Subgroups — #10624622

Order 729.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6895	C₃	1	1	yes	maximal, characteristic, complemented
2	C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₉×H(3,𝔽₃)	C₃	1	2	yes	maximal, complemented
4	#6861	C₃	1	1	yes	maximal, characteristic, complemented
5	#6861	C₃	1	2	yes	maximal, complemented
6	#6859	C₃	1	1	yes	maximal, characteristic, complemented
7	#6859	C₃	1	2	yes	maximal, complemented
8	#6856	C₃	1	2	yes	maximal, complemented
9	#6856	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×3₋¹⁺²	—	3	1	no	—
11	C₃×H(3,𝔽₃)	C₃²	1	2	yes	—
12	C₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃²×C₉	—	3	2	no	—
14	C₃²×C₉	C₃²	1	1	yes	characteristic
15	C₉⋊C₉	—	3	2	no	—
16	C₉⋊C₉	—	3	1	no	—
17	C₃²⋊C₉	—	3	2	no	—
18	C₃²⋊C₉	—	3	2	no	—
19	C₃²⋊C₉	—	3	2	no	—
20	C₃²⋊C₉	C₃²	1	1	yes	characteristic
21	C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃²⋊C₉	C₃²	1	2	yes	—
23	C₃²⋊C₉	C₃²	1	1	yes	characteristic
24	C₃²⋊C₉	C₃²	1	2	yes	—
25	C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	C₃³	—	3	2	no	—
27	C₃³	—	3	1	no	—
28	C₃³	C₃³	1	1	yes	characteristic
29	3₋¹⁺²	—	9	3	no	—
30	H(3,𝔽₃)	—	3	6	no	—
31	H(3,𝔽₃)	—	9	1	no	—
32	C₃×C₉	—	3	2	no	—
33	C₃×C₉	—	3	2	no	—
34	C₃×C₉	—	3	1	no	—
35	C₃×C₉	—	3	1	no	—
36	C₃×C₉	—	3	1	no	—
37	C₃×C₉	—	3	2	no	—
38	C₃×C₉	—	9	1	no	—
39	C₃×C₉	—	9	6	no	—
40	C₃×C₉	H(3,𝔽₃)	1	2	yes	—
41	C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
42	C₃²	—	3	1	no	—
43	C₃²	—	3	2	no	—
44	C₃²	—	3	6	no	—
45	C₃²	—	3	1	no	—
46	C₃²	—	9	2	no	—
47	C₃²	—	9	2	no	—
48	C₃²	—	9	2	no	—
49	C₃²	—	9	1	no	—
50	C₃²	—	9	2	no	—
51	C₃²	—	9	1	no	—
52	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₉	—	3	2	no	—
54	C₉	—	3	1	no	—
55	C₉	—	9	1	no	—
56	C₉	—	9	2	no	—
57	C₉	—	9	2	no	—
58	C₉	—	9	2	no	—
59	C₉	—	9	1	no	—
60	C₉	—	9	1	no	—
61	C₃	—	9	1	no	—
62	C₃	—	9	6	no	—
63	C₃	—	27	1	no	—
64	C₃	#6908	1	2	yes	—
65	C₃	#6911	1	1	yes	characteristic
66	C₃	#6913	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #6895	C₃	semidirect
2. C₉⋊H(3,𝔽₃)	C₃	semidirect
3. C₉×H(3,𝔽₃)	C₃	semidirect
4. #6861	C₃	semidirect
5. #6861	C₃	semidirect
6. #6859	C₃	semidirect
7. #6859	C₃	semidirect
8. #6856	C₃	semidirect
9. #6856	C₃	semidirect

Maximal subgroups

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

subgroup	index
#6895	3
C₉⋊H(3,𝔽₃)	3
C₉×H(3,𝔽₃)	3
#6861	3
#6859	3
#6856	3