Abstract group 2048.ctz: $C_2^5.C_2\wr C_2^2$

• Label: 2048.ctz
• Order: \( 2^{11} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{19} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \)
• Perm deg.: $36$
• Trans deg.: $64$
• Rank: $4$

Group information

Description:	$C_2^5.C_2\wr C_2^2$
Order:	\(2048\)\(\medspace = 2^{11} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$(C_{100}\times D_{25}):C_4$, of order \(524288\)\(\medspace = 2^{19} \)
Composition factors:	$C_2$ x 11
Nilpotency class:	$4$
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	223	672	1152	2048
Conjugacy classes	1	23	56	36	116
Divisions	1	23	47	27	98
Autjugacy classes	1	9	24	6	40

Dimension	1	2	4	8	16
Irr. complex chars.	16	28	56	16	0	116
Irr. rational chars.	16	24	42	8	8	98

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$64$
Rank:	$4$
Inequivalent generating quadruples:	$10321920$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{8}=d^{4}=e^{4}=f^{4}=[a,b]= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,2)(3,9)(4,5)(6,10)(7,11)(8,12)(13,20)(14,32)(15,28)(16,29)(17,24)(18,30) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^3.C_2^5)$ $\,\rtimes\,$ $D_4$	$(C_4^2:\OD_{16})$ $\,\rtimes\,$ $D_4$	$(C_4^2:C_4)$ $\,\rtimes\,$ $(D_8:C_2)$	$(C_4:\OD_{16})$ $\,\rtimes\,$ $(D_8:C_2)$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4^2.C_2^4)$ . $D_4$	$(C_4^2.C_2^4)$ . $D_4$ (2)	$(C_4^3:C_2^2)$ . $D_4$	$(C_4^2.C_4^2)$ . $D_4$ (2)	all 78
Aut. group:	$\Aut(C_8\times C_{16})$	$\Aut(C_{16}:C_8)$

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 72544 subgroups in 11959 conjugacy classes, 316 normal (70 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 2048: $C_2^5.C_2\wr C_2^2$

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

Order 512: $C_4.C_4^2.C_2^3$ x 16, $C_4.C_4^2.C_2^3$ x 8, $C_4.C_4^2.C_2^3$ x 4, $C_4.C_4^2.C_2^3$ x 4, $(C_2\times C_4^2).C_4^2$ x 4, $(C_2^2\times C_4^2).C_2^3$ x 4, $C_4.C_4^2.C_2^3$ x 4, $C_4.C_4^2.C_2^3$ x 4, $C_4^2.C_2^5$ x 3, $(C_2\times C_4^2).C_2^4$ x 2, $(C_2^3\times C_4).C_2^4$ x 2, $(C_2\times C_4).C_2^6$ x 2, $(C_2^3\times C_4).C_2^4$ x 2, $(C_2^3\times C_4).C_2^4$ x 2, $(C_2^3\times C_4).C_2^4$ x 2, $(C_2\times C_4^2).C_2^4$ x 2, $(C_2^3\times C_4).C_2^4$ x 2, $C_4.C_4^2.C_2^3$ x 2, $(C_2\times C_4^2).D_4.C_2$ x 2, $(C_2\times C_4^2).C_2^4$ x 2, $(C_2^3\times C_4).C_2^4$, $C_2\times C_2^3.C_2^5$, $(C_2\times C_4^2).C_2^4$, $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$, $(C_2\times C_4^2).C_2^4$, $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$

