Subgroups — C₉×#2499 #11767220

Order 1152.

726 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	C₂×C₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#117	C₂	1	1	yes	maximal, characteristic
3	C₉×#397	C₂	1	4	yes	maximal
4	C₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#387	C₂	1	4	yes	maximal, complemented
7	C₉×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₉×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
10	C₃×#2499	C₃	1	1	yes	maximal, characteristic
11	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₉×C₂²⋊Q₈	—	2	4	no	—
14	C₉×#117	—	2	4	no	—
15	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₉	C₂²	1	2	yes	—
26	C₉×#95	C₂²	1	2	yes	—
27	C₉×#95	C₂²	1	4	yes	—
28	C₉×#95	C₂²	1	4	yes	—
29	C₉×#95	C₂²	1	4	yes	—
30	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×#117	C₂×C₃	1	1	yes	characteristic
32	C₃×#397	C₂×C₃	1	4	yes	—
33	C₃×#391	C₂×C₃	1	1	yes	characteristic
34	C₃×#391	C₂×C₃	1	1	yes	characteristic
35	C₃×#387	C₂×C₃	1	4	yes	—
36	C₃×#384	C₂×C₃	1	1	yes	characteristic
37	C₃×#382	C₂×C₃	1	1	yes	characteristic
38	C₃×C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
39	C₂⁴×C₉	C₂³	1	1	yes	characteristic
40	C₂×Q₈×C₉	—	2	2	no	—
41	C₂×Q₈×C₉	—	2	2	no	—
42	C₂×Q₈×C₉	—	2	2	no	—
43	C₂²×C₄×C₉	—	2	2	no	—
44	C₂²×C₄×C₉	—	2	2	no	—
45	C₂²×C₄×C₉	D₄	1	4	yes	—
46	C₂²×C₄×C₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
54	C₉×C₄⋊C₄	—	2	2	no	—
55	C₉×C₄⋊C₄	—	2	2	no	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	—	2	4	no	—
58	C₉×C₄⋊C₄	—	2	8	no	—
59	C₉×C₂²⋊C₄	—	2	4	no	—
60	C₉×C₂²⋊C₄	—	2	4	no	—
61	C₉×C₂²⋊C₄	—	2	4	no	—
62	C₄²×C₉	—	2	4	no	—
63	C₄²×C₉	—	2	2	no	—
64	C₄²×C₉	—	2	2	no	—
65	#2499	C₉	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
68	C₃×C₂²⋊Q₈	—	2	4	no	—
69	C₃×#117	—	2	4	no	—
70	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
73	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
81	C₃×#95	C₂²×C₃	1	2	yes	—
82	C₃×#95	C₂²×C₃	1	4	yes	—
83	C₃×#95	C₂²×C₃	1	4	yes	—
84	C₃×#95	C₂²×C₃	1	4	yes	—
85	C₂³×C₉	—	2	2	no	—
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	—	2	1	no	—
88	C₂³×C₉	—	2	2	no	—
89	C₂³×C₉	C₂×D₄	1	2	yes	—
90	C₂³×C₉	C₂⁴	1	1	yes	characteristic
91	Q₈×C₉	—	4	4	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	8	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	4	no	—
104	C₂×C₄×C₉	—	2	4	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	4	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₉	#41	1	4	yes	—
112	C₂×C₄×C₉	#41	1	2	yes	—
113	C₂×C₄×C₉	#41	1	2	yes	—
114	C₂×C₂²⋊Q₈	C₂×C₉	1	1	yes	characteristic
115	C₂×#117	C₂×C₉	1	1	yes	characteristic
116	#397	C₂×C₉	1	4	yes	—
117	#391	C₂×C₉	1	1	yes	characteristic, complemented
118	#391	C₂×C₉	1	1	yes	characteristic, complemented
119	#387	C₂×C₉	1	4	yes	complemented
120	#384	C₂×C₉	1	1	yes	characteristic, complemented
121	#382	C₂×C₉	1	1	yes	characteristic, complemented
122	C₄×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic
123	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
124	C₂×C₃×Q₈	—	2	2	no	—
125	C₂×C₃×Q₈	—	2	2	no	—
126	C₂×C₃×Q₈	—	2	2	no	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	—	2	2	no	—
129	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
130	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
132	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
135	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
137	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	—	2	4	no	—
142	C₃×C₄⋊C₄	—	2	8	no	—
143	C₃×C₂²⋊C₄	—	2	4	no	—
144	C₃×C₂²⋊C₄	—	2	4	no	—
145	C₃×C₂²⋊C₄	—	2	4	no	—
146	C₃×C₄²	—	2	4	no	—
147	C₃×C₄²	—	2	2	no	—
148	C₃×C₄²	—	2	2	no	—
149	C₂²×C₉	—	2	2	no	—
150	C₂²×C₉	—	2	2	no	—
151	C₂²×C₉	—	2	2	no	—
152	C₂²×C₉	—	4	2	no	—
153	C₂²×C₉	—	4	2	no	—
154	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
155	C₂²×C₉	C₂×#41	1	1	yes	characteristic
156	C₂²×C₉	C₂×#41	1	1	yes	characteristic
157	C₂²×C₉	C₂×#41	1	2	yes	—
158	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
159	C₄×C₉	—	2	2	no	—
