Subgroups — #1667

Order 128.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#493	C₂	1	1	yes	maximal, characteristic, complemented
2	#459	C₂	1	2	yes	maximal, complemented
3	C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#335	C₂	1	1	yes	maximal, characteristic, complemented
5	#331	C₂	1	2	yes	maximal, complemented
6	C₂×D₈	—	2	2	no	—
7	C₂×D₈	—	2	1	no	—
8	C₂×D₈	—	2	1	no	—
9	C₂²×C₈	—	2	1	no	—
10	#127	C₂²	1	2	yes	—
11	C₄×D₄	C₂²	1	1	yes	characteristic
12	C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊C₈	C₂²	1	2	yes	—
14	#102	—	2	2	no	—
15	C₂²⋊C₈	—	2	1	no	—
16	C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×D₄	—	2	2	no	—
18	C₂×D₄	—	2	2	no	—
19	C₂×D₄	—	2	2	no	—
20	C₂×D₄	D₄	1	1	yes	characteristic
21	C₂²×C₄	—	2	1	no	—
22	D₈	—	4	2	no	—
23	D₈	—	4	4	no	—
24	D₈	—	4	2	no	—
25	C₂×C₈	—	2	1	no	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	4	2	no	—
28	C₂×C₈	D₄	1	1	yes	characteristic
29	C₂×C₈	D₄	1	1	yes	characteristic
30	C₄⋊C₄	D₄	1	1	yes	characteristic
31	C₂²⋊C₄	—	2	1	no	—
32	C₄²	C₂³	1	1	yes	characteristic
33	C₂³	—	2	1	no	—
34	C₂³	—	4	2	no	—
35	D₄	—	2	1	no	—
36	D₄	—	4	2	no	—
37	D₄	—	4	2	no	—
38	D₄	—	4	2	no	—
39	D₄	—	4	2	no	—
40	D₄	—	8	2	no	—
41	D₄	D₈	1	2	yes	complemented
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	4	1	no	—
45	C₂×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄	#41	1	1	yes	characteristic
48	C₈	—	2	1	no	—
49	C₈	—	4	1	no	—
50	C₈	—	4	2	no	—
51	C₈	D₈	1	2	yes	complemented
52	C₂²	—	2	2	no	—
53	C₂²	—	4	1	no	—
54	C₂²	—	4	1	no	—
55	C₂²	—	8	2	no	—
56	C₂²	—	8	2	no	—
57	C₂²	—	8	2	no	—
58	C₂²	#121	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	C₂×D₈	1	1	yes	characteristic
62	C₄	C₂×D₈	1	1	yes	characteristic
63	C₄	#135	1	1	yes	characteristic
64	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂	—	4	2	no	—
66	C₂	—	16	2	no	—
67	C₂	#459	1	1	yes	characteristic
68	C₂	#466	1	1	yes	characteristic
69	C₂	#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator, Frattini, cosocle	C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #493	C₂	semidirect
2. #459	C₂	semidirect
3. C₈×D₄	C₂	semidirect
4. #335	C₂	semidirect
5. #331	C₂	semidirect
41. D₄	D₈	semidirect
51. C₈	D₈	semidirect

Maximal subgroups

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

subgroup	index
#493	2
#459	2
C₈×D₄	2
#335	2
#331	2