Subgroups — C₃×#611 #6220

Order 216.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#67	—	3	1	no	maximal
5	#611	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×C₉	—	2	1	no	—
7	C₂×C₃×C₉	C₂²	1	1	yes	characteristic
8	C₃×D₉	—	2	1	no	—
9	C₂²×C₃²	D₃	1	1	yes	characteristic
10	C₂×C₃×D₃	—	3	1	no	—
11	C₃×Dic₃	—	3	1	no	—
12	C₂²×C₉	—	2	1	no	—
13	C₂²×C₉	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂×D₉	C₂×C₃	1	1	yes	characteristic, complemented
15	Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
16	C₃×C₉	D₄	1	1	yes	characteristic, complemented, Sylow
17	C₃×D₄	—	9	1	no	—
18	#67	—	3	1	no	—
19	C₂×C₃²	—	2	1	no	—
20	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
21	C₃×D₃	—	6	1	no	—
22	C₂×C₉	—	2	2	no	—
23	C₂×C₉	—	2	1	no	—
24	C₂×C₉	—	2	1	no	—
25	C₂×C₉	C₂²×C₃	1	1	yes	characteristic
26	D₉	—	2	1	no	—
27	C₂²×C₃	—	2	1	no	—
28	C₂²×C₃	—	9	1	no	—
29	C₂²×C₃	D₉	1	1	yes	characteristic, complemented
30	C₂²×C₃	C₃×D₃	1	1	yes	characteristic
31	C₂×D₃	—	3	1	no	—
32	C₃×C₄	—	9	1	no	—
33	Dic₃	—	3	1	no	—
34	C₃²	#67	1	1	yes	characteristic
35	C₉	—	2	1	no	—
36	C₉	C₃×D₄	1	1	yes	characteristic, complemented
37	D₄	—	9	1	no	Sylow
38	C₂×C₃	—	2	2	no	—
39	C₂×C₃	—	2	1	no	—
40	C₂×C₃	—	2	1	no	—
41	C₂×C₃	—	2	1	no	—
42	C₂×C₃	—	18	1	no	—
43	C₂×C₃	C₂×D₉	1	1	yes	characteristic
44	C₂×C₃	C₂×C₃×D₃	1	1	yes	characteristic
45	D₃	—	6	1	no	—
46	C₂²	—	9	1	no	—
47	C₂²	C₃×D₉	1	1	yes	characteristic, complemented
48	C₄	—	9	1	no	—
49	C₃	—	2	1	no	—
50	C₃	#611	1	1	yes	characteristic, direct product factor
51	C₃	C₃×#67	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	18	1	no	—
54	C₂	C₂×C₃×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₉
commutator, cosocle	C₂×C₉	C₂²×C₃
Frattini	C₂×C₃	C₂×C₃×D₃
Fitting	C₂²×C₃×C₉	C₂
socle	C₂×C₃²	C₂×D₃
hypercenter	C₂²×C₃	D₉
hypocenter	C₉	C₃×D₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₉	C₂	semidirect
2. C₂×C₃×D₉	C₂	semidirect
3. C₃×Dic₉	C₂	semidirect
5. #611	C₃	direct (N × H)
13. C₂²×C₉	C₂×C₃	semidirect
14. C₂×D₉	C₂×C₃	semidirect
15. Dic₉	C₂×C₃	semidirect
16. C₃×C₉	D₄	semidirect
29. C₂²×C₃	D₉	semidirect
36. C₉	C₃×D₄	semidirect
47. C₂²	C₃×D₉	semidirect
50. C₃	#611	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₉	2
C₂×C₃×D₉	2
C₃×Dic₉	2
C₃×#67	3
#611	3

Subgroups — C3×#611 #6220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#611 #6220