Abstract group 16384.mu: $C_2^7.C_2\wr D_4$

• Label: 16384.mu
• Order: \( 2^{14} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{19} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$C_2^7.C_2\wr D_4$
Order:	\(16384\)\(\medspace = 2^{14} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2\times C_2^9.C_2^6.C_2^3$, of order \(524288\)\(\medspace = 2^{19} \)
Composition factors:	$C_2$ x 14
Nilpotency class:	$7$
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	767	9472	6144	16384
Conjugacy classes	1	37	91	22	151
Divisions	1	37	81	16	135
Autjugacy classes	1	29	54	6	90

Minimal presentations

Permutation degree:	not computed
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	64	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid d^{4}=e^{2}=f^{4}=g^{4}=h^{2}=i^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $128$ $\langle(1,112,91,19)(2,52,104,24)(3,67,70,44)(4,102,84,88)(5,98,38,59)(6,63,115,66) \!\cdots\! \rangle$
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:	$C_2^7$ . $(C_2\wr D_4)$	$(C_2^6:C_4)$ . $D_4^2$ (2)	$C_2^7$ . $(C_2^4:D_4)$	$C_2^7$ . $(C_4^2:D_4)$	all 143
Aut. group:	$\Aut(C_2^6.\OD_{16})$

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:	$2$

Subgroups

There are 377 normal subgroups (211 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^5.C_2^5.C_2^3$
Commutator:	$G' \simeq$ $C_2^5.C_2^5$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^5.C_2^5$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^7.C_2\wr D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^7.C_2\wr D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2^5.C_2^5.C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2\wr D_4$

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 8 (order at least 2048) are shown, as well as all normal subgroups of any index.

Order 16384: $C_2^7.C_2\wr D_4$

Order 8192: $C_2^4.C_2^4.C_2^2:D_4$ x 2, $C_2^4.C_2^3.C_2^4.C_2^2$ x 2, $C_2^4.C_2^4.C_2^2:D_4$ x 2, $C_2^4.C_2^4.C_2^2:D_4$ x 2, $C_2^5.C_2^5.C_2^3$, $C_2^4.C_2^3.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^4$, $C_2^4.C_2^4.C_2^2:D_4$, $C_2^5.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^4$, $C_2^4.C_2^3.C_2.C_2^5$

Order 4096: $C_2^4.C_2^3.C_2.C_2^4$ x 4, $C_2^4.C_2^3.C_2.C_2^4$ x 4, $C_2^4.C_2^3.C_2^2:D_4$ x 4, $C_2^4.C_2^3.C_2.C_2^4$ x 4, $C_2^4.C_2^3.C_2^2:D_4$ x 4, $C_2^4.C_2^3.C_2^3.C_2^2$ x 4, $C_2^4.C_2^5.C_2^3$ x 2, $C_2^4.C_2^3.C_2^3.C_2^2$ x 2, $C_2^4.C_2^4.C_2^4$ x 2, $C_2^5.C_2^4.C_2^3$ x 2, $C_2^4.C_2^5.C_2^3$ x 2, $C_2^4.C_2^3.C_2.C_2^4$ x 2, $C_2^4.C_2^6.C_2^2$ x 2, $C_2^5.C_2^4.C_2^3$ x 2, $C_2^4.C_2^3.C_2^4.C_2$ x 2, $C_2^5.C_2^5.C_2^2$ x 2, $C_2^4.C_2^6.C_2^2$ x 2, $C_2^5.C_2^4.C_2^3$ x 2, $C_2^4.C_2^4.C_2^4$ x 2, $C_2^5.C_2^5.C_2^2$ x 2, $C_2^4.C_2^3.C_2.C_2^4$ x 2, $C_2^4.C_2^3.C_2^2:D_4$ x 2, $C_2.C_2^6.C_2^5$ x 2, $C_2^4.C_2^5.C_2^3$ x 2, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^5.C_2^2$, $C_2^4.C_2^4.C_2^4$, $C_2.C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^5.C_2^2$, $C_2^4.C_2^3.C_2^2:D_4$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^5.C_2^2$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^5.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^5.C_2^2$, $C_2.C_2^6.C_2^5$, $C_2.C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^4$, $C_2.C_2^6.C_2^5$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^3.C_2^2:D_4$, $C_2.C_2^6.C_2^5$, $C_2^4.C_2^4.C_2^4$, $C_2^6.C_2^6$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^4.C_2^3$, $C_2.C_2^6.C_2^5$

