Subgroups — C₃×C₉×#390 #13144928

Order 1728.

1290 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₉×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
4	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#390	C₃	1	1	yes	maximal, characteristic
6	C₉×#390	C₃	1	3	yes	maximal, direct product factor
7	C₂⁴×C₃×C₉	C₂²	1	2	yes	—
8	C₂×C₃×D₄×C₉	—	2	2	no	—
9	C₂×C₃×D₄×C₉	—	2	2	no	—
10	C₂×C₃×D₄×C₉	C₄	1	4	yes	complemented
11	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
12	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
13	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	—	2	2	no	—
15	C₃×C₉×C₂²⋊C₄	—	2	8	no	—
16	C₃×C₄²×C₉	—	2	2	no	—
17	C₂²×C₃²×D₄	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	4	yes	—
20	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
21	C₂²×D₄×C₉	C₂×C₃	1	3	yes	—
22	C₂×C₉×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
23	C₂×C₉×C₂²⋊C₄	C₂×C₃	1	12	yes	complemented
24	C₂×C₄²×C₉	C₂×C₃	1	3	yes	complemented
25	C₂³×C₃×C₉	—	2	2	no	—
26	C₂³×C₃×C₉	—	2	2	no	—
27	C₂³×C₃×C₉	—	2	4	no	—
28	C₂³×C₃×C₉	—	2	4	no	—
29	C₂³×C₃×C₉	C₂×C₄	1	4	yes	—
30	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
31	C₃×D₄×C₉	—	2	8	no	—
32	C₂×C₃×C₄×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	—	2	2	no	—
34	C₂×C₃×C₄×C₉	—	2	4	no	—
35	C₂×C₃×C₄×C₉	—	2	2	no	—
36	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
38	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
39	C₂×C₃×C₄×C₉	D₄	1	4	yes	complemented
40	C₃×#390	C₉	1	3	yes	direct product factor
41	C₃×#390	C₃²	1	1	yes	characteristic
42	C₂⁴×C₃²	C₂²×C₃	1	2	yes	—
43	C₂×C₃²×D₄	—	2	2	no	—
44	C₂×C₃²×D₄	—	2	2	no	—
45	C₂×C₃²×D₄	C₃×C₄	1	4	yes	—
46	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
47	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
48	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂⁴×C₉	C₂²×C₃	1	6	yes	—
50	C₃²×C₄⋊C₄	—	2	2	no	—
51	C₃²×C₂²⋊C₄	—	2	8	no	—
52	C₃²×C₄²	—	2	2	no	—
53	C₂×D₄×C₉	—	2	6	no	—
54	C₂×D₄×C₉	—	2	6	no	—
55	C₂×D₄×C₉	C₃×C₄	1	12	yes	complemented
56	C₂²×C₄×C₉	C₂²×C₃	1	6	yes	—
57	C₂²×C₄×C₉	C₂²×C₃	1	6	yes	—
58	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
59	C₉×C₄⋊C₄	—	2	6	no	—
60	C₉×C₂²⋊C₄	—	2	24	no	—
61	C₄²×C₉	—	2	6	no	—
62	C₂²×C₃×C₉	—	2	4	no	—
63	C₂²×C₃×C₉	—	2	8	no	—
64	C₂²×C₃×C₉	—	4	4	no	—
65	C₂²×C₃×C₉	—	4	2	no	—
66	C₂²×C₃×C₉	—	4	2	no	—
67	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
71	C₂²×C₃×C₉	#41	1	2	yes	—
72	C₃×C₄×C₉	—	2	4	no	—
73	C₃×C₄×C₉	—	4	2	no	—
74	C₃×C₄×C₉	C₂²⋊C₄	1	4	yes	complemented
75	C₂²×C₃×D₄	C₂×C₉	1	3	yes	—
76	C₂²×C₃×D₄	C₂×C₃²	1	1	yes	characteristic
77	C₂×C₃×C₄⋊C₄	C₂×C₉	1	3	yes	complemented
78	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic
79	C₂×C₃×C₂²⋊C₄	C₂×C₉	1	12	yes	complemented
80	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	4	yes	—
