Subgroups — #14554467

Order 2187.

765 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6913	C₃	1	1	yes	maximal, characteristic
2	#10624887	C₃	1	2	yes	maximal, complemented
3	C₃×#6896	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624864	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624803	C₃	1	2	yes	maximal, complemented
6	#10624803	C₃	1	2	yes	maximal, complemented
7	#10624802	C₃	1	2	yes	maximal, complemented
8	#10624802	C₃	1	2	yes	maximal, complemented
9	#10624801	C₃	1	2	yes	maximal, complemented
10	#10624801	C₃	1	2	yes	maximal, complemented
11	#10624800	C₃	1	2	yes	maximal, complemented
12	#10624799	C₃	1	2	yes	maximal, complemented
13	#10624798	C₃	1	2	yes	maximal, complemented
14	#10624797	C₃	1	1	yes	maximal, characteristic, complemented
15	#10624797	C₃	1	1	yes	maximal, characteristic, complemented
16	#10624796	C₃	1	1	yes	maximal, characteristic, complemented
17	#10624796	C₃	1	1	yes	maximal, characteristic, complemented
18	#10624795	C₃	1	1	yes	maximal, characteristic, complemented
19	#10624795	C₃	1	1	yes	maximal, characteristic, complemented
20	#10624794	C₃	1	1	yes	maximal, characteristic, complemented
21	#10624793	C₃	1	1	yes	maximal, characteristic, complemented
22	#10624792	C₃	1	1	yes	maximal, characteristic, complemented
23	#10624700	C₃	1	2	yes	maximal, complemented
24	#10624700	C₃	1	1	yes	maximal, characteristic, complemented
25	#10624697	C₃	1	2	yes	maximal, complemented
26	#10624697	C₃	1	1	yes	maximal, characteristic, complemented
27	#10624692	C₃	1	2	yes	maximal, complemented
28	C₃×#737	C₃²	1	2	yes	—
29	C₃×#737	C₃²	1	1	yes	characteristic
30	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
31	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
32	#6913	—	3	9	no	—
33	C₃×#733	C₃²	1	2	yes	—
34	C₃×#732	C₃²	1	1	yes	characteristic
35	C₃×#731	C₃²	1	1	yes	characteristic
36	C₃×#731	C₃²	1	2	yes	—
37	C₃×AΣL(1,𝔽₂₇)	C₃²	1	1	yes	characteristic
38	C₃×AΣL(1,𝔽₂₇)	C₃²	1	2	yes	—
39	#6900	—	3	2	no	—
40	#6899	—	3	2	no	—
41	#6899	—	3	2	no	—
42	#6896	—	3	1	no	—
43	#6896	—	3	2	no	—
44	#6896	—	3	2	no	—
45	#6896	—	3	2	no	—
46	#6896	—	3	2	no	—
47	#6896	—	3	2	no	—
48	#6896	—	3	2	no	—
49	#6896	—	3	2	no	—
50	#6894	—	3	2	no	—
51	#6894	—	3	2	no	—
52	#6894	—	3	2	no	—
53	#6894	C₃²	1	2	yes	—
54	#6894	C₃²	1	2	yes	—
55	#6894	C₃²	1	2	yes	—
56	#6894	C₃²	1	2	yes	—
57	#6894	C₃²	1	2	yes	—
58	#6894	C₃²	1	2	yes	—
59	C₉⋊H(3,𝔽₃)	C₃²	1	2	yes	—
60	C₉⋊H(3,𝔽₃)	C₃²	1	2	yes	—
61	C₉⋊H(3,𝔽₃)	C₃²	1	2	yes	—
62	C₉×3₋¹⁺²	—	3	2	no	—
63	C₉×3₋¹⁺²	—	3	2	no	—
