Abstract group 4096.brc: $C_2^5.(D_4\times C_2^4)$

• Label: 4096.brc
• Order: \( 2^{12} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{23} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{13} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $6$

Group information

Description:	$C_2^5.(D_4\times C_2^4)$
Order:	\(4096\)\(\medspace = 2^{12} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^{12}.C_2^6.C_2^5$, of order \(8388608\)\(\medspace = 2^{23} \)
Composition factors:	$C_2$ x 12
Nilpotency class:	$5$
Derived length:	$3$

This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).

Group statistics

Order	1	2	4	8
Elements	1	511	2560	1024	4096
Conjugacy classes	1	59	96	16	172
Divisions	1	59	88	16	164
Autjugacy classes	1	21	18	1	41

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$6$
Inequivalent generating 6-tuples:	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:	${\langle a, b, c, d, e, f, g, h, i \mid a^{2}=c^{4}=d^{2}=e^{2}=f^{4}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $34$ $\langle(33,34), (1,6,4,14)(3,10,12,25)(5,16,13,18)(8,21)(9,15,24,26)(17,27)(20,30) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^5:Q_8)$ . $C_2^4$ (16)	$(C_2^5:Q_8)$ . $C_2^4$ (6)	$(C_2^5:Q_8)$ . $C_2^4$ (4)	$(C_2^5:Q_8)$ . $C_2^4$ (16)	all 233
Aut. group:	$\Aut(C_2^5.D_8)$	$\Aut(C_2^5.D_8)$	$\Aut(C_2^5.\SD_{16})$	$\Aut(C_2^5.\SD_{16})$	all 7

Elements of the group are displayed as permutations of degree 34.

Homology

Abelianization:	$C_{2}^{6} $
Schur multiplier:	$C_{2}^{15}$
Commutator length:	$1$

Subgroups

There are 4075 normal subgroups (73 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2^2$
Commutator:	a subgroup isomorphic to $C_2^4:C_4$
Frattini:	a subgroup isomorphic to $C_2^4:C_4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.(D_4\times C_2^4)$

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 4 (order at least 1024) are shown, as well as all normal subgroups of any index.

Order 4096: $C_2^5.(D_4\times C_2^4)$

Order 2048: $C_2^5.C_2.C_2^5$ x 43, $C_2^5.C_2^5.C_2$ x 16, $(D_4\times C_2^5).D_4$ x 2, $C_2^7.C_2^4$, $C_2^6.C_2^5$

Order 1024: $C_2^5.C_2.C_2^4$ x 381, $C_2^5.C_2^4.C_2$ x 72, $C_2^6.(C_2\times D_4)$ x 64, $C_2^5.C_2^5$ x 60, $C_2^4.C_2^5.C_2$ x 57, $C_2^6.C_2^4$ x 52, $C_2^3.C_2^6.C_2$ x 39, $C_2^7.C_2^3$ x 13, $C_2^3.C_2^5.C_2^2$ x 7, $C_2^4.C_2^4.C_2^2$, $C_2.C_2^6.C_2^3$

