Subgroups — #14552357

Order 2187.

741 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6900	C₃	1	1	yes	maximal, characteristic
2	C₃×#6899	C₃	1	1	yes	maximal, characteristic
3	#10624575	C₃	1	2	yes	maximal, complemented
4	#10624575	C₃	1	2	yes	maximal, complemented
5	#10624575	C₃	1	2	yes	maximal, complemented
6	#10624574	C₃	1	2	yes	maximal, complemented
7	#10624574	C₃	1	2	yes	maximal, complemented
8	#10624574	C₃	1	2	yes	maximal, complemented
9	#10624570	C₃	1	2	yes	maximal
10	#10624565	C₃	1	2	yes	maximal
11	#10624565	C₃	1	2	yes	maximal
12	#10624561	C₃	1	1	yes	maximal, characteristic, complemented
13	#10624561	C₃	1	1	yes	maximal, characteristic, complemented
14	#10624561	C₃	1	1	yes	maximal, characteristic, complemented
15	#10624561	C₃	1	1	yes	maximal, characteristic, complemented
16	#10624561	C₃	1	1	yes	maximal, characteristic, complemented
17	#10624561	C₃	1	1	yes	maximal, characteristic, complemented
18	#10624560	C₃	1	1	yes	maximal, characteristic
19	#10624560	C₃	1	1	yes	maximal, characteristic
20	#10624560	C₃	1	1	yes	maximal, characteristic
21	#10624560	C₃	1	2	yes	maximal
22	#10624559	C₃	1	2	yes	maximal, complemented
23	#10624559	C₃	1	2	yes	maximal, complemented
24	#10624559	C₃	1	2	yes	maximal, complemented
25	#10624550	C₃	1	1	yes	maximal, characteristic, complemented
26	C₉×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
27	C₉×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
28	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
29	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
30	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
31	#6900	—	3	9	no	—
32	#6899	—	3	9	no	—
33	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
34	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
35	C₃×C₉⋊C₉	C₃²	1	2	yes	—
36	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
37	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
38	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
39	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
40	C₃×C₉⋊C₉	C₃²	1	2	yes	—
41	C₃×C₉⋊C₉	C₃²	1	2	yes	—
42	C₃×C₉⋊C₉	C₃²	1	2	yes	—
43	C₃×C₉⋊C₉	C₃²	1	2	yes	—
44	C₃×C₉⋊C₉	C₃²	1	2	yes	—
45	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
46	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
47	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
48	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
49	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
50	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
51	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
52	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
53	C₃×C₉⋊C₉	C₃²	1	2	yes	—
54	C₃×C₉⋊C₉	C₃²	1	2	yes	—
55	C₃×C₉⋊C₉	C₃²	1	2	yes	—
56	C₃×C₉⋊C₉	C₃²	1	2	yes	—
