Subgroups — #1666

Order 128.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#492	C₂	1	1	yes	maximal, characteristic
2	#475	C₂	1	1	yes	maximal, characteristic, complemented
3	#460	C₂	1	1	yes	maximal, characteristic
4	C₈×Q₈	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
8	C₂×SD₁₆	—	2	1	no	—
9	C₂×SD₁₆	—	2	1	no	—
10	C₂×SD₁₆	—	2	1	no	—
11	C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	#127	C₂²	1	1	yes	characteristic
13	C₄×Q₈	C₂²	1	1	yes	characteristic
14	#106	—	2	1	no	—
15	C₄⋊C₈	—	2	1	no	—
16	C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊C₈	C₂²	1	1	yes	characteristic
19	#103	—	2	1	no	—
20	#102	—	2	1	no	—
21	C₄×C₈	—	2	1	no	—
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈	—	2	1	no	—
24	C₂×Q₈	D₄	1	1	yes	characteristic
25	C₂×D₄	—	2	1	no	—
26	C₂×D₄	—	2	1	no	—
27	C₂×D₄	—	2	1	no	—
28	SD₁₆	—	4	2	no	—
29	SD₁₆	—	4	2	no	—
30	SD₁₆	—	4	2	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	D₄	1	1	yes	characteristic
35	C₂×C₈	D₄	1	1	yes	characteristic
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	—	2	1	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₄²	—	2	1	no	—
41	C₄²	C₂³	1	1	yes	characteristic
42	C₂³	—	4	1	no	—
43	Q₈	—	2	1	no	—
44	Q₈	—	4	1	no	—
45	Q₈	—	4	1	no	—
46	Q₈	SD₁₆	1	2	yes	complemented
47	D₄	—	4	1	no	—
48	D₄	—	4	1	no	—
49	D₄	—	4	1	no	—
50	D₄	—	4	1	no	—
51	D₄	—	8	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄	#41	1	1	yes	characteristic
58	C₈	—	2	1	no	—
59	C₈	—	4	1	no	—
60	C₈	—	4	1	no	—
61	C₈	—	4	1	no	—
62	C₈	SD₁₆	1	2	yes	complemented
63	C₂²	—	8	1	no	—
64	C₂²	—	8	1	no	—
65	C₂²	—	8	1	no	—
66	C₂²	#121	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	8	1	no	—
71	C₄	C₂×SD₁₆	1	1	yes	characteristic
72	C₄	C₂×SD₁₆	1	1	yes	characteristic
73	C₄	#135	1	1	yes	characteristic
74	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂	—	16	1	no	—
76	C₂	#460	1	1	yes	characteristic
77	C₂	#465	1	1	yes	characteristic
78	C₂	#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	#127	C₂²
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
2. #475	C₂	semidirect
4. C₈×Q₈	C₂	semidirect
5. #334	C₂	semidirect
6. #332	C₂	semidirect
46. Q₈	SD₁₆	semidirect
62. C₈	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#492	2
#475	2
#460	2
C₈×Q₈	2
#334	2
#332	2
#331	2