Subgroups — #1701

Order 128.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#486	C₂	1	1	yes	maximal, characteristic, complemented
2	#461	C₂	1	1	yes	maximal, characteristic, complemented
3	#459	C₂	1	1	yes	maximal, characteristic
4	#436	C₂	1	1	yes	maximal, characteristic, complemented
5	#334	C₂	1	1	yes	maximal, characteristic, complemented
6	#332	C₂	1	1	yes	maximal, characteristic, complemented
7	#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×SD₁₆	—	2	1	no	—
9	C₂×D₈	—	2	1	no	—
10	C₂×M₁₆	—	2	1	no	—
11	C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	#127	C₂²	1	1	yes	characteristic
13	C₄×D₄	C₂²	1	1	yes	characteristic
14	C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₈	C₂²	1	1	yes	characteristic
17	#103	—	2	1	no	—
18	#102	—	2	1	no	—
19	#102	—	2	1	no	—
20	#102	—	2	1	no	—
21	C₂²⋊C₈	—	2	1	no	—
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈	—	2	1	no	—
24	C₂×D₄	—	2	1	no	—
25	C₂×D₄	—	2	1	no	—
26	C₂×D₄	—	2	1	no	—
27	C₂×D₄	D₄	1	1	yes	characteristic
28	C₂²×C₄	—	2	1	no	—
29	SD₁₆	—	4	2	no	—
30	D₈	—	4	2	no	—
31	M₁₆	—	4	2	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	D₄	1	1	yes	characteristic
36	C₂×C₈	D₄	1	1	yes	characteristic
37	C₄⋊C₄	—	2	1	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₂²⋊C₄	—	2	1	no	—
41	C₄²	C₂³	1	1	yes	characteristic
42	C₂³	—	2	1	no	—
43	C₂³	—	4	1	no	—
44	Q₈	—	4	1	no	—
45	Q₈	—	4	1	no	—
46	D₄	—	2	1	no	—
47	D₄	—	2	1	no	—
48	D₄	—	4	1	no	—
49	D₄	—	4	1	no	—
50	D₄	—	4	1	no	—
51	D₄	—	4	1	no	—
52	D₄	—	8	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	4	1	no	—
57	C₂×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄	#41	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	—	2	1	no	—
62	C₈	—	4	1	no	—
63	C₈	—	4	1	no	—
64	C₈	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	8	1	no	—
69	C₂²	—	8	1	no	—
70	C₂²	—	8	1	no	—
71	C₂²	#121	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	8	1	no	—
75	C₄	#135	1	1	yes	characteristic
76	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂	—	8	1	no	—
80	C₂	—	16	1	no	—
81	C₂	#463	1	1	yes	characteristic
82	C₂	#469	1	1	yes	characteristic
83	C₂	#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	#459	C₂
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. #486	C₂	semidirect
2. #461	C₂	semidirect
4. #436	C₂	semidirect
5. #334	C₂	semidirect
6. #332	C₂	semidirect
7. #331	C₂	semidirect

Maximal subgroups

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

subgroup	index
#486	2
#461	2
#459	2
#436	2
#334	2
#332	2
#331	2