Order 256: $C_4^2.C_2^4$ x 40, $C_4^2:\SD_{16}$ x 32, $C_4^2:D_8$ x 32, $C_4^2.C_2^4$ x 32, $C_2^3.C_2^5$ x 21, $C_4^2:\OD_{16}$ x 16, $C_4^2.(C_2\times D_4)$ x 16, $C_4^2.C_2^4$ x 12, $C_4^2.C_4^2$ x 9, $C_4^2.Q_{16}$ x 8, $C_4^2.D_8$ x 8, $C_4^2.C_2^4$ x 6, $C_4^2.C_2^4$ x 6, $C_4^2.C_2^4$ x 6, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $(C_2\times C_4^2).D_4$ x 4, $C_4^2.C_4^2$ x 4, $C_2\times C_4^2:Q_8$ x 4, $C_2\times C_4^2:D_4$ x 4, $C_2^5.D_4$ x 4, $C_4^2.(C_2\times D_4)$ x 4, $C_2\times D_4.C_4^2$ x 4, $C_4^2:C_2^4$ x 3, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_2^5.D_4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_2^5.D_4$ x 2, $(C_2\times \OD_{16}):Q_8$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_2^5.D_4$ x 2, $(C_2^2\times C_8):D_4$ x 2, $(C_2\times \OD_{16}):D_4$ x 2, $(C_2\times D_4).C_4^2$ x 2, $(C_2\times D_4):C_4^2$ x 2, $\OD_{16}.C_4^2$ x 2, $(C_2\times C_8).C_4^2$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2^2\times C_4^2):C_4$ x 2, $C_2^5.D_4$ x 2, $C_4^2:C_4^2$ x 2, $C_2\times C_4^2.D_4$ x 2, $(C_2\times C_4^2):D_4$ x 2, $C_4^2.(C_2\times D_4)$ x 2, $C_2\times C_4^2.D_4$ x 2, $C_2\times C_4^2.D_4$ x 2, $(C_2\times C_4^2):D_4$ x 2, $C_4^2.(C_2\times D_4)$ x 2, $C_4^2.(C_2\times D_4)$ x 2, $C_4^2.(C_2\times D_4)$ x 2, $C_4^2.(C_2\times D_4)$ x 2, $(C_2\times D_4):C_4^2$ x 2, $(C_2\times C_4^2).D_4$, $(C_2\times C_4^2).D_4$, $C_2^5.Q_8$, $(C_2\times C_4^2).D_4$, $C_4^3:C_4$, $(C_2\times C_8).C_4^2$, $C_2^5.Q_8$, $C_2^5:D_4$, $C_4^3:C_2^2$, $C_2^3.C_2^5$, $C_2\times C_4^2.D_4$, $(C_2\times C_4^2):D_4$, $C_2^4.C_2^4$, $C_2^5.D_4$, $C_4^2.(C_2\times D_4)$, $C_4^3:C_2^2$, $C_2^5.D_4$, $C_2^5.D_4$

