3\times D_4$), $C_2\times C_4^2.C_2^4$ ($C_6:D_4$), $C_2^3\times (C_2\times C_4).C_2^3$ ($C_2\times S_4$)

Order 384: $(C_2\times C_4^2):A_4$ ($C_2^3:D_4$), $(C_2\times C_4^2):A_4$ ($C_2\wr C_2^2$), $C_2^5:A_4$ ($C_2\wr C_2^2$)

Order 256: $C_2^4.C_2^4$ ($C_2^2\times S_4$), $D_4^2:C_2^2$ ($D_4\times D_6$), $D_4^2:C_2^2$ ($D_6:D_4$), $D_4^2:C_2^2$ ($D_4\times D_6$), $D_4^2:C_2^2$ ($D_6:D_4$), $D_4^2:C_2^2$ ($C_2^3:D_6$), $D_4^2:C_2^2$ ($D_6:D_4$), $D_4^2:C_2^2$ ($D_6:D_4$), $C_4^2:C_2^4$ ($C_2^2\times S_4$), $C_4^2:C_2^4$ ($C_2^2\times S_4$), $C_4^2:C_2^4$ ($\GL(2,\mathbb{Z}/4)$), $C_4^3:C_2^2$ ($C_2^2\times S_4$), $C_4^3:C_2^2$ ($\GL(2,\mathbb{Z}/4)$), $C_2^2\times C_4^3$ ($\GL(2,\mathbb{Z}/4)$)

Order 192: $C_4^2:A_4$ ($C_2^4:D_4$)

Order 128: $C_4^2:C_2^3$ ($D_4\times S_4$) x 4, $C_4^2:C_2^3$ ($C_2\times \GL(2,\mathbb{Z}/4)$) x 2, $C_2\times C_4^3$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $C_2^7$ ($C_2^3:S_4$), $D_4^2:C_2$ ($C_2^4:D_6$), $D_4^2:C_2$ ($C_2^4:D_6$), $C_4^2:C_2^3$ ($C_2^4:D_6$), $C_4^2:C_2^3$ ($C_2^3\times S_4$), $C_2^3\times C_4^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 64: $C_2^6$ ($C_2^4:S_4$) x 2, $C_4^2:C_2^2$ ($\GL(2,\mathbb{Z}/4):C_2^2$) x 2, $C_4^2:C_2^2$ ($C_2^2:\GL(2,\mathbb{Z}/4)$) x 2, $C_2^2\times C_4^2$ ($C_2^2:\GL(2,\mathbb{Z}/4)$) x 2, $C_4^2:C_2^2$ ($C_2^5:D_6$), $C_2^2\times C_4^2$ ($C_2^2\times \GL(2,\mathbb{Z}/4)$)

Order 32: $C_2^5$ ($C_2\wr D_6$) x 4, $C_2^5$ ($C_2^3:\GL(2,\mathbb{Z}/4)$) x 2, $C_2\times C_4^2$ ($C_2^3:\GL(2,\mathbb{Z}/4)$) x 2, $C_2^5$ ($C_2^5:S_4$), $C_2\times C_4^2$ ($C_2^3:\GL(2,\mathbb{Z}/4)$)

Order 16: $C_2^4$ ($C_2^4:\GL(2,\mathbb{Z}/4)$) x 4, $C_2^4$ ($C_2^7:D_6$) x 2, $C_4^2$ ($C_2^4:\GL(2,\mathbb{Z}/4)$), $C_2^4$ ($C_2^4:\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_2^5:\GL(2,\mathbb{Z}/4)$) x 2, $C_2^3$ ($C_2\times C_2^4.C_2^3:D_6$)

Order 4: $C_2^2$ ($C_2\times C_2^4:C_3.C_2\wr C_2^2$)

Order 2: $C_2$ ($(C_2\times C_4^3).\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$)

Series

Derived series

$(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_4^2:A_4.C_2^3$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2\times C_4^2.C_2^4.C_6.C_2^2$

$\rhd$

$C_2\times C_4^2.C_2^4.C_6.C_2$

$\rhd$

$C_4^2.C_2^4.C_6.C_2$

$\rhd$

$C_4^2:A_4.C_2^3$

$\rhd$

$C_4^2:A_4.C_2^2$

$\rhd$

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

$\rhd$

$C_4^2:A_4$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_4^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_4^2:A_4.C_2^3$

$\rhd$

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

$\rhd$

$C_4^2:A_4$

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 1 larger groups in the database.

Character theory

Complex character table

The $156 \times 156$ character table is not available for this group.

Rational character table

The $148 \times 148$ rational character table is not available for this group.