Subgroups — #11757850

Order 972.

174 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, complemented
2	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#1094	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#1094	C₃	1	1	yes	maximal, characteristic, complemented
5	#66365	C₃	1	9	yes	maximal, complemented
6	#6910	—	4	1	no	maximal, Sylow
7	C₂×C₃×3₋¹⁺²	—	3	1	no	—
8	C₂²×C₃³	C₃²	1	1	yes	characteristic
9	C₃²×A₄	—	3	1	no	—
10	C₂²×3₋¹⁺²	—	3	1	no	—
11	C₂²×3₋¹⁺²	—	3	1	no	—
12	C₂²×3₋¹⁺²	C₃²	1	3	yes	complemented
13	C₂²×C₃×C₉	—	3	1	no	—
14	C₂²⋊H(3,𝔽₃)	—	3	1	no	—
15	C₂²⋊H(3,𝔽₃)	—	3	1	no	—
16	C₂²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
17	#1094	—	3	1	no	—
18	#1094	—	3	1	no	—
19	#1094	—	3	1	no	—
20	#1094	—	3	1	no	—
21	#1094	—	3	1	no	—
22	#1094	C₃²	1	3	yes	complemented
23	C₃×C₂²⋊C₉	—	3	1	no	—
24	C₃×C₂²⋊C₉	C₃²	1	3	yes	complemented
25	C₃×3₋¹⁺²	—	4	1	no	—
26	C₃×3₋¹⁺²	—	4	1	no	—
27	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic, complemented
28	C₃×H(3,𝔽₃)	—	4	1	no	—
29	#731	—	4	9	no	—
30	C₂×C₃³	—	3	1	no	—
31	C₂×3₋¹⁺²	—	3	3	no	—
32	C₂×3₋¹⁺²	—	3	3	no	—
33	C₂×3₋¹⁺²	—	3	3	no	—
34	C₂×C₃×C₉	—	3	3	no	—
35	C₂²×C₃²	—	9	1	no	—
36	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
37	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
38	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
39	C₂²×C₃²	C₃³	1	1	yes	characteristic
40	C₃×A₄	—	3	3	no	—
41	C₃×A₄	—	9	3	no	—
42	C₂²×C₉	—	3	3	no	—
43	C₂²⋊C₉	—	3	3	no	—
44	C₂²⋊C₉	—	3	3	no	—
45	C₃³	—	12	1	no	—
46	C₃³	C₃×A₄	1	1	yes	characteristic
47	3₋¹⁺²	—	3	1	no	—
48	3₋¹⁺²	—	3	1	no	—
49	3₋¹⁺²	—	4	3	no	—
50	3₋¹⁺²	—	12	1	no	—
51	3₋¹⁺²	—	12	1	no	—
52	3₋¹⁺²	—	12	1	no	—
53	3₋¹⁺²	—	12	1	no	—
54	3₋¹⁺²	—	12	1	no	—
55	3₋¹⁺²	C₃×A₄	1	3	yes	complemented
56	H(3,𝔽₃)	—	4	3	no	—
57	H(3,𝔽₃)	—	12	1	no	—
58	H(3,𝔽₃)	—	12	1	no	—
59	C₃×C₉	—	3	1	no	—
60	C₃×C₉	—	4	3	no	—
61	C₃×C₉	—	12	1	no	—
62	C₂×C₃²	—	3	1	no	—
63	C₂×C₃²	—	3	1	no	—
64	C₂×C₃²	—	3	1	no	—
65	C₂×C₃²	—	3	1	no	—
66	C₂×C₃²	—	9	3	no	—
67	C₂×C₉	—	3	9	no	—
68	C₂²×C₃	—	3	1	no	—
69	C₂²×C₃	—	3	1	no	—
70	C₂²×C₃	—	3	1	no	—
71	C₂²×C₃	—	3	1	no	—
72	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
73	A₄	—	9	3	no	—
74	C₃²	—	9	1	no	—
75	C₃²	—	12	3	no	—
76	C₃²	—	36	3	no	—
77	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
78	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
79	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
80	C₃²	C₃²×A₄	1	1	yes	characteristic
81	C₉	—	3	3	no	—
82	C₉	—	12	3	no	—
83	C₉	—	12	3	no	—
84	C₂×C₃	—	3	3	no	—
85	C₂×C₃	—	3	1	no	—
86	C₂×C₃	—	9	1	no	—
87	C₂×C₃	—	9	1	no	—
88	C₂×C₃	—	9	1	no	—
89	C₂²	#6910	1	1	yes	characteristic, complemented, Sylow
90	C₃	—	3	1	no	—
91	C₃	—	3	1	no	—
92	C₃	—	3	1	no	—
93	C₃	—	3	1	no	—
94	C₃	—	36	3	no	—
95	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
96	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃	C₃×C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃×3₋¹⁺²	C₃
socle	C₂²×C₃	C₃×H(3,𝔽₃)
prime radical	C₃×C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	#6910
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×3₋¹⁺²	C₃	semidirect
3. C₃×#1094	C₃	semidirect
4. C₃×#1094	C₃	semidirect
5. #66365	C₃	semidirect
12. C₂²×3₋¹⁺²	C₃²	semidirect
22. #1094	C₃²	semidirect
24. C₃×C₂²⋊C₉	C₃²	semidirect
27. C₃×3₋¹⁺²	A₄	semidirect
55. 3₋¹⁺²	C₃×A₄	semidirect
89. C₂²	#6910	semidirect

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×C₂²⋊H(3,𝔽₃)	3
C₃×#1094	3
#66365	3
#6910	4