64	C₉×3₋¹⁺²	—	3	2	no	—
65	C₉×3₋¹⁺²	C₃²	1	2	yes	—
66	C₉×3₋¹⁺²	C₃²	1	2	yes	—
67	C₉×3₋¹⁺²	C₃²	1	2	yes	—
68	C₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
69	C₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
70	C₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
71	#6887	—	3	2	no	—
72	#6887	C₃²	1	2	yes	—
73	#6887	C₃²	1	2	yes	—
74	#6887	C₃²	1	2	yes	—
75	C₃×C₉⋊C₉	—	3	1	no	—
76	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
77	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
78	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
79	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
80	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
81	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
82	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
83	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
84	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
85	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
86	C₃×C₉²	C₃²	1	1	yes	characteristic
87	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
88	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
89	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
90	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
91	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
92	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
93	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
94	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
95	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
96	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
97	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
98	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
99	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
100	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
101	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
102	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
103	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
104	3₋¹⁺²⋊C₉	C₃²	1	2	yes	—
105	H(3,𝔽₃)⋊C₉	C₃²	1	1	yes	characteristic
106	H(3,𝔽₃)⋊C₉	C₃²	1	1	yes	characteristic
107	H(3,𝔽₃)⋊C₉	C₃²	1	2	yes	—
108	H(3,𝔽₃)⋊C₉	C₃²	1	2	yes	—
109	H(3,𝔽₃)⋊C₉	C₃²	1	1	yes	characteristic
110	H(3,𝔽₃)⋊C₉	C₃²	1	1	yes	characteristic
111	H(3,𝔽₃)⋊C₉	C₃²	1	1	yes	characteristic
112	H(3,𝔽₃)⋊C₉	C₃²	1	1	yes	characteristic
113	H(3,𝔽₃)⋊C₉	C₃²	1	2	yes	—
114	H(3,𝔽₃)⋊C₉	C₃²	1	2	yes	—
115	H(3,𝔽₃)⋊C₉	C₃²	1	2	yes	—
116	H(3,𝔽₃)⋊C₉	C₃²	1	2	yes	—
117	#6868	C₃²	1	2	yes	—
118	#6868	C₃²	1	2	yes	—
