Subgroups — #14553760

Order 2187.

696 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624906	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624897	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624896	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624886	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624677	C₃	1	3	yes	maximal, complemented
6	#10624677	C₃	1	3	yes	maximal, complemented
7	#10624677	C₃	1	3	yes	maximal, complemented
8	#10624651	C₃	1	3	yes	maximal, complemented
9	#10624650	C₃	1	3	yes	maximal, complemented
10	#10624634	C₃	1	3	yes	maximal, complemented
11	#10624633	C₃	1	1	yes	maximal, characteristic, complemented
12	#10624629	C₃	1	1	yes	maximal, characteristic, complemented
13	#10624623	C₃	1	3	yes	maximal, complemented
14	#10624620	C₃	1	1	yes	maximal, characteristic, complemented
15	#10624613	C₃	1	3	yes	maximal, complemented
16	#10624612	C₃	1	1	yes	maximal, characteristic, complemented
17	#10624610	C₃	1	1	yes	maximal, characteristic, complemented
18	#10624605	C₃	1	1	yes	maximal, characteristic, complemented
19	#10624596	C₃	1	1	yes	maximal, characteristic
20	#10624595	C₃	1	3	yes	maximal, complemented
21	#10624591	C₃	1	1	yes	maximal, characteristic, complemented
22	C₃×#6857	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃×#737	C₃²	1	3	yes	complemented
24	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
25	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
26	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
27	#6898	—	3	3	no	—
28	#6897	—	3	3	no	—
29	#6895	—	3	1	no	—
30	#6895	—	3	1	no	—
31	#6895	—	3	1	no	—
32	#6894	—	3	3	no	—
33	#6894	—	3	3	no	—
34	#6894	—	3	1	no	—
35	#6894	—	3	1	no	—
36	#6894	C₃²	1	3	yes	complemented
37	#6894	C₃²	1	3	yes	complemented
38	#6894	C₃²	1	3	yes	complemented
39	#6894	C₃²	1	3	yes	complemented
40	#6894	C₃²	1	3	yes	complemented
41	#6893	C₃²	1	3	yes	complemented
42	#6893	C₃²	1	3	yes	complemented
43	C₉⋊H(3,𝔽₃)	—	3	1	no	—
44	C₉⋊H(3,𝔽₃)	—	3	1	no	—
45	C₉⋊H(3,𝔽₃)	—	3	3	no	—
46	C₉⋊H(3,𝔽₃)	—	3	3	no	—
47	C₉⋊H(3,𝔽₃)	—	3	3	no	—
48	C₉⋊H(3,𝔽₃)	—	3	3	no	—
49	C₉⋊H(3,𝔽₃)	C₃²	1	3	yes	—
50	C₉⋊H(3,𝔽₃)	C₃²	1	3	yes	—
51	#6891	—	3	1	no	—
52	#6891	—	3	1	no	—
53	#6891	C₃²	1	1	yes	characteristic
54	#6891	C₃²	1	1	yes	characteristic
