Subgroups — #14552380

Order 2187.

949 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6899	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6896	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624570	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624569	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624568	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624568	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624566	C₃	1	2	yes	maximal, complemented
10	#10624566	C₃	1	2	yes	maximal, complemented
11	#10624566	C₃	1	2	yes	maximal, complemented
12	#10624566	C₃	1	2	yes	maximal, complemented
13	#10624566	C₃	1	2	yes	maximal, complemented
14	#10624566	C₃	1	2	yes	maximal, complemented
15	#10624564	C₃	1	1	yes	maximal, characteristic
16	#10624563	C₃	1	1	yes	maximal, characteristic
17	#10624560	C₃	1	2	yes	maximal, complemented
18	#10624560	C₃	1	2	yes	maximal, complemented
19	#10624560	C₃	1	2	yes	maximal, complemented
20	#10624558	C₃	1	1	yes	maximal, characteristic, complemented
21	#10624557	C₃	1	2	yes	maximal, complemented
22	#10624557	C₃	1	2	yes	maximal, complemented
23	#10624557	C₃	1	2	yes	maximal, complemented
24	#10624555	C₃	1	1	yes	maximal, characteristic
25	#10624549	C₃	1	1	yes	maximal, characteristic, complemented
26	#10624548	C₃	1	2	yes	maximal, complemented
27	C₉×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
28	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
29	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
30	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
31	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
32	C₃³×C₉	C₃²	1	1	yes	characteristic
33	C₃³×C₉	C₃²	1	1	yes	characteristic
34	#6899	—	3	9	no	—
35	#6896	—	3	9	no	—
36	C₉⋊H(3,𝔽₃)	—	3	9	no	—
37	C₉×3₋¹⁺²	—	3	9	no	—
38	C₃×C₉⋊C₉	C₃²	1	2	yes	—
39	C₃×C₉⋊C₉	C₃²	1	2	yes	—
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×C₃²⋊C₉	C₃²	1	1	yes	characteristic
50	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
51	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
52	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
53	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
54	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
55	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
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	2	yes	—
63	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
64	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
65	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
66	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
67	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
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	2	yes	—
75	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
76	C₃×C₉²	C₃²	1	1	yes	characteristic
77	C₃×C₉²	C₃²	1	1	yes	characteristic
78	C₃×C₉²	C₃²	1	1	yes	characteristic
79	C₃×C₉²	C₃²	1	2	yes	—
80	#6855	C₃²	1	1	yes	characteristic
81	#6855	C₃²	1	1	yes	characteristic
82	#6855	C₃²	1	1	yes	characteristic
83	#6855	C₃²	1	1	yes	characteristic
84	#6855	C₃²	1	1	yes	characteristic
85	#6855	C₃²	1	1	yes	characteristic
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	#6855	C₃²	1	2	yes	—
109	#6855	C₃²	1	2	yes	—
110	C₃⁴	C₃³	1	2	yes	—
111	C₃⁴	C₃³	1	1	yes	characteristic
