Subgroups — #14548232

Order 2187.

500 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	#10624811	C₃	1	6	yes	maximal, complemented
3	#10624768	C₃	1	2	yes	maximal, complemented
4	#10624732	C₃	1	1	yes	maximal, characteristic
5	#10624731	C₃	1	3	yes	maximal, complemented
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃³×C₉	C₃²	1	1	yes	characteristic
8	C₃³×C₉	C₃²	1	2	yes	—
9	C₃×#732	—	3	3	no	—
10	C₃×#731	—	3	9	no	—
11	#6891	C₃²	1	6	yes	—
12	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
13	C₃×C₉⋊C₉	—	3	3	no	—
14	C₃×C₉⋊C₉	—	3	6	no	—
15	C₃×C₉⋊C₉	—	3	3	no	—
16	H(3,𝔽₃)⋊C₉	—	3	18	no	—
17	#6868	—	3	6	no	—
18	C₃⁴	C₃³	1	1	yes	characteristic
19	C₃×3₋¹⁺²	—	3	6	no	—
20	C₃×H(3,𝔽₃)	—	3	6	no	—
21	C₃×H(3,𝔽₃)	—	3	3	no	—
22	C₃×H(3,𝔽₃)	—	3	3	no	—
23	C₃²×C₉	—	3	2	no	—
24	C₃²×C₉	—	3	1	no	—
25	C₃²×C₉	—	3	9	no	—
26	C₃²×C₉	—	3	2	no	—
27	C₃²×C₉	—	3	1	no	—
28	C₃²×C₉	—	3	2	no	—
29	C₃²×C₉	—	3	2	no	—
30	C₃²×C₉	—	3	2	no	—
31	C₃²×C₉	—	9	3	no	—
32	C₃²×C₉	—	9	6	no	—
33	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
34	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
35	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
36	#732	—	9	9	no	—
37	#731	—	9	27	no	—
38	C₉⋊C₉	—	3	27	no	—
39	C₃²⋊C₉	—	3	6	no	—
40	C₃²⋊C₉	—	3	12	no	—
41	C₃³	—	3	6	no	—
42	C₃³	—	3	3	no	—
43	C₃³	—	3	3	no	—
44	C₃³	—	9	3	no	—
45	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
46	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
48	3₋¹⁺²	—	9	18	no	—
49	H(3,𝔽₃)	—	9	18	no	—
50	H(3,𝔽₃)	—	9	9	no	—
51	H(3,𝔽₃)	—	9	9	no	—
52	C₃×C₉	—	3	18	no	—
53	C₃×C₉	—	3	9	no	—
54	C₃×C₉	—	3	3	no	—
55	C₃×C₉	—	3	6	no	—
56	C₃×C₉	—	3	3	no	—
57	C₃×C₉	—	3	6	no	—
58	C₃×C₉	—	3	3	no	—
59	C₃×C₉	—	3	6	no	—
60	C₃×C₉	—	9	6	no	—
61	C₃×C₉	—	9	3	no	—
62	C₃×C₉	—	9	6	no	—
63	C₃×C₉	—	9	6	no	—
64	C₃×C₉	—	9	6	no	—
65	C₃×C₉	—	9	6	no	—
66	C₃×C₉	—	9	6	no	—
67	C₃×C₉	#732	1	9	yes	complemented
68	C₃²	—	3	27	no	—
69	C₃²	—	3	2	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	2	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	3	2	no	—
76	C₃²	—	27	3	no	—
77	C₃²	—	27	3	no	—
78	C₃²	—	27	6	no	—
79	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
80	C₃²	C₃×#732	1	3	yes	—
81	C₃²	#6910	1	3	yes	—
82	C₃²	#6913	1	6	yes	—
83	C₉	—	3	9	no	—
84	C₉	—	9	3	no	—
85	C₉	—	9	6	no	—
86	C₉	—	27	6	no	—
87	C₃	—	3	6	no	—
88	C₃	—	3	3	no	—
89	C₃	—	3	3	no	—
90	C₃	—	81	3	no	—
91	C₃	#10624732	1	1	yes	characteristic
92	C₃	#10624736	1	1	yes	characteristic
93	C₃	#10624739	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624732	C₃
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. #10624811	C₃	semidirect
3. #10624768	C₃	semidirect
5. #10624731	C₃	semidirect
67. C₃×C₉	#732	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
#10624811	3
#10624768	3
#10624732	3
#10624731	3