Subgroups — D₃×C₃⋊(C₃⋊D₃) #66476

Order 324.

436 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×D₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃⋊(C₃⋊D₃)	—	3	1	no	maximal
5	D₃×C₃⋊D₃	—	3	13	no	maximal
6	C₃⁴	C₂²	1	1	yes	characteristic, complemented, Sylow
7	C₂×C₃³	—	3	1	no	—
8	C₃⋊(C₃⋊D₃)	—	3	1	no	—
9	C₃⋊(C₃⋊D₃)	—	3	13	no	—
10	C₃⋊(C₃⋊D₃)	—	6	13	no	—
11	C₃⋊(C₃⋊D₃)	D₃	1	1	yes	characteristic, direct product factor
12	C₃×C₃⋊D₃	—	3	13	no	—
13	C₃²×D₃	D₃	1	13	yes	complemented
14	C₂×C₃⋊D₃	—	9	13	no	—
15	D₃²	—	9	13	no	—
16	C₃³	—	2	13	no	—
17	C₃³	C₂×D₃	1	13	yes	complemented
18	C₃³	C₂×D₃	1	1	yes	characteristic, complemented
19	C₂×C₃²	—	3	13	no	—
20	C₃⋊D₃	—	3	13	no	—
21	C₃⋊D₃	—	9	13	no	—
22	C₃⋊D₃	—	9	13	no	—
23	C₃⋊D₃	—	18	52	no	—
24	C₃×D₃	—	9	13	no	—
25	C₃×D₃	C₃⋊D₃	1	13	yes	complemented
26	C₂×D₃	—	27	13	no	—
27	C₂×D₃	—	27	1	no	—
28	C₃²	—	2	52	no	—
29	C₃²	D₃²	1	13	yes	complemented
30	C₃²	C₂×C₃⋊D₃	1	13	yes	complemented
31	C₂×C₃	—	3	13	no	—
32	C₂×C₃	—	27	1	no	—
33	D₃	—	9	13	no	—
34	D₃	—	27	13	no	—
35	D₃	—	27	1	no	—
36	D₃	—	54	13	no	—
37	D₃	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
38	C₂²	—	81	1	no	Sylow
39	C₃	—	2	13	no	—
40	C₃	D₃×C₃⋊D₃	1	13	yes	complemented
41	C₃	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
42	C₂	—	3	1	no	—
43	C₂	—	27	1	no	—
44	C₂	—	81	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃⁴	C₂²
LCS	G ▷ C₃⁴
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
2. C₃×C₃⋊(C₃⋊D₃)	C₂	semidirect
3. C₃³×D₃	C₂	semidirect
6. C₃⁴	C₂²	semidirect
11. C₃⋊(C₃⋊D₃)	D₃	direct (N × H)
13. C₃²×D₃	D₃	semidirect
17. C₃³	C₂×D₃	semidirect
18. C₃³	C₂×D₃	semidirect
25. C₃×D₃	C₃⋊D₃	semidirect
29. C₃²	D₃²	semidirect
30. C₃²	C₂×C₃⋊D₃	semidirect
37. D₃	C₃⋊(C₃⋊D₃)	direct (N × H)
40. C₃	D₃×C₃⋊D₃	semidirect
41. C₃	C₂×C₃⋊(C₃⋊D₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(C₃⋊(C₃⋊D₃))	2
C₃×C₃⋊(C₃⋊D₃)	2
C₃³×D₃	2
C₂×C₃⋊(C₃⋊D₃)	3
D₃×C₃⋊D₃	3

Subgroups — D3×C3⋊(C3⋊D3) #66476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₃⋊(C₃⋊D₃) #66476