Subgroups — #2246

Order 128.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆	C₂	1	4	yes	maximal, complemented
2	#493	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈	—	2	2	no	—
5	C₂×D₈	C₂²	1	4	yes	—
6	#127	—	2	2	no	—
7	D₁₆	—	2	8	no	—
8	C₂×C₁₆	C₂²	1	2	yes	—
9	C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×D₄	—	2	4	no	—
11	C₂×D₄	—	4	4	no	—
12	D₈	—	2	8	no	—
13	D₈	—	4	4	no	—
14	C₂×C₈	D₄	1	1	yes	characteristic
15	C₂×C₈	C₂³	1	1	yes	characteristic
16	C₄²	D₄	1	1	yes	characteristic
17	C₁₆	D₄	1	4	yes	complemented
18	C₂³	—	4	4	no	—
19	D₄	—	4	4	no	—
20	D₄	—	4	4	no	—
21	D₄	—	8	8	no	—
22	C₂×C₄	D₈	1	2	yes	—
23	C₂×C₄	C₂×D₄	1	1	yes	characteristic
24	C₈	D₈	1	2	yes	—
25	C₈	C₂×D₄	1	2	yes	—
26	C₂²	—	8	4	no	—
27	C₂²	—	8	8	no	—
28	C₂²	C₂×D₈	1	1	yes	characteristic
29	C₄	D₁₆	1	4	yes	complemented
30	C₄	#127	1	1	yes	characteristic
31	C₄	C₂×D₈	1	1	yes	characteristic
32	C₂	—	16	4	no	—
33	C₂	#493	1	1	yes	characteristic
34	C₂	C₂×D₁₆	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×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.