Subgroups — #14553767

Order 2187.

669 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624905	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624903	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	#10624629	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	#10624616	C₃	1	1	yes	maximal, characteristic, complemented
15	#10624613	C₃	1	3	yes	maximal, complemented
16	#10624611	C₃	1	1	yes	maximal, characteristic, complemented
17	#10624609	C₃	1	1	yes	maximal, characteristic, complemented
18	#10624605	C₃	1	1	yes	maximal, characteristic, complemented
19	#10624599	C₃	1	1	yes	maximal, characteristic
20	#10624598	C₃	1	1	yes	maximal, characteristic, complemented
21	#10624595	C₃	1	3	yes	maximal, complemented
22	C₃×#6860	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃×#737	C₃²	1	3	yes	—
24	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
25	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
26	#6899	—	3	3	no	—
27	#6897	—	3	3	no	—
28	#6895	—	3	1	no	—
29	#6895	—	3	1	no	—
30	#6895	—	3	1	no	—
31	#6895	—	3	1	no	—
32	#6895	—	3	1	no	—
33	#6894	—	3	3	no	—
34	#6894	—	3	3	no	—
35	#6894	C₃²	1	3	yes	—
36	#6894	C₃²	1	3	yes	—
37	#6894	C₃²	1	3	yes	—
38	#6894	C₃²	1	3	yes	—
39	#6894	C₃²	1	3	yes	—
40	#6893	—	3	3	no	—
41	#6893	C₃²	1	3	yes	—
42	#6893	C₃²	1	3	yes	—
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	1	no	—
49	C₉⋊H(3,𝔽₃)	—	3	1	no	—
50	C₉⋊H(3,𝔽₃)	C₃²	1	3	yes	—
51	C₉⋊H(3,𝔽₃)	C₃²	1	3	yes	—
52	#6891	C₃²	1	1	yes	characteristic
53	#6891	C₃²	1	1	yes	characteristic
54	#6891	C₃²	1	1	yes	characteristic
55	#6891	C₃²	1	1	yes	characteristic
56	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
57	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
58	C₉×3₋¹⁺²	C₃²	1	3	yes	—
59	C₉×H(3,𝔽₃)	—	3	1	no	—
60	C₉×H(3,𝔽₃)	—	3	1	no	—
61	C₉×H(3,𝔽₃)	C₃²	1	3	yes	—
62	#6887	—	3	1	no	—
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	1	yes	characteristic
68	#6860	C₃²	1	1	yes	characteristic
69	#6860	C₃²	1	1	yes	characteristic
70	#6860	C₃²	1	1	yes	characteristic
71	#6860	C₃²	1	1	yes	characteristic
72	#6860	C₃²	1	1	yes	characteristic
73	#6860	C₃²	1	1	yes	characteristic
74	#6860	C₃²	1	1	yes	characteristic
75	#6860	C₃²	1	1	yes	characteristic
76	#6860	C₃²	1	1	yes	characteristic
77	#6860	C₃²	1	3	yes	—
78	#6860	C₃²	1	3	yes	—
79	#6860	C₃²	1	3	yes	—
80	#6859	C₃²	1	3	yes	—
81	#6859	C₃²	1	3	yes	—
82	#6859	C₃²	1	3	yes	—
83	#6858	C₃²	1	3	yes	—
84	#6858	C₃²	1	3	yes	—
85	#6858	C₃²	1	3	yes	—
86	#6858	C₃²	1	3	yes	—
87	#6858	C₃²	1	3	yes	—
88	#6858	C₃²	1	3	yes	—
89	#6857	C₃²	1	3	yes	—
90	#6857	C₃²	1	3	yes	—
91	#6857	C₃²	1	3	yes	—
92	#6857	C₃²	1	3	yes	—
93	#6857	C₃²	1	3	yes	—
94	#6857	C₃²	1	3	yes	—
95	#6857	C₃²	1	3	yes	—
