Subgroups — #13741

Order 256.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2262	C₂	1	2	yes	maximal, complemented
2	Q₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₁₂₈	C₂	1	2	yes	maximal, complemented
4	D₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#508	—	2	2	no	—
7	Q₆₄	C₂²	1	2	yes	—
8	SD₆₄	—	2	2	no	—
9	D₃₂	C₂²	1	2	yes	—
10	C₂×C₃₂	C₂²	1	1	yes	characteristic
11	C₆₄	C₂²	1	1	yes	characteristic
12	C₆₄	C₂²	1	1	yes	characteristic
13	#135	—	4	2	no	—
14	Q₃₂	—	2	2	no	—
15	SD₃₂	—	4	2	no	—
16	D₁₆	—	2	2	no	—
17	C₂×C₁₆	D₄	1	1	yes	characteristic
18	C₃₂	D₄	1	1	yes	characteristic
19	C₃₂	C₂³	1	1	yes	characteristic
20	#41	—	8	2	no	—
21	Q₁₆	—	4	2	no	—
22	SD₁₆	—	8	2	no	—
23	D₈	—	4	2	no	—
24	C₂×C₈	D₈	1	1	yes	characteristic
25	C₁₆	D₈	1	1	yes	characteristic
26	C₁₆	C₂×D₄	1	1	yes	characteristic
27	Q₈	—	8	2	no	—
28	D₄	—	8	2	no	—
29	D₄	—	16	2	no	—
30	C₂×C₄	—	16	2	no	—
31	C₂×C₄	D₁₆	1	1	yes	characteristic
32	C₈	D₁₆	1	1	yes	characteristic
33	C₈	C₂×D₈	1	1	yes	characteristic
34	C₂²	—	16	2	no	—
35	C₂²	D₃₂	1	1	yes	characteristic
36	C₄	—	16	2	no	—
37	C₄	D₃₂	1	1	yes	characteristic
38	C₄	C₂×D₁₆	1	1	yes	characteristic
39	C₂	—	2	1	no	—
40	C₂	—	32	2	no	—
41	C₂	C₂×D₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³
socle	C₂	C₂×D₃₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.