Subgroups — C₉×#2549 #11767270

Order 1152.

816 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₂²⋊Q₈	C₂	1	2	yes	maximal
2	C₉×#398	C₂	1	2	yes	maximal, complemented
3	C₉×#395	C₂	1	2	yes	maximal, complemented
4	C₉×#391	C₂	1	2	yes	maximal, complemented
5	C₉×#389	C₂	1	2	yes	maximal, complemented
6	C₉×#385	C₂	1	2	yes	maximal
7	C₉×#380	C₂	1	1	yes	maximal, characteristic
8	C₄×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₉×#95	C₂	1	1	yes	maximal, characteristic
10	C₃×#2549	C₃	1	1	yes	maximal, characteristic
11	C₂²×Q₈×C₉	C₂²	1	2	yes	—
12	C₂³×C₄×C₉	C₂²	1	2	yes	—
13	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
14	C₉×C₂²⋊Q₈	—	2	8	no	—
15	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₉×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₉	C₂²	1	2	yes	—
25	C₉×#95	C₂²	1	2	yes	—
26	C₉×#95	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	1	yes	characteristic
31	C₉×#95	C₂²	1	2	yes	—
32	C₉×#95	C₂²	1	2	yes	—
33	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
34	C₃×#398	C₂×C₃	1	2	yes	—
35	C₃×#395	C₂×C₃	1	2	yes	—
36	C₃×#391	C₂×C₃	1	2	yes	—
37	C₃×#389	C₂×C₃	1	2	yes	—
38	C₃×#385	C₂×C₃	1	2	yes	—
39	C₃×#380	C₂×C₃	1	1	yes	characteristic
40	C₃×C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
41	C₂×C₃×#95	C₂×C₃	1	1	yes	characteristic
42	C₂⁴×C₉	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₉	—	2	4	no	—
44	C₂×Q₈×C₉	—	2	4	no	—
45	C₂×Q₈×C₉	—	4	2	no	—
46	C₂²×C₄×C₉	—	2	4	no	—
47	C₂²×C₄×C₉	—	2	4	no	—
48	C₂²×C₄×C₉	—	4	2	no	—
49	C₂²×C₄×C₉	—	4	2	no	—
50	C₂²×C₄×C₉	Q₈	1	4	yes	complemented
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	2	yes	—
55	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₉	C₂³	1	2	yes	—
59	C₉×C₄⋊C₄	—	2	4	no	—
60	C₉×C₄⋊C₄	—	2	4	no	—
61	C₉×C₄⋊C₄	—	2	4	no	—
62	C₉×C₄⋊C₄	—	2	4	no	—
63	C₉×C₄⋊C₄	—	4	2	no	—
64	C₉×C₄⋊C₄	—	4	2	no	—
65	C₉×C₂²⋊C₄	—	2	4	no	—
66	C₉×C₂²⋊C₄	D₄	1	4	yes	—
67	C₉×C₂²⋊C₄	Q₈	1	4	yes	complemented
68	C₄²×C₉	—	4	2	no	—
69	#2549	C₉	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
71	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
72	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
73	C₃×C₂²⋊Q₈	—	2	8	no	—
74	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
84	C₃×#95	C₂²×C₃	1	2	yes	—
85	C₃×#95	C₂²×C₃	1	1	yes	characteristic
86	C₃×#95	C₂²×C₃	1	1	yes	characteristic
87	C₃×#95	C₂²×C₃	1	2	yes	—
88	C₃×#95	C₂²×C₃	1	1	yes	characteristic
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	C₂³×C₉	—	2	2	no	—
93	C₂³×C₉	—	2	2	no	—
94	C₂³×C₉	C₂×D₄	1	2	yes	—
95	C₂³×C₉	C₂×Q₈	1	2	yes	—
96	C₂³×C₉	C₂×Q₈	1	2	yes	—
97	C₂³×C₉	C₂⁴	1	1	yes	characteristic
98	Q₈×C₉	—	4	8	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	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	2	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	—	4	4	no	—
117	C₂×C₄×C₉	—	4	8	no	—
118	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
119	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
120	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
121	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
122	C₂×C₄×C₉	#41	1	4	yes	—
123	C₂×C₂²⋊Q₈	C₂×C₉	1	2	yes	—
124	#398	C₂×C₉	1	2	yes	complemented
125	#395	C₂×C₉	1	2	yes	complemented
126	#391	C₂×C₉	1	2	yes	complemented
127	#389	C₂×C₉	1	2	yes	complemented
128	#385	C₂×C₉	1	2	yes	—