96	#6857	C₃²	1	3	yes	—
97	#6857	C₃²	1	3	yes	—
98	#6856	C₃²	1	3	yes	—
99	#6856	C₃²	1	3	yes	—
100	#6856	C₃²	1	3	yes	—
101	C₃⁴	C₃³	1	1	yes	characteristic
102	#737	—	9	9	no	—
103	C₃×3₋¹⁺²	—	3	1	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	3	no	—
110	C₃×3₋¹⁺²	—	3	3	no	—
111	C₃×3₋¹⁺²	—	3	3	no	—
112	C₃×3₋¹⁺²	—	3	3	no	—
113	C₃×3₋¹⁺²	—	9	3	no	—
114	C₃×3₋¹⁺²	—	9	3	no	—
115	C₃×3₋¹⁺²	C₃³	1	1	yes	characteristic
116	C₃×3₋¹⁺²	C₃³	1	1	yes	characteristic
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	3	yes	—
122	C₃×3₋¹⁺²	C₃³	1	3	yes	—
123	C₃×H(3,𝔽₃)	—	3	1	no	—
124	C₃×H(3,𝔽₃)	—	3	1	no	—
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)	C₃³	1	3	yes	—
130	C₃×H(3,𝔽₃)	C₃³	1	3	yes	—
131	C₃²×C₉	—	3	1	no	—
132	C₃²×C₉	—	3	1	no	—
133	C₃²×C₉	—	3	1	no	—
134	C₃²×C₉	—	3	1	no	—
135	C₃²×C₉	—	3	1	no	—
136	C₃²×C₉	—	3	1	no	—
137	C₃²×C₉	—	3	3	no	—
138	C₃²×C₉	—	3	3	no	—
139	C₃²×C₉	C₃³	1	3	yes	—
140	C₉⋊C₉	—	3	3	no	—
141	C₉⋊C₉	—	3	3	no	—
142	C₉⋊C₉	—	3	3	no	—
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	C₃²⋊C₉	—	3	1	no	—
158	C₃²⋊C₉	—	3	1	no	—
159	C₃²⋊C₉	—	3	1	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	3	no	—
176	C₃²⋊C₉	—	3	3	no	—
177	C₃²⋊C₉	—	3	3	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	H(3,𝔽₃)	1	1	yes	characteristic
188	C₃²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
189	C₃²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
190	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
191	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
192	C₃²⋊C₉	C₃³	1	1	yes	characteristic
193	C₃²⋊C₉	C₃³	1	1	yes	characteristic
194	C₃²⋊C₉	C₃³	1	3	yes	—
195	C₃²⋊C₉	C₃³	1	1	yes	characteristic
196	C₃²⋊C₉	C₃³	1	3	yes	—
197	C₃²⋊C₉	C₃³	1	3	yes	—
198	C₃²⋊C₉	C₃³	1	3	yes	—
199	C₃²⋊C₉	C₃³	1	1	yes	characteristic
200	C₃²⋊C₉	C₃³	1	1	yes	characteristic
201	C₃²⋊C₉	C₃³	1	1	yes	characteristic
202	C₉²	—	3	3	no	—
203	C₃³	—	3	3	no	—
204	C₃³	—	3	3	no	—
205	C₃³	—	3	3	no	—
206	C₃³	—	9	1	no	—
207	C₃³	—	9	1	no	—
208	C₃³	—	9	1	no	—
209	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
210	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
211	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
212	C₃³	C₃⁴	1	1	yes	characteristic
213	3₋¹⁺²	—	3	3	no	—
214	3₋¹⁺²	—	9	9	no	—
215	3₋¹⁺²	—	9	9	no	—
216	3₋¹⁺²	—	9	3	no	—
217	3₋¹⁺²	—	9	3	no	—
218	3₋¹⁺²	—	9	3	no	—
219	3₋¹⁺²	—	9	3	no	—
220	3₋¹⁺²	—	9	1	no	—
221	3₋¹⁺²	—	9	1	no	—
222	3₋¹⁺²	—	9	1	no	—
223	3₋¹⁺²	—	9	1	no	—
224	3₋¹⁺²	—	9	1	no	—
225	3₋¹⁺²	—	9	3	no	—
226	3₋¹⁺²	—	9	3	no	—
227	3₋¹⁺²	—	27	3	no	—
228	3₋¹⁺²	—	27	3	no	—
229	3₋¹⁺²	—	27	1	no	—
230	3₋¹⁺²	—	27	1	no	—
231	H(3,𝔽₃)	—	9	3	no	—
232	H(3,𝔽₃)	—	9	3	no	—
233	H(3,𝔽₃)	—	9	3	no	—
234	H(3,𝔽₃)	—	9	3	no	—