Order 512: $C_2^5.C_2^3.C_2$ x 128, $C_2^5.(C_2\times D_4)$ x 128, $C_2^5.C_2^3.C_2$ x 64, $C_2^5.C_4.C_2^2$ x 64, $C_2^5:\SD_{16}$ x 64, $C_2^5:D_8$ x 64, $C_2^5.C_2^3.C_2$ x 64, $C_2^6.D_4$ x 64, $C_2^4.C_2^4.C_2$ x 64, $C_2^5.C_2^4$ x 32, $C_2^4.C_2^5$ x 32, $C_2^5.C_2^4$ x 32, $C_2^5.C_2^3.C_2$ x 32, $C_2^4.C_2^4.C_2$ x 32, $C_2^5:\OD_{16}$ x 32, $C_2^4.C_2^4.C_2$ x 32, $C_2^4.C_2^4.C_2$ x 32, $C_2^4.(C_2^2\times D_4)$ x 32, $C_2^4.C_2^4.C_2$ x 32, $C_2^6.D_4$ x 32, $C_2^5.C_2^4$ x 32, $C_2^5.C_2^4$ x 16, $C_2^5.C_2^4$ x 16, $C_2\times C_2^5.D_4$ x 16, $C_2\times C_2^5.D_4$ x 16, $C_2^4.C_2^4.C_2$ x 16, $C_2^5.C_4.C_2^2$ x 16, $C_2^3.C_2^5.C_2$ x 16, $C_2^5.C_2^4$ x 16, $C_2^5.C_2^4$ x 16, $C_2^6.C_2^3$ x 16, $C_2^5.C_2^4$ x 16, $C_2\times C_2^5.C_2^3$ x 12, $C_2^6.C_2^3$ x 8, $C_2\times C_2^5.C_2^3$ x 8, $C_2^5.C_2^4$ x 8, $C_2\times C_2^4.C_2^4$ x 8, $C_2^5.C_2^4$ x 8, $C_2\times C_2^4.C_2^3.C_2$ x 8, $C_2\times C_2^4.C_2^3.C_2$ x 8, $C_2\times C_2^4.C_2^3.C_2$ x 8, $C_2\times C_2^4.C_2^3.C_2$ x 8, $C_2\times C_2^4.C_2^3.C_2$ x 8, $C_2\times C_2^4.C_2^3.C_2$ x 8, $C_2\times C_2^4.C_2^4$ x 4, $C_2\times C_2^4.C_2^4$ x 4, $C_2\times C_2^4.C_2^4$ x 4, $C_2^4.C_2^5$ x 4, $C_2\times C_2^5.C_2^3$ x 4, $C_2^4.C_2^5$ x 4, $C_2\times C_2^5.C_2^3$ x 4, $C_2^4.C_2^5$ x 4, $C_2\times C_2^3.C_2^4.C_2$ x 4, $C_2\times C_2^3.C_2^4.C_2$ x 4, $C_2\times C_2^3.C_2^4.C_2$ x 4, $C_2\times C_2^5.C_2^3$ x 4, $C_2\times C_2^5.C_2^3$ x 4, $C_2\times C_2^5.C_2^3$ x 4, $C_2^5.C_2^4$ x 4, $C_2\times C_2^5.C_2^3$ x 2, $C_2\times C_2^5.C_2^3$ x 2, $C_2\times C_2^6.C_2^2$ x 2, $C_2\times C_2^5.C_2^3$ x 2, $C_2^2\times C_2^4.C_2^3$ x 2, $C_2^4.C_2^5$ x 2, $C_2^2\times C_2^4.C_2^3$ x 2, $C_2^2\times C_2^5.C_2^2$, $C_2^3.C_2^6$, $C_2^3.C_2^6$, $C_2^3.C_2^6$, $C_4^2:C_2^5$

Order 256: $C_2^4.Q_{16}$ x 64, $C_2^4.D_8$ x 64, $C_2^4.C_2^4$ x 64, $C_2^4:Q_{16}$ x 32, $C_2^4:\SD_{16}$ x 32, $C_2^4:\SD_{16}$ x 32, $C_2^4:D_8$ x 32, $C_2^4.\OD_{16}$ x 32, $C_2^5:C_8$ x 32, $C_2^4.C_2^4$ x 32, $C_2^4.C_2^4$ x 32, $C_2^4.C_2^4$ x 32, $C_2^4.C_2^4$ x 32, $C_2^5:D_4$ x 32, $C_2^4.C_2^4$ x 32, $C_2^5.D_4$ x 32, $C_2^5.D_4$ x 32, $C_2^5.D_4$ x 26, $C_4^2:C_2^4$ x 24, $C_2^4:\OD_{16}$ x 16, $C_2^4:\OD_{16}$ x 16, $C_2^4.\OD_{16}$ x 16, $C_2^3.C_2^5$ x 16, $C_2^5.C_2^3$ x 16, $C_2^5:Q_8$ x 16, $C_2^5.D_4$ x 16, $C_2^5:D_4$ x 16, $C_2^5:Q_8$ x 16, $C_2^5:D_4$ x 16, $C_2^4.C_2^4$ x 16, $C_2^4.C_2^4$ x 16, $C_4^2:C_2^4$ x 8, $C_2^5:C_8$ x 8, $C_2^3.C_2^5$ x 8, $C_2^5.C_2^3$ x 8, $C_2^5.C_2^3$ x 8, $C_2^4.C_2^4$ x 8, $C_2^5:Q_8$ x 8, $C_2^5.D_4$ x 8, $C_2^5:D_4$ x 8, $C_2^6:C_4$ x 8, $C_2^4.C_4^2$ x 8, $C_2^5:Q_8$ x 6, $C_2^5:D_4$ x 6, $C_2^5:D_4$ x 4, $C_2^6:C_4$ x 4, $C_2^4.C_4^2$ x 4, $C_2^4.C_4^2$ x 4, $C_2^5:Q_8$ x 2, $C_2^5:D_4$ x 2, $C_2^6:C_4$ x 2, $C_2^3.C_2^5$ x 2, $C_2^5.D_4$ x 2, $D_4:C_2^5$, $D_4\times C_2^5$, $C_4^2:C_2^4$

