Subgroups — C₉×#1882 #11766603

Order 1152.

501 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#137	C₂	1	1	yes	maximal, characteristic
2	C₉×#520	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#104	C₂	1	1	yes	maximal, characteristic
4	C₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#343	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#339	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1882	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₉×#137	—	2	2	no	—
12	C₉×#137	C₄	1	4	yes	complemented
13	C₂×C₉×Q₁₆	—	2	1	no	—
14	C₂×C₉×SD₁₆	—	2	1	no	—
15	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₉	—	2	1	no	—
18	C₄×D₄×C₉	—	2	1	no	—
19	C₄×D₄×C₉	—	2	2	no	—
20	C₉×#117	—	2	1	no	—
21	C₉×#117	C₂²	1	1	yes	characteristic
22	C₂×C₉×C₄⋊C₄	—	2	1	no	—
23	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
25	C₉×#104	—	2	2	no	—
26	C₉×#103	—	2	1	no	—
27	C₉×#103	—	2	1	no	—
28	C₉×#95	—	2	2	no	—
29	C₂×C₃×#137	C₂×C₃	1	1	yes	characteristic
30	C₃×#520	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×#104	C₂×C₃	1	1	yes	characteristic
32	C₃×#419	C₂×C₃	1	1	yes	characteristic
33	C₃×#391	C₂×C₃	1	1	yes	characteristic
34	C₃×#343	C₂×C₃	1	1	yes	characteristic
35	C₃×#339	C₂×C₃	1	1	yes	characteristic
36	C₉×#41	—	2	1	no	—
37	C₉×#41	—	2	1	no	—
38	C₉×#41	C₂×C₄	1	2	yes	—
39	C₉×#41	D₄	1	2	yes	—
40	C₂×Q₈×C₉	—	2	1	no	—
41	C₂×Q₈×C₉	—	2	1	no	—
42	C₂×Q₈×C₉	—	2	1	no	—
43	C₂×Q₈×C₉	—	2	1	no	—
44	C₂×Q₈×C₉	—	2	1	no	—
45	C₂×Q₈×C₉	C₂×C₄	1	2	yes	complemented
46	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
47	C₂×D₄×C₉	—	2	1	no	—
48	C₂×D₄×C₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₉	—	2	1	no	—
50	C₂²×C₄×C₉	—	2	1	no	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	—	2	2	no	—
53	C₂²×C₄×C₉	D₄	1	2	yes	—
54	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
55	C₉×Q₁₆	—	2	4	no	—
56	C₉×SD₁₆	—	2	4	no	—
57	C₉×M₁₆	—	2	1	no	—
58	C₉×M₁₆	—	2	1	no	—
59	C₉×M₁₆	—	2	1	no	—
60	C₉×M₁₆	C₂×C₄	1	2	yes	complemented
61	C₂×C₈×C₉	—	2	1	no	—
62	C₂×C₈×C₉	—	2	1	no	—
63	C₉×C₄⋊C₄	—	2	2	no	—
64	C₉×C₄⋊C₄	—	2	1	no	—
65	C₉×C₄⋊C₄	—	2	1	no	—
66	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
67	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
68	C₉×C₂²⋊C₄	—	2	2	no	—
69	C₉×C₂²⋊C₄	—	2	1	no	—
70	C₄²×C₉	—	2	2	no	—
71	C₄²×C₉	—	2	1	no	—
72	C₄²×C₉	—	2	1	no	—
73	C₄²×C₉	—	2	1	no	—
74	#1882	C₉	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
76	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
77	C₃×#137	—	2	2	no	—
78	C₃×#137	C₃×C₄	1	4	yes	—
79	C₂×C₃×Q₁₆	—	2	1	no	—
80	C₂×C₃×SD₁₆	—	2	1	no	—
81	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
83	C₃×C₄×Q₈	—	2	1	no	—
84	C₃×C₄×D₄	—	2	1	no	—
85	C₃×C₄×D₄	—	2	2	no	—
86	C₃×#117	—	2	1	no	—
87	C₃×#117	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₃×C₄⋊C₄	—	2	1	no	—
89	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
91	C₃×#104	—	2	2	no	—
92	C₃×#103	—	2	1	no	—
93	C₃×#103	—	2	1	no	—
94	C₃×#95	—	2	2	no	—
95	C₂³×C₉	—	2	1	no	—
96	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₉	—	2	1	no	—
98	Q₈×C₉	—	2	2	no	—
99	Q₈×C₉	—	2	2	no	—
100	Q₈×C₉	—	4	1	no	—
101	Q₈×C₉	—	4	1	no	—
102	Q₈×C₉	C₂²⋊C₄	1	2	yes	—
103	D₄×C₉	—	2	2	no	—
104	D₄×C₉	—	2	2	no	—
105	D₄×C₉	—	2	1	no	—
106	D₄×C₉	C₂²⋊C₄	1	2	yes	—
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	2	no	—
112	C₂×C₄×C₉	—	2	1	no	—
113	C₂×C₄×C₉	—	2	2	no	—
114	C₂×C₄×C₉	—	2	1	no	—
115	C₂×C₄×C₉	—	4	1	no	—
116	C₂×C₄×C₉	—	4	1	no	—
117	C₂×C₄×C₉	—	4	1	no	—
118	C₂×C₄×C₉	—	4	1	no	—
119	C₂×C₄×C₉	—	4	2	no	—
120	C₂×C₄×C₉	—	8	2	no	—
121	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₉	#41	1	1	yes	characteristic