119	#6868	C₃²	1	2	yes	—
120	#6867	C₃²	1	2	yes	—
121	#6867	C₃²	1	2	yes	—
122	#6867	C₃²	1	2	yes	—
123	#6866	C₃²	1	2	yes	—
124	#6866	C₃²	1	2	yes	—
125	#6866	C₃²	1	2	yes	—
126	C₃⁴	C₃³	1	1	yes	characteristic
127	#737	—	3	6	no	—
128	#737	—	3	6	no	—
129	#737	—	3	12	no	—
130	#737	3₋¹⁺²	1	9	yes	complemented
131	C₃×3₋¹⁺²	—	3	2	no	—
132	C₃×3₋¹⁺²	—	3	2	no	—
133	C₃×3₋¹⁺²	—	3	1	no	—
134	C₃×3₋¹⁺²	—	3	1	no	—
135	C₃×3₋¹⁺²	—	3	2	no	—
136	C₃×3₋¹⁺²	—	3	2	no	—
137	C₃×3₋¹⁺²	—	3	2	no	—
138	C₃×3₋¹⁺²	—	3	2	no	—
139	C₃×3₋¹⁺²	—	3	2	no	—
140	C₃×3₋¹⁺²	—	3	2	no	—
141	C₃×3₋¹⁺²	—	3	3	no	—
142	C₃×3₋¹⁺²	—	3	6	no	—
143	C₃×3₋¹⁺²	—	9	3	no	—
144	C₃×3₋¹⁺²	C₃³	1	2	yes	—
145	C₃×3₋¹⁺²	C₃³	1	2	yes	—
146	C₃×3₋¹⁺²	C₃³	1	2	yes	—
147	C₃×H(3,𝔽₃)	C₃³	1	2	yes	—
148	C₃×H(3,𝔽₃)	C₃³	1	1	yes	characteristic
149	C₃²×C₉	—	3	1	no	—
150	C₃²×C₉	—	3	2	no	—
151	C₃²×C₉	—	3	1	no	—
152	C₃²×C₉	—	3	1	no	—
153	C₃²×C₉	—	3	1	no	—
154	C₃²×C₉	C₃³	1	1	yes	characteristic
155	C₃²×C₉	C₃³	1	1	yes	characteristic
156	C₃²×C₉	C₃³	1	1	yes	characteristic
157	C₃²×C₉	C₃³	1	1	yes	characteristic
158	C₃²×C₉	C₃³	1	1	yes	characteristic
159	C₃²×C₉	C₃³	1	1	yes	characteristic
160	C₃²×C₉	C₃³	1	1	yes	characteristic
161	C₃²×C₉	C₃³	1	1	yes	characteristic
162	C₃²×C₉	C₃³	1	1	yes	characteristic
163	C₃²×C₉	C₃³	1	1	yes	characteristic
164	C₃²×C₉	C₃³	1	1	yes	characteristic
165	C₃²×C₉	C₃³	1	1	yes	characteristic
166	#733	—	3	6	no	—
167	#732	—	3	3	no	—
168	#731	—	3	6	no	—
169	#731	—	3	3	no	—
170	AΣL(1,𝔽₂₇)	—	3	6	no	—
171	AΣL(1,𝔽₂₇)	—	3	3	no	—
172	C₉⋊C₉	—	3	2	no	—
173	C₉⋊C₉	—	3	2	no	—
174	C₉⋊C₉	—	3	2	no	—
175	C₉⋊C₉	—	3	2	no	—
176	C₉⋊C₉	—	3	2	no	—
177	C₉⋊C₉	—	3	2	no	—
178	C₉⋊C₉	—	3	1	no	—
179	C₉⋊C₉	—	3	1	no	—
180	C₉⋊C₉	—	3	2	no	—
181	C₉⋊C₉	—	3	2	no	—
182	C₉⋊C₉	—	3	1	no	—
183	C₉⋊C₉	—	3	1	no	—
184	C₉⋊C₉	—	3	2	no	—
185	C₉⋊C₉	—	3	2	no	—
186	C₉⋊C₉	—	3	1	no	—
187	C₉⋊C₉	—	3	1	no	—
188	C₉⋊C₉	—	3	1	no	—
189	C₉⋊C₉	—	3	2	no	—
190	C₉⋊C₉	—	3	2	no	—
191	C₉⋊C₉	—	3	2	no	—
192	C₉⋊C₉	—	3	2	no	—
193	C₉⋊C₉	—	3	2	no	—
194	C₉⋊C₉	—	3	2	no	—
195	C₉⋊C₉	—	3	2	no	—
196	C₉⋊C₉	—	3	2	no	—
197	C₉⋊C₉	—	3	2	no	—
198	C₉⋊C₉	—	3	2	no	—
199	C₉⋊C₉	—	3	2	no	—
200	C₉⋊C₉	—	3	2	no	—
201	C₉⋊C₉	—	3	2	no	—
202	C₉⋊C₉	—	3	2	no	—
203	C₉⋊C₉	—	3	2	no	—
204	C₉⋊C₉	—	3	2	no	—
205	C₉⋊C₉	—	3	2	no	—
206	C₉⋊C₉	—	3	2	no	—
207	C₉⋊C₉	—	3	2	no	—
208	C₉⋊C₉	—	3	2	no	—
209	C₉⋊C₉	—	3	2	no	—
210	C₉⋊C₉	—	3	2	no	—
211	C₉⋊C₉	—	3	2	no	—
212	C₉⋊C₉	—	3	2	no	—
213	C₉⋊C₉	—	3	2	no	—
214	C₉⋊C₉	—	3	2	no	—
215	C₉⋊C₉	—	3	2	no	—
216	C₉⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
217	C₉⋊C₉	H(3,𝔽₃)	1	2	yes	—
