Subgroups — #14553761

Order 2187.

639 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	#10624903	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624899	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624897	C₃	1	1	yes	maximal, characteristic
4	#10624886	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624676	C₃	1	3	yes	maximal, complemented
6	#10624676	C₃	1	3	yes	maximal, complemented
7	#10624676	C₃	1	3	yes	maximal, complemented
8	#10624653	C₃	1	3	yes	maximal, complemented
9	#10624652	C₃	1	3	yes	maximal, complemented
10	#10624645	C₃	1	3	yes	maximal, complemented
11	#10624631	C₃	1	1	yes	maximal, characteristic, complemented
12	#10624631	C₃	1	1	yes	maximal, characteristic, complemented
13	#10624624	C₃	1	3	yes	maximal, complemented
14	#10624621	C₃	1	3	yes	maximal, complemented
15	#10624619	C₃	1	1	yes	maximal, characteristic, complemented
16	#10624614	C₃	1	1	yes	maximal, characteristic, complemented
17	#10624611	C₃	1	1	yes	maximal, characteristic, complemented
18	#10624606	C₃	1	3	yes	maximal, complemented
19	#10624602	C₃	1	1	yes	maximal, characteristic
20	#10624593	C₃	1	1	yes	maximal, characteristic
21	#10624592	C₃	1	1	yes	maximal, characteristic, complemented
22	C₃×#6857	C₃	1	1	yes	maximal, characteristic
23	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
24	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
25	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
26	#6899	—	3	3	no	—
27	#6899	—	3	3	no	—
28	#6897	—	3	3	no	—
29	#6896	—	3	3	no	—
30	#6895	—	3	1	no	—
31	#6895	—	3	1	no	—
32	#6895	—	3	1	no	—
33	#6895	C₃²	1	3	yes	complemented
34	#6895	C₃²	1	3	yes	complemented
35	#6894	—	3	3	no	—
36	#6894	—	3	3	no	—
37	#6894	—	3	1	no	—
38	#6894	—	3	1	no	—
39	#6894	—	3	1	no	—
40	#6894	—	3	1	no	—
41	#6894	C₃²	1	3	yes	complemented
42	#6894	C₃²	1	3	yes	complemented
43	#6894	C₃²	1	3	yes	complemented
44	#6894	C₃²	1	3	yes	complemented
45	#6894	C₃²	1	3	yes	complemented
46	C₉⋊H(3,𝔽₃)	—	3	3	no	—
47	C₉⋊H(3,𝔽₃)	—	3	3	no	—
48	C₉⋊H(3,𝔽₃)	C₃²	1	3	yes	—
49	C₉⋊H(3,𝔽₃)	C₃²	1	3	yes	—
50	#6891	—	3	1	no	—
51	#6891	—	3	1	no	—
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	#6891	C₃²	1	1	yes	characteristic
57	#6891	C₃²	1	1	yes	characteristic
58	C₉×3₋¹⁺²	—	3	1	no	—
59	C₉×3₋¹⁺²	—	3	1	no	—
60	C₉×3₋¹⁺²	C₃²	1	3	yes	complemented
61	C₉×H(3,𝔽₃)	C₃²	1	3	yes	—
62	#6887	C₃²	1	3	yes	complemented
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	#6862	C₃²	1	3	yes	complemented
69	#6862	C₃²	1	3	yes	complemented
70	#6862	C₃²	1	3	yes	complemented
71	#6861	C₃²	1	3	yes	complemented
72	#6861	C₃²	1	3	yes	complemented
73	#6861	C₃²	1	3	yes	complemented
74	#6859	C₃²	1	3	yes	—
75	#6859	C₃²	1	3	yes	—
76	#6859	C₃²	1	3	yes	—
77	#6859	C₃²	1	3	yes	—
78	#6859	C₃²	1	3	yes	—
79	#6859	C₃²	1	3	yes	—
80	#6858	C₃²	1	3	yes	complemented
81	#6858	C₃²	1	3	yes	complemented
82	#6858	C₃²	1	3	yes	complemented
83	#6858	C₃²	1	3	yes	complemented
84	#6858	C₃²	1	3	yes	complemented
85	#6858	C₃²	1	3	yes	complemented
86	#6858	C₃²	1	3	yes	complemented
87	#6858	C₃²	1	3	yes	complemented
88	#6858	C₃²	1	3	yes	complemented
89	#6858	C₃²	1	3	yes	complemented
90	#6858	C₃²	1	3	yes	complemented
