Subgroups — C₂²×C₄⋊(C₃⋊D₃) #64357

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	2	yes	maximal, complemented
2	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊(C₃⋊D₃)	C₂	1	12	yes	maximal, direct product factor
4	C₂²×D₁₂	—	3	4	no	maximal
5	C₂³×C₃²	C₂²	1	1	yes	characteristic
6	C₂²×C₃⋊D₃	—	2	8	no	—
7	C₂²×C₃⋊D₃	C₂²	1	12	yes	complemented
8	C₂×C₃²×C₄	C₂²	1	6	yes	complemented
9	C₄⋊(C₃⋊D₃)	C₂²	1	16	yes	direct product factor
10	C₂³×D₃	—	3	8	no	—
11	C₂²×C₃×C₄	D₃	1	4	yes	complemented
12	C₂×D₁₂	—	3	48	no	—
13	C₂²×C₃²	D₄	1	4	yes	complemented
14	C₂²×C₃²	C₂³	1	3	yes	—
15	C₂×C₃⋊D₃	—	2	24	no	—
16	C₂×C₃⋊D₃	C₂³	1	8	yes	complemented
17	C₃²×C₄	C₂³	1	4	yes	complemented
18	C₂²×D₄	—	9	1	no	Sylow
19	C₂³×C₃	C₂×D₃	1	4	yes	—
20	C₂²×D₃	—	3	48	no	—
21	C₂²×D₃	—	6	32	no	—
22	C₂×C₃×C₄	C₂×D₃	1	24	yes	complemented
23	D₁₂	—	3	64	no	—
24	C₂×C₃²	C₂×D₄	1	6	yes	complemented
25	C₂×C₃²	C₂⁴	1	1	yes	characteristic
26	C₃⋊D₃	—	2	8	no	—
27	C₂⁴	—	9	2	no	—
28	C₂×D₄	—	9	12	no	—
29	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic, complemented
30	C₂²×C₃	D₁₂	1	16	yes	complemented
31	C₂²×C₃	C₂²×D₃	1	12	yes	—
32	C₂×D₃	—	3	32	no	—
33	C₂×D₃	—	6	96	no	—
34	C₃×C₄	C₂²×D₃	1	16	yes	complemented
35	C₃²	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
36	C₂³	—	9	12	no	—
37	C₂³	—	18	8	no	—
38	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
39	D₄	—	9	16	no	—
40	C₂×C₄	C₂×C₃⋊D₃	1	6	yes	complemented
41	C₂×C₃	C₂×D₁₂	1	24	yes	complemented
42	C₂×C₃	C₂³×D₃	1	4	yes	—
43	D₃	—	6	32	no	—
44	C₂²	—	9	8	no	—
45	C₂²	—	18	24	no	—
46	C₂²	C₄⋊(C₃⋊D₃)	1	4	yes	direct product factor
47	C₂²	C₂²×C₃⋊D₃	1	3	yes	—
48	C₄	C₂²×C₃⋊D₃	1	4	yes	complemented
49	C₃	C₂²×D₁₂	1	4	yes	complemented
50	C₂	—	18	8	no	—
51	C₂	C₂×C₄⋊(C₃⋊D₃)	1	6	yes	direct product factor
52	C₂	C₂³×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁴
Frattini	C₂	C₂³×C₃⋊D₃
Fitting	C₂²×C₃²×C₄	C₂
socle	C₂³×C₃²	C₂²
hypercenter	C₂²×C₄	C₃⋊D₃
hypocenter	C₃²	C₂²×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃⋊D₃	C₂	semidirect
2. C₂²×C₃²×C₄	C₂	semidirect
3. C₂×C₄⋊(C₃⋊D₃)	C₂	direct (N × H)
7. C₂²×C₃⋊D₃	C₂²	semidirect
8. C₂×C₃²×C₄	C₂²	semidirect
9. C₄⋊(C₃⋊D₃)	C₂²	direct (N × H)
11. C₂²×C₃×C₄	D₃	semidirect
13. C₂²×C₃²	D₄	semidirect
16. C₂×C₃⋊D₃	C₂³	semidirect
17. C₃²×C₄	C₂³	semidirect
22. C₂×C₃×C₄	C₂×D₃	semidirect
24. C₂×C₃²	C₂×D₄	semidirect
29. C₂²×C₄	C₃⋊D₃	semidirect
30. C₂²×C₃	D₁₂	semidirect
34. C₃×C₄	C₂²×D₃	semidirect
35. C₃²	C₂²×D₄	semidirect
40. C₂×C₄	C₂×C₃⋊D₃	semidirect
41. C₂×C₃	C₂×D₁₂	semidirect
46. C₂²	C₄⋊(C₃⋊D₃)	direct (N × H)
48. C₄	C₂²×C₃⋊D₃	semidirect
49. C₃	C₂²×D₁₂	semidirect
51. 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
C₂×C₄⋊(C₃⋊D₃)	2
C₂²×D₁₂	3

Subgroups — C22×C4⋊(C3⋊D3) #64357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊(C₃⋊D₃) #64357