Subgroups — C₂⁴×SD₃₂ #10585863

Order 512.

9106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₈	C₂	1	1	yes	maximal, characteristic
3	C₂³×SD₃₂	C₂	1	60	yes	maximal, direct product factor
4	C₂⁴×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×Q₈	—	2	1	no	—
6	C₂⁴×D₄	—	2	1	no	—
7	C₂³×Q₁₆	—	2	15	no	—
8	C₂³×Q₁₆	C₂²	1	30	yes	complemented
9	C₂³×D₈	—	2	15	no	—
10	C₂³×D₈	C₂²	1	30	yes	—
11	C₂⁴×C₈	C₂²	1	1	yes	characteristic
12	C₂²×SD₃₂	C₂²	1	560	yes	direct product factor
13	C₂³×C₁₆	C₂²	1	30	yes	complemented
14	C₂⁶	—	4	1	no	—
15	C₂³×Q₈	—	2	30	no	—
16	C₂³×Q₈	—	4	15	no	—
17	C₂³×D₄	—	2	30	no	—
18	C₂³×D₄	—	4	15	no	—
19	C₂⁴×C₄	—	4	1	no	—
20	C₂⁴×C₄	D₄	1	1	yes	characteristic
21	C₂²×Q₁₆	—	2	210	no	—
22	C₂²×Q₁₆	C₂³	1	140	yes	complemented
23	C₂²×D₈	—	2	210	no	—
24	C₂²×D₈	C₂³	1	140	yes	—
25	C₂³×C₈	D₄	1	15	yes	—
26	C₂³×C₈	C₂³	1	15	yes	—
27	C₂×SD₃₂	C₂³	1	960	yes	direct product factor
28	C₂²×C₁₆	C₂³	1	140	yes	complemented
29	C₂⁵	—	4	30	no	—
30	C₂⁵	—	8	16	no	—
31	C₂⁵	D₈	1	1	yes	characteristic
32	C₂²×Q₈	—	2	140	no	—
33	C₂²×Q₈	—	4	210	no	—
34	C₂²×D₄	—	2	140	no	—
35	C₂²×D₄	—	4	210	no	—
36	C₂³×C₄	—	4	30	no	—
37	C₂³×C₄	D₈	1	15	yes	—
38	C₂³×C₄	C₂×D₄	1	15	yes	—
39	C₂×Q₁₆	—	2	420	no	—
40	C₂×Q₁₆	C₂⁴	1	120	yes	complemented
41	C₂×D₈	—	2	420	no	—
42	C₂×D₈	C₂⁴	1	120	yes	—
43	C₂²×C₈	C₂×D₄	1	105	yes	—
44	C₂²×C₈	C₂⁴	1	35	yes	—
45	SD₃₂	C₂⁴	1	256	yes	direct product factor
46	C₂×C₁₆	C₂⁴	1	120	yes	complemented
47	C₂⁴	—	4	140	no	—
48	C₂⁴	—	8	240	no	—
49	C₂⁴	SD₃₂	1	16	yes	direct product factor
50	C₂⁴	C₂×D₈	1	15	yes	—
51	C₂×Q₈	—	2	120	no	—
52	C₂×Q₈	—	4	420	no	—
53	C₂×D₄	—	2	120	no	—
54	C₂×D₄	—	4	420	no	—
55	C₂²×C₄	—	4	140	no	—
56	C₂²×C₄	C₂×D₈	1	105	yes	—
57	C₂²×C₄	C₂²×D₄	1	35	yes	—
58	Q₁₆	—	2	120	no	—
59	Q₁₆	C₂⁵	1	16	yes	complemented
60	D₈	—	2	120	no	—
61	D₈	C₂⁵	1	16	yes	—
62	C₂×C₈	C₂²×D₄	1	105	yes	—
63	C₂×C₈	C₂⁵	1	15	yes	—
64	C₁₆	C₂⁵	1	16	yes	complemented
65	C₂³	—	4	120	no	—
66	C₂³	—	8	560	no	—
67	C₂³	C₂×SD₃₂	1	120	yes	direct product factor
68	C₂³	C₂²×D₈	1	35	yes	—
69	Q₈	—	2	16	no	—
70	Q₈	—	4	120	no	—
71	D₄	—	2	16	no	—
72	D₄	—	4	120	no	—
73	C₂×C₄	—	4	120	no	—
74	C₂×C₄	C₂²×D₈	1	105	yes	—
75	C₂×C₄	C₂³×D₄	1	15	yes	—
76	C₈	C₂³×D₄	1	15	yes	—
77	C₈	C₂⁶	1	1	yes	characteristic
78	C₂²	—	4	16	no	—
79	C₂²	—	8	240	no	—
80	C₂²	C₂²×SD₃₂	1	140	yes	direct product factor
81	C₂²	C₂³×D₈	1	15	yes	—
82	C₄	—	4	16	no	—
83	C₄	C₂³×D₈	1	15	yes	—
84	C₄	C₂⁴×D₄	1	1	yes	characteristic
85	C₂	—	8	16	no	—
86	C₂	C₂³×SD₃₂	1	30	yes	direct product factor
87	C₂	C₂⁴×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵	D₈
commutator, Frattini, cosocle	C₈	C₂⁶
prime radical	C₂⁴×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×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. C₂⁴×Q₁₆	C₂	semidirect
3. C₂³×SD₃₂	C₂	direct (N × H)
4. C₂⁴×C₁₆	C₂	semidirect
8. C₂³×Q₁₆	C₂²	semidirect
12. C₂²×SD₃₂	C₂²	direct (N × H)
13. C₂³×C₁₆	C₂²	semidirect
22. C₂²×Q₁₆	C₂³	semidirect
27. C₂×SD₃₂	C₂³	direct (N × H)
28. C₂²×C₁₆	C₂³	semidirect
40. C₂×Q₁₆	C₂⁴	semidirect
45. SD₃₂	C₂⁴	direct (N × H)
46. C₂×C₁₆	C₂⁴	semidirect
49. C₂⁴	SD₃₂	direct (N × H)
59. Q₁₆	C₂⁵	semidirect
64. C₁₆	C₂⁵	semidirect
67. C₂³	C₂×SD₃₂	direct (N × H)
80. C₂²	C₂²×SD₃₂	direct (N × H)
86. C₂	C₂³×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
C₂⁴×Q₁₆	2
C₂⁴×D₈	2
C₂³×SD₃₂	2
C₂⁴×C₁₆	2

Subgroups — C24×SD32 #10585863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×SD₃₂ #10585863