Subgroups — #184688

Order 512.

908 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Q₈	C₂	1	3	yes	maximal
2	C₈×C₄⋊C₈	C₂	1	9	yes	maximal
3	#8131	C₂	1	3	yes	maximal
4	C₂×C₈×Q₈	C₄	1	6	yes	—
5	C₄²×Q₈	C₂²	1	1	yes	characteristic
6	C₈×C₄⋊C₄	C₄	1	18	yes	—
7	C₄×C₄⋊C₈	C₂²	1	9	yes	—
8	C₄²×C₈	C₂²	1	9	yes	—
9	C₂×C₈²	C₂²	1	6	yes	—
10	#1273	C₂²	1	9	yes	—
11	#1273	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈	C₂×C₄	1	3	yes	—
13	C₈×Q₈	C₈	1	48	yes	complemented
14	C₂×C₄⋊C₈	C₂×C₄	1	18	yes	—
15	C₂×C₄×C₈	C₂×C₄	1	18	yes	—
16	C₂×C₄×C₈	C₂×C₄	1	18	yes	—
17	C₂×C₄×C₈	C₂³	1	9	yes	—
18	C₂×C₄×C₈	C₂³	1	3	yes	—
19	C₄×C₄⋊C₄	C₂×C₄	1	9	yes	—
20	C₄³	C₂³	1	3	yes	—
21	#336	C₂×C₄	1	18	yes	—
22	C₈²	—	4	6	no	—
23	C₂²×Q₈	C₄²	1	1	yes	characteristic
24	C₂²×C₈	C₂²×C₄	1	18	yes	—
25	C₂²×C₈	C₂²×C₄	1	6	yes	—
26	C₄×Q₈	C₂×C₈	1	24	yes	—
27	C₂×C₄⋊C₄	C₄²	1	9	yes	—
28	C₂×C₄²	C₂²×C₄	1	9	yes	—
29	C₂×C₄²	C₂²×C₄	1	9	yes	—
30	C₂×C₄²	C₂⁴	1	1	yes	characteristic
31	C₄⋊C₈	C₂×C₈	1	72	yes	—
32	C₄×C₈	—	2	9	no	—
33	C₄×C₈	—	2	6	no	—
34	C₄×C₈	—	2	9	no	—
35	C₄×C₈	—	4	18	no	—
36	C₄×C₈	C₂×C₈	1	72	yes	—
37	#95	C₄²	1	6	yes	—
38	C₂×Q₈	C₄×C₈	1	12	yes	—
39	C₂²×C₄	C₂×C₄²	1	9	yes	—
40	C₂²×C₄	C₂×C₄²	1	3	yes	—
41	C₂²×C₄	C₂³×C₄	1	3	yes	—
42	C₂×C₈	—	2	6	no	—
43	C₂×C₈	—	2	18	no	—
44	C₂×C₈	—	2	6	no	—
45	C₂×C₈	—	2	18	no	—
46	C₂×C₈	C₂²×C₈	1	12	yes	—
47	C₂×C₈	C₂²×C₈	1	36	yes	—
48	C₄⋊C₄	C₄×C₈	1	36	yes	—
49	C₄²	—	2	18	no	—
50	C₄²	C₂²×C₈	1	36	yes	—
51	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
52	C₄²	2₋¹⁺⁴	1	3	yes	—
53	C₂³	C₂²×C₄²	1	1	yes	characteristic
54	Q₈	C₈²	1	16	yes	complemented
55	C₂×C₄	—	2	18	no	—
56	C₂×C₄	C₂×C₄×C₈	1	18	yes	—
57	C₂×C₄	C₂×C₄×C₈	1	18	yes	—
58	C₂×C₄	#519	1	3	yes	—
59	C₂×C₄	#520	1	3	yes	—
60	C₂×C₄	C₂³×C₈	1	6	yes	—
61	C₂×C₄	#568	1	6	yes	—
62	C₈	—	2	12	no	—
63	C₈	—	2	36	no	—
64	C₂²	#2276	1	1	yes	characteristic
65	C₂²	C₂²×C₄×C₈	1	3	yes	—
66	C₂²	#2875	1	3	yes	—
67	C₄	—	2	18	no	—
68	C₄	C₂×C₈²	1	12	yes	—
69	C₄	#2982	1	6	yes	—
70	C₄	#2988	1	6	yes	—
71	C₂	C₂²×C₈²	1	1	yes	characteristic
72	C₂	#17316	1	3	yes	—
73	C₂	#20569	1	3	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator	C₂	C₂²×C₈²
socle, prime radical	C₂³	C₂²×C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
13. C₈×Q₈	C₈	semidirect
54. Q₈	C₈²	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Q₈	2
C₈×C₄⋊C₈	2
#8131	2