Subgroups — C₃×D₃×Dic₃ #6282

Order 216.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×Dic₃	—	3	1	no	maximal
5	C₃×C₄×D₃	—	3	1	no	maximal
6	D₃×Dic₃	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃³	C₂²	1	1	yes	characteristic
8	C₃²×D₃	C₄	1	2	yes	complemented
9	C₂²×C₃²	—	3	1	no	—
10	C₂×C₃×D₃	—	2	1	no	—
11	C₂×C₃×D₃	D₃	1	1	yes	characteristic
12	C₂×C₃×D₃	C₂×C₃	1	1	yes	characteristic
13	C₃²×C₄	—	3	1	no	—
14	C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
15	C₃×Dic₃	—	2	1	no	—
16	C₃×Dic₃	—	3	1	no	—
17	C₃×Dic₃	—	3	1	no	—
18	C₃×Dic₃	—	6	1	no	—
19	C₃×Dic₃	D₃	1	1	yes	characteristic, direct product factor
20	C₃×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
21	C₃³	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃×C₄	—	9	1	no	—
23	C₂×Dic₃	—	3	1	no	—
24	C₄×D₃	—	3	1	no	—
25	C₂×C₃²	—	2	1	no	—
26	C₂×C₃²	—	2	1	no	—
27	C₂×C₃²	—	2	1	no	—
28	C₂×C₃²	—	3	2	no	—
29	C₂×C₃²	—	4	1	no	—
30	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
31	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
32	C₂×C₃²	C₂²×C₃	1	1	yes	characteristic
33	C₃×D₃	—	2	2	no	—
34	C₃×D₃	Dic₃	1	2	yes	direct product factor
35	C₃×D₃	C₃×C₄	1	2	yes	complemented
36	C₂²×C₃	—	3	1	no	—
37	C₂²×C₃	—	3	1	no	—
38	C₂²×C₃	—	6	1	no	—
39	C₂×D₃	C₃×D₃	1	1	yes	characteristic
40	C₃×C₄	—	3	1	no	—
41	C₃×C₄	—	3	1	no	—
42	C₃×C₄	—	6	1	no	—
43	C₃×C₄	—	9	1	no	—
44	Dic₃	—	3	1	no	—
45	Dic₃	—	3	1	no	—
46	Dic₃	—	6	1	no	—
47	Dic₃	C₃×D₃	1	1	yes	characteristic, direct product factor
48	C₃²	—	2	1	no	—
49	C₃²	—	2	1	no	—
50	C₃²	—	2	1	no	—
51	C₃²	—	4	1	no	—
52	C₃²	C₄×D₃	1	1	yes	characteristic, complemented
53	C₃²	C₂×Dic₃	1	1	yes	characteristic, complemented
54	C₃²	C₂×C₃×C₄	1	1	yes	characteristic, complemented
55	C₂×C₄	—	9	1	no	Sylow
56	C₂×C₃	—	2	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	3	2	no	—
60	C₂×C₃	—	3	2	no	—
61	C₂×C₃	—	4	1	no	—
62	C₂×C₃	—	6	2	no	—
63	C₂×C₃	D₃²	1	1	yes	characteristic
64	C₂×C₃	C₂×C₃×D₃	1	1	yes	characteristic
65	C₂×C₃	C₂×C₃×D₃	1	1	yes	characteristic
66	D₃	C₃×Dic₃	1	2	yes	direct product factor
67	C₂²	—	3	1	no	—
68	C₄	—	3	1	no	—
69	C₄	—	9	1	no	—
70	C₃	—	2	1	no	—
71	C₃	—	2	1	no	—
72	C₃	—	2	1	no	—
73	C₃	—	4	1	no	—
74	C₃	D₃×Dic₃	1	1	yes	characteristic, direct product factor
75	C₃	C₃×C₄×D₃	1	1	yes	characteristic, complemented
76	C₃	C₂×C₃×Dic₃	1	1	yes	characteristic, complemented
77	C₂	—	3	2	no	—
78	C₂	C₃×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃²
commutator, hypocenter	C₃²	C₂×C₃×C₄
Frattini	C₂	C₃×D₃²
Fitting, socle	C₂×C₃³	C₂²
prime radical	C₂×C₃²×D₃	C₂
cosocle	C₂×C₃²	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
2. C₃×C₃⋊Dic₃	C₂	semidirect
3. C₃²×Dic₃	C₂	semidirect
6. D₃×Dic₃	C₃	direct (N × H)
8. C₃²×D₃	C₄	semidirect
14. C₃⋊Dic₃	C₂×C₃	semidirect
19. C₃×Dic₃	D₃	direct (N × H)
20. C₃×Dic₃	C₂×C₃	semidirect
21. C₃³	C₂×C₄	semidirect
34. C₃×D₃	Dic₃	direct (N × H)
35. C₃×D₃	C₃×C₄	semidirect
47. Dic₃	C₃×D₃	direct (N × H)
52. C₃²	C₄×D₃	semidirect
53. C₃²	C₂×Dic₃	semidirect
54. C₃²	C₂×C₃×C₄	semidirect
66. D₃	C₃×Dic₃	direct (N × H)
74. C₃	D₃×Dic₃	direct (N × H)
75. C₃	C₃×C₄×D₃	semidirect
76. C₃	C₂×C₃×Dic₃	semidirect

Maximal subgroups

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

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

Subgroups — C3×D3×Dic3 #6282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×Dic₃ #6282