Subgroups — #2206

Order 128.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆	C₂	1	2	yes	maximal, complemented
2	#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#357	C₂	1	2	yes	maximal, complemented
6	C₂×D₈	—	2	1	no	—
7	C₂×D₈	C₂²	1	2	yes	—
8	C₂×D₈	C₂²	1	1	yes	characteristic
9	#127	—	2	1	no	—
10	C₄×D₄	—	2	1	no	—
11	D₁₆	—	2	4	no	—
12	C₂×C₁₆	C₂²	1	2	yes	—
13	#107	C₂²	1	1	yes	characteristic
14	#102	—	2	1	no	—
15	C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×D₄	—	2	1	no	—
17	C₂×D₄	—	2	2	no	—
18	C₂×D₄	—	4	1	no	—
19	C₂×D₄	—	4	1	no	—
20	C₂²×C₄	—	4	1	no	—
21	D₈	—	2	2	no	—
22	D₈	—	2	1	no	—
23	D₈	—	2	2	no	—
24	D₈	—	4	2	no	—
25	D₈	D₄	1	2	yes	complemented
26	C₂×C₈	D₄	1	1	yes	characteristic
27	C₂×C₈	C₂³	1	1	yes	characteristic
28	C₄⋊C₄	—	2	1	no	—
29	C₂²⋊C₄	—	4	1	no	—
30	C₄²	D₄	1	1	yes	characteristic
31	C₁₆	—	2	2	no	—
32	C₂³	—	4	1	no	—
33	C₂³	—	4	2	no	—
34	D₄	—	2	2	no	—
35	D₄	—	4	2	no	—
36	D₄	—	4	1	no	—
37	D₄	—	4	2	no	—
38	D₄	—	8	2	no	—
39	D₄	—	8	2	no	—
40	C₂×C₄	—	4	1	no	—
41	C₂×C₄	—	8	2	no	—
42	C₂×C₄	D₈	1	1	yes	characteristic
43	C₂×C₄	D₈	1	1	yes	characteristic
44	C₂×C₄	C₂×D₄	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	C₂×D₄	1	1	yes	characteristic
47	C₈	#41	1	1	yes	characteristic
48	C₂²	—	4	2	no	—
49	C₂²	—	8	1	no	—
50	C₂²	—	8	1	no	—
51	C₂²	—	8	2	no	—
52	C₂²	—	8	2	no	—
53	C₂²	—	8	2	no	—
54	C₂²	C₂×D₈	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	—	8	1	no	—
57	C₄	D₁₆	1	2	yes	complemented
58	C₄	#121	1	1	yes	characteristic
59	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂	—	8	2	no	—
61	C₂	—	16	2	no	—
62	C₂	#459	1	1	yes	characteristic
63	C₂	C₂×D₁₆	1	1	yes	characteristic
64	C₂	#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	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

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₆	C₂	semidirect
2. #493	C₂	semidirect
3. C₄×D₈	C₂	semidirect
4. C₄⋊C₁₆	C₂	semidirect
5. #357	C₂	semidirect
25. D₈	D₄	semidirect
57. C₄	D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆	2
#493	2
C₄×D₈	2
C₄⋊C₁₆	2
#357	2