Subgroups — C₂⁴×C₄⋊Q₈ #10585896

Order 512.

62338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×Q₈	C₂	1	2	yes	maximal
2	C₂³×C₄⋊Q₈	C₂	1	120	yes	maximal, direct product factor
3	C₂⁴×C₄⋊C₄	C₂	1	4	yes	maximal
4	C₂⁴×C₄²	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Q₈	—	2	64	no	—
6	C₂⁴×Q₈	C₂²	1	120	yes	—
7	C₂⁵×C₄	C₂²	1	4	yes	—
8	C₂⁵×C₄	C₂²	1	1	yes	characteristic
9	C₂⁵×C₄	C₂²	1	2	yes	—
10	C₂²×C₄⋊Q₈	C₂²	1	2240	yes	direct product factor
11	C₂³×C₄⋊C₄	C₂²	1	240	yes	—
12	C₂³×C₄²	C₂²	1	60	yes	—
13	C₂⁶	C₂³	1	1	yes	characteristic
14	C₂³×Q₈	—	2	1920	no	—
15	C₂³×Q₈	C₂³	1	1120	yes	—
16	C₂⁴×C₄	—	2	64	no	—
17	C₂⁴×C₄	D₄	1	32	yes	—
18	C₂⁴×C₄	Q₈	1	64	yes	complemented
19	C₂⁴×C₄	C₂³	1	120	yes	—
20	C₂⁴×C₄	C₂³	1	30	yes	—
21	C₂⁴×C₄	C₂³	1	60	yes	—
22	C₂×C₄⋊Q₈	C₂³	1	7680	yes	direct product factor
23	C₂²×C₄⋊C₄	C₂³	1	2240	yes	—
24	C₂²×C₄²	C₂³	1	560	yes	—
25	C₂⁵	C₂×D₄	1	16	yes	—
26	C₂⁵	C₂×Q₈	1	32	yes	—
27	C₂⁵	C₂⁴	1	15	yes	—
28	C₂²×Q₈	—	2	8960	no	—
29	C₂²×Q₈	C₂⁴	1	1920	yes	—
30	C₂³×C₄	—	2	960	no	—
31	C₂³×C₄	C₂×D₄	1	480	yes	—
32	C₂³×C₄	C₂×Q₈	1	960	yes	complemented
33	C₂³×C₄	C₂⁴	1	560	yes	—
34	C₂³×C₄	C₂⁴	1	140	yes	—
35	C₂³×C₄	C₂⁴	1	280	yes	—
36	C₄⋊Q₈	C₂⁴	1	4096	yes	direct product factor
37	C₂×C₄⋊C₄	C₂⁴	1	3840	yes	—
38	C₂×C₄²	C₂⁴	1	960	yes	—
39	C₂⁴	C₄⋊Q₈	1	256	yes	direct product factor
40	C₂⁴	C₂²×D₄	1	120	yes	—
41	C₂⁴	C₂²×Q₈	1	240	yes	—
42	C₂⁴	C₂⁵	1	35	yes	—
43	C₂×Q₈	—	2	7680	no	—
44	C₂×Q₈	C₂⁵	1	512	yes	—
45	C₂²×C₄	—	2	2240	no	—
46	C₂²×C₄	C₂²×D₄	1	1120	yes	—
47	C₂²×C₄	C₂²×Q₈	1	2240	yes	complemented
48	C₂²×C₄	C₂⁵	1	480	yes	—
49	C₂²×C₄	C₂⁵	1	120	yes	—
50	C₂²×C₄	C₂⁵	1	240	yes	—
51	C₄⋊C₄	C₂⁵	1	1024	yes	—
52	C₄²	C₂⁵	1	256	yes	—
53	C₂³	C₂×C₄⋊Q₈	1	960	yes	direct product factor
54	C₂³	C₂³×D₄	1	140	yes	—
55	C₂³	C₂³×Q₈	1	280	yes	—
56	C₂³	C₂⁶	1	15	yes	—
57	Q₈	—	2	1024	no	—
58	C₂×C₄	—	2	960	no	—
59	C₂×C₄	C₂³×D₄	1	480	yes	—
60	C₂×C₄	C₂³×Q₈	1	960	yes	complemented
61	C₂×C₄	C₂⁶	1	64	yes	—
62	C₂×C₄	C₂⁶	1	16	yes	—
63	C₂×C₄	C₂⁶	1	32	yes	—
64	C₂²	C₂²×C₄⋊Q₈	1	560	yes	direct product factor
65	C₂²	C₂⁴×D₄	1	30	yes	—
66	C₂²	C₂⁴×Q₈	1	60	yes	—
67	C₂²	C₂⁷	1	1	yes	characteristic
68	C₄	—	2	64	no	—
69	C₄	C₂⁴×D₄	1	32	yes	—
70	C₄	C₂⁴×Q₈	1	64	yes	complemented
71	C₂	C₂³×C₄⋊Q₈	1	60	yes	direct product factor
72	C₂	C₂⁵×D₄	1	1	yes	characteristic
73	C₂	C₂⁵×Q₈	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁶	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁷
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊Q₈	C₂	direct (N × H)
10. C₂²×C₄⋊Q₈	C₂²	direct (N × H)
18. C₂⁴×C₄	Q₈	semidirect
22. C₂×C₄⋊Q₈	C₂³	direct (N × H)
32. C₂³×C₄	C₂×Q₈	semidirect
36. C₄⋊Q₈	C₂⁴	direct (N × H)
39. C₂⁴	C₄⋊Q₈	direct (N × H)
47. C₂²×C₄	C₂²×Q₈	semidirect
53. C₂³	C₂×C₄⋊Q₈	direct (N × H)
60. C₂×C₄	C₂³×Q₈	semidirect
64. C₂²	C₂²×C₄⋊Q₈	direct (N × H)
70. C₄	C₂⁴×Q₈	semidirect
71. C₂	C₂³×C₄⋊Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×Q₈	2
C₂³×C₄⋊Q₈	2
C₂⁴×C₄⋊C₄	2
C₂⁴×C₄²	2

Subgroups — C24×C4⋊Q8 #10585896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄⋊Q₈ #10585896