Subgroups — C₃²×#6899 #14555295

Order 2187.

4634 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃×#6899	C₃	1	108	yes	maximal, direct product factor
3	C₃²×C₉⋊C₉	C₃	1	6	yes	maximal, complemented
4	C₃²×C₉⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₃²×C₃²⋊C₉	C₃	1	3	yes	maximal
6	C₃²×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃⁵	C₃²	1	1	yes	characteristic
8	C₃²×3₋¹⁺²	—	3	27	no	—
9	C₃²×3₋¹⁺²	C₃²	1	36	yes	—
10	C₃³×C₉	C₃²	1	3	yes	—
11	C₃³×C₉	C₃²	1	6	yes	—
12	C₃³×C₉	C₃²	1	2	yes	—
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	#6899	C₃²	1	729	yes	direct product factor
15	C₃×C₉⋊C₉	C₃²	1	216	yes	complemented
16	C₃×C₉⋊C₉	C₃²	1	72	yes	complemented
17	C₃×C₃²⋊C₉	C₃²	1	108	yes	—
18	C₃×C₉²	C₃²	1	36	yes	complemented
19	C₃⁴	—	3	9	no	—
20	C₃⁴	—	3	27	no	—
21	C₃⁴	C₃³	1	12	yes	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃×3₋¹⁺²	—	3	324	no	—
24	C₃×3₋¹⁺²	C₃³	1	81	yes	—
25	C₃²×C₉	—	3	18	no	—
26	C₃²×C₉	—	3	27	no	—
27	C₃²×C₉	—	3	54	no	—
28	C₃²×C₉	3₋¹⁺²	1	27	yes	—
29	C₃²×C₉	C₃³	1	24	yes	—
30	C₃²×C₉	C₃³	1	36	yes	—
31	C₃²×C₉	C₃³	1	72	yes	—
32	C₃²×C₉	C₃³	1	12	yes	—
33	C₉⋊C₉	C₃³	1	486	yes	complemented
34	C₉⋊C₉	C₃³	1	162	yes	complemented
35	C₃²⋊C₉	C₃³	1	243	yes	—
36	C₉²	C₃³	1	81	yes	complemented
37	C₃³	—	3	36	no	—
38	C₃³	—	3	108	no	—
39	C₃³	—	9	81	no	—
40	C₃³	C₃×3₋¹⁺²	1	9	yes	—
41	C₃³	#737	1	27	yes	—
42	C₃³	C₃⁴	1	9	yes	—
43	C₃³	C₃⁴	1	4	yes	—
44	3₋¹⁺²	—	3	243	no	—
45	C₃×C₉	—	3	72	no	—
46	C₃×C₉	—	3	108	no	—
47	C₃×C₉	—	3	216	no	—
48	C₃×C₉	C₃×3₋¹⁺²	1	108	yes	—
49	C₃×C₉	C₃⁴	1	27	yes	—
50	C₃×C₉	C₃⁴	1	54	yes	—
51	C₃×C₉	C₃⁴	1	18	yes	—
52	C₃×C₉	C₃⁴	1	9	yes	—
53	C₃²	—	3	9	no	—
54	C₃²	—	3	27	no	—
55	C₃²	—	9	108	no	—
56	C₃²	#6899	1	81	yes	direct product factor
57	C₃²	C₃²×3₋¹⁺²	1	12	yes	—
58	C₃²	C₃×#737	1	36	yes	—
59	C₃²	C₃⁵	1	1	yes	characteristic
60	C₉	—	3	18	no	—
61	C₉	—	3	27	no	—
62	C₉	—	3	54	no	—
63	C₉	C₃²×3₋¹⁺²	1	27	yes	—
64	C₃	—	9	9	no	—
65	C₃	C₃×#6899	1	36	yes	direct product factor
66	C₃	C₃³×3₋¹⁺²	1	1	yes	characteristic
67	C₃	C₃²×#737	1	3	yes	—

	subgroup	G/H
center, socle	C₃⁴	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁵
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃⁴ ◁ G
LECS	G ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃×#6899	C₃	direct (N × H)
3. C₃²×C₉⋊C₉	C₃	semidirect
4. C₃²×C₉⋊C₉	C₃	semidirect
6. C₃²×C₉²	C₃	semidirect
14. #6899	C₃²	direct (N × H)
15. C₃×C₉⋊C₉	C₃²	semidirect
16. C₃×C₉⋊C₉	C₃²	semidirect
18. C₃×C₉²	C₃²	semidirect
33. C₉⋊C₉	C₃³	semidirect
34. C₉⋊C₉	C₃³	semidirect
36. C₉²	C₃³	semidirect
56. C₃²	#6899	direct (N × H)
65. C₃	C₃×#6899	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×3₋¹⁺²	3
C₃×#6899	3
C₃²×C₉⋊C₉	3
C₃²×C₃²⋊C₉	3
C₃²×C₉²	3

Subgroups — C32×#6899 #14555295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#6899 #14555295