Subgroups — C₃×#10624908 #14555301

Order 2187.

3458 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624908	C₃	1	81	yes	maximal, direct product factor
2	C₃×#6900	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#6899	C₃	1	2	yes	maximal, complemented
4	C₃×#6899	C₃	1	2	yes	maximal, complemented
5	C₃×#6899	C₃	1	2	yes	maximal, complemented
6	C₃×#6899	C₃	1	2	yes	maximal, complemented
7	C₃×#6898	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#6898	C₃	1	2	yes	maximal, complemented
9	C₃×#6897	C₃	1	2	yes	maximal, complemented
10	C₃×#6897	C₃	1	2	yes	maximal, complemented
11	C₃×#6897	C₃	1	2	yes	maximal, complemented
12	C₃×#6896	C₃	1	1	yes	maximal, characteristic, complemented
13	C₃×#6896	C₃	1	2	yes	maximal, complemented
14	C₃×#6896	C₃	1	2	yes	maximal, complemented
15	C₃×#6896	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃×#6895	C₃	1	1	yes	maximal, characteristic, complemented
17	C₃×#6894	C₃	1	2	yes	maximal, complemented
18	C₃×#6894	C₃	1	1	yes	maximal, characteristic, complemented
19	C₃×#6893	C₃	1	1	yes	maximal, characteristic
20	C₃×C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal
21	C₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
22	C₃×C₉×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
23	C₃×C₉×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
24	C₃×C₉×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
25	C₃×#6887	C₃	1	2	yes	maximal, complemented
26	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
27	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
28	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
29	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
30	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
31	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
32	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
33	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
34	C₃³×C₉	C₃²	1	2	yes	—
35	C₃³×C₉	C₃²	1	1	yes	characteristic
36	C₃³×C₉	C₃²	1	1	yes	characteristic
37	#6900	C₃²	1	27	yes	complemented
38	#6899	C₃²	1	54	yes	complemented
39	#6899	C₃²	1	54	yes	complemented
40	#6899	C₃²	1	54	yes	complemented
41	#6899	C₃²	1	54	yes	complemented
42	#6898	C₃²	1	27	yes	complemented
43	#6898	C₃²	1	54	yes	complemented
44	#6897	C₃²	1	54	yes	complemented
45	#6897	C₃²	1	54	yes	complemented
46	#6897	C₃²	1	54	yes	complemented
47	#6896	C₃²	1	27	yes	complemented
48	#6896	C₃²	1	54	yes	complemented
49	#6896	C₃²	1	54	yes	complemented
