Subgroups — #10624800

Order 729.

118 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	1	yes	maximal, characteristic, complemented
3	C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	3₋¹⁺²⋊C₉	C₃	1	2	yes	maximal, complemented
7	3₋¹⁺²⋊C₉	C₃	1	2	yes	maximal, complemented
8	H(3,𝔽₃)⋊C₉	C₃	1	2	yes	maximal
9	#6867	C₃	1	2	yes	maximal, complemented
10	C₃×3₋¹⁺²	—	3	1	no	—
11	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
13	C₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃²×C₉	C₃²	1	1	yes	characteristic
15	C₃²×C₉	C₃²	1	2	yes	—
16	C₃²×C₉	C₃²	1	1	yes	characteristic
17	#731	—	3	3	no	—
18	C₉⋊C₉	—	3	2	no	—
19	C₉⋊C₉	—	3	1	no	—
20	C₉⋊C₉	—	3	2	no	—
21	C₉⋊C₉	—	3	2	no	—
22	C₉⋊C₉	—	3	1	no	—
23	C₃²⋊C₉	—	3	2	no	—
24	C₃²⋊C₉	C₃²	1	2	yes	—
25	C₃²⋊C₉	C₃²	1	2	yes	—
26	C₃²⋊C₉	C₃²	1	2	yes	—
27	C₉²	—	3	1	no	—
28	C₃³	—	3	1	no	—
29	C₃³	C₃³	1	1	yes	characteristic
30	3₋¹⁺²	—	3	2	no	—
31	3₋¹⁺²	—	3	1	no	—
32	3₋¹⁺²	—	3	2	no	—
33	3₋¹⁺²	—	9	3	no	—
34	3₋¹⁺²	3₋¹⁺²	1	3	yes	complemented
35	H(3,𝔽₃)	—	3	1	no	—
36	H(3,𝔽₃)	—	3	2	no	—
37	C₃×C₉	—	3	1	no	—
38	C₃×C₉	—	3	3	no	—
39	C₃×C₉	—	3	2	no	—
40	C₃×C₉	—	3	2	no	—
41	C₃×C₉	—	3	2	no	—
42	C₃×C₉	—	3	2	no	—
43	C₃×C₉	—	3	1	no	—
44	C₃×C₉	—	3	1	no	—
45	C₃×C₉	—	3	1	no	—
46	C₃×C₉	—	3	2	no	—
47	C₃×C₉	—	9	2	no	—
48	C₃×C₉	H(3,𝔽₃)	1	2	yes	—
49	C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
50	C₃²	—	3	2	no	—
51	C₃²	—	3	1	no	—
52	C₃²	—	9	2	no	—
53	C₃²	—	9	1	no	—
54	C₃²	—	9	1	no	—
55	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²	C₃×3₋¹⁺²	1	1	yes	characteristic
57	C₃²	#737	1	2	yes	—
58	C₉	—	3	3	no	—
59	C₉	—	3	2	no	—
60	C₉	—	9	2	no	—
61	C₉	—	9	2	no	—
62	C₉	—	9	2	no	—
63	C₉	—	9	1	no	—
64	C₉	—	9	1	no	—
65	C₉	—	9	2	no	—
66	C₉	#731	1	3	yes	complemented
67	C₃	—	3	1	no	—
68	C₃	—	3	2	no	—
69	C₃	—	27	1	no	—
70	C₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
71	C₃	C₃×#731	1	1	yes	characteristic
72	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. C₉×3₋¹⁺²	C₃	semidirect
5. C₃×C₉⋊C₉	C₃	semidirect
6. 3₋¹⁺²⋊C₉	C₃	semidirect
7. 3₋¹⁺²⋊C₉	C₃	semidirect
9. #6867	C₃	semidirect
34. 3₋¹⁺²	3₋¹⁺²	semidirect
66. C₉	#731	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
C₉×3₋¹⁺²	3
C₃×C₉⋊C₉	3
3₋¹⁺²⋊C₉	3
H(3,𝔽₃)⋊C₉	3
#6867	3