Subgroups — #66409

Order 324.

63 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#737	C₂	1	1	yes	maximal, characteristic
2	C₃⋊Dic₉	C₃	1	1	yes	maximal, characteristic, complemented
3	3₋¹⁺²⋊C₄	—	3	2	no	maximal
4	(C₃⋊Dic₃)⋊C₃	—	3	1	no	maximal
5	C₃×Dic₉	—	3	1	no	maximal
6	#737	C₄	1	1	yes	characteristic, complemented, Sylow
7	C₂×3₋¹⁺²	—	2	1	no	—
8	C₂×3₋¹⁺²	—	2	2	no	—
9	C₂×3₋¹⁺²	D₃	1	2	yes	—
10	C₂×H(3,𝔽₃)	D₃	1	1	yes	characteristic
11	C₂×C₃×C₉	—	2	1	no	—
12	C₂×C₃×C₉	D₃	1	1	yes	characteristic
13	C₂×C₃×C₉	C₂×C₃	1	1	yes	characteristic
14	C₃⋊Dic₃	—	3	1	no	—
15	C₃×Dic₃	—	9	1	no	—
16	Dic₉	—	3	1	no	—
17	Dic₉	—	3	2	no	—
18	3₋¹⁺²	—	2	1	no	—
19	3₋¹⁺²	—	2	2	no	—
20	3₋¹⁺²	Dic₃	1	2	yes	complemented
21	H(3,𝔽₃)	Dic₃	1	1	yes	characteristic
22	C₃×C₉	—	2	1	no	—
23	C₃×C₉	Dic₃	1	1	yes	characteristic, complemented
24	C₃×C₉	C₃×C₄	1	1	yes	characteristic, complemented
25	C₂×C₃²	—	2	1	no	—
26	C₂×C₃²	C₃×D₃	1	1	yes	characteristic
27	C₂×C₃²	C₃⋊D₃	1	1	yes	characteristic
28	C₂×C₉	—	2	2	no	—
29	C₂×C₉	—	2	1	no	—
30	C₂×C₉	C₃×D₃	1	1	yes	characteristic
31	C₂×C₉	C₃×D₃	1	2	yes	—
32	C₃×C₄	—	27	1	no	—
33	Dic₃	—	9	1	no	—
34	Dic₃	—	27	1	no	—
35	C₃²	—	2	1	no	—
36	C₃²	C₃×Dic₃	1	1	yes	characteristic
37	C₃²	C₃⋊Dic₃	1	1	yes	characteristic
38	C₉	—	2	2	no	—
39	C₉	—	2	1	no	—
40	C₉	C₃×Dic₃	1	1	yes	characteristic
41	C₉	C₃×Dic₃	1	2	yes	—
42	C₂×C₃	—	3	1	no	—
43	C₂×C₃	—	3	1	no	—
44	C₂×C₃	—	6	1	no	—
45	C₂×C₃	C₃×C₃⋊D₃	1	1	yes	characteristic
46	C₄	—	27	1	no	Sylow
47	C₃	—	3	1	no	—
48	C₃	—	3	1	no	—
49	C₃	—	6	1	no	—
50	C₃	C₃×C₃⋊Dic₃	1	1	yes	characteristic
51	C₂	#4207	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#4207
commutator, hypocenter	C₃×C₉	C₃×C₄
Frattini, socle	C₂×C₃	C₃×C₃⋊D₃
Fitting	C₂×#737	C₂
prime radical	C₂×H(3,𝔽₃)	D₃
cosocle	C₂×C₃×C₉	C₂×C₃
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₃⋊Dic₉	C₃	semidirect
6. #737	C₄	semidirect
20. 3₋¹⁺²	Dic₃	semidirect
23. C₃×C₉	Dic₃	semidirect
24. C₃×C₉	C₃×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#737	2
C₃⋊Dic₉	3
3₋¹⁺²⋊C₄	3
(C₃⋊Dic₃)⋊C₃	3
C₃×Dic₉	3