Subgroups — #2262

Order 128.

53 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₄	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³
socle	C₂	C₂×D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ 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.