Subgroups — C₉×#1869 #11766590

Order 1152.

648 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₉×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₉×#336	C₂	1	2	yes	maximal, complemented
6	C₃×#1869	C₃	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₉	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	—	2	2	no	—
11	C₂²×C₈×C₉	—	2	4	no	—
12	C₂²×C₈×C₉	C₂²	1	2	yes	—
13	C₂²×C₈×C₉	C₂²	1	2	yes	—
14	C₄×D₄×C₉	—	2	2	no	—
15	C₄×D₄×C₉	—	2	2	no	—
16	C₂×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
17	C₂×C₉×C₂²⋊C₄	C₄	1	1	yes	characteristic
18	C₂×C₉×C₂²⋊C₄	C₄	1	1	yes	characteristic
19	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
20	C₉×C₂²⋊C₈	—	2	2	no	—
21	C₉×C₂²⋊C₈	—	2	4	no	—
22	C₂³×C₃×C₈	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	2	yes	—
26	C₃×#336	C₂×C₃	1	2	yes	—
27	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₉	—	2	2	no	—
30	C₂×D₄×C₉	—	2	2	no	—
31	C₂×D₄×C₉	C₈	1	4	yes	complemented
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	—	2	2	no	—
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	—	2	2	no	—
37	C₂²×C₄×C₉	—	2	2	no	—
38	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	D₄	1	2	yes	—
43	C₂²×C₄×C₉	D₄	1	2	yes	—
44	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₉	—	2	8	no	—
46	C₂×C₈×C₉	—	2	2	no	—
47	C₂×C₈×C₉	—	2	2	no	—
48	C₂×C₈×C₉	—	2	2	no	—
49	C₂×C₈×C₉	—	2	2	no	—
50	C₂×C₈×C₉	—	2	2	no	—
51	C₂×C₈×C₉	D₄	1	4	yes	complemented
52	C₉×C₄⋊C₄	—	2	2	no	—
53	C₉×C₂²⋊C₄	—	2	2	no	—
54	C₉×C₂²⋊C₄	—	2	2	no	—
55	C₄²×C₉	—	2	1	no	—
56	C₄²×C₉	—	2	1	no	—
57	#1869	C₉	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₃×D₄	C₃×C₄	1	1	yes	characteristic
59	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
60	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	—	2	2	no	—
62	C₂²×C₃×C₈	—	2	4	no	—
63	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
64	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
65	C₃×C₄×D₄	—	2	2	no	—
66	C₃×C₄×D₄	—	2	2	no	—
67	C₂×C₃×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
68	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
69	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
70	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
71	C₃×C₂²⋊C₈	—	2	2	no	—
72	C₃×C₂²⋊C₈	—	2	4	no	—
73	C₂³×C₉	—	2	2	no	—
74	C₂³×C₉	—	2	2	no	—
75	C₂³×C₉	—	2	1	no	—
76	C₂³×C₉	—	2	2	no	—
77	C₂³×C₉	—	2	1	no	—
78	C₂³×C₉	—	2	2	no	—
79	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
80	C₂³×C₉	C₂×C₈	1	2	yes	—
81	C₂³×C₉	C₂×C₈	1	2	yes	complemented
82	C₂³×C₉	M₁₆	1	2	yes	—
83	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
84	D₄×C₉	—	2	8	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	4	no	—
90	C₂×C₄×C₉	—	2	2	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	4	1	no	—
96	C₂×C₄×C₉	—	4	2	no	—
97	C₂×C₄×C₉	—	4	1	no	—
98	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₉	C₂×C₈	1	2	yes	complemented
100	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₉	#41	1	2	yes	—
104	C₈×C₉	—	2	4	no	—
105	C₈×C₉	—	4	2	no	—
106	C₂³×C₈	C₂×C₉	1	1	yes	characteristic, complemented
107	C₂×C₄×D₄	C₂×C₉	1	1	yes	characteristic
108	C₂×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
109	C₂×C₂²⋊C₈	C₂×C₉	1	2	yes	complemented