Order 128: $C_4^2:C_2^3$ x 102, $C_4^2:D_4$ x 76, $D_4:\OD_{16}$ x 64, $C_4^2.D_4$ x 64, $C_4^2.D_4$ x 64, $C_4^2.D_4$ x 64, $C_4^2.C_2^3$ x 49, $D_4:\OD_{16}$ x 48, $C_4^2.D_4$ x 48, $C_4^2.D_4$ x 40, $C_4^2.D_4$ x 40, $C_4^2.D_4$ x 32, $C_4^2.D_4$ x 32, $C_2^4.D_4$ x 32, $C_4^2.D_4$ x 32, $C_4^2.D_4$ x 32, $C_2^4.D_4$ x 32, $C_4^2:C_8$ x 26, $C_4^2.D_4$ x 24, $C_2^4.Q_8$ x 24, $C_2^4:D_4$ x 24, $C_4^2.C_2^3$ x 22, $C_4^2:D_4$ x 18, $C_4^2:D_4$ x 17, $C_4^2:Q_8$ x 16, $D_4.C_4^2$ x 16, $C_2^4.D_4$ x 16, $C_2^3.C_2^4$ x 15, $C_4^2.C_2^3$ x 15, $C_4^2.C_2^3$ x 9, $C_4^2.C_2^3$ x 9, $C_2^3.C_4^2$ x 9, $C_4^2.D_4$ x 8, $C_4^2.D_4$ x 8, $C_4^2.D_4$ x 8, $C_2^3.D_8$ x 8, $D_4:C_4^2$ x 8, $D_4:C_4^2$ x 8, $C_4^2.D_4$ x 8, $C_4^2.D_4$ x 8, $C_2^3:\SD_{16}$ x 8, $C_2^3:D_8$ x 8, $C_4^2.D_4$ x 8, $C_4^2.D_4$ x 8, $C_4^2.D_4$ x 8, $C_4^2.D_4$ x 8, $C_4^2:C_2^3$ x 8, $C_2^4.D_4$ x 8, $C_2^3.C_4^2$ x 6, $C_4^2.C_2^3$ x 6, $C_2^3:\OD_{16}$ x 6, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_2^3.Q_{16}$ x 4, $C_2^3.D_8$ x 4, $C_2^4.D_4$ x 4, $C_8:C_2^4$ x 4, $C_4^2.C_2^3$ x 4, $C_4^2.C_2^3$ x 4, $C_4^2:C_2^3$ x 4, $C_4^2.C_2^3$ x 4, $C_2^3.Q_{16}$ x 4, $C_2^4.D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^3.D_8$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_2^4.D_4$ x 4, $C_4^2:C_2^3$ x 4, $C_2^4.D_4$ x 4, $C_2^3.C_2^4$ x 4, $C_2^4.D_4$ x 3, $D_4\times C_2^4$ x 3, $C_2^4.D_4$ x 3, $C_2^5:C_4$ x 3, $C_2^3\times C_4^2$ x 3, $C_2^3.C_2^4$ x 3, $C_4^2:D_4$ x 2, $C_4^2:D_4$ x 2, $C_4^2:D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2:D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2:D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2:D_4$ x 2, $C_2^4.Q_8$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $\OD_{16}.C_2^3$ x 2, $C_2^3.C_2^4$ x 2, $(C_2^2\times C_8):C_2^2$ x 2, $C_2^3.C_2^4$ x 2, $C_4^3:C_2$ x 2, $C_2^3.C_4^2$, $C_2^4.D_4$, $C_2^4:D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^3:C_2$, $C_2^3.C_4^2$, $C_2^4.C_2^3$, $C_2^4.Q_8$, $C_2^5\times C_4$, $D_8:C_2^3$, $C_4^2.C_2^3$, $C_2^4.Q_8$, $C_4^2.C_2^3$

Order 64: $C_4^2:C_2^2$ x 564, $D_4:C_8$ x 96, $C_4^2.C_4$ x 82, $D_4\times C_2^3$ x 75, $C_4^2.C_2^2$ x 72, $C_8:C_2^3$ x 66, $C_2^3.Q_8$ x 64, $C_2^3:D_4$ x 64, $D_4.Q_8$ x 64, $D_4.D_4$ x 64, $C_2^2:\SD_{16}$ x 64, $D_4:D_4$ x 64, $D_8:C_4$ x 64, $\SD_{16}:C_4$ x 64, $C_4\times \SD_{16}$ x 64, $C_4\times D_8$ x 64, $C_2^4:C_4$ x 60, $C_4^2.C_4$ x 58, $C_2^2\times C_4^2$ x 53, $C_4^2:C_2^2$ x 52, $C_2^3.D_4$ x 51, $C_2^3.D_4$ x 50, $C_4\times \OD_{16}$ x 43, $C_2^4\times C_4$ x 41, $C_2^2.D_8$ x 40, $C_4^2:C_2^2$ x 40, $C_4:\OD_{16}$ x 40, $C_4^2.C_2^2$ x 38, $C_2^3.D_4$ x 36, $C_4^2.C_2^2$ x 34, $C_4^2:C_4$ x 34, $C_4^2.C_2^2$ x 33, $C_2^2:\OD_{16}$ x 32, $C_2^2.Q_{16}$ x 32, $C_2^3.D_4$ x 32, $C_2^3.D_4$ x 32, $C_2^2.D_8$ x 32, $D_4:Q_8$ x 32, $D_4:Q_8$ x 32, $C_4:\SD_{16}$ x 32, $C_4:D_8$ x 32, $\OD_{16}:C_4$ x 32, $C_4^2:C_4$ x 30, $C_4^2.C_4$ x 18, $C_4.C_4^2$ x 16, $C_2^3:C_8$ x 15, $C_2\times C_4\times C_8$ x 15, $C_2^2.C_4^2$ x 12, $C_2^2\times \OD_{16}$ x 12, $C_4^2:C_2^2$ x 12, $C_4^2:C_2^2$ x 10, $C_4.C_4^2$ x 9, $C_2^2.Q_{16}$ x 8, $C_2^2\times \SD_{16}$ x 8, $C_2^2\times D_8$ x 8, $\OD_{16}:C_4$ x 8, $C_2^2.D_8$ x 8, $C_2^3.D_4$ x 8, $\OD_{16}.C_2^2$ x 6, $C_2^3.C_2^3$ x 6, $C_2^3.D_4$ x 6, $C_4:C_4^2$ x 6, $C_2^3:Q_8$ x 6, $C_2^3:D_4$ x 6, $C_4^2.C_2^2$ x 6, $D_8:C_2^2$ x 5, $C_4^2:C_4$ x 5, $(C_2^2\times C_8):C_2$ x 4, $C_4^2:C_4$ x 4, $D_4:C_2^3$ x 4, $C_4^2.C_2^2$ x 4, $C_4^2.C_2^2$ x 4, $C_4^2:C_2^2$ x 4, $C_2^3.D_4$ x 4, $C_4^2:C_4$ x 3, $C_2^2:C_4^2$ x 3, $C_2^3.D_4$ x 2, $\OD_{16}.C_2^2$ x 2, $\OD_{16}:C_2^2$ x 2, $C_2^4:C_4$ x 2, $C_2^3.D_4$ x 2, $Q_{16}:C_2^2$ x 2, $\OD_{16}:C_4$ x 2, $C_4.C_4^2$ x 2, $C_2^2.C_2^4$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3:D_4$ x 2, $C_4.C_4^2$ x 2, $C_2^3.Q_8$ x 2, $C_4^2.C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_4^2:C_4$, $C_2^4:C_4$, $C_4^3$, $C_2^6$, $D_8:C_2^2$, $C_2^2.C_4^2$, $C_2^3:D_4$, $C_2^3.Q_8$

