Subgroups — #14552395

Order 2187.

1046 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6895	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6894	C₃	1	2	yes	maximal, complemented
3	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	#10624566	C₃	1	2	yes	maximal, complemented
6	#10624566	C₃	1	2	yes	maximal, complemented
7	#10624566	C₃	1	2	yes	maximal, complemented
8	#10624566	C₃	1	2	yes	maximal, complemented
9	#10624566	C₃	1	2	yes	maximal, complemented
10	#10624566	C₃	1	1	yes	maximal, characteristic, complemented
11	#10624565	C₃	1	2	yes	maximal, complemented
12	#10624565	C₃	1	2	yes	maximal, complemented
13	#10624563	C₃	1	1	yes	maximal, characteristic
14	#10624563	C₃	1	2	yes	maximal
15	#10624560	C₃	1	2	yes	maximal, complemented
16	#10624557	C₃	1	2	yes	maximal, complemented
17	#10624557	C₃	1	2	yes	maximal, complemented
18	#10624557	C₃	1	2	yes	maximal, complemented
19	#10624557	C₃	1	1	yes	maximal, characteristic, complemented
20	#10624556	C₃	1	1	yes	maximal, characteristic, complemented
21	#10624556	C₃	1	1	yes	maximal, characteristic, complemented
22	#10624556	C₃	1	2	yes	maximal, complemented
23	#10624556	C₃	1	2	yes	maximal, complemented
24	#10624556	C₃	1	2	yes	maximal, complemented
25	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
26	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
27	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
28	C₃³×C₉	C₃²	1	1	yes	characteristic
29	C₃³×C₉	C₃²	1	2	yes	—
30	C₃³×C₉	C₃²	1	1	yes	characteristic
31	#6895	—	3	9	no	—
32	#6894	—	3	18	no	—
33	C₉×3₋¹⁺²	—	3	9	no	—
34	C₉×H(3,𝔽₃)	—	3	9	no	—
35	C₃×C₉⋊C₉	C₃²	1	2	yes	—
36	C₃×C₉⋊C₉	C₃²	1	2	yes	—
37	C₃×C₉⋊C₉	C₃²	1	2	yes	—
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	2	yes	—
46	C₃×C₉⋊C₉	C₃²	1	2	yes	—
47	C₃×C₉⋊C₉	C₃²	1	2	yes	—
48	C₃×C₉⋊C₉	C₃²	1	2	yes	—
49	C₃×C₉⋊C₉	C₃²	1	2	yes	—
50	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
51	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
52	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
53	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
54	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
55	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
56	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
57	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
58	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
59	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
60	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
61	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
