Subgroups — #94604

Order 512.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13646	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#7510	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×M₆₄	—	2	1	no	—
5	D₄×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₆₄	—	2	1	no	—
7	C₂×C₆₄	—	2	1	no	—
8	C₄⋊C₃₂	C₄	1	1	yes	characteristic
9	C₂²⋊C₃₂	—	2	1	no	—
10	C₄×C₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	—	2	1	no	—
12	C₈×D₄	C₈	1	1	yes	characteristic
13	M₆₄	—	4	1	no	—
14	M₆₄	—	4	1	no	—
15	C₂×C₃₂	—	2	1	no	—
16	C₂×C₃₂	D₄	1	1	yes	characteristic
17	C₂×C₃₂	D₄	1	1	yes	characteristic
18	C₄⋊C₁₆	C₈	1	1	yes	characteristic
19	C₂²⋊C₁₆	—	2	1	no	—
20	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₆₄	—	2	2	no	—
22	C₆₄	—	4	1	no	—
23	C₂²×C₈	—	2	1	no	—
24	C₄×D₄	C₁₆	1	1	yes	characteristic
25	C₂×C₁₆	—	2	1	no	—
26	C₂×C₁₆	—	4	1	no	—
27	C₂×C₁₆	—	4	1	no	—
28	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₁₆	M₁₆	1	1	yes	characteristic
30	C₄⋊C₈	C₁₆	1	1	yes	characteristic
31	C₂²⋊C₈	—	2	1	no	—
32	C₄×C₈	C₂×C₈	1	1	yes	characteristic
33	C₃₂	—	2	1	no	—
34	C₃₂	—	4	1	no	—
35	C₃₂	D₈	1	1	yes	characteristic
36	C₃₂	SD₁₆	1	1	yes	characteristic
37	C₂×D₄	C₃₂	1	1	yes	characteristic
38	C₂²×C₄	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	4	1	no	—
41	C₂×C₈	—	4	1	no	—
42	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×C₈	M₃₂	1	1	yes	characteristic
44	C₄⋊C₄	C₃₂	1	1	yes	characteristic
45	C₂²⋊C₄	—	2	1	no	—
46	C₄²	C₂×C₁₆	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	—	4	1	no	—
49	C₁₆	#102	1	1	yes	characteristic
50	C₁₆	#104	1	1	yes	characteristic
51	C₂³	—	2	1	no	—
52	D₄	—	4	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
57	C₂×C₄	C₂×C₃₂	1	1	yes	characteristic
58	C₂×C₄	M₆₄	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	4	1	no	—
61	C₈	D₄⋊C₈	1	1	yes	characteristic
62	C₈	#350	1	1	yes	characteristic
63	C₂²	—	4	1	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	C₂²⋊C₃₂	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	D₄⋊C₁₆	1	1	yes	characteristic
71	C₄	#1401	1	1	yes	characteristic
72	C₂	—	8	1	no	—
73	C₂	D₄⋊C₃₂	1	1	yes	characteristic
74	C₂	#7513	1	1	yes	characteristic
75	C₂	#7514	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₆₄	C₂	semidirect
3. #7510	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13646	2
C₄⋊C₆₄	2
#7510	2