57	C₃×C₉⋊C₉	C₃²	1	2	yes	—
58	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
59	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
60	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
61	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
62	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
63	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
64	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
65	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
66	C₃×C₉²	C₃²	1	1	yes	characteristic
67	C₃×C₉²	C₃²	1	1	yes	characteristic
68	C₃×C₉²	C₃²	1	1	yes	characteristic
69	C₃×C₉²	C₃²	1	1	yes	characteristic
70	C₃×C₉²	C₃²	1	1	yes	characteristic
71	C₃×C₉²	C₃²	1	2	yes	—
72	#6855	C₃²	1	2	yes	—
73	#6855	C₃²	1	2	yes	—
74	#6855	C₃²	1	2	yes	—
75	#6855	C₃²	1	2	yes	—
76	#6855	C₃²	1	2	yes	—
77	#6855	C₃²	1	2	yes	—
78	#6855	C₃²	1	2	yes	—
79	#6855	C₃²	1	2	yes	—
80	#6855	C₃²	1	2	yes	—
81	#6855	C₃²	1	2	yes	—
82	#6855	C₃²	1	2	yes	—
83	#6855	C₃²	1	2	yes	—
84	#6855	C₃²	1	2	yes	—
85	#6855	C₃²	1	2	yes	—
86	#6855	C₃²	1	2	yes	—
87	#6855	C₃²	1	2	yes	—
88	#6855	C₃²	1	2	yes	—
89	#6855	C₃²	1	2	yes	—
90	#6855	C₃²	1	2	yes	—
91	#6855	C₃²	1	2	yes	—
92	#6855	C₃²	1	2	yes	—
93	#6855	C₃²	1	2	yes	—
94	#6855	C₃²	1	2	yes	—
95	#6855	C₃²	1	2	yes	—
96	#6855	C₃²	1	2	yes	—
97	#6855	C₃²	1	2	yes	—
98	#6855	C₃²	1	2	yes	—
99	#6855	C₃²	1	2	yes	—
100	#6855	C₃²	1	2	yes	—
101	#6855	C₃²	1	2	yes	—
102	#6855	C₃²	1	2	yes	—
103	#6855	C₃²	1	2	yes	—
104	#6855	C₃²	1	2	yes	—
105	#6855	C₃²	1	2	yes	—
106	#6855	C₃²	1	2	yes	—
107	#6855	C₃²	1	2	yes	—
108	C₃⁴	C₃³	1	1	yes	characteristic
109	C₃×3₋¹⁺²	—	3	3	no	—
110	C₃×3₋¹⁺²	—	3	6	no	—
111	C₃×3₋¹⁺²	—	3	3	no	—
112	C₃×3₋¹⁺²	—	3	3	no	—
113	C₃×3₋¹⁺²	—	3	6	no	—
114	C₃×3₋¹⁺²	—	3	3	no	—
115	C₃×3₋¹⁺²	—	3	3	no	—
116	C₃×3₋¹⁺²	—	3	3	no	—
117	C₃×3₋¹⁺²	—	9	2	no	—
118	C₃×3₋¹⁺²	—	9	2	no	—
119	C₃×3₋¹⁺²	—	9	2	no	—
120	C₃²×C₉	C₃³	1	2	yes	—
121	C₃²×C₉	C₃³	1	1	yes	characteristic
122	C₃²×C₉	C₃³	1	2	yes	—
123	C₃²×C₉	C₃³	1	2	yes	—
124	C₃²×C₉	C₃³	1	2	yes	—
125	C₃²×C₉	C₃³	1	2	yes	—
126	C₃²×C₉	C₃³	1	2	yes	—
127	C₃²×C₉	C₃³	1	2	yes	—
128	C₃²×C₉	C₃³	1	2	yes	—
129	C₃²×C₉	C₃³	1	2	yes	—
130	C₃²×C₉	C₃³	1	2	yes	—
131	C₃²×C₉	C₃³	1	2	yes	—
132	C₃²×C₉	C₃³	1	1	yes	characteristic
133	C₃²×C₉	C₃³	1	1	yes	characteristic
134	C₃²×C₉	C₃³	1	1	yes	characteristic
135	C₃²×C₉	C₃³	1	1	yes	characteristic
136	C₃²×C₉	C₃³	1	1	yes	characteristic
137	C₃²×C₉	C₃³	1	1	yes	characteristic
138	C₃²×C₉	C₃³	1	1	yes	characteristic
139	C₃²×C₉	C₃³	1	1	yes	characteristic
140	C₃²×C₉	C₃³	1	2	yes	—
141	C₃²×C₉	C₃³	1	1	yes	characteristic
142	C₃²×C₉	C₃³	1	1	yes	characteristic
143	C₃²×C₉	C₃³	1	1	yes	characteristic
144	C₃²×C₉	C₃³	1	2	yes	—
145	C₃²×C₉	C₃³	1	1	yes	characteristic
146	C₉⋊C₉	—	3	3	no	—
147	C₉⋊C₉	—	3	3	no	—
