Subgroups — C₉×#1806 #11766527

Order 1152.

468 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	C₄×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#104	C₂	1	2	yes	maximal
4	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#339	C₂	1	2	yes	maximal, complemented
6	C₃×#1806	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#41	C₂²	1	1	yes	characteristic
8	C₂×C₉×M₁₆	C₂²	1	2	yes	—
9	C₄×Q₈×C₉	C₄	1	1	yes	characteristic, complemented
10	C₄×Q₈×C₉	C₄	1	1	yes	characteristic, complemented
11	C₄×D₄×C₉	—	2	1	no	—
12	C₄×D₄×C₉	—	2	1	no	—
13	C₄×D₄×C₉	C₄	1	1	yes	characteristic, complemented
14	C₄×D₄×C₉	C₄	1	1	yes	characteristic, complemented
15	C₉×#117	—	2	1	no	—
16	C₉×#117	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₄⋊C₄	—	2	1	no	—
18	C₂×C₄²×C₉	—	2	1	no	—
19	C₂×C₄²×C₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
21	C₉×C₄⋊C₈	—	2	1	no	—
22	C₉×C₄⋊C₈	—	2	1	no	—
23	C₉×#104	—	2	4	no	—
24	C₉×#95	—	2	1	no	—
25	C₃×C₄×#41	C₂×C₃	1	1	yes	characteristic
26	C₃×#423	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×#104	C₂×C₃	1	2	yes	—
28	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
29	C₃×#339	C₂×C₃	1	2	yes	—
30	C₉×#41	—	2	1	no	—
31	C₉×#41	—	2	1	no	—
32	C₉×#41	D₄	1	2	yes	—
33	C₉×#41	Q₈	1	2	yes	—
34	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₉	—	2	1	no	—
36	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₉	—	2	1	no	—
38	C₂²×C₄×C₉	—	2	2	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₉×M₁₆	—	2	2	no	—
45	C₉×M₁₆	—	2	2	no	—
46	C₂×C₈×C₉	—	2	2	no	—
47	C₉×C₄⋊C₄	—	2	4	no	—
48	C₉×C₄⋊C₄	—	2	1	no	—
49	C₉×C₄⋊C₄	—	2	1	no	—
50	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₉×C₂²⋊C₄	—	2	1	no	—
53	C₉×C₂²⋊C₄	—	2	1	no	—
54	C₉×C₂²⋊C₄	—	2	1	no	—
55	C₄²×C₉	—	2	1	no	—
56	C₄²×C₉	—	2	1	no	—
57	C₄²×C₉	—	2	1	no	—
58	C₄²×C₉	—	2	2	no	—
59	C₄²×C₉	—	2	2	no	—
60	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
62	C₄²×C₉	D₄	1	1	yes	characteristic
63	C₄²×C₉	D₄	1	1	yes	characteristic
64	#1806	C₉	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
67	C₃×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
68	C₃×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
69	C₃×C₄×D₄	—	2	1	no	—
70	C₃×C₄×D₄	—	2	1	no	—
71	C₃×C₄×D₄	C₃×C₄	1	1	yes	characteristic
72	C₃×C₄×D₄	C₃×C₄	1	1	yes	characteristic
73	C₃×#117	—	2	1	no	—
74	C₃×#117	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₃×C₄⋊C₄	—	2	1	no	—
76	C₂×C₃×C₄²	—	2	1	no	—
77	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
78	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
79	C₃×C₄⋊C₈	—	2	1	no	—
80	C₃×C₄⋊C₈	—	2	1	no	—
81	C₃×#104	—	2	4	no	—
82	C₃×#95	—	2	1	no	—
83	C₂³×C₉	—	2	1	no	—
84	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
85	Q₈×C₉	—	2	1	no	—
86	Q₈×C₉	C₄⋊C₄	1	2	yes	complemented
87	D₄×C₉	—	2	2	no	—
88	D₄×C₉	—	2	2	no	—
89	D₄×C₉	—	2	1	no	—
90	D₄×C₉	C₄⋊C₄	1	2	yes	complemented
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	4	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	4	1	no	—
99	C₂×C₄×C₉	—	4	2	no	—
100	C₂×C₄×C₉	—	4	2	no	—
101	C₂×C₄×C₉	—	4	2	no	—
102	C₂×C₄×C₉	—	4	2	no	—
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	1	yes	characteristic
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	1	yes	characteristic