129	#380	C₂×C₉	1	1	yes	characteristic
130	C₄×C₂²⋊C₄	C₂×C₉	1	1	yes	characteristic
131	C₂×#95	C₂×C₉	1	1	yes	characteristic
132	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
133	C₂×C₃×Q₈	—	2	4	no	—
134	C₂×C₃×Q₈	—	2	4	no	—
135	C₂×C₃×Q₈	—	4	2	no	—
136	C₂²×C₃×C₄	—	2	4	no	—
137	C₂²×C₃×C₄	—	2	4	no	—
138	C₂²×C₃×C₄	—	4	2	no	—
139	C₂²×C₃×C₄	—	4	2	no	—
140	C₂²×C₃×C₄	C₃×Q₈	1	4	yes	—
141	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
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	C₃×C₄⋊C₄	—	2	4	no	—
150	C₃×C₄⋊C₄	—	2	4	no	—
151	C₃×C₄⋊C₄	—	2	4	no	—
152	C₃×C₄⋊C₄	—	2	4	no	—
153	C₃×C₄⋊C₄	—	4	2	no	—
154	C₃×C₄⋊C₄	—	4	2	no	—
155	C₃×C₂²⋊C₄	—	2	4	no	—
156	C₃×C₂²⋊C₄	C₃×D₄	1	4	yes	—
157	C₃×C₂²⋊C₄	C₃×Q₈	1	4	yes	—
158	C₃×C₄²	—	4	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	2	4	no	—
162	C₂²×C₉	—	2	4	no	—
163	C₂²×C₉	C₄⋊Q₈	1	4	yes	—
164	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
165	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
166	C₂²×C₉	C₂²×Q₈	1	1	yes	characteristic
167	C₂²×C₉	C₂×#41	1	2	yes	—
168	C₂²×C₉	H(5,𝔽₂)	1	2	yes	—
169	C₄×C₉	—	2	2	no	—
170	C₄×C₉	—	2	2	no	—
171	C₄×C₉	—	2	4	no	—
172	C₄×C₉	—	2	4	no	—
173	C₄×C₉	—	2	4	no	—
174	C₄×C₉	—	4	2	no	—
175	C₄×C₉	—	4	2	no	—
176	C₄×C₉	—	4	2	no	—
177	C₂²×Q₈	C₂²×C₉	1	2	yes	—
178	C₂³×C₄	C₂²×C₉	1	2	yes	—
179	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
180	C₂²⋊Q₈	—	2	8	no	—
181	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
182	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
183	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
184	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
185	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
186	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
188	C₂×C₂²⋊C₄	C₂²×C₉	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	C₂²×C₉	1	1	yes	characteristic
190	C₂×C₄²	C₂²×C₉	1	2	yes	—
191	#95	C₂²×C₉	1	2	yes	—
192	#95	C₂²×C₉	1	1	yes	characteristic
193	#95	C₂²×C₉	1	1	yes	characteristic
194	#95	C₂²×C₉	1	2	yes	—
195	#95	C₂²×C₉	1	1	yes	characteristic
196	#95	C₂²×C₉	1	1	yes	characteristic
197	#95	C₂²×C₉	1	2	yes	—
198	#95	C₂²×C₉	1	2	yes	—
199	C₂³×C₃	—	2	2	no	—
200	C₂³×C₃	—	2	2	no	—
201	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
202	C₂³×C₃	C₂×C₃×Q₈	1	2	yes	—
203	C₂³×C₃	C₂×C₃×Q₈	1	2	yes	—
204	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
205	C₃×Q₈	—	4	8	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	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	2	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	4	4	no	—
224	C₂×C₃×C₄	—	4	8	no	—
225	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
226	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
227	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
228	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
229	C₂×C₃×C₄	C₃×#41	1	4	yes	—
230	C₂×C₉	—	2	4	no	—
231	C₂×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
232	C₂×C₉	#533	1	1	yes	characteristic
233	C₂×C₉	#543	1	1	yes	characteristic
234	C₂×C₉	#546	1	2	yes	—
235	C₂×C₉	#554	1	2	yes	—
236	C₂⁴	C₂³×C₉	1	1	yes	characteristic
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	Q₈×C₉	1	4	yes	complemented
245	C₂²×C₄	C₂³×C₉	1	2	yes	—
246	C₂²×C₄	C₂³×C₉	1	2	yes	—
247	C₂²×C₄	C₂³×C₉	1	2	yes	—
248	C₂²×C₄	C₂³×C₉	1	2	yes	—
