Subgroups — Q₈×Q₆₄ #513283

Order 512.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₃₂	C₂	1	2	yes	maximal
2	#13731	C₂	1	3	yes	maximal
3	#13706	C₂	1	6	yes	maximal
4	Q₈×C₃₂	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₆₄	C₂	1	3	yes	maximal
6	Q₈×Q₁₆	—	2	2	no	—
7	C₂×Q₆₄	C₂²	1	1	yes	characteristic
8	#2252	C₂²	1	6	yes	—
9	#2227	—	2	6	no	—
10	Q₈×C₁₆	C₂²	1	1	yes	characteristic
11	C₄×Q₃₂	C₂²	1	6	yes	—
12	#1423	C₂²	1	3	yes	—
13	#1423	C₂²	1	6	yes	—
14	C₄⋊C₃₂	C₂²	1	3	yes	—
15	#1416	C₂²	1	6	yes	—
16	C₄×C₃₂	C₂²	1	3	yes	—
17	Q₈²	—	4	2	no	—
18	C₂×Q₃₂	C₂³	1	2	yes	—
19	#500	—	2	6	no	—
20	#477	—	4	6	no	—
21	C₈×Q₈	D₄	1	1	yes	characteristic
22	C₄×Q₁₆	—	2	6	no	—
23	Q₆₄	Q₈	1	4	yes	direct product factor
24	C₂×C₃₂	C₂³	1	3	yes	—
25	C₂×C₃₂	C₂³	1	1	yes	characteristic
26	#366	—	2	6	no	—
27	#366	C₂³	1	6	yes	—
28	C₄⋊C₁₆	D₄	1	3	yes	—
29	#358	—	2	6	no	—
30	C₄×C₁₆	C₂³	1	3	yes	—
31	C₂×Q₁₆	—	2	2	no	—
32	C₄⋊Q₈	—	4	6	no	—
33	C₄⋊Q₈	—	8	6	no	—
34	C₄×Q₈	—	4	6	no	—
35	C₄×Q₈	—	8	2	no	—
36	C₄×Q₈	D₈	1	1	yes	characteristic
37	Q₃₂	—	2	2	no	—
38	Q₃₂	C₂×Q₈	1	4	yes	—
39	C₂×C₁₆	C₂×D₄	1	3	yes	—
40	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
41	#107	—	2	6	no	—
42	#107	—	4	6	no	—
43	C₄⋊C₈	D₈	1	3	yes	—
44	#103	—	4	6	no	—
45	C₄×C₈	C₂×D₄	1	3	yes	—
46	C₃₂	—	2	3	no	—
47	C₃₂	C₂×Q₈	1	2	yes	—
48	C₂×Q₈	—	4	2	no	—
49	C₂×Q₈	—	8	6	no	—
50	C₂×Q₈	D₁₆	1	1	yes	characteristic
51	Q₁₆	—	2	4	no	—
52	Q₁₆	—	4	2	no	—
53	C₂×C₈	C₂×D₈	1	3	yes	—
54	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
55	C₄⋊C₄	—	4	6	no	—
56	C₄⋊C₄	—	8	6	no	—
57	C₄⋊C₄	—	8	6	no	—
58	C₄⋊C₄	D₁₆	1	3	yes	—
59	C₄²	—	8	6	no	—
60	C₄²	C₂×D₈	1	3	yes	—
61	C₁₆	—	2	3	no	—
62	C₁₆	C₂²×Q₈	1	1	yes	characteristic
63	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
64	Q₈	—	4	4	no	—
65	Q₈	—	8	2	no	—
66	Q₈	—	32	6	no	—
67	Q₈	Q₆₄	1	4	yes	direct product factor
68	C₂×C₄	—	8	2	no	—
69	C₂×C₄	—	8	6	no	—
70	C₂×C₄	C₂×D₁₆	1	3	yes	—
71	C₂×C₄	C₂×D₁₆	1	3	yes	—
72	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
73	C₈	—	2	3	no	—
74	C₈	D₄×Q₈	1	1	yes	characteristic
75	C₈	#578	1	1	yes	characteristic
76	C₂²	C₂²×D₁₆	1	1	yes	characteristic
77	C₄	—	2	3	no	—
78	C₄	—	8	4	no	—
79	C₄	—	16	6	no	—
80	C₄	C₂×Q₆₄	1	6	yes	—
81	C₄	Q₈×D₈	1	1	yes	characteristic
82	C₄	#3417	1	1	yes	characteristic
83	C₂	Q₈×D₁₆	1	1	yes	characteristic
84	C₂	C₂²×Q₆₄	1	1	yes	characteristic
85	C₂	#33982	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×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
23. Q₆₄	Q₈	direct (N × H)
67. Q₈	Q₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×Q₃₂	2
#13731	2
#13706	2
Q₈×C₃₂	2
C₄×Q₆₄	2

Subgroups — Q8×Q64 #513283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Q₆₄ #513283