Subgroups — #1622

Order 128.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#554	C₂	1	1	yes	maximal, characteristic, complemented
2	#492	C₂	1	1	yes	maximal, characteristic, complemented
3	#460	C₂	1	1	yes	maximal, characteristic, complemented
4	#459	C₂	1	1	yes	maximal, characteristic
5	#333	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×SD₁₆	—	2	1	no	—
9	C₂×SD₁₆	—	2	2	no	—
10	C₂×D₈	—	2	1	no	—
11	C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	#127	C₂²	1	1	yes	characteristic
13	#125	—	2	1	no	—
14	C₂²⋊Q₈	—	2	1	no	—
15	C₂²⋊Q₈	—	2	2	no	—
16	C₄×Q₈	C₂²	1	1	yes	characteristic
17	C₄×D₄	—	2	1	no	—
18	C₄×D₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄	—	2	1	no	—
20	C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊C₈	C₂²	1	1	yes	characteristic
22	#102	—	2	1	no	—
23	#102	—	2	1	no	—
24	C₄×C₈	C₂²	1	1	yes	characteristic
25	C₂×Q₈	D₄	1	1	yes	characteristic
26	C₂×Q₈	D₄	1	2	yes	—
27	C₂×D₄	—	2	1	no	—
28	C₂×D₄	—	2	1	no	—
29	C₂×D₄	—	2	1	no	—
30	C₂×D₄	D₄	1	1	yes	characteristic
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	—	2	1	no	—
33	SD₁₆	—	4	4	no	—
34	SD₁₆	—	4	2	no	—
35	D₈	—	4	2	no	—
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₄⋊C₄	—	2	2	no	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₂²⋊C₄	—	2	2	no	—
47	C₂²⋊C₄	—	2	1	no	—
48	C₄²	—	2	1	no	—
49	C₄²	C₂³	1	1	yes	characteristic
50	C₂³	—	2	1	no	—
51	C₂³	—	4	1	no	—
52	Q₈	—	2	1	no	—
53	Q₈	—	2	1	no	—
54	Q₈	—	4	2	no	—
55	D₄	—	2	1	no	—
56	D₄	—	4	2	no	—
57	D₄	—	4	1	no	—
58	D₄	—	4	1	no	—
59	D₄	—	4	1	no	—
60	D₄	—	4	1	no	—
61	D₄	SD₁₆	1	2	yes	complemented
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	4	1	no	—
68	C₂×C₄	—	8	2	no	—
69	C₂×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄	C₂×D₄	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	4	1	no	—
74	C₈	—	4	1	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	4	1	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	8	1	no	—
79	C₂²	—	8	1	no	—
80	C₂²	—	8	1	no	—
81	C₂²	#120	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	2	no	—
87	C₄	C₂×SD₁₆	1	1	yes	characteristic
88	C₄	#135	1	1	yes	characteristic
89	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₄	#137	1	1	yes	characteristic
91	C₂	—	4	2	no	—
92	C₂	—	16	1	no	—
93	C₂	#448	1	1	yes	characteristic
94	C₂	#449	1	1	yes	characteristic
95	C₂	#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#120
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. #554	C₂	semidirect
2. #492	C₂	semidirect
3. #460	C₂	semidirect
5. #333	C₂	semidirect
6. Q₈⋊C₈	C₂	semidirect
7. D₄⋊C₈	C₂	semidirect
61. D₄	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#554	2
#492	2
#460	2
#459	2
#333	2
Q₈⋊C₈	2
D₄⋊C₈	2