Order 2048: $C_2^4.C_2^3:C_8.C_2$ x 8, $C_2^4.C_2^3:C_8.C_2$ x 8, $C_2^4.C_2^3.C_2.C_2^3$ x 8, $C_2^4.C_2^2:\OD_{16}.C_2$ x 8, $C_2^5.C_2^4.C_2^2$ x 6, $C_2^4.C_2^3.C_2^3.C_2$ x 6, $C_2^4.C_2^2:\OD_{16}.C_2$ x 6, $C_2^5.C_2^4.C_2^2$ x 6, $C_2^5.C_2^4.C_2^2$ x 6, $C_2^4.C_2^2:\OD_{16}.C_2$ x 6, $C_2^5.C_2^4.C_2^2$ x 6, $C_2.C_2^6.C_2^4$ x 4, $C_2^4.C_2^5.C_2^2$ x 4, $C_2.C_2^6.C_2^4$ x 4, $C_2.C_2^6.C_2^4$ x 4, $C_2^4.C_2^3.C_2^4$ x 4, $C_2.C_2^6.C_2^4$ x 4, $C_2^4.C_2^3.C_2^4$ x 4, $C_2.C_2^6.C_2^4$ x 4, $C_2^4.C_2^3:C_8.C_2$ x 4, $C_2.C_2^6.C_2^4$ x 4, $C_2^4.C_2^4.C_2^3$ x 4, $C_2^4.C_2^3:C_8.C_2$ x 4, $C_2.C_2^6.C_2^4$ x 4, $C_2^4.C_2^4.C_2^3$ x 4, $C_2^4.C_2^3:C_8.C_2$ x 4, $C_2^4.C_2^3.C_2.Q_8$ x 4, $C_2^4.C_2^3:C_8.C_2$ x 4, $C_2^4.C_2^3.C_2^4$ x 4, $C_2^4.C_2^3.C_2^4$ x 4, $C_2^4.C_2^4.C_2^3$ x 3, $C_2^5.C_2^4.C_2^2$ x 3, $C_2^5.C_2^4.C_2^2$ x 3, $C_2^5.C_2^4.C_2^2$ x 3, $C_2^5.C_2^4.C_2^2$ x 3, $C_2^5.C_2^4.C_2^2$ x 3, $C_2^5.C_2^4.C_2^2$ x 3, $C_2^4.C_2^4.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^3.C_2.Q_8$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^6.C_2.C_2^4$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^3.C_2^3.C_2$ x 2, $C_2^4.C_2^3.C_2.Q_8$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^3.C_2$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^4.C_2^3.C_2^3.C_2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2^3.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2.C_2^5$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2^3.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^3.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2^3.C_2^3$, $C_2^5.C_2^4.C_2^2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^6$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^5.C_2^6$, $C_2^5.C_2^3.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^3.C_2^3$, $C_2^5.C_2^3.C_2^3$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^6$

Order 1024: $C_2^4.C_2^3.C_2^3$ x 4, $C_2^5.C_2^4.C_2$ x 4, $C_2^5.C_2^5$ x 4, $C_2^5.C_2^4.C_2$ x 4, $C_2^5.C_2^4.C_2$ x 4, $C_2^4.C_2^3.C_2^3$ x 4, $C_2^5.C_2^5$ x 4, $C_2^4.C_2^3.C_2^3$ x 2, $C_2^5.C_2^5$ x 2, $C_2^4.C_2^6$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^5.C_2^5$ x 2, $C_2^4.C_2^3.C_2^3$ x 2, $C_2^5.C_2^5$ x 2, $C_2^5.C_2^5$ x 2, $C_2^5.C_2^5$, $C_2^4.C_2^6$, $C_2^6.C_2^4$, $C_2^6.C_2^4$, $C_2^5.C_2^5$, $C_2^5.C_2^5$, $C_2^5.C_2^5$, $C_2^6.C_2^4$, $C_2^5.C_2^4.C_2$, $C_2.C_2^6.C_2^3$, $C_2^5.C_2^5$

Order 512: $C_2\times C_2^5.C_2^3$ x 8, $C_2^5.C_2^4$ x 8, $C_2\times C_2^5.C_2^3$ x 4, $C_2^5.C_2^4$ x 4, $C_2\times C_2^4.C_2^4$ x 4, $C_2^4.C_2^5$ x 4, $C_2^4.C_2^5$ x 4, $C_2^5.C_2^4$ x 4, $C_2^4.C_2^2:C_8$ x 4, $C_2^4.C_2^5$ x 2, $C_2^4.C_2^5$ x 2, $C_2\times C_2^5.C_2^3$ x 2, $C_2^3.C_2^6$ x 2, $C_2^3.C_2^6$, $D_4.C_2^6$, $C_2^3.C_2^6$

Order 256: $C_2^5.C_2^3$ x 8, $C_2^6:C_4$ x 8, $C_2^6:C_4$ x 8, $D_4:C_2^5$ x 4, $D_4\times C_2^5$ x 4, $C_2^5:C_2^3$ x 4, $C_2^4.C_2^4$ x 4, $C_2^4.C_2^4$ x 4, $C_2^5.D_4$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^6:C_4$

Order 128: $D_4\times C_2^4$ x 12, $C_2^5:C_4$ x 8, $C_2^7$ x 5, $D_4:C_2^4$ x 4, $C_2^5\times C_4$ x 4, $C_2^4:D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^5:C_4$ x 2

Order 64: $C_2^6$ x 15, $D_4\times C_2^3$ x 12, $C_2^4\times C_4$ x 6, $C_2^4:C_4$ x 2

Order 32: $C_2^5$ x 17, $C_2^3\times C_4$ x 6

Order 16: $C_2^4$ x 9

Order 8: $C_2^3$ x 3

Order 4: $C_2^2$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 8 (order at least 2048) are shown, as well as all normal subgroups of any index.

