Subgroups — #14552081

Order 2187.

998 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6894	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
4	#10624561	C₃	1	3	yes	maximal, complemented
5	#10624561	C₃	1	6	yes	maximal, complemented
6	#10624561	C₃	1	3	yes	maximal, complemented
7	#10624559	C₃	1	6	yes	maximal, complemented
8	#10624559	C₃	1	6	yes	maximal, complemented
9	#10624558	C₃	1	2	yes	maximal
10	#10624556	C₃	1	2	yes	maximal
11	#10624556	C₃	1	2	yes	maximal
12	#10624548	C₃	1	1	yes	maximal, characteristic
13	C₉×C₉⋊C₉	C₃	1	3	yes	maximal, complemented
14	C₉×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
15	C₉×C₃²⋊C₉	C₃	1	2	yes	maximal
16	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
17	C₃³×C₉	C₃²	1	1	yes	characteristic
18	C₃³×C₉	C₃²	1	2	yes	—
19	C₃³×C₉	C₃²	1	1	yes	characteristic
20	#6894	C₉	1	27	yes	complemented
21	C₉×3₋¹⁺²	—	3	9	no	—
22	C₃×C₉⋊C₉	—	3	9	no	—
23	C₃×C₉⋊C₉	C₃²	1	3	yes	—
24	C₃×C₉⋊C₉	C₃²	1	2	yes	—
25	C₃×C₉⋊C₉	C₃²	1	2	yes	—
26	C₃×C₉⋊C₉	C₃²	1	3	yes	—
27	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
28	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
29	C₃×C₉⋊C₉	C₃²	1	2	yes	—
30	C₃×C₉⋊C₉	C₃²	1	3	yes	—
31	C₃×C₉⋊C₉	C₃²	1	6	yes	—
32	C₃×C₉⋊C₉	C₃²	1	6	yes	—
33	C₃×C₉⋊C₉	C₃²	1	3	yes	—
34	C₃×C₉⋊C₉	C₃²	1	6	yes	—
35	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
36	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
37	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
38	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
39	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
40	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
41	C₃×C₉²	C₃²	1	3	yes	—
42	C₃×C₉²	C₃²	1	3	yes	—
43	C₃×C₉²	C₃²	1	6	yes	—
44	#6855	C₃²	1	6	yes	—
45	#6855	C₃²	1	6	yes	—
46	#6855	C₃²	1	6	yes	—
47	#6855	C₃²	1	6	yes	—
48	#6855	C₃²	1	6	yes	—
49	#6855	C₃²	1	6	yes	—
50	#6855	C₃²	1	6	yes	—
51	#6855	C₃²	1	6	yes	—
52	#6855	C₃²	1	6	yes	—
53	#6855	C₃²	1	6	yes	—
54	#6855	C₃²	1	6	yes	—
55	C₃⁴	C₃³	1	1	yes	characteristic
56	C₃×3₋¹⁺²	—	3	3	no	—
57	C₃×3₋¹⁺²	—	3	6	no	—
58	C₃×3₋¹⁺²	C₃×C₉	1	9	yes	—
59	C₃²×C₉	—	3	3	no	—
60	C₃²×C₉	—	3	3	no	—
61	C₃²×C₉	—	3	3	no	—
62	C₃²×C₉	—	3	3	no	—
