Subgroups — #1583

Order 128.

83 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic
3	#436	C₂	1	1	yes	maximal, characteristic
4	#334	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×SD₁₆	C₄	1	1	yes	characteristic
9	C₂×M₁₆	—	2	1	no	—
10	C₄×Q₈	C₂²	1	1	yes	characteristic
11	C₄×D₄	C₂²	1	1	yes	characteristic
12	#106	C₄	1	1	yes	characteristic
13	C₄⋊C₈	—	2	1	no	—
14	C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₈	C₂²	1	1	yes	characteristic
16	#103	C₄	1	1	yes	characteristic
17	#102	C₄	1	1	yes	characteristic
18	C₂²⋊C₈	—	2	1	no	—
19	C₄×C₈	—	2	1	no	—
20	C₄×C₈	C₂²	1	1	yes	characteristic
21	C₄×C₈	C₂²	1	1	yes	characteristic
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄	—	2	1	no	—
26	SD₁₆	—	2	2	no	—
27	M₁₆	—	4	2	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈	D₄	1	1	yes	characteristic
34	C₂×C₈	D₄	1	1	yes	characteristic
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₂²⋊C₄	—	2	1	no	—
39	C₄²	—	2	1	no	—
40	C₄²	C₂³	1	1	yes	characteristic
41	C₂³	—	2	1	no	—
42	Q₈	—	2	1	no	—
43	Q₈	M₁₆	1	2	yes	complemented
44	D₄	—	2	1	no	—
45	D₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄	#41	1	1	yes	characteristic
54	C₈	—	2	1	no	—
55	C₈	—	2	1	no	—
56	C₈	—	2	1	no	—
57	C₈	—	4	1	no	—
58	C₈	—	4	1	no	—
59	C₈	—	4	1	no	—
60	C₈	SD₁₆	1	2	yes	complemented
61	C₂²	—	4	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	1	no	—
64	C₂²	C₄×D₄	1	1	yes	characteristic
65	C₄	—	2	2	no	—
66	C₄	—	2	1	no	—
67	C₄	—	4	1	no	—
68	C₄	—	4	1	no	—
69	C₄	C₂×M₁₆	1	1	yes	characteristic
70	C₄	#131	1	1	yes	characteristic
71	C₄	C₂×SD₁₆	1	1	yes	characteristic
72	C₄	#135	1	1	yes	characteristic
73	C₂	—	8	1	no	—
74	C₂	#435	1	1	yes	characteristic
75	C₂	C₄×SD₁₆	1	1	yes	characteristic
76	C₂	#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₄²	C₂³
socle	C₂²	C₄×D₄
prime radical	C₂×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₈×Q₈	C₂	semidirect
4. #334	C₂	semidirect
5. Q₈⋊C₈	C₂	semidirect
7. #322	C₂	semidirect
43. Q₈	M₁₆	semidirect
60. C₈	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈	2
C₄×SD₁₆	2
#436	2
#334	2
Q₈⋊C₈	2
D₄⋊C₈	2
#322	2