Order 16384: $C_2^7.C_2\wr D_4$

Order 8192: $C_2^5.C_2^5.C_2^3$, $C_2^4.C_2^4.C_2^2:D_4$, $C_2^4.C_2^3.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^4$, $C_2^4.C_2^4.C_2^2:D_4$, $C_2^4.C_2^3.C_2^4.C_2^2$, $C_2^4.C_2^4.C_2^2:D_4$, $C_2^5.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^4$, $C_2^4.C_2^4.C_2^2:D_4$, $C_2^4.C_2^3.C_2.C_2^5$

Order 4096: $C_2^4.C_2^3.C_2.C_2^4$ x 2, $C_2^4.C_2^3.C_2^3.C_2^2$ x 2, $C_2^4.C_2^3.C_2.C_2^4$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^3.C_2^3.C_2^2$, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^5.C_2^2$, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^5.C_2^2$, $C_2^4.C_2^4.C_2^4$, $C_2.C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^3.C_2^2:D_4$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^3.C_2.C_2^4$, $C_2^4.C_2^3.C_2.C_2^4$, $C_2^5.C_2^5.C_2^2$, $C_2^4.C_2^6.C_2^2$, $C_2^4.C_2^3.C_2^2:D_4$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^3.C_2^4.C_2$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^5.C_2^2$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^3.C_2^2:D_4$, $C_2^5.C_2^5.C_2^2$, $C_2^4.C_2^6.C_2^2$, $C_2^4.C_2^5.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^5.C_2^2$, $C_2.C_2^6.C_2^5$, $C_2.C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^4.C_2^4$, $C_2.C_2^6.C_2^5$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^3.C_2^2:D_4$, $C_2^5.C_2^5.C_2^2$, $C_2.C_2^6.C_2^5$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^3.C_2.C_2^4$, $C_2^6.C_2^6$, $C_2^4.C_2^3.C_2^2:D_4$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^4.C_2^4$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^4.C_2^3$, $C_2.C_2^6.C_2^5$, $C_2^4.C_2^5.C_2^3$, $C_2.C_2^6.C_2^5$

Order 2048: $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^3.C_2^3.C_2$ x 2, $C_2^4.C_2^2:\OD_{16}.C_2$ x 2, $C_2^4.C_2^3.C_2.C_2^3$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^2:\OD_{16}.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^4.C_2^2:\OD_{16}.C_2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^4.C_2^3.C_2^4$ x 2, $C_2^4.C_2^4.C_2^3$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2.C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^3:C_8.C_2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^3.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2.C_2^5$, $C_2^5.C_2^3.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^5.C_2^3.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^3:C_8.C_2$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^5.C_2^2$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2^3.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^3.C_2.Q_8$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^4.C_2^3:C_8.C_2$, $C_2^4.C_2^4.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^5.C_2^2$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2^3.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^3.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2^3.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^5.C_2^2$, $C_2^5.C_2^3.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^3:C_8.C_2$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^3.C_2^3.C_2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^3.C_2.Q_8$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^3:C_8.C_2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^3.C_2.Q_8$, $C_2.C_2^6.C_2^3.C_2$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^3.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^3.C_2^3$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^5.C_2^3.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^4.C_2^3:C_8.C_2$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^6$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^5.C_2^6$, $C_2^5.C_2^3.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^3.C_2^3$, $C_2^5.C_2^3.C_2^3$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^3.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^6$, $C_2^4.C_2^5.C_2^2$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^3.C_2^3.C_2$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^6$

Order 1024: $C_2^5.C_2^5$ x 4, $C_2^5.C_2^5$ x 4, $C_2^4.C_2^3.C_2^3$ x 2, $C_2^5.C_2^5$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^4.C_2^3.C_2^3$ x 2, $C_2^5.C_2^5$ x 2, $C_2^4.C_2^3.C_2^3$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^5$, $C_2^4.C_2^6$, $C_2^5.C_2^4.C_2$, $C_2^6.C_2^4$, $C_2^6.C_2^4$, $C_2^4.C_2^6$, $C_2^5.C_2^5$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^5$, $C_2^5.C_2^5$, $C_2^5.C_2^5$, $C_2^6.C_2^4$, $C_2^5.C_2^4.C_2$, $C_2.C_2^6.C_2^3$, $C_2^4.C_2^3.C_2^3$, $C_2^5.C_2^5$, $C_2^5.C_2^5$

Order 512: $C_2\times C_2^5.C_2^3$ x 8, $C_2\times C_2^5.C_2^3$ x 4, $C_2^5.C_2^4$ x 4, $C_2\times C_2^4.C_2^4$ x 4, $C_2^4.C_2^5$ x 4, $C_2^4.C_2^5$ x 4, $C_2^5.C_2^4$ x 4, $C_2\times C_2^5.C_2^3$ x 2, $C_2^3.C_2^6$ x 2, $C_2^5.C_2^4$ x 2, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $C_2^3.C_2^6$, $C_2^4.C_2^2:C_8$, $D_4.C_2^6$, $C_2^3.C_2^6$

