Subgroups — #94606

Order 512.

73 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13647	C₂	1	1	yes	maximal, characteristic
2	#7544	C₂	1	1	yes	maximal, characteristic, complemented
3	#7510	C₂	1	1	yes	maximal, characteristic, complemented
4	#2258	—	2	1	no	—
5	#2170	C₄	1	1	yes	characteristic
6	M₁₂₈	—	2	1	no	—
7	C₂×C₆₄	—	2	1	no	—
8	#1422	C₄	1	1	yes	characteristic
9	#1401	—	2	1	no	—
10	C₄×C₃₂	C₂²	1	1	yes	characteristic
11	#504	—	2	1	no	—
12	#443	C₈	1	1	yes	characteristic
13	M₆₄	—	2	1	no	—
14	M₆₄	—	4	1	no	—
15	C₂×C₃₂	—	4	1	no	—
16	C₂×C₃₂	D₄	1	1	yes	characteristic
17	C₂×C₃₂	D₄	1	1	yes	characteristic
18	#364	C₈	1	1	yes	characteristic
19	#350	—	2	1	no	—
20	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₆₄	—	2	2	no	—
22	C₆₄	—	4	1	no	—
23	#135	C₁₆	1	1	yes	characteristic
24	#131	—	2	1	no	—
25	M₃₂	—	2	1	no	—
26	M₃₂	—	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	#108	C₁₆	1	1	yes	characteristic
31	#104	—	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	#41	—	2	1	no	—
38	Q₁₆	C₃₂	1	1	yes	characteristic
39	SD₁₆	—	2	1	no	—
40	D₈	C₃₂	1	1	yes	characteristic
41	M₁₆	—	2	1	no	—
42	M₁₆	—	4	1	no	—
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
46	C₄²	M₃₂	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	Q₈	—	2	1	no	—
52	D₄	—	2	1	no	—
53	D₄	—	4	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
57	C₈	—	4	1	no	—
58	C₈	D₄⋊C₈	1	1	yes	characteristic
59	C₈	#350	1	1	yes	characteristic
60	C₈	C₂×C₃₂	1	1	yes	characteristic
61	C₈	M₆₄	1	1	yes	characteristic
62	C₂²	—	4	1	no	—
63	C₂²	#1401	1	1	yes	characteristic
64	C₄	—	4	1	no	—
65	C₄	—	4	1	no	—
66	C₄	D₄⋊C₁₆	1	1	yes	characteristic
67	C₄	C₂²⋊C₃₂	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	8	1	no	—
70	C₂	D₄⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#102
commutator	C₈	C₂×C₃₂
Frattini, cosocle	C₄×C₃₂	C₂²
socle	C₂	D₄⋊C₃₂
prime radical	#135	C₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ 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. #7544	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
#13647	2
#7544	2
#7510	2