Subgroups — #14553768

Order 2187.

592 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624907	C₃	1	2	yes	maximal, complemented
2	#10624900	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624887	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624677	C₃	1	2	yes	maximal, complemented
5	#10624677	C₃	1	2	yes	maximal, complemented
6	#10624676	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624674	C₃	1	2	yes	maximal, complemented
8	#10624674	C₃	1	2	yes	maximal, complemented
9	#10624655	C₃	1	2	yes	maximal, complemented
10	#10624654	C₃	1	2	yes	maximal, complemented
11	#10624652	C₃	1	2	yes	maximal, complemented
12	#10624645	C₃	1	2	yes	maximal, complemented
13	#10624634	C₃	1	2	yes	maximal, complemented
14	#10624634	C₃	1	1	yes	maximal, characteristic, complemented
15	#10624623	C₃	1	2	yes	maximal, complemented
16	#10624622	C₃	1	1	yes	maximal, characteristic, complemented
17	#10624621	C₃	1	2	yes	maximal, complemented
18	#10624621	C₃	1	2	yes	maximal, complemented
19	#10624618	C₃	1	2	yes	maximal, complemented
20	#10624606	C₃	1	2	yes	maximal, complemented
21	#10624606	C₃	1	1	yes	maximal, characteristic, complemented
22	#10624601	C₃	1	1	yes	maximal, characteristic, complemented
23	#10624595	C₃	1	2	yes	maximal, complemented
24	#10624583	C₃	1	1	yes	maximal, characteristic, complemented
25	C₃×#737	C₃²	1	1	yes	characteristic
26	C₃×#737	C₃²	1	1	yes	characteristic
27	#6900	—	3	1	no	—
28	#6899	—	3	2	no	—
29	#6899	—	3	2	no	—
30	#6899	—	3	2	no	—
31	#6898	—	3	2	no	—
32	#6897	—	3	2	no	—
33	#6897	—	3	2	no	—
34	#6896	—	3	2	no	—
35	#6896	—	3	1	no	—
36	#6896	—	3	2	no	—
37	#6895	—	3	1	no	—
38	#6895	—	3	1	no	—
39	#6895	—	3	2	no	—
40	#6895	C₃²	1	1	yes	characteristic
41	#6895	C₃²	1	1	yes	characteristic
42	#6895	C₃²	1	2	yes	—
43	#6894	—	3	1	no	—
44	#6894	—	3	2	no	—
45	#6894	—	3	2	no	—
46	#6894	—	3	2	no	—
47	#6894	—	3	2	no	—
48	#6894	—	3	2	no	—
49	#6894	C₃²	1	2	yes	—
50	#6894	C₃²	1	2	yes	—
51	#6894	C₃²	1	2	yes	—
52	#6894	C₃²	1	2	yes	—
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	#6893	C₃²	1	2	yes	—
60	C₉⋊H(3,𝔽₃)	C₃²	1	2	yes	—
61	C₉⋊H(3,𝔽₃)	C₃²	1	2	yes	—
62	C₉⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
63	C₉⋊H(3,𝔽₃)	C₃²	1	2	yes	—
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×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
68	C₉×H(3,𝔽₃)	C₃²	1	2	yes	—
69	C₉×H(3,𝔽₃)	C₃²	1	2	yes	—
70	#6887	C₃²	1	1	yes	characteristic
71	#6887	C₃²	1	1	yes	characteristic
