Subgroups — #1711

Order 128.

89 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#487	C₂	1	3	yes	maximal, complemented
2	#486	C₂	1	3	yes	maximal
3	C₈²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈	C₂²	1	3	yes	—
5	#127	C₂²	1	1	yes	characteristic
6	#103	—	2	6	no	—
7	#102	—	2	6	no	—
8	C₄×C₈	C₂²	1	3	yes	—
9	C₂×Q₈	—	2	3	no	—
10	C₂×D₄	—	2	3	no	—
11	C₂×C₈	D₄	1	6	yes	—
12	C₄⋊C₄	—	2	3	no	—
13	C₄⋊C₄	—	2	3	no	—
14	C₄²	C₂³	1	1	yes	characteristic
15	C₂³	—	4	1	no	—
16	Q₈	—	4	6	no	—
17	D₄	—	4	6	no	—
18	C₂×C₄	—	4	3	no	—
19	C₂×C₄	C₂×D₄	1	3	yes	—
20	C₈	—	2	6	no	—
21	C₂²	—	8	3	no	—
22	C₂²	#127	1	1	yes	characteristic
23	C₄	—	8	3	no	—
24	C₄	#135	1	6	yes	—
25	C₂	—	16	1	no	—
26	C₂	#495	1	3	yes	—

	subgroup	G/H
center, socle	C₂²	#127
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	#127	C₂²
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
1. #487	C₂	semidirect
3. C₈²	C₂	semidirect

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