55	#6891	C₃²	1	1	yes	characteristic
56	#6891	C₃²	1	1	yes	characteristic
57	#6891	C₃²	1	1	yes	characteristic
58	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
59	C₉×3₋¹⁺²	—	3	1	no	—
60	C₉×3₋¹⁺²	C₃²	1	3	yes	complemented
61	C₉×H(3,𝔽₃)	—	3	1	no	—
62	C₉×H(3,𝔽₃)	C₃²	1	3	yes	—
63	C₃×C₃²⋊C₉	—	3	1	no	—
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	1	yes	characteristic
68	#6860	C₃²	1	3	yes	complemented
69	#6860	C₃²	1	3	yes	complemented
70	#6860	C₃²	1	3	yes	complemented
71	#6860	C₃²	1	3	yes	complemented
72	#6860	C₃²	1	3	yes	complemented
73	#6860	C₃²	1	3	yes	complemented
74	#6859	C₃²	1	3	yes	complemented
75	#6859	C₃²	1	3	yes	complemented
76	#6859	C₃²	1	3	yes	complemented
77	#6858	C₃²	1	3	yes	complemented
78	#6858	C₃²	1	3	yes	complemented
79	#6858	C₃²	1	3	yes	complemented
80	#6858	C₃²	1	3	yes	complemented
81	#6858	C₃²	1	3	yes	complemented
82	#6858	C₃²	1	3	yes	complemented
83	#6857	C₃²	1	1	yes	characteristic
84	#6857	C₃²	1	1	yes	characteristic
85	#6857	C₃²	1	1	yes	characteristic
86	#6857	C₃²	1	1	yes	characteristic
87	#6857	C₃²	1	1	yes	characteristic
88	#6857	C₃²	1	1	yes	characteristic
89	#6857	C₃²	1	1	yes	characteristic
90	#6857	C₃²	1	1	yes	characteristic
91	#6857	C₃²	1	1	yes	characteristic
92	#6857	C₃²	1	3	yes	—
93	#6857	C₃²	1	3	yes	—
94	#6857	C₃²	1	3	yes	—
95	#6857	C₃²	1	3	yes	complemented
96	#6857	C₃²	1	3	yes	complemented
97	#6857	C₃²	1	3	yes	complemented
98	#6856	C₃²	1	3	yes	—
99	#6856	C₃²	1	3	yes	—
100	#6856	C₃²	1	3	yes	—
101	C₃⁴	—	3	1	no	—
102	C₃⁴	C₃³	1	1	yes	characteristic
103	#737	—	9	9	no	—
104	C₃×3₋¹⁺²	—	3	1	no	—
105	C₃×3₋¹⁺²	—	3	1	no	—
106	C₃×3₋¹⁺²	—	3	1	no	—
107	C₃×3₋¹⁺²	—	3	1	no	—
108	C₃×3₋¹⁺²	—	3	1	no	—
109	C₃×3₋¹⁺²	—	3	1	no	—
110	C₃×3₋¹⁺²	—	3	1	no	—
111	C₃×3₋¹⁺²	—	3	3	no	—
112	C₃×3₋¹⁺²	—	3	3	no	—
113	C₃×3₋¹⁺²	—	3	3	no	—
114	C₃×3₋¹⁺²	—	3	3	no	—
115	C₃×3₋¹⁺²	—	9	3	no	—
116	C₃×3₋¹⁺²	—	9	3	no	—
117	C₃×3₋¹⁺²	C₃³	1	1	yes	characteristic
118	C₃×3₋¹⁺²	C₃³	1	1	yes	characteristic
119	C₃×3₋¹⁺²	C₃³	1	1	yes	characteristic
120	C₃×3₋¹⁺²	C₃³	1	1	yes	characteristic
121	C₃×3₋¹⁺²	C₃³	1	1	yes	characteristic
122	C₃×3₋¹⁺²	C₃³	1	1	yes	characteristic
123	C₃×3₋¹⁺²	C₃³	1	3	yes	—
124	C₃×3₋¹⁺²	C₃³	1	3	yes	—
125	C₃×H(3,𝔽₃)	—	3	1	no	—
126	C₃×H(3,𝔽₃)	—	3	1	no	—
127	C₃×H(3,𝔽₃)	—	3	1	no	—
128	C₃×H(3,𝔽₃)	—	3	1	no	—
