Subgroups — #353047

Order 512.

906 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33312	C₂	1	6	yes	maximal
2	#8818	C₂	1	8	yes	maximal
3	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
4	C₂×#500	—	2	12	no	—
5	#2868	C₂²	1	4	yes	—
6	#1845	—	2	12	no	—
7	#1845	C₂²	1	24	yes	—
8	C₄²×C₈	C₂²	1	3	yes	—
9	C₂×C₈²	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈	—	2	12	no	—
11	#500	—	4	48	no	—
12	C₂×#107	—	2	48	no	—
13	C₂×C₄×C₈	D₄	1	6	yes	—
14	C₂×C₄×C₈	D₄	1	6	yes	—
15	C₂×C₄×C₈	C₂³	1	6	yes	—
16	#384	—	2	12	no	—
17	#384	C₂³	1	8	yes	—
18	C₄³	C₂³	1	1	yes	characteristic
19	C₈²	Q₈	1	16	yes	complemented
20	C₂²×Q₈	—	4	4	no	—
21	C₂²×C₈	C₂×D₄	1	12	yes	—
22	C₄⋊Q₈	—	4	48	no	—
23	C₂×C₄⋊C₄	—	2	24	no	—
24	C₂×C₄⋊C₄	—	4	12	no	—
25	C₂×C₄²	C₂×D₄	1	3	yes	—
26	C₂×C₄²	C₂×D₄	1	3	yes	—
27	C₂×C₄²	C₂⁴	1	1	yes	characteristic
28	#107	—	4	96	no	—
29	C₄×C₈	D₈	1	24	yes	—
30	C₄×C₈	Q₁₆	1	24	yes	—
31	C₄×C₈	C₂×Q₈	1	24	yes	—
32	C₂×Q₈	—	8	24	no	—
33	C₂²×C₄	—	4	8	no	—
34	C₂²×C₄	#127	1	1	yes	characteristic
35	C₂²×C₄	#127	1	3	yes	—
36	C₂²×C₄	C₂²×D₄	1	3	yes	—
37	C₂×C₈	C₄⋊Q₈	1	24	yes	—
38	C₂×C₈	C₂×D₈	1	24	yes	—
39	C₂×C₈	C₂×Q₁₆	1	24	yes	—
40	C₄⋊C₄	—	4	48	no	—
41	C₄⋊C₄	—	8	24	no	—
42	C₄²	C₂×D₈	1	12	yes	—
43	C₄²	C₂×Q₁₆	1	12	yes	—
44	C₄²	C₂²×Q₈	1	4	yes	—
45	C₂³	C₂×#127	1	1	yes	characteristic
46	Q₈	—	16	16	no	—
47	C₂×C₄	—	8	24	no	—
48	C₂×C₄	#493	1	6	yes	—
49	C₂×C₄	#493	1	6	yes	—
50	C₂×C₄	#494	1	12	yes	—
51	C₂×C₄	C₂×C₄⋊Q₈	1	6	yes	—
52	C₂×C₄	C₂²×D₈	1	6	yes	—
53	C₂×C₄	C₂²×Q₁₆	1	6	yes	—
54	C₈	#500	1	48	yes	complemented
55	C₂²	#2868	1	1	yes	characteristic
56	C₂²	C₂×#493	1	3	yes	—
57	C₂²	C₂×#494	1	3	yes	—
58	C₄	—	16	8	no	—
59	C₄	#1712	1	4	yes	—
60	C₄	#1713	1	12	yes	—
61	C₄	C₂×#500	1	12	yes	—
62	C₂	C₂×#1712	1	1	yes	characteristic
63	C₂	C₂×#1713	1	3	yes	—
64	C₂	#33312	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	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
19. C₈²	Q₈	semidirect
54. C₈	#500	semidirect

Maximal subgroups

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

subgroup	index
#33312	2
#8818	2
C₄×C₈²	2