Subgroups — C₉×#2065 #11766786

Order 1152.

426 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#102	C₂	1	1	yes	maximal, characteristic
4	C₂×C₉×#102	C₂	1	1	yes	maximal, characteristic
5	C₉×#390	C₂	1	1	yes	maximal, characteristic
6	C₉×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#340	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2065	C₃	1	1	yes	maximal, characteristic
9	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	—	2	1	no	—
13	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
17	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
18	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
19	C₉×#106	—	2	2	no	—
20	C₉×C₄⋊C₈	—	2	2	no	—
21	C₉×#102	—	2	2	no	—
22	C₉×#102	—	2	2	no	—
23	C₉×#95	—	2	1	no	—
24	C₂×C₃×#106	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
26	C₂×C₃×#102	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×#102	C₂×C₃	1	1	yes	characteristic
28	C₃×#390	C₂×C₃	1	1	yes	characteristic
29	C₃×#389	C₂×C₃	1	1	yes	characteristic
30	C₃×#340	C₂×C₃	1	1	yes	characteristic
31	C₂⁴×C₉	—	2	1	no	—
32	C₂×D₄×C₉	—	2	1	no	—
33	C₂×D₄×C₉	—	2	1	no	—
34	C₂×D₄×C₉	—	2	1	no	—
35	C₂×D₄×C₉	—	2	1	no	—
36	C₂×D₄×C₉	—	2	1	no	—
37	C₂×D₄×C₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₉	—	2	1	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	C₂×C₈×C₉	—	2	1	no	—
45	C₂×C₈×C₉	—	2	1	no	—
46	C₂×C₈×C₉	—	2	1	no	—
47	C₂×C₈×C₉	—	2	1	no	—
48	C₂×C₈×C₉	—	2	1	no	—
49	C₂×C₈×C₉	D₄	1	2	yes	—
50	C₉×C₄⋊C₄	—	2	1	no	—
51	C₉×C₄⋊C₄	—	2	1	no	—
52	C₉×C₄⋊C₄	—	2	1	no	—
53	C₉×C₄⋊C₄	—	2	1	no	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	—	4	2	no	—
56	C₉×C₄⋊C₄	D₄	1	2	yes	—
57	C₉×C₂²⋊C₄	—	4	2	no	—
58	C₉×C₂²⋊C₄	—	4	2	no	—
59	C₄²×C₉	—	2	1	no	—
60	C₄²×C₉	—	2	1	no	—
61	#2065	C₉	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₃×C₄⋊C₄	—	2	1	no	—
66	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
70	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
71	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
72	C₃×#106	—	2	2	no	—
73	C₃×C₄⋊C₈	—	2	2	no	—
74	C₃×#102	—	2	2	no	—
75	C₃×#102	—	2	2	no	—
76	C₃×#95	—	2	1	no	—
77	C₂³×C₉	—	2	2	no	—
78	C₂³×C₉	—	4	1	no	—
79	C₂³×C₉	—	4	1	no	—
80	C₂³×C₉	—	4	1	no	—
81	C₂³×C₉	—	4	1	no	—
82	C₂³×C₉	—	4	1	no	—
83	C₂³×C₉	—	4	1	no	—
84	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
85	D₄×C₉	—	2	2	no	—
86	D₄×C₉	—	2	2	no	—
87	D₄×C₉	—	4	1	no	—
88	D₄×C₉	—	4	1	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	4	1	no	—
96	C₂×C₄×C₉	—	4	1	no	—
97	C₂×C₄×C₉	—	4	1	no	—
98	C₂×C₄×C₉	—	4	1	no	—
99	C₂×C₄×C₉	—	4	1	no	—
100	C₂×C₄×C₉	—	4	1	no	—
101	C₂×C₄×C₉	—	4	1	no	—
102	C₂×C₄×C₉	—	4	1	no	—
103	C₂×C₄×C₉	SD₁₆	1	2	yes	—
104	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₉	#41	1	1	yes	characteristic
108	C₂×C₄×C₉	#41	1	1	yes	characteristic
109	C₂×C₄×C₉	#41	1	1	yes	characteristic
110	C₈×C₉	—	2	2	no	—
111	C₈×C₉	—	4	1	no	—
112	C₂×#106	C₂×C₉	1	1	yes	characteristic, complemented
113	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
114	C₂×#102	C₂×C₉	1	1	yes	characteristic
115	C₂×#102	C₂×C₉	1	1	yes	characteristic
116	#390	C₂×C₉	1	1	yes	characteristic
117	#389	C₂×C₉	1	1	yes	characteristic, complemented
