Subgroups — #10624792

Order 729.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#732	C₃	1	1	yes	maximal, characteristic
2	C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal, complemented
3	C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	H(3,𝔽₃)⋊C₉	C₃	1	4	yes	maximal, complemented
6	H(3,𝔽₃)⋊C₉	C₃	1	2	yes	maximal, complemented
7	#6868	C₃	1	2	yes	maximal, complemented
8	C₃×3₋¹⁺²	—	3	2	no	—
9	C₃×H(3,𝔽₃)	C₃²	1	2	yes	—
10	C₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃²×C₉	—	3	1	no	—
12	C₃²×C₉	C₃²	1	1	yes	characteristic
13	C₃²×C₉	C₃²	1	2	yes	—
14	C₃²×C₉	C₃²	1	1	yes	characteristic
15	#732	—	3	3	no	—
16	C₉⋊C₉	—	3	2	no	—
17	C₉⋊C₉	—	3	1	no	—
18	C₃²⋊C₉	—	3	2	no	—
19	C₃²⋊C₉	—	3	4	no	—
20	C₃²⋊C₉	C₃²	1	2	yes	—
21	C₃²⋊C₉	C₃²	1	4	yes	—
22	C₃³	—	3	1	no	—
23	C₃³	—	3	2	no	—
24	C₃³	C₃³	1	1	yes	characteristic
25	3₋¹⁺²	—	9	6	no	—
26	H(3,𝔽₃)	—	3	2	no	—
27	H(3,𝔽₃)	—	3	2	no	—
28	H(3,𝔽₃)	—	3	4	no	—
29	H(3,𝔽₃)	3₋¹⁺²	1	3	yes	complemented
30	C₃×C₉	—	3	3	no	—
31	C₃×C₉	—	3	2	no	—
32	C₃×C₉	—	3	4	no	—
33	C₃×C₉	—	3	2	no	—
34	C₃×C₉	—	3	1	no	—
35	C₃×C₉	—	3	1	no	—
36	C₃×C₉	—	3	2	no	—
37	C₃×C₉	—	9	3	no	—
38	C₃×C₉	—	9	2	no	—
39	C₃×C₉	H(3,𝔽₃)	1	2	yes	—
40	C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
41	C₃²	—	3	2	no	—
42	C₃²	—	3	1	no	—
43	C₃²	—	3	3	no	—
44	C₃²	—	9	2	no	—
45	C₃²	—	9	1	no	—
46	C₃²	—	9	4	no	—
47	C₃²	—	9	2	no	—
48	C₃²	—	9	2	no	—
49	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃²	C₃×3₋¹⁺²	1	1	yes	characteristic
51	C₃²	#737	1	2	yes	—
52	C₉	—	3	2	no	—
53	C₉	—	9	2	no	—
54	C₉	—	9	1	no	—
55	C₉	—	9	2	no	—
56	C₉	—	9	4	no	—
57	C₉	#732	1	3	yes	complemented
58	C₃	—	3	1	no	—
59	C₃	—	3	2	no	—
60	C₃	—	9	3	no	—
61	C₃	—	27	2	no	—
62	C₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
63	C₃	C₃×#732	1	1	yes	characteristic
64	C₃	#6913	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³
prime radical	C₃×#732	C₃
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. C₉⋊H(3,𝔽₃)	C₃	semidirect
3. C₉×H(3,𝔽₃)	C₃	semidirect
4. C₃×C₉⋊C₉	C₃	semidirect
5. H(3,𝔽₃)⋊C₉	C₃	semidirect
6. H(3,𝔽₃)⋊C₉	C₃	semidirect
7. #6868	C₃	semidirect
29. H(3,𝔽₃)	3₋¹⁺²	semidirect
57. C₉	#732	semidirect

Maximal subgroups

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

subgroup	index
C₃×#732	3
C₉⋊H(3,𝔽₃)	3
C₉×H(3,𝔽₃)	3
C₃×C₉⋊C₉	3
H(3,𝔽₃)⋊C₉	3
#6868	3