Abstract group 24576.bei: $C_2^6.C_2\wr S_3$

• Label: 24576.bei
• Order: \( 2^{13} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{18} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$C_2^6.C_2\wr S_3$
Order:	\(24576\)\(\medspace = 2^{13} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_{1158}:C_{16}$, of order \(786432\)\(\medspace = 2^{18} \cdot 3 \)
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	1183	512	9056	3584	6144	4096	24576
Conjugacy classes	1	42	1	75	7	20	8	154
Divisions	1	42	1	72	5	14	2	137
Autjugacy classes	1	21	1	43	4	7	1	78

Minimal presentations

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

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid c^{2}=d^{4}=e^{2}=f^{4}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $96$ $\langle(1,73)(3,27)(4,28)(6,30)(7,31)(9,33)(10,34)(12,36)(13,37)(15,39)(16,40) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^6$ . $(C_2\wr S_3)$ (2)	$(C_2^6.C_2^6)$ . $S_3$	$C_2^7$ . $(C_2^3:S_4)$ (2)	$C_2^5$ . $(C_2^5:S_4)$	all 38
Aut. group:	$\Aut(C_2\times C_8^2)$

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$2$

Subgroups

There are 75 normal subgroups (43 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_4^2.C_2^4:S_4$
Commutator:	$G' \simeq$ $C_2^6.C_2^4.C_6$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2^5$	$G/\Phi \simeq$ $C_2^5:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6.C_2^6$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^6.C_2\wr S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2\times C_2^5.S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^4\wr C_2$
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: $C_2^6.C_2\wr S_3$

Order 12288: $C_2^4.C_2^6.D_6$, $C_2^6.C_2^4.C_6.C_2$, $C_2^4.C_2^6.C_6.C_2$

Order 8192: $C_2^4.C_2^4\wr C_2$

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

Order 4096: $C_2^5.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2$ x 2, $C_2^5.C_2^6.C_2$ x 2, $(D_4\times C_2^3).D_4^2$ x 2, $C_2^6.D_4^2$ x 2, $(D_4\times C_2^3).D_4^2$ x 2, $C_2^6.C_2^6$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^4.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2.C_2^6.C_2^5$, $C_2.C_2^6.C_2^5$, $C_2^5.C_2^6.C_2$, $C_2.C_2^6.C_2^5$, $C_2^5.C_2^6.C_2$, $(D_4\times C_2^3).D_4^2$

Order 3072: $C_2.C_4^2.A_4.D_4$ x 4, $C_2.C_4^2.A_4.D_4$ x 4, $C_2^7:S_4$ x 3, $C_2.C_4^2.A_4.C_4.C_2$ x 2, $C_2^4.C_2^4.C_6.C_2$ x 2, $C_2.C_4^2.A_4.D_4$ x 2, $C_2^6.A_4.C_2^2$ x 2, $C_2^4.C_2^4.C_6.C_2$ x 2, $C_2^4.C_2^4.C_6.C_2$ x 2, $C_2^5.S_4.C_2^2$ x 2, $C_2.C_4^2.A_4.D_4$ x 2, $(C_2^6\times C_4).A_4$ x 2, $C_2^3.C_2^4.D_6.C_2$, $C_2^4.C_2^4.C_6.C_2$, $C_2^4.C_2^4.C_6.C_2$, $C_2^4.C_2^4.C_6.C_2$, $C_2^6.C_2^4.C_3$, $C_2^6.A_4.C_2^2$, $C_2^6.A_4.C_4$, $C_2^4.C_2^4.C_6.C_2$, $C_2^4.C_2^4.C_6.C_2$, $C_2^4.C_2^4.D_6$, $C_2.C_4^2.A_4.C_4.C_2$, $C_2.C_4^2.A_4.C_2^3$

Order 2048: $(C_2^2\times D_4).D_4^2$ x 16, $C_2^4.C_2^6.C_2$ x 12, $C_2^5.C_2^6$ x 12, $C_2^6.C_2^5$ x 10, $C_2^5.C_2^5.C_2$ x 8, $C_2^5.C_2^5.C_2$ x 8, $C_2^4.C_2^6.C_2$ x 8, $C_2^4.C_2^6.C_2$ x 8, $(D_4\times C_2^5):D_4$ x 8, $C_2^4.C_2^6.C_2$ x 6, $C_2^5.C_2^5.C_2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^4.C_2^6.C_2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^6$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^6$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^3.C_2^6.C_2^2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^8.D_4$ x 2, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2\times C_2^6.C_2^4$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^3.C_2^6.C_2^2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^3.C_2^6.C_2^2$, $C_2^5.C_2^6$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2\times C_2^6.C_2^4$, $C_2^6.C_2^5$

