Subgroups — #10624791

Order 729.

107 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#733	C₃	1	1	yes	maximal, characteristic
2	#6894	C₃	1	3	yes	maximal
3	C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
4	3₋¹⁺²⋊C₉	C₃	1	6	yes	maximal
5	#6868	C₃	1	2	yes	maximal
6	C₃×3₋¹⁺²	C₃²	1	3	yes	—
7	C₃²×C₉	C₃²	1	1	yes	characteristic
8	C₃²×C₉	C₃²	1	2	yes	—
9	C₃²×C₉	C₃²	1	1	yes	characteristic
10	#733	—	3	3	no	—
11	C₉⋊C₉	—	3	3	no	—
12	C₉⋊C₉	—	3	9	no	—
13	C₃²⋊C₉	C₃²	1	6	yes	—
14	C₃³	C₃³	1	1	yes	characteristic
15	3₋¹⁺²	—	3	3	no	—
16	3₋¹⁺²	—	3	6	no	—
17	C₃×C₉	—	3	1	no	—
18	C₃×C₉	—	3	2	no	—
19	C₃×C₉	—	3	6	no	—
20	C₃×C₉	—	3	2	no	—
21	C₃×C₉	—	3	3	no	—
22	C₃×C₉	—	9	1	no	—
23	C₃×C₉	—	9	2	no	—
24	C₃×C₉	H(3,𝔽₃)	1	3	yes	—
25	C₃×C₉	3₋¹⁺²	1	3	yes	—
26	C₃²	—	3	1	no	—
27	C₃²	—	3	2	no	—
28	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₃²	C₃×3₋¹⁺²	1	1	yes	characteristic
30	C₃²	#737	1	2	yes	—
31	C₉	—	3	3	no	—
32	C₉	—	3	3	no	—
33	C₉	—	9	6	no	—
34	C₉	—	9	2	no	—
35	C₉	—	9	3	no	—
36	C₃	—	3	1	no	—
37	C₃	—	3	2	no	—
38	C₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
39	C₃	#6912	1	1	yes	characteristic
40	C₃	#6913	1	2	yes	—

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

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