148	C₉⋊C₉	—	3	3	no	—
149	C₉⋊C₉	—	3	3	no	—
150	C₉⋊C₉	—	3	3	no	—
151	C₉⋊C₉	—	3	3	no	—
152	C₉⋊C₉	—	3	3	no	—
153	C₉⋊C₉	—	3	6	no	—
154	C₉⋊C₉	—	3	6	no	—
155	C₉⋊C₉	—	3	6	no	—
156	C₉⋊C₉	—	3	3	no	—
157	C₉⋊C₉	—	3	6	no	—
158	C₉⋊C₉	—	3	6	no	—
159	C₉⋊C₉	—	3	6	no	—
160	C₉⋊C₉	—	3	3	no	—
161	C₉⋊C₉	—	3	3	no	—
162	C₉⋊C₉	—	3	3	no	—
163	C₉⋊C₉	—	3	3	no	—
164	C₉⋊C₉	—	3	3	no	—
165	C₉⋊C₉	—	3	3	no	—
166	C₉⋊C₉	—	9	2	no	—
167	C₉⋊C₉	—	9	2	no	—
168	C₉⋊C₉	—	9	2	no	—
169	C₉⋊C₉	—	9	2	no	—
170	C₉⋊C₉	—	9	2	no	—
171	C₉⋊C₉	3₋¹⁺²	1	9	yes	—
172	C₉⋊C₉	3₋¹⁺²	1	9	yes	complemented
173	C₃²⋊C₉	—	3	3	no	—
174	C₃²⋊C₉	—	3	3	no	—
175	C₃²⋊C₉	—	3	3	no	—
176	C₃²⋊C₉	—	9	2	no	—
177	C₃²⋊C₉	—	9	2	no	—
178	C₃²⋊C₉	—	9	2	no	—
179	C₉²	—	3	3	no	—
180	C₉²	—	3	3	no	—
181	C₉²	—	3	3	no	—
182	C₉²	—	3	3	no	—
183	C₉²	—	3	3	no	—
184	C₉²	—	9	2	no	—
185	C₃³	—	3	1	no	—
186	C₃³	—	3	1	no	—
187	C₃³	—	3	1	no	—
188	C₃³	—	3	1	no	—
189	C₃³	—	3	1	no	—
190	C₃³	—	3	2	no	—
191	C₃³	—	3	2	no	—
192	C₃³	—	3	2	no	—
193	C₃³	—	3	2	no	—
194	C₃³	C₃⁴	1	1	yes	characteristic
195	3₋¹⁺²	—	9	9	no	—
196	3₋¹⁺²	—	9	18	no	—
197	3₋¹⁺²	—	9	9	no	—
198	C₃×C₉	—	3	1	no	—
199	C₃×C₉	—	3	1	no	—
200	C₃×C₉	—	3	2	no	—
201	C₃×C₉	—	3	1	no	—
202	C₃×C₉	—	3	1	no	—
203	C₃×C₉	—	3	1	no	—
204	C₃×C₉	—	3	2	no	—
205	C₃×C₉	—	3	2	no	—
206	C₃×C₉	—	3	2	no	—
207	C₃×C₉	—	3	2	no	—
208	C₃×C₉	—	3	2	no	—
209	C₃×C₉	—	3	2	no	—
210	C₃×C₉	—	3	2	no	—
211	C₃×C₉	—	3	2	no	—
212	C₃×C₉	—	3	2	no	—
213	C₃×C₉	—	3	2	no	—
214	C₃×C₉	—	3	1	no	—
215	C₃×C₉	—	3	1	no	—
216	C₃×C₉	—	3	1	no	—
217	C₃×C₉	—	3	1	no	—
218	C₃×C₉	—	3	1	no	—
219	C₃×C₉	—	3	1	no	—
220	C₃×C₉	—	3	2	no	—
221	C₃×C₉	—	3	1	no	—
222	C₃×C₉	—	3	1	no	—
223	C₃×C₉	—	3	2	no	—
224	C₃×C₉	—	3	2	no	—
225	C₃×C₉	—	3	2	no	—
226	C₃×C₉	—	3	2	no	—
227	C₃×C₉	—	3	1	no	—
228	C₃×C₉	—	3	1	no	—
229	C₃×C₉	—	3	2	no	—
230	C₃×C₉	—	3	2	no	—
231	C₃×C₉	—	3	2	no	—
232	C₃×C₉	—	3	2	no	—
233	C₃×C₉	—	3	1	no	—
234	C₃×C₉	—	3	1	no	—
235	C₃×C₉	—	3	1	no	—
236	C₃×C₉	—	3	1	no	—
237	C₃×C₉	—	3	2	no	—
238	C₃×C₉	—	3	2	no	—
239	C₃×C₉	—	3	2	no	—
240	C₃×C₉	—	3	2	no	—
241	C₃×C₉	—	3	2	no	—
242	C₃×C₉	—	3	2	no	—
243	C₃×C₉	—	3	2	no	—
244	C₃×C₉	—	3	2	no	—
245	C₃×C₉	—	3	2	no	—
246	C₃×C₉	—	3	2	no	—
247	C₃×C₉	—	3	2	no	—
248	C₃×C₉	—	3	2	no	—
249	C₃×C₉	—	3	2	no	—
250	C₃×C₉	—	3	2	no	—
251	C₃×C₉	—	3	2	no	—
252	C₃×C₉	—	3	2	no	—
253	C₃×C₉	—	3	2	no	—
254	C₃×C₉	—	3	2	no	—
255	C₃×C₉	—	3	2	no	—
256	C₃×C₉	—	3	2	no	—
257	C₃×C₉	—	3	2	no	—
258	C₃×C₉	—	3	2	no	—
259	C₃×C₉	—	3	2	no	—
260	C₃×C₉	—	3	2	no	—
261	C₃×C₉	—	3	2	no	—
262	C₃×C₉	—	3	2	no	—
263	C₃×C₉	—	3	2	no	—
264	C₃×C₉	—	3	2	no	—
265	C₃×C₉	—	3	2	no	—
266	C₃×C₉	—	3	2	no	—
267	C₃×C₉	—	3	2	no	—
268	C₃×C₉	—	3	2	no	—
269	C₃×C₉	—	3	2	no	—
270	C₃×C₉	—	3	2	no	—
271	C₃×C₉	—	3	2	no	—
272	C₃×C₉	—	3	2	no	—
273	C₃×C₉	—	3	2	no	—
274	C₃×C₉	—	3	2	no	—