118	#340	C₂×C₉	1	1	yes	characteristic, complemented
119	C₂⁴×C₃	—	2	1	no	—
120	C₂×C₃×D₄	—	2	1	no	—
121	C₂×C₃×D₄	—	2	1	no	—
122	C₂×C₃×D₄	—	2	1	no	—
123	C₂×C₃×D₄	—	2	1	no	—
124	C₂×C₃×D₄	—	2	1	no	—
125	C₂×C₃×D₄	C₃×Q₈	1	2	yes	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	2	1	no	—
128	C₂²×C₃×C₄	—	2	1	no	—
129	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄⋊C₄	—	2	1	no	—
141	C₃×C₄⋊C₄	—	2	1	no	—
142	C₃×C₄⋊C₄	—	2	1	no	—
143	C₃×C₄⋊C₄	—	4	2	no	—
144	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
145	C₃×C₂²⋊C₄	—	4	2	no	—
146	C₃×C₂²⋊C₄	—	4	2	no	—
147	C₃×C₄²	—	2	1	no	—
148	C₃×C₄²	—	2	1	no	—
149	C₂²×C₉	—	4	2	no	—
150	C₂²×C₉	—	4	2	no	—
151	C₂²×C₉	—	4	2	no	—
152	C₂²×C₉	—	8	1	no	—
153	C₂²×C₉	—	8	1	no	—
154	C₂²×C₉	—	8	1	no	—
155	C₂²×C₉	—	8	1	no	—
156	C₂²×C₉	#121	1	1	yes	characteristic
157	C₂²×C₉	#121	1	1	yes	characteristic
158	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
160	C₂²×C₉	#135	1	1	yes	characteristic
161	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₄×C₉	—	2	2	no	—
164	C₄×C₉	—	4	2	no	—
165	C₄×C₉	—	4	1	no	—
166	C₄×C₉	—	8	1	no	—
167	C₄×C₉	—	8	1	no	—
168	C₄×C₉	#121	1	1	yes	characteristic
169	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
170	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
171	C₄×C₉	#126	1	1	yes	characteristic
172	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
173	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
174	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	—	2	1	no	—
180	C₂×C₂²⋊C₄	—	2	1	no	—
181	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
182	#106	—	2	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	#102	—	2	2	no	—
185	#102	—	2	2	no	—
186	#95	—	2	1	no	—
187	C₂³×C₃	—	2	2	no	—
188	C₂³×C₃	—	4	1	no	—
189	C₂³×C₃	—	4	1	no	—
190	C₂³×C₃	—	4	1	no	—
191	C₂³×C₃	—	4	1	no	—
192	C₂³×C₃	—	4	1	no	—
193	C₂³×C₃	—	4	1	no	—
194	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
195	C₃×D₄	—	2	2	no	—
196	C₃×D₄	—	2	2	no	—
197	C₃×D₄	—	4	1	no	—
198	C₃×D₄	—	4	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	—	4	1	no	—
209	C₂×C₃×C₄	—	4	1	no	—
210	C₂×C₃×C₄	—	4	1	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	4	1	no	—
213	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
214	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
217	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
218	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
219	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
220	C₃×C₈	—	2	2	no	—
221	C₃×C₈	—	4	1	no	—
222	C₂×C₉	—	8	2	no	—
223	C₂×C₉	#396	1	1	yes	characteristic
224	C₂×C₉	#460	1	1	yes	characteristic
225	C₂×C₉	#463	1	1	yes	characteristic
226	C₂×C₉	#465	1	1	yes	characteristic
227	C₂×C₉	#469	1	1	yes	characteristic
228	C₂×C₉	#476	1	1	yes	characteristic
229	C₂×C₉	#478	1	1	yes	characteristic
230	C₂⁴	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	Q₈×C₉	1	2	yes	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
241	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
242	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	D₄×C₉	1	2	yes	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	D₄×C₉	1	2	yes	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₂²×C₃	—	4	2	no	—