249	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
250	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
251	C₂²×C₄	C₂³×C₉	1	2	yes	—
252	C₂²×C₄	C₂³×C₉	1	2	yes	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	D₄×C₉	1	4	yes	—
261	C₂²⋊C₄	Q₈×C₉	1	4	yes	complemented
262	C₄²	—	4	2	no	—
263	C₂²×C₃	—	2	2	no	—
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	2	4	no	—
266	C₂²×C₃	—	2	4	no	—
267	C₂²×C₃	C₃×C₄⋊Q₈	1	4	yes	—
268	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
269	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
270	C₂²×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
271	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
272	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
273	C₃×C₄	—	2	2	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	4	no	—
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	—	2	4	no	—
278	C₃×C₄	—	4	2	no	—
279	C₃×C₄	—	4	2	no	—
280	C₃×C₄	—	4	2	no	—
281	C₉	#2549	1	1	yes	characteristic, direct product factor, Sylow
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	C₂×D₄×C₉	1	2	yes	—
285	C₂³	C₂×Q₈×C₉	1	2	yes	—
286	C₂³	C₂×Q₈×C₉	1	2	yes	—
287	C₂³	C₂⁴×C₉	1	1	yes	characteristic
288	Q₈	—	4	8	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	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	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	8	no	—
308	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
309	C₂×C₄	C₂×Q₈×C₉	1	4	yes	—
310	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
311	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
312	C₂×C₄	C₉×#41	1	4	yes	—
313	C₂×C₃	—	2	4	no	—
314	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
315	C₂×C₃	C₃×#533	1	1	yes	characteristic
316	C₂×C₃	C₃×#543	1	1	yes	characteristic
317	C₂×C₃	C₃×#546	1	2	yes	—
318	C₂×C₃	C₃×#554	1	2	yes	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	2	4	no	—
323	C₂²	C₉×C₄⋊Q₈	1	4	yes	—
324	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
325	C₂²	C₂²×Q₈×C₉	1	1	yes	characteristic
326	C₂²	C₂²×Q₈×C₉	1	1	yes	characteristic
327	C₂²	C₂×C₉×#41	1	2	yes	—
328	C₂²	C₉×H(5,𝔽₂)	1	2	yes	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	4	no	—
332	C₄	—	2	4	no	—
333	C₄	—	2	4	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₃	C₃×#2549	1	1	yes	characteristic
338	C₂	—	2	4	no	—
339	C₂	C₂×C₉×C₄⋊Q₈	1	1	yes	characteristic
340	C₂	C₉×#533	1	1	yes	characteristic
341	C₂	C₉×#543	1	1	yes	characteristic
342	C₂	C₉×#546	1	2	yes	—
343	C₂	C₉×#554	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂³	C₂⁴×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³×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
2. C₉×#398	C₂	semidirect
3. C₉×#395	C₂	semidirect
4. C₉×#391	C₂	semidirect
5. C₉×#389	C₂	semidirect
50. C₂²×C₄×C₉	Q₈	semidirect
67. C₉×C₂²⋊C₄	Q₈	semidirect
69. #2549	C₉	direct (N × H)
124. #398	C₂×C₉	semidirect
125. #395	C₂×C₉	semidirect
126. #391	C₂×C₉	semidirect
127. #389	C₂×C₉	semidirect
244. C₂²×C₄	Q₈×C₉	semidirect
261. C₂²⋊C₄	Q₈×C₉	semidirect
281. C₉	#2549	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×C₂²⋊Q₈	2
C₉×#398	2
C₉×#395	2
C₉×#391	2
C₉×#389	2
C₉×#385	2
C₉×#380	2
C₄×C₉×C₂²⋊C₄	2
C₂×C₉×#95	2
C₃×#2549	3

Subgroups — C9×#2549 #11767270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2549 #11767270