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

Order 1024: $C_2^6.C_2^4$ x 2, $C_2^4\times C_2.C_4:D_4$ x 2, $C_2\times C_2^4.C_2^5$, $C_2^6.C_2^4$, $C_2^2\times C_2^3.C_2^5$, $C_2^6.C_2^4$

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

Order 256: $C_2^6:C_2^2$ x 3, $C_2^5:D_4$ x 3, $C_2^6\times C_4$ x 2, $C_2^4.C_4^2$ x 2, $C_2^6:C_4$, $C_2^4:C_4^2$, $C_2^3.C_2^5$, $C_2^2.Q_8^2$

Order 128: $C_2^3.C_4^2$ x 4, $C_2^7$ x 3, $C_2^4:D_4$ x 2

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

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

Order 16: $C_2^4$ x 3

Order 8: $C_2^3$ x 4

Order 4: $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 3

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: $C_2^6.C_2\wr S_3$

Order 12288: $C_2^4.C_2^6.D_6$, $C_2^6.C_2^4.C_6.C_2$, $C_2^4.C_2^6.C_6.C_2$

Order 8192: $C_2^4.C_2^4\wr C_2$

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

Order 4096: $C_2^6.C_2^6$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^4.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2.C_2^6.C_2^5$, $C_2.C_2^6.C_2^5$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2.C_2^6.C_2^5$, $C_2^5.C_2^6.C_2$, $(D_4\times C_2^3).D_4^2$, $C_2^6.D_4^2$, $(D_4\times C_2^3).D_4^2$, $(D_4\times C_2^3).D_4^2$

Order 3072: $C_2^7:S_4$ x 2, $C_2^3.C_2^4.D_6.C_2$, $C_2^4.C_2^4.C_6.C_2$, $C_2.C_4^2.A_4.C_4.C_2$, $C_2^4.C_2^4.C_6.C_2$, $C_2^4.C_2^4.C_6.C_2$, $C_2^6.C_2^4.C_3$, $C_2^4.C_2^4.C_6.C_2$, $C_2^6.A_4.C_2^2$, $C_2.C_4^2.A_4.D_4$, $C_2^6.A_4.C_2^2$, $C_2.C_4^2.A_4.D_4$, $C_2^6.A_4.C_4$, $C_2^4.C_2^4.C_6.C_2$, $C_2^4.C_2^4.C_6.C_2$, $C_2.C_4^2.A_4.D_4$, $C_2^4.C_2^4.C_6.C_2$, $C_2^4.C_2^4.C_6.C_2$, $C_2^4.C_2^4.D_6$, $C_2^5.S_4.C_2^2$, $C_2.C_4^2.A_4.D_4$, $C_2.C_4^2.A_4.C_4.C_2$, $C_2.C_4^2.A_4.C_2^3$, $(C_2^6\times C_4).A_4$

Order 2048: $C_2^5.C_2^5.C_2$ x 6, $C_2^5.C_2^6$ x 6, $C_2^6.C_2^5$ x 5, $C_2^5.C_2^5.C_2$ x 4, $C_2^4.C_2^6.C_2$ x 4, $(C_2^2\times D_4).D_4^2$ x 4, $C_2^4.C_2^6.C_2$ x 3, $C_2^4.C_2^6.C_2$ x 3, $C_2^4.C_2^6.C_2$ x 3, $(D_4\times C_2^5):D_4$ x 3, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2\times C_2^6.C_2^4$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^3.C_2^6.C_2^2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^6$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^6.C_2$, $C_2^3.C_2^6.C_2^2$, $C_2^5.C_2^6$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^6.C_2$, $C_2^3.C_2^6.C_2^2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2\times C_2^6.C_2^4$, $C_2^4.C_2^6.C_2$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^8.D_4$

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

Order 1024: $C_2^6.C_2^4$, $C_2\times C_2^4.C_2^5$, $C_2^6.C_2^4$, $C_2^2\times C_2^3.C_2^5$, $C_2^6.C_2^4$, $C_2^4\times C_2.C_4:D_4$

