Subgroups — #14548238

Order 2187.

464 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
2	#10624768	C₃	1	3	yes	maximal
3	#10624768	C₃	1	6	yes	maximal
4	#10624733	C₃	1	3	yes	maximal
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₃³×C₉	C₃²	1	2	yes	—
7	C₃³×C₉	C₃²	1	1	yes	characteristic
8	C₃×#733	—	3	9	no	—
9	#6891	C₃²	1	9	yes	—
10	C₃×C₉⋊C₉	—	3	6	no	—
11	C₃×C₉⋊C₉	—	3	3	no	—
12	C₃×C₉⋊C₉	—	3	3	no	—
13	#6868	—	3	18	no	—
14	#6868	—	3	9	no	—
15	C₃⁴	C₃³	1	1	yes	characteristic
16	C₃×3₋¹⁺²	—	3	9	no	—
17	C₃²×C₉	—	3	2	no	—
18	C₃²×C₉	—	3	1	no	—
19	C₃²×C₉	—	3	9	no	—
20	C₃²×C₉	—	3	2	no	—
21	C₃²×C₉	—	3	2	no	—
22	C₃²×C₉	—	3	2	no	—
23	C₃²×C₉	—	3	1	no	—
24	C₃²×C₉	—	3	2	no	—
25	C₃²×C₉	—	9	6	no	—
26	C₃²×C₉	—	9	3	no	—
27	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
28	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
29	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
30	#733	—	9	27	no	—
31	C₉⋊C₉	—	3	27	no	—
32	C₃²⋊C₉	—	3	18	no	—
33	C₃²⋊C₉	—	3	9	no	—
34	C₃³	—	3	6	no	—
35	C₃³	—	3	3	no	—
36	C₃³	—	3	3	no	—
37	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
38	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	3₋¹⁺²	—	9	27	no	—
41	C₃×C₉	—	3	18	no	—
42	C₃×C₉	—	3	9	no	—
43	C₃×C₉	—	3	3	no	—
44	C₃×C₉	—	3	6	no	—
45	C₃×C₉	—	3	18	no	—
46	C₃×C₉	—	3	3	no	—
47	C₃×C₉	—	3	6	no	—
48	C₃×C₉	—	3	9	no	—
49	C₃×C₉	—	9	6	no	—
50	C₃×C₉	—	9	3	no	—
51	C₃×C₉	—	9	6	no	—
52	C₃×C₉	—	9	6	no	—
53	C₃×C₉	—	9	6	no	—
54	C₃×C₉	—	9	9	no	—
55	C₃×C₉	#733	1	9	yes	complemented
56	C₃²	—	3	27	no	—
57	C₃²	—	3	2	no	—
58	C₃²	—	3	1	no	—
59	C₃²	—	3	2	no	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	2	no	—
62	C₃²	—	3	2	no	—
63	C₃²	—	3	2	no	—
64	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
65	C₃²	C₃×#733	1	3	yes	—
66	C₃²	#6911	1	3	yes	—
67	C₃²	#6912	1	6	yes	—
68	C₉	—	3	9	no	—
69	C₉	—	9	3	no	—
70	C₉	—	9	6	no	—
71	C₉	—	27	9	no	—
72	C₃	—	3	6	no	—
73	C₃	—	3	3	no	—
74	C₃	—	3	3	no	—
75	C₃	#10624733	1	1	yes	characteristic
76	C₃	#10624737	1	1	yes	characteristic
77	C₃	#10624738	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, prime radical, cosocle	C₃⁴	C₃³
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
55. C₃×C₉	#733	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
#10624768	3
#10624733	3