Subgroups — C₉×#2508 #11767229

Order 1152.

570 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#400	C₂	1	2	yes	maximal
2	C₉×#398	C₂	1	2	yes	maximal
3	C₉×#389	C₂	1	1	yes	maximal, characteristic
4	C₉×#389	C₂	1	1	yes	maximal, characteristic
5	C₉×#389	C₂	1	1	yes	maximal, characteristic
6	C₉×#389	C₂	1	1	yes	maximal, characteristic
7	C₉×#387	C₂	1	2	yes	maximal
8	C₉×#382	C₂	1	2	yes	maximal
9	C₉×#382	C₂	1	1	yes	maximal, characteristic
10	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₃×#2508	C₃	1	1	yes	maximal, characteristic
13	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
27	C₉×#95	C₂²	1	1	yes	characteristic
28	C₉×#95	C₂²	1	2	yes	—
29	C₉×#95	C₂²	1	1	yes	characteristic
30	C₉×#95	C₂²	1	2	yes	—
31	C₉×#95	C₂²	1	2	yes	—
32	C₉×#95	C₂²	1	2	yes	—
33	C₉×#95	C₂²	1	2	yes	—
34	C₉×#95	C₂²	1	2	yes	—
35	C₉×#95	C₂²	1	2	yes	—
36	C₃×#400	C₂×C₃	1	2	yes	—
37	C₃×#398	C₂×C₃	1	2	yes	—
38	C₃×#389	C₂×C₃	1	1	yes	characteristic
39	C₃×#389	C₂×C₃	1	1	yes	characteristic
40	C₃×#389	C₂×C₃	1	1	yes	characteristic
41	C₃×#389	C₂×C₃	1	1	yes	characteristic
42	C₃×#387	C₂×C₃	1	2	yes	—
43	C₃×#382	C₂×C₃	1	2	yes	—
44	C₃×#382	C₂×C₃	1	1	yes	characteristic
45	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
46	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
47	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
58	C₉×C₄⋊C₄	—	2	2	no	—
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₉×C₄⋊C₄	—	2	4	no	—
61	C₉×C₄⋊C₄	—	2	2	no	—
62	C₉×C₄⋊C₄	—	2	2	no	—
63	C₉×C₄⋊C₄	—	2	2	no	—
64	C₉×C₄⋊C₄	—	4	2	no	—
65	C₉×C₄⋊C₄	—	4	2	no	—
66	C₉×C₄⋊C₄	—	4	2	no	—
67	C₉×C₄⋊C₄	Q₈	1	4	yes	—
68	C₄²×C₉	—	2	4	no	—
69	C₄²×C₉	—	2	2	no	—
70	C₄²×C₉	—	2	2	no	—
71	C₄²×C₉	—	2	2	no	—
72	#2508	C₉	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
85	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
87	C₃×#95	C₂²×C₃	1	1	yes	characteristic
88	C₃×#95	C₂²×C₃	1	2	yes	—
89	C₃×#95	C₂²×C₃	1	1	yes	characteristic
90	C₃×#95	C₂²×C₃	1	2	yes	—
91	C₃×#95	C₂²×C₃	1	2	yes	—
92	C₃×#95	C₂²×C₃	1	2	yes	—
93	C₃×#95	C₂²×C₃	1	2	yes	—
94	C₃×#95	C₂²×C₃	1	2	yes	—
95	C₃×#95	C₂²×C₃	1	2	yes	—
96	C₂³×C₉	C₂⁴	1	1	yes	characteristic
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	1	no	—
105	C₂×C₄×C₉	—	2	1	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	2	1	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	2	1	no	—
112	C₂×C₄×C₉	—	2	2	no	—
113	C₂×C₄×C₉	—	2	1	no	—
114	C₂×C₄×C₉	—	2	1	no	—
115	C₂×C₄×C₉	—	2	2	no	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	—	2	2	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	2	2	no	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
125	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
126	C₂×C₄×C₉	#41	1	2	yes	—
127	C₂×C₄×C₉	#41	1	2	yes	—
128	C₂×C₄×C₉	#41	1	2	yes	—
129	C₂×C₄×C₉	#41	1	2	yes	—
130	#400	C₂×C₉	1	2	yes	—
131	#398	C₂×C₉	1	2	yes	—
132	#389	C₂×C₉	1	1	yes	characteristic
133	#389	C₂×C₉	1	1	yes	characteristic
134	#389	C₂×C₉	1	1	yes	characteristic
135	#389	C₂×C₉	1	1	yes	characteristic
136	#387	C₂×C₉	1	2	yes	—
137	#382	C₂×C₉	1	2	yes	—
138	#382	C₂×C₉	1	1	yes	characteristic
139	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
140	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
143	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
145	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
146	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
148	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
149	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
151	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
152	C₃×C₄⋊C₄	—	2	2	no	—
153	C₃×C₄⋊C₄	—	2	2	no	—
154	C₃×C₄⋊C₄	—	2	4	no	—
155	C₃×C₄⋊C₄	—	2	2	no	—
156	C₃×C₄⋊C₄	—	2	2	no	—
157	C₃×C₄⋊C₄	—	2	2	no	—
158	C₃×C₄⋊C₄	—	4	2	no	—
159	C₃×C₄⋊C₄	—	4	2	no	—
160	C₃×C₄⋊C₄	—	4	2	no	—
161	C₃×C₄⋊C₄	C₃×Q₈	1	4	yes	—
162	C₃×C₄²	—	2	4	no	—
163	C₃×C₄²	—	2	2	no	—
164	C₃×C₄²	—	2	2	no	—
165	C₃×C₄²	—	2	2	no	—
166	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
167	C₂²×C₉	C₂×#41	1	1	yes	characteristic
168	C₂²×C₉	C₂×#41	1	1	yes	characteristic
169	C₂²×C₉	C₂×#41	1	1	yes	characteristic
170	C₂²×C₉	C₂×#41	1	1	yes	characteristic
171	C₂²×C₉	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂²×C₉	2₋¹⁺⁴	1	1	yes	characteristic
173	C₄×C₉	—	2	2	no	—
174	C₄×C₉	—	2	2	no	—
175	C₄×C₉	—	2	4	no	—
176	C₄×C₉	—	4	2	no	—
177	C₄×C₉	—	4	1	no	—
178	C₄×C₉	—	4	2	no	—
179	C₄×C₉	—	4	1	no	—
180	C₄×C₉	—	4	1	no	—
