Subgroups — C₉×#1859 #11766580

Order 1152.

453 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#520	C₂	1	1	yes	maximal, characteristic
2	C₉×#428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#104	C₂	1	1	yes	maximal, characteristic
4	C₉×#417	C₂	1	1	yes	maximal, characteristic
5	C₉×#384	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×#1859	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
12	C₄×Q₈×C₉	—	2	1	no	—
13	C₄×D₄×C₉	—	2	1	no	—
14	C₄×D₄×C₉	—	2	2	no	—
15	C₉×#117	—	2	1	no	—
16	C₉×#117	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₄⋊C₄	—	2	1	no	—
18	C₂×C₉×C₄⋊C₄	—	2	2	no	—
19	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
22	C₉×#107	—	2	1	no	—
23	C₉×#106	—	2	1	no	—
24	C₉×#104	C₄	1	4	yes	complemented
25	C₉×#103	—	2	1	no	—
26	C₉×#102	—	2	1	no	—
27	C₃×#520	C₂×C₃	1	1	yes	characteristic
28	C₃×#428	C₂×C₃	1	1	yes	characteristic
29	C₂×C₃×#104	C₂×C₃	1	1	yes	characteristic
30	C₃×#417	C₂×C₃	1	1	yes	characteristic
31	C₃×#384	C₂×C₃	1	1	yes	characteristic
32	C₃×#343	C₂×C₃	1	1	yes	characteristic
33	C₃×#339	C₂×C₃	1	1	yes	characteristic
34	C₉×#41	—	2	1	no	—
35	C₉×#41	—	2	1	no	—
36	C₉×#41	C₂×C₄	1	2	yes	—
37	C₉×#41	Q₈	1	2	yes	—
38	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
39	C₂×D₄×C₉	—	2	1	no	—
40	C₂×D₄×C₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	—	2	1	no	—
43	C₂²×C₄×C₉	—	2	2	no	—
44	C₂²×C₄×C₉	—	2	1	no	—
45	C₂²×C₄×C₉	D₄	1	2	yes	—
46	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₉×M₁₆	—	2	1	no	—
48	C₉×M₁₆	—	2	1	no	—
49	C₉×M₁₆	—	2	1	no	—
50	C₉×M₁₆	C₂×C₄	1	2	yes	complemented
51	C₂×C₈×C₉	—	2	1	no	—
52	C₂×C₈×C₉	—	2	1	no	—
53	C₉×C₄⋊C₄	—	2	2	no	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	—	2	1	no	—
56	C₉×C₄⋊C₄	—	2	1	no	—
57	C₉×C₄⋊C₄	—	2	1	no	—
58	C₉×C₄⋊C₄	—	4	4	no	—
59	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
60	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₉×C₂²⋊C₄	—	2	2	no	—
62	C₉×C₂²⋊C₄	—	2	1	no	—
63	C₄²×C₉	—	2	2	no	—
64	C₄²×C₉	—	2	1	no	—
65	C₄²×C₉	—	2	1	no	—
66	C₄²×C₉	C₂×C₄	1	2	yes	complemented
67	#1859	C₉	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
71	C₃×C₄×Q₈	—	2	1	no	—
72	C₃×C₄×D₄	—	2	1	no	—
73	C₃×C₄×D₄	—	2	2	no	—
74	C₃×#117	—	2	1	no	—
75	C₃×#117	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×C₄⋊C₄	—	2	1	no	—
77	C₂×C₃×C₄⋊C₄	—	2	2	no	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
81	C₃×#107	—	2	1	no	—
82	C₃×#106	—	2	1	no	—
83	C₃×#104	C₃×C₄	1	4	yes	—
84	C₃×#103	—	2	1	no	—
85	C₃×#102	—	2	1	no	—
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₉	—	2	1	no	—
89	Q₈×C₉	C₄⋊C₄	1	2	yes	complemented
90	D₄×C₉	—	2	2	no	—
91	D₄×C₉	—	2	2	no	—
92	D₄×C₉	—	2	1	no	—
93	D₄×C₉	C₄⋊C₄	1	2	yes	complemented
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	2	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	—	4	2	no	—
104	C₂×C₄×C₉	—	4	2	no	—
105	C₂×C₄×C₉	—	4	1	no	—
106	C₂×C₄×C₉	—	8	2	no	—
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	—	2	2	no	—
114	C₈×C₉	—	4	1	no	—
115	#520	C₂×C₉	1	1	yes	characteristic
116	#428	C₂×C₉	1	1	yes	characteristic, complemented
