Subgroups — #14555252

Order 2187.

6239 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₉)	C₃	1	27	yes	maximal, complemented
2	H(3,𝔽₃)²	C₃	1	54	yes	maximal, complemented
3	#10624880	C₃	1	36	yes	maximal, complemented
4	C₃×C₃²⋊H(3,𝔽₃)	C₃	1	4	yes	maximal, complemented
5	C₃⁵	C₃²	1	1	yes	characteristic, complemented
6	C₃²×H(3,𝔽₃)	C₃²	1	12	yes	complemented
7	C₃²×H(3,𝔽₃)	C₃²	1	36	yes	complemented
8	C₃²×H(3,𝔽₃)	C₃²	1	108	yes	complemented
9	C₃²⋊H(3,𝔽₃)	C₃²	1	36	yes	complemented
10	C₃²⋊H(3,𝔽₃)	C₃²	1	36	yes	complemented
11	C₃²⋊H(3,𝔽₃)	C₃²	1	36	yes	complemented
12	C₃²⋊H(3,𝔽₃)	C₃²	1	486	yes	complemented
13	C₃²⋊H(3,𝔽₃)	C₃²	1	243	yes	complemented
14	C₃²⋊H(3,𝔽₃)	C₃²	1	216	yes	complemented
15	C₃⁴	—	9	12	no	—
16	C₃⁴	C₃³	1	6	yes	—
17	C₃⁴	C₃³	1	36	yes	—
18	C₃⁴	C₃³	1	4	yes	—
19	C₃⁴	C₃³	1	3	yes	—
20	C₃⁴	C₃³	1	81	yes	complemented
21	C₃×H(3,𝔽₃)	—	9	108	no	—
22	C₃×H(3,𝔽₃)	—	9	324	no	—
23	C₃×H(3,𝔽₃)	—	9	36	no	—
24	C₃×H(3,𝔽₃)	C₃³	1	108	yes	—
25	C₃×H(3,𝔽₃)	C₃³	1	108	yes	—
26	C₃×H(3,𝔽₃)	C₃³	1	108	yes	—
27	C₃×H(3,𝔽₃)	C₃³	1	108	yes	—
28	C₃×H(3,𝔽₃)	C₃³	1	648	yes	complemented
29	C₃³	—	3	36	no	—
30	C₃³	—	3	12	no	—
31	C₃³	—	3	108	no	—
32	C₃³	—	9	12	no	—
33	C₃³	—	9	12	no	—
34	C₃³	—	9	12	no	—
35	C₃³	—	9	108	no	—
36	C₃³	—	9	324	no	—
37	C₃³	—	9	81	no	—
38	C₃³	C₃⁴	1	3	yes	—
39	C₃³	C₃⁴	1	6	yes	—
40	C₃³	C₃⁴	1	108	yes	—
41	C₃³	C₃⁴	1	4	yes	—
42	H(3,𝔽₃)	—	3	324	no	—
43	H(3,𝔽₃)	—	9	108	no	—
44	H(3,𝔽₃)	—	9	108	no	—
45	H(3,𝔽₃)	—	9	108	no	—
46	H(3,𝔽₃)	—	9	648	no	—
47	C₃²	—	3	12	no	—
48	C₃²	—	3	324	no	—
49	C₃²	—	3	12	no	—
50	C₃²	—	3	12	no	—
51	C₃²	—	3	12	no	—
52	C₃²	—	3	108	no	—
53	C₃²	—	9	54	no	—
54	C₃²	—	9	36	no	—
55	C₃²	—	9	27	no	—
56	C₃²	—	27	108	no	—
57	C₃²	—	27	243	no	—
58	C₃²	H(5,𝔽₃)	1	12	yes	—
59	C₃²	C₃⁵	1	1	yes	characteristic
60	C₃	—	3	12	no	—
61	C₃	—	9	3	no	—
62	C₃	—	9	6	no	—
63	C₃	—	9	108	no	—
64	C₃	C₃×H(5,𝔽₃)	1	4	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃²	C₃⁵
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ G
LECS	G ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₉)	C₃	semidirect
2. H(3,𝔽₃)²	C₃	semidirect
3. #10624880	C₃	semidirect
4. C₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
5. C₃⁵	C₃²	semidirect
6. C₃²×H(3,𝔽₃)	C₃²	semidirect
7. C₃²×H(3,𝔽₃)	C₃²	semidirect
8. C₃²×H(3,𝔽₃)	C₃²	semidirect
9. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
10. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
11. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
12. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
13. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
14. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
20. C₃⁴	C₃³	semidirect
28. C₃×H(3,𝔽₃)	C₃³	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₉)	3
H(3,𝔽₃)²	3
#10624880	3
C₃×C₃²⋊H(3,𝔽₃)	3