Subgroups — C₃×D₃×D₉ #66422

Order 324.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₃²	—	3	1	no	maximal
5	C₂×C₃×D₉	—	3	1	no	maximal
6	D₃×D₉	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃²×C₉	C₂²	1	1	yes	characteristic, complemented, Sylow
8	C₃×C₃⋊D₃	—	3	1	no	—
9	C₃²×D₃	—	3	1	no	—
10	C₃²×D₃	D₃	1	1	yes	characteristic
11	C₂×C₃×C₉	—	3	1	no	—
12	C₃⋊D₉	C₂×C₃	1	1	yes	characteristic, complemented
13	D₃×C₉	—	2	1	no	—
14	D₃×C₉	C₂×C₃	1	1	yes	characteristic, complemented
15	C₃×D₉	—	2	1	no	—
16	C₃×D₉	—	3	1	no	—
17	C₃×D₉	—	6	1	no	—
18	C₃×D₉	D₃	1	1	yes	characteristic, direct product factor
19	C₃×D₉	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×C₃×D₃	—	9	1	no	—
21	C₂×C₃×D₃	—	9	1	no	—
22	D₃²	—	3	1	no	—
23	C₂×D₉	—	3	1	no	—
24	C₃³	C₂×D₃	1	1	yes	characteristic
25	C₃×C₉	—	2	1	no	—
26	C₃×C₉	—	2	1	no	—
27	C₃×C₉	—	2	1	no	—
28	C₃×C₉	—	4	1	no	—
29	C₃×C₉	C₂×D₃	1	1	yes	characteristic, complemented
30	C₃×C₉	C₂²×C₃	1	1	yes	characteristic, complemented
31	C₂×C₃²	—	3	1	no	—
32	C₂×C₃²	—	9	1	no	—
33	C₃⋊D₃	—	3	1	no	—
34	C₃×D₃	—	2	1	no	—
35	C₃×D₃	—	3	1	no	—
36	C₃×D₃	—	3	1	no	—
37	C₃×D₃	—	6	1	no	—
38	C₃×D₃	—	9	1	no	—
39	C₃×D₃	—	9	1	no	—
40	C₃×D₃	—	18	1	no	—
41	C₃×D₃	D₉	1	1	yes	characteristic, direct product factor
42	C₃×D₃	C₃×D₃	1	1	yes	characteristic
43	C₂×C₉	—	3	1	no	—
44	C₂×C₉	—	6	1	no	—
45	D₉	—	3	1	no	—
46	D₉	—	6	1	no	—
47	D₉	C₃×D₃	1	1	yes	characteristic, direct product factor
48	C₂²×C₃	—	27	1	no	—
49	C₂×D₃	—	9	1	no	—
50	C₂×D₃	—	9	1	no	—
51	C₃²	—	2	1	no	—
52	C₃²	—	2	1	no	—
53	C₃²	—	2	1	no	—
54	C₃²	—	4	1	no	—
55	C₃²	C₂×D₉	1	1	yes	characteristic, complemented
56	C₃²	D₃²	1	1	yes	characteristic
57	C₃²	C₂×C₃×D₃	1	1	yes	characteristic
58	C₉	—	2	1	no	—
59	C₉	—	2	1	no	—
60	C₉	—	4	1	no	—
61	C₉	C₂×C₃×D₃	1	1	yes	characteristic, complemented
62	C₂×C₃	—	3	1	no	—
63	C₂×C₃	—	3	1	no	—
64	C₂×C₃	—	6	1	no	—
65	C₂×C₃	—	9	1	no	—
66	C₂×C₃	—	9	1	no	—
67	C₂×C₃	—	18	1	no	—
68	C₂×C₃	—	27	1	no	—
69	D₃	—	3	1	no	—
70	D₃	—	9	1	no	—
71	D₃	—	9	1	no	—
72	D₃	—	18	1	no	—
73	D₃	C₃×D₉	1	1	yes	characteristic, direct product factor
74	C₂²	—	27	1	no	Sylow
75	C₃	—	2	1	no	—
76	C₃	—	2	1	no	—
77	C₃	—	2	1	no	—
78	C₃	—	4	1	no	—
79	C₃	D₃×D₉	1	1	yes	characteristic, direct product factor
80	C₃	C₂×C₃×D₉	1	1	yes	characteristic, complemented
81	C₃	C₃×D₃²	1	1	yes	characteristic
82	C₂	—	3	1	no	—
83	C₂	—	9	1	no	—
84	C₂	—	27	1	no	—

	subgroup	G/H
center, hypercenter	C₃	D₃×D₉
commutator, cosocle, hypocenter	C₃×C₉	C₂²×C₃
Frattini	C₃	C₃×D₃²
Fitting	C₃²×C₉	C₂²
socle	C₃³	C₂×D₃
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₃⋊D₉	C₂	semidirect
2. C₃×D₃×C₉	C₂	semidirect
3. C₃²×D₉	C₂	semidirect
6. D₃×D₉	C₃	direct (N × H)
7. C₃²×C₉	C₂²	semidirect
12. C₃⋊D₉	C₂×C₃	semidirect
14. D₃×C₉	C₂×C₃	semidirect
18. C₃×D₉	D₃	direct (N × H)
19. C₃×D₉	C₂×C₃	semidirect
29. C₃×C₉	C₂×D₃	semidirect
30. C₃×C₉	C₂²×C₃	semidirect
41. C₃×D₃	D₉	direct (N × H)
47. D₉	C₃×D₃	direct (N × H)
55. C₃²	C₂×D₉	semidirect
61. C₉	C₂×C₃×D₃	semidirect
73. D₃	C₃×D₉	direct (N × H)
79. C₃	D₃×D₉	direct (N × H)
80. C₃	C₂×C₃×D₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃⋊D₉	2
C₃×D₃×C₉	2
C₃²×D₉	2
C₃×D₃²	3
C₂×C₃×D₉	3
D₃×D₉	3

Subgroups — C3×D3×D9 #66422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×D₉ #66422