109	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₉	#41	1	1	yes	characteristic
112	C₂×C₄×C₉	#41	1	1	yes	characteristic
113	C₈×C₉	—	4	2	no	—
114	C₄×#41	C₂×C₉	1	1	yes	characteristic
115	#423	C₂×C₉	1	1	yes	characteristic, complemented
116	C₂×#104	C₂×C₉	1	2	yes	—
117	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic, complemented
118	#339	C₂×C₉	1	2	yes	complemented
119	C₃×#41	—	2	1	no	—
120	C₃×#41	—	2	1	no	—
121	C₃×#41	C₃×D₄	1	2	yes	—
122	C₃×#41	C₃×Q₈	1	2	yes	—
123	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
124	C₂×C₃×D₄	—	2	1	no	—
125	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	—	2	1	no	—
129	C₂²×C₃×C₄	—	2	1	no	—
130	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
133	C₃×M₁₆	—	2	2	no	—
134	C₃×M₁₆	—	2	2	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₃×C₄⋊C₄	—	2	4	no	—
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
140	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
141	C₃×C₂²⋊C₄	—	2	1	no	—
142	C₃×C₂²⋊C₄	—	2	1	no	—
143	C₃×C₂²⋊C₄	—	2	1	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	—	2	1	no	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	—	2	2	no	—
148	C₃×C₄²	—	2	2	no	—
149	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
150	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
151	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
152	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
153	C₂²×C₉	—	2	1	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	2	no	—
156	C₂²×C₉	—	4	1	no	—
157	C₂²×C₉	—	4	1	no	—
158	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₉	#121	1	1	yes	characteristic
160	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₉	—	2	2	no	—
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	4	4	no	—
166	C₄×C₉	—	4	1	no	—
167	C₄×C₉	—	4	1	no	—
168	C₄×C₉	#104	1	2	yes	complemented
169	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₉	#117	1	1	yes	characteristic
171	C₄×C₉	#121	1	1	yes	characteristic
172	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×#41	C₂²×C₉	1	1	yes	characteristic
174	C₂×M₁₆	C₂²×C₉	1	2	yes	—
175	C₄×Q₈	C₄×C₉	1	1	yes	characteristic, complemented
176	C₄×Q₈	C₄×C₉	1	1	yes	characteristic, complemented
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	C₄×C₉	1	1	yes	characteristic, complemented
180	C₄×D₄	C₄×C₉	1	1	yes	characteristic, complemented
181	#117	—	2	1	no	—
182	#117	C₂²×C₉	1	1	yes	characteristic
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	C₂²×C₉	1	2	yes	—
186	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	2	1	no	—
189	#104	—	2	4	no	—
190	#95	—	2	1	no	—
191	C₂³×C₃	—	2	1	no	—
192	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
193	C₃×Q₈	—	2	1	no	—
194	C₃×Q₈	C₃×C₄⋊C₄	1	2	yes	—
195	C₃×D₄	—	2	2	no	—
196	C₃×D₄	—	2	2	no	—
197	C₃×D₄	—	2	1	no	—
198	C₃×D₄	C₃×C₄⋊C₄	1	2	yes	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	2	no	—
208	C₂×C₃×C₄	—	4	2	no	—
209	C₂×C₃×C₄	—	4	2	no	—
210	C₂×C₃×C₄	—	4	2	no	—
211	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
219	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
220	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
221	C₃×C₈	—	4	2	no	—
222	C₂×C₉	—	2	1	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	4	2	no	—
225	C₂×C₉	#380	1	1	yes	characteristic
226	C₂×C₉	C₂×#104	1	1	yes	characteristic
