Subgroups — C₂³×S₄ #5978

Order 192.

701 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×S₄	C₂	1	14	yes	maximal, direct product factor
3	C₂³×D₄	—	3	1	no	maximal, Sylow
4	C₂³×D₃	—	4	1	no	maximal
5	C₂²×A₄	C₂²	1	7	yes	complemented
6	C₂×S₄	C₂²	1	28	yes	direct product factor
7	C₂⁵	—	3	1	no	—
8	C₂⁵	D₃	1	1	yes	characteristic, complemented
9	C₂²×D₄	—	3	14	no	—
10	C₂²×D₄	—	3	14	no	—
11	C₂³×C₄	—	3	1	no	—
12	C₂³×C₃	—	4	1	no	—
13	C₂²×D₃	—	4	14	no	—
14	C₂×A₄	C₂³	1	7	yes	complemented
15	S₄	C₂³	1	8	yes	direct product factor
16	C₂⁴	—	3	1	no	—
17	C₂⁴	—	3	7	no	—
18	C₂⁴	—	3	14	no	—
19	C₂⁴	—	6	7	no	—
20	C₂⁴	—	6	1	no	—
21	C₂⁴	C₂×D₃	1	7	yes	complemented
22	C₂×D₄	—	3	84	no	—
23	C₂×D₄	—	3	28	no	—
24	C₂²×C₄	—	3	14	no	—
25	C₂²×C₃	—	4	7	no	—
26	C₂×D₃	—	4	28	no	—
27	A₄	C₂⁴	1	1	yes	characteristic, complemented
28	C₂³	—	3	7	no	—
29	C₂³	—	3	7	no	—
30	C₂³	—	3	21	no	—
31	C₂³	—	3	28	no	—
32	C₂³	—	6	7	no	—
33	C₂³	—	6	42	no	—
34	C₂³	—	6	14	no	—
35	C₂³	S₄	1	1	yes	characteristic, direct product factor
36	C₂³	C₂²×D₃	1	7	yes	complemented
37	D₄	—	3	56	no	—
38	D₄	—	3	8	no	—
39	C₂×C₄	—	3	28	no	—
40	C₂×C₃	—	4	7	no	—
41	D₃	—	4	8	no	—
42	C₂²	—	3	7	no	—
43	C₂²	—	3	7	no	—
44	C₂²	—	3	21	no	—
45	C₂²	—	3	8	no	—
46	C₂²	—	6	7	no	—
47	C₂²	—	6	28	no	—
48	C₂²	—	6	28	no	—
49	C₂²	C₂×S₄	1	7	yes	direct product factor
50	C₂²	C₂³×D₃	1	1	yes	characteristic, complemented
51	C₄	—	3	8	no	—
52	C₃	—	4	1	no	Sylow
53	C₂	—	3	1	no	—
54	C₂	—	3	7	no	—
55	C₂	—	6	8	no	—
56	C₂	C₂²×S₄	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	S₄
commutator, cosocle, hypocenter	A₄	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂⁵	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂³×A₄	C₂	semidirect
2. C₂²×S₄	C₂	direct (N × H)
5. C₂²×A₄	C₂²	semidirect
6. C₂×S₄	C₂²	direct (N × H)
8. C₂⁵	D₃	semidirect
14. C₂×A₄	C₂³	semidirect
15. S₄	C₂³	direct (N × H)
21. C₂⁴	C₂×D₃	semidirect
27. A₄	C₂⁴	semidirect
35. C₂³	S₄	direct (N × H)
36. C₂³	C₂²×D₃	semidirect
49. C₂²	C₂×S₄	direct (N × H)
50. C₂²	C₂³×D₃	semidirect
56. C₂	C₂²×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
C₂³×A₄	2
C₂²×S₄	2
C₂³×D₄	3
C₂³×D₃	4

Subgroups — C23×S4 #5978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×S₄ #5978