Subgroups — #11757857

Order 972.

204 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×C₉⋊A₄	C₃	1	2	yes	maximal, complemented
4	C₃×C₉×A₄	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
6	(C₃×A₄)⋊C₉	C₃	1	4	yes	maximal, complemented
7	(C₃×A₄)⋊C₉	C₃	1	2	yes	maximal, complemented
8	C₉⋊H(3,𝔽₃)	—	4	1	no	maximal, Sylow
9	C₂×C₃×3₋¹⁺²	—	3	1	no	—
10	C₂²×C₃³	C₃²	1	1	yes	characteristic
11	C₃²×A₄	C₃²	1	2	yes	—
12	C₃²×A₄	C₃²	1	1	yes	characteristic
13	C₂²×3₋¹⁺²	—	3	3	no	—
14	C₂²×C₃×C₉	C₃²	1	2	yes	—
15	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
16	C₂²⋊H(3,𝔽₃)	—	3	3	no	—
17	C₃×C₂²⋊C₉	C₃²	1	2	yes	—
18	C₃×C₂²⋊C₉	C₃²	1	4	yes	—
19	C₉⋊A₄	—	3	6	no	—
20	C₉×A₄	—	3	3	no	—
21	C₃×3₋¹⁺²	—	4	2	no	—
22	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic, complemented
23	C₃×H(3,𝔽₃)	—	4	1	no	—
24	C₃²×C₉	—	4	1	no	—
25	C₃²⋊C₉	—	4	2	no	—
26	C₃²⋊C₉	—	4	4	no	—
27	C₃²⋊C₉	—	4	2	no	—
28	C₂×C₃³	—	3	1	no	—
29	C₂×3₋¹⁺²	—	3	9	no	—
30	C₂×C₃×C₉	—	3	2	no	—
31	C₂×C₃×C₉	—	3	1	no	—
32	C₂²×C₃²	—	3	1	no	—
33	C₂²×C₃²	—	3	1	no	—
34	C₂²×C₃²	—	3	2	no	—
35	C₂²×C₃²	C₃³	1	1	yes	characteristic
36	C₃×A₄	—	3	1	no	—
37	C₃×A₄	—	3	2	no	—
38	C₃×A₄	—	3	2	no	—
39	C₃×A₄	—	3	2	no	—
40	C₃×A₄	—	3	4	no	—
41	C₃×A₄	3₋¹⁺²	1	3	yes	complemented
42	C₂²×C₉	—	3	2	no	—
43	C₂²×C₉	H(3,𝔽₃)	1	3	yes	complemented
44	C₂²⋊C₉	—	3	2	no	—
45	C₂²⋊C₉	—	3	4	no	—
46	C₃³	—	4	1	no	—
47	C₃³	—	4	2	no	—
48	C₃³	C₃×A₄	1	1	yes	characteristic
49	3₋¹⁺²	—	3	3	no	—
50	3₋¹⁺²	—	12	6	no	—
51	H(3,𝔽₃)	—	12	3	no	—
52	C₃×C₉	—	4	2	no	—
53	C₃×C₉	—	4	4	no	—
54	C₃×C₉	—	12	3	no	—
55	C₃×C₉	C₃×A₄	1	2	yes	—
56	C₃×C₉	C₃×A₄	1	1	yes	characteristic
57	C₂×C₃²	—	3	1	no	—
58	C₂×C₃²	—	3	3	no	—
59	C₂×C₃²	—	9	1	no	—
60	C₂×C₃²	—	9	2	no	—
61	C₂×C₉	—	3	6	no	—
62	C₂×C₉	—	3	3	no	—
63	C₂²×C₃	—	9	1	no	—
64	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₂²×C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
66	C₂²×C₃	#737	1	2	yes	—
67	A₄	—	3	3	no	—
68	A₄	—	9	2	no	—
69	C₃²	—	3	1	no	—
70	C₃²	—	3	2	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	4	3	no	—
73	C₃²	—	12	2	no	—
74	C₃²	—	12	1	no	—
75	C₃²	—	12	2	no	—
76	C₃²	—	12	4	no	—
77	C₃²	—	12	2	no	—
78	C₃²	C₃²×A₄	1	1	yes	characteristic
79	C₉	—	3	2	no	—
80	C₉	—	12	2	no	—
81	C₉	—	12	4	no	—
82	C₉	C₂²⋊H(3,𝔽₃)	1	3	yes	complemented
83	C₂×C₃	—	3	1	no	—
84	C₂×C₃	—	3	2	no	—
85	C₂×C₃	—	3	1	no	—
86	C₂×C₃	—	9	3	no	—
87	C₂²	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
88	C₃	—	9	1	no	—
89	C₃	—	12	3	no	—
90	C₃	—	36	2	no	—
91	C₃	A₄×3₋¹⁺²	1	1	yes	characteristic
92	C₃	#66440	1	2	yes	—
93	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
94	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃²	C₃³
Fitting	C₂²×C₃×3₋¹⁺²	C₃
prime radical	C₃×C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	C₉⋊H(3,𝔽₃)
LCS	G ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ 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×C₉⋊A₄	C₃	semidirect
4. C₃×C₉×A₄	C₃	semidirect
5. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
6. (C₃×A₄)⋊C₉	C₃	semidirect
7. (C₃×A₄)⋊C₉	C₃	semidirect
22. C₃×3₋¹⁺²	A₄	semidirect
41. C₃×A₄	3₋¹⁺²	semidirect
43. C₂²×C₉	H(3,𝔽₃)	semidirect
82. C₉	C₂²⋊H(3,𝔽₃)	semidirect
87. 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×C₃×3₋¹⁺²	3
C₃×C₂²⋊H(3,𝔽₃)	3
C₃×C₉⋊A₄	3
C₃×C₉×A₄	3
C₃²⋊(C₂²⋊C₉)	3
(C₃×A₄)⋊C₉	3
C₉⋊H(3,𝔽₃)	4