Subgroups — #352885

Order 512.

1922 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33301	C₂	1	6	yes	maximal, complemented
2	C₂×#1712	C₂	1	8	yes	maximal, complemented
3	C₄×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#493	—	2	24	no	—
5	C₂×#493	C₂²	1	24	yes	—
6	#2867	C₂²	1	4	yes	—
7	C₄²×C₈	C₂²	1	3	yes	—
8	#1712	—	2	32	no	—
9	C₂×C₈²	C₂²	1	4	yes	—
10	C₂²×D₈	—	2	48	no	—
11	C₂×#127	—	2	24	no	—
12	C₂×#127	C₂³	1	8	yes	—
13	#493	—	2	96	no	—
14	#493	—	4	96	no	—
15	C₂×C₄×C₈	D₄	1	12	yes	—
16	C₂×C₄×C₈	C₂³	1	6	yes	—
17	C₄³	C₂³	1	1	yes	characteristic
18	C₈²	D₄	1	16	yes	complemented
19	C₂²×D₄	—	2	24	no	—
20	C₂²×D₄	—	4	16	no	—
21	C₂×D₈	—	4	96	no	—
22	C₂×D₈	—	4	192	no	—
23	C₂²×C₈	C₂×D₄	1	12	yes	—
24	#127	—	2	32	no	—
25	#127	—	4	96	no	—
26	C₂×C₄²	C₂×D₄	1	6	yes	—
27	C₂×C₄²	C₂⁴	1	1	yes	characteristic
28	C₄×C₈	D₈	1	48	yes	complemented
29	C₄×C₈	C₂×D₄	1	24	yes	—
30	C₂⁴	—	4	8	no	—
31	C₂×D₄	—	4	48	no	—
32	C₂×D₄	—	4	96	no	—
33	C₂×D₄	—	8	96	no	—
34	C₂²×C₄	#127	1	4	yes	—
35	C₂²×C₄	C₂²×D₄	1	3	yes	—
36	D₈	—	8	192	no	—
37	C₂×C₈	#127	1	24	yes	—
38	C₂×C₈	C₂×D₈	1	48	yes	—
39	C₄²	C₂×D₈	1	24	yes	—
40	C₄²	C₂²×D₄	1	4	yes	—
41	C₂³	—	8	8	no	—
42	C₂³	—	8	48	no	—
43	C₂³	C₂×#127	1	1	yes	characteristic
44	D₄	—	8	96	no	—
45	D₄	—	16	64	no	—
46	C₂×C₄	#493	1	24	yes	—
47	C₂×C₄	C₂×#127	1	6	yes	—
48	C₂×C₄	C₂²×D₈	1	12	yes	—
49	C₈	#493	1	48	yes	complemented
50	C₂²	—	16	24	no	—
51	C₂²	—	16	32	no	—
52	C₂²	#2867	1	1	yes	characteristic
53	C₂²	C₂×#493	1	6	yes	—
54	C₄	#1712	1	16	yes	complemented
55	C₄	C₂×#493	1	12	yes	—
56	C₂	—	32	8	no	—
57	C₂	C₂×#1712	1	4	yes	—
58	C₂	#33301	1	3	yes	—

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33301	C₂	semidirect
2. C₂×#1712	C₂	semidirect
3. C₄×C₈²	C₂	semidirect
18. C₈²	D₄	semidirect
28. C₄×C₈	D₈	semidirect
49. C₈	#493	semidirect
54. C₄	#1712	semidirect

Maximal subgroups

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

subgroup	index
#33301	2
C₂×#1712	2
C₄×C₈²	2