72	#6887	C₃²	1	2	yes	—
73	C₃×C₉⋊C₉	—	3	1	no	—
74	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
75	#6862	C₃²	1	2	yes	—
76	#6862	C₃²	1	1	yes	characteristic
77	#6861	C₃²	1	2	yes	—
78	#6861	C₃²	1	1	yes	characteristic
79	#6861	C₃²	1	2	yes	—
80	#6861	C₃²	1	1	yes	characteristic
81	#6861	C₃²	1	2	yes	—
82	#6861	C₃²	1	2	yes	—
83	#6861	C₃²	1	2	yes	—
84	#6860	C₃²	1	2	yes	—
85	#6860	C₃²	1	2	yes	—
86	#6860	C₃²	1	2	yes	—
87	#6860	C₃²	1	2	yes	—
88	#6860	C₃²	1	2	yes	—
89	#6860	C₃²	1	2	yes	—
90	#6859	C₃²	1	1	yes	characteristic
91	#6859	C₃²	1	2	yes	—
92	#6859	C₃²	1	1	yes	characteristic
93	#6859	C₃²	1	2	yes	—
94	#6859	C₃²	1	2	yes	—
95	#6859	C₃²	1	2	yes	—
96	#6859	C₃²	1	2	yes	—
97	#6858	C₃²	1	2	yes	—
98	#6858	C₃²	1	2	yes	—
99	#6858	C₃²	1	2	yes	—
100	#6858	C₃²	1	2	yes	—
101	#6858	C₃²	1	2	yes	—
102	#6858	C₃²	1	2	yes	—
103	#6858	C₃²	1	2	yes	—
104	#6858	C₃²	1	2	yes	—
105	#6858	C₃²	1	2	yes	—
106	#6858	C₃²	1	2	yes	—
107	#6858	C₃²	1	2	yes	—
108	#6858	C₃²	1	2	yes	—
109	#6857	C₃²	1	2	yes	—
110	#6857	C₃²	1	1	yes	characteristic
111	#6857	C₃²	1	2	yes	—
112	#6857	C₃²	1	2	yes	—
113	#6857	C₃²	1	1	yes	characteristic
114	#6857	C₃²	1	2	yes	—
115	#6857	C₃²	1	2	yes	—
116	#6856	C₃²	1	1	yes	characteristic
117	#6856	C₃²	1	2	yes	—
118	#6856	C₃²	1	1	yes	characteristic
119	#6856	C₃²	1	2	yes	—
120	#737	—	9	3	no	—
121	#737	—	9	3	no	—
122	C₃×3₋¹⁺²	—	3	2	no	—
123	C₃×3₋¹⁺²	—	3	1	no	—
124	C₃×3₋¹⁺²	—	3	1	no	—
125	C₃×3₋¹⁺²	—	3	2	no	—
126	C₃×3₋¹⁺²	—	3	2	no	—
127	C₃×3₋¹⁺²	—	3	2	no	—
128	C₃×3₋¹⁺²	—	3	1	no	—
129	C₃×3₋¹⁺²	C₃³	1	2	yes	—
130	C₃×3₋¹⁺²	C₃³	1	2	yes	—
131	C₃×3₋¹⁺²	C₃³	1	2	yes	—
132	C₃×3₋¹⁺²	C₃³	1	2	yes	—
133	C₃×H(3,𝔽₃)	C₃³	1	1	yes	characteristic
134	C₃×H(3,𝔽₃)	C₃³	1	1	yes	characteristic
135	C₃×H(3,𝔽₃)	C₃³	1	2	yes	—
136	C₃²×C₉	—	3	1	no	—
137	C₃²×C₉	—	3	1	no	—
138	C₃²×C₉	—	3	1	no	—
139	C₃²×C₉	—	3	2	no	—
140	C₃²×C₉	—	3	2	no	—
141	C₃²×C₉	C₃³	1	1	yes	characteristic
142	C₃²×C₉	C₃³	1	1	yes	characteristic
143	C₃²×C₉	C₃³	1	2	yes	—
144	C₉⋊C₉	—	3	2	no	—
145	C₉⋊C₉	—	3	2	no	—
146	C₉⋊C₉	—	3	2	no	—
147	C₉⋊C₉	—	3	1	no	—
148	C₉⋊C₉	—	3	1	no	—
149	C₉⋊C₉	—	3	1	no	—
150	C₉⋊C₉	—	3	1	no	—
151	C₉⋊C₉	—	3	2	no	—
152	C₉⋊C₉	—	3	2	no	—
153	C₉⋊C₉	—	3	1	no	—
154	C₉⋊C₉	—	3	1	no	—
155	C₉⋊C₉	—	3	2	no	—
156	C₉⋊C₉	—	3	2	no	—
157	C₉⋊C₉	—	3	2	no	—
158	C₉⋊C₉	—	3	2	no	—
159	C₉⋊C₉	—	3	2	no	—
160	C₉⋊C₉	—	3	2	no	—
161	C₉⋊C₉	—	3	2	no	—
162	C₉⋊C₉	—	3	2	no	—
163	C₉⋊C₉	—	3	2	no	—
164	C₉⋊C₉	—	3	2	no	—
165	C₉⋊C₉	—	3	2	no	—
