Subgroups — C₂²×C₄×C₃⋊D₃ #64356

Order 288.

708 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₃⋊D₃	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₄×D₃	—	3	4	no	maximal
6	C₂³×C₃²	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊D₃	C₄	1	8	yes	direct product factor
8	C₂²×C₃⋊D₃	C₂²	1	6	yes	—
9	C₂×C₃²×C₄	C₂²	1	6	yes	complemented
10	C₂×C₃⋊Dic₃	C₂²	1	6	yes	complemented
11	C₄×C₃⋊D₃	C₂²	1	16	yes	direct product factor
12	C₂³×D₃	—	3	4	no	—
13	C₂²×C₃×C₄	D₃	1	4	yes	complemented
14	C₂²×Dic₃	—	3	4	no	—
15	C₂×C₄×D₃	—	3	48	no	—
16	C₂²×C₃²	C₂×C₄	1	4	yes	complemented
17	C₂²×C₃²	C₂³	1	3	yes	—
18	C₂×C₃⋊D₃	C₂×C₄	1	24	yes	direct product factor
19	C₂×C₃⋊D₃	C₂³	1	4	yes	—
20	C₃²×C₄	C₂³	1	4	yes	complemented
21	C₃⋊Dic₃	C₂³	1	4	yes	complemented
22	C₂³×C₄	—	9	1	no	Sylow
23	C₂³×C₃	C₂×D₃	1	4	yes	—
24	C₂²×D₃	—	3	32	no	—
25	C₂²×D₃	—	3	24	no	—
26	C₂×C₃×C₄	C₂×D₃	1	24	yes	complemented
27	C₂×Dic₃	—	3	24	no	—
28	C₄×D₃	—	3	64	no	—
29	C₂×C₃²	C₂²×C₄	1	6	yes	complemented
30	C₂×C₃²	C₂⁴	1	1	yes	characteristic
31	C₃⋊D₃	C₂²×C₄	1	8	yes	direct product factor
32	C₂⁴	—	9	1	no	—
33	C₂²×C₄	—	9	1	no	—
34	C₂²×C₄	—	9	12	no	—
35	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic, direct product factor
36	C₂²×C₃	C₄×D₃	1	16	yes	complemented
37	C₂²×C₃	C₂²×D₃	1	12	yes	—
38	C₂×D₃	—	3	96	no	—
39	C₂×D₃	—	3	16	no	—
40	C₃×C₄	C₂²×D₃	1	16	yes	complemented
41	Dic₃	—	3	16	no	—
42	C₃²	C₂³×C₄	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	9	8	no	—
44	C₂³	—	9	6	no	—
45	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
46	C₂×C₄	—	9	6	no	—
47	C₂×C₄	—	9	16	no	—
48	C₂×C₄	C₂×C₃⋊D₃	1	6	yes	direct product factor
49	C₂×C₃	C₂×C₄×D₃	1	24	yes	complemented
50	C₂×C₃	C₂³×D₃	1	4	yes	—
51	D₃	—	3	32	no	—
52	C₂²	—	9	4	no	—
53	C₂²	—	9	24	no	—
54	C₂²	C₄×C₃⋊D₃	1	4	yes	direct product factor
55	C₂²	C₂²×C₃⋊D₃	1	3	yes	—
56	C₄	—	9	4	no	—
57	C₄	C₂²×C₃⋊D₃	1	4	yes	direct product factor
58	C₃	C₂²×C₄×D₃	1	4	yes	complemented
59	C₂	—	9	8	no	—
60	C₂	C₂×C₄×C₃⋊D₃	1	6	yes	direct product factor
61	C₂	C₂³×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	C₃⋊D₃
commutator, hypocenter	C₃²	C₂³×C₄
Frattini	C₂	C₂³×C₃⋊D₃
Fitting	C₂²×C₃²×C₄	C₂
socle	C₂³×C₃²	C₂²
prime radical	C₂³×C₃⋊D₃	C₂
cosocle	C₂×C₃²	C₂⁴
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₄	C₂	semidirect
3. C₂²×C₃⋊Dic₃	C₂	semidirect
4. C₂×C₄×C₃⋊D₃	C₂	direct (N × H)
7. C₂²×C₃⋊D₃	C₄	direct (N × H)
9. C₂×C₃²×C₄	C₂²	semidirect
10. C₂×C₃⋊Dic₃	C₂²	semidirect
11. C₄×C₃⋊D₃	C₂²	direct (N × H)
13. C₂²×C₃×C₄	D₃	semidirect
16. C₂²×C₃²	C₂×C₄	semidirect
18. C₂×C₃⋊D₃	C₂×C₄	direct (N × H)
20. C₃²×C₄	C₂³	semidirect
21. C₃⋊Dic₃	C₂³	semidirect
26. C₂×C₃×C₄	C₂×D₃	semidirect
29. C₂×C₃²	C₂²×C₄	semidirect
31. C₃⋊D₃	C₂²×C₄	direct (N × H)
35. C₂²×C₄	C₃⋊D₃	direct (N × H)
36. C₂²×C₃	C₄×D₃	semidirect
40. C₃×C₄	C₂²×D₃	semidirect
42. C₃²	C₂³×C₄	semidirect
48. C₂×C₄	C₂×C₃⋊D₃	direct (N × H)
49. C₂×C₃	C₂×C₄×D₃	semidirect
54. C₂²	C₄×C₃⋊D₃	direct (N × H)
57. C₄	C₂²×C₃⋊D₃	direct (N × H)
58. C₃	C₂²×C₄×D₃	semidirect
60. C₂	C₂×C₄×C₃⋊D₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃⋊D₃	2
C₂²×C₃²×C₄	2
C₂²×C₃⋊Dic₃	2
C₂×C₄×C₃⋊D₃	2
C₂²×C₄×D₃	3

Subgroups — C22×C4×C3⋊D3 #64356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₃⋊D₃ #64356