Subgroups — C₃×#626 #6285

Order 216.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#67	—	3	1	no	maximal
5	C₃×D₁₂	—	3	1	no	maximal
6	#626	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃³	C₂²	1	1	yes	characteristic
8	C₃×C₃⋊D₃	—	2	1	no	—
9	C₃²×D₃	—	2	1	no	—
10	C₂²×C₃²	—	3	1	no	—
11	C₂×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂×C₃×D₃	—	2	1	no	—
13	C₂×C₃×D₃	—	3	1	no	—
14	C₂×C₃×D₃	—	3	1	no	—
15	C₂×C₃×D₃	—	6	1	no	—
16	C₂×C₃×D₃	D₃	1	1	yes	characteristic, complemented
17	C₂×C₃×D₃	C₂×C₃	1	1	yes	characteristic, complemented
18	C₃²×C₄	—	3	1	no	—
19	C₃×Dic₃	—	2	1	no	—
20	C₃×Dic₃	D₃	1	1	yes	characteristic, complemented
21	C₃×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
22	C₃³	D₄	1	1	yes	characteristic, complemented, Sylow
23	C₃×D₄	—	9	1	no	—
24	#67	—	3	1	no	—
25	D₁₂	—	3	1	no	—
26	C₂×C₃²	—	2	1	no	—
27	C₂×C₃²	—	2	1	no	—
28	C₂×C₃²	—	2	1	no	—
29	C₂×C₃²	—	4	1	no	—
30	C₂×C₃²	—	6	1	no	—
31	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
32	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃²	C₂²×C₃	1	1	yes	characteristic
34	C₃⋊D₃	—	2	1	no	—
35	C₃×D₃	—	2	2	no	—
36	C₃×D₃	—	2	1	no	—
37	C₃×D₃	—	2	1	no	—
38	C₃×D₃	—	6	2	no	—
39	C₃×D₃	—	6	1	no	—
40	C₃×D₃	—	6	1	no	—
41	C₂²×C₃	—	3	1	no	—
42	C₂²×C₃	—	3	1	no	—
43	C₂²×C₃	—	6	1	no	—
44	C₂²×C₃	—	9	1	no	—
45	C₂×D₃	—	3	1	no	—
46	C₂×D₃	—	3	1	no	—
47	C₂×D₃	—	6	1	no	—
48	C₂×D₃	C₃×D₃	1	1	yes	characteristic, complemented
49	C₃×C₄	—	3	1	no	—
50	C₃×C₄	—	3	1	no	—
51	C₃×C₄	—	6	1	no	—
52	Dic₃	C₃×D₃	1	1	yes	characteristic, complemented
53	C₃²	—	2	1	no	—
54	C₃²	—	2	1	no	—
55	C₃²	—	2	1	no	—
56	C₃²	—	4	1	no	—
57	C₃²	D₁₂	1	1	yes	characteristic, complemented
58	C₃²	#67	1	1	yes	characteristic, complemented
59	C₃²	C₃×D₄	1	1	yes	characteristic, complemented
60	D₄	—	9	1	no	Sylow
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	—	2	1	no	—
63	C₂×C₃	—	2	1	no	—
64	C₂×C₃	—	4	1	no	—
65	C₂×C₃	—	6	2	no	—
66	C₂×C₃	—	6	1	no	—
67	C₂×C₃	—	6	1	no	—
68	C₂×C₃	—	18	1	no	—
69	C₂×C₃	D₃²	1	1	yes	characteristic
70	C₂×C₃	C₂×C₃×D₃	1	1	yes	characteristic
71	C₂×C₃	C₂×C₃×D₃	1	1	yes	characteristic
72	D₃	—	2	1	no	—
73	D₃	—	6	2	no	—
74	D₃	—	6	1	no	—
75	D₃	—	6	1	no	—
76	C₂²	—	3	1	no	—
77	C₂²	—	9	1	no	—
78	C₄	—	3	1	no	—
79	C₃	—	2	1	no	—
80	C₃	—	2	1	no	—
81	C₃	—	2	1	no	—
82	C₃	—	4	1	no	—
83	C₃	#626	1	1	yes	characteristic, direct product factor
84	C₃	C₃×D₁₂	1	1	yes	characteristic, complemented
85	C₃	C₃×#67	1	1	yes	characteristic, complemented
86	C₂	—	6	1	no	—
87	C₂	—	18	1	no	—
88	C₂	C₃×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃²
commutator, cosocle	C₂×C₃²	C₂²×C₃
Frattini	C₂	C₃×D₃²
Fitting, socle	C₂×C₃³	C₂²
hypocenter	C₃²	C₃×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₃⋊D₃	C₂	semidirect
2. C₂×C₃²×D₃	C₂	semidirect
3. C₃²×Dic₃	C₂	semidirect
6. #626	C₃	direct (N × H)
11. C₂×C₃⋊D₃	C₂×C₃	semidirect
16. C₂×C₃×D₃	D₃	semidirect
17. C₂×C₃×D₃	C₂×C₃	semidirect
20. C₃×Dic₃	D₃	semidirect
21. C₃×Dic₃	C₂×C₃	semidirect
22. C₃³	D₄	semidirect
48. C₂×D₃	C₃×D₃	semidirect
52. Dic₃	C₃×D₃	semidirect
57. C₃²	D₁₂	semidirect
58. C₃²	#67	semidirect
59. C₃²	C₃×D₄	semidirect
83. C₃	#626	direct (N × H)
84. C₃	C₃×D₁₂	semidirect
85. C₃	C₃×#67	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₃⋊D₃	2
C₂×C₃²×D₃	2
C₃²×Dic₃	2
C₃×#67	3
C₃×D₁₂	3
#626	3

Subgroups — C3×#626 #6285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#626 #6285