110	#336	C₂×C₉	1	2	yes	complemented
111	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₃×D₄	—	2	2	no	—
114	C₂×C₃×D₄	—	2	2	no	—
115	C₂×C₃×D₄	C₃×C₈	1	4	yes	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	—	2	2	no	—
118	C₂²×C₃×C₄	—	2	2	no	—
119	C₂²×C₃×C₄	—	2	1	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	2	2	no	—
122	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
127	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
128	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₂×C₃×C₈	—	2	8	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₂²⋊C₄	—	2	2	no	—
138	C₃×C₂²⋊C₄	—	2	2	no	—
139	C₃×C₄²	—	2	1	no	—
140	C₃×C₄²	—	2	1	no	—
141	C₂²×C₉	—	2	2	no	—
142	C₂²×C₉	—	2	2	no	—
143	C₂²×C₉	—	2	4	no	—
144	C₂²×C₉	—	2	4	no	—
145	C₂²×C₉	—	2	2	no	—
146	C₂²×C₉	—	2	4	no	—
147	C₂²×C₉	—	4	1	no	—
148	C₂²×C₉	—	4	1	no	—
149	C₂²×C₉	—	4	2	no	—
150	C₂²×C₉	C₂²⋊C₈	1	4	yes	complemented
151	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₉	C₄×D₄	1	2	yes	—
153	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
154	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
155	C₂²×C₉	#131	1	2	yes	—
156	C₄×C₉	—	2	2	no	—
157	C₄×C₉	—	2	2	no	—
158	C₄×C₉	—	4	1	no	—
159	C₄×C₉	—	4	1	no	—
160	C₄×C₉	#120	1	1	yes	characteristic
161	C₄×C₉	#121	1	2	yes	—
162	C₄×C₉	#123	1	1	yes	characteristic
163	C₂²×D₄	C₄×C₉	1	1	yes	characteristic
164	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
165	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	C₂²×C₉	1	2	yes	—
169	C₂²×C₈	C₂²×C₉	1	2	yes	—
170	C₄×D₄	—	2	2	no	—
171	C₄×D₄	—	2	2	no	—
172	C₂×C₄⋊C₄	C₄×C₉	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	C₄×C₉	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	C₄×C₉	1	1	yes	characteristic
175	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	2	4	no	—
178	C₂³×C₃	—	2	2	no	—
179	C₂³×C₃	—	2	2	no	—
180	C₂³×C₃	—	2	1	no	—
181	C₂³×C₃	—	2	2	no	—
182	C₂³×C₃	—	2	1	no	—
183	C₂³×C₃	—	2	2	no	—
184	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
185	C₂³×C₃	C₂×C₃×C₈	1	2	yes	—
186	C₂³×C₃	C₂×C₃×C₈	1	2	yes	—
187	C₂³×C₃	C₃×M₁₆	1	2	yes	—
188	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
189	C₃×D₄	—	2	8	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	4	1	no	—
201	C₂×C₃×C₄	—	4	2	no	—
202	C₂×C₃×C₄	—	4	1	no	—
203	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
204	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
205	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
206	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₃×#41	1	2	yes	—
209	C₃×C₈	—	2	4	no	—
210	C₃×C₈	—	4	2	no	—
211	C₂×C₉	—	2	4	no	—
212	C₂×C₉	—	4	2	no	—
213	C₂×C₉	#386	1	1	yes	characteristic
214	C₂×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
215	C₂×C₉	#408	1	1	yes	characteristic
216	C₂×C₉	C₈×D₄	1	2	yes	—
217	C₂×C₉	#435	1	2	yes	—
218	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
219	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	C₈×C₉	1	4	yes	complemented
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
233	C₂²×C₄	D₄×C₉	1	2	yes	—
234	C₂²×C₄	D₄×C₉	1	2	yes	—
235	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	D₄×C₉	1	4	yes	complemented