Order 32: $C_4\times D_4$ x 566, $C_2^2\times D_4$ x 506, $C_2^3\times C_4$ x 318, $C_2^3:C_4$ x 206, $C_2\times C_4^2$ x 202, $C_2^2.D_4$ x 157, $D_8:C_2$ x 134, $D_4:C_4$ x 132, $C_2\times \OD_{16}$ x 81, $C_2\times \SD_{16}$ x 66, $C_2\times D_8$ x 65, $C_4^2:C_2$ x 62, $C_4:C_8$ x 60, $C_2^2:C_8$ x 56, $D_4:C_2^2$ x 52, $C_4:D_4$ x 42, $C_2^2:Q_8$ x 40, $C_4\wr C_2$ x 40, $C_2^5$ x 39, $C_4\times C_8$ x 38, $C_4\times Q_8$ x 36, $Q_8:C_4$ x 36, $C_8:C_4$ x 32, $C_8:C_4$ x 32, $C_8:C_4$ x 26, $C_2^2.D_4$ x 26, $C_2.C_4^2$ x 22, $C_4.Q_8$ x 20, $C_4^2:C_2$ x 20, $C_2^2\times C_8$ x 18, $C_4:Q_8$ x 17, $C_4^2:C_2$ x 16, $C_2^2\wr C_2$ x 10, $C_4:D_4$ x 7, $Q_{16}:C_2$ x 6, $C_4.D_4$ x 4, $\OD_{16}:C_2$ x 4, $C_2^3:C_4$ x 4, $C_2^2\times Q_8$ x 4, $D_8:C_2$ x 4, $C_8.C_4$ x 2, $C_2\times Q_{16}$

Order 16: $C_2\times D_4$ x 860, $C_2^2\times C_4$ x 621, $C_2^4$ x 275, $C_2^2:C_4$ x 146, $C_4^2$ x 130, $C_4:C_4$ x 118, $D_4:C_2$ x 80, $\SD_{16}$ x 68, $D_8$ x 66, $\OD_{16}$ x 66, $C_2\times C_8$ x 63, $C_2\times Q_8$ x 26, $Q_{16}$ x 2