Order 128: $C_2^4.D_4$ x 72, $C_2^4.D_4$ x 64, $C_2^5:C_4$ x 36, $C_2^4.C_2^3$ x 32, $C_2^4:Q_8$ x 32, $C_2^4.D_4$ x 32, $C_2^4:D_4$ x 32, $C_2^5:C_4$ x 32, $C_4^2:C_2^3$ x 32, $C_2^5:C_4$ x 30, $C_2^3.C_4^2$ x 24, $C_2^4.D_4$ x 24, $C_2^4:D_4$ x 20, $C_2^4:C_8$ x 16, $D_4\times C_2^4$ x 14, $C_2^4:Q_8$ x 12, $C_2^3.C_4^2$ x 8, $C_2^4:D_4$ x 8, $C_2^3.C_2^4$ x 8, $D_4:C_2^4$ x 4, $C_4^2:C_2^3$ x 4, $C_2^4:Q_8$ x 4, $C_2^4:D_4$ x 4, $C_2^7$ x 2, $C_2^5\times C_4$

Order 64: $C_2^4:C_4$ x 60, $C_2^4:C_4$ x 40, $C_2^3.D_4$ x 32, $D_4\times C_2^3$ x 24, $C_2^6$ x 23, $C_2^2.C_4^2$ x 16, $C_2^4\times C_4$ x 8, $C_2^3:Q_8$ x 8, $C_2^3:D_4$ x 8

Order 32: $C_2^5$ x 51, $C_2^3:C_4$ x 16, $C_2^3\times C_4$ x 8

Order 16: $C_2^4$ x 35

Order 8: $C_2^3$ x 19

Order 4: $C_2^2$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 4096: $C_2^5.(D_4\times C_2^4)$ ($C_1$)

Order 2048: $C_2^5.C_2.C_2^5$ ($C_2$) x 43, $C_2^5.C_2^5.C_2$ ($C_2$) x 16, $(D_4\times C_2^5).D_4$ ($C_2$) x 2, $C_2^7.C_2^4$ ($C_2$), $C_2^6.C_2^5$ ($C_2$)

Order 1024: $C_2^5.C_2.C_2^4$ ($C_2^2$) x 381, $C_2^5.C_2^4.C_2$ ($C_2^2$) x 72, $C_2^6.(C_2\times D_4)$ ($C_2^2$) x 64, $C_2^4.C_2^5.C_2$ ($C_2^2$) x 43, $C_2^6.C_2^4$ ($C_2^2$) x 32, $C_2^5.C_2^5$ ($C_2^2$) x 24, $C_2^3.C_2^6.C_2$ ($C_2^2$) x 23, $C_2^3.C_2^5.C_2^2$ ($C_2^2$) x 6, $C_2^7.C_2^3$ ($C_2^2$) x 5, $C_2^4.C_2^4.C_2^2$ ($C_2^2$)

