Subgroups — #2247

Order 128.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂	C₂	1	4	yes	maximal
2	#494	C₂	1	2	yes	maximal
3	C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆	—	2	2	no	—
5	C₂×Q₁₆	C₂²	1	4	yes	—
6	C₄⋊Q₈	—	2	2	no	—
7	Q₃₂	—	2	8	no	—
8	C₂×C₁₆	C₂²	1	2	yes	—
9	C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×Q₈	—	2	4	no	—
11	Q₁₆	—	2	8	no	—
12	Q₁₆	—	4	4	no	—
13	C₂×C₈	D₄	1	1	yes	characteristic
14	C₂×C₈	C₂³	1	1	yes	characteristic
15	C₄⋊C₄	—	4	4	no	—
16	C₄²	D₄	1	1	yes	characteristic
17	C₁₆	D₄	1	4	yes	—
18	Q₈	—	4	4	no	—
19	Q₈	—	4	4	no	—
20	C₂×C₄	—	4	4	no	—
21	C₂×C₄	D₈	1	2	yes	—
22	C₂×C₄	C₂×D₄	1	1	yes	characteristic
23	C₈	D₈	1	2	yes	—
24	C₈	C₂×D₄	1	2	yes	—
25	C₂²	C₂×D₈	1	1	yes	characteristic
26	C₄	—	8	4	no	—
27	C₄	Q₃₂	1	4	yes	complemented
28	C₄	#127	1	1	yes	characteristic
29	C₄	C₂×D₈	1	1	yes	characteristic
30	C₂	#493	1	1	yes	characteristic
31	C₂	C₂×Q₃₂	1	2	yes	—

	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
27. C₄	Q₃₂	semidirect

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