Subgroups — #10624789

Order 729.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#731	C₃	1	1	yes	maximal, characteristic
2	#6894	C₃	1	2	yes	maximal, complemented
3	C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²⋊C₉	C₃	1	4	yes	maximal, complemented
6	H(3,𝔽₃)⋊C₉	C₃	1	2	yes	maximal
7	#6867	C₃	1	2	yes	maximal, complemented
8	C₃×3₋¹⁺²	—	3	1	no	—
9	C₃×3₋¹⁺²	C₃²	1	2	yes	—
10	C₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃²×C₉	C₃²	1	1	yes	characteristic
12	C₃²×C₉	C₃²	1	2	yes	—
13	C₃²×C₉	C₃²	1	1	yes	characteristic
14	#731	—	3	3	no	—
15	C₉⋊C₉	—	3	2	no	—
16	C₉⋊C₉	—	3	1	no	—
17	C₉⋊C₉	—	3	4	no	—
18	C₉⋊C₉	—	3	2	no	—
19	C₃²⋊C₉	—	3	2	no	—
20	C₃²⋊C₉	C₃²	1	4	yes	—
21	C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃³	—	3	1	no	—
23	C₃³	C₃³	1	1	yes	characteristic
24	3₋¹⁺²	—	3	2	no	—
25	3₋¹⁺²	—	3	4	no	—
26	3₋¹⁺²	—	9	3	no	—
27	H(3,𝔽₃)	—	3	1	no	—
28	H(3,𝔽₃)	—	3	2	no	—
29	C₃×C₉	—	3	1	no	—
30	C₃×C₉	—	3	2	no	—
31	C₃×C₉	—	3	2	no	—
32	C₃×C₉	—	3	4	no	—
33	C₃×C₉	—	3	2	no	—
34	C₃×C₉	—	3	2	no	—
35	C₃×C₉	—	9	1	no	—
36	C₃×C₉	—	9	2	no	—
37	C₃×C₉	H(3,𝔽₃)	1	2	yes	—
38	C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
39	C₃×C₉	3₋¹⁺²	1	3	yes	complemented
40	C₃²	—	3	2	no	—
41	C₃²	—	3	1	no	—
42	C₃²	—	9	2	no	—
43	C₃²	—	9	1	no	—
44	C₃²	—	9	1	no	—
45	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
46	C₃²	C₃×3₋¹⁺²	1	1	yes	characteristic
47	C₃²	#737	1	2	yes	—
48	C₉	—	3	3	no	—
49	C₉	—	3	2	no	—
50	C₉	—	3	1	no	—
51	C₉	—	9	4	no	—
52	C₉	—	9	2	no	—
53	C₉	—	9	2	no	—
54	C₉	—	9	2	no	—
55	C₃	—	3	1	no	—
56	C₃	—	3	2	no	—
57	C₃	—	27	1	no	—
58	C₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
59	C₃	#6910	1	1	yes	characteristic
60	C₃	#6913	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³
prime radical	C₃×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #6894	C₃	semidirect
4. C₃×C₉⋊C₉	C₃	semidirect
5. 3₋¹⁺²⋊C₉	C₃	semidirect
7. #6867	C₃	semidirect
39. C₃×C₉	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
C₃×#731	3
#6894	3
C₉⋊H(3,𝔽₃)	3
C₃×C₉⋊C₉	3
3₋¹⁺²⋊C₉	3
H(3,𝔽₃)⋊C₉	3
#6867	3