Subgroups — C₂×C₃²×S₄ #89113

Order 432.

230 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	C₃²×S₄	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×S₄	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×D₄	—	3	1	no	maximal
5	C₂×C₃²×D₃	—	4	1	no	maximal
6	C₃²×A₄	C₂²	1	1	yes	characteristic, complemented
7	C₂³×C₃²	—	3	1	no	—
8	C₂³×C₃²	D₃	1	1	yes	characteristic, complemented
9	C₂×C₃×A₄	—	2	4	no	—
10	C₂×C₃×A₄	C₂×C₃	1	4	yes	complemented
11	C₃×S₄	C₂×C₃	1	8	yes	direct product factor
12	C₃²×D₄	—	3	2	no	—
13	C₃²×D₄	—	3	2	no	—
14	C₂×C₃²×C₄	—	3	1	no	—
15	C₂×C₃³	—	4	1	no	—
16	C₃²×D₃	—	4	2	no	—
17	C₂×S₄	C₃²	1	1	yes	characteristic, direct product factor
18	C₂×C₃×D₄	—	3	4	no	—
19	C₂²×C₃²	—	3	1	no	—
20	C₂²×C₃²	—	3	1	no	—
21	C₂²×C₃²	—	3	2	no	—
22	C₂²×C₃²	—	6	1	no	—
23	C₂²×C₃²	—	6	1	no	—
24	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic, complemented
25	C₂×C₃×D₃	—	4	4	no	—
26	C₃×A₄	—	2	4	no	—
27	C₃×A₄	C₂²×C₃	1	4	yes	complemented
28	C₃²×C₄	—	3	2	no	—
29	C₃³	—	4	1	no	Sylow
30	C₂³×C₃	—	3	4	no	—
31	C₂³×C₃	C₃×D₃	1	4	yes	complemented
32	C₂×A₄	—	2	4	no	—
33	C₂×A₄	C₂×C₃²	1	1	yes	characteristic, complemented
34	S₄	C₂×C₃²	1	2	yes	direct product factor
35	C₃×D₄	—	3	8	no	—
36	C₃×D₄	—	3	8	no	—
37	C₂×C₃×C₄	—	3	4	no	—
38	C₂×C₃²	—	3	1	no	—
39	C₂×C₃²	—	3	1	no	—
40	C₂×C₃²	—	4	4	no	—
41	C₂×C₃²	—	6	2	no	—
42	C₂×C₃²	—	8	4	no	—
43	C₂×C₃²	S₄	1	1	yes	characteristic, direct product factor
44	C₃×D₃	—	4	8	no	—
45	C₂×D₄	—	3	1	no	Sylow
46	C₂²×C₃	—	3	4	no	—
47	C₂²×C₃	—	3	4	no	—
48	C₂²×C₃	—	3	8	no	—
49	C₂²×C₃	—	6	4	no	—
50	C₂²×C₃	—	6	4	no	—
51	C₂²×C₃	C₂×C₃×D₃	1	4	yes	complemented
52	C₂×D₃	—	4	1	no	—
53	A₄	—	2	4	no	—
54	A₄	C₂²×C₃²	1	1	yes	characteristic, complemented
55	C₃×C₄	—	3	8	no	—
56	C₃²	—	4	4	no	—
57	C₃²	—	8	4	no	—
58	C₃²	C₂×S₄	1	1	yes	characteristic, direct product factor
59	C₂³	—	3	1	no	—
60	C₂³	C₃²×D₃	1	1	yes	characteristic, complemented
61	D₄	—	3	2	no	—
62	D₄	—	3	2	no	—
63	C₂×C₄	—	3	1	no	—
64	C₂×C₃	—	3	4	no	—
65	C₂×C₃	—	3	4	no	—
66	C₂×C₃	—	4	1	no	—
67	C₂×C₃	—	6	8	no	—
68	C₂×C₃	—	8	4	no	—
69	C₂×C₃	C₃×S₄	1	4	yes	direct product factor
70	D₃	—	4	2	no	—
71	C₂²	—	3	1	no	—
72	C₂²	—	3	1	no	—
73	C₂²	—	3	2	no	—
74	C₂²	—	6	1	no	—
75	C₂²	—	6	1	no	—
76	C₂²	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
77	C₄	—	3	2	no	—
78	C₃	—	4	1	no	—
79	C₃	—	8	4	no	—
80	C₃	C₂×C₃×S₄	1	4	yes	direct product factor
81	C₂	—	3	1	no	—
82	C₂	—	3	1	no	—
83	C₂	—	6	2	no	—
84	C₂	C₃²×S₄	1	1	yes	characteristic, direct product factor

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×A₄	C₂	semidirect
2. C₃²×S₄	C₂	direct (N × H)
3. C₂×C₃×S₄	C₃	direct (N × H)
6. C₃²×A₄	C₂²	semidirect
8. C₂³×C₃²	D₃	semidirect
10. C₂×C₃×A₄	C₂×C₃	semidirect
11. C₃×S₄	C₂×C₃	direct (N × H)
17. C₂×S₄	C₃²	direct (N × H)
24. C₂²×C₃²	C₂×D₃	semidirect
27. C₃×A₄	C₂²×C₃	semidirect
31. C₂³×C₃	C₃×D₃	semidirect
33. C₂×A₄	C₂×C₃²	semidirect
34. S₄	C₂×C₃²	direct (N × H)
43. C₂×C₃²	S₄	direct (N × H)
51. C₂²×C₃	C₂×C₃×D₃	semidirect
54. A₄	C₂²×C₃²	semidirect
58. C₃²	C₂×S₄	direct (N × H)
60. C₂³	C₃²×D₃	semidirect
69. C₂×C₃	C₃×S₄	direct (N × H)
76. C₂²	C₂×C₃²×D₃	semidirect
80. C₃	C₂×C₃×S₄	direct (N × H)
84. C₂	C₃²×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
C₃²×S₄	2
C₂×C₃×S₄	3
C₂×C₃²×D₄	3
C₂×C₃²×D₃	4

Subgroups — C2×C32×S4 #89113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×S₄ #89113