63	C₃²×C₉	—	3	6	no	—
64	C₃²×C₉	—	3	6	no	—
65	C₃²×C₉	—	3	3	no	—
66	C₃²×C₉	—	9	2	no	—
67	C₃²×C₉	—	9	2	no	—
68	C₃²×C₉	—	9	2	no	—
69	C₃²×C₉	C₃×C₉	1	9	yes	—
70	C₃²×C₉	C₃³	1	2	yes	—
71	C₃²×C₉	C₃³	1	1	yes	characteristic
72	C₃²×C₉	C₃³	1	2	yes	—
73	C₃²×C₉	C₃³	1	1	yes	characteristic
74	C₃²×C₉	C₃³	1	6	yes	—
75	C₃²×C₉	C₃³	1	6	yes	—
76	C₃²×C₉	C₃³	1	3	yes	—
77	C₃²×C₉	C₃³	1	2	yes	—
78	C₃²×C₉	C₃³	1	3	yes	—
79	C₃²×C₉	C₃³	1	6	yes	—
80	C₃²×C₉	C₃³	1	1	yes	characteristic
81	C₃²×C₉	C₃³	1	2	yes	—
82	C₃²×C₉	C₃³	1	3	yes	—
83	C₃²×C₉	C₃³	1	1	yes	characteristic
84	C₉⋊C₉	—	3	6	no	—
85	C₉⋊C₉	—	3	6	no	—
86	C₉⋊C₉	—	3	6	no	—
87	C₉⋊C₉	—	3	3	no	—
88	C₉⋊C₉	—	3	3	no	—
89	C₉⋊C₉	—	3	9	no	—
90	C₉⋊C₉	—	3	9	no	—
91	C₉⋊C₉	—	3	9	no	—
92	C₉⋊C₉	—	9	6	no	—
93	C₉⋊C₉	—	9	6	no	—
94	C₉⋊C₉	C₃×C₉	1	54	yes	complemented
95	C₉⋊C₉	C₃×C₉	1	27	yes	complemented
96	C₉⋊C₉	3₋¹⁺²	1	9	yes	complemented
97	C₃²⋊C₉	—	3	3	no	—
98	C₃²⋊C₉	—	3	6	no	—
99	C₃²⋊C₉	—	3	6	no	—
100	C₃²⋊C₉	C₃×C₉	1	18	yes	—
101	C₃²⋊C₉	3₋¹⁺²	1	9	yes	—
102	C₉²	—	3	9	no	—
103	C₉²	—	3	9	no	—
104	C₉²	—	9	6	no	—
105	C₃³	—	3	1	no	—
106	C₃³	—	3	1	no	—
107	C₃³	—	3	1	no	—
108	C₃³	—	3	2	no	—
109	C₃³	—	3	2	no	—
110	C₃³	—	3	2	no	—
111	C₃³	C₃²×C₉	1	3	yes	—
112	C₃³	C₃×3₋¹⁺²	1	3	yes	—
113	C₃³	#737	1	6	yes	—
114	C₃³	C₃⁴	1	1	yes	characteristic
115	3₋¹⁺²	—	3	27	no	—
116	C₃×C₉	—	3	1	no	—
117	C₃×C₉	—	3	3	no	—
118	C₃×C₉	—	3	2	no	—
119	C₃×C₉	—	3	3	no	—
120	C₃×C₉	—	3	2	no	—
121	C₃×C₉	—	3	2	no	—
122	C₃×C₉	—	3	2	no	—
123	C₃×C₉	—	3	6	no	—
124	C₃×C₉	—	3	2	no	—
125	C₃×C₉	—	3	6	no	—
126	C₃×C₉	—	3	6	no	—
127	C₃×C₉	—	3	6	no	—
128	C₃×C₉	—	3	6	no	—
129	C₃×C₉	—	3	2	no	—
130	C₃×C₉	—	3	2	no	—
131	C₃×C₉	—	3	27	no	—
132	C₃×C₉	—	3	3	no	—
133	C₃×C₉	—	3	1	no	—
134	C₃×C₉	—	3	1	no	—
135	C₃×C₉	—	3	1	no	—
136	C₃×C₉	—	3	1	no	—
137	C₃×C₉	—	3	2	no	—
138	C₃×C₉	—	3	2	no	—
139	C₃×C₉	—	3	6	no	—
140	C₃×C₉	—	3	6	no	—
141	C₃×C₉	—	3	6	no	—
142	C₃×C₉	—	3	3	no	—
143	C₃×C₉	—	3	6	no	—
144	C₃×C₉	—	3	2	no	—
145	C₃×C₉	—	3	2	no	—
146	C₃×C₉	—	3	6	no	—
147	C₃×C₉	—	3	2	no	—
148	C₃×C₉	—	3	2	no	—
149	C₃×C₉	—	3	6	no	—
150	C₃×C₉	—	3	6	no	—
151	C₃×C₉	—	3	6	no	—
152	C₃×C₉	—	3	6	no	—
153	C₃×C₉	—	3	2	no	—
154	C₃×C₉	—	3	2	no	—
155	C₃×C₉	—	3	2	no	—
156	C₃×C₉	—	3	2	no	—