Order 512: $C_2^4.C_2^5$ x 2, $C_2^2\times C_2^5.C_2^2$, $C_2^4\times C_2.C_4^2$, $C_2\times C_2^4.C_2^4$

Order 256: $C_2^6:C_2^2$ x 2, $C_2^5:D_4$ x 2, $C_2^6\times C_4$, $C_2^6:C_4$, $C_2^4:C_4^2$, $C_2^3.C_2^5$, $C_2^4.C_4^2$, $C_2^2.Q_8^2$

Order 128: $C_2^3.C_4^2$ x 3, $C_2^7$ x 2, $C_2^4:D_4$

Order 64: $C_2^2.C_4^2$ x 5, $C_2^6$ x 2, $C_2^4\times C_4$

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

Order 16: $C_2^4$ x 3

Order 8: $C_2^3$ x 3

Order 4: $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 24576: $C_2^6.C_2\wr S_3$ ($C_1$)

Order 12288: $C_2^4.C_2^6.D_6$ ($C_2$), $C_2^6.C_2^4.C_6.C_2$ ($C_2$), $C_2^4.C_2^6.C_6.C_2$ ($C_2$)

Order 6144: $C_2^6.C_2^4.C_6$ ($C_2^2$)

Order 4096: $C_2^6.C_2^6$ ($S_3$)

Order 3072: $C_2^6.C_2^4.C_3$ ($D_4$)

Order 2048: $C_2^6.C_2^5$ ($D_6$)

Order 1024: $C_2^6.C_2^4$ ($S_4$) x 2, $C_2^4\times C_2.C_4:D_4$ ($S_4$) x 2, $C_2\times C_2^4.C_2^5$ ($S_4$), $C_2^6.C_2^4$ ($S_4$), $C_2^2\times C_2^3.C_2^5$ ($S_4$), $C_2^6.C_2^4$ ($C_3:D_4$)

Order 512: $C_2^4.C_2^5$ ($C_2\times S_4$) x 3, $C_2^4\times C_2.C_4^2$ ($C_2\times S_4$) x 2, $C_2^2\times C_2^5.C_2^2$ ($C_2\times S_4$), $C_2\times C_2^4.C_2^4$ ($C_2\times S_4$)

Order 256: $C_2^6:C_2^2$ ($\GL(2,\mathbb{Z}/4)$) x 3, $C_2^5:D_4$ ($C_2^2:S_4$) x 3, $C_2^6\times C_4$ ($C_2^2:S_4$) x 2, $C_2^4.C_4^2$ ($\GL(2,\mathbb{Z}/4)$) x 2, $C_2^6:C_4$ ($C_2^2:S_4$), $C_2^4:C_4^2$ ($\GL(2,\mathbb{Z}/4)$), $C_2^3.C_2^5$ ($C_2^2:S_4$), $C_2^2.Q_8^2$ ($\GL(2,\mathbb{Z}/4)$)

Order 128: $C_2^3.C_4^2$ ($C_2^3:S_4$) x 4, $C_2^7$ ($C_2^3:S_4$) x 3, $C_2^4:D_4$ ($C_2^3:S_4$) x 2

Order 64: $C_2^2.C_4^2$ ($C_2\wr S_3$) x 4, $C_2^6$ ($C_2\wr S_3$) x 3, $C_2^2.C_4^2$ ($C_2^4:S_4$), $C_2^2.C_4^2$ ($C_2^4:S_4$), $C_2^4\times C_4$ ($C_2^4:S_4$)

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

Order 16: $C_2^4$ ($C_2^4:\GL(2,\mathbb{Z}/4)$), $C_2^4$ ($C_2.C_2^6.D_6$), $C_2^4$ ($C_2\times C_2^5.S_4$)

Order 8: $C_2^3$ ($C_2^4.C_2^4.D_6$) x 2, $C_2^3$ ($C_2^5.A_4.D_4$), $C_2^3$ ($C_2^4.C_2^4.D_6$)

Order 4: $C_2^2$ ($C_2^2.C_2^6.C_3.D_4$), $C_2^2$ ($C_4^2.C_2^4.C_3.D_4$), $C_2^2$ ($C_4^2.C_2^4:S_4$), $C_2^2$ ($C_4^2.C_2^4:S_4$)

Order 2: $C_2$ ($C_2^4.C_2^6.D_6$) x 2, $C_2$ ($C_2^3.C_2^6.C_3.D_4$)