243	C₄⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	2	no	—
250	C₂²×C₃	—	2	4	no	—
251	C₂²×C₃	—	2	4	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	4	no	—
254	C₂²×C₃	—	4	1	no	—
255	C₂²×C₃	—	4	1	no	—
256	C₂²×C₃	—	4	2	no	—
257	C₂²×C₃	C₃×C₂²⋊C₈	1	4	yes	—
258	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
259	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
260	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
261	C₂²×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
262	C₂²×C₃	C₃×#131	1	2	yes	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	4	1	no	—
266	C₃×C₄	—	4	1	no	—
267	C₃×C₄	C₃×#120	1	1	yes	characteristic
268	C₃×C₄	C₃×#121	1	2	yes	—
269	C₃×C₄	C₃×#123	1	1	yes	characteristic
270	C₉	#1869	1	1	yes	characteristic, direct product factor, Sylow
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
278	C₂³	C₂×C₈×C₉	1	2	yes	—
279	C₂³	C₂×C₈×C₉	1	2	yes	complemented
280	C₂³	C₉×M₁₆	1	2	yes	—
281	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
282	D₄	—	2	8	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
297	C₂×C₄	C₂×C₈×C₉	1	2	yes	complemented
298	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
299	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
301	C₂×C₄	C₉×#41	1	2	yes	—
302	C₈	—	2	4	no	—
303	C₈	—	4	2	no	—
304	C₂×C₃	—	2	4	no	—
305	C₂×C₃	—	4	2	no	—
306	C₂×C₃	C₃×#386	1	1	yes	characteristic
307	C₂×C₃	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
308	C₂×C₃	C₃×#408	1	1	yes	characteristic
309	C₂×C₃	C₃×C₈×D₄	1	2	yes	—
310	C₂×C₃	C₃×#435	1	2	yes	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	C₉×C₂²⋊C₈	1	4	yes	complemented
321	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
322	C₂²	C₄×D₄×C₉	1	2	yes	—
323	C₂²	C₂²×C₈×C₉	1	1	yes	characteristic
324	C₂²	C₂×C₉×M₁₆	1	1	yes	characteristic
325	C₂²	C₉×#131	1	2	yes	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	C₉×#120	1	1	yes	characteristic
331	C₄	C₉×#121	1	2	yes	—
332	C₄	C₉×#123	1	1	yes	characteristic
333	C₃	C₃×#1869	1	1	yes	characteristic
334	C₂	—	2	4	no	—
335	C₂	—	4	2	no	—
336	C₂	C₉×#386	1	1	yes	characteristic
337	C₂	C₂×C₉×C₂²⋊C₈	1	1	yes	characteristic
338	C₂	C₉×#408	1	1	yes	characteristic
339	C₂	C₈×D₄×C₉	1	2	yes	—
340	C₂	C₉×#435	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂²	C₂²×C₈×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂²×C₃×C₄
prime radical	C₂²×C₃×D₄	C₃×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₈×C₉	C₂	semidirect
3. C₂×C₉×C₂²⋊C₈	C₂	semidirect
4. C₂×C₉×C₂²⋊C₈	C₂	semidirect
5. C₉×#336	C₂	semidirect
31. C₂×D₄×C₉	C₈	semidirect
51. C₂×C₈×C₉	D₄	semidirect
57. #1869	C₉	direct (N × H)
81. C₂³×C₉	C₂×C₈	semidirect
99. C₂×C₄×C₉	C₂×C₈	semidirect
106. C₂³×C₈	C₂×C₉	semidirect
108. C₂×C₂²⋊C₈	C₂×C₉	semidirect
109. C₂×C₂²⋊C₈	C₂×C₉	semidirect
110. #336	C₂×C₉	semidirect
150. C₂²×C₉	C₂²⋊C₈	semidirect
222. C₂×D₄	C₈×C₉	semidirect
242. C₂×C₈	D₄×C₉	semidirect
270. C₉	#1869	direct (N × H)
279. C₂³	C₂×C₈×C₉	semidirect
297. C₂×C₄	C₂×C₈×C₉	semidirect
320. C₂²	C₉×C₂²⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₂×C₄×D₄×C₉	2
C₂×C₉×C₂²⋊C₈	2
C₉×#336	2
C₃×#1869	3

Subgroups — C9×#1869 #11766590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1869 #11766590