Subgroups — #93753

Order 512.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₆×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#7458	C₂	1	1	yes	maximal, characteristic
3	C₈×C₃₂	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₁₆	—	2	1	no	—
5	C₈×Q₁₆	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	Q₈⋊C₁₆	—	2	1	no	—
10	C₈×C₁₆	C₂²	1	1	yes	characteristic
11	C₈×Q₈	—	2	1	no	—
12	C₄×Q₁₆	C₈	1	1	yes	characteristic
13	C₂×C₃₂	—	2	2	no	—
14	C₂×C₃₂	—	4	1	no	—
15	C₄⋊C₁₆	—	2	1	no	—
16	C₄⋊C₁₆	—	4	1	no	—
17	C₄×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	Q₈⋊C₈	—	2	1	no	—
22	C₈²	C₂×C₄	1	1	yes	characteristic
23	C₂×Q₁₆	C₁₆	1	1	yes	characteristic
24	C₄×Q₈	—	2	1	no	—
25	C₂×C₁₆	—	2	1	no	—
26	C₂×C₁₆	—	4	1	no	—
27	C₂×C₁₆	D₈	1	1	yes	characteristic
28	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
29	#107	C₁₆	1	1	yes	characteristic
30	C₄⋊C₈	—	2	1	no	—
31	C₄⋊C₈	—	4	1	no	—
32	#103	—	2	1	no	—
33	C₄×C₈	—	4	1	no	—
34	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
35	C₄×C₈	C₂×C₈	1	1	yes	characteristic
36	C₄×C₈	M₁₆	1	1	yes	characteristic
37	C₃₂	—	2	4	no	—
38	C₃₂	—	8	1	no	—
39	C₂×Q₈	—	2	1	no	—
40	Q₁₆	—	2	1	no	—
41	Q₁₆	C₃₂	1	2	yes	complemented
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	#102	1	1	yes	characteristic
45	C₂×C₈	#104	1	1	yes	characteristic
46	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₈	M₃₂	1	1	yes	characteristic
48	C₄⋊C₄	—	2	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄²	—	4	1	no	—
51	C₄²	C₂²⋊C₈	1	1	yes	characteristic
52	C₁₆	—	2	2	no	—
53	C₁₆	—	2	1	no	—
54	C₁₆	—	8	1	no	—
55	C₁₆	SD₃₂	1	1	yes	characteristic
56	C₁₆	Q₃₂	1	1	yes	characteristic
57	Q₈	—	2	2	no	—
58	Q₈	—	4	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
63	C₂×C₄	#350	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₈	—	2	1	no	—
66	C₈	—	2	1	no	—
67	C₈	—	8	1	no	—
68	C₈	#358	1	1	yes	characteristic
69	C₈	#359	1	1	yes	characteristic
70	C₈	C₂×C₃₂	1	1	yes	characteristic
71	C₈	M₆₄	1	1	yes	characteristic
72	C₂²	D₄⋊C₁₆	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	4	2	no	—
75	C₄	—	8	1	no	—
76	C₄	#1332	1	1	yes	characteristic
77	C₄	#1335	1	1	yes	characteristic
78	C₄	C₂²⋊C₃₂	1	1	yes	characteristic
79	C₄	#1401	1	1	yes	characteristic
80	C₂	#7069	1	1	yes	characteristic
81	C₂	D₄⋊C₃₂	1	1	yes	characteristic
82	C₂	#7381	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₈
commutator	C₈	C₂×C₃₂
Frattini, cosocle	C₈×C₁₆	C₂²
socle, prime radical	C₂²	D₄⋊C₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ 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
41. Q₁₆	C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₁₆×Q₁₆	2
#7458	2
C₈×C₃₂	2