126	C₂×C₄×C₉	#41	1	1	yes	characteristic
127	C₈×C₉	—	2	2	no	—
128	C₈×C₉	—	4	1	no	—
129	C₂×#137	C₂×C₉	1	1	yes	characteristic
130	#520	C₂×C₉	1	1	yes	characteristic
131	C₂×#104	C₂×C₉	1	1	yes	characteristic
132	#419	C₂×C₉	1	1	yes	characteristic, complemented
133	#391	C₂×C₉	1	1	yes	characteristic, complemented
134	#343	C₂×C₉	1	1	yes	characteristic, complemented
135	#339	C₂×C₉	1	1	yes	characteristic, complemented
136	C₃×#41	—	2	1	no	—
137	C₃×#41	—	2	1	no	—
138	C₃×#41	C₂×C₃×C₄	1	2	yes	—
139	C₃×#41	C₃×D₄	1	2	yes	—
140	C₂×C₃×Q₈	—	2	1	no	—
141	C₂×C₃×Q₈	—	2	1	no	—
142	C₂×C₃×Q₈	—	2	1	no	—
143	C₂×C₃×Q₈	—	2	1	no	—
144	C₂×C₃×Q₈	—	2	1	no	—
145	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
146	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
147	C₂×C₃×D₄	—	2	1	no	—
148	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
149	C₂²×C₃×C₄	—	2	1	no	—
150	C₂²×C₃×C₄	—	2	1	no	—
151	C₂²×C₃×C₄	—	2	1	no	—
152	C₂²×C₃×C₄	—	2	2	no	—
153	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
154	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
155	C₃×Q₁₆	—	2	4	no	—
156	C₃×SD₁₆	—	2	4	no	—
157	C₃×M₁₆	—	2	1	no	—
158	C₃×M₁₆	—	2	1	no	—
159	C₃×M₁₆	—	2	1	no	—
160	C₃×M₁₆	C₂×C₃×C₄	1	2	yes	—
161	C₂×C₃×C₈	—	2	1	no	—
162	C₂×C₃×C₈	—	2	1	no	—
163	C₃×C₄⋊C₄	—	2	2	no	—
164	C₃×C₄⋊C₄	—	2	1	no	—
165	C₃×C₄⋊C₄	—	2	1	no	—
166	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
167	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
168	C₃×C₂²⋊C₄	—	2	2	no	—
169	C₃×C₂²⋊C₄	—	2	1	no	—
170	C₃×C₄²	—	2	2	no	—
171	C₃×C₄²	—	2	1	no	—
172	C₃×C₄²	—	2	1	no	—
173	C₃×C₄²	—	2	1	no	—
174	C₂²×C₉	—	2	1	no	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	2	2	no	—
177	C₂²×C₉	—	4	1	no	—
178	C₂²×C₉	—	4	1	no	—
179	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
180	C₂²×C₉	#120	1	1	yes	characteristic
181	C₂²×C₉	#121	1	1	yes	characteristic
182	C₄×C₉	—	2	2	no	—
183	C₄×C₉	—	4	2	no	—
184	C₄×C₉	—	4	2	no	—
185	C₄×C₉	—	4	2	no	—
186	C₄×C₉	—	4	1	no	—
187	C₄×C₉	—	4	1	no	—
188	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₄×C₉	C₄×D₄	1	1	yes	characteristic
190	C₄×C₉	#121	1	1	yes	characteristic
191	C₄×C₉	#123	1	1	yes	characteristic
192	C₂×#41	C₂²×C₉	1	1	yes	characteristic
193	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
194	#137	—	2	2	no	—
195	#137	C₄×C₉	1	4	yes	complemented
196	C₂×Q₁₆	—	2	1	no	—
197	C₂×SD₁₆	—	2	1	no	—
198	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
199	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
200	C₄×Q₈	—	2	1	no	—
201	C₄×D₄	—	2	1	no	—
202	C₄×D₄	—	2	2	no	—
203	#117	—	2	1	no	—
204	#117	C₂²×C₉	1	1	yes	characteristic
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
207	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
208	#104	—	2	2	no	—
209	#103	—	2	1	no	—
210	#103	—	2	1	no	—
211	#95	—	2	2	no	—
212	C₂³×C₃	—	2	1	no	—
213	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
214	C₃×Q₈	—	2	1	no	—
215	C₃×Q₈	—	2	2	no	—
216	C₃×Q₈	—	2	2	no	—
217	C₃×Q₈	—	4	1	no	—
218	C₃×Q₈	—	4	1	no	—
219	C₃×Q₈	C₃×C₂²⋊C₄	1	2	yes	—
220	C₃×D₄	—	2	2	no	—
221	C₃×D₄	—	2	2	no	—
222	C₃×D₄	—	2	1	no	—
223	C₃×D₄	C₃×C₂²⋊C₄	1	2	yes	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	4	1	no	—
233	C₂×C₃×C₄	—	4	1	no	—
234	C₂×C₃×C₄	—	4	1	no	—
235	C₂×C₃×C₄	—	4	1	no	—
236	C₂×C₃×C₄	—	4	2	no	—
237	C₂×C₃×C₄	—	8	2	no	—
238	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
241	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
242	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
243	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
244	C₃×C₈	—	2	2	no	—
245	C₃×C₈	—	4	1	no	—
246	C₂×C₉	—	2	1	no	—
247	C₂×C₉	—	2	1	no	—
248	C₂×C₉	—	4	2	no	—
