Subgroups — #512520

Order 512.

1098 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2247	C₂	1	12	yes	maximal
2	#33302	C₂	1	2	yes	maximal
3	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₃₂	C₂²	1	16	yes	—
5	C₂×#494	—	2	6	no	—
6	C₂×#494	C₂²	1	12	yes	—
7	#2868	—	2	2	no	—
8	#2247	—	2	48	no	—
9	C₂×C₄×C₁₆	C₂²	1	6	yes	—
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂²×Q₁₆	—	2	12	no	—
12	C₂²×Q₁₆	C₂³	1	8	yes	—
13	C₂×C₄⋊Q₈	—	2	12	no	—
14	C₂×Q₃₂	—	2	96	no	—
15	C₂²×C₁₆	C₂³	1	4	yes	—
16	#494	—	2	48	no	—
17	#494	—	4	24	no	—
18	C₂×C₄×C₈	D₄	1	3	yes	—
19	C₂×C₄×C₈	C₂³	1	3	yes	—
20	#384	—	4	8	no	—
21	C₄³	D₄	1	1	yes	characteristic
22	C₄×C₁₆	D₄	1	24	yes	—
23	C₂²×Q₈	—	2	8	no	—
24	C₂×Q₁₆	—	2	48	no	—
25	C₂×Q₁₆	—	4	24	no	—
26	C₂×Q₁₆	—	4	48	no	—
27	C₂²×C₈	C₂×D₄	1	3	yes	—
28	C₂²×C₈	C₂⁴	1	1	yes	characteristic
29	C₄⋊Q₈	—	4	24	no	—
30	C₄⋊Q₈	—	4	24	no	—
31	C₂×C₄⋊C₄	—	4	24	no	—
32	C₂×C₄²	D₈	1	4	yes	—
33	C₂×C₄²	C₂×D₄	1	3	yes	—
34	Q₃₂	—	4	64	no	—
35	C₂×C₁₆	C₂×D₄	1	24	yes	—
36	C₄×C₈	D₈	1	12	yes	—
37	C₄×C₈	C₂×D₄	1	12	yes	—
38	C₂×Q₈	—	4	24	no	—
39	C₂×Q₈	—	4	24	no	—
40	C₂²×C₄	—	4	8	no	—
41	C₂²×C₄	C₂×D₈	1	6	yes	—
42	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
43	Q₁₆	—	4	32	no	—
44	Q₁₆	—	8	48	no	—
45	C₂×C₈	#127	1	6	yes	—
46	C₂×C₈	C₂×D₈	1	12	yes	—
47	C₂×C₈	C₂²×D₄	1	6	yes	—
48	C₄⋊C₄	—	8	48	no	—
49	C₄²	Q₃₂	1	16	yes	complemented
50	C₄²	#127	1	6	yes	—
51	C₄²	C₂×D₈	1	6	yes	—
52	C₁₆	#127	1	16	yes	—
53	C₂³	C₂²×D₈	1	1	yes	characteristic
54	Q₈	—	8	8	no	—
55	Q₈	—	8	24	no	—
56	C₂×C₄	—	8	24	no	—
57	C₂×C₄	#493	1	12	yes	—
58	C₂×C₄	C₂×Q₃₂	1	24	yes	—
59	C₂×C₄	C₂×#127	1	3	yes	—
60	C₂×C₄	C₂²×D₈	1	3	yes	—
61	C₈	#493	1	12	yes	—
62	C₈	C₂×#127	1	4	yes	—
63	C₂²	C₂×#493	1	3	yes	—
64	C₂²	C₂²×Q₃₂	1	4	yes	—
65	C₄	—	16	8	no	—
66	C₄	#2247	1	24	yes	complemented
67	C₄	#2867	1	1	yes	characteristic
68	C₄	C₂×#493	1	3	yes	—
69	C₂	#33301	1	1	yes	characteristic
70	C₂	C₂×#2247	1	6	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
49. C₄²	Q₃₂	semidirect
66. C₄	#2247	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2247	2
#33302	2
C₄²×C₁₆	2