Subgroups — #11757833

Order 972.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×#1094	C₃	1	2	yes	maximal, complemented
3	C₃²×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²⋊(C₂²⋊C₉)	C₃	1	9	yes	maximal, complemented
5	#6893	—	4	1	no	maximal, Sylow
6	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	3	yes	—
8	C₂²×C₃³	C₃²	1	1	yes	characteristic
9	C₂²×H(3,𝔽₃)	—	3	3	no	—
10	#1094	—	3	6	no	—
11	C₃×C₂²⋊C₉	—	3	3	no	—
12	C₃×C₂²⋊C₉	C₃²	1	3	yes	—
13	C₃×C₂²⋊C₉	C₃²	1	6	yes	—
14	C₃×3₋¹⁺²	—	4	2	no	—
15	C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic
16	C₃²×C₉	—	4	1	no	—
17	C₃²⋊C₉	—	4	9	no	—
18	C₂×C₃³	—	3	3	no	—
19	C₂×C₃³	—	3	1	no	—
20	C₂×H(3,𝔽₃)	—	3	9	no	—
21	C₂²×C₃²	—	3	3	no	—
22	C₂²×C₃²	—	3	3	no	—
23	C₂²×C₃²	—	3	9	no	—
24	C₂²×C₃²	H(3,𝔽₃)	1	3	yes	—
25	C₂²×C₃²	C₃³	1	1	yes	characteristic
26	C₂²⋊C₉	—	3	3	no	—
27	C₂²⋊C₉	—	3	6	no	—
28	C₃³	C₃×A₄	1	3	yes	—
29	C₃³	C₃×A₄	1	1	yes	characteristic
30	3₋¹⁺²	—	12	6	no	—
31	H(3,𝔽₃)	—	3	3	no	—
32	C₃×C₉	—	4	3	no	—
33	C₃×C₉	—	4	6	no	—
34	C₃×C₉	—	12	3	no	—
35	C₂×C₃²	—	3	1	no	—
36	C₂×C₃²	—	3	9	no	—
37	C₂×C₃²	—	3	3	no	—
38	C₂×C₃²	—	9	3	no	—
39	C₂×C₃²	—	9	9	no	—
40	C₂²×C₃	—	3	3	no	—
41	C₂²×C₃	—	9	3	no	—
42	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₂²×C₃	#737	1	3	yes	—
44	C₃²	—	3	3	no	—
45	C₃²	—	3	9	no	—
46	C₃²	—	3	3	no	—
47	C₃²	C₂²⋊H(3,𝔽₃)	1	3	yes	—
48	C₃²	C₃²×A₄	1	1	yes	characteristic
49	C₉	—	12	3	no	—
50	C₉	—	12	6	no	—
51	C₂×C₃	—	3	3	no	—
52	C₂×C₃	—	3	1	no	—
53	C₂×C₃	—	9	3	no	—
54	C₂×C₃	—	9	9	no	—
55	C₂²	#6893	1	1	yes	characteristic, complemented, Sylow
56	C₃	—	3	3	no	—
57	C₃	—	9	3	no	—
58	C₃	#66437	1	3	yes	—
59	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
60	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃²	C₃³
Fitting, prime radical	C₂²×C₃×H(3,𝔽₃)	C₃
hypercenter	C₃×H(3,𝔽₃)	A₄
hypocenter	C₂²	#6893
LCS	G ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. C₃×#1094	C₃	semidirect
3. C₃²×C₂²⋊C₉	C₃	semidirect
4. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
55. C₂²	#6893	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×H(3,𝔽₃)	3
C₃×#1094	3
C₃²×C₂²⋊C₉	3
C₃²⋊(C₂²⋊C₉)	3
#6893	4