Subgroups — #14548227

Order 2187.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624767	C₃	1	2	yes	maximal, complemented
3	#10624767	C₃	1	6	yes	maximal, complemented
4	#10624731	C₃	1	1	yes	maximal, characteristic
5	#10624731	C₃	1	3	yes	maximal
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃³×C₉	C₃²	1	3	yes	—
8	C₃×#731	—	3	9	no	—
9	C₃×#731	—	3	3	no	—
10	#6891	C₃²	1	6	yes	—
11	#6891	C₃²	1	2	yes	—
12	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃×C₉⋊C₉	—	3	9	no	—
14	C₃×C₉⋊C₉	—	3	3	no	—
15	#6867	—	3	18	no	—
16	#6867	—	3	6	no	—
17	C₃⁴	C₃³	1	1	yes	characteristic
18	C₃×3₋¹⁺²	—	3	2	no	—
19	C₃×3₋¹⁺²	—	3	6	no	—
20	C₃×H(3,𝔽₃)	—	3	1	no	—
21	C₃×H(3,𝔽₃)	—	3	3	no	—
22	C₃²×C₉	—	3	3	no	—
23	C₃²×C₉	—	3	9	no	—
24	C₃²×C₉	—	3	3	no	—
25	C₃²×C₉	—	3	6	no	—
26	C₃²×C₉	—	9	9	no	—
27	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
28	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
29	#731	—	9	27	no	—
30	#731	—	9	9	no	—
31	C₉⋊C₉	—	3	27	no	—
32	C₃²⋊C₉	—	3	6	no	—
33	C₃²⋊C₉	—	3	6	no	—
34	C₃²⋊C₉	—	3	12	no	—
35	C₃³	—	3	3	no	—
36	C₃³	—	3	9	no	—
37	C₃³	—	9	1	no	—
38	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
40	3₋¹⁺²	—	9	6	no	—
41	3₋¹⁺²	—	9	18	no	—
42	H(3,𝔽₃)	—	9	3	no	—
43	H(3,𝔽₃)	—	9	9	no	—
44	C₃×C₉	—	3	27	no	—
45	C₃×C₉	—	3	3	no	—
46	C₃×C₉	—	3	6	no	—
47	C₃×C₉	—	3	9	no	—
48	C₃×C₉	—	3	9	no	—
49	C₃×C₉	—	3	18	no	—
50	C₃×C₉	—	9	9	no	—
51	C₃×C₉	—	9	18	no	—
52	C₃×C₉	—	9	2	no	—
53	C₃×C₉	—	9	6	no	—
54	C₃×C₉	#731	1	9	yes	complemented
55	C₃²	—	3	27	no	—
56	C₃²	—	3	3	no	—
57	C₃²	—	3	3	no	—
58	C₃²	—	3	6	no	—
59	C₃²	—	27	1	no	—
60	C₃²	—	27	3	no	—
61	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
62	C₃²	C₃×#731	1	3	yes	—
63	C₃²	#6910	1	9	yes	—
64	C₉	—	3	9	no	—
65	C₉	—	9	9	no	—
66	C₉	—	27	2	no	—
67	C₉	—	27	6	no	—
68	C₃	—	3	3	no	—
69	C₃	—	3	9	no	—
70	C₃	—	81	1	no	—
71	C₃	#10624731	1	1	yes	characteristic
72	C₃	#10624736	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃²×C₉⋊C₉	C₃	semidirect
2. #10624767	C₃	semidirect
3. #10624767	C₃	semidirect
54. C₃×C₉	#731	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₉⋊C₉	3
#10624767	3
#10624731	3