Order 256: $C_2^6:C_4$ x 8, $C_2^5.C_2^3$ x 6, $D_4:C_2^5$ x 4, $D_4\times C_2^5$ x 4, $C_2^6:C_4$ x 4, $C_2^5.D_4$ x 2, $C_2^5:D_4$ x 2, $C_2^5:C_2^3$ x 2, $C_2^5.D_4$ x 2, $C_2^4.C_2^4$ x 2, $C_2^4.C_2^4$ x 2, $C_2^6:C_4$

Order 128: $D_4\times C_2^4$ x 12, $C_2^5:C_4$ x 6, $C_2^7$ x 5, $C_2^5\times C_4$ x 4, $C_2^4.D_4$ x 4, $D_4:C_2^4$ x 2, $C_2^4:D_4$ x 2, $C_2^5:C_4$ x 2

Order 64: $C_2^6$ x 14, $D_4\times C_2^3$ x 8, $C_2^4\times C_4$ x 6, $C_2^4:C_4$

Order 32: $C_2^5$ x 15, $C_2^3\times C_4$ x 4

Order 16: $C_2^4$ x 9

Order 8: $C_2^3$ x 3

Order 4: $C_2^2$ x 3

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 16384: $C_2^7.C_2\wr D_4$ ($C_1$)

Order 8192: $C_2^4.C_2^4.C_2^2:D_4$ ($C_2$) x 2, $C_2^4.C_2^3.C_2^4.C_2^2$ ($C_2$) x 2, $C_2^4.C_2^4.C_2^2:D_4$ ($C_2$) x 2, $C_2^4.C_2^4.C_2^2:D_4$ ($C_2$) x 2, $C_2^5.C_2^5.C_2^3$ ($C_2$), $C_2^4.C_2^3.C_2^4.C_2^2$ ($C_2$), $C_2^4.C_2^5.C_2^4$ ($C_2$), $C_2^4.C_2^4.C_2^2:D_4$ ($C_2$), $C_2^5.C_2^5.C_2^3$ ($C_2$), $C_2^4.C_2^5.C_2^4$ ($C_2$), $C_2^4.C_2^3.C_2.C_2^5$ ($C_2$)

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

Order 2048: $C_2^4.C_2^2:\OD_{16}.C_2$ ($D_4$) x 4, $C_2^4.C_2^2:\OD_{16}.C_2$ ($D_4$) x 4, $C_2^4.C_2^4.C_2^3$ ($D_4$) x 2, $C_2^5.C_2^4.C_2^2$ ($D_4$) x 2, $C_2^4.C_2^6.C_2$ ($D_4$) x 2, $C_2^4.C_2^4.C_2^3$ ($C_2^3$) x 2, $C_2^5.C_2^5.C_2$ ($C_2^3$) x 2, $C_2.C_2^6.C_2^4$ ($C_2^3$) x 2, $C_2^4.C_2^4.C_2^3$ ($C_2^3$) x 2, $C_2^5.C_2^5.C_2$ ($C_2^3$) x 2, $C_2^5.C_2^4.C_2^2$ ($D_4$) x 2, $C_2.C_2^6.C_2^4$ ($D_4$) x 2, $C_2^5.C_2^5.C_2$ ($D_4$), $C_2.C_2^6.C_2^4$ ($D_4$), $C_2^6.C_2^5$ ($D_4$), $C_2.C_2^6.C_2^4$ ($D_4$), $C_2^4.C_2^4.C_2^3$ ($D_4$), $C_2^6.C_2^5$ ($D_4$), $C_2^6.C_2^5$ ($D_4$), $C_2^6.C_2^5$ ($D_4$), $C_2^6.C_2^5$ ($D_4$), $C_2.C_2^6.C_2^4$ ($D_4$), $C_2^6.C_2^5$ ($D_4$), $C_2^4.C_2^4.C_2^3$ ($C_2^3$), $C_2.C_2^6.C_2^4$ ($C_2^3$), $C_2^4.C_2^4.C_2^3$ ($D_4$), $C_2^5.C_2^6$ ($C_2^3$), $C_2^5.C_2^6$ ($C_2^3$), $C_2^4.C_2^6.C_2$ ($D_4$), $C_2^6.C_2^5$ ($C_2^3$), $C_2^5.C_2^6$ ($D_4$)