50	#6896	C₃²	1	27	yes	complemented
51	#6895	C₃²	1	27	yes	complemented
52	#6894	C₃²	1	27	yes	complemented
53	#6894	C₃²	1	54	yes	complemented
54	#6893	C₃²	1	27	yes	—
55	C₉⋊H(3,𝔽₃)	C₃²	1	27	yes	—
56	C₉⋊H(3,𝔽₃)	C₃²	1	54	yes	—
57	C₉×3₋¹⁺²	C₃²	1	54	yes	complemented
58	C₉×3₋¹⁺²	C₃²	1	54	yes	complemented
59	C₉×3₋¹⁺²	C₃²	1	54	yes	complemented
60	#6887	C₃²	1	54	yes	complemented
61	C₃×C₉⋊C₉	C₃²	1	2	yes	—
62	C₃×C₉⋊C₉	C₃²	1	2	yes	—
63	C₃×C₉⋊C₉	C₃²	1	2	yes	—
64	C₃×C₉⋊C₉	C₃²	1	2	yes	—
65	C₃×C₉⋊C₉	C₃²	1	2	yes	—
66	C₃×C₉⋊C₉	C₃²	1	2	yes	—
67	C₃×C₉⋊C₉	C₃²	1	2	yes	—
68	C₃×C₉⋊C₉	C₃²	1	2	yes	—
69	C₃×C₉⋊C₉	C₃²	1	2	yes	—
70	C₃×C₉⋊C₉	C₃²	1	2	yes	—
71	C₃×C₉⋊C₉	C₃²	1	2	yes	—
72	C₃×C₉⋊C₉	C₃²	1	2	yes	—
73	C₃×C₉⋊C₉	C₃²	1	2	yes	—
74	C₃×C₉⋊C₉	C₃²	1	2	yes	—
75	C₃×C₉⋊C₉	C₃²	1	2	yes	—
76	C₃×C₉⋊C₉	C₃²	1	2	yes	—
77	C₃×C₉⋊C₉	C₃²	1	2	yes	—
78	C₃×C₉⋊C₉	C₃²	1	2	yes	—
79	C₃×C₉⋊C₉	C₃²	1	2	yes	—
80	C₃×C₉⋊C₉	C₃²	1	2	yes	—
81	C₃×C₉⋊C₉	C₃²	1	2	yes	—
82	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
83	C₃×C₉⋊C₉	C₃²	1	2	yes	—
84	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
85	C₃×C₉⋊C₉	C₃²	1	2	yes	—
86	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
87	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
88	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
89	C₃×C₉⋊C₉	C₃²	1	2	yes	—
90	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
91	C₃×C₉⋊C₉	C₃²	1	2	yes	—
92	C₃×C₉⋊C₉	C₃²	1	2	yes	—
93	C₃×C₉⋊C₉	C₃²	1	2	yes	—
94	C₃×C₉⋊C₉	C₃²	1	2	yes	—
95	C₃×C₉⋊C₉	C₃²	1	2	yes	—
96	C₃×C₉⋊C₉	C₃²	1	2	yes	—
97	C₃×C₉⋊C₉	C₃²	1	2	yes	—
98	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
99	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
100	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
101	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
102	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
103	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
104	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
105	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
106	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
107	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
108	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
109	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
110	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
111	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
112	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
