Subgroups — C₂³×C₃×A₄ #64393

Order 288.

420 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×A₄	C₂	1	7	yes	maximal, direct product factor
2	C₂⁵×C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂³×A₄	C₃	1	3	yes	maximal, direct product factor
4	C₂³×C₃²	—	4	1	no	maximal
5	C₂×C₃×A₄	C₂²	1	7	yes	direct product factor
6	C₂⁴×C₃	—	3	7	no	—
7	C₂⁴×C₃	—	3	1	no	—
8	C₂⁴×C₃	C₂×C₃	1	7	yes	complemented
9	C₂²×A₄	C₂×C₃	1	21	yes	direct product factor
10	C₂²×C₃²	—	4	7	no	—
11	C₃×A₄	C₂³	1	1	yes	characteristic, direct product factor
12	C₂⁵	C₃²	1	1	yes	characteristic, complemented, Sylow
13	C₂³×C₃	—	3	21	no	—
14	C₂³×C₃	—	3	14	no	—
15	C₂³×C₃	—	3	7	no	—
16	C₂³×C₃	—	3	7	no	—
17	C₂³×C₃	—	4	3	no	—
18	C₂³×C₃	A₄	1	1	yes	characteristic, direct product factor
19	C₂³×C₃	C₂²×C₃	1	7	yes	complemented
20	C₂×A₄	C₂²×C₃	1	21	yes	direct product factor
21	C₂×C₃²	—	4	7	no	—
22	C₂⁴	—	3	7	no	—
23	C₂⁴	—	3	1	no	—
24	C₂⁴	C₂×C₃²	1	7	yes	complemented
25	C₂²×C₃	—	3	7	no	—
26	C₂²×C₃	—	3	14	no	—
27	C₂²×C₃	—	3	21	no	—
28	C₂²×C₃	—	3	7	no	—
29	C₂²×C₃	—	4	21	no	—
30	C₂²×C₃	C₂×A₄	1	7	yes	direct product factor
31	C₂²×C₃	C₂³×C₃	1	1	yes	characteristic, complemented
32	A₄	C₂³×C₃	1	3	yes	direct product factor
33	C₃²	—	4	1	no	Sylow
34	C₂³	—	3	21	no	—
35	C₂³	—	3	14	no	—
36	C₂³	—	3	7	no	—
37	C₂³	—	3	7	no	—
38	C₂³	C₃×A₄	1	1	yes	characteristic, direct product factor
39	C₂³	C₂²×C₃²	1	7	yes	complemented
40	C₂×C₃	—	3	7	no	—
41	C₂×C₃	—	3	1	no	—
42	C₂×C₃	—	4	21	no	—
43	C₂×C₃	C₂²×A₄	1	7	yes	direct product factor
44	C₂²	—	3	7	no	—
45	C₂²	—	3	14	no	—
46	C₂²	—	3	7	no	—
47	C₂²	—	3	21	no	—
48	C₂²	C₂×C₃×A₄	1	7	yes	direct product factor
49	C₂²	C₂³×C₃²	1	1	yes	characteristic, complemented
50	C₃	—	4	3	no	—
51	C₃	C₂³×A₄	1	1	yes	characteristic, direct product factor
52	C₂	—	3	1	no	—
53	C₂	—	3	7	no	—
54	C₂	C₂²×C₃×A₄	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₃	A₄
commutator, cosocle, hypocenter	C₂²	C₂³×C₃²
Frattini	C₁	G
Fitting, socle	C₂⁵×C₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×A₄	C₂	direct (N × H)
2. C₂⁵×C₃	C₃	semidirect
3. C₂³×A₄	C₃	direct (N × H)
5. C₂×C₃×A₄	C₂²	direct (N × H)
8. C₂⁴×C₃	C₂×C₃	semidirect
9. C₂²×A₄	C₂×C₃	direct (N × H)
11. C₃×A₄	C₂³	direct (N × H)
12. C₂⁵	C₃²	semidirect
18. C₂³×C₃	A₄	direct (N × H)
19. C₂³×C₃	C₂²×C₃	semidirect
20. C₂×A₄	C₂²×C₃	direct (N × H)
24. C₂⁴	C₂×C₃²	semidirect
30. C₂²×C₃	C₂×A₄	direct (N × H)
31. C₂²×C₃	C₂³×C₃	semidirect
32. A₄	C₂³×C₃	direct (N × H)
38. C₂³	C₃×A₄	direct (N × H)
39. C₂³	C₂²×C₃²	semidirect
43. C₂×C₃	C₂²×A₄	direct (N × H)
48. C₂²	C₂×C₃×A₄	direct (N × H)
49. C₂²	C₂³×C₃²	semidirect
51. C₃	C₂³×A₄	direct (N × H)
54. C₂	C₂²×C₃×A₄	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
C₂⁵×C₃	3
C₂³×A₄	3
C₂³×C₃²	4

Subgroups — C23×C3×A4 #64393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×A₄ #64393