Subgroups — C₄⋊(C₃⋊S₄) #64261

Order 288.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊S₄	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×A₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊S₄	—	3	3	no	maximal
4	#930	—	3	1	no	maximal
5	C₂×C₃×A₄	C₂²	1	1	yes	characteristic
6	C₃⋊S₄	—	2	2	no	—
7	C₄⋊(C₃⋊D₃)	—	4	1	no	maximal
8	C₂×S₄	—	3	6	no	—
9	C₂²×C₃×C₄	D₃	1	1	yes	characteristic, complemented
10	C₂×#67	—	3	2	no	—
11	C₂×D₁₂	—	3	1	no	—
12	C₄×A₄	D₃	1	3	yes	complemented
13	(C₂×D₃)⋊C₄	—	3	2	no	—
14	C₄⋊Dic₃	—	3	1	no	—
15	C₂×C₃⋊D₃	—	4	2	no	—
16	C₃×A₄	D₄	1	1	yes	characteristic, complemented
17	C₃²×C₄	—	4	1	no	—
18	#121	—	9	1	no	Sylow
19	C₂³×C₃	C₂×D₃	1	1	yes	characteristic
20	C₂²×D₃	—	3	2	no	—
21	C₂×A₄	C₂×D₃	1	3	yes	—
22	S₄	—	6	6	no	—
23	C₂×C₃×C₄	—	3	1	no	—
24	C₂×C₃×C₄	—	3	1	no	—
25	#67	—	6	2	no	—
26	#67	—	6	2	no	—
27	C₂×Dic₃	—	3	2	no	—
28	D₁₂	—	6	1	no	—
29	D₁₂	—	6	1	no	—
30	D₁₂	—	12	3	no	—
31	C₂×C₃²	—	4	1	no	—
32	C₃⋊D₃	—	8	2	no	—
33	C₂×D₄	—	9	1	no	—
34	C₂×D₄	—	9	2	no	—
35	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic, complemented
36	C₄⋊C₄	—	9	1	no	—
37	C₂²⋊C₄	—	9	2	no	—
38	C₂²×C₃	—	3	1	no	—
39	C₂²×C₃	—	3	1	no	—
40	C₂²×C₃	D₁₂	1	1	yes	characteristic, complemented
41	C₂×D₃	—	6	2	no	—
42	C₂×D₃	—	6	2	no	—
43	C₂×D₃	—	6	2	no	—
44	C₂×D₃	—	12	6	no	—
45	A₄	D₁₂	1	3	yes	complemented
46	C₃×C₄	—	3	1	no	—
47	C₃×C₄	—	4	3	no	—
48	C₃×C₄	S₄	1	1	yes	characteristic, complemented
49	Dic₃	—	6	2	no	—
50	C₃²	—	4	1	no	Sylow
51	C₂³	—	9	2	no	—
52	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
53	D₄	—	18	1	no	—
54	D₄	—	18	2	no	—
55	D₄	—	18	2	no	—
56	D₄	—	18	1	no	—
57	C₂×C₄	—	3	1	no	—
58	C₂×C₄	—	3	1	no	—
59	C₂×C₄	—	9	2	no	—
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	3	1	no	—
62	C₂×C₃	—	4	3	no	—
63	C₂×C₃	C₂×S₄	1	1	yes	characteristic
64	D₃	—	12	2	no	—
65	D₃	—	24	6	no	—
66	C₂²	—	3	1	no	—
67	C₂²	—	3	1	no	—
68	C₂²	—	18	2	no	—
69	C₂²	—	18	2	no	—
70	C₂²	—	18	2	no	—
71	C₂²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
72	C₄	—	3	1	no	—
73	C₄	—	18	2	no	—
74	C₄	C₃⋊S₄	1	1	yes	characteristic, complemented
75	C₃	—	4	3	no	—
76	C₃	C₄⋊S₄	1	1	yes	characteristic, complemented
77	C₂	—	3	1	no	—
78	C₂	—	3	1	no	—
79	C₂	—	36	2	no	—
80	C₂	C₂×C₃⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊S₄
commutator, cosocle	C₂×C₃×A₄	C₂²
Fitting	C₂²×C₃×C₄	D₃
socle	C₂³×C₃	C₂×D₃
hypercenter	C₄	C₃⋊S₄
hypocenter	C₃×A₄	D₄
LCS	G ▷ C₂×C₃×A₄ ▷ C₃×A₄
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃⋊S₄	C₂	semidirect
2. C₃×C₄×A₄	C₂	semidirect
9. C₂²×C₃×C₄	D₃	semidirect
12. C₄×A₄	D₃	semidirect
16. C₃×A₄	D₄	semidirect
35. C₂²×C₄	C₃⋊D₃	semidirect
40. C₂²×C₃	D₁₂	semidirect
45. A₄	D₁₂	semidirect
48. C₃×C₄	S₄	semidirect
71. C₂²	C₄⋊(C₃⋊D₃)	semidirect
74. C₄	C₃⋊S₄	semidirect
76. C₃	C₄⋊S₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊S₄	2
C₃×C₄×A₄	2
C₄⋊S₄	3
#930	3
C₄⋊(C₃⋊D₃)	4

Subgroups — C4⋊(C3⋊S4) #64261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊(C₃⋊S₄) #64261