81	C₂×C₃×C₄²	C₂×C₉	1	3	yes	complemented
82	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
83	C₂³×C₃²	—	2	4	no	—
84	C₂³×C₃²	—	2	2	no	—
85	C₂³×C₃²	—	2	2	no	—
86	C₂³×C₃²	—	2	4	no	—
87	C₂³×C₃²	C₂×C₃×C₄	1	4	yes	—
88	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
89	C₃²×D₄	—	2	8	no	—
90	C₂×C₃²×C₄	—	2	2	no	—
91	C₂×C₃²×C₄	—	2	2	no	—
92	C₂×C₃²×C₄	—	2	4	no	—
93	C₂×C₃²×C₄	—	2	2	no	—
94	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
95	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
96	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
97	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
98	C₂³×C₉	—	2	12	no	—
99	C₂³×C₉	—	2	6	no	—
100	C₂³×C₉	—	2	6	no	—
101	C₂³×C₉	—	2	12	no	—
102	C₂³×C₉	C₂×C₃×C₄	1	12	yes	—
103	C₂³×C₉	C₂³×C₃	1	3	yes	—
104	D₄×C₉	—	2	24	no	—
105	C₂×C₄×C₉	—	2	6	no	—
106	C₂×C₄×C₉	—	2	6	no	—
107	C₂×C₄×C₉	—	2	12	no	—
108	C₂×C₄×C₉	—	2	6	no	—
109	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
110	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
111	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
112	C₂×C₄×C₉	C₃×D₄	1	12	yes	complemented
113	#390	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×C₃×C₉	—	4	4	no	—
115	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₃×C₉	C₄×D₄	1	2	yes	—
117	C₂×C₃×C₉	#121	1	2	yes	—
118	C₂×C₃×C₉	#124	1	1	yes	characteristic
119	C₂×C₃×C₉	#127	1	1	yes	characteristic
120	C₂⁴×C₃	C₂²×C₉	1	6	yes	—
121	C₂⁴×C₃	C₂²×C₃²	1	2	yes	—
122	C₂×C₃×D₄	—	2	6	no	—
123	C₂×C₃×D₄	—	2	6	no	—
124	C₂×C₃×D₄	—	2	2	no	—
125	C₂×C₃×D₄	—	2	2	no	—
126	C₂×C₃×D₄	C₄×C₉	1	12	yes	complemented
127	C₂×C₃×D₄	C₃²×C₄	1	4	yes	—
128	C₂²×C₃×C₄	C₂²×C₉	1	6	yes	—
129	C₂²×C₃×C₄	C₂²×C₉	1	6	yes	—
130	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
131	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	—
132	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	—
133	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	2	6	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₂²⋊C₄	—	2	24	no	—
137	C₃×C₂²⋊C₄	—	2	8	no	—
138	C₃×C₄²	—	2	6	no	—
139	C₃×C₄²	—	2	2	no	—
140	C₂²×C₃²	—	2	4	no	—
141	C₂²×C₃²	—	2	8	no	—
142	C₂²×C₃²	—	4	4	no	—
143	C₂²×C₃²	—	4	2	no	—
144	C₂²×C₃²	—	4	2	no	—
145	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
147	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
149	C₂²×C₃²	C₃×#41	1	2	yes	—
150	C₃²×C₄	—	2	4	no	—
151	C₃²×C₄	—	4	2	no	—
152	C₃²×C₄	C₃×C₂²⋊C₄	1	4	yes	—
153	C₂²×C₉	—	2	12	no	—
154	C₂²×C₉	—	2	24	no	—
155	C₂²×C₉	—	4	12	no	—
156	C₂²×C₉	—	4	6	no	—
157	C₂²×C₉	—	4	6	no	—
158	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
159	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
160	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
161	C₂²×C₉	C₂×C₃×D₄	1	6	yes	—
162	C₂²×C₉	C₃×#41	1	6	yes	—
163	C₄×C₉	—	2	12	no	—
164	C₄×C₉	—	4	6	no	—
165	C₄×C₉	C₃×C₂²⋊C₄	1	12	yes	complemented
