Subgroups — C₉×#2935 #11767656

Order 1152.

738 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#131	C₂	1	2	yes	maximal, complemented
2	C₉×#532	C₂	1	1	yes	maximal, characteristic
3	C₉×#436	C₂	1	2	yes	maximal, complemented
4	C₉×#435	C₂	1	4	yes	maximal, complemented
5	C₈×D₄×C₉	C₂	1	2	yes	maximal, complemented
6	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#408	C₂	1	2	yes	maximal, complemented
8	C₄×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#2935	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	C₂²	1	2	yes	—
11	C₉×#131	—	2	8	no	—
12	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₉×M₁₆	C₂²	1	4	yes	—
14	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₉	C₂²	1	2	yes	—
16	C₂²×C₈×C₉	C₂²	1	4	yes	—
17	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
18	C₉×#127	C₄	1	1	yes	characteristic
19	C₉×#124	C₄	1	2	yes	—
20	C₉×#121	C₄	1	2	yes	—
21	C₉×#121	C₄	1	2	yes	—
22	C₄×D₄×C₉	C₂²	1	2	yes	—
23	C₄×D₄×C₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
25	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₉×C₄⋊C₈	C₂²	1	2	yes	—
27	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
29	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
30	C₉×#97	C₂²	1	2	yes	—
31	C₄×C₈×C₉	C₂²	1	2	yes	—
32	C₂×C₃×#131	C₂×C₃	1	2	yes	—
33	C₃×#532	C₂×C₃	1	1	yes	characteristic
34	C₃×#436	C₂×C₃	1	2	yes	—
35	C₃×#435	C₂×C₃	1	4	yes	—
36	C₃×C₈×D₄	C₂×C₃	1	2	yes	—
37	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
38	C₃×#408	C₂×C₃	1	2	yes	—
39	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
40	C₉×#41	—	2	4	no	—
41	C₉×#41	—	2	4	no	—
42	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
44	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
45	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
46	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	C₂³	1	4	yes	—
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₉×M₁₆	—	2	2	no	—
52	C₉×M₁₆	—	2	8	no	—
53	C₉×M₁₆	D₄	1	4	yes	complemented
54	C₂×C₈×C₉	—	2	4	no	—
55	C₂×C₈×C₉	—	2	8	no	—
56	C₂×C₈×C₉	C₂³	1	2	yes	—
57	C₂×C₈×C₉	C₂³	1	2	yes	—
58	C₂×C₈×C₉	C₂³	1	2	yes	—
59	C₂×C₈×C₉	C₂³	1	2	yes	—
60	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
62	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
63	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
64	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₉	C₂³	1	1	yes	characteristic
67	C₄²×C₉	C₂³	1	1	yes	characteristic
68	#2935	C₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
70	C₃×#131	—	2	8	no	—
71	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
72	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
75	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
76	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
77	C₃×#127	C₃×C₄	1	1	yes	characteristic
78	C₃×#124	C₃×C₄	1	2	yes	—
79	C₃×#121	C₃×C₄	1	2	yes	—
80	C₃×#121	C₃×C₄	1	2	yes	—
81	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
82	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
83	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
84	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
85	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
86	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
87	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
88	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
89	C₃×#97	C₂²×C₃	1	2	yes	—
90	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
91	C₂³×C₉	C₂²×C₄	1	4	yes	—
92	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
93	Q₈×C₉	—	2	4	no	—
94	D₄×C₉	—	2	8	no	—
95	D₄×C₉	—	2	4	no	—
96	D₄×C₉	—	2	4	no	—
97	D₄×C₉	—	4	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	4	no	—
101	C₂×C₄×C₉	—	4	1	no	—
102	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
106	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₉	#41	1	2	yes	—
110	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
111	C₈×C₉	—	2	2	no	—
112	C₈×C₉	—	2	2	no	—
113	C₈×C₉	—	2	2	no	—
114	C₈×C₉	C₂×D₄	1	4	yes	—
115	C₂×#131	C₂×C₉	1	2	yes	complemented
116	#532	C₂×C₉	1	1	yes	characteristic
117	#436	C₂×C₉	1	2	yes	complemented
118	#435	C₂×C₉	1	4	yes	complemented
119	C₈×D₄	C₂×C₉	1	2	yes	complemented
120	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
121	#408	C₂×C₉	1	2	yes	complemented
122	C₄×M₁₆	C₂×C₉	1	1	yes	characteristic, complemented
123	C₃×#41	—	2	4	no	—
124	C₃×#41	—	2	4	no	—
125	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
126	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
127	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
128	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
129	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
130	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
133	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
134	C₃×M₁₆	—	2	2	no	—
135	C₃×M₁₆	—	2	8	no	—
136	C₃×M₁₆	C₃×D₄	1	4	yes	—
137	C₂×C₃×C₈	—	2	4	no	—
138	C₂×C₃×C₈	—	2	8	no	—
139	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
140	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
141	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
142	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
143	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
144	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
145	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
146	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
147	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
148	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
149	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
150	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
