Subgroups — #10624809

Order 729.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6891	C₃	1	1	yes	maximal, characteristic
2	#6891	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
5	#6867	C₃	1	9	yes	maximal, complemented
6	C₃⁴	C₃²	1	1	yes	characteristic
7	C₃×3₋¹⁺²	—	3	1	no	—
8	C₃×3₋¹⁺²	—	3	1	no	—
9	C₃²×C₉	—	3	1	no	—
10	C₃²×C₉	C₃²	1	3	yes	—
11	C₃²⋊C₉	—	3	1	no	—
12	C₃²⋊C₉	—	3	1	no	—
13	C₃²⋊C₉	—	3	1	no	—
14	C₃²⋊C₉	—	3	1	no	—
15	C₃²⋊C₉	—	3	1	no	—
16	C₃²⋊C₉	—	3	1	no	—
17	C₃²⋊C₉	—	3	1	no	—
18	C₃²⋊C₉	—	3	1	no	—
19	C₃²⋊C₉	—	3	1	no	—
20	C₃²⋊C₉	C₃²	1	3	yes	—
21	C₃²⋊C₉	C₃²	1	3	yes	—
22	C₃²⋊C₉	C₃²	1	3	yes	—
23	C₃³	—	3	3	no	—
24	C₃³	—	3	3	no	—
25	C₃³	—	9	1	no	—
26	C₃³	—	9	1	no	—
27	C₃³	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃³	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³	C₃³	1	1	yes	characteristic
31	3₋¹⁺²	—	9	3	no	—
32	3₋¹⁺²	—	9	3	no	—
33	C₃×C₉	—	3	3	no	—
34	C₃×C₉	—	3	3	no	—
35	C₃×C₉	—	3	3	no	—
36	C₃×C₉	—	3	3	no	—
37	C₃×C₉	—	3	9	no	—
38	C₃×C₉	—	9	3	no	—
39	C₃²	—	3	1	no	—
40	C₃²	—	3	1	no	—
41	C₃²	—	3	1	no	—
42	C₃²	—	3	1	no	—
43	C₃²	—	3	1	no	—
44	C₃²	—	3	1	no	—
45	C₃²	—	3	1	no	—
46	C₃²	—	3	1	no	—
47	C₃²	—	3	1	no	—
48	C₃²	—	3	1	no	—
49	C₃²	—	3	1	no	—
50	C₃²	—	3	1	no	—
51	C₃²	—	3	1	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	9	9	no	—
54	C₃²	#731	1	3	yes	—
55	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²	#737	1	3	yes	—
57	C₉	—	9	3	no	—
58	C₉	—	9	3	no	—
59	C₉	—	9	3	no	—
60	C₉	—	9	3	no	—
61	C₃	—	3	3	no	—
62	C₃	—	3	3	no	—
63	C₃	—	9	1	no	—
64	C₃	—	9	1	no	—
65	C₃	#6893	1	1	yes	characteristic
66	C₃	C₃×#731	1	1	yes	characteristic
67	C₃	#6910	1	1	yes	characteristic
68	C₃	#6910	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³
prime radical	C₃⁴	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #6867	C₃	semidirect

Maximal subgroups

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

subgroup	index
#6891	3
C₃×C₃²⋊C₉	3
#6867	3