Subgroups — C₃×#66364 #11757846

Order 972.

170 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
2	C₃×#1094	C₃	1	2	yes	maximal
3	C₃²×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	#66364	C₃	1	9	yes	maximal, direct product factor
5	C₃×#733	—	4	1	no	maximal, Sylow
6	C₂×C₃×3₋¹⁺²	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	1	yes	characteristic
8	C₂²×3₋¹⁺²	—	3	2	no	—
9	C₂²×3₋¹⁺²	C₃²	1	3	yes	—
10	C₂²×C₃×C₉	—	3	1	no	—
11	#1094	—	3	4	no	—
12	#1094	C₃²	1	6	yes	—
13	C₃×C₂²⋊C₉	—	3	2	no	—
14	C₃×C₂²⋊C₉	—	3	1	no	—
15	C₃×C₂²⋊C₉	—	3	2	no	—
16	C₃×C₂²⋊C₉	C₃²	1	3	yes	—
17	C₃×3₋¹⁺²	—	4	2	no	—
18	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
19	C₃²×C₉	—	4	1	no	—
20	#733	—	4	9	no	—
21	C₂×C₃³	—	3	1	no	—
22	C₂×3₋¹⁺²	—	3	6	no	—
23	C₂×3₋¹⁺²	—	3	3	no	—
24	C₂×C₃×C₉	—	3	3	no	—
25	C₂²×C₃²	—	3	3	no	—
26	C₂²×C₃²	H(3,𝔽₃)	1	2	yes	—
27	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
28	C₂²×C₃²	C₃³	1	1	yes	characteristic
29	C₂²×C₉	—	3	3	no	—
30	C₂²⋊C₉	—	3	3	no	—
31	C₂²⋊C₉	—	3	6	no	—
32	C₃³	C₃×A₄	1	1	yes	characteristic
33	3₋¹⁺²	—	3	2	no	—
34	3₋¹⁺²	—	4	6	no	—
35	3₋¹⁺²	—	12	4	no	—
36	3₋¹⁺²	C₃×A₄	1	3	yes	—
37	C₃×C₉	—	3	1	no	—
38	C₃×C₉	—	4	3	no	—
39	C₃×C₉	—	12	2	no	—
40	C₃×C₉	—	12	1	no	—
41	C₃×C₉	—	12	2	no	—
42	C₂×C₃²	—	3	1	no	—
43	C₂×C₃²	—	3	2	no	—
44	C₂×C₃²	—	3	1	no	—
45	C₂×C₃²	—	9	3	no	—
46	C₂×C₉	—	3	9	no	—
47	C₂²×C₃	—	3	2	no	—
48	C₂²×C₃	—	3	1	no	—
49	C₂²×C₃	#733	1	3	yes	complemented
50	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃²	—	3	3	no	—
52	C₃²	C₂²⋊H(3,𝔽₃)	1	2	yes	—
53	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²	C₃²×A₄	1	1	yes	characteristic
55	C₉	—	3	3	no	—
56	C₉	—	12	3	no	—
57	C₉	—	12	6	no	—
58	C₂×C₃	—	3	3	no	—
59	C₂×C₃	—	3	1	no	—
60	C₂×C₃	—	9	2	no	—
61	C₂×C₃	—	9	1	no	—
62	C₂²	C₃×#733	1	1	yes	characteristic, complemented, Sylow
63	C₃	—	3	2	no	—
64	C₃	—	3	1	no	—
65	C₃	#66364	1	3	yes	direct product factor
66	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
67	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃×3₋¹⁺²	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
prime radical	C₂²×C₃³	C₃²
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	C₃×#733
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
4. #66364	C₃	direct (N × H)
49. C₂²×C₃	#733	semidirect
62. C₂²	C₃×#733	semidirect
65. C₃	#66364	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×C₃×3₋¹⁺²	3
C₃×#1094	3
C₃²×C₂²⋊C₉	3
#66364	3
C₃×#733	4

Subgroups — C3×#66364 #11757846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66364 #11757846