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

Order 512: $C_2\times C_2^5.C_2^3$ ($C_4:D_4$) x 4, $C_2^5.C_2^4$ ($C_4:D_4$) x 4, $C_2^4.C_2^2:C_8$ ($C_2^2\wr C_2$) x 4, $C_2^5.C_2^4$ ($D_4:C_2^2$) x 4, $C_2\times C_2^5.C_2^3$ ($C_4:D_4$) x 2, $C_2^5.C_2^4$ ($C_2^2\times D_4$) x 2, $C_2^5.C_2^4$ ($C_2^2\wr C_2$) x 2, $C_2^4.C_2^5$ ($C_4:D_4$) x 2, $C_2^4.C_2^5$ ($C_4:D_4$) x 2, $C_2\times C_2^4.C_2^4$ ($C_2^2\wr C_2$) x 2, $C_2^4.C_2^5$ ($C_4:D_4$) x 2, $C_2^5.C_2^4$ ($D_4:C_2^2$) x 2, $C_2^5.C_2^4$ ($C_2^2\times D_4$) x 2, $C_2\times C_2^5.C_2^3$ ($D_4:C_2^2$), $C_2\times C_2^5.C_2^3$ ($C_2^2\times D_4$), $C_2\times C_2^5.C_2^3$ ($C_4:D_4$), $C_2\times C_2^5.C_2^3$ ($C_2^2\wr C_2$), $C_2\times C_2^5.C_2^3$ ($C_4:D_4$), $C_2\times C_2^5.C_2^3$ ($C_2^2\wr C_2$), $C_2\times C_2^5.C_2^3$ ($C_4:D_4$), $C_2\times C_2^5.C_2^3$ ($C_2^2\wr C_2$), $C_2^3.C_2^6$ ($C_2^2\times D_4$), $C_2^3.C_2^6$ ($C_2^2\wr C_2$), $C_2^3.C_2^6$ ($C_4:D_4$), $C_2\times C_2^4.C_2^4$ ($C_2^2\times D_4$), $C_2\times C_2^4.C_2^4$ ($C_4:D_4$), $C_2^4.C_2^5$ ($C_2^2\times D_4$), $C_2^4.C_2^5$ ($C_4:D_4$), $C_2^4.C_2^5$ ($C_4:D_4$), $C_2^4.C_2^5$ ($C_2^2\wr C_2$), $C_2^4.C_2^5$ ($C_4:D_4$), $C_2^4.C_2^5$ ($C_2^2\wr C_2$), $D_4.C_2^6$ ($C_2^2\wr C_2$), $C_2^3.C_2^6$ ($C_2^2\wr C_2$)

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

Order 128: $C_2^5:C_4$ ($C_2.D_4^2$) x 4, $D_4\times C_2^4$ ($C_2^4:D_4$) x 3, $D_4:C_2^4$ ($C_2^4:D_4$) x 2, $D_4:C_2^4$ ($C_2^4:D_4$) x 2, $D_4\times C_2^4$ ($C_2\wr D_4$) x 2, $D_4\times C_2^4$ ($C_2^4:D_4$) x 2, $C_2^4:D_4$ ($C_4^2:C_2^3$) x 2, $C_2^4:D_4$ ($C_2^4:D_4$) x 2, $C_2^4.D_4$ ($C_4^2:D_4$) x 2, $C_2^5:C_4$ ($C_2^3.C_2^4$), $C_2^5:C_4$ ($C_2.D_4^2$), $C_2^5:C_4$ ($C_2^3.C_2^4$), $C_2^5:C_4$ ($C_2^4:D_4$), $C_2^7$ ($C_4^2:D_4$), $C_2^7$ ($C_2\wr D_4$), $C_2^7$ ($C_2^4:D_4$), $C_2^7$ ($C_4^2:D_4$), $C_2^7$ ($C_4^2:C_2^3$), $D_4\times C_2^4$ ($C_4^2:D_4$), $D_4\times C_2^4$ ($C_4^2:D_4$), $D_4\times C_2^4$ ($C_4^2:D_4$), $D_4\times C_2^4$ ($C_4^2:D_4$), $D_4\times C_2^4$ ($C_2^4:D_4$), $C_2^5\times C_4$ ($C_4^2:D_4$), $C_2^5\times C_4$ ($C_2\wr D_4$), $C_2^5\times C_4$ ($C_4^2:D_4$), $C_2^5\times C_4$ ($C_4^2:D_4$), $C_2^5:C_4$ ($C_4^2:D_4$), $C_2^5:C_4$ ($C_4^2:C_2^3$), $C_2^4.D_4$ ($C_2^4:D_4$), $C_2^4.D_4$ ($C_2^4:D_4$)

Order 64: $D_4\times C_2^3$ ($D_4^2:C_2^2$) x 4, $D_4\times C_2^3$ ($C_2^5:D_4$) x 4, $C_2^6$ ($C_2^5.D_4$) x 3, $C_2^6$ ($C_2^5:D_4$) x 3, $C_2^6$ ($C_2^5.D_4$) x 3, $C_2^6$ ($C_4.D_4^2$) x 2, $C_2^4:C_4$ ($C_4.D_4^2$) x 2, $C_2^6$ ($C_2^5:D_4$), $C_2^6$ ($D_4^2:C_2^2$), $C_2^6$ ($C_2^5:D_4$), $C_2^6$ ($C_2^5.D_4$), $D_4\times C_2^3$ ($C_2^5:D_4$), $D_4\times C_2^3$ ($C_2^5:D_4$), $D_4\times C_2^3$ ($C_2^5.D_4$), $D_4\times C_2^3$ ($C_2^5.D_4$), $C_2^4\times C_4$ ($C_2^5.D_4$), $C_2^4\times C_4$ ($C_2^5:D_4$), $C_2^4\times C_4$ ($C_2^5.D_4$), $C_2^4\times C_4$ ($C_2^5.D_4$), $C_2^4\times C_4$ ($D_4^2:C_2^2$), $C_2^4\times C_4$ ($C_2^5.D_4$)