Order 512: $C_2^5.C_2^3.C_2$ ($C_2^3$) x 128, $C_2^5.(C_2\times D_4)$ ($C_2^3$) x 128, $C_2^5.C_2^3.C_2$ ($C_2^3$) x 64, $C_2^5.C_4.C_2^2$ ($C_2^3$) x 64, $C_2^5:\SD_{16}$ ($C_2^3$) x 64, $C_2^5:D_8$ ($C_2^3$) x 64, $C_2^5.C_2^3.C_2$ ($C_2^3$) x 64, $C_2^6.D_4$ ($C_2^3$) x 64, $C_2^4.C_2^4.C_2$ ($C_2^3$) x 64, $C_2^5.C_2^4$ ($C_2^3$) x 32, $C_2^4.C_2^5$ ($C_2^3$) x 32, $C_2^5.C_2^3.C_2$ ($C_2^3$) x 32, $C_2^4.C_2^4.C_2$ ($C_2^3$) x 32, $C_2^5:\OD_{16}$ ($C_2^3$) x 32, $C_2^4.C_2^4.C_2$ ($C_2^3$) x 32, $C_2^4.C_2^4.C_2$ ($C_2^3$) x 32, $C_2^4.(C_2^2\times D_4)$ ($C_2^3$) x 32, $C_2^4.C_2^4.C_2$ ($C_2^3$) x 32, $C_2^6.D_4$ ($C_2^3$) x 32, $C_2^5.C_2^4$ ($C_2^3$) x 32, $C_2^5.C_2^4$ ($C_2^3$) x 16, $C_2^5.C_2^4$ ($C_2^3$) x 16, $C_2^5.C_2^4$ ($D_4$) x 16, $C_2\times C_2^5.D_4$ ($C_2^3$) x 16, $C_2\times C_2^5.D_4$ ($C_2^3$) x 16, $C_2^4.C_2^4.C_2$ ($C_2^3$) x 16, $C_2^5.C_4.C_2^2$ ($C_2^3$) x 16, $C_2^3.C_2^5.C_2$ ($C_2^3$) x 16, $C_2^5.C_2^4$ ($C_2^3$) x 16, $C_2^5.C_2^4$ ($C_2^3$) x 16, $C_2^6.C_2^3$ ($C_2^3$) x 16, $C_2^5.C_2^4$ ($C_2^3$) x 16, $C_2\times C_2^5.C_2^3$ ($C_2^3$) x 12, $C_2^6.C_2^3$ ($C_2^3$) x 8, $C_2\times C_2^5.C_2^3$ ($C_2^3$) x 8, $C_2^5.C_2^4$ ($C_2^3$) x 8, $C_2^5.C_2^4$ ($C_2^3$) x 8, $C_2^5.C_2^4$ ($C_2^3$) x 8, $C_2^5.C_2^4$ ($D_4$) x 8, $C_2\times C_2^4.C_2^3.C_2$ ($C_2^3$) x 8, $C_2\times C_2^4.C_2^3.C_2$ ($C_2^3$) x 8, $C_2\times C_2^4.C_2^3.C_2$ ($C_2^3$) x 8, $C_2\times C_2^4.C_2^3.C_2$ ($C_2^3$) x 8, $C_2\times C_2^4.C_2^3.C_2$ ($C_2^3$) x 8, $C_2\times C_2^4.C_2^3.C_2$ ($C_2^3$) x 8, $C_2\times C_2^4.C_2^4$ ($C_2^3$) x 4, $C_2\times C_2^4.C_2^4$ ($C_2^3$) x 4, $C_2\times C_2^4.C_2^4$ ($D_4$) x 4, $C_2\times C_2^4.C_2^4$ ($D_4$) x 4, $C_2\times C_2^4.C_2^4$ ($C_2^3$) x 4, $C_2^4.C_2^5$ ($C_2^3$) x 4, $C_2\times C_2^3.C_2^4.C_2$ ($C_2^3$) x 4, $C_2\times C_2^3.C_2^4.C_2$ ($C_2^3$) x 4, $C_2\times C_2^3.C_2^4.C_2$ ($C_2^3$) x 4, $C_2\times C_2^5.C_2^3$ ($C_2^3$) x 4, $C_2\times C_2^5.C_2^3$ ($C_2^3$) x 4, $C_2\times C_2^5.C_2^3$ ($C_2^3$) x 4, $C_2^5.C_2^4$ ($C_2^3$) x 4, $C_2\times C_2^5.C_2^3$ ($C_2^3$) x 2, $C_2\times C_2^5.C_2^3$ ($C_2^3$) x 2, $C_2\times C_2^6.C_2^2$ ($C_2^3$) x 2, $C_2\times C_2^5.C_2^3$ ($C_2^3$) x 2, $C_2\times C_2^5.C_2^3$ ($D_4$) x 2, $C_2^4.C_2^5$ ($C_2^3$) x 2, $C_2^4.C_2^5$ ($D_4$) x 2, $C_2\times C_2^5.C_2^3$ ($C_2^3$) x 2, $C_2\times C_2^5.C_2^3$ ($D_4$) x 2, $C_2^4.C_2^5$ ($C_2^3$) x 2, $C_2^4.C_2^5$ ($D_4$) x 2, $C_2^2\times C_2^4.C_2^3$ ($D_4$) x 2, $C_2^2\times C_2^4.C_2^3$ ($D_4$) x 2, $C_2^2\times C_2^5.C_2^2$ ($C_2^3$), $C_2^3.C_2^6$ ($C_2^3$), $C_2^3.C_2^6$ ($D_4$), $C_2^3.C_2^6$ ($C_2^3$), $C_2\times C_2^5.C_2^3$ ($C_2^3$), $C_2\times C_2^5.C_2^3$ ($D_4$), $C_2^4.C_2^5$ ($C_2^3$), $C_2^4.C_2^5$ ($D_4$), $C_4^2:C_2^5$ ($D_4$)