166	C₉⋊C₉	—	3	2	no	—
167	C₉⋊C₉	—	3	2	no	—
168	C₉⋊C₉	—	3	2	no	—
169	C₉⋊C₉	—	3	2	no	—
170	C₉⋊C₉	—	3	2	no	—
171	C₉⋊C₉	—	3	2	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	4	no	—
178	C₉⋊C₉	—	3	2	no	—
179	C₉⋊C₉	—	3	2	no	—
180	C₉⋊C₉	—	3	2	no	—
181	C₉⋊C₉	—	3	2	no	—
182	C₉⋊C₉	—	3	2	no	—
183	C₉⋊C₉	—	3	2	no	—
184	C₉⋊C₉	—	3	2	no	—
185	C₉⋊C₉	—	3	2	no	—
186	C₉⋊C₉	—	3	2	no	—
187	C₉⋊C₉	—	3	2	no	—
188	C₉⋊C₉	—	3	2	no	—
189	C₉⋊C₉	—	3	2	no	—
190	C₉⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
191	C₉⋊C₉	H(3,𝔽₃)	1	2	yes	—
192	C₉⋊C₉	H(3,𝔽₃)	1	2	yes	—
193	C₉⋊C₉	H(3,𝔽₃)	1	2	yes	—
194	C₉⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
195	C₉⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
196	C₃²⋊C₉	—	3	1	no	—
197	C₃²⋊C₉	—	3	1	no	—
198	C₃²⋊C₉	—	3	2	no	—
199	C₃²⋊C₉	—	3	2	no	—
200	C₃²⋊C₉	—	3	2	no	—
201	C₃²⋊C₉	—	3	2	no	—
202	C₃²⋊C₉	—	3	2	no	—
203	C₃²⋊C₉	—	3	2	no	—
204	C₃²⋊C₉	—	3	2	no	—
205	C₃²⋊C₉	—	3	2	no	—
206	C₃²⋊C₉	—	3	2	no	—
207	C₃²⋊C₉	—	3	2	no	—
208	C₃²⋊C₉	—	3	2	no	—
209	C₃²⋊C₉	—	3	2	no	—
210	C₃²⋊C₉	—	3	2	no	—
211	C₃²⋊C₉	—	3	2	no	—
212	C₃²⋊C₉	C₃³	1	2	yes	—
213	C₃²⋊C₉	C₃³	1	1	yes	characteristic
214	C₃²⋊C₉	C₃³	1	1	yes	characteristic
215	C₃²⋊C₉	C₃³	1	2	yes	—
216	C₃²⋊C₉	C₃³	1	2	yes	—
217	C₃²⋊C₉	C₃³	1	2	yes	—
218	C₃²⋊C₉	C₃³	1	2	yes	—
219	C₃²⋊C₉	C₃³	1	2	yes	—
220	C₃²⋊C₉	C₃³	1	2	yes	—
221	C₃²⋊C₉	C₃³	1	2	yes	—
222	C₃²⋊C₉	C₃³	1	1	yes	characteristic
223	C₃²⋊C₉	C₃³	1	2	yes	—
224	C₃²⋊C₉	C₃³	1	1	yes	characteristic
225	C₃²⋊C₉	C₃³	1	2	yes	—
226	C₉²	—	3	2	no	—
227	C₉²	—	3	1	no	—
228	C₉²	—	3	1	no	—
229	C₉²	—	3	2	no	—
230	C₉²	—	3	2	no	—
231	C₃³	—	3	1	no	—
232	C₃³	—	3	2	no	—
233	C₃³	—	3	1	no	—
234	C₃³	C₃⁴	1	1	yes	characteristic
235	3₋¹⁺²	—	9	6	no	—
236	3₋¹⁺²	—	9	3	no	—
237	3₋¹⁺²	—	9	3	no	—
238	3₋¹⁺²	—	9	6	no	—
239	3₋¹⁺²	—	9	2	no	—
240	3₋¹⁺²	—	9	2	no	—
241	3₋¹⁺²	—	9	2	no	—
242	3₋¹⁺²	—	9	2	no	—
243	3₋¹⁺²	—	27	1	no	—
244	3₋¹⁺²	—	27	2	no	—
245	3₋¹⁺²	—	27	2	no	—
246	H(3,𝔽₃)	—	3	3	no	—
247	H(3,𝔽₃)	—	9	2	no	—
248	H(3,𝔽₃)	—	9	1	no	—
249	C₃×C₉	—	3	3	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	1	no	—
256	C₃×C₉	—	3	1	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	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	2	no	—
267	C₃×C₉	—	3	2	no	—
268	C₃×C₉	—	9	2	no	—
269	C₃×C₉	—	9	1	no	—
270	C₃×C₉	—	9	3	no	—
271	C₃×C₉	—	9	3	no	—
272	C₃×C₉	—	9	3	no	—
273	C₃×C₉	—	27	2	no	—
274	C₃×C₉	—	27	2	no	—
275	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