Order 32: $C_2^5$ ($C_2^6.D_4$) x 4, $C_2^5$ ($C_2^4.C_2^4.C_2$) x 2, $C_2^5$ ($C_2^6:D_4$) x 2, $C_2^3\times C_4$ ($C_2^4.C_2^4.C_2$) x 2, $C_2^3\times C_4$ ($C_2^6:D_4$) x 2, $C_2^5$ ($C_2^4.C_2^4.C_2$), $C_2^5$ ($C_2^6.D_4$), $C_2^5$ ($(C_2^2\times D_4).C_2^4$), $C_2^5$ ($C_2^6:D_4$), $C_2^5$ ($C_2.C_2^6.C_2^2$), $C_2^5$ ($C_2^6.C_2^3$), $C_2^5$ ($C_2^4.C_2^4.C_2$), $C_2^5$ ($C_2^4.C_2^4.C_2$), $C_2^5$ ($C_2\times C_2^4.C_2^3.C_2$), $C_2^3\times C_4$ ($(C_2^5\times C_4).C_2^2$), $C_2^3\times C_4$ ($C_2^4.C_2^4.C_2$)

Order 16: $C_2^4$ ($C_2^4.C_2^4.C_2^2$) x 2, $C_2^4$ ($C_2^5.C_2^3.C_2^2$), $C_2^4$ ($C_2^4.C_2^4.C_2^2$), $C_2^4$ ($C_2^4.C_2^4.C_2^2$), $C_2^4$ ($C_2^5.C_2^3.C_2^2$), $C_2^4$ ($C_2^5.C_2^3.C_2^2$), $C_2^4$ ($C_2^5.C_2^3.C_2^2$), $C_2^4$ ($C_2^4.C_2^4.C_2^2$)

Order 8: $C_2^3$ ($C_2^4.C_2^4.C_2^3$), $C_2^3$ ($C_2^5.C_2^4.C_2^2$), $C_2^3$ ($C_2^4.C_2^4.C_2^3$)

Order 4: $C_2^2$ ($C_2^5.C_2^4.C_2^3$), $C_2^2$ ($C_2^4.C_2^5.C_2^3$), $C_2^2$ ($C_2^5.C_2^4.C_2^3$)

Order 2: $C_2$ ($C_2^5.C_2^5.C_2^3$)

Order 1: $C_1$ ($C_2^7.C_2\wr D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 16384: $C_2^7.C_2\wr D_4$ ($C_1$)

Order 8192: $C_2^5.C_2^5.C_2^3$ ($C_2$), $C_2^4.C_2^4.C_2^2:D_4$ ($C_2$), $C_2^4.C_2^3.C_2^4.C_2^2$ ($C_2$), $C_2^4.C_2^5.C_2^4$ ($C_2$), $C_2^4.C_2^4.C_2^2:D_4$ ($C_2$), $C_2^4.C_2^3.C_2^4.C_2^2$ ($C_2$), $C_2^4.C_2^4.C_2^2:D_4$ ($C_2$), $C_2^5.C_2^5.C_2^3$ ($C_2$), $C_2^4.C_2^5.C_2^4$ ($C_2$), $C_2^4.C_2^4.C_2^2:D_4$ ($C_2$), $C_2^4.C_2^3.C_2.C_2^5$ ($C_2$)

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

Order 2048: $C_2^5.C_2^5.C_2$ ($C_2^3$) x 2, $C_2^5.C_2^5.C_2$ ($D_4$), $C_2.C_2^6.C_2^4$ ($D_4$), $C_2^4.C_2^4.C_2^3$ ($D_4$), $C_2^6.C_2^5$ ($D_4$), $C_2.C_2^6.C_2^4$ ($D_4$), $C_2^4.C_2^4.C_2^3$ ($D_4$), $C_2^5.C_2^4.C_2^2$ ($D_4$), $C_2^4.C_2^6.C_2$ ($D_4$), $C_2^4.C_2^4.C_2^3$ ($C_2^3$), $C_2^6.C_2^5$ ($D_4$), $C_2.C_2^6.C_2^4$ ($C_2^3$), $C_2^4.C_2^4.C_2^3$ ($C_2^3$), $C_2^6.C_2^5$ ($D_4$), $C_2^5.C_2^5.C_2$ ($C_2^3$), $C_2^6.C_2^5$ ($D_4$), $C_2^4.C_2^2:\OD_{16}.C_2$ ($D_4$), $C_2^6.C_2^5$ ($D_4$), $C_2^5.C_2^4.C_2^2$ ($D_4$), $C_2^4.C_2^2:\OD_{16}.C_2$ ($D_4$), $C_2.C_2^6.C_2^4$ ($D_4$), $C_2^6.C_2^5$ ($D_4$), $C_2^4.C_2^4.C_2^3$ ($C_2^3$), $C_2.C_2^6.C_2^4$ ($C_2^3$), $C_2^4.C_2^4.C_2^3$ ($D_4$), $C_2^5.C_2^6$ ($C_2^3$), $C_2^5.C_2^6$ ($C_2^3$), $C_2^4.C_2^6.C_2$ ($D_4$), $C_2.C_2^6.C_2^4$ ($D_4$), $C_2^6.C_2^5$ ($C_2^3$), $C_2^5.C_2^6$ ($D_4$)