227	C₂×C₉	#421	1	1	yes	characteristic
228	#41	—	2	1	no	—
229	#41	—	2	1	no	—
230	#41	D₄×C₉	1	2	yes	—
231	#41	Q₈×C₉	1	2	yes	—
232	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
240	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
241	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂×C₄×C₉	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
259	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
260	C₄²	D₄×C₉	1	1	yes	characteristic
261	C₄²	D₄×C₉	1	1	yes	characteristic
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	4	1	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
268	C₂²×C₃	C₃×#121	1	1	yes	characteristic
269	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	4	4	no	—
275	C₃×C₄	—	4	1	no	—
276	C₃×C₄	—	4	1	no	—
277	C₃×C₄	C₃×#104	1	2	yes	—
278	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
279	C₃×C₄	C₃×#117	1	1	yes	characteristic
280	C₃×C₄	C₃×#121	1	1	yes	characteristic
281	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
282	C₉	#1806	1	1	yes	characteristic, direct product factor, Sylow
283	C₂³	—	2	1	no	—
284	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
285	Q₈	—	2	1	no	—
286	Q₈	C₉×C₄⋊C₄	1	2	yes	complemented
287	D₄	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	2	1	no	—
290	D₄	C₉×C₄⋊C₄	1	2	yes	complemented
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
304	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
310	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
311	C₂×C₄	C₉×#41	1	1	yes	characteristic
312	C₂×C₄	C₉×#41	1	1	yes	characteristic
313	C₈	—	4	2	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	1	no	—
316	C₂×C₃	—	4	2	no	—
317	C₂×C₃	C₃×#380	1	1	yes	characteristic
318	C₂×C₃	C₂×C₃×#104	1	1	yes	characteristic
319	C₂×C₃	C₃×#421	1	1	yes	characteristic
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
326	C₂²	C₉×#121	1	1	yes	characteristic
327	C₂²	C₉×C₂²⋊Q₈	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	C₉×#104	1	2	yes	complemented
336	C₄	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
337	C₄	C₉×#117	1	1	yes	characteristic
338	C₄	C₉×#121	1	1	yes	characteristic
339	C₄	C₉×C₂²⋊Q₈	1	1	yes	characteristic
340	C₃	C₃×#1806	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	4	2	no	—
344	C₂	C₉×#380	1	1	yes	characteristic
345	C₂	C₂×C₉×#104	1	1	yes	characteristic
346	C₂	C₉×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₂²⋊C₄
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#423	C₂	semidirect
4. C₄×C₉×C₄⋊C₄	C₂	semidirect
5. C₉×#339	C₂	semidirect
9. C₄×Q₈×C₉	C₄	semidirect
10. C₄×Q₈×C₉	C₄	semidirect
13. C₄×D₄×C₉	C₄	semidirect
14. C₄×D₄×C₉	C₄	semidirect
64. #1806	C₉	direct (N × H)
86. Q₈×C₉	C₄⋊C₄	semidirect
90. D₄×C₉	C₄⋊C₄	semidirect
115. #423	C₂×C₉	semidirect
117. C₄×C₄⋊C₄	C₂×C₉	semidirect
118. #339	C₂×C₉	semidirect
168. C₄×C₉	#104	semidirect
175. C₄×Q₈	C₄×C₉	semidirect
176. C₄×Q₈	C₄×C₉	semidirect
179. C₄×D₄	C₄×C₉	semidirect
180. C₄×D₄	C₄×C₉	semidirect
282. C₉	#1806	direct (N × H)
286. Q₈	C₉×C₄⋊C₄	semidirect
290. D₄	C₉×C₄⋊C₄	semidirect
335. C₄	C₉×#104	semidirect

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×#423	2
C₂×C₉×#104	2
C₄×C₉×C₄⋊C₄	2
C₉×#339	2
C₃×#1806	3

Subgroups — C9×#1806 #11766527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1806 #11766527