Subgroups — #10624799

Order 729.

111 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	2	yes	maximal
3	C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
5	3₋¹⁺²⋊C₉	C₃	1	4	yes	maximal
6	3₋¹⁺²⋊C₉	C₃	1	2	yes	maximal
7	#6868	C₃	1	2	yes	maximal
8	C₃×3₋¹⁺²	C₃²	1	2	yes	—
9	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
10	C₃²×C₉	C₃²	1	1	yes	characteristic
11	C₃²×C₉	C₃²	1	2	yes	—
12	C₃²×C₉	C₃²	1	1	yes	characteristic
13	#733	—	3	3	no	—
14	C₉⋊C₉	—	3	2	no	—
15	C₉⋊C₉	—	3	1	no	—
16	C₉⋊C₉	—	3	2	no	—
17	C₉⋊C₉	—	3	4	no	—
18	C₉⋊C₉	—	3	2	no	—
19	C₃²⋊C₉	C₃²	1	2	yes	—
20	C₃²⋊C₉	C₃²	1	4	yes	—
21	C₉²	—	3	1	no	—
22	C₃³	C₃³	1	1	yes	characteristic
23	3₋¹⁺²	—	3	2	no	—
24	3₋¹⁺²	—	3	2	no	—
25	3₋¹⁺²	—	3	4	no	—
26	3₋¹⁺²	3₋¹⁺²	1	3	yes	—
27	C₃×C₉	—	3	1	no	—
28	C₃×C₉	—	3	3	no	—
29	C₃×C₉	—	3	2	no	—
30	C₃×C₉	—	3	4	no	—
31	C₃×C₉	—	3	2	no	—
32	C₃×C₉	—	3	1	no	—
33	C₃×C₉	—	3	2	no	—
34	C₃×C₉	—	3	1	no	—
35	C₃×C₉	—	3	2	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	C₃²	—	3	2	no	—
40	C₃²	—	3	1	no	—
41	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₃²	C₃×3₋¹⁺²	1	1	yes	characteristic
43	C₃²	#737	1	2	yes	—
44	C₉	—	3	3	no	—
45	C₉	—	3	2	no	—
46	C₉	—	9	2	no	—
47	C₉	—	9	4	no	—
48	C₉	—	9	2	no	—
49	C₉	—	9	1	no	—
50	C₉	—	9	2	no	—
51	C₉	#733	1	3	yes	complemented
52	C₃	—	3	1	no	—
53	C₃	—	3	2	no	—
54	C₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
55	C₃	C₃×#733	1	1	yes	characteristic
56	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

Complements

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

N	H ≅ G/N	type
51. C₉	#733	semidirect

Maximal subgroups

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

subgroup	index
C₃×#733	3
#6894	3
C₉×3₋¹⁺²	3
C₃×C₉⋊C₉	3
3₋¹⁺²⋊C₉	3
#6868	3