Subgroups — #3147

Order 128.

286 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135	C₂	1	4	yes	maximal, complemented
2	#532	C₂	1	2	yes	maximal, complemented
3	#495	C₂	1	4	yes	maximal, complemented
4	#494	C₂	1	1	yes	maximal, characteristic, complemented
5	#493	C₂	1	1	yes	maximal, characteristic, complemented
6	#492	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41	C₂²	1	4	yes	—
9	#135	—	2	16	no	—
10	C₂×Q₁₆	C₂²	1	4	yes	—
11	C₂×SD₁₆	C₂²	1	8	yes	—
12	C₂×D₈	C₂²	1	4	yes	—
13	C₂²×C₈	C₂²	1	2	yes	—
14	C₄⋊Q₈	C₂²	1	2	yes	—
15	#127	C₂²	1	2	yes	—
16	#124	C₂²	1	4	yes	—
17	#121	—	2	4	no	—
18	C₄×D₄	—	2	4	no	—
19	C₂×C₄²	C₂²	1	1	yes	characteristic
20	C₄×C₈	C₂²	1	1	yes	characteristic
21	C₄×C₈	C₂²	1	1	yes	characteristic
22	C₄×C₈	C₂²	1	2	yes	—
23	#41	—	2	16	no	—
24	C₂×Q₈	C₂³	1	4	yes	—
25	C₂×D₄	—	2	4	no	—
26	C₂×D₄	—	2	4	no	—
27	C₂×D₄	C₂³	1	4	yes	—
28	C₂²×C₄	—	2	4	no	—
29	C₂²×C₄	D₄	1	2	yes	—
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	Q₁₆	—	2	8	no	—
32	SD₁₆	—	2	16	no	—
33	D₈	—	2	8	no	—
34	C₂×C₈	D₄	1	8	yes	complemented
35	C₂×C₈	C₂³	1	2	yes	—
36	C₂×C₈	C₂³	1	2	yes	—
37	C₄⋊C₄	—	2	4	no	—
38	C₂²⋊C₄	—	2	8	no	—
39	C₄²	D₄	1	2	yes	—
40	C₄²	C₂³	1	1	yes	characteristic
41	C₄²	C₂³	1	1	yes	characteristic
42	C₂³	—	2	4	no	—
43	C₂³	C₂×D₄	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	D₄	—	2	8	no	—
46	D₄	—	4	8	no	—
47	D₄	—	4	8	no	—
48	C₂×C₄	—	2	4	no	—
49	C₂×C₄	—	2	4	no	—
50	C₂×C₄	—	4	8	no	—
51	C₂×C₄	C₂×D₄	1	2	yes	—
52	C₂×C₄	C₂×D₄	1	4	yes	—
53	C₂×C₄	C₂×D₄	1	2	yes	—
54	C₂×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄	C₂⁴	1	1	yes	characteristic
56	C₈	C₂×D₄	1	4	yes	—
57	C₈	C₂×D₄	1	4	yes	—
58	C₂²	—	2	2	no	—
59	C₂²	—	4	4	no	—
60	C₂²	—	4	8	no	—
61	C₂²	#127	1	2	yes	—
62	C₂²	C₂²×D₄	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	4	4	no	—
65	C₄	#127	1	2	yes	—
66	C₄	#135	1	4	yes	complemented
67	C₄	C₂²×D₄	1	2	yes	—
68	C₂	—	2	2	no	—
69	C₂	—	8	4	no	—
70	C₂	C₂×#127	1	1	yes	characteristic
71	C₂	C₂×#135	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂²	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#135	C₂	semidirect
2. #532	C₂	semidirect
3. #495	C₂	semidirect
4. #494	C₂	semidirect
5. #493	C₂	semidirect
6. #492	C₂	semidirect
7. C₂×C₄×C₈	C₂	semidirect
34. C₂×C₈	D₄	semidirect
66. C₄	#135	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135	2
#532	2
#495	2
#494	2
#493	2
#492	2
C₂×C₄×C₈	2