Subgroups — #93757

Order 512.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12045	C₂	1	1	yes	maximal, characteristic
2	#7332	C₂	1	1	yes	maximal, characteristic, complemented
3	#7099	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₁₆	C₄	1	1	yes	characteristic
5	C₈×D₈	—	2	1	no	—
6	#1397	—	2	1	no	—
7	#1372	C₄	1	1	yes	characteristic
8	#1367	—	2	1	no	—
9	#1331	—	2	1	no	—
10	#1311	C₂²	1	1	yes	characteristic
11	C₂×D₁₆	C₈	1	1	yes	characteristic
12	C₄×D₈	—	2	1	no	—
13	C₈×D₄	—	4	1	no	—
14	C₂×C₃₂	—	2	2	no	—
15	#366	C₈	1	1	yes	characteristic
16	C₄⋊C₁₆	—	4	1	no	—
17	#357	—	2	1	no	—
18	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₄×C₁₆	D₄	1	1	yes	characteristic
20	#335	—	2	1	no	—
21	D₄⋊C₈	—	4	1	no	—
22	C₈²	D₄	1	1	yes	characteristic
23	C₂×D₈	—	2	1	no	—
24	C₂²×C₈	—	8	1	no	—
25	C₄×D₄	—	4	1	no	—
26	D₁₆	—	2	1	no	—
27	D₁₆	C₁₆	1	2	yes	complemented
28	C₂×C₁₆	—	2	1	no	—
29	C₂×C₁₆	—	8	1	no	—
30	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
31	C₂×C₁₆	M₁₆	1	1	yes	characteristic
32	#107	—	2	1	no	—
33	C₄⋊C₈	—	4	1	no	—
34	#102	—	4	1	no	—
35	C₂²⋊C₈	—	8	1	no	—
36	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×C₈	D₈	1	1	yes	characteristic
38	C₄×C₈	SD₁₆	1	1	yes	characteristic
39	C₃₂	—	4	2	no	—
40	C₂×D₄	—	4	1	no	—
41	C₂²×C₄	—	8	1	no	—
42	D₈	—	2	2	no	—
43	D₈	—	4	1	no	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	8	1	no	—
47	C₂×C₈	—	16	2	no	—
48	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₈	#104	1	1	yes	characteristic
50	C₄⋊C₄	—	4	1	no	—
51	C₂²⋊C₄	—	8	1	no	—
52	C₄²	#102	1	1	yes	characteristic
53	C₁₆	—	2	2	no	—
54	C₁₆	—	2	1	no	—
55	C₁₆	—	16	1	no	—
56	C₁₆	C₂×C₁₆	1	1	yes	characteristic
57	C₁₆	M₃₂	1	1	yes	characteristic
58	C₂³	—	8	1	no	—
59	D₄	—	4	2	no	—
60	D₄	—	8	1	no	—
61	C₂×C₄	—	8	1	no	—
62	C₂×C₄	—	16	1	no	—
63	C₂×C₄	—	16	1	no	—
64	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄	#360	1	1	yes	characteristic
66	C₂×C₄	#361	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	4	1	no	—
69	C₈	—	4	1	no	—
70	C₈	—	16	1	no	—
71	C₈	C₂²⋊C₁₆	1	1	yes	characteristic
72	C₈	#350	1	1	yes	characteristic
73	C₂²	—	8	2	no	—
74	C₂²	—	16	1	no	—
75	C₂²	—	16	1	no	—
76	C₂²	#1333	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	1	no	—
79	C₄	—	16	1	no	—
80	C₄	D₄⋊C₁₆	1	1	yes	characteristic
81	C₄	#1336	1	1	yes	characteristic
82	C₂	—	16	2	no	—
83	C₂	#7070	1	1	yes	characteristic
84	C₂	#7389	1	1	yes	characteristic
85	C₂	#7390	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1333
commutator	C₁₆	C₂×C₁₆
Frattini, cosocle	#1311	C₂²
prime radical	C₂×D₁₆	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #7332	C₂	semidirect
3. #7099	C₂	semidirect
27. D₁₆	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#12045	2
#7332	2
#7099	2