Subgroups — C₂³×SD₃₂ #62626

Order 256.

1167 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×SD₃₂	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×Q₈	—	2	1	no	—
6	C₂³×D₄	—	2	1	no	—
7	C₂²×Q₁₆	—	2	7	no	—
8	C₂²×Q₁₆	C₂²	1	14	yes	complemented
9	C₂²×D₈	—	2	7	no	—
10	C₂²×D₈	C₂²	1	14	yes	—
11	C₂³×C₈	C₂²	1	1	yes	characteristic
12	C₂×SD₃₂	C₂²	1	112	yes	direct product factor
13	C₂²×C₁₆	C₂²	1	14	yes	complemented
14	C₂⁵	—	4	1	no	—
15	C₂²×Q₈	—	2	14	no	—
16	C₂²×Q₈	—	4	7	no	—
17	C₂²×D₄	—	2	14	no	—
18	C₂²×D₄	—	4	7	no	—
19	C₂³×C₄	—	4	1	no	—
20	C₂³×C₄	D₄	1	1	yes	characteristic
21	C₂×Q₁₆	—	2	42	no	—
22	C₂×Q₁₆	C₂³	1	28	yes	complemented
23	C₂×D₈	—	2	42	no	—
24	C₂×D₈	C₂³	1	28	yes	—
25	C₂²×C₈	D₄	1	7	yes	—
26	C₂²×C₈	C₂³	1	7	yes	—
27	SD₃₂	C₂³	1	64	yes	direct product factor
28	C₂×C₁₆	C₂³	1	28	yes	complemented
29	C₂⁴	—	4	14	no	—
30	C₂⁴	—	8	8	no	—
31	C₂⁴	D₈	1	1	yes	characteristic
32	C₂×Q₈	—	2	28	no	—
33	C₂×Q₈	—	4	42	no	—
34	C₂×D₄	—	2	28	no	—
35	C₂×D₄	—	4	42	no	—
36	C₂²×C₄	—	4	14	no	—
37	C₂²×C₄	D₈	1	7	yes	—
38	C₂²×C₄	C₂×D₄	1	7	yes	—
39	Q₁₆	—	2	28	no	—
40	Q₁₆	C₂⁴	1	8	yes	complemented
41	D₈	—	2	28	no	—
42	D₈	C₂⁴	1	8	yes	—
43	C₂×C₈	C₂×D₄	1	21	yes	—
44	C₂×C₈	C₂⁴	1	7	yes	—
45	C₁₆	C₂⁴	1	8	yes	complemented
46	C₂³	—	4	28	no	—
47	C₂³	—	8	56	no	—
48	C₂³	SD₃₂	1	8	yes	direct product factor
49	C₂³	C₂×D₈	1	7	yes	—
50	Q₈	—	2	8	no	—
51	Q₈	—	4	28	no	—
52	D₄	—	2	8	no	—
53	D₄	—	4	28	no	—
54	C₂×C₄	—	4	28	no	—
55	C₂×C₄	C₂×D₈	1	21	yes	—
56	C₂×C₄	C₂²×D₄	1	7	yes	—
57	C₈	C₂²×D₄	1	7	yes	—
58	C₈	C₂⁵	1	1	yes	characteristic
59	C₂²	—	4	8	no	—
60	C₂²	—	8	56	no	—
61	C₂²	C₂×SD₃₂	1	28	yes	direct product factor
62	C₂²	C₂²×D₈	1	7	yes	—
63	C₄	—	4	8	no	—
64	C₄	C₂²×D₈	1	7	yes	—
65	C₄	C₂³×D₄	1	1	yes	characteristic
66	C₂	—	8	8	no	—
67	C₂	C₂²×SD₃₂	1	14	yes	direct product factor
68	C₂	C₂³×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴	D₈
commutator, Frattini, cosocle	C₈	C₂⁵
prime radical	C₂³×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆	C₂	semidirect
3. C₂²×SD₃₂	C₂	direct (N × H)
4. C₂³×C₁₆	C₂	semidirect
8. C₂²×Q₁₆	C₂²	semidirect
12. C₂×SD₃₂	C₂²	direct (N × H)
13. C₂²×C₁₆	C₂²	semidirect
22. C₂×Q₁₆	C₂³	semidirect
27. SD₃₂	C₂³	direct (N × H)
28. C₂×C₁₆	C₂³	semidirect
40. Q₁₆	C₂⁴	semidirect
45. C₁₆	C₂⁴	semidirect
48. C₂³	SD₃₂	direct (N × H)
61. C₂²	C₂×SD₃₂	direct (N × H)
67. C₂	C₂²×SD₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆	2
C₂³×D₈	2
C₂²×SD₃₂	2
C₂³×C₁₆	2

Subgroups — C23×SD32 #62626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×SD₃₂ #62626