Order 256: $C_2^4.Q_{16}$ ($C_2^4$) x 64, $C_2^4.D_8$ ($C_2^4$) x 64, $C_2^4.C_2^4$ ($C_2\times D_4$) x 48, $C_2^4:Q_{16}$ ($C_2^4$) x 32, $C_2^4:\SD_{16}$ ($C_2^4$) x 32, $C_2^4:\SD_{16}$ ($C_2^4$) x 32, $C_2^4:D_8$ ($C_2^4$) x 32, $C_2^4.\OD_{16}$ ($C_2^4$) x 32, $C_2^5:C_8$ ($C_2^4$) x 32, $C_2^4.C_2^4$ ($C_2^4$) x 32, $C_2^4.C_2^4$ ($C_2\times D_4$) x 32, $C_2^5.D_4$ ($C_2\times D_4$) x 30, $C_4^2:C_2^4$ ($C_2\times D_4$) x 24, $C_2^4.C_2^4$ ($C_2\times D_4$) x 24, $C_2^5:D_4$ ($C_2\times D_4$) x 24, $C_2^4.C_2^4$ ($C_2\times D_4$) x 24, $C_2^5.D_4$ ($C_2\times D_4$) x 24, $C_2^5.D_4$ ($C_2\times D_4$) x 20, $C_2^4:\OD_{16}$ ($C_2^4$) x 16, $C_2^4:\OD_{16}$ ($C_2^4$) x 16, $C_2^4.\OD_{16}$ ($C_2^4$) x 16, $C_2^3.C_2^5$ ($C_2^4$) x 16, $C_2^5.C_2^3$ ($C_2^4$) x 16, $C_2^5:Q_8$ ($C_2^4$) x 16, $C_2^5.D_4$ ($C_2^4$) x 16, $C_2^5:D_4$ ($C_2^4$) x 16, $C_2^5:Q_8$ ($C_2^4$) x 16, $C_2^4.C_2^4$ ($C_2^4$) x 16, $C_2^4.C_2^4$ ($C_2^4$) x 16, $C_2^4.C_2^4$ ($C_2\times D_4$) x 16, $C_2^5:D_4$ ($C_2^4$) x 16, $C_2^4.C_2^4$ ($C_2\times D_4$) x 12, $C_2^4.C_2^4$ ($C_2\times D_4$) x 12, $C_4^2:C_2^4$ ($C_2\times D_4$) x 8, $C_2^5:C_8$ ($C_2^4$) x 8, $C_2^3.C_2^5$ ($C_2\times D_4$) x 8, $C_2^5.C_2^3$ ($C_2^4$) x 8, $C_2^5.C_2^3$ ($C_2^4$) x 8, $C_2^4.C_2^4$ ($C_2^4$) x 8, $C_2^4.C_2^4$ ($C_2\times D_4$) x 8, $C_2^5:D_4$ ($C_2^4$) x 8, $C_2^5.D_4$ ($C_2\times D_4$) x 8, $C_2^4.C_2^4$ ($C_2^4$) x 8, $C_2^5.D_4$ ($C_2^4$) x 8, $C_2^4.C_4^2$ ($C_2^4$) x 8, $C_2^5:Q_8$ ($C_2^4$) x 6, $C_2^5:D_4$ ($C_2^4$) x 6, $C_2^5.D_4$ ($C_2^4$) x 6, $C_2^6:C_4$ ($C_2\times D_4$) x 6, $C_2^5:D_4$ ($C_2\times D_4$) x 4, $C_2^5:Q_8$ ($C_2^4$) x 4, $C_2^5:Q_8$ ($C_2\times D_4$) x 4, $C_2^5:D_4$ ($C_2^4$) x 4, $C_2^5:D_4$ ($C_2\times D_4$) x 4, $C_2^4.C_2^4$ ($C_2^4$) x 4, $C_2^4.C_2^4$ ($C_2^4$) x 4, $C_2^4.C_4^2$ ($C_2\times D_4$) x 4, $C_2^4.C_4^2$ ($C_2^4$) x 4, $C_2^6:C_4$ ($C_2\times D_4$) x 3, $C_2^5:Q_8$ ($C_2\times D_4$) x 2, $C_2^5:D_4$ ($C_2\times D_4$) x 2, $C_2^6:C_4$ ($C_2\times D_4$) x 2, $C_2^3.C_2^5$ ($C_2\times D_4$) x 2, $C_2^5.D_4$ ($C_2\times D_4$) x 2, $C_2^5.D_4$ ($C_2^4$) x 2, $C_2^6:C_4$ ($C_2^4$) x 2, $D_4:C_2^5$ ($C_2\times D_4$), $D_4\times C_2^5$ ($C_2\times D_4$), $C_4^2:C_2^4$ ($C_2\times D_4$), $C_2^6:C_4$ ($C_2^4$)

