Subgroups — C₉×#2925 #11767646

Order 1152.

588 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	C₄×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₉×#433	C₂	1	4	yes	maximal, complemented
3	C₉×#432	C₂	1	2	yes	maximal, complemented
4	C₉×#431	C₂	1	2	yes	maximal, complemented
5	C₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#408	C₂	1	2	yes	maximal
8	C₉×#405	C₂	1	2	yes	maximal, complemented
9	C₃×#2925	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	C₂²	1	1	yes	characteristic
11	C₂×C₉×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₉	C₂²	1	2	yes	—
13	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
15	C₄×D₄×C₉	C₄	1	2	yes	—
16	C₄×D₄×C₉	C₄	1	1	yes	characteristic
17	C₄×D₄×C₉	C₄	1	2	yes	—
18	C₄×D₄×C₉	C₄	1	1	yes	characteristic
19	C₉×#117	C₂²	1	1	yes	characteristic
20	C₉×#117	C₂²	1	2	yes	—
21	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₉	C₂²	1	2	yes	—
23	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₉×C₄⋊C₈	C₂²	1	2	yes	—
26	C₉×C₄⋊C₈	C₂²	1	2	yes	—
27	C₉×C₄⋊C₈	C₂²	1	2	yes	—
28	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
29	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
30	C₉×#97	C₂²	1	4	yes	—
31	C₄×C₈×C₉	C₂²	1	4	yes	—
32	C₃×C₄×#41	C₂×C₃	1	1	yes	characteristic
33	C₃×#433	C₂×C₃	1	4	yes	—
34	C₃×#432	C₂×C₃	1	2	yes	—
35	C₃×#431	C₂×C₃	1	2	yes	—
36	C₃×#423	C₂×C₃	1	1	yes	characteristic
37	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
38	C₃×#408	C₂×C₃	1	2	yes	—
39	C₃×#405	C₂×C₃	1	2	yes	—
40	C₉×#41	—	2	2	no	—
41	C₉×#41	—	2	2	no	—
42	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
44	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
48	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₉×M₁₆	—	2	4	no	—
52	C₂×C₈×C₉	—	2	4	no	—
53	C₂×C₈×C₉	C₂³	1	2	yes	—
54	C₂×C₈×C₉	C₂³	1	2	yes	—
55	C₂×C₈×C₉	C₂³	1	4	yes	—
56	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
57	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
58	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
59	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
61	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
62	C₉×C₂²⋊C₄	C₂×C₄	1	2	yes	—
63	C₄²×C₉	C₂×C₄	1	2	yes	—
64	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₉	C₂×C₄	1	2	yes	—
66	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
67	C₄²×C₉	C₂³	1	1	yes	characteristic
68	C₄²×C₉	C₂³	1	2	yes	—
69	C₄²×C₉	C₂³	1	1	yes	characteristic
70	#2925	C₉	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
73	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
74	C₃×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
75	C₃×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
76	C₃×C₄×D₄	C₃×C₄	1	2	yes	—
77	C₃×C₄×D₄	C₃×C₄	1	1	yes	characteristic
78	C₃×C₄×D₄	C₃×C₄	1	2	yes	—
79	C₃×C₄×D₄	C₃×C₄	1	1	yes	characteristic
80	C₃×#117	C₂²×C₃	1	1	yes	characteristic
81	C₃×#117	C₂²×C₃	1	2	yes	—
82	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
84	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
85	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
86	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
87	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
88	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
89	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
90	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
91	C₃×#97	C₂²×C₃	1	4	yes	—
92	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
93	C₂³×C₉	C₂²×C₄	1	2	yes	—
94	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
95	Q₈×C₉	—	2	2	no	—
96	D₄×C₉	—	2	4	no	—
97	D₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	4	1	no	—
103	C₂×C₄×C₉	—	4	2	no	—
104	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₉	#41	1	2	yes	—
113	C₂×C₄×C₉	#41	1	2	yes	—
114	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
115	C₈×C₉	—	2	2	no	—
116	C₈×C₉	—	2	2	no	—
117	C₈×C₉	—	2	4	no	—
118	C₄×#41	C₂×C₉	1	1	yes	characteristic
119	#433	C₂×C₉	1	4	yes	complemented
120	#432	C₂×C₉	1	2	yes	complemented
121	#431	C₂×C₉	1	2	yes	complemented
122	#423	C₂×C₉	1	1	yes	characteristic, complemented
123	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
124	#408	C₂×C₉	1	2	yes	—
125	#405	C₂×C₉	1	2	yes	complemented
126	C₃×#41	—	2	2	no	—
127	C₃×#41	—	2	2	no	—
128	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
129	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
130	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
132	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
134	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₃×M₁₆	—	2	4	no	—
138	C₂×C₃×C₈	—	2	4	no	—
139	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
140	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
141	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
142	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
143	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
144	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
145	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
146	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
147	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
148	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
149	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
150	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
151	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
152	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
