Subgroups — C₂×C₃⋊(C₃⋊S₄) #89115

Order 432.

358 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊(C₃⋊S₄)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⋊S₄	—	3	12	no	maximal
4	C₂×#638	—	3	1	no	maximal
5	C₂×C₃⋊(C₃⋊D₃)	—	4	1	no	maximal
6	C₃²×A₄	C₂²	1	1	yes	characteristic, complemented
7	C₂³×C₃²	D₃	1	1	yes	characteristic, complemented
8	C₂²×C₃⋊D₃	—	3	1	no	—
9	C₂×C₃×A₄	D₃	1	12	yes	complemented
10	C₃⋊S₄	—	3	24	no	—
11	#638	—	3	2	no	—
12	#638	—	3	2	no	—
13	C₂×C₃⋊Dic₃	—	3	1	no	—
14	C₂×C₃³	—	4	1	no	—
15	C₃⋊(C₃⋊D₃)	—	4	2	no	—
16	C₂×S₄	—	9	9	no	—
17	C₂×#67	—	9	4	no	—
18	C₂²×C₃²	—	3	1	no	—
19	C₂²×C₃²	—	3	1	no	—
20	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic, complemented
21	C₂×C₃⋊D₃	—	3	2	no	—
22	C₂×C₃⋊D₃	—	6	1	no	—
23	C₂×C₃⋊D₃	—	6	1	no	—
24	C₂×C₃⋊D₃	—	12	12	no	—
25	C₃×A₄	C₂×D₃	1	12	yes	complemented
26	C₃⋊Dic₃	—	3	2	no	—
27	C₃³	—	4	1	no	Sylow
28	C₂³×C₃	C₃⋊D₃	1	4	yes	complemented
29	C₂²×D₃	—	9	4	no	—
30	C₂×A₄	C₃⋊D₃	1	9	yes	complemented
31	S₄	—	9	18	no	—
32	#67	—	9	8	no	—
33	#67	—	9	8	no	—
34	C₂×Dic₃	—	9	4	no	—
35	C₂×C₃²	—	3	1	no	—
36	C₂×C₃²	—	3	1	no	—
37	C₂×C₃²	—	4	12	no	—
38	C₂×C₃²	S₄	1	1	yes	characteristic, complemented
39	C₃⋊D₃	—	6	2	no	—
40	C₃⋊D₃	—	12	24	no	—
41	C₂×D₄	—	27	1	no	Sylow
42	C₂²×C₃	—	3	4	no	—
43	C₂²×C₃	—	3	4	no	—
44	C₂²×C₃	C₂×C₃⋊D₃	1	4	yes	complemented
45	C₂×D₃	—	9	8	no	—
46	C₂×D₃	—	18	4	no	—
47	C₂×D₃	—	18	4	no	—
48	C₂×D₃	—	36	9	no	—
49	A₄	C₂×C₃⋊D₃	1	9	yes	complemented
50	Dic₃	—	9	8	no	—
51	C₃²	—	4	12	no	—
52	C₃²	C₂×S₄	1	1	yes	characteristic, complemented
53	C₂³	—	27	1	no	—
54	C₂³	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
55	D₄	—	27	2	no	—
56	D₄	—	27	2	no	—
57	C₂×C₄	—	27	1	no	—
58	C₂×C₃	—	3	4	no	—
59	C₂×C₃	—	3	4	no	—
60	C₂×C₃	—	4	9	no	—
61	C₂×C₃	C₃⋊S₄	1	4	yes	complemented
62	D₃	—	18	8	no	—
63	D₃	—	36	18	no	—
64	C₂²	—	3	1	no	—
65	C₂²	—	3	1	no	—
66	C₂²	—	27	2	no	—
67	C₂²	—	54	1	no	—
68	C₂²	—	54	1	no	—
69	C₂²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
70	C₄	—	27	2	no	—
71	C₃	—	4	9	no	—
72	C₃	C₂×C₃⋊S₄	1	4	yes	complemented
73	C₂	—	3	1	no	—
74	C₂	—	3	1	no	—
75	C₂	—	54	2	no	—
76	C₂	C₃⋊(C₃⋊S₄)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₃⋊(C₃⋊S₄)
commutator, cosocle, hypocenter	C₃²×A₄	C₂²
Frattini	C₁	G
Fitting, socle	C₂³×C₃²	D₃
LCS	G ▷ C₃²×A₄
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×A₄	C₂	semidirect
2. C₃⋊(C₃⋊S₄)	C₂	direct (N × H)
6. C₃²×A₄	C₂²	semidirect
7. C₂³×C₃²	D₃	semidirect
9. C₂×C₃×A₄	D₃	semidirect
20. C₂²×C₃²	C₂×D₃	semidirect
25. C₃×A₄	C₂×D₃	semidirect
28. C₂³×C₃	C₃⋊D₃	semidirect
30. C₂×A₄	C₃⋊D₃	semidirect
38. C₂×C₃²	S₄	semidirect
44. C₂²×C₃	C₂×C₃⋊D₃	semidirect
49. A₄	C₂×C₃⋊D₃	semidirect
52. C₃²	C₂×S₄	semidirect
54. C₂³	C₃⋊(C₃⋊D₃)	semidirect
61. C₂×C₃	C₃⋊S₄	semidirect
69. C₂²	C₂×C₃⋊(C₃⋊D₃)	semidirect
72. C₃	C₂×C₃⋊S₄	semidirect
76. C₂	C₃⋊(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×C₃²×A₄	2
C₃⋊(C₃⋊S₄)	2
C₂×C₃⋊S₄	3
C₂×#638	3
C₂×C₃⋊(C₃⋊D₃)	4

Subgroups — C2×C3⋊(C3⋊S4) #89115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊(C₃⋊S₄) #89115