113	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
114	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
115	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
116	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
117	C₃×C₉²	C₃²	1	1	yes	characteristic
118	C₃×C₉²	C₃²	1	2	yes	—
119	C₃×C₉²	C₃²	1	2	yes	—
120	C₃×C₉²	C₃²	1	2	yes	—
121	C₃×C₉²	C₃²	1	2	yes	—
122	C₃×C₉²	C₃²	1	1	yes	characteristic
123	C₃×C₉²	C₃²	1	2	yes	—
124	C₃⁴	C₃³	1	1	yes	characteristic
125	C₃⁴	C₃³	1	2	yes	—
126	C₃⁴	C₃³	1	1	yes	characteristic
127	C₃×3₋¹⁺²	—	3	18	no	—
128	C₃×3₋¹⁺²	—	3	18	no	—
129	C₃×3₋¹⁺²	—	3	18	no	—
130	C₃×3₋¹⁺²	—	9	6	no	—
131	C₃×3₋¹⁺²	—	9	3	no	—
132	C₃×3₋¹⁺²	—	9	3	no	—
133	C₃×3₋¹⁺²	—	9	3	no	—
134	C₃×3₋¹⁺²	C₃³	1	18	yes	—
135	C₃×3₋¹⁺²	C₃³	1	18	yes	—
136	C₃×3₋¹⁺²	C₃³	1	18	yes	—
137	C₃×3₋¹⁺²	C₃³	1	18	yes	—
138	C₃×3₋¹⁺²	C₃³	1	9	yes	—
139	C₃×3₋¹⁺²	C₃³	1	9	yes	—
140	C₃×3₋¹⁺²	C₃³	1	9	yes	—
141	C₃×H(3,𝔽₃)	—	9	3	no	—
142	C₃×H(3,𝔽₃)	C₃³	1	9	yes	—
143	C₃²×C₉	—	3	18	no	—
144	C₃²×C₉	—	9	3	no	—
145	C₃²×C₉	—	9	3	no	—
146	C₃²×C₉	C₃³	1	18	yes	—
147	C₃²×C₉	C₃³	1	1	yes	characteristic
148	C₃²×C₉	C₃³	1	2	yes	—
149	C₃²×C₉	C₃³	1	2	yes	—
150	C₃²×C₉	C₃³	1	2	yes	—
151	C₃²×C₉	C₃³	1	2	yes	—
152	C₃²×C₉	C₃³	1	1	yes	characteristic
153	C₃²×C₉	C₃³	1	2	yes	—
154	C₃²×C₉	C₃³	1	2	yes	—
155	C₃²×C₉	C₃³	1	2	yes	—
156	C₃²×C₉	C₃³	1	2	yes	—
157	C₃²×C₉	C₃³	1	2	yes	—
158	C₃²×C₉	C₃³	1	2	yes	—
159	C₃²×C₉	C₃³	1	2	yes	—
160	C₃²×C₉	C₃³	1	2	yes	—
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	2	yes	—
165	C₃²×C₉	C₃³	1	2	yes	—
166	C₃²×C₉	C₃³	1	1	yes	characteristic
167	C₃²×C₉	C₃³	1	9	yes	—
168	C₃²×C₉	C₃³	1	9	yes	—
169	C₃²×C₉	C₃³	1	2	yes	—
170	C₉⋊C₉	C₃³	1	18	yes	—
171	C₉⋊C₉	C₃³	1	18	yes	—
172	C₉⋊C₉	C₃³	1	18	yes	—
173	C₉⋊C₉	C₃³	1	18	yes	—
174	C₉⋊C₉	C₃³	1	18	yes	—
175	C₉⋊C₉	C₃³	1	18	yes	—
176	C₉⋊C₉	C₃³	1	18	yes	—
177	C₉⋊C₉	C₃³	1	18	yes	—
178	C₉⋊C₉	C₃³	1	18	yes	—
179	C₉⋊C₉	C₃³	1	18	yes	—
180	C₉⋊C₉	C₃³	1	18	yes	—
181	C₉⋊C₉	C₃³	1	18	yes	—
182	C₉⋊C₉	C₃³	1	18	yes	—
183	C₉⋊C₉	C₃³	1	18	yes	—
184	C₉⋊C₉	C₃³	1	18	yes	—
185	C₉⋊C₉	C₃³	1	18	yes	—
186	C₉⋊C₉	C₃³	1	18	yes	—
187	C₉⋊C₉	C₃³	1	18	yes	—
188	C₉⋊C₉	C₃³	1	18	yes	—
189	C₉⋊C₉	C₃³	1	18	yes	—
190	C₉⋊C₉	C₃³	1	18	yes	—
191	C₉⋊C₉	C₃³	1	9	yes	—
192	C₉⋊C₉	C₃³	1	9	yes	—
193	C₉⋊C₉	C₃³	1	18	yes	—
194	C₉⋊C₉	C₃³	1	9	yes	—
195	C₉⋊C₉	C₃³	1	18	yes	—
196	C₉⋊C₉	C₃³	1	9	yes	—
197	C₉⋊C₉	C₃³	1	9	yes	—
198	C₉⋊C₉	C₃³	1	18	yes	—
199	C₉⋊C₉	C₃³	1	9	yes	—
200	C₉⋊C₉	C₃³	1	18	yes	—
201	C₉⋊C₉	C₃³	1	18	yes	—
202	C₉⋊C₉	C₃³	1	18	yes	—