275	C₃×C₉	—	3	2	no	—
276	C₃×C₉	—	3	2	no	—
277	C₃×C₉	—	3	2	no	—
278	C₃×C₉	—	3	2	no	—
279	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
280	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
281	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
282	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
283	C₃×C₉	C₃×3₋¹⁺²	1	6	yes	—
284	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
285	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
286	C₃×C₉	#737	1	6	yes	—
287	C₃×C₉	#737	1	3	yes	—
288	C₃×C₉	#737	1	3	yes	—
289	C₃×C₉	#737	1	3	yes	—
290	C₃²	—	9	1	no	—
291	C₃²	—	9	1	no	—
292	C₃²	—	9	2	no	—
293	C₃²	—	9	1	no	—
294	C₃²	—	9	2	no	—
295	C₃²	—	9	1	no	—
296	C₃²	—	9	1	no	—
297	C₃²	—	9	2	no	—
298	C₃²	—	9	2	no	—
299	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
300	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
301	C₃²	C₃×#737	1	1	yes	characteristic
302	C₃²	C₃×#737	1	1	yes	characteristic
303	C₃²	C₃×#737	1	1	yes	characteristic
304	C₃²	C₃×#737	1	2	yes	—
305	C₃²	3₋¹⁺⁴	1	2	yes	—
306	C₃²	3₋¹⁺⁴	1	2	yes	—
307	C₃²	3₋¹⁺⁴	1	2	yes	—
308	C₉	—	3	3	no	—
309	C₉	—	3	3	no	—
310	C₉	—	3	3	no	—
311	C₉	—	3	6	no	—
312	C₉	—	3	6	no	—
313	C₉	—	3	3	no	—
314	C₉	—	3	3	no	—
315	C₉	—	9	2	no	—
316	C₉	—	9	1	no	—
317	C₉	—	9	1	no	—
318	C₉	—	9	2	no	—
319	C₉	—	9	2	no	—
320	C₉	—	9	2	no	—
321	C₉	—	9	1	no	—
322	C₉	—	9	2	no	—
323	C₉	—	9	1	no	—
324	C₉	—	9	1	no	—
325	C₉	—	9	1	no	—
326	C₉	—	9	1	no	—
327	C₉	—	9	2	no	—
328	C₉	—	9	2	no	—
329	C₉	—	9	2	no	—
330	C₉	—	9	2	no	—
331	C₉	—	9	2	no	—
332	C₉	—	9	2	no	—
333	C₉	#6899	1	9	yes	complemented
334	C₃	—	27	1	no	—
335	C₃	C₃×#6899	1	1	yes	characteristic
336	C₃	#10624879	1	1	yes	characteristic
337	C₃	#10624892	1	2	yes	—
338	C₃	3₋¹⁺²²	1	1	yes	characteristic
339	C₃	#10624899	1	2	yes	—
340	C₃	#10624901	1	1	yes	characteristic
341	C₃	#10624901	1	1	yes	characteristic
342	C₃	#10624908	1	2	yes	—
343	C₃	#10624908	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃³ ▷ 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
3. #10624575	C₃	semidirect
4. #10624575	C₃	semidirect
5. #10624575	C₃	semidirect
6. #10624574	C₃	semidirect
7. #10624574	C₃	semidirect
8. #10624574	C₃	semidirect
12. #10624561	C₃	semidirect
13. #10624561	C₃	semidirect
14. #10624561	C₃	semidirect
15. #10624561	C₃	semidirect
16. #10624561	C₃	semidirect
17. #10624561	C₃	semidirect
22. #10624559	C₃	semidirect
23. #10624559	C₃	semidirect
24. #10624559	C₃	semidirect
25. #10624550	C₃	semidirect
26. C₉×C₉⋊C₉	C₃	semidirect
27. C₉×C₉⋊C₉	C₃	semidirect
172. C₉⋊C₉	3₋¹⁺²	semidirect
333. C₉	#6899	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6900	3
C₃×#6899	3
#10624575	3
#10624574	3
#10624570	3
#10624565	3
#10624561	3
#10624560	3
#10624559	3
#10624550	3
C₉×C₉⋊C₉	3