160	C₄×C₉	—	2	4	no	—
161	C₄×C₉	—	4	2	no	—
162	C₄×C₉	—	4	2	no	—
163	C₄×C₉	—	4	2	no	—
164	C₄×C₉	—	4	2	no	—
165	C₄×C₉	—	4	2	no	—
166	C₄×C₉	#123	1	4	yes	—
167	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
168	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
169	C₂²⋊Q₈	—	2	4	no	—
170	#117	—	2	4	no	—
171	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
175	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
179	C₂×C₄²	C₂²×C₉	1	2	yes	—
180	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
181	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
182	#95	C₂²×C₉	1	2	yes	—
183	#95	C₂²×C₉	1	4	yes	—
184	#95	C₂²×C₉	1	4	yes	—
185	#95	C₂²×C₉	1	4	yes	—
186	C₂³×C₃	—	2	2	no	—
187	C₂³×C₃	—	2	1	no	—
188	C₂³×C₃	—	2	1	no	—
189	C₂³×C₃	—	2	2	no	—
190	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
191	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
192	C₃×Q₈	—	4	4	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	8	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	4	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
211	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
212	C₂×C₃×C₄	C₃×#41	1	4	yes	—
213	C₂×C₃×C₄	C₃×#41	1	2	yes	—
214	C₂×C₃×C₄	C₃×#41	1	2	yes	—
215	C₂×C₉	—	4	2	no	—
216	C₂×C₉	C₂×#123	1	1	yes	characteristic
217	C₂×C₉	#532	1	1	yes	characteristic
218	C₂×C₉	#536	1	1	yes	characteristic
219	C₂×C₉	#552	1	2	yes	—
220	C₂×C₉	#556	1	2	yes	—
221	C₂⁴	C₂³×C₉	1	1	yes	characteristic
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	D₄×C₉	1	4	yes	—
228	C₂²×C₄	C₂³×C₉	1	2	yes	—
229	C₂²×C₄	C₂³×C₉	1	2	yes	—
230	C₂²×C₄	C₂³×C₉	1	2	yes	—
231	C₂²×C₄	C₂³×C₉	1	2	yes	—
232	C₂²×C₄	C₂³×C₉	1	2	yes	—
233	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
234	C₂²×C₄	C₂³×C₉	1	2	yes	—
235	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	2	no	—
250	C₂²×C₃	—	4	2	no	—
251	C₂²×C₃	—	4	2	no	—
252	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
253	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
254	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
255	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
256	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	—	4	2	no	—
260	C₃×C₄	—	4	2	no	—
261	C₃×C₄	—	4	2	no	—
262	C₃×C₄	—	4	2	no	—
263	C₃×C₄	—	4	2	no	—
264	C₃×C₄	C₃×#123	1	4	yes	—
265	C₉	#2499	1	1	yes	characteristic, direct product factor, Sylow
266	C₂³	—	2	2	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	2	no	—
270	C₂³	C₂×D₄×C₉	1	2	yes	—
271	C₂³	C₂⁴×C₉	1	1	yes	characteristic
272	Q₈	—	4	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
291	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
292	C₂×C₄	C₉×#41	1	4	yes	—
293	C₂×C₄	C₉×#41	1	2	yes	—
294	C₂×C₄	C₉×#41	1	2	yes	—
295	C₂×C₃	—	4	2	no	—
296	C₂×C₃	C₂×C₃×#123	1	1	yes	characteristic
297	C₂×C₃	C₃×#532	1	1	yes	characteristic
298	C₂×C₃	C₃×#536	1	1	yes	characteristic
299	C₂×C₃	C₃×#552	1	2	yes	—
300	C₂×C₃	C₃×#556	1	2	yes	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
307	C₂²	C₂×C₉×#41	1	1	yes	characteristic
308	C₂²	C₂×C₉×#41	1	1	yes	characteristic
309	C₂²	C₂×C₉×#41	1	2	yes	—
310	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	C₉×#123	1	4	yes	—
319	C₃	C₃×#2499	1	1	yes	characteristic
320	C₂	—	4	2	no	—
321	C₂	C₂×C₉×#123	1	1	yes	characteristic
322	C₂	C₉×#532	1	1	yes	characteristic
323	C₂	C₉×#536	1	1	yes	characteristic
324	C₂	C₉×#552	1	2	yes	—
325	C₂	C₉×#556	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×#391	C₂	semidirect
5. C₉×#391	C₂	semidirect
6. C₉×#387	C₂	semidirect
7. C₉×#384	C₂	semidirect
8. C₉×#382	C₂	semidirect
65. #2499	C₉	direct (N × H)
117. #391	C₂×C₉	semidirect
118. #391	C₂×C₉	semidirect
119. #387	C₂×C₉	semidirect
120. #384	C₂×C₉	semidirect
121. #382	C₂×C₉	semidirect
265. C₉	#2499	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×C₂²⋊Q₈	2
C₂×C₉×#117	2
C₉×#397	2
C₉×#391	2
C₉×#387	2
C₉×#384	2
C₉×#382	2
C₄×C₉×C₂²⋊C₄	2
C₃×#2499	3

Subgroups — C9×#2499 #11767220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2499 #11767220