249	C₂×C₉	#386	1	1	yes	characteristic
250	C₂×C₉	#455	1	1	yes	characteristic
251	C₂×C₉	#456	1	1	yes	characteristic
252	#41	—	2	1	no	—
253	#41	—	2	1	no	—
254	#41	C₂×C₄×C₉	1	2	yes	—
255	#41	D₄×C₉	1	2	yes	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	C₂×C₄×C₉	1	2	yes	complemented
262	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	D₄×C₉	1	1	yes	characteristic
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	D₄×C₉	1	2	yes	—
270	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
271	Q₁₆	—	2	4	no	—
272	SD₁₆	—	2	4	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	2	1	no	—
276	M₁₆	C₂×C₄×C₉	1	2	yes	complemented
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
283	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
284	C₂²⋊C₄	—	2	2	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	2	1	no	—
290	C₂²×C₃	—	2	1	no	—
291	C₂²×C₃	—	2	1	no	—
292	C₂²×C₃	—	2	2	no	—
293	C₂²×C₃	—	4	1	no	—
294	C₂²×C₃	—	4	1	no	—
295	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
296	C₂²×C₃	C₃×#120	1	1	yes	characteristic
297	C₂²×C₃	C₃×#121	1	1	yes	characteristic
298	C₃×C₄	—	2	2	no	—
299	C₃×C₄	—	4	2	no	—
300	C₃×C₄	—	4	2	no	—
301	C₃×C₄	—	4	2	no	—
302	C₃×C₄	—	4	1	no	—
303	C₃×C₄	—	4	1	no	—
304	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
305	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
306	C₃×C₄	C₃×#121	1	1	yes	characteristic
307	C₃×C₄	C₃×#123	1	1	yes	characteristic
308	C₉	#1882	1	1	yes	characteristic, direct product factor, Sylow
309	C₂³	—	2	1	no	—
310	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
311	Q₈	—	2	1	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	4	1	no	—
315	Q₈	—	4	1	no	—
316	Q₈	C₉×C₂²⋊C₄	1	2	yes	—
317	D₄	—	2	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	2	1	no	—
320	D₄	C₉×C₂²⋊C₄	1	2	yes	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
336	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
337	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
338	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
339	C₂×C₄	C₉×#41	1	1	yes	characteristic
340	C₂×C₄	C₉×#41	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	4	1	no	—
343	C₂×C₃	—	2	1	no	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	4	2	no	—
346	C₂×C₃	C₃×#386	1	1	yes	characteristic
347	C₂×C₃	C₃×#455	1	1	yes	characteristic
348	C₂×C₃	C₃×#456	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
355	C₂²	C₉×#120	1	1	yes	characteristic
356	C₂²	C₉×#121	1	1	yes	characteristic
357	C₄	—	2	2	no	—
358	C₄	—	4	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	1	no	—
362	C₄	—	4	1	no	—
363	C₄	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
364	C₄	C₄×D₄×C₉	1	1	yes	characteristic
365	C₄	C₉×#121	1	1	yes	characteristic
366	C₄	C₉×#123	1	1	yes	characteristic
367	C₃	C₃×#1882	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	4	2	no	—
371	C₂	C₉×#386	1	1	yes	characteristic
372	C₂	C₉×#455	1	1	yes	characteristic
373	C₂	C₉×#456	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×#419	C₂	semidirect
5. C₉×#391	C₂	semidirect
6. C₉×#343	C₂	semidirect
7. C₉×#339	C₂	semidirect
12. C₉×#137	C₄	semidirect
45. C₂×Q₈×C₉	C₂×C₄	semidirect
60. C₉×M₁₆	C₂×C₄	semidirect
74. #1882	C₉	direct (N × H)
132. #419	C₂×C₉	semidirect
133. #391	C₂×C₉	semidirect
134. #343	C₂×C₉	semidirect
135. #339	C₂×C₉	semidirect
195. #137	C₄×C₉	semidirect
261. C₂×Q₈	C₂×C₄×C₉	semidirect
276. M₁₆	C₂×C₄×C₉	semidirect
308. C₉	#1882	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×#137	2
C₉×#520	2
C₂×C₉×#104	2
C₉×#419	2
C₉×#391	2
C₉×#343	2
C₉×#339	2
C₃×#1882	3

Subgroups — C9×#1882 #11766603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1882 #11766603