Subgroups — C₃×#66358 #11757824

Order 972.

218 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	3	yes	maximal, complemented
3	#66358	C₃	1	9	yes	maximal, direct product factor
4	C₃×#732	—	4	1	no	maximal, Sylow
5	C₂×C₃²×C₉	—	3	1	no	—
6	C₂²×C₃³	C₃²	1	1	yes	characteristic
7	C₃²×A₄	—	3	3	no	—
8	C₂²×C₃×C₉	—	3	2	no	—
9	C₂²×C₃×C₉	—	3	1	no	—
10	C₂²×C₃×C₉	C₃²	1	3	yes	complemented
11	C₂²⋊H(3,𝔽₃)	—	3	6	no	—
12	C₂²⋊H(3,𝔽₃)	C₃²	1	9	yes	complemented
13	C₃×H(3,𝔽₃)	—	4	3	no	—
14	C₃²×C₉	A₄	1	1	yes	characteristic, complemented
15	#732	—	4	9	no	—
16	C₂×C₃³	—	3	1	no	—
17	C₂×C₃×C₉	—	3	6	no	—
18	C₂×C₃×C₉	—	3	3	no	—
19	C₂×C₃×C₉	—	3	3	no	—
20	C₂²×C₃²	—	3	3	no	—
21	C₂²×C₃²	H(3,𝔽₃)	1	2	yes	—
22	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
23	C₂²×C₃²	C₃³	1	1	yes	characteristic
24	C₃×A₄	—	3	9	no	—
25	C₃×A₄	—	9	9	no	—
26	C₂²×C₉	—	3	3	no	—
27	C₃³	—	12	3	no	—
28	C₃³	C₃×A₄	1	1	yes	characteristic
29	H(3,𝔽₃)	—	4	9	no	—
30	H(3,𝔽₃)	—	12	6	no	—
31	C₃×C₉	—	3	2	no	—
32	C₃×C₉	—	3	1	no	—
33	C₃×C₉	C₃×A₄	1	3	yes	complemented
34	C₂×C₃²	—	3	9	no	—
35	C₂×C₃²	—	3	2	no	—
36	C₂×C₃²	—	3	1	no	—
37	C₂×C₃²	—	3	1	no	—
38	C₂×C₉	—	3	9	no	—
39	C₂²×C₃	—	3	2	no	—
40	C₂²×C₃	—	3	1	no	—
41	C₂²×C₃	#732	1	3	yes	complemented
42	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	A₄	—	9	9	no	—
44	C₃²	—	3	3	no	—
45	C₃²	—	12	9	no	—
46	C₃²	—	36	9	no	—
47	C₃²	C₂²⋊H(3,𝔽₃)	1	2	yes	—
48	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
49	C₃²	C₃²×A₄	1	1	yes	characteristic
50	C₉	—	3	3	no	—
51	C₂×C₃	—	3	6	no	—
52	C₂×C₃	—	3	3	no	—
53	C₂×C₃	—	3	3	no	—
54	C₂×C₃	—	3	1	no	—
55	C₂²	C₃×#732	1	1	yes	characteristic, complemented, Sylow
56	C₃	—	3	2	no	—
57	C₃	—	3	1	no	—
58	C₃	—	36	9	no	—
59	C₃	#66358	1	3	yes	direct product factor
60	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
61	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃²×C₉	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
hypercenter	C₃²×C₉	A₄
hypocenter	C₂²	C₃×#732
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₉	C₃	semidirect
2. C₃×C₂²⋊H(3,𝔽₃)	C₃	semidirect
3. #66358	C₃	direct (N × H)
10. C₂²×C₃×C₉	C₃²	semidirect
12. C₂²⋊H(3,𝔽₃)	C₃²	semidirect
14. C₃²×C₉	A₄	semidirect
33. C₃×C₉	C₃×A₄	semidirect
41. C₂²×C₃	#732	semidirect
55. C₂²	C₃×#732	semidirect
59. C₃	#66358	direct (N × H)

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
#66358	3
C₃×#732	4

Subgroups — C3×#66358 #11757824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66358 #11757824