Subgroups — #1713

Order 128.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500	C₂	1	4	yes	maximal
2	#494	C₂	1	2	yes	maximal
3	C₈²	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆	—	2	4	no	—
5	C₄⋊Q₈	C₂²	1	4	yes	—
6	#107	—	2	8	no	—
7	C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄×C₈	C₂²	1	2	yes	—
9	C₂×Q₈	—	2	4	no	—
10	Q₁₆	—	4	8	no	—
11	C₂×C₈	D₄	1	2	yes	—
12	C₂×C₈	D₄	1	4	yes	—
13	C₄⋊C₄	—	2	8	no	—
14	C₄²	C₂³	1	1	yes	characteristic
15	Q₈	—	4	8	no	—
16	C₂×C₄	—	4	4	no	—
17	C₂×C₄	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄	C₂×D₄	1	2	yes	—
19	C₈	D₈	1	4	yes	—
20	C₈	Q₁₆	1	8	yes	complemented
21	C₂²	#127	1	1	yes	characteristic
22	C₄	—	8	4	no	—
23	C₄	C₂×D₈	1	2	yes	—
24	C₄	C₂×Q₁₆	1	4	yes	—
25	C₂	#493	1	1	yes	characteristic
26	C₂	#494	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²	#127
commutator, Frattini, cosocle	C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

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

N	H ≅ G/N	type
20. C₈	Q₁₆	semidirect

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