Subgroups — #93755

Order 512.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₆×D₈	C₂	1	1	yes	maximal, characteristic
2	#7455	C₂	1	1	yes	maximal, characteristic, complemented
3	#7331	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₁₆	—	2	1	no	—
5	C₈×D₈	C₄	1	1	yes	characteristic
6	C₄⋊C₃₂	—	2	1	no	—
7	C₄×C₃₂	—	2	1	no	—
8	#1370	C₄	1	1	yes	characteristic
9	D₄⋊C₁₆	—	2	1	no	—
10	C₈×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	—	4	1	no	—
12	C₄×D₈	C₈	1	1	yes	characteristic
13	C₈×D₄	—	2	1	no	—
14	C₂×C₃₂	—	2	2	no	—
15	C₂×C₃₂	—	4	1	no	—
16	C₄⋊C₁₆	—	2	1	no	—
17	C₂²⋊C₁₆	—	4	1	no	—
18	C₄×C₁₆	D₄	1	1	yes	characteristic
19	C₄×C₁₆	D₄	1	1	yes	characteristic
20	#335	C₈	1	1	yes	characteristic
21	D₄⋊C₈	—	2	1	no	—
22	C₈²	C₂×C₄	1	1	yes	characteristic
23	C₂×D₈	C₁₆	1	1	yes	characteristic
24	C₂²×C₈	—	4	1	no	—
25	C₄×D₄	—	2	1	no	—
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	—	4	1	no	—
28	C₂×C₁₆	—	8	2	no	—
29	C₂×C₁₆	D₈	1	1	yes	characteristic
30	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
31	#107	C₁₆	1	1	yes	characteristic
32	C₄⋊C₈	—	2	1	no	—
33	#102	—	2	1	no	—
34	C₂²⋊C₈	—	4	1	no	—
35	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₈	C₂×C₈	1	1	yes	characteristic
37	C₄×C₈	M₁₆	1	1	yes	characteristic
38	C₃₂	—	4	2	no	—
39	C₃₂	—	8	1	no	—
40	C₂×D₄	—	2	1	no	—
41	C₂²×C₄	—	4	1	no	—
42	D₈	—	2	1	no	—
43	D₈	C₃₂	1	2	yes	complemented
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	4	1	no	—
46	C₂×C₈	—	8	1	no	—
47	C₂×C₈	—	8	1	no	—
48	C₂×C₈	#102	1	1	yes	characteristic
49	C₂×C₈	#104	1	1	yes	characteristic
50	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₈	M₃₂	1	1	yes	characteristic
52	C₄⋊C₄	—	2	1	no	—
53	C₂²⋊C₄	—	4	1	no	—
54	C₄²	C₂²⋊C₈	1	1	yes	characteristic
55	C₁₆	—	2	2	no	—
56	C₁₆	—	2	1	no	—
57	C₁₆	—	2	1	no	—
58	C₁₆	—	8	1	no	—
59	C₂³	—	4	1	no	—
60	D₄	—	2	2	no	—
61	D₄	—	4	1	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	—	8	1	no	—
64	C₂×C₄	—	8	1	no	—
65	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
67	C₂×C₄	#350	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	2	1	no	—
70	C₈	—	4	1	no	—
71	C₈	—	8	1	no	—
72	C₈	#360	1	1	yes	characteristic
73	C₈	#361	1	1	yes	characteristic
74	C₈	C₂×C₃₂	1	1	yes	characteristic
75	C₈	M₆₄	1	1	yes	characteristic
76	C₂²	—	4	2	no	—
77	C₂²	—	8	1	no	—
78	C₂²	—	8	1	no	—
79	C₂²	D₄⋊C₁₆	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	8	1	no	—
82	C₄	#1333	1	1	yes	characteristic
83	C₄	#1336	1	1	yes	characteristic
84	C₄	C₂²⋊C₃₂	1	1	yes	characteristic
85	C₄	#1401	1	1	yes	characteristic
86	C₂	—	8	2	no	—
87	C₂	#7070	1	1	yes	characteristic
88	C₂	D₄⋊C₃₂	1	1	yes	characteristic
89	C₂	#7382	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₂×C₃₂
Frattini, cosocle	C₈×C₁₆	C₂²
socle	C₂²	D₄⋊C₁₆
prime radical	C₂×D₈	C₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #7455	C₂	semidirect
3. #7331	C₂	semidirect
43. D₈	C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₁₆×D₈	2
#7455	2
#7331	2