112	C₃⁴	C₃³	1	1	yes	characteristic
113	C₃×3₋¹⁺²	—	3	3	no	—
114	C₃×3₋¹⁺²	—	3	6	no	—
115	C₃×3₋¹⁺²	—	3	3	no	—
116	C₃×3₋¹⁺²	—	3	6	no	—
117	C₃×3₋¹⁺²	—	3	3	no	—
118	C₃×3₋¹⁺²	—	3	6	no	—
119	C₃×3₋¹⁺²	—	3	6	no	—
120	C₃×3₋¹⁺²	—	3	6	no	—
121	C₃×3₋¹⁺²	—	9	2	no	—
122	C₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
123	C₃×H(3,𝔽₃)	—	3	3	no	—
124	C₃×H(3,𝔽₃)	—	3	3	no	—
125	C₃×H(3,𝔽₃)	—	9	2	no	—
126	C₃²×C₉	—	3	3	no	—
127	C₃²×C₉	—	3	3	no	—
128	C₃²×C₉	—	3	3	no	—
129	C₃²×C₉	—	3	3	no	—
130	C₃²×C₉	—	3	6	no	—
131	C₃²×C₉	—	9	2	no	—
132	C₃²×C₉	C₃³	1	1	yes	characteristic
133	C₃²×C₉	C₃³	1	1	yes	characteristic
134	C₃²×C₉	C₃³	1	2	yes	—
135	C₃²×C₉	C₃³	1	2	yes	—
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	1	yes	characteristic
140	C₃²×C₉	C₃³	1	1	yes	characteristic
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	1	yes	characteristic
145	C₃²×C₉	C₃³	1	2	yes	—
146	C₃²×C₉	C₃³	1	2	yes	—
147	C₃²×C₉	C₃³	1	2	yes	—
148	C₃²×C₉	C₃³	1	2	yes	—
149	C₃²×C₉	C₃³	1	2	yes	—
150	C₃²×C₉	C₃³	1	2	yes	—
151	C₃²×C₉	C₃³	1	2	yes	—
152	C₃²×C₉	C₃³	1	2	yes	—
153	C₃²×C₉	C₃³	1	1	yes	characteristic
154	C₃²×C₉	C₃³	1	1	yes	characteristic
155	C₃²×C₉	C₃³	1	1	yes	characteristic
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	6	no	—
161	C₉⋊C₉	—	3	6	no	—
162	C₉⋊C₉	—	3	6	no	—
163	C₉⋊C₉	—	3	6	no	—
164	C₉⋊C₉	—	3	6	no	—
165	C₉⋊C₉	—	3	6	no	—
166	C₉⋊C₉	—	3	6	no	—
167	C₃²⋊C₉	—	3	3	no	—
168	C₃²⋊C₉	—	3	3	no	—
169	C₃²⋊C₉	—	3	3	no	—
170	C₃²⋊C₉	—	3	3	no	—
171	C₃²⋊C₉	—	3	3	no	—
172	C₃²⋊C₉	—	3	3	no	—
173	C₃²⋊C₉	—	3	3	no	—
174	C₃²⋊C₉	—	3	3	no	—
175	C₃²⋊C₉	—	3	3	no	—
176	C₃²⋊C₉	—	3	3	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	3	no	—
182	C₃²⋊C₉	—	3	6	no	—
183	C₃²⋊C₉	—	3	3	no	—
184	C₃²⋊C₉	—	3	6	no	—
185	C₃²⋊C₉	—	9	2	no	—
186	C₃²⋊C₉	—	9	2	no	—
187	C₃²⋊C₉	—	9	2	no	—
188	C₃²⋊C₉	—	9	2	no	—
189	C₃²⋊C₉	—	9	2	no	—
190	C₃²⋊C₉	—	9	2	no	—
191	C₃²⋊C₉	—	9	2	no	—
192	C₃²⋊C₉	—	9	2	no	—
193	C₃²⋊C₉	—	9	2	no	—
194	C₉²	—	3	6	no	—
195	C₉²	—	3	3	no	—
196	C₉²	—	3	3	no	—
197	C₉²	—	3	3	no	—
198	C₃³	—	3	1	no	—
199	C₃³	—	3	1	no	—
200	C₃³	—	3	1	no	—
201	C₃³	—	3	1	no	—
202	C₃³	—	3	1	no	—
203	C₃³	—	3	2	no	—
204	C₃³	—	3	1	no	—
205	C₃³	—	3	1	no	—
206	C₃³	—	3	2	no	—
207	C₃³	—	3	1	no	—
208	C₃³	—	3	2	no	—
209	C₃³	—	3	2	no	—
210	C₃³	—	3	2	no	—
211	C₃³	—	3	2	no	—
212	C₃³	—	3	2	no	—
213	C₃³	—	3	2	no	—
214	C₃³	—	3	2	no	—
215	C₃³	—	3	2	no	—
216	C₃³	—	3	2	no	—
217	C₃³	—	3	2	no	—
218	C₃³	—	3	2	no	—
219	C₃³	—	3	2	no	—
220	C₃³	—	3	2	no	—
221	C₃³	—	3	2	no	—
222	C₃³	—	3	2	no	—
223	C₃³	—	3	2	no	—
224	C₃³	—	3	2	no	—
225	C₃³	—	3	2	no	—
226	C₃³	—	3	2	no	—
227	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
228	C₃³	#737	1	3	yes	—
229	C₃³	C₃⁴	1	1	yes	characteristic
230	3₋¹⁺²	—	3	27	no	—
231	3₋¹⁺²	—	9	9	no	—
232	3₋¹⁺²	—	9	18	no	—
233	H(3,𝔽₃)	—	9	9	no	—
234	C₃×C₉	—	3	1	no	—
235	C₃×C₉	—	3	1	no	—
236	C₃×C₉	—	3	2	no	—
237	C₃×C₉	—	3	4	no	—
238	C₃×C₉	—	3	2	no	—
239	C₃×C₉	—	3	2	no	—
240	C₃×C₉	—	3	2	no	—
241	C₃×C₉	—	3	1	no	—
242	C₃×C₉	—	3	2	no	—
243	C₃×C₉	—	3	1	no	—
244	C₃×C₉	—	3	1	no	—
245	C₃×C₉	—	3	1	no	—
246	C₃×C₉	—	3	1	no	—
247	C₃×C₉	—	3	1	no	—
248	C₃×C₉	—	3	1	no	—
249	C₃×C₉	—	3	1	no	—
250	C₃×C₉	—	3	1	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	1	no	—
261	C₃×C₉	—	3	1	no	—
262	C₃×C₉	—	3	1	no	—
263	C₃×C₉	—	3	2	no	—
264	C₃×C₉	—	3	2	no	—
265	C₃×C₉	—	3	2	no	—