Order 128: $C_2^4.D_4$ ($C_2^2\times D_4$) x 60, $C_2^4.D_4$ ($C_2^2\times D_4$) x 44, $C_2^4.C_2^3$ ($C_2^2\times D_4$) x 32, $C_2^4.D_4$ ($C_2^2\times D_4$) x 32, $C_4^2:C_2^3$ ($C_2^2\times D_4$) x 32, $C_2^5:C_4$ ($C_2^2\times D_4$) x 28, $C_2^5:C_4$ ($C_2^2\times D_4$) x 27, $C_2^4:Q_8$ ($C_2^2\times D_4$) x 24, $C_2^4:D_4$ ($C_2^2\times D_4$) x 24, $C_2^3.C_4^2$ ($C_2^2\times D_4$) x 24, $C_2^4.D_4$ ($C_2^2\wr C_2$) x 24, $C_2^4:D_4$ ($C_2^2\times D_4$) x 20, $C_2^4:C_8$ ($C_2^5$) x 16, $C_2^5:C_4$ ($C_2^2\times D_4$) x 16, $C_2^5:C_4$ ($C_2^2\wr C_2$) x 14, $C_2^4:Q_8$ ($C_2^2\times D_4$) x 12, $C_2^4.D_4$ ($C_2^2\times D_4$) x 12, $C_2^4.D_4$ ($C_2^2\wr C_2$) x 12, $C_2^4:Q_8$ ($C_2^5$) x 8, $C_2^4:D_4$ ($C_2^5$) x 8, $C_2^3.C_4^2$ ($C_2^5$) x 8, $D_4\times C_2^4$ ($C_2^2\times D_4$) x 8, $C_2^4:D_4$ ($C_2^2\times D_4$) x 8, $C_2^3.C_2^4$ ($C_2^2\times D_4$) x 8, $D_4\times C_2^4$ ($C_2^2\wr C_2$) x 6, $C_2^5:C_4$ ($C_2^2\wr C_2$) x 6, $C_2^4.D_4$ ($C_2^5$) x 4, $C_2^5:C_4$ ($C_2^5$) x 4, $D_4:C_2^4$ ($C_2^2\times D_4$) x 4, $C_4^2:C_2^3$ ($C_2^2\times D_4$) x 4, $C_2^4:Q_8$ ($C_2^5$) x 4, $C_2^4:D_4$ ($C_2^5$) x 4, $C_2^4.D_4$ ($C_2^5$) x 4, $C_2^5:C_4$ ($C_2^5$) x 3, $C_2^7$ ($C_2^2\times D_4$), $C_2^7$ ($C_2^2\wr C_2$), $C_2^5\times C_4$ ($C_2^2\wr C_2$)