218	C₉⋊C₉	H(3,𝔽₃)	1	2	yes	—
219	C₉⋊C₉	H(3,𝔽₃)	1	2	yes	—
220	C₉⋊C₉	H(3,𝔽₃)	1	2	yes	—
221	C₃²⋊C₉	—	3	2	no	—
222	C₃²⋊C₉	—	3	2	no	—
223	C₃²⋊C₉	—	3	1	no	—
224	C₃²⋊C₉	—	3	1	no	—
225	C₃²⋊C₉	—	3	2	no	—
226	C₃²⋊C₉	—	3	2	no	—
227	C₃²⋊C₉	—	3	2	no	—
228	C₃²⋊C₉	—	3	2	no	—
229	C₃²⋊C₉	—	3	2	no	—
230	C₃²⋊C₉	—	3	2	no	—
231	C₃²⋊C₉	—	3	2	no	—
232	C₃²⋊C₉	—	3	2	no	—
233	C₃²⋊C₉	—	3	2	no	—
234	C₃²⋊C₉	C₃³	1	2	yes	—
235	C₃²⋊C₉	C₃³	1	2	yes	—
236	C₃²⋊C₉	C₃³	1	2	yes	—
237	C₃²⋊C₉	C₃³	1	2	yes	—
238	C₃²⋊C₉	C₃³	1	2	yes	—
239	C₃²⋊C₉	C₃³	1	2	yes	—
240	C₃²⋊C₉	C₃³	1	2	yes	—
241	C₃²⋊C₉	C₃³	1	2	yes	—
242	C₃²⋊C₉	C₃³	1	2	yes	—
243	C₉²	—	3	1	no	—
244	C₉²	—	3	2	no	—
245	C₉²	—	3	2	no	—
246	C₉²	—	3	2	no	—
247	C₉²	—	3	2	no	—
248	C₉²	—	3	2	no	—
249	C₉²	—	3	2	no	—
250	C₉²	—	3	2	no	—
251	C₃³	—	3	1	no	—
252	C₃³	—	3	2	no	—
253	C₃³	—	3	1	no	—
254	C₃³	—	3	1	no	—
255	C₃³	—	3	2	no	—
256	C₃³	—	9	1	no	—
257	C₃³	—	9	1	no	—
258	C₃³	—	9	1	no	—
259	C₃³	C₃⁴	1	1	yes	characteristic
260	3₋¹⁺²	—	3	4	no	—
261	3₋¹⁺²	—	3	18	no	—
262	3₋¹⁺²	—	3	2	no	—
263	3₋¹⁺²	—	3	2	no	—
264	3₋¹⁺²	—	3	4	no	—
265	3₋¹⁺²	—	3	4	no	—
266	3₋¹⁺²	—	9	3	no	—
267	3₋¹⁺²	—	9	6	no	—
268	3₋¹⁺²	—	9	3	no	—
269	3₋¹⁺²	—	9	6	no	—
270	3₋¹⁺²	—	9	6	no	—
271	3₋¹⁺²	—	9	6	no	—
272	3₋¹⁺²	—	27	2	no	—
273	3₋¹⁺²	—	27	2	no	—
274	3₋¹⁺²	—	27	2	no	—
275	3₋¹⁺²	C₃×3₋¹⁺²	1	6	yes	—
276	H(3,𝔽₃)	—	3	2	no	—
277	H(3,𝔽₃)	—	3	2	no	—
278	H(3,𝔽₃)	—	3	4	no	—
279	H(3,𝔽₃)	C₃×3₋¹⁺²	1	3	yes	—
280	C₃×C₉	—	3	2	no	—
281	C₃×C₉	—	3	1	no	—
282	C₃×C₉	—	3	9	no	—
283	C₃×C₉	—	3	3	no	—
284	C₃×C₉	—	3	1	no	—
285	C₃×C₉	—	3	1	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	1	no	—
296	C₃×C₉	—	3	1	no	—
297	C₃×C₉	—	3	2	no	—
298	C₃×C₉	—	3	2	no	—
299	C₃×C₉	—	3	1	no	—
300	C₃×C₉	—	3	1	no	—
301	C₃×C₉	—	3	1	no	—
302	C₃×C₉	—	3	1	no	—
303	C₃×C₉	—	3	1	no	—
304	C₃×C₉	—	3	2	no	—
305	C₃×C₉	—	3	2	no	—
306	C₃×C₉	—	9	1	no	—
307	C₃×C₉	—	9	1	no	—
308	C₃×C₉	—	9	1	no	—
309	C₃×C₉	—	9	6	no	—
310	C₃×C₉	—	9	1	no	—
311	C₃×C₉	—	9	1	no	—
312	C₃×C₉	—	9	1	no	—
313	C₃×C₉	—	9	1	no	—
314	C₃×C₉	—	9	1	no	—
315	C₃×C₉	—	27	1	no	—
316	C₃×C₉	—	27	1	no	—
317	C₃×C₉	—	27	1	no	—
318	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
319	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
320	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
321	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
322	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
323	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
324	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
