Subgroups — #6200

Order 216.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#272	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×(C₃⋊D₃)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₃⋊Dic₃)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#626	—	3	1	no	maximal
5	#625	—	3	1	no	maximal
6	C₂×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	#272	—	2	1	no	—
8	(C₃⋊D₃)⋊C₃	—	2	1	no	—
9	C₂×C₃⋊D₃	D₃	1	1	yes	characteristic, complemented
10	C₂×C₃×D₃	—	3	1	no	—
11	C₂×C₃×D₃	—	3	1	no	—
12	C₂×C₃×D₃	—	3	1	no	—
13	C₂×C₃×D₃	—	6	1	no	—
14	C₃⋊Dic₃	D₃	1	1	yes	characteristic, complemented
15	C₃×Dic₃	—	3	1	no	—
16	H(3,𝔽₃)	D₄	1	1	yes	characteristic, complemented, Sylow
17	#67	—	9	1	no	—
18	#67	—	9	1	no	—
19	D₁₂	—	9	1	no	—
20	C₂×C₃²	—	2	1	no	—
21	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
22	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
23	C₃⋊D₃	—	2	1	no	—
24	C₃×D₃	—	6	2	no	—
25	C₃×D₃	—	6	1	no	—
26	C₃×D₃	—	6	1	no	—
27	C₃×D₃	—	6	1	no	—
28	C₂²×C₃	—	9	1	no	—
29	C₂²×C₃	—	9	1	no	—
30	C₂×D₃	—	3	1	no	—
31	C₂×D₃	—	3	1	no	—
32	C₂×D₃	—	3	1	no	—
33	C₂×D₃	—	6	1	no	—
34	C₂×D₃	—	9	1	no	—
35	C₃×C₄	—	9	1	no	—
36	Dic₃	—	3	1	no	—
37	Dic₃	—	9	1	no	—
38	C₃²	—	2	1	no	—
39	C₃²	D₁₂	1	1	yes	characteristic, complemented
40	C₃²	#67	1	1	yes	characteristic, complemented
41	D₄	—	27	1	no	Sylow
42	C₂×C₃	—	3	1	no	—
43	C₂×C₃	—	3	1	no	—
44	C₂×C₃	—	6	1	no	—
45	C₂×C₃	—	18	1	no	—
46	C₂×C₃	—	18	1	no	—
47	C₂×C₃	D₃²	1	1	yes	characteristic
48	D₃	—	6	1	no	—
49	D₃	—	6	1	no	—
50	D₃	—	6	2	no	—
51	D₃	—	6	1	no	—
52	D₃	—	18	1	no	—
53	C₂²	—	9	1	no	—
54	C₂²	—	9	1	no	—
55	C₄	—	9	1	no	—
56	C₃	—	3	1	no	—
57	C₃	—	3	1	no	—
58	C₃	—	6	1	no	—
59	C₃	#626	1	1	yes	characteristic
60	C₂	—	18	1	no	—
61	C₂	—	18	1	no	—
62	C₂	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, Fitting, cosocle	C₂×H(3,𝔽₃)	C₂²
Frattini, socle	C₂×C₃	D₃²
hypocenter	H(3,𝔽₃)	D₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×#272	C₂	semidirect
2. C₂×(C₃⋊D₃)⋊C₃	C₂	semidirect
3. (C₃⋊Dic₃)⋊C₃	C₂	semidirect
9. C₂×C₃⋊D₃	D₃	semidirect
14. C₃⋊Dic₃	D₃	semidirect
16. H(3,𝔽₃)	D₄	semidirect
39. C₃²	D₁₂	semidirect
40. C₃²	#67	semidirect

Maximal subgroups

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

subgroup	index
C₂×#272	2
C₂×(C₃⋊D₃)⋊C₃	2
(C₃⋊Dic₃)⋊C₃	2
#626	3
#625	3