Subgroups — C₃²×#6891 #14555253

Order 2187.

24308 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁶	C₃	1	1	yes	maximal, characteristic
2	C₃³×3₋¹⁺²	C₃	1	3	yes	maximal, complemented
3	C₃×#6891	C₃	1	108	yes	maximal, direct product factor
4	C₃²×C₃²⋊C₉	C₃	1	9	yes	maximal, complemented
5	C₃⁵	—	3	27	no	—
6	C₃⁵	—	3	81	no	—
7	C₃⁵	C₃²	1	36	yes	—
8	C₃⁵	C₃²	1	1	yes	characteristic
9	C₃⁵	C₃²	1	3	yes	—
10	C₃²×3₋¹⁺²	—	3	81	no	—
11	C₃²×3₋¹⁺²	C₃²	1	108	yes	complemented
12	C₃³×C₉	C₃²	1	9	yes	complemented
13	#6891	C₃²	1	729	yes	direct product factor
14	C₃×C₃²⋊C₉	C₃²	1	324	yes	complemented
15	C₃⁴	—	3	2187	no	—
16	C₃⁴	—	3	324	no	—
17	C₃⁴	—	3	27	no	—
18	C₃⁴	—	3	27	no	—
19	C₃⁴	—	3	54	no	—
20	C₃⁴	—	3	972	no	—
21	C₃⁴	H(3,𝔽₃)	1	27	yes	—
22	C₃⁴	3₋¹⁺²	1	81	yes	complemented
23	C₃⁴	C₃³	1	81	yes	—
24	C₃⁴	C₃³	1	12	yes	—
25	C₃⁴	C₃³	1	36	yes	—
26	C₃⁴	C₃³	1	1	yes	characteristic
27	C₃×3₋¹⁺²	—	3	972	no	—
28	C₃×3₋¹⁺²	C₃³	1	243	yes	complemented
29	C₃²×C₉	—	3	81	no	—
30	C₃²×C₉	C₃³	1	108	yes	complemented
31	C₃²⋊C₉	C₃³	1	729	yes	complemented
32	C₃³	—	3	8748	no	—
33	C₃³	—	3	243	no	—
34	C₃³	—	3	108	no	—
35	C₃³	—	3	108	no	—
36	C₃³	—	3	216	no	—
37	C₃³	—	3	729	no	—
38	C₃³	—	3	243	no	—
39	C₃³	—	3	729	no	—
40	C₃³	C₃×H(3,𝔽₃)	1	108	yes	—
41	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
42	C₃³	C₃×3₋¹⁺²	1	27	yes	—
43	C₃³	C₃×3₋¹⁺²	1	324	yes	complemented
44	C₃³	C₃⁴	1	9	yes	—
45	C₃³	C₃⁴	1	27	yes	—
46	C₃³	C₃⁴	1	4	yes	—
47	3₋¹⁺²	—	3	729	no	—
48	C₃×C₉	—	3	324	no	—
49	C₃×C₉	C₃⁴	1	81	yes	complemented
50	C₃²	—	3	2187	no	—
51	C₃²	—	3	324	no	—
52	C₃²	—	3	972	no	—
53	C₃²	—	3	27	no	—
54	C₃²	—	3	27	no	—
55	C₃²	—	3	54	no	—
56	C₃²	#6891	1	81	yes	direct product factor
57	C₃²	C₃²×H(3,𝔽₃)	1	27	yes	—
58	C₃²	C₃²×H(3,𝔽₃)	1	12	yes	—
59	C₃²	C₃²×3₋¹⁺²	1	36	yes	—
60	C₃²	C₃²×3₋¹⁺²	1	81	yes	complemented
61	C₃²	C₃⁵	1	1	yes	characteristic
62	C₉	—	3	81	no	—
63	C₃	—	3	27	no	—
64	C₃	—	3	81	no	—
65	C₃	C₃×#6891	1	36	yes	direct product factor
66	C₃	C₃³×H(3,𝔽₃)	1	1	yes	characteristic
67	C₃	C₃³×3₋¹⁺²	1	3	yes	—

	subgroup	G/H
center, socle	C₃⁴	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁵
prime radical	C₃⁶	C₃
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. C₃³×3₋¹⁺²	C₃	semidirect
3. C₃×#6891	C₃	direct (N × H)
4. C₃²×C₃²⋊C₉	C₃	semidirect
11. C₃²×3₋¹⁺²	C₃²	semidirect
12. C₃³×C₉	C₃²	semidirect
13. #6891	C₃²	direct (N × H)
14. C₃×C₃²⋊C₉	C₃²	semidirect
22. C₃⁴	3₋¹⁺²	semidirect
28. C₃×3₋¹⁺²	C₃³	semidirect
30. C₃²×C₉	C₃³	semidirect
31. C₃²⋊C₉	C₃³	semidirect
43. C₃³	C₃×3₋¹⁺²	semidirect
49. C₃×C₉	C₃⁴	semidirect
56. C₃²	#6891	direct (N × H)
60. C₃²	C₃²×3₋¹⁺²	semidirect
65. C₃	C₃×#6891	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁶	3
C₃³×3₋¹⁺²	3
C₃×#6891	3
C₃²×C₃²⋊C₉	3

Subgroups — C32×#6891 #14555253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#6891 #14555253