Subgroups — #11757852

Order 972.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
3	#66358	C₃	1	9	yes	maximal, complemented
4	#6911	—	4	1	no	maximal, Sylow
5	C₂×C₃×3₋¹⁺²	—	3	1	no	—
6	C₂²×C₃³	C₃²	1	1	yes	characteristic
7	C₃²×A₄	—	3	3	no	—
8	C₂²×3₋¹⁺²	—	3	3	no	—
9	C₂²×C₃×C₉	C₃²	1	3	yes	complemented
10	C₂²⋊H(3,𝔽₃)	—	3	6	no	—
11	C₂²⋊H(3,𝔽₃)	C₃²	1	9	yes	complemented
12	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic, complemented
13	C₃×H(3,𝔽₃)	—	4	3	no	—
14	#732	—	4	9	no	—
15	C₂×C₃³	—	3	1	no	—
16	C₂×3₋¹⁺²	—	3	9	no	—
17	C₂×C₃×C₉	—	3	3	no	—
18	C₂²×C₃²	—	9	1	no	—
19	C₂²×C₃²	H(3,𝔽₃)	1	3	yes	—
20	C₂²×C₃²	C₃³	1	1	yes	characteristic
21	C₃×A₄	—	3	9	no	—
22	C₃×A₄	—	9	9	no	—
23	C₂²×C₉	—	3	3	no	—
24	C₃³	—	12	3	no	—
25	C₃³	C₃×A₄	1	1	yes	characteristic
26	3₋¹⁺²	—	3	3	no	—
27	H(3,𝔽₃)	—	4	9	no	—
28	H(3,𝔽₃)	—	12	6	no	—
29	C₃×C₉	C₃×A₄	1	3	yes	complemented
30	C₂×C₃²	—	3	1	no	—
31	C₂×C₃²	—	3	3	no	—
32	C₂×C₃²	—	9	3	no	—
33	C₂×C₉	—	3	9	no	—
34	C₂²×C₃	—	3	1	no	—
35	C₂²×C₃	—	3	3	no	—
36	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	A₄	—	9	9	no	—
38	C₃²	—	9	1	no	—
39	C₃²	—	12	9	no	—
40	C₃²	—	36	9	no	—
41	C₃²	C₂²⋊H(3,𝔽₃)	1	3	yes	—
42	C₃²	C₃²×A₄	1	1	yes	characteristic
43	C₉	—	3	3	no	—
44	C₂×C₃	—	3	3	no	—
45	C₂×C₃	—	3	1	no	—
46	C₂×C₃	—	9	3	no	—
47	C₂²	#6911	1	1	yes	characteristic, complemented, Sylow
48	C₃	—	3	1	no	—
49	C₃	—	3	3	no	—
50	C₃	—	36	9	no	—
51	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
52	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃	C₃×C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃×3₋¹⁺²	C₃
socle	C₂²×C₃	C₃×H(3,𝔽₃)
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	#6911
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×3₋¹⁺²	C₃	semidirect
2. C₃×C₂²⋊H(3,𝔽₃)	C₃	semidirect
3. #66358	C₃	semidirect
9. C₂²×C₃×C₉	C₃²	semidirect
11. C₂²⋊H(3,𝔽₃)	C₃²	semidirect
12. C₃×3₋¹⁺²	A₄	semidirect
29. C₃×C₉	C₃×A₄	semidirect
47. C₂²	#6911	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×3₋¹⁺²	3
C₃×C₂²⋊H(3,𝔽₃)	3
#66358	3
#6911	4