Subgroups — #6290

Order 216.

120 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	C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	#638	—	3	1	no	maximal
5	#625	—	3	4	no	maximal
6	C₂×C₃³	C₂²	1	1	yes	characteristic
7	C₃×C₃⋊D₃	—	2	1	no	—
8	C₃²×D₃	—	2	1	no	—
9	C₂²×C₃²	—	3	1	no	—
10	C₂×C₃⋊D₃	D₃	1	1	yes	characteristic, complemented
11	C₂×C₃×D₃	—	3	4	no	—
12	C₂×C₃×D₃	D₃	1	4	yes	complemented
13	C₃⋊Dic₃	—	3	1	no	—
14	C₃⋊Dic₃	—	3	4	no	—
15	C₃⋊Dic₃	—	6	4	no	—
16	C₃³	D₄	1	1	yes	characteristic, complemented, Sylow
17	#67	—	9	4	no	—
18	#67	—	9	1	no	—
19	C₂×C₃²	—	2	4	no	—
20	C₂×C₃²	—	6	1	no	—
21	C₂×C₃²	C₂×D₃	1	4	yes	—
22	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
23	C₃⋊D₃	—	2	1	no	—
24	C₃×D₃	—	2	4	no	—
25	C₃×D₃	—	6	4	no	—
26	C₂²×C₃	—	3	4	no	—
27	C₂²×C₃	—	9	1	no	—
28	C₂×D₃	—	3	4	no	—
29	C₂×D₃	C₃⋊D₃	1	1	yes	characteristic, complemented
30	Dic₃	—	9	4	no	—
31	Dic₃	—	9	1	no	—
32	Dic₃	—	18	4	no	—
33	C₃²	—	2	4	no	—
34	C₃²	#67	1	4	yes	complemented
35	C₃²	#67	1	1	yes	characteristic, complemented
36	D₄	—	27	1	no	Sylow
37	C₂×C₃	—	2	4	no	—
38	C₂×C₃	—	6	4	no	—
39	C₂×C₃	—	18	1	no	—
40	C₂×C₃	D₃²	1	4	yes	—
41	C₂×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
42	D₃	—	2	1	no	—
43	D₃	—	6	4	no	—
44	C₂²	—	3	1	no	—
45	C₂²	—	9	1	no	—
46	C₄	—	27	1	no	—
47	C₃	—	2	4	no	—
48	C₃	#625	1	4	yes	complemented
49	C₃	#638	1	1	yes	characteristic, complemented
50	C₂	—	6	1	no	—
51	C₂	—	18	1	no	—
52	C₂	D₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×C₃⋊D₃
commutator, Fitting, socle, cosocle	C₂×C₃³	C₂²
hypocenter	C₃³	D₄
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂

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. C₃⋊(C₃⋊Dic₃)	C₂	semidirect
10. C₂×C₃⋊D₃	D₃	semidirect
12. C₂×C₃×D₃	D₃	semidirect
16. C₃³	D₄	semidirect
29. C₂×D₃	C₃⋊D₃	semidirect
34. C₃²	#67	semidirect
35. C₃²	#67	semidirect
48. C₃	#625	semidirect
49. C₃	#638	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
C₃⋊(C₃⋊Dic₃)	2
#638	3
#625	3