Subgroups — C₉×C₂²⋊H(3,𝔽₃) #11757828

Order 972.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉×A₄	C₃	1	3	yes	maximal, complemented
4	C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
5	(C₃×A₄)⋊C₉	C₃	1	6	yes	maximal, complemented
6	C₉×H(3,𝔽₃)	—	4	1	no	maximal, Sylow
7	C₂×C₃²×C₉	—	3	1	no	—
8	C₂²×C₃³	C₃²	1	1	yes	characteristic
9	C₃²×A₄	C₃²	1	3	yes	—
10	C₂²×C₃×C₉	—	3	3	no	—
11	C₂²×C₃×C₉	C₃²	1	2	yes	—
12	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
13	C₂²⋊H(3,𝔽₃)	C₉	1	9	yes	direct product factor
14	C₃×C₂²⋊C₉	C₃²	1	6	yes	—
15	C₉×A₄	—	3	9	no	—
16	C₃×H(3,𝔽₃)	—	4	1	no	—
17	C₃²×C₉	—	4	3	no	—
18	C₃²×C₉	A₄	1	1	yes	characteristic, complemented
19	C₃²⋊C₉	—	4	2	no	—
20	C₃²⋊C₉	—	4	6	no	—
21	C₂×C₃³	—	3	1	no	—
22	C₂×C₃×C₉	—	3	9	no	—
23	C₂×C₃×C₉	—	3	2	no	—
24	C₂×C₃×C₉	—	3	1	no	—
25	C₂²×C₃²	—	3	1	no	—
26	C₂²×C₃²	—	3	2	no	—
27	C₂²×C₃²	C₃×C₉	1	3	yes	complemented
28	C₂²×C₃²	C₃³	1	1	yes	characteristic
29	C₃×A₄	—	3	3	no	—
30	C₃×A₄	—	3	6	no	—
31	C₃×A₄	C₃×C₉	1	9	yes	complemented
32	C₂²×C₉	—	3	2	no	—
33	C₂²×C₉	H(3,𝔽₃)	1	3	yes	complemented
34	C₂²⋊C₉	—	3	6	no	—
35	C₃³	—	4	3	no	—
36	C₃³	C₃×A₄	1	1	yes	characteristic
37	H(3,𝔽₃)	—	4	9	no	—
38	C₃×C₉	—	3	3	no	—
39	C₃×C₉	—	4	6	no	—
40	C₃×C₉	—	12	9	no	—
41	C₃×C₉	C₃×A₄	1	2	yes	—
42	C₃×C₉	C₃×A₄	1	1	yes	characteristic
43	C₂×C₃²	—	3	3	no	—
44	C₂×C₃²	—	3	6	no	—
45	C₂×C₃²	—	3	3	no	—
46	C₂×C₃²	—	3	1	no	—
47	C₂×C₉	—	3	6	no	—
48	C₂×C₉	—	3	3	no	—
49	C₂²×C₃	—	3	3	no	—
50	C₂²×C₃	C₃²×C₉	1	1	yes	characteristic
51	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	C₂²×C₃	#737	1	2	yes	—
53	A₄	—	3	9	no	—
54	C₃²	—	3	2	no	—
55	C₃²	—	3	1	no	—
56	C₃²	—	4	9	no	—
57	C₃²	—	12	6	no	—
58	C₃²	—	12	3	no	—
59	C₃²	C₉×A₄	1	3	yes	complemented
60	C₃²	C₃²×A₄	1	1	yes	characteristic
61	C₉	—	3	2	no	—
62	C₉	—	12	6	no	—
63	C₉	C₂²⋊H(3,𝔽₃)	1	3	yes	direct product factor
64	C₂×C₃	—	3	9	no	—
65	C₂×C₃	—	3	1	no	—
66	C₂×C₃	—	3	2	no	—
67	C₂×C₃	—	3	1	no	—
68	C₂²	C₉×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
69	C₃	—	3	3	no	—
70	C₃	—	12	9	no	—
71	C₃	C₃×C₉×A₄	1	1	yes	characteristic
72	C₃	#66436	1	2	yes	—
73	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
74	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃×C₉	C₃×A₄
commutator	C₂²×C₃	C₃²×C₉
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃²×C₉	C₃
socle, cosocle	C₂²×C₃²	C₃³
prime radical	C₃×C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃²×C₉	A₄
hypocenter	C₂²	C₉×H(3,𝔽₃)
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₉	C₃	semidirect
3. C₃×C₉×A₄	C₃	semidirect
4. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
5. (C₃×A₄)⋊C₉	C₃	semidirect
13. C₂²⋊H(3,𝔽₃)	C₉	direct (N × H)
18. C₃²×C₉	A₄	semidirect
27. C₂²×C₃²	C₃×C₉	semidirect
31. C₃×A₄	C₃×C₉	semidirect
33. C₂²×C₉	H(3,𝔽₃)	semidirect
59. C₃²	C₉×A₄	semidirect
63. C₉	C₂²⋊H(3,𝔽₃)	direct (N × H)
68. 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×C₃²×C₉	3
C₃×C₂²⋊H(3,𝔽₃)	3
C₃×C₉×A₄	3
C₃²⋊(C₂²⋊C₉)	3
(C₃×A₄)⋊C₉	3
C₉×H(3,𝔽₃)	4

Subgroups — C9×C22⋊H(3,𝔽3) #11757828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×C₂²⋊H(3,𝔽₃) #11757828