117	C₂×#104	C₂×C₉	1	1	yes	characteristic
118	#417	C₂×C₉	1	1	yes	characteristic
119	#384	C₂×C₉	1	1	yes	characteristic, complemented
120	#343	C₂×C₉	1	1	yes	characteristic, complemented
121	#339	C₂×C₉	1	1	yes	characteristic, complemented
122	C₃×#41	—	2	1	no	—
123	C₃×#41	—	2	1	no	—
124	C₃×#41	C₂×C₃×C₄	1	2	yes	—
125	C₃×#41	C₃×Q₈	1	2	yes	—
126	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
127	C₂×C₃×D₄	—	2	1	no	—
128	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	—	2	1	no	—
130	C₂²×C₃×C₄	—	2	1	no	—
131	C₂²×C₃×C₄	—	2	2	no	—
132	C₂²×C₃×C₄	—	2	1	no	—
133	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
134	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
135	C₃×M₁₆	—	2	1	no	—
136	C₃×M₁₆	—	2	1	no	—
137	C₃×M₁₆	—	2	1	no	—
138	C₃×M₁₆	C₂×C₃×C₄	1	2	yes	—
139	C₂×C₃×C₈	—	2	1	no	—
140	C₂×C₃×C₈	—	2	1	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	1	no	—
143	C₃×C₄⋊C₄	—	2	1	no	—
144	C₃×C₄⋊C₄	—	2	1	no	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	—	4	4	no	—
147	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
148	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
149	C₃×C₂²⋊C₄	—	2	2	no	—
150	C₃×C₂²⋊C₄	—	2	1	no	—
151	C₃×C₄²	—	2	2	no	—
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	—	2	1	no	—
154	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	—	4	1	no	—
159	C₂²×C₉	—	4	1	no	—
160	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
161	C₂²×C₉	#120	1	1	yes	characteristic
162	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
163	C₄×C₉	—	2	4	no	—
164	C₄×C₉	—	2	2	no	—
165	C₄×C₉	—	4	2	no	—
166	C₄×C₉	—	4	1	no	—
167	C₄×C₉	—	4	1	no	—
168	C₄×C₉	—	8	1	no	—
169	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₉	C₄×D₄	1	1	yes	characteristic
171	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
172	C₄×C₉	#123	1	1	yes	characteristic
173	C₂×#41	C₂²×C₉	1	1	yes	characteristic
174	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
175	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
176	C₄×Q₈	—	2	1	no	—
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	2	no	—
179	#117	—	2	1	no	—
180	#117	C₂²×C₉	1	1	yes	characteristic
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
185	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
186	#107	—	2	1	no	—
187	#106	—	2	1	no	—
188	#104	C₄×C₉	1	4	yes	complemented
189	#103	—	2	1	no	—
190	#102	—	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	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	4	2	no	—
209	C₂×C₃×C₄	—	4	2	no	—
210	C₂×C₃×C₄	—	4	1	no	—
211	C₂×C₃×C₄	—	8	2	no	—
212	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
216	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
217	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
218	C₃×C₈	—	2	2	no	—
219	C₃×C₈	—	4	1	no	—
220	C₂×C₉	—	2	1	no	—
221	C₂×C₉	—	2	1	no	—
222	C₂×C₉	—	4	2	no	—
223	C₂×C₉	#385	1	1	yes	characteristic
224	C₂×C₉	#453	1	1	yes	characteristic
225	C₂×C₉	#456	1	1	yes	characteristic
226	#41	—	2	1	no	—
227	#41	—	2	1	no	—
228	#41	C₂×C₄×C₉	1	2	yes	—
