Subgroups — #2212

Order 128.

77 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#510	C₂	1	1	yes	maximal, characteristic, complemented
2	#509	C₂	1	1	yes	maximal, characteristic
3	#495	C₂	1	1	yes	maximal, characteristic, complemented
4	#443	C₂	1	1	yes	maximal, characteristic, complemented
5	#364	C₂	1	1	yes	maximal, characteristic, complemented
6	#362	C₂	1	1	yes	maximal, characteristic, complemented
7	#361	C₂	1	1	yes	maximal, characteristic, complemented
8	#135	C₂²	1	1	yes	characteristic
9	C₂×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×SD₁₆	—	2	1	no	—
11	C₂×D₈	C₂²	1	1	yes	characteristic
12	#131	—	2	1	no	—
13	#124	—	2	1	no	—
14	Q₃₂	—	2	1	no	—
15	SD₃₂	—	2	1	no	—
16	SD₃₂	—	2	1	no	—
17	D₁₆	—	2	1	no	—
18	M₃₂	C₂²	1	1	yes	characteristic
19	M₃₂	C₂²	1	1	yes	characteristic
20	#108	C₂²	1	1	yes	characteristic
21	#104	—	2	1	no	—
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	#41	—	2	1	no	—
24	C₂×Q₈	—	2	1	no	—
25	C₂×D₄	—	2	1	no	—
26	Q₁₆	—	2	1	no	—
27	Q₁₆	—	2	1	no	—
28	Q₁₆	D₄	1	1	yes	characteristic
29	SD₁₆	—	2	1	no	—
30	SD₁₆	—	4	1	no	—
31	SD₁₆	—	4	1	no	—
32	D₈	—	2	1	no	—
33	D₈	—	2	1	no	—
34	D₈	D₄	1	1	yes	characteristic
35	M₁₆	—	2	1	no	—
36	M₁₆	—	4	1	no	—
37	C₂×C₈	—	4	1	no	—
38	C₂×C₈	D₄	1	1	yes	characteristic
39	C₂×C₈	C₂³	1	1	yes	characteristic
40	C₂²⋊C₄	—	4	2	no	—
41	C₄²	D₄	1	1	yes	characteristic
42	C₁₆	—	2	1	no	—
43	C₁₆	—	2	1	no	—
44	C₂³	—	4	1	no	—
45	Q₈	—	2	1	no	—
46	Q₈	—	4	1	no	—
47	Q₈	—	4	1	no	—
48	D₄	—	2	1	no	—
49	D₄	—	4	1	no	—
50	D₄	—	4	1	no	—
51	D₄	—	4	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	C₂×D₄	1	1	yes	characteristic
56	C₈	—	4	1	no	—
57	C₈	D₈	1	1	yes	characteristic
58	C₈	D₈	1	1	yes	characteristic
59	C₈	C₂×D₄	1	1	yes	characteristic
60	C₈	#41	1	1	yes	characteristic
61	C₂²	—	4	1	no	—
62	C₂²	—	8	2	no	—
63	C₂²	—	8	1	no	—
64	C₂²	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄	—	4	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	8	1	no	—
68	C₄	#121	1	1	yes	characteristic
69	C₄	C₂×D₈	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	8	1	no	—
72	C₂	—	16	1	no	—
73	C₂	#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#459
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 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. #510	C₂	semidirect
3. #495	C₂	semidirect
4. #443	C₂	semidirect
5. #364	C₂	semidirect
6. #362	C₂	semidirect
7. #361	C₂	semidirect

Maximal subgroups

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

subgroup	index
#510	2
#509	2
#495	2
#443	2
#364	2
#362	2
#361	2