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×C₃²⋊C₉	C₃²	1	2	yes	—
67	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
68	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
69	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
70	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
71	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
72	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
73	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
74	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
75	C₃×C₉²	C₃²	1	1	yes	characteristic
76	C₃×C₉²	C₃²	1	2	yes	—
77	#6855	C₃²	1	1	yes	characteristic
78	#6855	C₃²	1	2	yes	—
79	#6855	C₃²	1	2	yes	—
80	#6855	C₃²	1	1	yes	characteristic
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	1	yes	characteristic
100	#6855	C₃²	1	2	yes	—
101	#6855	C₃²	1	2	yes	—
102	#6855	C₃²	1	1	yes	characteristic
103	C₃⁴	C₃³	1	1	yes	characteristic
104	C₃⁴	C₃³	1	2	yes	—
105	C₃⁴	C₃³	1	1	yes	characteristic
106	C₃×3₋¹⁺²	—	3	3	no	—
107	C₃×3₋¹⁺²	—	3	6	no	—
108	C₃×3₋¹⁺²	—	3	3	no	—
109	C₃×3₋¹⁺²	—	3	6	no	—
110	C₃×3₋¹⁺²	—	3	6	no	—
111	C₃×3₋¹⁺²	—	9	2	no	—
112	C₃×3₋¹⁺²	—	9	2	no	—
113	C₃×H(3,𝔽₃)	—	3	3	no	—
114	C₃×H(3,𝔽₃)	—	3	3	no	—
115	C₃×H(3,𝔽₃)	—	3	6	no	—
116	C₃²×C₉	—	3	3	no	—
117	C₃²×C₉	—	3	3	no	—
118	C₃²×C₉	—	3	6	no	—
119	C₃²×C₉	—	3	3	no	—
120	C₃²×C₉	—	3	3	no	—
121	C₃²×C₉	—	3	6	no	—
122	C₃²×C₉	—	9	1	no	—
123	C₃²×C₉	—	9	1	no	—
124	C₃²×C₉	—	9	2	no	—
125	C₃²×C₉	—	9	2	no	—
126	C₃²×C₉	—	9	1	no	—
127	C₃²×C₉	—	9	1	no	—
128	C₃²×C₉	C₃³	1	1	yes	characteristic
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	2	yes	—
133	C₃²×C₉	C₃³	1	2	yes	—
134	C₃²×C₉	C₃³	1	2	yes	—
135	C₃²×C₉	C₃³	1	1	yes	characteristic
136	C₃²×C₉	C₃³	1	2	yes	—
137	C₃²×C₉	C₃³	1	2	yes	—
138	C₃²×C₉	C₃³	1	1	yes	characteristic
139	C₃²×C₉	C₃³	1	2	yes	—
140	C₃²×C₉	C₃³	1	2	yes	—
141	C₃²×C₉	C₃³	1	2	yes	—
142	C₃²×C₉	C₃³	1	2	yes	—
143	C₃²×C₉	C₃³	1	1	yes	characteristic
144	C₃²×C₉	C₃³	1	1	yes	characteristic
145	C₃²×C₉	C₃³	1	2	yes	—
146	C₃²×C₉	C₃³	1	1	yes	characteristic
147	C₃²×C₉	C₃³	1	2	yes	—
148	C₃²×C₉	C₃³	1	2	yes	—
149	C₉⋊C₉	—	3	6	no	—
150	C₉⋊C₉	—	3	6	no	—
151	C₉⋊C₉	—	3	3	no	—
152	C₉⋊C₉	—	3	6	no	—
153	C₉⋊C₉	—	3	6	no	—
154	C₉⋊C₉	—	3	6	no	—
155	C₉⋊C₉	—	3	6	no	—
156	C₉⋊C₉	—	3	6	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	6	no	—
163	C₉⋊C₉	—	3	6	no	—
164	C₉⋊C₉	—	3	6	no	—
165	C₉⋊C₉	H(3,𝔽₃)	1	9	yes	complemented
166	C₃²⋊C₉	—	3	3	no	—
167	C₃²⋊C₉	—	3	6	no	—
168	C₃²⋊C₉	—	3	6	no	—
169	C₃²⋊C₉	—	3	3	no	—
170	C₃²⋊C₉	—	3	3	no	—
171	C₃²⋊C₉	—	3	6	no	—
172	C₃²⋊C₉	—	3	6	no	—
173	C₃²⋊C₉	—	3	6	no	—
174	C₃²⋊C₉	—	3	3	no	—
