Subgroups — C₂²×SD₆₄ #33976

Order 256.

251 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×SD₆₄	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₁₆	—	2	1	no	—
6	C₂²×D₈	—	2	1	no	—
7	C₂×Q₃₂	—	2	3	no	—
8	C₂×Q₃₂	C₂²	1	6	yes	complemented
9	C₂×D₁₆	—	2	3	no	—
10	C₂×D₁₆	C₂²	1	6	yes	—
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	SD₆₄	C₂²	1	16	yes	direct product factor
13	C₂×C₃₂	C₂²	1	6	yes	complemented
14	C₂²×Q₈	—	4	1	no	—
15	C₂²×D₄	—	4	1	no	—
16	C₂×Q₁₆	—	2	6	no	—
17	C₂×Q₁₆	—	4	3	no	—
18	C₂×D₈	—	2	6	no	—
19	C₂×D₈	—	4	3	no	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	Q₃₂	—	2	6	no	—
22	Q₃₂	C₂³	1	4	yes	complemented
23	D₁₆	—	2	6	no	—
24	D₁₆	C₂³	1	4	yes	—
25	C₂×C₁₆	D₄	1	3	yes	—
26	C₂×C₁₆	C₂³	1	3	yes	—
27	C₃₂	C₂³	1	4	yes	complemented
28	C₂⁴	—	8	1	no	—
29	C₂×Q₈	—	4	6	no	—
30	C₂×Q₈	—	8	3	no	—
31	C₂×D₄	—	4	6	no	—
32	C₂×D₄	—	8	3	no	—
33	C₂²×C₄	—	8	1	no	—
34	C₂²×C₄	D₈	1	1	yes	characteristic
35	Q₁₆	—	2	4	no	—
36	Q₁₆	—	4	6	no	—
37	D₈	—	2	4	no	—
38	D₈	—	4	6	no	—
39	C₂×C₈	D₈	1	3	yes	—
40	C₂×C₈	C₂×D₄	1	3	yes	—
41	C₁₆	C₂×D₄	1	3	yes	—
42	C₁₆	C₂⁴	1	1	yes	characteristic
43	C₂³	—	8	6	no	—
44	C₂³	—	16	4	no	—
45	C₂³	D₁₆	1	1	yes	characteristic
46	Q₈	—	4	4	no	—
47	Q₈	—	8	6	no	—
48	D₄	—	4	4	no	—
49	D₄	—	8	6	no	—
50	C₂×C₄	—	8	6	no	—
51	C₂×C₄	D₁₆	1	3	yes	—
52	C₂×C₄	C₂×D₈	1	3	yes	—
53	C₈	C₂×D₈	1	3	yes	—
54	C₈	C₂²×D₄	1	1	yes	characteristic
55	C₂²	—	8	4	no	—
56	C₂²	—	16	12	no	—
57	C₂²	SD₆₄	1	4	yes	direct product factor
58	C₂²	C₂×D₁₆	1	3	yes	—
59	C₄	—	8	4	no	—
60	C₄	C₂×D₁₆	1	3	yes	—
61	C₄	C₂²×D₈	1	1	yes	characteristic
62	C₂	—	16	4	no	—
63	C₂	C₂×SD₆₄	1	6	yes	direct product factor
64	C₂	C₂²×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴
prime radical	C₂²×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₃₂	C₂	semidirect
3. C₂×SD₆₄	C₂	direct (N × H)
4. C₂²×C₃₂	C₂	semidirect
8. C₂×Q₃₂	C₂²	semidirect
12. SD₆₄	C₂²	direct (N × H)
13. C₂×C₃₂	C₂²	semidirect
22. Q₃₂	C₂³	semidirect
27. C₃₂	C₂³	semidirect
57. C₂²	SD₆₄	direct (N × H)
63. C₂	C₂×SD₆₄	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×Q₃₂	2
C₂²×D₁₆	2
C₂×SD₆₄	2
C₂²×C₃₂	2

Subgroups — C22×SD64 #33976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×SD₆₄ #33976