203	C₉⋊C₉	C₃³	1	18	yes	—
204	C₉⋊C₉	C₃³	1	18	yes	—
205	C₉⋊C₉	C₃³	1	18	yes	—
206	C₉⋊C₉	C₃³	1	18	yes	—
207	C₉⋊C₉	C₃³	1	9	yes	—
208	C₃²⋊C₉	C₃³	1	18	yes	—
209	C₃²⋊C₉	C₃³	1	18	yes	—
210	C₃²⋊C₉	C₃³	1	18	yes	—
211	C₃²⋊C₉	C₃³	1	18	yes	—
212	C₃²⋊C₉	C₃³	1	18	yes	—
213	C₃²⋊C₉	C₃³	1	18	yes	—
214	C₃²⋊C₉	C₃³	1	18	yes	—
215	C₃²⋊C₉	C₃³	1	18	yes	—
216	C₃²⋊C₉	C₃³	1	18	yes	—
217	C₃²⋊C₉	C₃³	1	18	yes	—
218	C₃²⋊C₉	C₃³	1	18	yes	—
219	C₃²⋊C₉	C₃³	1	18	yes	—
220	C₃²⋊C₉	C₃³	1	18	yes	—
221	C₃²⋊C₉	C₃³	1	9	yes	—
222	C₃²⋊C₉	C₃³	1	18	yes	—
223	C₃²⋊C₉	C₃³	1	9	yes	—
224	C₃²⋊C₉	C₃³	1	18	yes	—
225	C₃²⋊C₉	C₃³	1	9	yes	—
226	C₉²	C₃³	1	9	yes	—
227	C₉²	C₃³	1	18	yes	—
228	C₉²	C₃³	1	18	yes	—
229	C₉²	C₃³	1	18	yes	—
230	C₉²	C₃³	1	18	yes	—
231	C₉²	C₃³	1	9	yes	—
232	C₉²	C₃³	1	18	yes	—
233	C₃³	—	3	6	no	—
234	C₃³	—	3	6	no	—
235	C₃³	—	3	6	no	—
236	C₃³	—	3	6	no	—
237	C₃³	—	3	3	no	—
238	C₃³	—	3	3	no	—
239	C₃³	—	3	6	no	—
240	C₃³	—	3	3	no	—
241	C₃³	—	3	3	no	—
242	C₃³	—	3	6	no	—
243	C₃³	C₃⁴	1	6	yes	—
244	C₃³	C₃⁴	1	3	yes	—
245	C₃³	C₃⁴	1	3	yes	—
246	C₃³	C₃⁴	1	1	yes	characteristic
247	3₋¹⁺²	—	3	54	no	—
248	3₋¹⁺²	—	3	54	no	—
249	3₋¹⁺²	—	3	54	no	—
250	3₋¹⁺²	—	9	18	no	—
251	3₋¹⁺²	—	9	9	no	—
252	3₋¹⁺²	—	9	9	no	—
253	3₋¹⁺²	—	9	9	no	—
254	H(3,𝔽₃)	—	9	9	no	—
255	C₃×C₉	—	3	6	no	—
256	C₃×C₉	—	3	6	no	—
257	C₃×C₉	—	3	6	no	—
258	C₃×C₉	—	3	6	no	—
259	C₃×C₉	—	3	6	no	—
260	C₃×C₉	—	3	6	no	—
261	C₃×C₉	—	3	6	no	—
262	C₃×C₉	—	3	6	no	—
263	C₃×C₉	—	3	6	no	—
264	C₃×C₉	—	3	6	no	—
265	C₃×C₉	—	3	3	no	—
266	C₃×C₉	—	3	3	no	—
267	C₃×C₉	—	3	3	no	—
268	C₃×C₉	—	3	6	no	—
269	C₃×C₉	—	3	6	no	—
270	C₃×C₉	—	3	3	no	—
271	C₃×C₉	—	3	6	no	—
272	C₃×C₉	—	3	6	no	—
273	C₃×C₉	—	3	3	no	—
274	C₃×C₉	—	3	3	no	—
275	C₃×C₉	—	3	54	no	—
276	C₃×C₉	—	9	9	no	—
277	C₃×C₉	—	9	9	no	—
278	C₃×C₉	#737	1	18	yes	—
279	C₃×C₉	C₃⁴	1	6	yes	—
280	C₃×C₉	C₃⁴	1	6	yes	—
281	C₃×C₉	C₃⁴	1	6	yes	—
282	C₃×C₉	C₃⁴	1	6	yes	—
283	C₃×C₉	C₃⁴	1	6	yes	—
284	C₃×C₉	C₃⁴	1	6	yes	—
285	C₃×C₉	C₃⁴	1	6	yes	—
286	C₃×C₉	C₃⁴	1	6	yes	—
287	C₃×C₉	C₃⁴	1	6	yes	—
288	C₃×C₉	C₃⁴	1	3	yes	—
289	C₃×C₉	C₃⁴	1	3	yes	—
290	C₃×C₉	C₃⁴	1	6	yes	—
291	C₃×C₉	C₃⁴	1	6	yes	—
292	C₃×C₉	C₃⁴	1	6	yes	—
293	C₃×C₉	C₃⁴	1	6	yes	—
294	C₃×C₉	C₃⁴	1	3	yes	—
295	C₃×C₉	C₃⁴	1	3	yes	—
296	C₃×C₉	C₃⁴	1	6	yes	—
297	C₃×C₉	C₃⁴	1	3	yes	—
298	C₃×C₉	C₃⁴	1	3	yes	—
299	C₃×C₉	C₃⁴	1	6	yes	—
300	C₃²	—	3	6	no	—
301	C₃²	—	3	6	no	—
302	C₃²	—	3	6	no	—
303	C₃²	—	3	6	no	—
304	C₃²	—	3	3	no	—
305	C₃²	—	3	3	no	—
306	C₃²	—	3	6	no	—
307	C₃²	—	3	3	no	—
308	C₃²	—	3	6	no	—
309	C₃²	—	3	3	no	—
