Subgroups — #6192

Order 216.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#626	—	3	1	no	maximal
5	D₃₆	—	3	1	no	maximal
6	C₂×C₃×C₉	C₂²	1	1	yes	characteristic
7	C₃⋊D₉	—	2	1	no	—
8	C₃×D₉	—	2	1	no	—
9	C₂×C₃⋊D₃	—	3	1	no	—
10	C₂×C₃×D₃	—	3	1	no	—
11	C₃×Dic₃	D₃	1	1	yes	characteristic
12	C₂×D₉	—	3	1	no	—
13	C₂×D₉	—	6	1	no	—
14	C₂×D₉	D₃	1	1	yes	characteristic, complemented
15	C₄×C₉	—	3	1	no	—
16	C₃×C₉	D₄	1	1	yes	characteristic, complemented, Sylow
17	#67	—	9	1	no	—
18	D₁₂	—	9	1	no	—
19	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
20	C₃⋊D₃	—	6	1	no	—
21	C₃×D₃	—	6	1	no	—
22	C₂×C₉	—	2	1	no	—
23	C₂×C₉	C₂×D₃	1	1	yes	characteristic
24	D₉	—	2	1	no	—
25	D₉	—	6	2	no	—
26	D₉	—	6	1	no	—
27	C₂²×C₃	—	9	1	no	—
28	C₂×D₃	—	3	1	no	—
29	C₂×D₃	—	9	1	no	—
30	C₂×D₃	—	9	1	no	—
31	C₂×D₃	—	18	1	no	—
32	C₃×C₄	—	3	1	no	—
33	Dic₃	D₉	1	1	yes	characteristic, complemented
34	C₃²	D₁₂	1	1	yes	characteristic
35	C₉	—	2	1	no	—
36	C₉	#67	1	1	yes	characteristic, complemented
37	D₄	—	27	1	no	Sylow
38	C₂×C₃	—	2	1	no	—
39	C₂×C₃	—	18	1	no	—
40	C₂×C₃	C₂×D₉	1	1	yes	characteristic
41	C₂×C₃	D₃²	1	1	yes	characteristic
42	D₃	—	6	1	no	—
43	D₃	—	18	2	no	—
44	D₃	—	18	1	no	—
45	D₃	—	18	1	no	—
46	C₂²	—	9	1	no	—
47	C₂²	—	27	1	no	—
48	C₄	—	3	1	no	—
49	C₃	—	2	1	no	—
50	C₃	D₃₆	1	1	yes	characteristic, complemented
51	C₃	#626	1	1	yes	characteristic
52	C₂	—	18	1	no	—
53	C₂	—	54	1	no	—
54	C₂	D₃×D₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	D₃×D₉
commutator, Fitting, cosocle	C₂×C₃×C₉	C₂²
Frattini	C₂×C₃	D₃²
socle	C₂×C₃²	C₂×D₃
hypocenter	C₃×C₉	D₄
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃⋊D₉	C₂	semidirect
2. C₂×C₃×D₉	C₂	semidirect
3. C₉×Dic₃	C₂	semidirect
14. C₂×D₉	D₃	semidirect
16. C₃×C₉	D₄	semidirect
33. Dic₃	D₉	semidirect
36. C₉	#67	semidirect
50. C₃	D₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊D₉	2
C₂×C₃×D₉	2
C₉×Dic₃	2
#626	3
D₃₆	3