166	C₂²×D₄	C₂×C₃×C₉	1	1	yes	characteristic
167	C₂×C₄⋊C₄	C₂×C₃×C₉	1	1	yes	characteristic, complemented
168	C₂×C₂²⋊C₄	C₂×C₃×C₉	1	4	yes	complemented
169	C₂×C₄²	C₂×C₃×C₉	1	1	yes	characteristic, complemented
170	C₃×C₉	#390	1	1	yes	characteristic, direct product factor, Sylow
171	C₂³×C₃	—	2	12	no	—
172	C₂³×C₃	—	2	6	no	—
173	C₂³×C₃	—	2	6	no	—
174	C₂³×C₃	—	2	12	no	—
175	C₂³×C₃	—	2	4	no	—
176	C₂³×C₃	—	2	2	no	—
177	C₂³×C₃	—	2	2	no	—
178	C₂³×C₃	—	2	4	no	—
179	C₂³×C₃	C₂×C₄×C₉	1	12	yes	—
180	C₂³×C₃	C₂³×C₉	1	3	yes	—
181	C₂³×C₃	C₂×C₃²×C₄	1	4	yes	—
182	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
183	C₃×D₄	—	2	24	no	—
184	C₃×D₄	—	2	8	no	—
185	C₂×C₃×C₄	—	2	6	no	—
186	C₂×C₃×C₄	—	2	6	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	12	no	—
190	C₂×C₃×C₄	—	2	6	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
194	C₂×C₃×C₄	D₄×C₉	1	6	yes	—
195	C₂×C₃×C₄	D₄×C₉	1	6	yes	—
196	C₂×C₃×C₄	D₄×C₉	1	12	yes	complemented
197	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
198	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
199	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
200	C₂×C₃×C₄	C₃²×D₄	1	4	yes	—
201	C₂×C₃²	—	4	4	no	—
202	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₃²	C₃×C₄×D₄	1	2	yes	—
204	C₂×C₃²	C₃×#121	1	2	yes	—
205	C₂×C₃²	C₃×#124	1	1	yes	characteristic
206	C₂×C₃²	C₃×#127	1	1	yes	characteristic
207	C₂×C₉	—	4	12	no	—
208	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	3	yes	—
209	C₂×C₉	C₃×C₄×D₄	1	6	yes	—
210	C₂×C₉	C₃×#121	1	6	yes	—
211	C₂×C₉	C₃×#124	1	3	yes	—
212	C₂×C₉	C₃×#127	1	3	yes	—
213	C₂⁴	C₂²×C₃×C₉	1	2	yes	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	C₃×C₄×C₉	1	4	yes	complemented
217	C₂²×C₄	C₂²×C₃×C₉	1	2	yes	—
218	C₂²×C₄	C₂²×C₃×C₉	1	2	yes	—
219	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
220	C₄⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	8	no	—
222	C₄²	—	2	2	no	—
223	C₂²×C₃	—	2	12	no	—
224	C₂²×C₃	—	2	24	no	—
225	C₂²×C₃	—	2	4	no	—
226	C₂²×C₃	—	2	8	no	—
227	C₂²×C₃	—	4	12	no	—
228	C₂²×C₃	—	4	6	no	—
229	C₂²×C₃	—	4	6	no	—
230	C₂²×C₃	—	4	4	no	—
231	C₂²×C₃	—	4	2	no	—
232	C₂²×C₃	—	4	2	no	—
233	C₂²×C₃	C₂²×C₄×C₉	1	3	yes	—
234	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
235	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
236	C₂²×C₃	C₂×D₄×C₉	1	6	yes	—
237	C₂²×C₃	C₉×#41	1	6	yes	—
238	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
239	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
240	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
241	C₂²×C₃	C₂×C₃²×D₄	1	2	yes	—
242	C₂²×C₃	C₃²×#41	1	2	yes	—
243	C₃×C₄	—	2	12	no	—
244	C₃×C₄	—	2	4	no	—
245	C₃×C₄	—	4	6	no	—
246	C₃×C₄	—	4	2	no	—
247	C₃×C₄	C₉×C₂²⋊C₄	1	12	yes	complemented
248	C₃×C₄	C₃²×C₂²⋊C₄	1	4	yes	—
249	C₃²	C₃×#390	1	1	yes	characteristic
250	C₉	C₃×#390	1	3	yes	direct product factor