91	#6858	C₃²	1	3	yes	complemented
92	#6857	C₃²	1	1	yes	characteristic
93	#6857	C₃²	1	1	yes	characteristic
94	#6857	C₃²	1	1	yes	characteristic
95	#6857	C₃²	1	1	yes	characteristic
96	#6857	C₃²	1	1	yes	characteristic
97	#6857	C₃²	1	1	yes	characteristic
98	#6857	C₃²	1	1	yes	characteristic
99	#6857	C₃²	1	1	yes	characteristic
100	#6857	C₃²	1	1	yes	characteristic
101	C₃⁴	—	3	1	no	—
102	C₃⁴	C₃³	1	1	yes	characteristic
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	1	no	—
110	C₃×3₋¹⁺²	—	3	1	no	—
111	C₃×3₋¹⁺²	—	3	1	no	—
112	C₃×3₋¹⁺²	—	3	1	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×H(3,𝔽₃)	—	3	1	no	—
124	C₃×H(3,𝔽₃)	—	3	1	no	—
125	C₃×H(3,𝔽₃)	—	9	3	no	—
126	C₃×H(3,𝔽₃)	C₃³	1	1	yes	characteristic
127	C₃×H(3,𝔽₃)	C₃³	1	1	yes	characteristic
128	C₃×H(3,𝔽₃)	C₃³	1	1	yes	characteristic
129	C₃²×C₉	—	3	1	no	—
130	C₃²×C₉	—	3	1	no	—
131	C₃²×C₉	—	3	1	no	—
132	C₃²×C₉	—	3	1	no	—
133	C₃²×C₉	—	3	1	no	—
134	C₃²×C₉	—	3	3	no	—
135	C₃²×C₉	C₃³	1	3	yes	—
136	C₉⋊C₉	—	3	3	no	—
137	C₉⋊C₉	—	3	3	no	—
138	C₉⋊C₉	—	3	3	no	—
139	C₉⋊C₉	—	3	3	no	—
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	C₉⋊C₉	—	3	3	no	—
158	C₉⋊C₉	—	3	3	no	—
159	C₉⋊C₉	—	3	3	no	—
160	C₉⋊C₉	—	3	3	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	1	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	H(3,𝔽₃)	1	1	yes	characteristic
186	C₃²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
187	C₃²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
188	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
189	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
190	C₃²⋊C₉	C₃³	1	1	yes	characteristic
191	C₃²⋊C₉	C₃³	1	1	yes	characteristic
192	C₃²⋊C₉	C₃³	1	1	yes	characteristic
193	C₃²⋊C₉	C₃³	1	3	yes	—
194	C₃²⋊C₉	C₃³	1	3	yes	—
195	C₃²⋊C₉	C₃³	1	3	yes	—
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	3	yes	—
200	C₃²⋊C₉	C₃³	1	3	yes	—
201	C₉²	—	3	3	no	—
202	C₉²	—	3	3	no	—
203	C₃³	—	3	1	no	—
204	C₃³	—	3	3	no	—
205	C₃³	—	3	1	no	—
206	C₃³	—	3	1	no	—
207	C₃³	—	9	1	no	—
208	C₃³	—	9	1	no	—
209	C₃³	—	9	1	no	—
210	C₃³	—	9	3	no	—
211	C₃³	—	9	3	no	—
212	C₃³	—	27	1	no	—
213	C₃³	—	27	1	no	—
214	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
215	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
216	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
217	C₃³	C₃⁴	1	1	yes	characteristic
218	3₋¹⁺²	—	3	3	no	—
219	3₋¹⁺²	—	9	3	no	—
220	3₋¹⁺²	—	9	3	no	—
221	3₋¹⁺²	—	9	3	no	—
222	3₋¹⁺²	—	9	3	no	—
223	3₋¹⁺²	—	9	3	no	—
224	3₋¹⁺²	—	9	3	no	—
225	3₋¹⁺²	—	9	1	no	—
226	3₋¹⁺²	—	9	1	no	—
227	3₋¹⁺²	—	9	1	no	—
228	3₋¹⁺²	—	9	1	no	—
229	3₋¹⁺²	—	9	1	no	—
230	3₋¹⁺²	—	27	3	no	—
231	3₋¹⁺²	—	27	3	no	—
232	3₋¹⁺²	—	27	1	no	—
233	3₋¹⁺²	—	27	1	no	—
234	3₋¹⁺²	—	27	1	no	—
235	3₋¹⁺²	—	27	1	no	—
236	H(3,𝔽₃)	—	3	3	no	—
237	H(3,𝔽₃)	—	9	3	no	—
238	H(3,𝔽₃)	—	9	3	no	—
239	H(3,𝔽₃)	—	9	1	no	—
240	H(3,𝔽₃)	—	9	1	no	—
241	C₃×C₉	—	3	1	no	—
242	C₃×C₉	—	3	1	no	—
243	C₃×C₉	—	3	3	no	—
244	C₃×C₉	—	3	3	no	—
245	C₃×C₉	—	3	3	no	—
246	C₃×C₉	—	3	3	no	—
247	C₃×C₉	—	3	3	no	—
248	C₃×C₉	—	3	3	no	—
249	C₃×C₉	—	3	3	no	—
250	C₃×C₉	—	3	3	no	—
251	C₃×C₉	—	3	1	no	—
252	C₃×C₉	—	3	1	no	—
253	C₃×C₉	—	3	1	no	—
254	C₃×C₉	—	3	1	no	—