Order 64: $C_2^4:C_4$ ($C_2^3:D_4$) x 42, $C_2^3.D_4$ ($C_2^3:D_4$) x 24, $C_2^4:C_4$ ($D_4\times C_2^3$) x 21, $C_2^4:C_4$ ($C_2^3:D_4$) x 18, $C_2^4:C_4$ ($D_4\times C_2^3$) x 18, $C_2^2.C_4^2$ ($D_4\times C_2^3$) x 16, $D_4\times C_2^3$ ($C_2^3:D_4$) x 12, $C_2^6$ ($C_2^3:D_4$) x 10, $C_2^3.D_4$ ($D_4\times C_2^3$) x 8, $C_2^6$ ($C_2\wr C_2^2$) x 8, $D_4\times C_2^3$ ($D_4\times C_2^3$) x 8, $C_2^3:Q_8$ ($D_4\times C_2^3$) x 8, $C_2^3:D_4$ ($D_4\times C_2^3$) x 8, $C_2^4\times C_4$ ($C_2^3:D_4$) x 6, $D_4\times C_2^3$ ($D_8:C_2^2$) x 4, $C_2^6$ ($D_4\times C_2^3$) x 3, $C_2^6$ ($D_8:C_2^2$) x 2, $C_2^4\times C_4$ ($D_8:C_2^2$) x 2, $C_2^4:C_4$ ($C_2^6$)

Order 32: $C_2^5$ ($C_2^4:D_4$) x 28, $C_2^3:C_4$ ($C_2^4:D_4$) x 14, $C_2^5$ ($C_2^4:D_4$) x 10, $C_2^5$ ($D_8:C_2^3$) x 8, $C_2^5$ ($C_4^2:C_2^3$) x 4, $C_2^3\times C_4$ ($D_8:C_2^3$) x 4, $C_2^3\times C_4$ ($C_2^4:D_4$) x 4, $C_2^3:C_4$ ($D_4\times C_2^4$) x 2, $C_2^5$ ($D_4\times C_2^4$)

Order 16: $C_2^4$ ($C_2^5:D_4$) x 14, $C_2^4$ ($D_4^2:C_2^2$) x 8, $C_2^4$ ($C_4^2:C_2^4$) x 6, $C_2^4$ ($C_2^5.D_4$) x 4, $C_2^4$ ($D_8:C_2^4$) x 2, $C_2^4$ ($C_2^5:D_4$)

Order 8: $C_2^3$ ($D_4^2:C_2^3$) x 12, $C_2^3$ ($C_2^3.C_2^5.C_2$) x 4, $C_2^3$ ($C_2^6:D_4$), $C_2^3$ ($C_2\times C_2.C_2^6.C_2$), $C_2^3$ ($C_2^2\times (C_2\times C_4).C_2^4$)

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

Order 2: $C_2$ ($C_2^5.C_2^5.C_2$) x 2, $C_2$ ($(D_4\times C_2^5).D_4$)

Order 1: $C_1$ ($C_2^5.(D_4\times C_2^4)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

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

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

Character theory

Complex character table

The $172 \times 172$ character table is not available for this group.

Rational character table

The $164 \times 164$ rational character table is not available for this group.