325	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
326	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
327	C₃×C₉	#737	1	6	yes	—
328	C₃²	—	3	1	no	—
329	C₃²	—	3	1	no	—
330	C₃²	—	3	1	no	—
331	C₃²	—	3	3	no	—
332	C₃²	—	9	1	no	—
333	C₃²	—	9	1	no	—
334	C₃²	—	9	1	no	—
335	C₃²	—	9	2	no	—
336	C₃²	—	9	1	no	—
337	C₃²	—	9	1	no	—
338	C₃²	—	9	2	no	—
339	C₃²	—	9	2	no	—
340	C₃²	—	9	1	no	—
341	C₃²	—	9	2	no	—
342	C₃²	—	9	1	no	—
343	C₃²	—	27	1	no	—
344	C₃²	—	27	2	no	—
345	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
346	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
347	C₃²	C₃×#737	1	2	yes	—
348	C₉	—	3	6	no	—
349	C₉	—	3	3	no	—
350	C₉	—	3	1	no	—
351	C₉	—	3	2	no	—
352	C₉	—	3	1	no	—
353	C₉	—	3	2	no	—
354	C₉	—	3	1	no	—
355	C₉	—	3	2	no	—
356	C₉	—	9	1	no	—
357	C₉	—	9	1	no	—
358	C₉	—	9	2	no	—
359	C₉	—	9	2	no	—
360	C₉	—	9	1	no	—
361	C₉	—	9	2	no	—
362	C₉	—	9	2	no	—
363	C₉	—	9	2	no	—
364	C₉	—	9	1	no	—
365	C₉	—	9	1	no	—
366	C₉	—	9	2	no	—
367	C₉	—	9	2	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	C₉⋊H(3,𝔽₃)	1	3	yes	—
375	C₉	C₉⋊H(3,𝔽₃)	1	6	yes	—
376	C₃	—	3	1	no	—
377	C₃	—	3	2	no	—
378	C₃	—	9	3	no	—
379	C₃	—	27	2	no	—
380	C₃	—	27	1	no	—
381	C₃	C₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
382	C₃	#10624934	1	1	yes	characteristic
383	C₃	#10624935	1	1	yes	characteristic
384	C₃	#10624936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃×#6913	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃×C₉⋊C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624887	C₃	semidirect
3. C₃×#6896	C₃	semidirect
4. #10624864	C₃	semidirect
5. #10624803	C₃	semidirect
6. #10624803	C₃	semidirect
7. #10624802	C₃	semidirect
8. #10624802	C₃	semidirect
9. #10624801	C₃	semidirect
10. #10624801	C₃	semidirect
11. #10624800	C₃	semidirect
12. #10624799	C₃	semidirect
13. #10624798	C₃	semidirect
14. #10624797	C₃	semidirect
15. #10624797	C₃	semidirect
16. #10624796	C₃	semidirect
17. #10624796	C₃	semidirect
18. #10624795	C₃	semidirect
19. #10624795	C₃	semidirect
20. #10624794	C₃	semidirect
21. #10624793	C₃	semidirect
22. #10624792	C₃	semidirect
23. #10624700	C₃	semidirect
24. #10624700	C₃	semidirect
25. #10624697	C₃	semidirect
26. #10624697	C₃	semidirect
27. #10624692	C₃	semidirect
130. #737	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×#6913	3
#10624887	3
C₃×#6896	3
#10624864	3
#10624803	3
#10624802	3
#10624801	3
#10624800	3
#10624799	3
#10624798	3
#10624797	3
#10624796	3
#10624795	3
#10624794	3
#10624793	3
#10624792	3
#10624700	3
#10624697	3
#10624692	3