251	C₂³	—	2	4	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	4	no	—
255	C₂³	C₂×C₃×C₄×C₉	1	4	yes	—
256	C₂³	C₂³×C₃×C₉	1	1	yes	characteristic
257	D₄	—	2	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	C₂×C₃×C₄×C₉	1	2	yes	—
263	C₂×C₄	C₃×D₄×C₉	1	2	yes	—
264	C₂×C₄	C₃×D₄×C₉	1	2	yes	—
265	C₂×C₄	C₃×D₄×C₉	1	4	yes	complemented
266	C₂×C₃	—	4	12	no	—
267	C₂×C₃	—	4	4	no	—
268	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	3	yes	—
269	C₂×C₃	C₄×D₄×C₉	1	6	yes	—
270	C₂×C₃	C₉×#121	1	6	yes	—
271	C₂×C₃	C₉×#124	1	3	yes	—
272	C₂×C₃	C₉×#127	1	3	yes	—
273	C₂×C₃	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₃	C₃²×C₄×D₄	1	2	yes	—
275	C₂×C₃	C₃²×#121	1	2	yes	—
276	C₂×C₃	C₃²×#124	1	1	yes	characteristic
277	C₂×C₃	C₃²×#127	1	1	yes	characteristic
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	C₂²×C₃×C₄×C₉	1	1	yes	characteristic
284	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
285	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
286	C₂²	C₂×C₃×D₄×C₉	1	2	yes	—
287	C₂²	C₃×C₉×#41	1	2	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	4	2	no	—
290	C₄	C₃×C₉×C₂²⋊C₄	1	4	yes	complemented
291	C₃	C₉×#390	1	3	yes	direct product factor
292	C₃	C₃²×#390	1	1	yes	characteristic
293	C₂	—	4	4	no	—
294	C₂	C₂×C₃×C₉×C₂²⋊C₄	1	1	yes	characteristic
295	C₂	C₃×C₄×D₄×C₉	1	2	yes	—
296	C₂	C₃×C₉×#121	1	2	yes	—
297	C₂	C₃×C₉×#124	1	1	yes	characteristic
298	C₂	C₃×C₉×#127	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃×C₉	C₂³
commutator	C₂²	C₂²×C₃×C₄×C₉
Frattini, cosocle	C₂³×C₃	C₂³×C₃²
socle	C₂³×C₃²	C₂³×C₃
prime radical	C₂²×C₃²×D₄	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₉×C₄⋊C₄	C₂	semidirect
3. C₂×C₃×C₉×C₂²⋊C₄	C₂	semidirect
4. C₂×C₃×C₄²×C₉	C₂	semidirect
6. C₉×#390	C₃	direct (N × H)
10. C₂×C₃×D₄×C₉	C₄	semidirect
22. C₂×C₉×C₄⋊C₄	C₂×C₃	semidirect
23. C₂×C₉×C₂²⋊C₄	C₂×C₃	semidirect
24. C₂×C₄²×C₉	C₂×C₃	semidirect
39. C₂×C₃×C₄×C₉	D₄	semidirect
40. C₃×#390	C₉	direct (N × H)
55. C₂×D₄×C₉	C₃×C₄	semidirect
74. C₃×C₄×C₉	C₂²⋊C₄	semidirect
77. C₂×C₃×C₄⋊C₄	C₂×C₉	semidirect
79. C₂×C₃×C₂²⋊C₄	C₂×C₉	semidirect
81. C₂×C₃×C₄²	C₂×C₉	semidirect
112. C₂×C₄×C₉	C₃×D₄	semidirect
113. #390	C₃×C₉	direct (N × H)
126. C₂×C₃×D₄	C₄×C₉	semidirect
165. C₄×C₉	C₃×C₂²⋊C₄	semidirect
167. C₂×C₄⋊C₄	C₂×C₃×C₉	semidirect
168. C₂×C₂²⋊C₄	C₂×C₃×C₉	semidirect
169. C₂×C₄²	C₂×C₃×C₉	semidirect
170. C₃×C₉	#390	direct (N × H)
196. C₂×C₃×C₄	D₄×C₉	semidirect
216. C₂×D₄	C₃×C₄×C₉	semidirect
247. C₃×C₄	C₉×C₂²⋊C₄	semidirect
250. C₉	C₃×#390	direct (N × H)
265. C₂×C₄	C₃×D₄×C₉	semidirect
290. C₄	C₃×C₉×C₂²⋊C₄	semidirect
291. C₃	C₉×#390	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₄×C₉	2
C₂×C₃×C₉×C₄⋊C₄	2
C₂×C₃×C₉×C₂²⋊C₄	2
C₂×C₃×C₄²×C₉	2
C₃²×#390	3
C₉×#390	3

Subgroups — C3×C9×#390 #13144928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#390 #13144928