129	C₃×H(3,𝔽₃)	—	3	1	no	—
130	C₃×H(3,𝔽₃)	—	9	3	no	—
131	C₃×H(3,𝔽₃)	C₃³	1	1	yes	characteristic
132	C₃×H(3,𝔽₃)	C₃³	1	1	yes	characteristic
133	C₃×H(3,𝔽₃)	C₃³	1	1	yes	characteristic
134	C₃×H(3,𝔽₃)	C₃³	1	3	yes	—
135	C₃²×C₉	—	3	1	no	—
136	C₃²×C₉	—	3	1	no	—
137	C₃²×C₉	—	3	1	no	—
138	C₃²×C₉	—	3	1	no	—
139	C₃²×C₉	—	3	1	no	—
140	C₃²×C₉	—	3	3	no	—
141	C₃²×C₉	—	3	3	no	—
142	C₃²×C₉	C₃³	1	3	yes	—
143	C₉⋊C₉	—	3	3	no	—
144	C₉⋊C₉	—	3	3	no	—
145	C₉⋊C₉	—	3	3	no	—
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	3	no	—
154	C₉⋊C₉	—	3	3	no	—
155	C₉⋊C₉	—	3	3	no	—
156	C₉⋊C₉	—	3	3	no	—
157	C₉⋊C₉	—	3	3	no	—
158	C₉⋊C₉	—	3	3	no	—
159	C₉⋊C₉	—	3	3	no	—
160	C₃²⋊C₉	—	3	1	no	—
161	C₃²⋊C₉	—	3	1	no	—
162	C₃²⋊C₉	—	3	1	no	—
163	C₃²⋊C₉	—	3	1	no	—
164	C₃²⋊C₉	—	3	1	no	—
165	C₃²⋊C₉	—	3	1	no	—
166	C₃²⋊C₉	—	3	1	no	—
167	C₃²⋊C₉	—	3	1	no	—
168	C₃²⋊C₉	—	3	1	no	—
169	C₃²⋊C₉	—	3	1	no	—
170	C₃²⋊C₉	—	3	1	no	—
171	C₃²⋊C₉	—	3	1	no	—
172	C₃²⋊C₉	—	3	1	no	—
173	C₃²⋊C₉	—	3	1	no	—
174	C₃²⋊C₉	—	3	1	no	—
175	C₃²⋊C₉	—	3	1	no	—
176	C₃²⋊C₉	—	3	1	no	—
177	C₃²⋊C₉	—	3	1	no	—
178	C₃²⋊C₉	—	3	3	no	—
179	C₃²⋊C₉	—	3	3	no	—
180	C₃²⋊C₉	—	3	3	no	—
181	C₃²⋊C₉	—	3	3	no	—
182	C₃²⋊C₉	—	3	3	no	—
183	C₃²⋊C₉	—	3	3	no	—
184	C₃²⋊C₉	—	3	3	no	—
185	C₃²⋊C₉	—	3	3	no	—
186	C₃²⋊C₉	—	3	3	no	—
187	C₃²⋊C₉	—	3	3	no	—
188	C₃²⋊C₉	—	3	3	no	—
189	C₃²⋊C₉	—	3	3	no	—
190	C₃²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
191	C₃²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
192	C₃²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
193	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
194	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
195	C₃²⋊C₉	C₃³	1	3	yes	—
196	C₃²⋊C₉	C₃³	1	3	yes	—
197	C₃²⋊C₉	C₃³	1	1	yes	characteristic
198	C₃²⋊C₉	C₃³	1	1	yes	characteristic
199	C₃²⋊C₉	C₃³	1	1	yes	characteristic
200	C₃²⋊C₉	C₃³	1	3	yes	—
201	C₃²⋊C₉	C₃³	1	3	yes	—
202	C₃²⋊C₉	C₃³	1	3	yes	—
203	C₉²	—	3	3	no	—
204	C₃³	—	3	1	no	—
205	C₃³	—	3	3	no	—
206	C₃³	—	3	3	no	—
207	C₃³	—	3	1	no	—
208	C₃³	—	3	1	no	—
209	C₃³	—	9	1	no	—
210	C₃³	—	9	1	no	—
211	C₃³	—	9	1	no	—
212	C₃³	—	9	3	no	—
213	C₃³	—	9	3	no	—
214	C₃³	—	27	1	no	—
215	C₃³	—	27	1	no	—
216	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
217	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