181	C₄×C₉	—	4	1	no	—
182	C₄×C₉	—	4	2	no	—
183	C₄×C₉	#125	1	4	yes	—
184	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
189	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
191	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
192	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
193	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
194	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
195	C₂×C₄²	C₂²×C₉	1	2	yes	—
196	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
197	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
198	#95	C₂²×C₉	1	1	yes	characteristic
199	#95	C₂²×C₉	1	2	yes	—
200	#95	C₂²×C₉	1	1	yes	characteristic
201	#95	C₂²×C₉	1	2	yes	—
202	#95	C₂²×C₉	1	2	yes	—
203	#95	C₂²×C₉	1	2	yes	—
204	#95	C₂²×C₉	1	2	yes	—
205	#95	C₂²×C₉	1	2	yes	—
206	#95	C₂²×C₉	1	2	yes	—
207	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	2	1	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	—	2	1	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
236	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
237	C₂×C₃×C₄	C₃×#41	1	2	yes	—
238	C₂×C₃×C₄	C₃×#41	1	2	yes	—
239	C₂×C₃×C₄	C₃×#41	1	2	yes	—
240	C₂×C₃×C₄	C₃×#41	1	2	yes	—
241	C₂×C₉	C₂×#125	1	1	yes	characteristic
242	C₂×C₉	#529	1	1	yes	characteristic
243	C₂×C₉	#542	1	1	yes	characteristic
244	C₂×C₉	#552	1	1	yes	characteristic
245	C₂×C₉	#554	1	1	yes	characteristic
246	C₂×C₉	#555	1	1	yes	characteristic
247	C₂×C₉	#557	1	1	yes	characteristic
248	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
249	C₂²×C₄	C₂³×C₉	1	2	yes	—
250	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
251	C₂²×C₄	C₂³×C₉	1	2	yes	—
252	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
253	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
254	C₂²×C₄	C₂³×C₉	1	2	yes	—
255	C₂²×C₄	C₂³×C₉	1	2	yes	—
256	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
257	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
258	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	Q₈×C₉	1	4	yes	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
274	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
275	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
276	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
277	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
278	C₂²×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
279	C₂²×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	2	4	no	—
283	C₃×C₄	—	4	2	no	—
284	C₃×C₄	—	4	1	no	—
285	C₃×C₄	—	4	2	no	—
286	C₃×C₄	—	4	1	no	—
287	C₃×C₄	—	4	1	no	—
288	C₃×C₄	—	4	1	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	C₃×#125	1	4	yes	—
291	C₉	#2508	1	1	yes	characteristic, direct product factor, Sylow
292	C₂³	C₂⁴×C₉	1	1	yes	characteristic
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	C₂×Q₈×C₉	1	4	yes	—
321	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
322	C₂×C₄	C₉×#41	1	2	yes	—
323	C₂×C₄	C₉×#41	1	2	yes	—
324	C₂×C₄	C₉×#41	1	2	yes	—
325	C₂×C₄	C₉×#41	1	2	yes	—
326	C₂×C₃	C₂×C₃×#125	1	1	yes	characteristic
327	C₂×C₃	C₃×#529	1	1	yes	characteristic
328	C₂×C₃	C₃×#542	1	1	yes	characteristic
329	C₂×C₃	C₃×#552	1	1	yes	characteristic
330	C₂×C₃	C₃×#554	1	1	yes	characteristic
331	C₂×C₃	C₃×#555	1	1	yes	characteristic
332	C₂×C₃	C₃×#557	1	1	yes	characteristic
333	C₂²	C₂²×Q₈×C₉	1	1	yes	characteristic
334	C₂²	C₂×C₉×#41	1	1	yes	characteristic
335	C₂²	C₂×C₉×#41	1	1	yes	characteristic
336	C₂²	C₂×C₉×#41	1	1	yes	characteristic
337	C₂²	C₂×C₉×#41	1	1	yes	characteristic
338	C₂²	C₉×H(5,𝔽₂)	1	1	yes	characteristic
339	C₂²	C₉×2₋¹⁺⁴	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	4	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	2	no	—
350	C₄	C₉×#125	1	4	yes	—
351	C₃	C₃×#2508	1	1	yes	characteristic
352	C₂	C₂×C₉×#125	1	1	yes	characteristic
353	C₂	C₉×#529	1	1	yes	characteristic
354	C₂	C₉×#542	1	1	yes	characteristic
355	C₂	C₉×#552	1	1	yes	characteristic
356	C₂	C₉×#554	1	1	yes	characteristic
357	C₂	C₉×#555	1	1	yes	characteristic
358	C₂	C₉×#557	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, prime radical, cosocle	C₂³×C₃	C₂⁴×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
72. #2508	C₉	direct (N × H)
291. C₉	#2508	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#400	2
C₉×#398	2
C₉×#389	2
C₉×#387	2
C₉×#382	2
C₄×C₉×C₄⋊C₄	2
C₃×#2508	3

Subgroups — C9×#2508 #11767229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2508 #11767229