Subgroups — C₂²×C₃×S₄ #64385

Order 288.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×S₄	C₂	1	6	yes	maximal, direct product factor
3	C₂²×S₄	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂²×C₃×D₄	—	3	1	no	maximal
5	C₂²×C₃×D₃	—	4	1	no	maximal
6	C₂×C₃×A₄	C₂²	1	3	yes	complemented
7	C₃×S₄	C₂²	1	4	yes	direct product factor
8	C₂⁴×C₃	—	3	1	no	—
9	C₂⁴×C₃	D₃	1	1	yes	characteristic, complemented
10	C₂²×A₄	—	2	1	no	—
11	C₂²×A₄	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂×S₄	C₂×C₃	1	6	yes	direct product factor
13	C₂×C₃×D₄	—	3	6	no	—
14	C₂×C₃×D₄	—	3	6	no	—
15	C₂²×C₃×C₄	—	3	1	no	—
16	C₂²×C₃²	—	4	1	no	—
17	C₂×C₃×D₃	—	4	6	no	—
18	C₃×A₄	C₂³	1	1	yes	characteristic, complemented
19	C₂²×D₄	—	3	1	no	Sylow
20	C₂³×C₃	—	3	1	no	—
21	C₂³×C₃	—	3	3	no	—
22	C₂³×C₃	—	3	6	no	—
23	C₂³×C₃	—	6	3	no	—
24	C₂³×C₃	—	6	1	no	—
25	C₂³×C₃	C₂×D₃	1	3	yes	complemented
26	C₂²×D₃	—	4	1	no	—
27	C₂×A₄	—	2	3	no	—
28	C₂×A₄	C₂²×C₃	1	3	yes	complemented
29	S₄	C₂²×C₃	1	4	yes	direct product factor
30	C₃×D₄	—	3	12	no	—
31	C₃×D₄	—	3	4	no	—
32	C₂×C₃×C₄	—	3	6	no	—
33	C₂×C₃²	—	4	3	no	—
34	C₃×D₃	—	4	4	no	—
35	C₂⁴	—	3	1	no	—
36	C₂⁴	C₃×D₃	1	1	yes	characteristic, complemented
37	C₂×D₄	—	3	6	no	—
38	C₂×D₄	—	3	6	no	—
39	C₂²×C₄	—	3	1	no	—
40	C₂²×C₃	—	3	3	no	—
41	C₂²×C₃	—	3	3	no	—
42	C₂²×C₃	—	3	3	no	—
43	C₂²×C₃	—	3	4	no	—
44	C₂²×C₃	—	4	1	no	—
45	C₂²×C₃	—	6	1	no	—
46	C₂²×C₃	—	6	6	no	—
47	C₂²×C₃	—	6	6	no	—
48	C₂²×C₃	—	8	1	no	—
49	C₂²×C₃	S₄	1	1	yes	characteristic, direct product factor
50	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic, complemented
51	C₂×D₃	—	4	6	no	—
52	A₄	—	2	1	no	—
53	A₄	C₂³×C₃	1	1	yes	characteristic, complemented
54	C₃×C₄	—	3	4	no	—
55	C₃²	—	4	1	no	Sylow
56	C₂³	—	3	1	no	—
57	C₂³	—	3	3	no	—
58	C₂³	—	3	6	no	—
59	C₂³	—	6	3	no	—
60	C₂³	—	6	1	no	—
61	C₂³	C₂×C₃×D₃	1	3	yes	complemented
62	D₄	—	3	12	no	—
63	D₄	—	3	4	no	—
64	C₂×C₄	—	3	6	no	—
65	C₂×C₃	—	3	1	no	—
66	C₂×C₃	—	3	3	no	—
67	C₂×C₃	—	4	3	no	—
68	C₂×C₃	—	6	4	no	—
69	C₂×C₃	—	8	3	no	—
70	C₂×C₃	C₂×S₄	1	3	yes	direct product factor
71	D₃	—	4	4	no	—
72	C₂²	—	3	3	no	—
73	C₂²	—	3	3	no	—
74	C₂²	—	3	3	no	—
75	C₂²	—	3	4	no	—
76	C₂²	—	6	1	no	—
77	C₂²	—	6	6	no	—
78	C₂²	—	6	6	no	—
79	C₂²	C₃×S₄	1	1	yes	characteristic, direct product factor
80	C₂²	C₂²×C₃×D₃	1	1	yes	characteristic, complemented
81	C₄	—	3	4	no	—
82	C₃	—	4	1	no	—
83	C₃	—	8	1	no	—
84	C₃	C₂²×S₄	1	1	yes	characteristic, direct product factor
85	C₂	—	3	1	no	—
86	C₂	—	3	3	no	—
87	C₂	—	6	4	no	—
88	C₂	C₂×C₃×S₄	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	S₄
commutator, cosocle, hypocenter	A₄	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×A₄	C₂	semidirect
2. C₂×C₃×S₄	C₂	direct (N × H)
3. C₂²×S₄	C₃	direct (N × H)
6. C₂×C₃×A₄	C₂²	semidirect
7. C₃×S₄	C₂²	direct (N × H)
9. C₂⁴×C₃	D₃	semidirect
11. C₂²×A₄	C₂×C₃	semidirect
12. C₂×S₄	C₂×C₃	direct (N × H)
18. C₃×A₄	C₂³	semidirect
25. C₂³×C₃	C₂×D₃	semidirect
28. C₂×A₄	C₂²×C₃	semidirect
29. S₄	C₂²×C₃	direct (N × H)
36. C₂⁴	C₃×D₃	semidirect
49. C₂²×C₃	S₄	direct (N × H)
50. C₂²×C₃	C₂²×D₃	semidirect
53. A₄	C₂³×C₃	semidirect
61. C₂³	C₂×C₃×D₃	semidirect
70. C₂×C₃	C₂×S₄	direct (N × H)
79. C₂²	C₃×S₄	direct (N × H)
80. C₂²	C₂²×C₃×D₃	semidirect
84. C₃	C₂²×S₄	direct (N × H)
88. C₂	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
C₂²×C₃×A₄	2
C₂×C₃×S₄	2
C₂²×S₄	3
C₂²×C₃×D₄	3
C₂²×C₃×D₃	4

Subgroups — C22×C3×S4 #64385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×S₄ #64385