175	C₃²⋊C₉	—	3	6	no	—
176	C₃²⋊C₉	—	3	6	no	—
177	C₃²⋊C₉	—	3	6	no	—
178	C₃²⋊C₉	—	3	6	no	—
179	C₃²⋊C₉	—	3	6	no	—
180	C₃²⋊C₉	—	3	6	no	—
181	C₃²⋊C₉	—	3	6	no	—
182	C₃²⋊C₉	—	9	2	no	—
183	C₃²⋊C₉	—	9	2	no	—
184	C₃²⋊C₉	—	9	2	no	—
185	C₃²⋊C₉	—	9	2	no	—
186	C₃²⋊C₉	—	9	2	no	—
187	C₃²⋊C₉	—	9	2	no	—
188	C₃²⋊C₉	3₋¹⁺²	1	9	yes	complemented
189	C₉²	—	3	6	no	—
190	C₉²	—	3	3	no	—
191	C₃³	—	3	1	no	—
192	C₃³	—	3	1	no	—
193	C₃³	—	3	2	no	—
194	C₃³	—	3	2	no	—
195	C₃³	—	3	1	no	—
196	C₃³	—	3	2	no	—
197	C₃³	—	3	2	no	—
198	C₃³	—	3	2	no	—
199	C₃³	—	3	1	no	—
200	C₃³	—	3	2	no	—
201	C₃³	—	3	2	no	—
202	C₃³	—	3	2	no	—
203	C₃³	—	3	2	no	—
204	C₃³	—	3	2	no	—
205	C₃³	—	3	2	no	—
206	C₃³	—	3	2	no	—
207	C₃³	—	3	2	no	—
208	C₃³	—	3	2	no	—
209	C₃³	—	3	2	no	—
210	C₃³	—	3	2	no	—
211	C₃³	—	3	1	no	—
212	C₃³	—	3	1	no	—
213	C₃³	—	3	1	no	—
214	C₃³	—	3	1	no	—
215	C₃³	—	3	2	no	—
216	C₃³	—	3	2	no	—
217	C₃³	—	3	2	no	—
218	C₃³	—	3	2	no	—
219	C₃³	C₃×3₋¹⁺²	1	3	yes	—
220	C₃³	#737	1	3	yes	—
221	C₃³	#737	1	6	yes	—
222	C₃³	C₃⁴	1	1	yes	characteristic
223	3₋¹⁺²	—	3	27	no	—
224	3₋¹⁺²	—	9	18	no	—
225	H(3,𝔽₃)	—	3	27	no	—
226	C₃×C₉	—	3	1	no	—
227	C₃×C₉	—	3	2	no	—
228	C₃×C₉	—	3	2	no	—
229	C₃×C₉	—	3	1	no	—
230	C₃×C₉	—	3	1	no	—
231	C₃×C₉	—	3	2	no	—
232	C₃×C₉	—	3	2	no	—
233	C₃×C₉	—	3	2	no	—
234	C₃×C₉	—	3	2	no	—
235	C₃×C₉	—	3	1	no	—
236	C₃×C₉	—	3	1	no	—
237	C₃×C₉	—	3	1	no	—
238	C₃×C₉	—	3	1	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	1	no	—
248	C₃×C₉	—	3	1	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	1	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	1	no	—
260	C₃×C₉	—	3	2	no	—
261	C₃×C₉	—	3	1	no	—
262	C₃×C₉	—	3	2	no	—
263	C₃×C₉	—	3	2	no	—
264	C₃×C₉	—	3	1	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	—	3	2	no	—
280	C₃×C₉	—	3	2	no	—
281	C₃×C₉	—	3	2	no	—
282	C₃×C₉	—	3	2	no	—
283	C₃×C₉	—	3	2	no	—
284	C₃×C₉	—	3	2	no	—
285	C₃×C₉	—	3	2	no	—
286	C₃×C₉	—	3	2	no	—
287	C₃×C₉	—	3	2	no	—
288	C₃×C₉	—	3	2	no	—
289	C₃×C₉	—	3	2	no	—
290	C₃×C₉	—	3	2	no	—
291	C₃×C₉	—	3	2	no	—
292	C₃×C₉	—	3	2	no	—
293	C₃×C₉	—	3	1	no	—
294	C₃×C₉	—	3	1	no	—
295	C₃×C₉	—	3	2	no	—
296	C₃×C₉	—	3	2	no	—
297	C₃×C₉	—	3	2	no	—
298	C₃×C₉	—	3	2	no	—
299	C₃×C₉	—	3	2	no	—
300	C₃×C₉	—	3	2	no	—
301	C₃×C₉	—	9	9	no	—
302	C₃×C₉	—	9	18	no	—
303	C₃×C₉	—	9	9	no	—
304	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
305	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
306	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
307	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
308	C₃×C₉	C₃×3₋¹⁺²	1	6	yes	—
309	C₃×C₉	#737	1	6	yes	—
310	C₃²	—	3	3	no	—
311	C₃²	—	3	3	no	—