255	C₃×C₉	—	3	1	no	—
256	C₃×C₉	—	3	1	no	—
257	C₃×C₉	—	3	1	no	—
258	C₃×C₉	—	9	3	no	—
259	C₃×C₉	—	9	3	no	—
260	C₃×C₉	—	9	3	no	—
261	C₃×C₉	—	27	3	no	—
262	C₃×C₉	—	27	1	no	—
263	C₃×C₉	—	27	1	no	—
264	C₃×C₉	—	27	1	no	—
265	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
266	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
267	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
268	C₃²	—	3	1	no	—
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	3	no	—
281	C₃²	—	3	1	no	—
282	C₃²	—	3	1	no	—
283	C₃²	—	3	1	no	—
284	C₃²	—	9	1	no	—
285	C₃²	—	9	1	no	—
286	C₃²	—	9	1	no	—
287	C₃²	—	9	1	no	—
288	C₃²	—	9	1	no	—
289	C₃²	—	9	1	no	—
290	C₃²	—	9	1	no	—
291	C₃²	—	9	1	no	—
292	C₃²	—	9	1	no	—
293	C₃²	—	9	1	no	—
294	C₃²	—	9	1	no	—
295	C₃²	—	27	3	no	—
296	C₃²	—	27	9	no	—
297	C₃²	#6913	1	3	yes	—
298	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
299	C₉	—	3	3	no	—
300	C₉	—	3	3	no	—
301	C₉	—	3	3	no	—
302	C₉	—	3	3	no	—
303	C₉	—	9	1	no	—
304	C₉	—	9	1	no	—
305	C₉	—	9	1	no	—
306	C₉	—	9	3	no	—
307	C₉	—	9	3	no	—
308	C₉	—	9	3	no	—
309	C₉	—	9	3	no	—
310	C₉	—	9	1	no	—
311	C₉	—	9	1	no	—
312	C₉	—	9	3	no	—
313	C₉	—	9	1	no	—
314	C₉	—	9	3	no	—
315	C₉	—	9	1	no	—
316	C₉	—	9	1	no	—
317	C₉	—	9	3	no	—
318	C₉	—	9	3	no	—
319	C₃	—	3	3	no	—
320	C₃	—	9	1	no	—
321	C₃	—	9	1	no	—
322	C₃	—	9	1	no	—
323	C₃	—	9	3	no	—
324	C₃	—	27	1	no	—
325	C₃	—	27	1	no	—
326	C₃	C₃×#6913	1	1	yes	characteristic
327	C₃	#10624931	1	1	yes	characteristic
328	C₃	#10624934	1	1	yes	characteristic
329	C₃	#10624934	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃²×H(3,𝔽₃)	C₃²
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. #10624903	C₃	semidirect
2. #10624899	C₃	semidirect
4. #10624886	C₃	semidirect
5. #10624676	C₃	semidirect
6. #10624676	C₃	semidirect
7. #10624676	C₃	semidirect
8. #10624653	C₃	semidirect
9. #10624652	C₃	semidirect
10. #10624645	C₃	semidirect
11. #10624631	C₃	semidirect
12. #10624631	C₃	semidirect
13. #10624624	C₃	semidirect
14. #10624621	C₃	semidirect
15. #10624619	C₃	semidirect
16. #10624614	C₃	semidirect
17. #10624611	C₃	semidirect
18. #10624606	C₃	semidirect
21. #10624592	C₃	semidirect
33. #6895	C₃²	semidirect
34. #6895	C₃²	semidirect
41. #6894	C₃²	semidirect
42. #6894	C₃²	semidirect
43. #6894	C₃²	semidirect
44. #6894	C₃²	semidirect
45. #6894	C₃²	semidirect
60. C₉×3₋¹⁺²	C₃²	semidirect
62. #6887	C₃²	semidirect
68. #6862	C₃²	semidirect
69. #6862	C₃²	semidirect
70. #6862	C₃²	semidirect
71. #6861	C₃²	semidirect
72. #6861	C₃²	semidirect
73. #6861	C₃²	semidirect
80. #6858	C₃²	semidirect
81. #6858	C₃²	semidirect
82. #6858	C₃²	semidirect
83. #6858	C₃²	semidirect
84. #6858	C₃²	semidirect
85. #6858	C₃²	semidirect
86. #6858	C₃²	semidirect
87. #6858	C₃²	semidirect
88. #6858	C₃²	semidirect
89. #6858	C₃²	semidirect
90. #6858	C₃²	semidirect
91. #6858	C₃²	semidirect

Maximal subgroups

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

subgroup	index
#10624903	3
#10624899	3
#10624897	3
#10624886	3
#10624676	3
#10624653	3
#10624652	3
#10624645	3
#10624631	3
#10624624	3
#10624621	3
#10624619	3
#10624614	3
#10624611	3
#10624606	3
#10624602	3
#10624593	3
#10624592	3
C₃×#6857	3