Order 8: $C_2^3$ x 463, $C_2\times C_4$ x 343, $D_4$ x 262, $C_8$ x 27, $Q_8$ x 20

Order 4: $C_2^2$ x 201, $C_4$ x 47

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2048: $C_2^5.C_2\wr C_2^2$

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

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

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

Order 128: $C_4^2:C_2^3$ x 40, $C_4^2.C_2^3$ x 14, $C_4^2:D_4$ x 13, $C_2^3.C_2^4$ x 8, $C_2^4.D_4$ x 8, $D_4:\OD_{16}$ x 8, $C_4^2:D_4$ x 7, $C_4^2:Q_8$ x 6, $C_4^2.C_2^3$ x 6, $C_4^2.C_2^3$ x 6, $C_4^2.C_2^3$ x 6, $C_4^2.C_2^3$ x 6, $C_2^4.Q_8$ x 6, $C_2^4:D_4$ x 6, $C_4^2:D_4$ x 5, $C_4^2:C_8$ x 5, $D_4.C_4^2$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2:C_2^3$ x 4, $C_2^3:\OD_{16}$ x 4, $C_2^3.C_4^2$ x 4, $C_2^3.D_8$ x 3, $C_2^3.C_4^2$ x 3, $C_4^2.C_2^3$ x 3, $C_2^3.C_2^4$ x 3, $C_2^3.C_2^4$ x 3, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_2^3.D_8$ x 2, $C_4^2.D_4$ x 2, $D_4:C_4^2$ x 2, $C_2^4.D_4$ x 2, $D_4\times C_2^4$ x 2, $D_4:\OD_{16}$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2:C_2^3$ x 2, $C_4^2.C_2^3$ x 2, $C_2^4.D_4$ x 2, $C_2^5:C_4$ x 2, $C_2^3\times C_4^2$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_2^4.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:D_8$ x 2, $C_4^2.D_4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_2^4$ x 2, $C_4^3:C_2$ x 2, $C_2^3.C_4^2$, $C_4^2:D_4$, $C_4^2:D_4$, $C_2^4.D_4$, $C_2^4:D_4$, $C_4^2:D_4$, $C_4^2.D_4$, $C_4^2:D_4$, $C_4^2.D_4$, $C_4^2:D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^3:C_2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2:D_4$, $C_2^3.Q_{16}$, $C_4^2.D_4$, $D_4:C_4^2$, $C_2^3.C_4^2$, $C_2^4.C_2^3$, $C_2^4.Q_8$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^4.Q_8$, $C_2^5\times C_4$, $D_8:C_2^3$, $C_8:C_2^4$, $C_4^2.C_2^3$, $C_2^3.Q_{16}$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^3.D_8$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_2^4.Q_8$, $C_2^4.D_4$, $C_4^2.C_2^3$, $C_2^4.D_4$, $C_2^4.D_4$, $\OD_{16}.C_2^3$, $C_2^3.C_2^4$, $(C_2^2\times C_8):C_2^2$

