Subgroups — #11757776

Order 972.

275 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×A₄	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉⋊A₄	C₃	1	3	yes	maximal, complemented
3	C₂²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
5	(C₃×A₄)⋊C₉	C₃	1	6	yes	maximal, complemented
6	#6891	—	4	1	no	maximal, Sylow
7	C₂×C₃²⋊C₉	—	3	1	no	—
8	C₂²×C₃³	C₃²	1	1	yes	characteristic
9	C₃²×A₄	—	3	3	no	—
10	C₃²×A₄	—	3	3	no	—
11	C₃²×A₄	—	3	6	no	—
12	C₃²×A₄	C₃²	1	1	yes	characteristic
13	C₃²×A₄	C₃²	1	2	yes	—
14	C₂²×C₃×C₉	C₃²	1	3	yes	complemented
15	C₃×C₂²⋊C₉	C₃²	1	6	yes	complemented
16	C₉⋊A₄	—	3	9	no	—
17	C₃⁴	—	4	1	no	—
18	C₃×3₋¹⁺²	—	4	3	no	—
19	C₃²⋊C₉	—	4	2	no	—
20	C₃²⋊C₉	—	4	6	no	—
21	C₃²⋊C₉	A₄	1	1	yes	characteristic, complemented
22	C₂×C₃³	—	3	1	no	—
23	C₂×C₃×C₉	—	3	3	no	—
24	C₂²×C₃²	—	3	1	no	—
25	C₂²×C₃²	—	3	2	no	—
26	C₂²×C₃²	3₋¹⁺²	1	3	yes	complemented
27	C₂²×C₃²	C₃³	1	1	yes	characteristic
28	C₃×A₄	—	3	27	no	—
29	C₃×A₄	—	3	2	no	—
30	C₃×A₄	—	3	1	no	—
31	C₃×A₄	—	3	2	no	—
32	C₃×A₄	—	3	2	no	—
33	C₃×A₄	—	3	2	no	—
34	C₃×A₄	H(3,𝔽₃)	1	3	yes	—
35	C₃×A₄	3₋¹⁺²	1	6	yes	complemented
36	C₂²×C₉	—	3	3	no	—
37	C₂²⋊C₉	—	3	6	no	—
38	C₃³	—	4	2	no	—
39	C₃³	—	4	1	no	—
40	C₃³	—	12	3	no	—
41	C₃³	—	12	6	no	—
42	C₃³	—	12	3	no	—
43	C₃³	C₃×A₄	1	1	yes	characteristic
44	3₋¹⁺²	—	12	9	no	—
45	C₃×C₉	—	4	6	no	—
46	C₃×C₉	C₃×A₄	1	3	yes	complemented
47	C₂×C₃²	—	3	1	no	—
48	C₂×C₃²	—	3	3	no	—
49	C₂×C₃²	—	9	1	no	—
50	C₂×C₃²	—	9	2	no	—
51	C₂×C₉	—	9	3	no	—
52	C₂²×C₃	—	3	3	no	—
53	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₂²×C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
55	C₂²×C₃	C₃×3₋¹⁺²	1	2	yes	—
56	A₄	—	3	3	no	—
57	A₄	—	3	6	no	—
58	C₃²	—	3	1	no	—
59	C₃²	—	3	2	no	—
60	C₃²	—	4	3	no	—
61	C₃²	—	4	6	no	—
62	C₃²	—	12	27	no	—
63	C₃²	—	12	2	no	—
64	C₃²	—	12	1	no	—
65	C₃²	—	12	2	no	—
66	C₃²	—	12	2	no	—
67	C₃²	—	12	2	no	—
68	C₃²	C₉⋊A₄	1	3	yes	complemented
69	C₃²	C₃²×A₄	1	1	yes	characteristic
70	C₉	—	3	3	no	—
71	C₉	—	12	6	no	—
72	C₂×C₃	—	3	1	no	—
73	C₂×C₃	—	3	1	no	—
74	C₂×C₃	—	3	2	no	—
75	C₂×C₃	—	9	3	no	—
76	C₂²	#6891	1	1	yes	characteristic, complemented, Sylow
77	C₃	—	3	3	no	—
78	C₃	—	12	3	no	—
79	C₃	—	12	6	no	—
80	C₃	C₃×C₉⋊A₄	1	1	yes	characteristic
81	C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
82	C₃	A₄×3₋¹⁺²	1	2	yes	—
83	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃²	C₃³
Fitting	C₂²×C₃²⋊C₉	C₃
prime radical	C₃³×A₄	C₃
hypercenter	C₃²⋊C₉	A₄
hypocenter	C₂²	#6891
LCS	G ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₃×C₉⋊A₄	C₃	semidirect
3. C₂²×C₃²⋊C₉	C₃	semidirect
4. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
5. (C₃×A₄)⋊C₉	C₃	semidirect
14. C₂²×C₃×C₉	C₃²	semidirect
15. C₃×C₂²⋊C₉	C₃²	semidirect
21. C₃²⋊C₉	A₄	semidirect
26. C₂²×C₃²	3₋¹⁺²	semidirect
35. C₃×A₄	3₋¹⁺²	semidirect
46. C₃×C₉	C₃×A₄	semidirect
68. C₃²	C₉⋊A₄	semidirect
76. C₂²	#6891	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃³×A₄	3
C₃×C₉⋊A₄	3
C₂²×C₃²⋊C₉	3
C₃²⋊(C₂²⋊C₉)	3
(C₃×A₄)⋊C₉	3
#6891	4