Subgroups — #1409

Order 128.

55 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#492	C₂	1	1	yes	maximal, characteristic
2	#353	C₂	1	2	yes	maximal
3	C₂×SD₁₆	—	2	2	no	—
4	C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
5	#127	C₂²	1	1	yes	characteristic
6	#99	—	2	2	no	—
7	C₄×C₈	C₄	1	1	yes	characteristic, complemented
8	C₂×Q₈	—	2	1	no	—
9	C₂×D₄	—	4	1	no	—
10	C₂×D₄	D₄	1	2	yes	—
11	SD₁₆	—	4	4	no	—
12	C₂×C₈	—	2	1	no	—
13	C₄⋊C₄	—	4	1	no	—
14	C₂²⋊C₄	—	4	2	no	—
15	C₄²	C₂×C₄	1	1	yes	characteristic
16	C₂³	—	2	2	no	—
17	Q₈	—	4	2	no	—
18	D₄	—	4	2	no	—
19	D₄	—	8	2	no	—
20	C₂×C₄	—	2	1	no	—
21	C₂×C₄	—	4	1	no	—
22	C₂×C₄	—	8	2	no	—
23	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
24	C₈	—	2	2	no	—
25	C₂²	—	4	2	no	—
26	C₂²	—	8	2	no	—
27	C₂²	#99	1	1	yes	characteristic
28	C₄	—	2	2	no	—
29	C₄	—	2	1	no	—
30	C₄	—	8	1	no	—
31	C₄	—	16	2	no	—
32	C₂	—	2	1	no	—
33	C₂	—	8	2	no	—
34	C₂	#353	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#353
commutator	C₄²	C₂×C₄
Frattini, prime radical, cosocle	#127	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

N	H ≅ G/N	type
4. C₄⋊Q₈	C₄	semidirect
7. C₄×C₈	C₄	semidirect

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

subgroup	index
#492	2
#353	2