310	C₃²	—	9	18	no	—
311	C₃²	—	9	9	no	—
312	C₃²	—	9	9	no	—
313	C₃²	C₃×#737	1	6	yes	—
314	C₃²	3₋¹⁺⁴	1	3	yes	—
315	C₃²	3₋¹⁺⁴	1	3	yes	—
316	C₃²	C₃⁵	1	1	yes	characteristic
317	C₉	—	3	6	no	—
318	C₉	—	3	6	no	—
319	C₉	—	3	6	no	—
320	C₉	—	3	6	no	—
321	C₉	—	3	6	no	—
322	C₉	—	3	6	no	—
323	C₉	—	3	6	no	—
324	C₉	—	3	6	no	—
325	C₉	—	3	6	no	—
326	C₉	—	3	6	no	—
327	C₉	—	3	3	no	—
328	C₉	—	3	3	no	—
329	C₉	—	3	3	no	—
330	C₉	—	3	6	no	—
331	C₉	—	3	6	no	—
332	C₉	—	3	3	no	—
333	C₉	—	3	3	no	—
334	C₉	—	3	6	no	—
335	C₉	—	3	6	no	—
336	C₉	—	3	3	no	—
337	C₉	C₃×#737	1	18	yes	—
338	C₃	—	9	6	no	—
339	C₃	—	9	3	no	—
340	C₃	—	9	3	no	—
341	C₃	#10624908	1	9	yes	direct product factor
342	C₃	C₃²×#737	1	2	yes	—
343	C₃	C₃×3₋¹⁺⁴	1	1	yes	characteristic
344	C₃	C₃×3₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃³	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁵
prime radical	C₃²×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #10624908	C₃	direct (N × H)
2. C₃×#6900	C₃	semidirect
3. C₃×#6899	C₃	semidirect
4. C₃×#6899	C₃	semidirect
5. C₃×#6899	C₃	semidirect
6. C₃×#6899	C₃	semidirect
7. C₃×#6898	C₃	semidirect
8. C₃×#6898	C₃	semidirect
9. C₃×#6897	C₃	semidirect
10. C₃×#6897	C₃	semidirect
11. C₃×#6897	C₃	semidirect
12. C₃×#6896	C₃	semidirect
13. C₃×#6896	C₃	semidirect
14. C₃×#6896	C₃	semidirect
15. C₃×#6896	C₃	semidirect
16. C₃×#6895	C₃	semidirect
17. C₃×#6894	C₃	semidirect
18. C₃×#6894	C₃	semidirect
22. C₃×C₉×3₋¹⁺²	C₃	semidirect
23. C₃×C₉×3₋¹⁺²	C₃	semidirect
24. C₃×C₉×3₋¹⁺²	C₃	semidirect
25. C₃×#6887	C₃	semidirect
37. #6900	C₃²	semidirect
38. #6899	C₃²	semidirect
39. #6899	C₃²	semidirect
40. #6899	C₃²	semidirect
41. #6899	C₃²	semidirect
42. #6898	C₃²	semidirect
43. #6898	C₃²	semidirect
44. #6897	C₃²	semidirect
45. #6897	C₃²	semidirect
46. #6897	C₃²	semidirect
47. #6896	C₃²	semidirect
48. #6896	C₃²	semidirect
49. #6896	C₃²	semidirect
50. #6896	C₃²	semidirect
51. #6895	C₃²	semidirect
52. #6894	C₃²	semidirect
53. #6894	C₃²	semidirect
57. C₉×3₋¹⁺²	C₃²	semidirect
58. C₉×3₋¹⁺²	C₃²	semidirect
59. C₉×3₋¹⁺²	C₃²	semidirect
60. #6887	C₃²	semidirect
341. C₃	#10624908	direct (N × H)

Maximal subgroups

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

subgroup	index
#10624908	3
C₃×#6900	3
C₃×#6899	3
C₃×#6898	3
C₃×#6897	3
C₃×#6896	3
C₃×#6895	3
C₃×#6894	3
C₃×#6893	3
C₃×C₉⋊H(3,𝔽₃)	3
C₃×C₉×3₋¹⁺²	3
C₃×#6887	3

Subgroups — C3×#10624908 #14555301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10624908 #14555301