Subgroups — #6292

Order 216.

136 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×C₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊(C₃⋊D₃)	—	3	1	no	maximal
5	#626	—	3	4	no	maximal
6	C₂×C₃³	C₂²	1	1	yes	characteristic
7	C₃⋊(C₃⋊D₃)	—	2	1	no	—
8	C₃×C₃⋊D₃	—	2	1	no	—
9	C₂×C₃⋊D₃	—	3	1	no	—
10	C₂×C₃⋊D₃	—	3	4	no	—
11	C₂×C₃⋊D₃	—	6	4	no	—
12	C₂×C₃⋊D₃	D₃	1	1	yes	characteristic, complemented
13	C₂×C₃×D₃	—	3	4	no	—
14	C₃²×C₄	—	3	1	no	—
15	C₃×Dic₃	D₃	1	4	yes	complemented
16	C₃³	D₄	1	1	yes	characteristic, complemented, Sylow
17	#67	—	9	1	no	—
18	D₁₂	—	9	4	no	—
19	C₂×C₃²	—	2	4	no	—
20	C₂×C₃²	C₂×D₃	1	4	yes	—
21	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
22	C₃⋊D₃	—	2	1	no	—
23	C₃⋊D₃	—	6	8	no	—
24	C₃⋊D₃	—	6	4	no	—
25	C₃⋊D₃	—	6	1	no	—
26	C₃×D₃	—	6	4	no	—
27	C₂²×C₃	—	9	1	no	—
28	C₂×D₃	—	3	4	no	—
29	C₂×D₃	—	9	4	no	—
30	C₂×D₃	—	9	1	no	—
31	C₂×D₃	—	18	4	no	—
32	C₃×C₄	—	3	4	no	—
33	Dic₃	C₃⋊D₃	1	1	yes	characteristic, complemented
34	C₃²	—	2	4	no	—
35	C₃²	D₁₂	1	4	yes	complemented
36	C₃²	#67	1	1	yes	characteristic, complemented
37	D₄	—	27	1	no	Sylow
38	C₂×C₃	—	2	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	—	6	4	no	—
43	D₃	—	18	8	no	—
44	D₃	—	18	4	no	—
45	D₃	—	18	1	no	—
46	C₂²	—	9	1	no	—
47	C₂²	—	27	1	no	—
48	C₄	—	3	1	no	—
49	C₃	—	2	4	no	—
50	C₃	#626	1	4	yes	complemented
51	C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
52	C₂	—	18	1	no	—
53	C₂	—	54	1	no	—
54	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×C₃×C₃⋊D₃	C₂	semidirect
3. C₃²×Dic₃	C₂	semidirect
12. C₂×C₃⋊D₃	D₃	semidirect
15. C₃×Dic₃	D₃	semidirect
16. C₃³	D₄	semidirect
33. Dic₃	C₃⋊D₃	semidirect
35. C₃²	D₁₂	semidirect
36. C₃²	#67	semidirect
50. C₃	#626	semidirect
51. C₃	C₄⋊(C₃⋊D₃)	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×C₃×C₃⋊D₃	2
C₃²×Dic₃	2
C₄⋊(C₃⋊D₃)	3
#626	3