Subgroups — C₂²×D₆₄ #513309

Order 512.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×D₆₄	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₁₆	—	2	2	no	—
5	C₂×D₃₂	—	2	6	no	—
6	C₂×D₃₂	C₂²	1	12	yes	complemented
7	C₂²×C₃₂	C₂²	1	1	yes	characteristic
8	D₆₄	C₂²	1	16	yes	direct product factor
9	C₂×C₆₄	C₂²	1	6	yes	complemented
10	C₂²×D₈	—	4	2	no	—
11	C₂×D₁₆	—	2	12	no	—
12	C₂×D₁₆	—	4	6	no	—
13	C₂²×C₁₆	D₄	1	1	yes	characteristic
14	D₃₂	—	2	12	no	—
15	D₃₂	C₂³	1	8	yes	complemented
16	C₂×C₃₂	D₄	1	3	yes	—
17	C₂×C₃₂	C₂³	1	3	yes	—
18	C₆₄	C₂³	1	4	yes	complemented
19	C₂²×D₄	—	8	2	no	—
20	C₂×D₈	—	4	12	no	—
21	C₂×D₈	—	8	6	no	—
22	C₂²×C₈	D₈	1	1	yes	characteristic
23	D₁₆	—	2	8	no	—
24	D₁₆	—	4	12	no	—
25	C₂×C₁₆	D₈	1	3	yes	—
26	C₂×C₁₆	C₂×D₄	1	3	yes	—
27	C₃₂	C₂×D₄	1	3	yes	—
28	C₃₂	C₂⁴	1	1	yes	characteristic
29	C₂⁴	—	16	2	no	—
30	C₂×D₄	—	8	12	no	—
31	C₂×D₄	—	16	6	no	—
32	C₂²×C₄	D₁₆	1	1	yes	characteristic
33	D₈	—	4	8	no	—
34	D₈	—	8	12	no	—
35	C₂×C₈	D₁₆	1	3	yes	—
36	C₂×C₈	C₂×D₈	1	3	yes	—
37	C₁₆	C₂×D₈	1	3	yes	—
38	C₁₆	C₂²×D₄	1	1	yes	characteristic
39	C₂³	—	16	12	no	—
40	C₂³	—	32	8	no	—
41	C₂³	D₃₂	1	1	yes	characteristic
42	D₄	—	8	8	no	—
43	D₄	—	16	12	no	—
44	C₂×C₄	D₃₂	1	3	yes	—
45	C₂×C₄	C₂×D₁₆	1	3	yes	—
46	C₈	C₂×D₁₆	1	3	yes	—
47	C₈	C₂²×D₈	1	1	yes	characteristic
48	C₂²	—	16	8	no	—
49	C₂²	—	32	24	no	—
50	C₂²	D₆₄	1	4	yes	direct product factor
51	C₂²	C₂×D₃₂	1	3	yes	—
52	C₄	C₂×D₃₂	1	3	yes	—
53	C₄	C₂²×D₁₆	1	1	yes	characteristic
54	C₂	—	32	8	no	—
55	C₂	C₂×D₆₄	1	6	yes	direct product factor
56	C₂	C₂²×D₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂⁴
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×D₃₂	C₂	semidirect
2. C₂×D₆₄	C₂	direct (N × H)
3. C₂²×C₆₄	C₂	semidirect
6. C₂×D₃₂	C₂²	semidirect
8. D₆₄	C₂²	direct (N × H)
9. C₂×C₆₄	C₂²	semidirect
15. D₃₂	C₂³	semidirect
18. C₆₄	C₂³	semidirect
50. C₂²	D₆₄	direct (N × H)
55. C₂	C₂×D₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₃₂	2
C₂×D₆₄	2
C₂²×C₆₄	2

Subgroups — C22×D64 #513309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₆₄ #513309