153	C₃×C₄²	C₂³×C₃	1	2	yes	—
154	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
155	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	2	1	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	#117	1	2	yes	—
162	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
163	C₄×C₉	—	2	2	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	2	2	no	—
166	C₄×C₉	—	2	1	no	—
167	C₄×C₉	—	2	1	no	—
168	C₄×C₉	—	2	2	no	—
169	C₄×C₉	#117	1	2	yes	—
170	C₄×C₉	#131	1	2	yes	—
171	C₄×C₉	C₂×#41	1	1	yes	characteristic
172	C₄×C₉	C₂×#41	1	1	yes	characteristic
173	C₂×#41	C₂²×C₉	1	1	yes	characteristic
174	C₂×M₁₆	C₂²×C₉	1	2	yes	—
175	C₂²×C₈	C₂²×C₉	1	2	yes	—
176	C₄×Q₈	C₄×C₉	1	1	yes	characteristic
177	C₄×Q₈	C₄×C₉	1	1	yes	characteristic
178	C₄×D₄	C₄×C₉	1	2	yes	—
179	C₄×D₄	C₄×C₉	1	1	yes	characteristic
180	C₄×D₄	C₄×C₉	1	2	yes	—
181	C₄×D₄	C₄×C₉	1	1	yes	characteristic
182	#117	C₂²×C₉	1	1	yes	characteristic
183	#117	C₂²×C₉	1	2	yes	—
184	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
185	C₂×C₄²	C₂²×C₉	1	2	yes	—
186	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
187	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
188	C₄⋊C₈	C₂²×C₉	1	2	yes	—
189	C₄⋊C₈	C₂²×C₉	1	2	yes	—
190	C₄⋊C₈	C₂²×C₉	1	2	yes	—
191	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
192	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
193	#97	C₂²×C₉	1	4	yes	—
194	C₄×C₈	C₂²×C₉	1	4	yes	—
195	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
196	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
197	C₃×Q₈	—	2	2	no	—
198	C₃×D₄	—	2	4	no	—
199	C₃×D₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	4	1	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
208	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
209	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
211	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₃×#41	1	2	yes	—
215	C₂×C₃×C₄	C₃×#41	1	2	yes	—
216	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
217	C₃×C₈	—	2	2	no	—
218	C₃×C₈	—	2	2	no	—
219	C₃×C₈	—	2	4	no	—
220	C₂×C₉	—	2	2	no	—
221	C₂×C₉	—	4	2	no	—
222	C₂×C₉	C₂×#117	1	1	yes	characteristic
223	C₂×C₉	C₂×#131	1	1	yes	characteristic
224	C₂×C₉	#568	1	1	yes	characteristic
225	#41	—	2	2	no	—
226	#41	—	2	2	no	—
227	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
228	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
229	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
231	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
233	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
234	C₂²×C₄	C₂³×C₉	1	2	yes	—
235	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂³×C₉	1	2	yes	—
239	C₂×C₈	C₂³×C₉	1	4	yes	—
240	C₂×C₈	C₂³×C₉	1	2	yes	—
241	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
242	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	—
243	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
244	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
245	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
246	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
247	C₂²⋊C₄	C₂×C₄×C₉	1	2	yes	—
248	C₄²	C₂×C₄×C₉	1	2	yes	—
249	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
250	C₄²	C₂×C₄×C₉	1	2	yes	—
251	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
252	C₄²	C₂³×C₉	1	2	yes	—
253	C₄²	C₂³×C₉	1	1	yes	characteristic
254	C₄²	C₂³×C₉	1	1	yes	characteristic
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	C₃×#117	1	2	yes	—
261	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	C₃×#117	1	2	yes	—
269	C₃×C₄	C₃×#131	1	2	yes	—
270	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
271	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
272	C₉	#2925	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³	C₂²×C₄×C₉	1	2	yes	—
274	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
286	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
287	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
289	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
292	C₂×C₄	C₉×#41	1	2	yes	—
293	C₂×C₄	C₉×#41	1	2	yes	—
294	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	4	no	—
298	C₂×C₃	—	2	2	no	—
299	C₂×C₃	—	4	2	no	—
300	C₂×C₃	C₂×C₃×#117	1	1	yes	characteristic
301	C₂×C₃	C₂×C₃×#131	1	1	yes	characteristic
302	C₂×C₃	C₃×#568	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	C₉×#117	1	2	yes	—
309	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	C₉×#117	1	2	yes	—
317	C₄	C₉×#131	1	2	yes	—
318	C₄	C₂×C₉×#41	1	1	yes	characteristic
319	C₄	C₂×C₉×#41	1	1	yes	characteristic
320	C₃	C₃×#2925	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	—	4	2	no	—
323	C₂	C₂×C₉×#117	1	1	yes	characteristic
324	C₂	C₂×C₉×#131	1	1	yes	characteristic
325	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	C₂×C₃×#41	C₂²×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
2. C₉×#433	C₂	semidirect
3. C₉×#432	C₂	semidirect
4. C₉×#431	C₂	semidirect
5. C₉×#423	C₂	semidirect
6. C₂×C₉×C₄⋊C₈	C₂	semidirect
8. C₉×#405	C₂	semidirect
70. #2925	C₉	direct (N × H)
119. #433	C₂×C₉	semidirect
120. #432	C₂×C₉	semidirect
121. #431	C₂×C₉	semidirect
122. #423	C₂×C₉	semidirect
123. C₂×C₄⋊C₈	C₂×C₉	semidirect
125. #405	C₂×C₉	semidirect
272. C₉	#2925	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₉×#41	2
C₉×#433	2
C₉×#432	2
C₉×#431	2
C₉×#423	2
C₂×C₉×C₄⋊C₈	2
C₉×#408	2
C₉×#405	2
C₃×#2925	3

Subgroups — C9×#2925 #11767646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2925 #11767646