Subgroups — #2252

Order 128.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500	C₂	1	2	yes	maximal
2	#366	C₂	1	4	yes	maximal
3	C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈	—	2	2	no	—
5	C₂×C₁₆	C₂²	1	2	yes	—
6	#107	—	2	2	no	—
7	#107	C₂²	1	4	yes	—
8	C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×Q₈	—	4	2	no	—
10	C₂×C₈	D₄	1	1	yes	characteristic
11	C₂×C₈	C₂³	1	1	yes	characteristic
12	C₄⋊C₄	—	2	4	no	—
13	C₄⋊C₄	—	4	2	no	—
14	C₄²	D₄	1	1	yes	characteristic
15	C₁₆	Q₈	1	4	yes	complemented
16	Q₈	—	8	4	no	—
17	C₂×C₄	—	4	4	no	—
18	C₂×C₄	D₈	1	1	yes	characteristic
19	C₂×C₄	D₈	1	1	yes	characteristic
20	C₂×C₄	C₂×D₄	1	1	yes	characteristic
21	C₈	Q₁₆	1	2	yes	—
22	C₈	C₂×Q₈	1	2	yes	—
23	C₂²	C₂×D₈	1	1	yes	characteristic
24	C₄	—	8	4	no	—
25	C₄	D₁₆	1	2	yes	—
26	C₄	Q₃₂	1	2	yes	—
27	C₄	C₄⋊Q₈	1	1	yes	characteristic
28	C₄	C₂×Q₁₆	1	1	yes	characteristic
29	C₂	#500	1	1	yes	characteristic
30	C₂	C₂×D₁₆	1	1	yes	characteristic
31	C₂	C₂×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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.

N	H ≅ G/N	type
15. C₁₆	Q₈	semidirect

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