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

Order 512: $C_2\times C_2^5.C_2^3$ ($C_4:D_4$) x 4, $C_2\times C_2^5.C_2^3$ ($C_4:D_4$) x 2, $C_2^5.C_2^4$ ($C_2^2\times D_4$) x 2, $C_2^5.C_2^4$ ($C_2^2\wr C_2$) x 2, $C_2\times C_2^4.C_2^4$ ($C_2^2\wr C_2$) x 2, $C_2^4.C_2^5$ ($C_4:D_4$) x 2, $C_2^5.C_2^4$ ($C_4:D_4$) x 2, $C_2^5.C_2^4$ ($D_4:C_2^2$) x 2, $C_2\times C_2^5.C_2^3$ ($D_4:C_2^2$), $C_2\times C_2^5.C_2^3$ ($C_2^2\times D_4$), $C_2\times C_2^5.C_2^3$ ($C_4:D_4$), $C_2\times C_2^5.C_2^3$ ($C_2^2\wr C_2$), $C_2\times C_2^5.C_2^3$ ($C_4:D_4$), $C_2\times C_2^5.C_2^3$ ($C_2^2\wr C_2$), $C_2^4.C_2^5$ ($C_4:D_4$), $C_2^4.C_2^5$ ($C_4:D_4$), $C_2\times C_2^5.C_2^3$ ($C_4:D_4$), $C_2\times C_2^5.C_2^3$ ($C_2^2\wr C_2$), $C_2^3.C_2^6$ ($C_2^2\times D_4$), $C_2^3.C_2^6$ ($C_2^2\wr C_2$), $C_2^3.C_2^6$ ($C_4:D_4$), $C_2\times C_2^4.C_2^4$ ($C_2^2\times D_4$), $C_2\times C_2^4.C_2^4$ ($C_4:D_4$), $C_2^4.C_2^5$ ($C_2^2\times D_4$), $C_2^4.C_2^5$ ($C_4:D_4$), $C_2^4.C_2^5$ ($C_4:D_4$), $C_2^4.C_2^5$ ($C_2^2\wr C_2$), $C_2^4.C_2^5$ ($C_4:D_4$), $C_2^4.C_2^5$ ($C_2^2\wr C_2$), $C_2^4.C_2^2:C_8$ ($C_2^2\wr C_2$), $C_2^5.C_2^4$ ($D_4:C_2^2$), $C_2^5.C_2^4$ ($C_2^2\times D_4$), $D_4.C_2^6$ ($C_2^2\wr C_2$), $C_2^3.C_2^6$ ($C_2^2\wr C_2$)

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

Order 128: $D_4\times C_2^4$ ($C_2^4:D_4$) x 3, $C_2^5:C_4$ ($C_2.D_4^2$) x 2, $D_4\times C_2^4$ ($C_2\wr D_4$) x 2, $D_4\times C_2^4$ ($C_2^4:D_4$) x 2, $C_2^4.D_4$ ($C_4^2:D_4$) x 2, $C_2^5:C_4$ ($C_2^3.C_2^4$), $C_2^5:C_4$ ($C_2.D_4^2$), $C_2^5:C_4$ ($C_2^3.C_2^4$), $C_2^5:C_4$ ($C_2^4:D_4$), $C_2^7$ ($C_4^2:D_4$), $C_2^7$ ($C_2\wr D_4$), $C_2^7$ ($C_2^4:D_4$), $C_2^7$ ($C_4^2:D_4$), $C_2^7$ ($C_4^2:C_2^3$), $D_4:C_2^4$ ($C_2^4:D_4$), $D_4:C_2^4$ ($C_2^4:D_4$), $D_4\times C_2^4$ ($C_4^2:D_4$), $D_4\times C_2^4$ ($C_4^2:D_4$), $D_4\times C_2^4$ ($C_4^2:D_4$), $D_4\times C_2^4$ ($C_4^2:D_4$), $D_4\times C_2^4$ ($C_2^4:D_4$), $C_2^5\times C_4$ ($C_4^2:D_4$), $C_2^5\times C_4$ ($C_2\wr D_4$), $C_2^5\times C_4$ ($C_4^2:D_4$), $C_2^5\times C_4$ ($C_4^2:D_4$), $C_2^4:D_4$ ($C_4^2:C_2^3$), $C_2^4:D_4$ ($C_2^4:D_4$), $C_2^5:C_4$ ($C_4^2:D_4$), $C_2^5:C_4$ ($C_4^2:C_2^3$), $C_2^4.D_4$ ($C_2^4:D_4$), $C_2^4.D_4$ ($C_2^4:D_4$)

