Subgroups — #2191

Order 128.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#575	C₂	1	1	yes	maximal, characteristic
2	#510	C₂	1	2	yes	maximal, complemented
3	#470	C₂	1	1	yes	maximal, characteristic, complemented
4	#360	C₂	1	2	yes	maximal, complemented
5	#349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41	—	2	1	no	—
7	#137	—	2	2	no	—
8	AGL(1,ℤ/8)	—	2	2	no	—
9	#135	—	2	1	no	—
10	#135	C₂²	1	2	yes	—
11	C₂×Q₁₆	C₂²	1	1	yes	characteristic, complemented
12	C₂×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²⋊Q₈	—	2	1	no	—
14	Q₃₂	—	2	2	no	—
15	SD₃₂	—	2	2	no	—
16	M₃₂	C₂²	1	2	yes	complemented
17	#106	C₂²	1	1	yes	characteristic, complemented
18	#103	—	2	1	no	—
19	#41	—	2	2	no	—
20	#41	—	2	2	no	—
21	#41	—	4	1	no	—
22	#41	—	4	1	no	—
23	C₂×Q₈	—	2	1	no	—
24	C₂×Q₈	—	2	1	no	—
25	C₂×D₄	—	2	1	no	—
26	C₂×D₄	—	4	1	no	—
27	C₂²×C₄	—	4	1	no	—
28	C₂²×C₄	D₄	1	1	yes	characteristic
29	Q₁₆	—	2	1	no	—
30	Q₁₆	—	2	2	no	—
31	Q₁₆	D₄	1	2	yes	—
32	SD₁₆	—	2	2	no	—
33	SD₁₆	—	2	2	no	—
34	D₈	—	2	1	no	—
35	D₈	D₄	1	2	yes	—
36	M₁₆	—	2	2	no	—
37	C₂×C₈	D₄	1	1	yes	characteristic
38	C₂×C₈	C₂³	1	1	yes	characteristic
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	4	1	no	—
41	C₂²⋊C₄	—	4	1	no	—
42	C₁₆	—	2	2	no	—
43	C₂³	—	4	1	no	—
44	C₂³	D₈	1	1	yes	characteristic
45	Q₈	—	2	2	no	—
46	Q₈	—	4	1	no	—
47	Q₈	—	4	1	no	—
48	Q₈	—	4	1	no	—
49	D₄	—	2	2	no	—
50	D₄	—	4	1	no	—
51	D₄	—	4	2	no	—
52	D₄	—	4	2	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	4	1	no	—
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	D₈	1	1	yes	characteristic
61	C₂×C₄	C₂×D₄	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	C₂×D₄	1	2	yes	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	8	2	no	—
68	C₂²	C₂×D₈	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	8	1	no	—
72	C₄	—	8	1	no	—
73	C₄	#120	1	1	yes	characteristic
74	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	4	1	no	—
77	C₂	—	8	2	no	—
78	C₂	#447	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#447
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	#575	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #510	C₂	semidirect
3. #470	C₂	semidirect
4. #360	C₂	semidirect
5. #349	C₂	semidirect
11. C₂×Q₁₆	C₂²	semidirect
16. M₃₂	C₂²	semidirect
17. #106	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#575	2
#510	2
#470	2
#360	2
#349	2