Subgroups — #3391

Order 128.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#550	C₂	1	2	yes	maximal, complemented
2	#493	C₂	1	3	yes	maximal, complemented
3	#459	C₂	1	6	yes	maximal, complemented
4	C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈	C₂	1	3	yes	maximal, complemented
6	C₂×#41	—	2	2	no	—
7	C₂×D₈	C₂²	1	3	yes	—
8	C₂×D₈	C₂²	1	6	yes	—
9	#127	C₂²	1	6	yes	—
10	#121	—	2	6	no	—
11	C₄×Q₈	C₂²	1	1	yes	characteristic
12	C₄×D₄	C₂²	1	6	yes	—
13	#107	C₂²	1	1	yes	characteristic
14	C₄⋊C₈	C₂²	1	3	yes	—
15	#102	C₂²	1	6	yes	—
16	C₄×C₈	C₂²	1	3	yes	—
17	#41	—	4	8	no	—
18	C₂×Q₈	D₄	1	1	yes	characteristic
19	C₂×D₄	—	2	6	no	—
20	C₂×D₄	—	2	6	no	—
21	C₂×D₄	C₂³	1	6	yes	—
22	C₂²×C₄	—	2	6	no	—
23	D₈	—	2	6	no	—
24	D₈	—	4	6	no	—
25	C₂×C₈	C₂³	1	3	yes	—
26	C₂×C₈	C₂³	1	1	yes	characteristic
27	C₄⋊C₄	D₄	1	3	yes	—
28	C₄⋊C₄	C₂³	1	2	yes	—
29	C₂²⋊C₄	—	2	6	no	—
30	C₄²	C₂³	1	3	yes	—
31	C₂³	—	2	6	no	—
32	Q₈	D₈	1	4	yes	complemented
33	D₄	—	2	6	no	—
34	D₄	—	2	6	no	—
35	D₄	—	4	12	no	—
36	D₄	—	8	6	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	4	12	no	—
39	C₂×C₄	C₂×D₄	1	3	yes	—
40	C₂×C₄	C₂×D₄	1	3	yes	—
41	C₂×C₄	C₂⁴	1	1	yes	characteristic
42	C₈	—	2	3	no	—
43	C₈	#41	1	2	yes	complemented
44	C₂²	—	4	6	no	—
45	C₂²	—	4	6	no	—
46	C₂²	—	4	6	no	—
47	C₂²	C₂²×D₄	1	1	yes	characteristic
48	C₄	—	2	3	no	—
49	C₄	—	4	2	no	—
50	C₄	C₂×D₈	1	6	yes	—
51	C₄	C₂×#41	1	1	yes	characteristic
52	C₄	H(5,𝔽₂)	1	1	yes	characteristic
53	C₂	—	8	6	no	—
54	C₂	#550	1	1	yes	characteristic
55	C₂	C₂²×D₈	1	1	yes	characteristic
56	C₂	#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

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