Subgroups — C₂⁴×S₄ #87586

Order 384.

5127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×S₄	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×D₄	—	3	1	no	maximal, Sylow
4	C₂⁴×D₃	—	4	1	no	maximal
5	C₂³×A₄	C₂²	1	15	yes	complemented
6	C₂²×S₄	C₂²	1	140	yes	direct product factor
7	C₂⁶	—	3	1	no	—
8	C₂⁶	D₃	1	1	yes	characteristic, complemented
9	C₂³×D₄	—	3	30	no	—
10	C₂³×D₄	—	3	30	no	—
11	C₂⁴×C₄	—	3	1	no	—
12	C₂⁴×C₃	—	4	1	no	—
13	C₂³×D₃	—	4	30	no	—
14	C₂²×A₄	C₂³	1	35	yes	complemented
15	C₂×S₄	C₂³	1	120	yes	direct product factor
16	C₂⁵	—	3	1	no	—
17	C₂⁵	—	3	15	no	—
18	C₂⁵	—	3	30	no	—
19	C₂⁵	—	6	15	no	—
20	C₂⁵	—	6	1	no	—
21	C₂⁵	C₂×D₃	1	15	yes	complemented
22	C₂²×D₄	—	3	420	no	—
23	C₂²×D₄	—	3	140	no	—
24	C₂³×C₄	—	3	30	no	—
25	C₂³×C₃	—	4	15	no	—
26	C₂²×D₃	—	4	140	no	—
27	C₂×A₄	C₂⁴	1	15	yes	complemented
28	S₄	C₂⁴	1	16	yes	direct product factor
29	C₂⁴	—	3	15	no	—
30	C₂⁴	—	3	15	no	—
31	C₂⁴	—	3	105	no	—
32	C₂⁴	—	3	140	no	—
33	C₂⁴	—	6	35	no	—
34	C₂⁴	—	6	210	no	—
35	C₂⁴	—	6	30	no	—
36	C₂⁴	S₄	1	1	yes	characteristic, direct product factor
37	C₂⁴	C₂²×D₃	1	35	yes	complemented
38	C₂×D₄	—	3	840	no	—
39	C₂×D₄	—	3	120	no	—
40	C₂²×C₄	—	3	140	no	—
41	C₂²×C₃	—	4	35	no	—
42	C₂×D₃	—	4	120	no	—
43	A₄	C₂⁵	1	1	yes	characteristic, complemented
44	C₂³	—	3	35	no	—
45	C₂³	—	3	105	no	—
46	C₂³	—	3	105	no	—
47	C₂³	—	3	120	no	—
48	C₂³	—	6	105	no	—
49	C₂³	—	6	420	no	—
50	C₂³	—	6	140	no	—
51	C₂³	C₂×S₄	1	15	yes	direct product factor
52	C₂³	C₂³×D₃	1	15	yes	complemented
53	D₄	—	3	240	no	—
54	D₄	—	3	16	no	—
55	C₂×C₄	—	3	120	no	—
56	C₂×C₃	—	4	15	no	—
57	D₃	—	4	16	no	—
58	C₂²	—	3	15	no	—
59	C₂²	—	3	15	no	—
60	C₂²	—	3	105	no	—
61	C₂²	—	3	16	no	—
62	C₂²	—	6	35	no	—
63	C₂²	—	6	120	no	—
64	C₂²	—	6	120	no	—
65	C₂²	C₂²×S₄	1	35	yes	direct product factor
66	C₂²	C₂⁴×D₃	1	1	yes	characteristic, complemented
67	C₄	—	3	16	no	—
68	C₃	—	4	1	no	Sylow
69	C₂	—	3	1	no	—
70	C₂	—	3	15	no	—
71	C₂	—	6	16	no	—
72	C₂	C₂³×S₄	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	S₄
commutator, cosocle, hypocenter	A₄	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂⁶	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×A₄	C₂	semidirect
2. C₂³×S₄	C₂	direct (N × H)
5. C₂³×A₄	C₂²	semidirect
6. C₂²×S₄	C₂²	direct (N × H)
8. C₂⁶	D₃	semidirect
14. C₂²×A₄	C₂³	semidirect
15. C₂×S₄	C₂³	direct (N × H)
21. C₂⁵	C₂×D₃	semidirect
27. C₂×A₄	C₂⁴	semidirect
28. S₄	C₂⁴	direct (N × H)
36. C₂⁴	S₄	direct (N × H)
37. C₂⁴	C₂²×D₃	semidirect
43. A₄	C₂⁵	semidirect
51. C₂³	C₂×S₄	direct (N × H)
52. C₂³	C₂³×D₃	semidirect
65. C₂²	C₂²×S₄	direct (N × H)
66. C₂²	C₂⁴×D₃	semidirect
72. C₂	C₂³×S₄	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×A₄	2
C₂³×S₄	2
C₂⁴×D₄	3
C₂⁴×D₃	4

Subgroups — C24×S4 #87586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×S₄ #87586