Subgroups — #10624905

Order 729.

394 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6893	C₃	1	9	yes	maximal, complemented
2	C₉⋊H(3,𝔽₃)	C₃	1	12	yes	maximal, complemented
3	C₉⋊H(3,𝔽₃)	C₃	1	9	yes	maximal, complemented
4	#6891	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
6	C₉×H(3,𝔽₃)	C₃	1	6	yes	maximal, complemented
7	C₃×C₃²⋊C₉	C₃	1	2	yes	maximal, complemented
8	C₃⁴	C₃²	1	1	yes	characteristic
9	C₃×3₋¹⁺²	C₃²	1	3	yes	—
10	C₃×3₋¹⁺²	C₃²	1	18	yes	complemented
11	C₃×H(3,𝔽₃)	C₃²	1	6	yes	—
12	C₃×H(3,𝔽₃)	C₃²	1	3	yes	—
13	C₃×H(3,𝔽₃)	C₃²	1	3	yes	—
14	C₃²×C₉	C₃²	1	6	yes	—
15	C₃²×C₉	C₃²	1	9	yes	complemented
16	C₃²⋊C₉	C₃²	1	2	yes	—
17	C₃²⋊C₉	C₃²	1	6	yes	—
18	C₃²⋊C₉	C₃²	1	4	yes	—
19	C₃²⋊C₉	C₃²	1	6	yes	—
20	C₃²⋊C₉	C₃²	1	6	yes	—
21	C₃²⋊C₉	C₃²	1	3	yes	—
22	C₃²⋊C₉	C₃²	1	36	yes	complemented
23	C₃²⋊C₉	C₃²	1	18	yes	complemented
24	C₃³	—	3	6	no	—
25	C₃³	—	9	1	no	—
26	C₃³	—	9	1	no	—
27	C₃³	C₃³	1	1	yes	characteristic
28	C₃³	C₃³	1	1	yes	characteristic
29	C₃³	C₃³	1	9	yes	—
30	C₃³	C₃³	1	2	yes	—
31	3₋¹⁺²	—	9	3	no	—
32	3₋¹⁺²	—	9	18	no	—
33	H(3,𝔽₃)	—	3	18	no	—
34	H(3,𝔽₃)	—	9	3	no	—
35	H(3,𝔽₃)	—	9	3	no	—
36	C₃×C₉	—	9	6	no	—
37	C₃×C₉	—	9	9	no	—
38	C₃×C₉	C₃³	1	6	yes	—
39	C₃×C₉	C₃³	1	12	yes	—
40	C₃×C₉	C₃³	1	9	yes	—
41	C₃²	—	3	2	no	—
42	C₃²	—	3	2	no	—
43	C₃²	—	3	2	no	—
44	C₃²	—	3	18	no	—
45	C₃²	—	3	2	no	—
46	C₃²	—	3	2	no	—
47	C₃²	—	3	9	no	—
48	C₃²	—	3	1	no	—
49	C₃²	—	3	1	no	—
50	C₃²	—	3	1	no	—
51	C₃²	—	3	9	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	9	9	no	—
54	C₃²	#737	1	6	yes	—
55	C₃²	C₃⁴	1	1	yes	characteristic
56	C₉	—	3	6	no	—
57	C₉	—	3	12	no	—
58	C₉	—	3	9	no	—
59	C₃	—	3	6	no	—
60	C₃	—	9	9	no	—
61	C₃	—	9	1	no	—
62	C₃	—	9	1	no	—
63	C₃	C₃×#737	1	2	yes	—
64	C₃	H(5,𝔽₃)	1	1	yes	characteristic
65	C₃	3₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃²	C₃⁴
prime radical	C₃²⋊H(3,𝔽₃)	C₃
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
1. #6893	C₃	semidirect
2. C₉⋊H(3,𝔽₃)	C₃	semidirect
3. C₉⋊H(3,𝔽₃)	C₃	semidirect
4. #6891	C₃	semidirect
6. C₉×H(3,𝔽₃)	C₃	semidirect
7. C₃×C₃²⋊C₉	C₃	semidirect
10. C₃×3₋¹⁺²	C₃²	semidirect
15. C₃²×C₉	C₃²	semidirect
22. C₃²⋊C₉	C₃²	semidirect
23. C₃²⋊C₉	C₃²	semidirect

Maximal subgroups

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

subgroup	index
#6893	3
C₉⋊H(3,𝔽₃)	3
#6891	3
C₃²⋊H(3,𝔽₃)	3
C₉×H(3,𝔽₃)	3
C₃×C₃²⋊C₉	3