Subgroups — #512806

Order 512.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃₂	C₂	1	8	yes	maximal, complemented
3	#13639	C₂	1	6	yes	maximal
4	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
5	Q₈×C₁₆	C₄	1	4	yes	—
6	Q₈×C₁₆	C₂²	1	4	yes	—
7	C₂×C₄⋊C₁₆	C₄	1	3	yes	—
8	C₂×C₄×C₁₆	C₂²	1	3	yes	—
9	C₄⋊C₃₂	C₂²	1	12	yes	—
10	C₂²⋊C₃₂	C₂²	1	4	yes	—
11	C₄×C₃₂	C₂²	1	12	yes	—
12	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
13	C₂²×C₁₆	C₂×C₄	1	3	yes	—
14	C₂²×C₁₆	C₂³	1	1	yes	characteristic
15	C₈×Q₈	C₈	1	4	yes	—
16	C₈×Q₈	C₂×C₄	1	4	yes	—
17	C₂×C₄⋊C₈	C₈	1	3	yes	—
18	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
19	C₂×C₃₂	C₂³	1	8	yes	—
20	C₄⋊C₁₆	C₂×C₄	1	12	yes	—
21	C₄×C₁₆	C₂×C₄	1	6	yes	—
22	C₄×C₁₆	C₂³	1	6	yes	—
23	C₂²×Q₈	C₁₆	1	1	yes	characteristic
24	C₂²×C₈	C₂×C₈	1	3	yes	—
25	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
26	C₄×Q₈	C₁₆	1	4	yes	—
27	C₄×Q₈	C₂×C₈	1	4	yes	—
28	C₂×C₄⋊C₄	C₁₆	1	3	yes	—
29	C₂×C₄²	C₂×C₈	1	3	yes	—
30	C₂×C₁₆	—	2	2	no	—
31	C₂×C₁₆	—	4	3	no	—
32	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₁₆	C₂²×C₄	1	6	yes	—
34	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
35	C₄⋊C₈	C₂×C₈	1	12	yes	—
36	C₄×C₈	C₂×C₈	1	6	yes	—
37	C₄×C₈	C₂²×C₄	1	6	yes	—
38	C₃₂	—	2	8	no	—
39	C₂×Q₈	C₃₂	1	8	yes	complemented
40	C₂×Q₈	C₂×C₁₆	1	4	yes	—
41	C₂²×C₄	C₂×C₁₆	1	3	yes	—
42	C₂²×C₄	C₂×C₁₆	1	3	yes	—
43	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	4	3	no	—
47	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₈	C₂²×C₈	1	6	yes	—
49	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
50	C₄⋊C₄	C₂×C₁₆	1	12	yes	—
51	C₄²	C₂×C₁₆	1	6	yes	—
52	C₄²	C₂²×C₈	1	6	yes	—
53	C₁₆	—	2	1	no	—
54	C₁₆	—	2	6	no	—
55	C₁₆	2₋¹⁺⁴	1	2	yes	—
56	C₂³	C₂²×C₁₆	1	1	yes	characteristic
57	Q₈	C₂×C₃₂	1	16	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	4	3	no	—
61	C₂×C₄	C₂×C₃₂	1	12	yes	—
62	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
63	C₂×C₄	C₂²×C₁₆	1	6	yes	—
64	C₂×C₄	C₂²×C₁₆	1	6	yes	—
65	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
66	C₈	—	2	6	no	—
67	C₈	—	2	1	no	—
68	C₈	#519	1	1	yes	characteristic
69	C₈	#568	1	1	yes	characteristic
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	C₂²×C₃₂	1	2	yes	—
73	C₂²	C₂³×C₁₆	1	1	yes	characteristic
74	C₄	—	2	6	no	—
75	C₄	—	2	1	no	—
76	C₄	C₂²×C₃₂	1	12	yes	—
77	C₄	#2982	1	1	yes	characteristic
78	C₄	#3407	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	#33496	1	1	yes	characteristic
81	C₂	C₂³×C₃₂	1	1	yes	characteristic
82	C₂	#33974	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator	C₂	C₂³×C₃₂
socle	C₂²	C₂³×C₁₆
prime radical	C₂³	C₂²×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃₂	C₂	semidirect
39. C₂×Q₈	C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₆	2
Q₈×C₃₂	2
#13639	2