Subgroups — C₂×(C₃⋊D₃)⋊C₉ #66370

Order 324.

67 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₃⋊D₃)⋊C₉	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic
4	C₂×D₃×C₉	—	3	1	no	maximal
5	C₃²⋊C₉	C₂²	1	1	yes	characteristic, complemented, Sylow
6	C₂×C₃³	C₂×C₃	1	1	yes	characteristic
7	C₃×C₃⋊D₃	C₂×C₃	1	2	yes	—
8	C₂×C₃×C₉	—	2	1	no	—
9	C₂×C₃×C₉	D₃	1	1	yes	characteristic, complemented
10	D₃×C₉	—	3	2	no	—
11	C₂×C₃⋊D₃	C₉	1	1	yes	characteristic, complemented
12	C₂×C₃×D₃	—	3	1	no	—
13	C₂×C₃×D₃	—	9	1	no	—
14	C₂²×C₉	—	9	1	no	—
15	C₃³	C₂²×C₃	1	1	yes	characteristic
16	C₃×C₉	—	2	1	no	—
17	C₃×C₉	C₂×D₃	1	1	yes	characteristic, complemented
18	C₂×C₃²	—	2	1	no	—
19	C₂×C₃²	—	3	1	no	—
20	C₂×C₃²	—	6	1	no	—
21	C₂×C₃²	C₂×C₉	1	1	yes	characteristic, complemented
22	C₂×C₃²	C₃×D₃	1	1	yes	characteristic
23	C₃⋊D₃	C₂×C₉	1	2	yes	complemented
24	C₃×D₃	—	3	2	no	—
25	C₃×D₃	—	9	2	no	—
26	C₂×C₉	—	3	1	no	—
27	C₂×C₉	—	6	1	no	—
28	C₂×C₉	—	9	2	no	—
29	C₂²×C₃	—	9	1	no	—
30	C₂×D₃	—	3	1	no	—
31	C₂×D₃	—	9	1	no	—
32	C₃²	—	2	1	no	—
33	C₃²	—	3	1	no	—
34	C₃²	—	6	1	no	—
35	C₃²	C₂²×C₉	1	1	yes	characteristic, complemented
36	C₃²	C₂×C₃×D₃	1	1	yes	characteristic
37	C₉	—	3	1	no	—
38	C₉	—	6	1	no	—
39	C₂×C₃	—	2	1	no	—
40	C₂×C₃	—	3	1	no	—
41	C₂×C₃	—	6	1	no	—
42	C₂×C₃	—	9	2	no	—
43	C₂×C₃	D₃×C₉	1	1	yes	characteristic
44	C₂×C₃	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
45	D₃	—	3	2	no	—
46	D₃	—	9	2	no	—
47	C₂²	—	9	1	no	Sylow
48	C₃	—	2	1	no	—
49	C₃	—	3	1	no	—
50	C₃	—	6	1	no	—
51	C₃	C₂×D₃×C₉	1	1	yes	characteristic
52	C₃	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
53	C₂	—	9	2	no	—
54	C₂	(C₃⋊D₃)⋊C₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₃⋊D₃)⋊C₃
commutator, hypocenter	C₃²	C₂²×C₉
Frattini	C₃²	C₂×C₃×D₃
Fitting	C₂×C₃²⋊C₉	C₂
socle	C₂×C₃²	C₃×D₃
prime radical	C₂×C₃×C₃⋊D₃	C₃
cosocle	C₃³	C₂²×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²⋊C₉	C₂	semidirect
2. (C₃⋊D₃)⋊C₉	C₂	direct (N × H)
5. C₃²⋊C₉	C₂²	semidirect
9. C₂×C₃×C₉	D₃	semidirect
11. C₂×C₃⋊D₃	C₉	semidirect
17. C₃×C₉	C₂×D₃	semidirect
21. C₂×C₃²	C₂×C₉	semidirect
23. C₃⋊D₃	C₂×C₉	semidirect
35. C₃²	C₂²×C₉	semidirect
54. C₂	(C₃⋊D₃)⋊C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²⋊C₉	2
(C₃⋊D₃)⋊C₉	2
C₂×C₃×C₃⋊D₃	3
C₂×D₃×C₉	3

Subgroups — C2×(C3⋊D3)⋊C9 #66370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₃⋊D₃)⋊C₉ #66370