261	C₂²×C₃	—	4	2	no	—
262	C₂²×C₃	—	4	2	no	—
263	C₂²×C₃	—	8	1	no	—
264	C₂²×C₃	—	8	1	no	—
265	C₂²×C₃	—	8	1	no	—
266	C₂²×C₃	—	8	1	no	—
267	C₂²×C₃	C₃×#121	1	1	yes	characteristic
268	C₂²×C₃	C₃×#121	1	1	yes	characteristic
269	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
270	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
271	C₂²×C₃	C₃×#135	1	1	yes	characteristic
272	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	4	2	no	—
276	C₃×C₄	—	4	1	no	—
277	C₃×C₄	—	8	1	no	—
278	C₃×C₄	—	8	1	no	—
279	C₃×C₄	C₃×#121	1	1	yes	characteristic
280	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
281	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
282	C₃×C₄	C₃×#126	1	1	yes	characteristic
283	C₉	#2065	1	1	yes	characteristic, direct product factor, Sylow
284	C₂³	—	2	2	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
292	D₄	—	2	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	C₉×SD₁₆	1	2	yes	—
311	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
313	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
314	C₂×C₄	C₉×#41	1	1	yes	characteristic
315	C₂×C₄	C₉×#41	1	1	yes	characteristic
316	C₂×C₄	C₉×#41	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	4	1	no	—
319	C₂×C₃	—	8	2	no	—
320	C₂×C₃	C₃×#396	1	1	yes	characteristic
321	C₂×C₃	C₃×#460	1	1	yes	characteristic
322	C₂×C₃	C₃×#463	1	1	yes	characteristic
323	C₂×C₃	C₃×#465	1	1	yes	characteristic
324	C₂×C₃	C₃×#469	1	1	yes	characteristic
325	C₂×C₃	C₃×#476	1	1	yes	characteristic
326	C₂×C₃	C₃×#478	1	1	yes	characteristic
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	8	1	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	1	no	—
334	C₂²	C₉×#121	1	1	yes	characteristic
335	C₂²	C₉×#121	1	1	yes	characteristic
336	C₂²	C₉×C₂²⋊Q₈	1	1	yes	characteristic
337	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
338	C₂²	C₉×#135	1	1	yes	characteristic
339	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
340	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	4	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	C₉×#121	1	1	yes	characteristic
347	C₄	C₉×C₂²⋊Q₈	1	1	yes	characteristic
348	C₄	C₉×C₂²⋊Q₈	1	1	yes	characteristic
349	C₄	C₉×#126	1	1	yes	characteristic
350	C₃	C₃×#2065	1	1	yes	characteristic
351	C₂	—	8	2	no	—
352	C₂	C₉×#396	1	1	yes	characteristic
353	C₂	C₉×#460	1	1	yes	characteristic
354	C₂	C₉×#463	1	1	yes	characteristic
355	C₂	C₉×#465	1	1	yes	characteristic
356	C₂	C₉×#469	1	1	yes	characteristic
357	C₂	C₉×#476	1	1	yes	characteristic
358	C₂	C₉×#478	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#106	C₂	semidirect
2. C₂×C₉×C₄⋊C₈	C₂	semidirect
6. C₉×#389	C₂	semidirect
7. C₉×#340	C₂	semidirect
61. #2065	C₉	direct (N × H)
112. C₂×#106	C₂×C₉	semidirect
113. C₂×C₄⋊C₈	C₂×C₉	semidirect
117. #389	C₂×C₉	semidirect
118. #340	C₂×C₉	semidirect
283. C₉	#2065	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×#106	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×#102	2
C₉×#390	2
C₉×#389	2
C₉×#340	2
C₃×#2065	3

Subgroups — C9×#2065 #11766786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2065 #11766786