Order 64: $C_4^2:C_2^2$ x 142, $C_2^2\times C_4^2$ x 27, $D_4\times C_2^3$ x 22, $C_4^2:C_2^2$ x 22, $C_2^4\times C_4$ x 20, $C_2^3.D_4$ x 20, $C_2^4:C_4$ x 20, $C_4:\OD_{16}$ x 17, $C_4^2.C_4$ x 16, $C_4^2:C_4$ x 15, $C_4^2:C_4$ x 14, $C_4^2.C_2^2$ x 12, $C_4^2.C_4$ x 12, $C_4^2.C_2^2$ x 11, $C_4^2.C_2^2$ x 10, $C_2^2:\OD_{16}$ x 9, $C_4^2:C_2^2$ x 9, $C_4\times \OD_{16}$ x 8, $C_4^2.C_2^2$ x 8, $C_4^2.C_4$ x 8, $C_2^3.Q_8$ x 7, $C_2^3:D_4$ x 7, $C_4^2:C_2^2$ x 7, $C_2^3:C_8$ x 6, $C_2^2.C_4^2$ x 6, $C_2^2\times \OD_{16}$ x 6, $C_2^2.D_8$ x 5, $C_4.C_4^2$ x 4, $C_2\times C_4\times C_8$ x 4, $C_2^3.D_4$ x 4, $D_4:C_8$ x 4, $C_4:C_4^2$ x 4, $C_4^2:C_2^2$ x 4, $C_2^3.D_4$ x 3, $C_4^2:C_4$ x 3, $C_2^2:C_4^2$ x 3, $D_8:C_2^2$ x 3, $C_8:C_2^3$ x 3, $\OD_{16}:C_4$ x 3, $C_2^3:Q_8$ x 3, $C_2^3:D_4$ x 3, $C_4^2.C_2^2$ x 3, $C_2^2:\SD_{16}$ x 3, $D_4:D_4$ x 3, $C_2^3.D_4$ x 2, $\OD_{16}.C_2^2$ x 2, $C_2^3.C_2^3$ x 2, $(C_2^2\times C_8):C_2$ x 2, $C_4^2:C_4$ x 2, $C_2^3.D_4$ x 2, $D_4:C_2^3$ x 2, $C_4^2.C_2^2$ x 2, $C_4.C_4^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_4^2:C_4$ x 2, $C_2^3.D_4$ x 2, $C_2^2.D_8$ x 2, $D_4:Q_8$ x 2, $D_4:Q_8$ x 2, $C_4:\SD_{16}$ x 2, $C_4:D_8$ x 2, $C_4\times \SD_{16}$ x 2, $C_4\times D_8$ x 2, $C_2^3.D_4$, $C_2^2.Q_{16}$, $\OD_{16}.C_2^2$, $\OD_{16}:C_2^2$, $C_2^4:C_4$, $C_4^2:C_4$, $C_2^4:C_4$, $C_4^3$, $C_2^6$, $Q_{16}:C_2^2$, $D_8:C_2^2$, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $\OD_{16}:C_4$, $C_2^2.C_4^2$, $C_4.C_4^2$, $C_2^2.C_2^4$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4.C_4^2$, $C_2^2.Q_{16}$, $C_2^3.D_4$, $D_4.Q_8$, $D_4.D_4$, $D_8:C_4$, $\SD_{16}:C_4$, $C_2^3.Q_8$, $C_2^3.Q_8$, $\OD_{16}:C_4$, $C_2^2.D_8$, $C_2^3.D_4$, $C_4^2.C_2^2$, $C_2^3.D_4$

Order 32: $C_2^3\times C_4$ x 101, $C_4\times D_4$ x 95, $C_2\times C_4^2$ x 87, $C_2^2\times D_4$ x 64, $C_2^2.D_4$ x 46, $C_2^3:C_4$ x 44, $C_2\times \OD_{16}$ x 24, $D_4:C_2^2$ x 15, $C_4^2:C_2$ x 15, $C_4:C_8$ x 15, $C_4\wr C_2$ x 14, $C_2^5$ x 12, $C_2.C_4^2$ x 11, $C_4:D_4$ x 10, $C_2^2:C_8$ x 9, $C_2^2:Q_8$ x 9, $C_4:Q_8$ x 8, $C_4\times Q_8$ x 8, $D_4:C_4$ x 6, $C_2^2\times C_8$ x 6, $C_4^2:C_2$ x 6, $C_4\times C_8$ x 6, $C_4.Q_8$ x 5, $C_2^2.D_4$ x 5, $C_8:C_4$ x 4, $C_4:D_4$ x 4, $D_8:C_2$ x 3, $C_2\times \SD_{16}$ x 3, $C_2\times D_8$ x 3, $C_2^2\wr C_2$ x 3, $C_2^2\times Q_8$ x 2, $Q_{16}:C_2$ x 2, $D_8:C_2$ x 2, $C_4^2:C_2$ x 2, $Q_8:C_4$ x 2, $C_4.D_4$, $\OD_{16}:C_2$, $C_2^3:C_4$, $C_2\times Q_{16}$, $C_8.C_4$, $C_8:C_4$, $C_8:C_4$

