Subgroups — C₉×#2545 #11767266

Order 1152.

864 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#124	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#396	C₂	1	4	yes	maximal, complemented
6	C₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#390	C₂	1	1	yes	maximal, characteristic
8	C₉×#388	C₂	1	4	yes	maximal, complemented
9	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#2545	C₃	1	1	yes	maximal, characteristic
11	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊Q₈	—	2	4	no	—
15	C₉×#124	—	2	4	no	—
16	C₉×#124	—	2	4	no	—
17	C₄×Q₈×C₉	—	2	4	no	—
18	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₉	C₂²	1	2	yes	—
29	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
30	C₉×#95	C₂²	1	2	yes	—
31	C₉×#95	C₂²	1	4	yes	—
32	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
33	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
34	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
35	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
36	C₃×#396	C₂×C₃	1	4	yes	—
37	C₃×#391	C₂×C₃	1	1	yes	characteristic
38	C₃×#390	C₂×C₃	1	1	yes	characteristic
39	C₃×#388	C₂×C₃	1	4	yes	—
40	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
41	C₂⁴×C₉	C₂³	1	2	yes	—
42	C₂×Q₈×C₉	—	2	2	no	—
43	C₂×Q₈×C₉	—	2	2	no	—
44	C₂×Q₈×C₉	—	2	2	no	—
45	C₂×Q₈×C₉	—	2	2	no	—
46	C₂×Q₈×C₉	—	2	2	no	—
47	C₂×Q₈×C₉	D₄	1	4	yes	—
48	C₂×D₄×C₉	—	2	2	no	—
49	C₂×D₄×C₉	—	2	2	no	—
50	C₂×D₄×C₉	—	2	2	no	—
51	C₂²×C₄×C₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
59	C₉×C₄⋊C₄	—	2	4	no	—
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	—	2	8	no	—
62	C₉×C₄⋊C₄	—	2	4	no	—
63	C₉×C₂²⋊C₄	—	2	8	no	—
64	C₉×C₂²⋊C₄	—	2	4	no	—
65	C₉×C₂²⋊C₄	—	2	4	no	—
66	C₉×C₂²⋊C₄	—	2	8	no	—
67	C₄²×C₉	—	2	4	no	—
68	C₄²×C₉	—	2	2	no	—
69	C₄²×C₉	—	2	2	no	—
70	C₄²×C₉	D₄	1	4	yes	—
71	#2545	C₉	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
73	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
74	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄⋊Q₈	—	2	4	no	—
76	C₃×#124	—	2	4	no	—
77	C₃×#124	—	2	4	no	—
78	C₃×C₄×Q₈	—	2	4	no	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
81	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
85	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
86	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
90	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
91	C₃×#95	C₂²×C₃	1	2	yes	—
92	C₃×#95	C₂²×C₃	1	4	yes	—
93	C₂³×C₉	—	2	2	no	—
94	C₂³×C₉	—	2	2	no	—
95	C₂³×C₉	—	2	4	no	—
96	C₂³×C₉	—	2	2	no	—
97	C₂³×C₉	—	2	4	no	—
98	C₂³×C₉	C₂⁴	1	1	yes	characteristic
99	Q₈×C₉	—	2	8	no	—
100	Q₈×C₉	—	4	4	no	—
101	D₄×C₉	—	4	4	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	2	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	2	no	—
112	C₂×C₄×C₉	—	2	4	no	—
113	C₂×C₄×C₉	—	2	2	no	—
114	C₂×C₄×C₉	—	2	2	no	—
115	C₂×C₄×C₉	—	2	2	no	—
116	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
119	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₉	#41	1	4	yes	—
122	C₂×C₄×C₉	#41	1	2	yes	—
123	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
124	C₂×#124	C₂×C₉	1	1	yes	characteristic
125	C₂×#124	C₂×C₉	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
127	#396	C₂×C₉	1	4	yes	complemented
128	#391	C₂×C₉	1	1	yes	characteristic, complemented
129	#390	C₂×C₉	1	1	yes	characteristic
130	#388	C₂×C₉	1	4	yes	complemented
131	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
132	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
133	C₂×C₃×Q₈	—	2	2	no	—
134	C₂×C₃×Q₈	—	2	2	no	—
135	C₂×C₃×Q₈	—	2	2	no	—
136	C₂×C₃×Q₈	—	2	2	no	—
137	C₂×C₃×Q₈	—	2	2	no	—
138	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
139	C₂×C₃×D₄	—	2	2	no	—
140	C₂×C₃×D₄	—	2	2	no	—
141	C₂×C₃×D₄	—	2	2	no	—
142	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
143	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
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	C₃×C₄⋊C₄	—	2	4	no	—
151	C₃×C₄⋊C₄	—	2	2	no	—
152	C₃×C₄⋊C₄	—	2	8	no	—
153	C₃×C₄⋊C₄	—	2	4	no	—
154	C₃×C₂²⋊C₄	—	2	8	no	—
155	C₃×C₂²⋊C₄	—	2	4	no	—
156	C₃×C₂²⋊C₄	—	2	4	no	—
157	C₃×C₂²⋊C₄	—	2	8	no	—
158	C₃×C₄²	—	2	4	no	—
159	C₃×C₄²	—	2	2	no	—
160	C₃×C₄²	—	2	2	no	—
161	C₃×C₄²	C₃×D₄	1	4	yes	—
162	C₂²×C₉	—	4	2	no	—
163	C₂²×C₉	—	4	2	no	—
164	C₂²×C₉	—	4	2	no	—
165	C₂²×C₉	—	4	4	no	—
166	C₂²×C₉	—	4	4	no	—
167	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
168	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
169	C₂²×C₉	C₂×#41	1	1	yes	characteristic
170	C₂²×C₉	C₂×#41	1	2	yes	—
171	C₂²×C₉	2₋¹⁺⁴	1	2	yes	—
172	C₄×C₉	—	2	4	no	—
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	2	no	—
178	C₄×C₉	—	4	2	no	—
179	C₄×C₉	#121	1	4	yes	—
180	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
181	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