235	H(3,𝔽₃)	—	27	1	no	—
236	H(3,𝔽₃)	—	27	1	no	—
237	H(3,𝔽₃)	—	27	1	no	—
238	H(3,𝔽₃)	—	27	1	no	—
239	C₃×C₉	—	3	1	no	—
240	C₃×C₉	—	3	1	no	—
241	C₃×C₉	—	3	3	no	—
242	C₃×C₉	—	3	3	no	—
243	C₃×C₉	—	3	3	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	3	no	—
250	C₃×C₉	—	3	1	no	—
251	C₃×C₉	—	3	1	no	—
252	C₃×C₉	—	3	3	no	—
253	C₃×C₉	—	3	3	no	—
254	C₃×C₉	—	3	1	no	—
255	C₃×C₉	—	3	1	no	—
256	C₃×C₉	—	3	1	no	—
257	C₃×C₉	—	9	3	no	—
258	C₃×C₉	—	9	9	no	—
259	C₃×C₉	—	9	3	no	—
260	C₃×C₉	—	9	3	no	—
261	C₃×C₉	—	9	3	no	—
262	C₃×C₉	—	27	3	no	—
263	C₃×C₉	—	27	1	no	—
264	C₃×C₉	—	27	1	no	—
265	C₃×C₉	—	27	1	no	—
266	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
267	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
268	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
269	C₃²	—	3	1	no	—
270	C₃²	—	3	1	no	—
271	C₃²	—	3	1	no	—
272	C₃²	—	3	1	no	—
273	C₃²	—	3	1	no	—
274	C₃²	—	3	1	no	—
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	—	9	3	no	—
285	C₃²	—	9	3	no	—
286	C₃²	—	9	3	no	—
287	C₃²	—	9	3	no	—
288	C₃²	—	9	3	no	—
289	C₃²	—	9	3	no	—
290	C₃²	—	9	3	no	—
291	C₃²	—	9	3	no	—
292	C₃²	—	27	3	no	—
293	C₃²	#6913	1	3	yes	—
294	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
295	C₉	—	3	3	no	—
296	C₉	—	3	3	no	—
297	C₉	—	3	3	no	—
298	C₉	—	3	3	no	—
299	C₉	—	9	1	no	—
300	C₉	—	9	1	no	—
301	C₉	—	9	1	no	—
302	C₉	—	9	1	no	—
303	C₉	—	9	1	no	—
304	C₉	—	9	1	no	—
305	C₉	—	9	3	no	—
306	C₉	—	9	1	no	—
307	C₉	—	9	1	no	—
308	C₉	—	9	1	no	—
309	C₉	—	9	3	no	—
310	C₉	—	9	1	no	—
311	C₉	—	9	1	no	—
312	C₉	—	9	3	no	—
313	C₉	—	9	3	no	—
314	C₉	—	9	3	no	—
315	C₉	—	9	3	no	—
316	C₃	—	3	3	no	—
317	C₃	—	9	1	no	—
318	C₃	—	9	1	no	—
319	C₃	—	9	1	no	—
320	C₃	—	27	3	no	—
321	C₃	—	27	3	no	—
322	C₃	C₃×#6913	1	1	yes	characteristic
323	C₃	#10624930	1	1	yes	characteristic
324	C₃	#10624936	1	1	yes	characteristic
325	C₃	#10624936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	#10624599	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. #10624905	C₃	semidirect
2. #10624903	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. #10624629	C₃	semidirect
12. #10624629	C₃	semidirect
13. #10624623	C₃	semidirect
14. #10624616	C₃	semidirect
15. #10624613	C₃	semidirect
16. #10624611	C₃	semidirect
17. #10624609	C₃	semidirect
18. #10624605	C₃	semidirect
20. #10624598	C₃	semidirect
21. #10624595	C₃	semidirect
22. C₃×#6860	C₃	semidirect

Maximal subgroups

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

subgroup	index
#10624905	3
#10624903	3
#10624896	3
#10624886	3
#10624677	3
#10624651	3
#10624650	3
#10624634	3
#10624629	3
#10624623	3
#10624616	3
#10624613	3
#10624611	3
#10624609	3
#10624605	3
#10624599	3
#10624598	3
#10624595	3
C₃×#6860	3