Order 16: $C_2^2\times C_4$ x 166, $C_2\times D_4$ x 74, $C_4^2$ x 56, $C_2^4$ x 42, $C_4:C_4$ x 27, $C_2^2:C_4$ x 23, $D_4:C_2$ x 18, $C_2\times C_8$ x 16, $\OD_{16}$ x 14, $C_2\times Q_8$ x 9, $\SD_{16}$ x 2, $D_8$ x 2, $Q_{16}$

Order 8: $C_2\times C_4$ x 108, $C_2^3$ x 56, $D_4$ x 28, $Q_8$ x 6, $C_8$ x 6

Order 4: $C_2^2$ x 33, $C_4$ x 24

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2048: $C_2^5.C_2\wr C_2^2$ ($C_1$)

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

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

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

Order 128: $C_2^3.C_2^4$ ($C_2\times D_4$) x 9, $C_4^2:C_8$ ($C_2\times D_4$) x 8, $C_4^2.C_2^3$ ($C_2\times D_4$) x 6, $C_4^2:D_4$ ($C_2\times D_4$) x 4, $C_2^3\times C_4^2$ ($C_2\times D_4$) x 3, $C_2^3.C_2^4$ ($D_4:C_2$) x 2, $C_4^2.C_2^3$ ($C_2\times D_4$) x 2, $C_4^2.C_2^3$ ($C_2\times D_4$) x 2, $C_2^3.C_2^4$ ($C_2\times D_4$) x 2, $C_2^3.C_2^4$ ($C_2^4$)

Order 64: $C_2^2\times C_4^2$ ($C_2^2\wr C_2$) x 7, $C_4^2:C_4$ ($C_4:D_4$) x 6, $C_2^2\times C_4^2$ ($C_4:D_4$) x 6, $C_4^2:C_4$ ($C_2^2\wr C_2$) x 4, $C_4^2.C_2^2$ ($D_8:C_2$) x 4, $C_4^2:C_4$ ($C_2^2\times D_4$) x 3, $C_2^2\times C_4^2$ ($C_2^2\times D_4$) x 3, $C_4^2:C_4$ ($D_8:C_2$) x 2, $C_4^2:C_4$ ($C_4:D_4$) x 2, $C_4^2.C_2^2$ ($D_8:C_2$) x 2, $C_4^2:C_2^2$ ($D_8:C_2$) x 2, $C_2^2\times C_4^2$ ($C_4:D_4$) x 2, $C_4:\OD_{16}$ ($D_8:C_2$) x 2, $C_4^2:C_4$ ($D_4:C_2^2$), $C_2^4\times C_4$ ($C_2^2\wr C_2$)

Order 32: $C_2\times C_4^2$ ($D_4:D_4$) x 10, $C_2\times C_4^2$ ($C_2^3:D_4$) x 6, $C_2\times C_4^2$ ($D_8:C_2^2$) x 6, $C_2\times C_4^2$ ($C_8:C_2^3$) x 6, $C_2^3\times C_4$ ($C_2\wr C_2^2$) x 3, $C_2\times C_4^2$ ($C_2^3:D_4$) x 3, $C_2^3\times C_4$ ($C_2^3:D_4$) x 2, $C_2\times C_4^2$ ($C_2^3:D_4$) x 2, $C_2\times C_4^2$ ($D_4.D_4$) x 2, $C_2^5$ ($C_2\wr C_2^2$), $C_2^3\times C_4$ ($C_2^3:D_4$), $C_2\times C_4^2$ ($C_4^2:C_2^2$)

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

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

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

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

Order 1: $C_1$ ($C_2^5.C_2\wr C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2048: $C_2^5.C_2\wr C_2^2$ ($C_1$)

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

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

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

Order 128: $C_2^3.C_2^4$ ($C_2\times D_4$) x 4, $C_4^2.C_2^3$ ($C_2\times D_4$) x 4, $C_2^3\times C_4^2$ ($C_2\times D_4$) x 2, $C_2^3.C_2^4$ ($C_2\times D_4$) x 2, $C_2^3.C_2^4$ ($C_2^4$), $C_2^3.C_2^4$ ($D_4:C_2$), $C_4^2:C_8$ ($C_2\times D_4$), $C_4^2.C_2^3$ ($C_2\times D_4$), $C_4^2.C_2^3$ ($C_2\times D_4$), $C_4^2:D_4$ ($C_2\times D_4$)