276	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
277	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
278	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
279	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
280	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
281	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
282	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
283	C₃²	—	3	1	no	—
284	C₃²	—	3	1	no	—
285	C₃²	—	3	3	no	—
286	C₃²	—	3	2	no	—
287	C₃²	—	9	1	no	—
288	C₃²	—	9	2	no	—
289	C₃²	—	9	1	no	—
290	C₃²	—	9	2	no	—
291	C₃²	—	9	2	no	—
292	C₃²	—	9	1	no	—
293	C₃²	—	9	2	no	—
294	C₃²	—	9	2	no	—
295	C₃²	—	9	2	no	—
296	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
297	C₉	—	3	1	no	—
298	C₉	—	3	1	no	—
299	C₉	—	3	1	no	—
300	C₉	—	3	2	no	—
301	C₉	—	3	2	no	—
302	C₉	—	3	2	no	—
303	C₉	—	3	2	no	—
304	C₉	—	9	1	no	—
305	C₉	—	9	1	no	—
306	C₉	—	9	2	no	—
307	C₉	—	9	2	no	—
308	C₉	—	9	2	no	—
309	C₉	—	9	2	no	—
310	C₉	—	9	2	no	—
311	C₉	—	9	2	no	—
312	C₉	—	9	1	no	—
313	C₉	—	9	1	no	—
314	C₉	—	9	2	no	—
315	C₉	—	9	2	no	—
316	C₉	—	9	2	no	—
317	C₉	—	9	2	no	—
318	C₉	—	9	2	no	—
319	C₉	—	9	2	no	—
320	C₉	—	9	2	no	—
321	C₉	—	9	2	no	—
322	C₉	#6913	1	3	yes	—
323	C₃	—	9	1	no	—
324	C₃	—	9	3	no	—
325	C₃	—	27	1	no	—
326	C₃	—	27	2	no	—
327	C₃	C₃×#6913	1	1	yes	characteristic
328	C₃	#10624934	1	1	yes	characteristic
329	C₃	#10624935	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. #10624907	C₃	semidirect
2. #10624900	C₃	semidirect
3. #10624887	C₃	semidirect
4. #10624677	C₃	semidirect
5. #10624677	C₃	semidirect
6. #10624676	C₃	semidirect
7. #10624674	C₃	semidirect
8. #10624674	C₃	semidirect
9. #10624655	C₃	semidirect
10. #10624654	C₃	semidirect
11. #10624652	C₃	semidirect
12. #10624645	C₃	semidirect
13. #10624634	C₃	semidirect
14. #10624634	C₃	semidirect
15. #10624623	C₃	semidirect
16. #10624622	C₃	semidirect
17. #10624621	C₃	semidirect
18. #10624621	C₃	semidirect
19. #10624618	C₃	semidirect
20. #10624606	C₃	semidirect
21. #10624606	C₃	semidirect
22. #10624601	C₃	semidirect
23. #10624595	C₃	semidirect
24. #10624583	C₃	semidirect

Maximal subgroups

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

subgroup	index
#10624907	3
#10624900	3
#10624887	3
#10624677	3
#10624676	3
#10624674	3
#10624655	3
#10624654	3
#10624652	3
#10624645	3
#10624634	3
#10624623	3
#10624622	3
#10624621	3
#10624618	3
#10624606	3
#10624601	3
#10624595	3
#10624583	3