218	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
219	C₃³	C₃⁴	1	1	yes	characteristic
220	3₋¹⁺²	—	3	3	no	—
221	3₋¹⁺²	—	9	9	no	—
222	3₋¹⁺²	—	9	9	no	—
223	3₋¹⁺²	—	9	3	no	—
224	3₋¹⁺²	—	9	3	no	—
225	3₋¹⁺²	—	9	3	no	—
226	3₋¹⁺²	—	9	3	no	—
227	3₋¹⁺²	—	9	3	no	—
228	3₋¹⁺²	—	9	1	no	—
229	3₋¹⁺²	—	9	1	no	—
230	3₋¹⁺²	—	9	1	no	—
231	3₋¹⁺²	—	9	1	no	—
232	3₋¹⁺²	—	9	1	no	—
233	3₋¹⁺²	—	9	3	no	—
234	3₋¹⁺²	—	9	3	no	—
235	3₋¹⁺²	—	27	3	no	—
236	3₋¹⁺²	—	27	3	no	—
237	3₋¹⁺²	—	27	1	no	—
238	3₋¹⁺²	—	27	1	no	—
239	H(3,𝔽₃)	—	3	3	no	—
240	H(3,𝔽₃)	—	9	3	no	—
241	H(3,𝔽₃)	—	9	3	no	—
242	H(3,𝔽₃)	—	9	1	no	—
243	H(3,𝔽₃)	—	9	1	no	—
244	H(3,𝔽₃)	—	9	3	no	—
245	H(3,𝔽₃)	—	9	3	no	—
246	H(3,𝔽₃)	—	27	1	no	—
247	H(3,𝔽₃)	—	27	1	no	—
248	C₃×C₉	—	3	1	no	—
249	C₃×C₉	—	3	1	no	—
250	C₃×C₉	—	3	3	no	—
251	C₃×C₉	—	3	3	no	—
252	C₃×C₉	—	3	3	no	—
253	C₃×C₉	—	3	1	no	—
254	C₃×C₉	—	3	1	no	—
255	C₃×C₉	—	3	3	no	—
256	C₃×C₉	—	3	3	no	—
257	C₃×C₉	—	3	3	no	—
258	C₃×C₉	—	3	1	no	—
259	C₃×C₉	—	3	1	no	—
260	C₃×C₉	—	3	3	no	—
261	C₃×C₉	—	3	1	no	—
262	C₃×C₉	—	3	1	no	—
263	C₃×C₉	—	3	1	no	—
264	C₃×C₉	—	9	3	no	—
265	C₃×C₉	—	9	9	no	—
266	C₃×C₉	—	9	3	no	—
267	C₃×C₉	—	9	3	no	—
268	C₃×C₉	—	27	3	no	—
269	C₃×C₉	—	27	1	no	—
270	C₃×C₉	—	27	1	no	—
271	C₃×C₉	—	27	1	no	—
272	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
273	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
274	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
275	C₃²	—	3	1	no	—
276	C₃²	—	3	1	no	—
277	C₃²	—	3	1	no	—
278	C₃²	—	3	1	no	—
279	C₃²	—	3	1	no	—
280	C₃²	—	3	1	no	—
281	C₃²	—	3	1	no	—
282	C₃²	—	3	1	no	—
283	C₃²	—	3	1	no	—
284	C₃²	—	3	1	no	—
285	C₃²	—	3	1	no	—
286	C₃²	—	3	3	no	—
287	C₃²	—	3	1	no	—
288	C₃²	—	3	1	no	—
289	C₃²	—	3	1	no	—
290	C₃²	—	3	1	no	—
291	C₃²	—	9	1	no	—
292	C₃²	—	9	1	no	—
293	C₃²	—	9	1	no	—
294	C₃²	—	9	3	no	—
295	C₃²	—	9	1	no	—
296	C₃²	—	9	1	no	—
297	C₃²	—	9	3	no	—
298	C₃²	—	9	1	no	—
299	C₃²	—	9	1	no	—
300	C₃²	—	9	1	no	—
301	C₃²	—	9	1	no	—
302	C₃²	—	9	3	no	—
303	C₃²	—	9	3	no	—
304	C₃²	—	9	1	no	—
305	C₃²	—	9	1	no	—
306	C₃²	—	27	3	no	—
307	C₃²	—	27	9	no	—
308	C₃²	#6913	1	3	yes	—
309	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
310	C₉	—	3	3	no	—
311	C₉	—	3	3	no	—
312	C₉	—	3	3	no	—
313	C₉	—	3	3	no	—
314	C₉	—	9	1	no	—
315	C₉	—	9	1	no	—
316	C₉	—	9	1	no	—
317	C₉	—	9	3	no	—
318	C₉	—	9	3	no	—