Order 64: $C_2^6$ ($C_2^5.D_4$) x 3, $C_2^6$ ($C_2^5:D_4$) x 3, $C_2^6$ ($C_2^5.D_4$) x 3, $D_4\times C_2^3$ ($D_4^2:C_2^2$) x 2, $D_4\times C_2^3$ ($C_2^5:D_4$) x 2, $C_2^6$ ($C_2^5:D_4$), $C_2^6$ ($D_4^2:C_2^2$), $C_2^6$ ($C_4.D_4^2$), $C_2^6$ ($C_2^5:D_4$), $C_2^6$ ($C_2^5.D_4$), $D_4\times C_2^3$ ($C_2^5:D_4$), $D_4\times C_2^3$ ($C_2^5:D_4$), $D_4\times C_2^3$ ($C_2^5.D_4$), $D_4\times C_2^3$ ($C_2^5.D_4$), $C_2^4\times C_4$ ($C_2^5.D_4$), $C_2^4\times C_4$ ($C_2^5:D_4$), $C_2^4\times C_4$ ($C_2^5.D_4$), $C_2^4\times C_4$ ($C_2^5.D_4$), $C_2^4\times C_4$ ($D_4^2:C_2^2$), $C_2^4\times C_4$ ($C_2^5.D_4$), $C_2^4:C_4$ ($C_4.D_4^2$)

Order 32: $C_2^5$ ($C_2^6.D_4$) x 4, $C_2^5$ ($C_2^4.C_2^4.C_2$), $C_2^5$ ($C_2^6:D_4$), $C_2^5$ ($C_2^4.C_2^4.C_2$), $C_2^5$ ($C_2^6.D_4$), $C_2^5$ ($(C_2^2\times D_4).C_2^4$), $C_2^5$ ($C_2^6:D_4$), $C_2^5$ ($C_2.C_2^6.C_2^2$), $C_2^5$ ($C_2^6.C_2^3$), $C_2^5$ ($C_2^4.C_2^4.C_2$), $C_2^5$ ($C_2^4.C_2^4.C_2$), $C_2^5$ ($C_2\times C_2^4.C_2^3.C_2$), $C_2^3\times C_4$ ($C_2^4.C_2^4.C_2$), $C_2^3\times C_4$ ($C_2^6:D_4$), $C_2^3\times C_4$ ($(C_2^5\times C_4).C_2^2$), $C_2^3\times C_4$ ($C_2^4.C_2^4.C_2$)

Order 16: $C_2^4$ ($C_2^4.C_2^4.C_2^2$) x 2, $C_2^4$ ($C_2^5.C_2^3.C_2^2$), $C_2^4$ ($C_2^4.C_2^4.C_2^2$), $C_2^4$ ($C_2^4.C_2^4.C_2^2$), $C_2^4$ ($C_2^5.C_2^3.C_2^2$), $C_2^4$ ($C_2^5.C_2^3.C_2^2$), $C_2^4$ ($C_2^5.C_2^3.C_2^2$), $C_2^4$ ($C_2^4.C_2^4.C_2^2$)

Order 8: $C_2^3$ ($C_2^4.C_2^4.C_2^3$), $C_2^3$ ($C_2^5.C_2^4.C_2^2$), $C_2^3$ ($C_2^4.C_2^4.C_2^3$)

Order 4: $C_2^2$ ($C_2^5.C_2^4.C_2^3$), $C_2^2$ ($C_2^4.C_2^5.C_2^3$), $C_2^2$ ($C_2^5.C_2^4.C_2^3$)

Order 2: $C_2$ ($C_2^5.C_2^5.C_2^3$)

Order 1: $C_1$ ($C_2^7.C_2\wr D_4$)

Series

Derived series

$C_2^7.C_2\wr D_4$

$\rhd$

$C_2^5.C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2^7.C_2\wr D_4$

$\rhd$

$C_2^4.C_2^4.C_2^2:D_4$

$\rhd$

$C_2.C_2^6.C_2^5$

$\rhd$

$C_2^4.C_2^4.C_2^3$

$\rhd$

$C_2^5.C_2^5$

$\rhd$

$C_2^5.C_2^4$

$\rhd$

$C_2^5:C_2^3$

$\rhd$

$C_2^4.D_4$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^7.C_2\wr D_4$

$\rhd$

$C_2^5.C_2^5$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^5$

$\lhd$

$C_2^7$

$\lhd$

$D_4.C_2^6$

$\lhd$

$C_2^5.C_2^6$

$\lhd$

$C_2^7.C_2\wr D_4$

Character theory

Complex character table

The $151 \times 151$ character table is not available for this group.

Rational character table

The $135 \times 135$ rational character table is not available for this group.