229	#41	Q₈×C₉	1	2	yes	—
230	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	D₄×C₉	1	1	yes	characteristic
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	D₄×C₉	1	2	yes	—
238	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
239	M₁₆	—	2	1	no	—
240	M₁₆	—	2	1	no	—
241	M₁₆	—	2	1	no	—
242	M₁₆	C₂×C₄×C₉	1	2	yes	complemented
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
252	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	C₂×C₄×C₉	1	2	yes	complemented
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	4	1	no	—
263	C₂²×C₃	—	4	1	no	—
264	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
265	C₂²×C₃	C₃×#120	1	1	yes	characteristic
266	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	4	2	no	—
270	C₃×C₄	—	4	1	no	—
271	C₃×C₄	—	4	1	no	—
272	C₃×C₄	—	8	1	no	—
273	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
274	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
275	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
276	C₃×C₄	C₃×#123	1	1	yes	characteristic
277	C₉	#1859	1	1	yes	characteristic, direct product factor, Sylow
278	C₂³	—	2	1	no	—
279	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
280	Q₈	—	2	1	no	—
281	Q₈	C₉×C₄⋊C₄	1	2	yes	complemented
282	D₄	—	2	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	2	1	no	—
285	D₄	C₉×C₄⋊C₄	1	2	yes	complemented
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
303	C₂×C₄	C₉×#41	1	1	yes	characteristic
304	C₂×C₄	C₉×#41	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	4	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	4	2	no	—
310	C₂×C₃	C₃×#385	1	1	yes	characteristic
311	C₂×C₃	C₃×#453	1	1	yes	characteristic
312	C₂×C₃	C₃×#456	1	1	yes	characteristic
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
319	C₂²	C₉×#120	1	1	yes	characteristic
320	C₂²	C₉×C₂²⋊Q₈	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	8	1	no	—
327	C₄	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
328	C₄	C₄×D₄×C₉	1	1	yes	characteristic
329	C₄	C₉×C₂²⋊Q₈	1	1	yes	characteristic
330	C₄	C₉×#123	1	1	yes	characteristic
331	C₃	C₃×#1859	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	4	2	no	—
335	C₂	C₉×#385	1	1	yes	characteristic
336	C₂	C₉×#453	1	1	yes	characteristic
337	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
2. C₉×#428	C₂	semidirect
5. C₉×#384	C₂	semidirect
6. C₉×#343	C₂	semidirect
7. C₉×#339	C₂	semidirect
24. C₉×#104	C₄	semidirect
50. C₉×M₁₆	C₂×C₄	semidirect
66. C₄²×C₉	C₂×C₄	semidirect
67. #1859	C₉	direct (N × H)
89. Q₈×C₉	C₄⋊C₄	semidirect
93. D₄×C₉	C₄⋊C₄	semidirect
116. #428	C₂×C₉	semidirect
119. #384	C₂×C₉	semidirect
120. #343	C₂×C₉	semidirect
121. #339	C₂×C₉	semidirect
188. #104	C₄×C₉	semidirect
242. M₁₆	C₂×C₄×C₉	semidirect
258. C₄²	C₂×C₄×C₉	semidirect
277. C₉	#1859	direct (N × H)
281. Q₈	C₉×C₄⋊C₄	semidirect
285. D₄	C₉×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₉×#520	2
C₉×#428	2
C₂×C₉×#104	2
C₉×#417	2
C₉×#384	2
C₉×#343	2
C₉×#339	2
C₃×#1859	3

Subgroups — C9×#1859 #11766580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1859 #11766580