157	C₃×C₉	—	9	9	no	—
158	C₃×C₉	—	9	18	no	—
159	C₃×C₉	C₃²×C₉	1	18	yes	—
160	C₃×C₉	C₃²×C₉	1	6	yes	—
161	C₃×C₉	C₃²×C₉	1	9	yes	—
162	C₃×C₉	C₃²×C₉	1	3	yes	—
163	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
164	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
165	C₃×C₉	C₃×3₋¹⁺²	1	6	yes	—
166	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
167	C₃×C₉	#737	1	3	yes	—
168	C₃×C₉	#737	1	3	yes	—
169	C₃²	—	3	3	no	—
170	C₃²	—	3	6	no	—
171	C₃²	—	3	3	no	—
172	C₃²	—	3	6	no	—
173	C₃²	—	3	3	no	—
174	C₃²	—	3	3	no	—
175	C₃²	—	3	6	no	—
176	C₃²	—	3	6	no	—
177	C₃²	C₉×3₋¹⁺²	1	9	yes	—
178	C₃²	C₃³×C₉	1	1	yes	characteristic
179	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
180	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
181	C₃²	C₃×#737	1	2	yes	—
182	C₃²	C₃×#737	1	1	yes	characteristic
183	C₃²	C₃×#737	1	1	yes	characteristic
184	C₃²	3₋¹⁺⁴	1	2	yes	—
185	C₃²	3₋¹⁺⁴	1	2	yes	—
186	C₃²	3₋¹⁺⁴	1	2	yes	—
187	C₉	—	3	6	no	—
188	C₉	—	3	9	no	—
189	C₉	—	3	18	no	—
190	C₉	—	3	3	no	—
191	C₉	—	3	6	no	—
192	C₉	—	3	9	no	—
193	C₉	—	9	1	no	—
194	C₉	—	9	1	no	—
195	C₉	—	9	2	no	—
196	C₉	—	9	2	no	—
197	C₉	—	9	3	no	—
198	C₉	—	9	6	no	—
199	C₉	—	9	6	no	—
200	C₉	C₉×3₋¹⁺²	1	9	yes	complemented
201	C₃	—	3	9	no	—
202	C₃	C₃×C₉×3₋¹⁺²	1	1	yes	characteristic
203	C₃	C₃×C₉×3₋¹⁺²	1	1	yes	characteristic
204	C₃	#10624864	1	2	yes	—
205	C₃	3₋¹⁺²²	1	1	yes	characteristic
206	C₃	#10624899	1	2	yes	—
207	C₃	#10624900	1	1	yes	characteristic
208	C₃	#10624901	1	1	yes	characteristic
209	C₃	#10624904	1	2	yes	—
210	C₃	#10624907	1	2	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₃³	C₃⁴
commutator	C₃²	C₃³×C₉
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃³ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #10624561	C₃	semidirect
5. #10624561	C₃	semidirect
6. #10624561	C₃	semidirect
7. #10624559	C₃	semidirect
8. #10624559	C₃	semidirect
13. C₉×C₉⋊C₉	C₃	semidirect
20. #6894	C₉	semidirect
94. C₉⋊C₉	C₃×C₉	semidirect
95. C₉⋊C₉	C₃×C₉	semidirect
96. C₉⋊C₉	3₋¹⁺²	semidirect
200. C₉	C₉×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6894	3
C₃×C₉×3₋¹⁺²	3
C₃²×C₉⋊C₉	3
#10624561	3
#10624559	3
#10624558	3
#10624556	3
#10624548	3
C₉×C₉⋊C₉	3
C₉×C₃²⋊C₉	3