266	C₃×C₉	—	3	1	no	—
267	C₃×C₉	—	3	1	no	—
268	C₃×C₉	—	3	1	no	—
269	C₃×C₉	—	3	1	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	1	no	—
275	C₃×C₉	—	3	1	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	2	no	—
294	C₃×C₉	—	3	2	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	—	3	2	no	—
302	C₃×C₉	—	3	2	no	—
303	C₃×C₉	—	3	2	no	—
304	C₃×C₉	—	3	2	no	—
305	C₃×C₉	—	3	2	no	—
306	C₃×C₉	—	3	2	no	—
307	C₃×C₉	—	3	2	no	—
308	C₃×C₉	—	3	2	no	—
309	C₃×C₉	—	3	2	no	—
310	C₃×C₉	—	3	2	no	—
311	C₃×C₉	—	9	9	no	—
312	C₃×C₉	—	9	9	no	—
313	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
314	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
315	C₃×C₉	#737	1	3	yes	—
316	C₃×C₉	#737	1	3	yes	—
317	C₃×C₉	#737	1	3	yes	—
318	C₃×C₉	#737	1	6	yes	—
319	C₃²	—	3	3	no	—
320	C₃²	—	3	3	no	—
321	C₃²	—	3	3	no	—
322	C₃²	—	3	3	no	—
323	C₃²	—	3	3	no	—
324	C₃²	—	3	3	no	—
325	C₃²	—	3	6	no	—
326	C₃²	—	9	1	no	—
327	C₃²	—	9	1	no	—
328	C₃²	—	9	2	no	—
329	C₃²	—	9	2	no	—
330	C₃²	—	9	1	no	—
331	C₃²	—	9	1	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	—	9	2	no	—
342	C₃²	—	9	2	no	—
343	C₃²	—	9	2	no	—
344	C₃²	—	9	2	no	—
345	C₃²	—	9	2	no	—
346	C₃²	—	9	2	no	—
347	C₃²	—	9	2	no	—
348	C₃²	—	9	2	no	—
349	C₃²	—	9	2	no	—
350	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
351	C₃²	C₃×#737	1	1	yes	characteristic
352	C₃²	C₃×#737	1	2	yes	—
353	C₃²	C₃×#737	1	1	yes	characteristic
354	C₃²	C₃×#737	1	1	yes	characteristic
355	C₃²	C₃×#737	1	1	yes	characteristic
356	C₃²	3₋¹⁺⁴	1	2	yes	—
357	C₃²	3₋¹⁺⁴	1	2	yes	—
358	C₃²	3₋¹⁺⁴	1	2	yes	—
359	C₉	—	3	6	no	—
360	C₉	—	3	3	no	—
361	C₉	—	3	3	no	—
362	C₉	—	9	1	no	—
363	C₉	—	9	1	no	—
364	C₉	—	9	2	no	—
365	C₉	—	9	2	no	—
366	C₉	—	9	1	no	—
367	C₉	—	9	1	no	—
368	C₉	—	9	2	no	—
369	C₉	—	9	1	no	—
370	C₉	—	9	1	no	—
371	C₉	—	9	2	no	—
372	C₉	—	9	2	no	—
373	C₉	—	9	2	no	—
374	C₉	—	9	2	no	—
375	C₉	—	9	1	no	—
376	C₉	—	9	1	no	—
377	C₉	—	9	1	no	—
378	C₉	—	9	2	no	—
379	C₉	—	9	2	no	—
380	C₉	—	9	2	no	—
381	C₉	—	9	2	no	—
382	C₉	#6893	1	9	yes	—
383	C₃	—	9	3	no	—
384	C₃	—	9	3	no	—
385	C₃	—	27	2	no	—
386	C₃	C₃×#6893	1	1	yes	characteristic
387	C₃	#10624877	1	1	yes	characteristic
388	C₃	#10624892	1	2	yes	—
389	C₃	#10624896	1	1	yes	characteristic
390	C₃	#10624896	1	1	yes	characteristic
391	C₃	#10624896	1	1	yes	characteristic
392	C₃	#10624907	1	2	yes	—
393	C₃	#10624908	1	2	yes	—
394	C₃	#10624908	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×#6899	C₃	semidirect
2. C₃×#6896	C₃	semidirect
4. C₃×C₉×3₋¹⁺²	C₃	semidirect
5. #10624570	C₃	semidirect
6. #10624569	C₃	semidirect
7. #10624568	C₃	semidirect
8. #10624568	C₃	semidirect
9. #10624566	C₃	semidirect
10. #10624566	C₃	semidirect
11. #10624566	C₃	semidirect
12. #10624566	C₃	semidirect
13. #10624566	C₃	semidirect
14. #10624566	C₃	semidirect
17. #10624560	C₃	semidirect
18. #10624560	C₃	semidirect
19. #10624560	C₃	semidirect
20. #10624558	C₃	semidirect
21. #10624557	C₃	semidirect
22. #10624557	C₃	semidirect
23. #10624557	C₃	semidirect
25. #10624549	C₃	semidirect
26. #10624548	C₃	semidirect
27. C₉×C₃²⋊C₉	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6899	3
C₃×#6896	3
C₃×C₉⋊H(3,𝔽₃)	3
C₃×C₉×3₋¹⁺²	3
#10624570	3
#10624569	3
#10624568	3
#10624566	3
#10624564	3
#10624563	3
#10624560	3
#10624558	3
#10624557	3
#10624555	3
#10624549	3
#10624548	3
C₉×C₃²⋊C₉	3