Order 1: $C_1$ ($C_2^6.C_2\wr S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 24576: $C_2^6.C_2\wr S_3$ ($C_1$)

Order 12288: $C_2^4.C_2^6.D_6$ ($C_2$), $C_2^6.C_2^4.C_6.C_2$ ($C_2$), $C_2^4.C_2^6.C_6.C_2$ ($C_2$)

Order 6144: $C_2^6.C_2^4.C_6$ ($C_2^2$)

Order 4096: $C_2^6.C_2^6$ ($S_3$)

Order 3072: $C_2^6.C_2^4.C_3$ ($D_4$)

Order 2048: $C_2^6.C_2^5$ ($D_6$)

Order 1024: $C_2^6.C_2^4$ ($S_4$), $C_2\times C_2^4.C_2^5$ ($S_4$), $C_2^6.C_2^4$ ($S_4$), $C_2^2\times C_2^3.C_2^5$ ($S_4$), $C_2^6.C_2^4$ ($C_3:D_4$), $C_2^4\times C_2.C_4:D_4$ ($S_4$)

Order 512: $C_2^4.C_2^5$ ($C_2\times S_4$) x 2, $C_2^2\times C_2^5.C_2^2$ ($C_2\times S_4$), $C_2^4\times C_2.C_4^2$ ($C_2\times S_4$), $C_2\times C_2^4.C_2^4$ ($C_2\times S_4$)

Order 256: $C_2^6:C_2^2$ ($\GL(2,\mathbb{Z}/4)$) x 2, $C_2^5:D_4$ ($C_2^2:S_4$) x 2, $C_2^6\times C_4$ ($C_2^2:S_4$), $C_2^6:C_4$ ($C_2^2:S_4$), $C_2^4:C_4^2$ ($\GL(2,\mathbb{Z}/4)$), $C_2^3.C_2^5$ ($C_2^2:S_4$), $C_2^4.C_4^2$ ($\GL(2,\mathbb{Z}/4)$), $C_2^2.Q_8^2$ ($\GL(2,\mathbb{Z}/4)$)

Order 128: $C_2^3.C_4^2$ ($C_2^3:S_4$) x 3, $C_2^7$ ($C_2^3:S_4$) x 2, $C_2^4:D_4$ ($C_2^3:S_4$)

Order 64: $C_2^2.C_4^2$ ($C_2\wr S_3$) x 3, $C_2^6$ ($C_2\wr S_3$) x 2, $C_2^2.C_4^2$ ($C_2^4:S_4$), $C_2^2.C_4^2$ ($C_2^4:S_4$), $C_2^4\times C_4$ ($C_2^4:S_4$)

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

Order 16: $C_2^4$ ($C_2^4:\GL(2,\mathbb{Z}/4)$), $C_2^4$ ($C_2.C_2^6.D_6$), $C_2^4$ ($C_2\times C_2^5.S_4$)

Order 8: $C_2^3$ ($C_2^4.C_2^4.D_6$), $C_2^3$ ($C_2^5.A_4.D_4$), $C_2^3$ ($C_2^4.C_2^4.D_6$)

Order 4: $C_2^2$ ($C_2^2.C_2^6.C_3.D_4$), $C_2^2$ ($C_4^2.C_2^4.C_3.D_4$), $C_2^2$ ($C_4^2.C_2^4:S_4$), $C_2^2$ ($C_4^2.C_2^4:S_4$)

Order 2: $C_2$ ($C_2^4.C_2^6.D_6$), $C_2$ ($C_2^3.C_2^6.C_3.D_4$)

Order 1: $C_1$ ($C_2^6.C_2\wr S_3$)

Series

Derived series

$C_2^6.C_2\wr S_3$

$\rhd$

$C_2^6.C_2^4.C_6$

$\rhd$

$C_2^6.C_2^4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2^6.C_2\wr S_3$

$\rhd$

$C_2^4.C_2^6.C_6.C_2$

$\rhd$

$C_2^6.C_2^4.C_6$

$\rhd$

$C_2^6.C_2^4.C_3$

$\rhd$

$C_2^6.C_2^4$

$\rhd$

$C_2^4.C_4^2$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6.C_2\wr S_3$

$\rhd$

$C_2^6.C_2^4.C_6$

$\rhd$

$C_2^6.C_2^4.C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Character theory

Complex character table

The $154 \times 154$ character table is not available for this group.

Rational character table

The $137 \times 137$ rational character table is not available for this group.