Subgroups — H(3,𝔽₃)×C₂²⋊C₉ #11757831

Order 972.

258 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	C₃²×C₂²⋊C₉	C₃	1	4	yes	maximal, complemented
3	C₃²⋊(C₂²⋊C₉)	C₃	1	8	yes	maximal, complemented
4	C₉×H(3,𝔽₃)	—	4	1	no	maximal, Sylow
5	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
6	C₂²×C₃³	C₃²	1	4	yes	—
7	C₂²×H(3,𝔽₃)	C₉	1	9	yes	complemented
8	C₃×C₂²⋊C₉	—	3	12	no	—
9	C₃×C₂²⋊C₉	C₃²	1	8	yes	—
10	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
11	C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic
12	C₃²×C₉	—	4	4	no	—
13	C₃²⋊C₉	—	4	8	no	—
14	C₂×C₃³	—	3	4	no	—
15	C₂×H(3,𝔽₃)	—	3	9	no	—
16	C₂²×C₃²	—	3	4	no	—
17	C₂²×C₃²	—	3	8	no	—
18	C₂²×C₃²	C₃×C₉	1	12	yes	complemented
19	C₂²×C₃²	C₃³	1	1	yes	characteristic
20	C₂²⋊C₉	—	3	8	no	—
21	C₂²⋊C₉	H(3,𝔽₃)	1	3	yes	direct product factor
22	C₃³	C₃×A₄	1	4	yes	—
23	H(3,𝔽₃)	C₂²⋊C₉	1	9	yes	direct product factor
24	C₃×C₉	—	4	1	no	—
25	C₃×C₉	—	4	8	no	—
26	C₃×C₉	—	12	12	no	—
27	C₂×C₃²	—	3	1	no	—
28	C₂×C₃²	—	3	12	no	—
29	C₂×C₃²	—	9	8	no	—
30	C₂×C₃²	—	9	4	no	—
31	C₂²×C₃	—	3	12	no	—
32	C₂²×C₃	C₃²×C₉	1	1	yes	characteristic
33	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₂²×C₃	#737	1	2	yes	—
35	C₃²	—	3	8	no	—
36	C₃²	—	3	4	no	—
37	C₃²	C₃×C₂²⋊C₉	1	12	yes	complemented
38	C₃²	C₃²×A₄	1	1	yes	characteristic
39	C₉	—	4	3	no	—
40	C₉	—	12	8	no	—
41	C₂×C₃	—	3	2	no	—
42	C₂×C₃	—	3	1	no	—
43	C₂×C₃	—	3	1	no	—
44	C₂×C₃	—	9	12	no	—
45	C₂²	C₉×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
46	C₃	—	3	12	no	—
47	C₃	#66437	1	2	yes	—
48	C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃	C₃²×C₂²⋊C₉	1	1	yes	characteristic
50	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator	C₂²×C₃	C₃²×C₉
Fitting, prime radical	C₂²×C₃×H(3,𝔽₃)	C₃
socle, cosocle	C₂²×C₃²	C₃³
hypercenter	C₃×H(3,𝔽₃)	A₄
hypocenter	C₂²	C₉×H(3,𝔽₃)
LCS	G ▷ C₂²×C₃ ▷ 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. C₃²×C₂²⋊C₉	C₃	semidirect
3. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
7. C₂²×H(3,𝔽₃)	C₉	semidirect
18. C₂²×C₃²	C₃×C₉	semidirect
21. C₂²⋊C₉	H(3,𝔽₃)	direct (N × H)
23. H(3,𝔽₃)	C₂²⋊C₉	direct (N × H)
37. C₃²	C₃×C₂²⋊C₉	semidirect
45. C₂²	C₉×H(3,𝔽₃)	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
C₃²×C₂²⋊C₉	3
C₃²⋊(C₂²⋊C₉)	3
C₉×H(3,𝔽₃)	4

Subgroups — H(3,𝔽3)×C22⋊C9 #11757831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×C₂²⋊C₉ #11757831