151	C₂²×C₉	—	2	1	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	—	2	4	no	—
154	C₂²×C₉	—	2	4	no	—
155	C₂²×C₉	—	2	4	no	—
156	C₂²×C₉	C₄×D₄	1	2	yes	—
157	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
158	C₄×C₉	—	2	4	no	—
159	C₄×C₉	—	2	1	no	—
160	C₄×C₉	—	2	1	no	—
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	C₄×D₄	1	2	yes	—
163	C₄×C₉	#131	1	2	yes	complemented
164	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
165	C₄×C₉	C₂×#41	1	1	yes	characteristic
166	C₂×#41	C₂²×C₉	1	2	yes	—
167	#131	—	2	8	no	—
168	C₂×M₁₆	C₂²×C₉	1	4	yes	—
169	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
170	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
171	C₂²×C₈	C₂²×C₉	1	2	yes	—
172	C₂²×C₈	C₂²×C₉	1	4	yes	—
173	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic
174	#127	C₄×C₉	1	1	yes	characteristic
175	#124	C₄×C₉	1	2	yes	—
176	#121	C₄×C₉	1	2	yes	—
177	#121	C₄×C₉	1	2	yes	—
178	C₄×D₄	C₂²×C₉	1	2	yes	—
179	C₄×D₄	C₂²×C₉	1	2	yes	—
180	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
181	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
182	C₄⋊C₈	C₂²×C₉	1	2	yes	—
183	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
184	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
185	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
186	#97	C₂²×C₉	1	2	yes	—
187	C₄×C₈	C₂²×C₉	1	2	yes	—
188	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
189	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
190	C₃×Q₈	—	2	4	no	—
191	C₃×D₄	—	2	8	no	—
192	C₃×D₄	—	2	4	no	—
193	C₃×D₄	—	2	4	no	—
194	C₃×D₄	—	4	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	4	1	no	—
199	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
203	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
206	C₂×C₃×C₄	C₃×#41	1	2	yes	—
207	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
208	C₃×C₈	—	2	2	no	—
209	C₃×C₈	—	2	2	no	—
210	C₃×C₈	—	2	2	no	—
211	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
212	C₂×C₉	—	2	2	no	—
213	C₂×C₉	—	4	4	no	—
214	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂×C₉	C₂×#131	1	1	yes	characteristic
216	C₂×C₉	#568	1	1	yes	characteristic
217	#41	—	2	4	no	—
218	#41	—	2	4	no	—
219	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
220	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
221	C₂×D₄	C₂×C₄×C₉	1	4	yes	—
222	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
223	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
224	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
226	C₂²×C₄	C₂³×C₉	1	4	yes	—
227	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	8	no	—
230	M₁₆	D₄×C₉	1	4	yes	complemented
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	8	no	—
233	C₂×C₈	C₂³×C₉	1	2	yes	—
234	C₂×C₈	C₂³×C₉	1	2	yes	—
235	C₂×C₈	C₂³×C₉	1	2	yes	—
236	C₂×C₈	C₂³×C₉	1	2	yes	—
237	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
238	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
239	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
240	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
241	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
242	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
243	C₄²	C₂³×C₉	1	1	yes	characteristic
244	C₄²	C₂³×C₉	1	1	yes	characteristic
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	2	4	no	—
248	C₂²×C₃	—	2	4	no	—
249	C₂²×C₃	—	2	4	no	—
250	C₂²×C₃	C₃×C₄×D₄	1	2	yes	—
251	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
257	C₃×C₄	C₃×#131	1	2	yes	—
258	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
259	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
260	C₉	#2935	1	1	yes	characteristic, direct product factor, Sylow
261	C₂³	C₂²×C₄×C₉	1	4	yes	—
262	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
263	Q₈	—	2	4	no	—
264	D₄	—	2	8	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
276	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
279	C₂×C₄	C₉×#41	1	2	yes	—
280	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	2	2	no	—
283	C₈	—	2	2	no	—
284	C₈	C₂×D₄×C₉	1	4	yes	—
285	C₂×C₃	—	2	2	no	—
286	C₂×C₃	—	4	4	no	—
287	C₂×C₃	C₂×C₃×C₄×D₄	1	1	yes	characteristic
288	C₂×C₃	C₂×C₃×#131	1	1	yes	characteristic
289	C₂×C₃	C₃×#568	1	1	yes	characteristic
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	C₄×D₄×C₉	1	2	yes	—
296	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	4	no	—
301	C₄	C₄×D₄×C₉	1	2	yes	—
302	C₄	C₉×#131	1	2	yes	complemented
303	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
304	C₄	C₂×C₉×#41	1	1	yes	characteristic
305	C₃	C₃×#2935	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	4	4	no	—
308	C₂	C₂×C₄×D₄×C₉	1	1	yes	characteristic
309	C₂	C₂×C₉×#131	1	1	yes	characteristic
310	C₂	C₉×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₃×C₄
prime radical	C₃×#532	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#131	C₂	semidirect
3. C₉×#436	C₂	semidirect
4. C₉×#435	C₂	semidirect
5. C₈×D₄×C₉	C₂	semidirect
6. C₂×C₉×C₄⋊C₈	C₂	semidirect
7. C₉×#408	C₂	semidirect
8. C₄×C₉×M₁₆	C₂	semidirect
53. C₉×M₁₆	D₄	semidirect
68. #2935	C₉	direct (N × H)
115. C₂×#131	C₂×C₉	semidirect
117. #436	C₂×C₉	semidirect
118. #435	C₂×C₉	semidirect
119. C₈×D₄	C₂×C₉	semidirect
120. C₂×C₄⋊C₈	C₂×C₉	semidirect
121. #408	C₂×C₉	semidirect
122. C₄×M₁₆	C₂×C₉	semidirect
163. C₄×C₉	#131	semidirect
230. M₁₆	D₄×C₉	semidirect
260. C₉	#2935	direct (N × H)
302. C₄	C₉×#131	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#131	2
C₉×#532	2
C₉×#436	2
C₉×#435	2
C₈×D₄×C₉	2
C₂×C₉×C₄⋊C₈	2
C₉×#408	2
C₄×C₉×M₁₆	2
C₃×#2935	3

Subgroups — C9×#2935 #11767656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2935 #11767656