Subgroups — #10624899

Order 729.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	#6899	C₃	1	9	yes	maximal, complemented
3	#6896	C₃	1	9	yes	maximal, complemented
4	#6894	C₃	1	6	yes	maximal, complemented
5	#6894	C₃	1	6	yes	maximal, complemented
6	#6891	C₃	1	1	yes	maximal, characteristic
7	#6891	C₃	1	1	yes	maximal, characteristic
8	C₉×3₋¹⁺²	C₃	1	3	yes	maximal, complemented
9	C₉×3₋¹⁺²	C₃	1	3	yes	maximal, complemented
10	C₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
11	C₃⁴	C₃²	1	1	yes	characteristic
12	C₃×3₋¹⁺²	—	3	9	no	—
13	C₃×3₋¹⁺²	C₃²	1	2	yes	—
14	C₃×3₋¹⁺²	C₃²	1	3	yes	—
15	C₃×3₋¹⁺²	C₃²	1	3	yes	—
16	C₃×3₋¹⁺²	C₃²	1	3	yes	—
17	C₃×3₋¹⁺²	C₃²	1	3	yes	—
18	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
19	C₃²×C₉	C₃²	1	3	yes	—
20	C₃²×C₉	C₃²	1	3	yes	—
21	C₉⋊C₉	C₃²	1	18	yes	complemented
22	C₉⋊C₉	C₃²	1	18	yes	complemented
23	C₉⋊C₉	C₃²	1	36	yes	complemented
24	C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	C₃²⋊C₉	C₃²	1	2	yes	—
26	C₃²⋊C₉	C₃²	1	3	yes	—
27	C₃²⋊C₉	C₃²	1	3	yes	—
28	C₃²⋊C₉	C₃²	1	3	yes	—
29	C₃²⋊C₉	C₃²	1	3	yes	—
30	C₃²⋊C₉	C₃²	1	3	yes	—
31	C₃²⋊C₉	C₃²	1	3	yes	—
32	C₃²⋊C₉	C₃²	1	3	yes	—
33	C₃²⋊C₉	C₃²	1	3	yes	—
34	C₉²	C₃²	1	9	yes	complemented
35	C₃³	—	3	3	no	—
36	C₃³	—	3	3	no	—
37	C₃³	—	9	1	no	—
38	C₃³	—	9	1	no	—
39	C₃³	C₃³	1	1	yes	characteristic
40	C₃³	C₃³	1	1	yes	characteristic
41	C₃³	C₃³	1	1	yes	characteristic
42	C₃³	C₃³	1	1	yes	characteristic
43	3₋¹⁺²	—	3	6	no	—
44	3₋¹⁺²	—	3	9	no	—
45	3₋¹⁺²	—	3	9	no	—
46	3₋¹⁺²	—	3	9	no	—
47	3₋¹⁺²	—	9	3	no	—
48	3₋¹⁺²	3₋¹⁺²	1	9	yes	complemented
49	C₃×C₉	—	3	9	no	—
50	C₃×C₉	—	9	3	no	—
51	C₃×C₉	C₃³	1	3	yes	—
52	C₃×C₉	C₃³	1	9	yes	—
53	C₃×C₉	C₃³	1	9	yes	—
54	C₃×C₉	C₃³	1	6	yes	—
55	C₃×C₉	C₃³	1	6	yes	—
56	C₃×C₉	C₃³	1	3	yes	—
57	C₃²	—	3	1	no	—
58	C₃²	—	3	1	no	—
59	C₃²	—	3	1	no	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	1	no	—
62	C₃²	—	3	1	no	—
63	C₃²	—	3	1	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	3	1	no	—
68	C₃²	—	3	1	no	—
69	C₃²	—	3	1	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	9	9	no	—
72	C₃²	C₃×3₋¹⁺²	1	3	yes	—
73	C₃²	#737	1	3	yes	—
74	C₃²	C₃⁴	1	1	yes	characteristic
75	C₉	—	3	6	no	—
76	C₉	—	3	9	no	—
77	C₉	—	3	3	no	—
78	C₉	—	3	6	no	—
79	C₉	—	3	9	no	—
80	C₉	C₃×3₋¹⁺²	1	9	yes	—
81	C₃	—	3	3	no	—
82	C₃	—	3	3	no	—
83	C₃	—	9	1	no	—
84	C₃	—	9	1	no	—
85	C₃	C₃²×3₋¹⁺²	1	1	yes	characteristic
86	C₃	C₃×#737	1	1	yes	characteristic
87	C₃	3₋¹⁺⁴	1	1	yes	characteristic
88	C₃	3₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃²	C₃⁴
prime radical	C₃⁴	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ G
LECS	G ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #6899	C₃	semidirect
3. #6896	C₃	semidirect
4. #6894	C₃	semidirect
5. #6894	C₃	semidirect
8. C₉×3₋¹⁺²	C₃	semidirect
9. C₉×3₋¹⁺²	C₃	semidirect
21. C₉⋊C₉	C₃²	semidirect
22. C₉⋊C₉	C₃²	semidirect
23. C₉⋊C₉	C₃²	semidirect
34. C₉²	C₃²	semidirect
48. 3₋¹⁺²	3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃²×3₋¹⁺²	3
#6899	3
#6896	3
#6894	3
#6891	3
C₉×3₋¹⁺²	3
C₃×C₃²⋊C₉	3