312	C₃²	—	3	3	no	—
313	C₃²	—	3	6	no	—
314	C₃²	—	3	3	no	—
315	C₃²	—	3	3	no	—
316	C₃²	—	3	3	no	—
317	C₃²	—	3	6	no	—
318	C₃²	—	3	6	no	—
319	C₃²	—	3	6	no	—
320	C₃²	—	3	6	no	—
321	C₃²	—	9	1	no	—
322	C₃²	—	9	2	no	—
323	C₃²	—	9	1	no	—
324	C₃²	—	9	1	no	—
325	C₃²	—	9	2	no	—
326	C₃²	—	9	2	no	—
327	C₃²	—	9	1	no	—
328	C₃²	—	9	2	no	—
329	C₃²	—	9	2	no	—
330	C₃²	—	9	2	no	—
331	C₃²	—	9	2	no	—
332	C₃²	—	9	2	no	—
333	C₃²	—	9	2	no	—
334	C₃²	—	9	2	no	—
335	C₃²	—	9	2	no	—
336	C₃²	—	9	2	no	—
337	C₃²	—	9	2	no	—
338	C₃²	—	9	2	no	—
339	C₃²	—	9	2	no	—
340	C₃²	—	9	2	no	—
341	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
342	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
343	C₃²	C₃×#737	1	1	yes	characteristic
344	C₃²	C₃×#737	1	2	yes	—
345	C₃²	C₃×#737	1	2	yes	—
346	C₃²	3₋¹⁺⁴	1	1	yes	characteristic
347	C₃²	3₋¹⁺⁴	1	1	yes	characteristic
348	C₃²	3₋¹⁺⁴	1	2	yes	—
349	C₃²	3₋¹⁺⁴	1	2	yes	—
350	C₉	—	3	3	no	—
351	C₉	—	3	6	no	—
352	C₉	—	3	3	no	—
353	C₉	—	3	3	no	—
354	C₉	—	3	6	no	—
355	C₉	—	3	6	no	—
356	C₉	—	9	2	no	—
357	C₉	—	9	2	no	—
358	C₉	—	9	1	no	—
359	C₉	—	9	1	no	—
360	C₉	—	9	1	no	—
361	C₉	—	9	2	no	—
362	C₉	—	9	2	no	—
363	C₉	—	9	2	no	—
364	C₉	—	9	2	no	—
365	C₉	—	9	2	no	—
366	C₉	—	9	2	no	—
367	C₉	—	9	2	no	—
368	C₉	—	9	2	no	—
369	C₉	—	9	2	no	—
370	C₉	—	9	2	no	—
371	C₃	—	9	3	no	—
372	C₃	—	9	3	no	—
373	C₃	—	9	6	no	—
374	C₃	#10624869	1	1	yes	characteristic
375	C₃	#10624872	1	1	yes	characteristic
376	C₃	#10624891	1	1	yes	characteristic
377	C₃	#10624891	1	1	yes	characteristic
378	C₃	H(3,𝔽₃)×3₋¹⁺²	1	1	yes	characteristic
379	C₃	#10624896	1	2	yes	—
380	C₃	#10624901	1	2	yes	—
381	C₃	#10624907	1	2	yes	—
382	C₃	#10624907	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃²×H(3,𝔽₃)	C₃²
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
1. C₃×#6895	C₃	semidirect
2. C₃×#6894	C₃	semidirect
3. C₃×C₉×3₋¹⁺²	C₃	semidirect
5. #10624566	C₃	semidirect
6. #10624566	C₃	semidirect
7. #10624566	C₃	semidirect
8. #10624566	C₃	semidirect
9. #10624566	C₃	semidirect
10. #10624566	C₃	semidirect
11. #10624565	C₃	semidirect
12. #10624565	C₃	semidirect
15. #10624560	C₃	semidirect
16. #10624557	C₃	semidirect
17. #10624557	C₃	semidirect
18. #10624557	C₃	semidirect
19. #10624557	C₃	semidirect
20. #10624556	C₃	semidirect
21. #10624556	C₃	semidirect
22. #10624556	C₃	semidirect
23. #10624556	C₃	semidirect
24. #10624556	C₃	semidirect
165. C₉⋊C₉	H(3,𝔽₃)	semidirect
188. C₃²⋊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×#6895	3
C₃×#6894	3
C₃×C₉×3₋¹⁺²	3
C₃×C₉×H(3,𝔽₃)	3
#10624566	3
#10624565	3
#10624563	3
#10624560	3
#10624557	3
#10624556	3