Subgroups — #352882

Order 512.

986 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33312	C₂	1	2	yes	maximal
2	#33225	C₂	1	4	yes	maximal
3	C₂×#1710	C₂	1	8	yes	maximal
4	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	C₂²	1	8	yes	—
6	C₂×#487	C₂²	1	16	yes	—
7	#2868	C₂²	1	4	yes	—
8	#1956	—	2	16	no	—
9	#1845	—	2	8	no	—
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₄²×C₈	C₂²	1	2	yes	—
12	#1710	—	2	32	no	—
13	C₂×C₈²	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈	—	2	8	no	—
15	C₂×C₄⋊Q₈	C₂³	1	8	yes	—
16	#500	—	2	32	no	—
17	#487	—	2	64	no	—
18	C₂×#107	—	2	16	no	—
19	C₂×#103	—	2	32	no	—
20	C₂×C₄×C₈	D₄	1	8	yes	—
21	C₂×C₄×C₈	D₄	1	4	yes	—
22	C₂×C₄×C₈	C₂³	1	4	yes	—
23	C₂×C₄×C₈	C₂³	1	2	yes	—
24	#384	—	2	8	no	—
25	#384	—	2	8	no	—
26	C₄³	C₂³	1	1	yes	characteristic
27	C₈²	D₄	1	16	yes	—
28	C₂²×Q₈	—	2	8	no	—
29	C₂²×C₈	C₂×D₄	1	8	yes	—
30	C₂²×C₈	C₂×D₄	1	4	yes	—
31	C₄⋊Q₈	—	2	32	no	—
32	C₄⋊Q₈	—	4	32	no	—
33	C₂×C₄⋊C₄	—	2	8	no	—
34	C₂×C₄⋊C₄	—	2	8	no	—
35	C₂×C₄⋊C₄	—	4	16	no	—
36	C₂×C₄²	C₂×D₄	1	2	yes	—
37	C₂×C₄²	C₂×D₄	1	4	yes	—
38	C₂×C₄²	C₂⁴	1	1	yes	characteristic
39	#107	—	4	32	no	—
40	#103	—	4	64	no	—
41	C₄×C₈	—	2	16	no	—
42	C₄×C₈	Q₁₆	1	16	yes	—
43	C₄×C₈	C₂×D₄	1	16	yes	—
44	C₄×C₈	C₂×D₄	1	8	yes	—
45	C₂×Q₈	—	4	16	no	—
46	C₂×Q₈	—	4	32	no	—
47	C₂²×C₄	—	4	8	no	—
48	C₂²×C₄	#127	1	4	yes	—
49	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₄	C₂²×D₄	1	2	yes	—
51	C₂×C₈	—	2	16	no	—
52	C₂×C₈	#127	1	16	yes	—
53	C₂×C₈	#127	1	8	yes	—
54	C₂×C₈	C₂×Q₁₆	1	16	yes	—
55	C₄⋊C₄	—	4	16	no	—
56	C₄⋊C₄	—	4	16	no	—
57	C₄⋊C₄	—	8	32	no	—
58	C₄²	C₂×Q₁₆	1	8	yes	—
59	C₄²	#135	1	16	yes	—
60	C₄²	C₂²×D₄	1	4	yes	—
61	C₂³	C₂×#127	1	1	yes	characteristic
62	Q₈	—	8	32	no	—
63	C₂×C₄	—	8	8	no	—
64	C₂×C₄	—	8	16	no	—
65	C₂×C₄	#494	1	8	yes	—
66	C₂×C₄	#495	1	16	yes	—
67	C₂×C₄	C₂×#127	1	2	yes	—
68	C₂×C₄	C₂×#127	1	4	yes	—
69	C₂×C₄	C₂²×Q₁₆	1	4	yes	—
70	C₂×C₄	C₂×#135	1	8	yes	—
71	C₈	—	2	16	no	—
72	C₈	#494	1	16	yes	—
73	C₂²	#2867	1	1	yes	characteristic
74	C₂²	C₂×#494	1	2	yes	—
75	C₂²	C₂×#495	1	4	yes	—
76	C₄	—	16	8	no	—
77	C₄	#1710	1	16	yes	complemented
78	C₄	C₂×#494	1	4	yes	—
79	C₄	#3147	1	8	yes	—
80	C₂	C₂×#1710	1	4	yes	—
81	C₂	#33302	1	1	yes	characteristic
82	C₂	#33303	1	2	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
77. C₄	#1710	semidirect

Maximal subgroups

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

subgroup	index
#33312	2
#33225	2
C₂×#1710	2
C₄×C₈²	2