182	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	4	no	—
184	#124	—	2	4	no	—
185	#124	—	2	4	no	—
186	C₄×Q₈	—	2	4	no	—
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	4	yes	—
190	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
193	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
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	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
199	#95	C₂²×C₉	1	2	yes	—
200	#95	C₂²×C₉	1	4	yes	—
201	C₂³×C₃	—	2	2	no	—
202	C₂³×C₃	—	2	2	no	—
203	C₂³×C₃	—	2	4	no	—
204	C₂³×C₃	—	2	2	no	—
205	C₂³×C₃	—	2	4	no	—
206	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
207	C₃×Q₈	—	2	8	no	—
208	C₃×Q₈	—	4	4	no	—
209	C₃×D₄	—	4	4	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
225	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
226	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
227	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
228	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
229	C₂×C₃×C₄	C₃×#41	1	4	yes	—
230	C₂×C₃×C₄	C₃×#41	1	2	yes	—
231	C₂×C₉	—	8	2	no	—
232	C₂×C₉	C₂×#121	1	1	yes	characteristic
233	C₂×C₉	#532	1	1	yes	characteristic
234	C₂×C₉	#536	1	1	yes	characteristic
235	C₂×C₉	#548	1	2	yes	—
236	C₂×C₉	#556	1	2	yes	—
237	C₂⁴	C₂³×C₉	1	2	yes	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	D₄×C₉	1	4	yes	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂²×C₄	C₂³×C₉	1	2	yes	—
248	C₂²×C₄	C₂³×C₉	1	2	yes	—
249	C₂²×C₄	C₂³×C₉	1	2	yes	—
250	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
251	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
252	C₂²×C₄	C₂³×C₉	1	2	yes	—
253	C₂²×C₄	C₂³×C₉	1	2	yes	—
254	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	8	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	D₄×C₉	1	4	yes	—
267	C₂²×C₃	—	4	2	no	—
268	C₂²×C₃	—	4	2	no	—
269	C₂²×C₃	—	4	2	no	—
270	C₂²×C₃	—	4	4	no	—
271	C₂²×C₃	—	4	4	no	—
272	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
273	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
274	C₂²×C₃	C₂×C₃×#41	1	1	yes	characteristic
275	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
276	C₂²×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
277	C₃×C₄	—	2	4	no	—
278	C₃×C₄	—	2	2	no	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	2	4	no	—
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	—	4	2	no	—
283	C₃×C₄	—	4	2	no	—
284	C₃×C₄	C₃×#121	1	4	yes	—
285	C₉	#2545	1	1	yes	characteristic, direct product factor, Sylow
286	C₂³	—	2	2	no	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	4	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	4	no	—
291	C₂³	C₂⁴×C₉	1	1	yes	characteristic
292	Q₈	—	2	8	no	—
293	Q₈	—	4	4	no	—
294	D₄	—	4	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
310	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
311	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
312	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
313	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
314	C₂×C₄	C₉×#41	1	4	yes	—
315	C₂×C₄	C₉×#41	1	2	yes	—
316	C₂×C₃	—	8	2	no	—
317	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
318	C₂×C₃	C₃×#532	1	1	yes	characteristic
319	C₂×C₃	C₃×#536	1	1	yes	characteristic
320	C₂×C₃	C₃×#548	1	2	yes	—
321	C₂×C₃	C₃×#556	1	2	yes	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	4	4	no	—
327	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
328	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
329	C₂²	C₂×C₉×#41	1	1	yes	characteristic
330	C₂²	C₂×C₉×#41	1	2	yes	—
331	C₂²	C₉×2₋¹⁺⁴	1	2	yes	—
332	C₄	—	2	4	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	4	no	—
336	C₄	—	4	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	2	no	—
339	C₄	C₉×#121	1	4	yes	—
340	C₃	C₃×#2545	1	1	yes	characteristic
341	C₂	—	8	2	no	—
342	C₂	C₂×C₉×#121	1	1	yes	characteristic
343	C₂	C₉×#532	1	1	yes	characteristic
344	C₂	C₉×#536	1	1	yes	characteristic
345	C₂	C₉×#548	1	2	yes	—
346	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	C₂²×C₃×D₄	C₂²×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
1. C₂×C₉×C₄⋊Q₈	C₂	semidirect
4. C₂×C₄×Q₈×C₉	C₂	semidirect
5. C₉×#396	C₂	semidirect
6. C₉×#391	C₂	semidirect
8. C₉×#388	C₂	semidirect
9. C₄×C₉×C₄⋊C₄	C₂	semidirect
71. #2545	C₉	direct (N × H)
123. C₂×C₄⋊Q₈	C₂×C₉	semidirect
126. C₂×C₄×Q₈	C₂×C₉	semidirect
127. #396	C₂×C₉	semidirect
128. #391	C₂×C₉	semidirect
130. #388	C₂×C₉	semidirect
131. C₄×C₄⋊C₄	C₂×C₉	semidirect
285. C₉	#2545	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×C₄⋊Q₈	2
C₂×C₉×#124	2
C₂×C₄×Q₈×C₉	2
C₉×#396	2
C₉×#391	2
C₉×#390	2
C₉×#388	2
C₄×C₉×C₄⋊C₄	2
C₃×#2545	3

Subgroups — C9×#2545 #11767266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2545 #11767266