319	C₉	—	9	1	no	—
320	C₉	—	9	1	no	—
321	C₉	—	9	3	no	—
322	C₉	—	9	3	no	—
323	C₉	—	9	1	no	—
324	C₉	—	9	3	no	—
325	C₉	—	9	1	no	—
326	C₉	—	9	1	no	—
327	C₉	—	9	3	no	—
328	C₉	—	9	3	no	—
329	C₃	—	3	3	no	—
330	C₃	—	9	1	no	—
331	C₃	—	9	1	no	—
332	C₃	—	9	1	no	—
333	C₃	—	9	3	no	—
334	C₃	—	27	3	no	—
335	C₃	—	27	1	no	—
336	C₃	—	27	1	no	—
337	C₃	C₃×#6913	1	1	yes	characteristic
338	C₃	#10624930	1	1	yes	characteristic
339	C₃	#10624934	1	1	yes	characteristic
340	C₃	#10624935	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	#10624596	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃×C₃²⋊C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #10624906	C₃	semidirect
2. #10624897	C₃	semidirect
3. #10624896	C₃	semidirect
4. #10624886	C₃	semidirect
5. #10624677	C₃	semidirect
6. #10624677	C₃	semidirect
7. #10624677	C₃	semidirect
8. #10624651	C₃	semidirect
9. #10624650	C₃	semidirect
10. #10624634	C₃	semidirect
11. #10624633	C₃	semidirect
12. #10624629	C₃	semidirect
13. #10624623	C₃	semidirect
14. #10624620	C₃	semidirect
15. #10624613	C₃	semidirect
16. #10624612	C₃	semidirect
17. #10624610	C₃	semidirect
18. #10624605	C₃	semidirect
20. #10624595	C₃	semidirect
21. #10624591	C₃	semidirect
22. C₃×#6857	C₃	semidirect
23. C₃×#737	C₃²	semidirect
36. #6894	C₃²	semidirect
37. #6894	C₃²	semidirect
38. #6894	C₃²	semidirect
39. #6894	C₃²	semidirect
40. #6894	C₃²	semidirect
41. #6893	C₃²	semidirect
42. #6893	C₃²	semidirect
60. C₉×3₋¹⁺²	C₃²	semidirect
68. #6860	C₃²	semidirect
69. #6860	C₃²	semidirect
70. #6860	C₃²	semidirect
71. #6860	C₃²	semidirect
72. #6860	C₃²	semidirect
73. #6860	C₃²	semidirect
74. #6859	C₃²	semidirect
75. #6859	C₃²	semidirect
76. #6859	C₃²	semidirect
77. #6858	C₃²	semidirect
78. #6858	C₃²	semidirect
79. #6858	C₃²	semidirect
80. #6858	C₃²	semidirect
81. #6858	C₃²	semidirect
82. #6858	C₃²	semidirect
95. #6857	C₃²	semidirect
96. #6857	C₃²	semidirect
97. #6857	C₃²	semidirect

Maximal subgroups

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

subgroup	index
#10624906	3
#10624897	3
#10624896	3
#10624886	3
#10624677	3
#10624651	3
#10624650	3
#10624634	3
#10624633	3
#10624629	3
#10624623	3
#10624620	3
#10624613	3
#10624612	3
#10624610	3
#10624605	3
#10624596	3
#10624595	3
#10624591	3
C₃×#6857	3