Order 64: $C_2^2\times C_4^2$ ($C_2^2\wr C_2$) x 3, $C_4^2:C_4$ ($C_2^2\times D_4$) x 2, $C_4^2:C_4$ ($C_4:D_4$) x 2, $C_2^2\times C_4^2$ ($C_2^2\times D_4$) x 2, $C_2^2\times C_4^2$ ($C_4:D_4$) x 2, $C_4^2:C_4$ ($D_8:C_2$), $C_4^2:C_4$ ($D_4:C_2^2$), $C_4^2:C_4$ ($C_4:D_4$), $C_4^2:C_4$ ($C_2^2\wr C_2$), $C_2^4\times C_4$ ($C_2^2\wr C_2$), $C_4^2.C_2^2$ ($D_8:C_2$), $C_4^2.C_2^2$ ($D_8:C_2$), $C_4^2:C_2^2$ ($D_8:C_2$), $C_2^2\times C_4^2$ ($C_4:D_4$), $C_4:\OD_{16}$ ($D_8:C_2$)

Order 32: $C_2\times C_4^2$ ($D_4:D_4$) x 5, $C_2\times C_4^2$ ($D_8:C_2^2$) x 4, $C_2\times C_4^2$ ($C_8:C_2^3$) x 4, $C_2^3\times C_4$ ($C_2\wr C_2^2$) x 2, $C_2\times C_4^2$ ($C_2^3:D_4$) x 2, $C_2\times C_4^2$ ($C_2^3:D_4$) x 2, $C_2^5$ ($C_2\wr C_2^2$), $C_2^3\times C_4$ ($C_2^3:D_4$), $C_2^3\times C_4$ ($C_2^3:D_4$), $C_2\times C_4^2$ ($C_4^2:C_2^2$), $C_2\times C_4^2$ ($C_2^3:D_4$), $C_2\times C_4^2$ ($D_4.D_4$)

Order 16: $C_2^2\times C_4$ ($C_2^4.D_4$) x 4, $C_2^2\times C_4$ ($C_4^2:C_2^3$) x 3, $C_4^2$ ($\OD_{16}:D_4$) x 2, $C_4^2$ ($C_4^2:D_4$) x 2, $C_4^2$ ($\OD_{16}:D_4$) x 2, $C_4^2$ ($C_4^2.D_4$) x 2, $C_4^2$ ($\OD_{16}:D_4$), $C_4^2$ ($\OD_{16}:D_4$), $C_4^2$ ($C_2^4.D_4$), $C_4^2$ ($C_2^4.D_4$), $C_2^4$ ($C_2^4:D_4$), $C_2^4$ ($C_2^4:D_4$), $C_2^2\times C_4$ ($C_2^4:D_4$), $C_2^2\times C_4$ ($C_2^4:D_4$), $C_2^2\times C_4$ ($C_4^2.C_2^3$), $C_2^2\times C_4$ ($C_2^4:D_4$)

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

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

Order 2: $C_2$ ($(C_2^2\times C_4^2).C_2^4$), $C_2$ ($C_4.C_4^2.C_2^4$)

Order 1: $C_1$ ($C_2^5.C_2\wr C_2^2$)

Series

Derived series

$C_2^5.C_2\wr C_2^2$

$\rhd$

$C_2^3.C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2^5.C_2\wr C_2^2$

$\rhd$

$(C_2\times C_4^2).C_4:D_4$

$\rhd$

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

$\rhd$

$C_4^2.C_4^2$

$\rhd$

$C_2^3.C_2^4$

$\rhd$

$C_2^2\times C_4^2$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^5.C_2\wr C_2^2$

$\rhd$

$C_2^3.C_2^4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2^3\times C_4$

$\lhd$

$C_2^3.C_2^5$

$\lhd$

$C_2^5.C_2\wr C_2^2$

Character theory

Complex character table

See the $116 \times 116$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $98 \times 98$ rational character table.