Subgroups — C₂×C₃×S₄ #3845

Order 144.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×S₄	C₂	1	2	yes	maximal, direct product factor
3	C₂×S₄	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃×D₄	—	3	1	no	maximal
5	C₂×C₃×D₃	—	4	1	no	maximal
6	C₃×A₄	C₂²	1	1	yes	characteristic, complemented
7	C₂³×C₃	—	3	1	no	—
8	C₂³×C₃	D₃	1	1	yes	characteristic, complemented
9	C₂×A₄	—	2	1	no	—
10	C₂×A₄	C₂×C₃	1	1	yes	characteristic, complemented
11	S₄	C₂×C₃	1	2	yes	direct product factor
12	C₃×D₄	—	3	2	no	—
13	C₃×D₄	—	3	2	no	—
14	C₂×C₃×C₄	—	3	1	no	—
15	C₂×C₃²	—	4	1	no	—
16	C₃×D₃	—	4	2	no	—
17	C₂×D₄	—	3	1	no	Sylow
18	C₂²×C₃	—	3	1	no	—
19	C₂²×C₃	—	3	1	no	—
20	C₂²×C₃	—	3	2	no	—
21	C₂²×C₃	—	6	1	no	—
22	C₂²×C₃	—	6	1	no	—
23	C₂²×C₃	C₂×D₃	1	1	yes	characteristic, complemented
24	C₂×D₃	—	4	1	no	—
25	A₄	—	2	1	no	—
26	A₄	C₂²×C₃	1	1	yes	characteristic, complemented
27	C₃×C₄	—	3	2	no	—
28	C₃²	—	4	1	no	Sylow
29	C₂³	—	3	1	no	—
30	C₂³	C₃×D₃	1	1	yes	characteristic, complemented
31	D₄	—	3	2	no	—
32	D₄	—	3	2	no	—
33	C₂×C₄	—	3	1	no	—
34	C₂×C₃	—	3	1	no	—
35	C₂×C₃	—	3	1	no	—
36	C₂×C₃	—	4	1	no	—
37	C₂×C₃	—	6	2	no	—
38	C₂×C₃	—	8	1	no	—
39	C₂×C₃	S₄	1	1	yes	characteristic, direct product factor
40	D₃	—	4	2	no	—
41	C₂²	—	3	1	no	—
42	C₂²	—	3	1	no	—
43	C₂²	—	3	2	no	—
44	C₂²	—	6	1	no	—
45	C₂²	—	6	1	no	—
46	C₂²	C₂×C₃×D₃	1	1	yes	characteristic, complemented
47	C₄	—	3	2	no	—
48	C₃	—	4	1	no	—
49	C₃	—	8	1	no	—
50	C₃	C₂×S₄	1	1	yes	characteristic, direct product factor
51	C₂	—	3	1	no	—
52	C₂	—	3	1	no	—
53	C₂	—	6	2	no	—
54	C₂	C₃×S₄	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	S₄
commutator, cosocle, hypocenter	A₄	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₃	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×A₄	C₂	semidirect
2. C₃×S₄	C₂	direct (N × H)
3. C₂×S₄	C₃	direct (N × H)
6. C₃×A₄	C₂²	semidirect
8. C₂³×C₃	D₃	semidirect
10. C₂×A₄	C₂×C₃	semidirect
11. S₄	C₂×C₃	direct (N × H)
23. C₂²×C₃	C₂×D₃	semidirect
26. A₄	C₂²×C₃	semidirect
30. C₂³	C₃×D₃	semidirect
39. C₂×C₃	S₄	direct (N × H)
46. C₂²	C₂×C₃×D₃	semidirect
50. C₃	C₂×S₄	direct (N × H)
54. C₂	C₃×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
C₂×C₃×A₄	2
C₃×S₄	2
C₂×S₄	3
C₂×C₃×D₄	3
C₂×C₃×D₃	4

Subgroups — C2×C3×S4 #3845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×S₄ #3845