Subgroups — C₉×#3082 #11767803

Order 1152.

543 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#544	C₂	1	1	yes	maximal, characteristic
2	C₉×#519	C₂	1	1	yes	maximal, characteristic
3	C₉×#479	C₂	1	2	yes	maximal, complemented
4	C₉×#479	C₂	1	2	yes	maximal, complemented
5	C₉×#477	C₂	1	2	yes	maximal, complemented
6	C₉×#475	C₂	1	2	yes	maximal, complemented
7	C₉×#428	C₂	1	1	yes	maximal, characteristic
8	C₉×#427	C₂	1	1	yes	maximal, characteristic
9	C₉×#424	C₂	1	1	yes	maximal, characteristic
10	C₉×#419	C₂	1	1	yes	maximal, characteristic
11	C₂×C₉×#103	C₂	1	1	yes	maximal, characteristic
12	C₃×#3082	C₃	1	1	yes	maximal, characteristic
13	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊Q₈	—	2	1	no	—
17	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₉×#125	—	2	1	no	—
19	C₉×#125	C₂²	1	2	yes	—
20	C₉×C₂²⋊Q₈	—	2	2	no	—
21	C₄×Q₈×C₉	—	2	1	no	—
22	C₄×Q₈×C₉	—	2	1	no	—
23	C₄×Q₈×C₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₉	C₂²	1	2	yes	—
25	C₉×#117	—	2	2	no	—
26	C₉×#117	C₂²	1	1	yes	characteristic
27	C₉×#117	C₂²	1	1	yes	characteristic
28	C₉×#117	C₂²	1	1	yes	characteristic
29	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₉×#107	C₂²	1	1	yes	characteristic
31	C₉×#107	C₂²	1	1	yes	characteristic
32	C₉×#107	C₂²	1	2	yes	—
33	C₉×#106	C₂²	1	1	yes	characteristic
34	C₉×#106	C₂²	1	1	yes	characteristic
35	C₉×#106	C₂²	1	2	yes	—
36	C₉×C₄⋊C₈	C₂²	1	2	yes	—
37	C₉×C₄⋊C₈	C₂²	1	2	yes	—
38	C₉×#103	C₂²	1	2	yes	—
39	C₉×#103	C₂²	1	2	yes	—
40	C₉×#103	C₂²	1	2	yes	—
41	C₉×#103	C₂²	1	2	yes	—
42	C₃×#544	C₂×C₃	1	1	yes	characteristic
43	C₃×#519	C₂×C₃	1	1	yes	characteristic
44	C₃×#479	C₂×C₃	1	2	yes	—
45	C₃×#479	C₂×C₃	1	2	yes	—
46	C₃×#477	C₂×C₃	1	2	yes	—
47	C₃×#475	C₂×C₃	1	2	yes	—
48	C₃×#428	C₂×C₃	1	1	yes	characteristic
49	C₃×#427	C₂×C₃	1	1	yes	characteristic
50	C₃×#424	C₂×C₃	1	1	yes	characteristic
51	C₃×#419	C₂×C₃	1	1	yes	characteristic
52	C₂×C₃×#103	C₂×C₃	1	1	yes	characteristic
53	C₂×Q₈×C₉	—	2	1	no	—
54	C₂×Q₈×C₉	—	2	2	no	—
55	C₂×Q₈×C₉	—	2	2	no	—
56	C₂×Q₈×C₉	Q₈	1	4	yes	complemented
57	C₂×Q₈×C₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₉	—	2	1	no	—
59	C₂²×C₄×C₉	—	2	1	no	—
60	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
61	C₉×M₁₆	—	2	2	no	—
62	C₂×C₈×C₉	—	2	2	no	—
63	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₉	C₂³	1	2	yes	—
66	C₉×C₄⋊C₄	—	2	1	no	—
67	C₉×C₄⋊C₄	—	2	2	no	—
68	C₉×C₄⋊C₄	—	2	2	no	—
69	C₉×C₄⋊C₄	—	2	2	no	—
70	C₉×C₄⋊C₄	—	2	2	no	—
71	C₉×C₄⋊C₄	—	2	2	no	—
72	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
73	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₉×C₄⋊C₄	C₂³	1	2	yes	—
75	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₉×C₂²⋊C₄	—	2	1	no	—
80	C₉×C₂²⋊C₄	—	2	1	no	—
81	C₉×C₂²⋊C₄	D₄	1	2	yes	—
82	C₄²×C₉	—	2	2	no	—
83	C₄²×C₉	—	2	2	no	—
84	C₄²×C₉	—	2	1	no	—
85	C₄²×C₉	—	2	1	no	—
86	C₄²×C₉	C₂³	1	2	yes	—
87	#3082	C₉	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
90	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
91	C₃×C₄⋊Q₈	—	2	1	no	—
92	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
93	C₃×#125	—	2	1	no	—
94	C₃×#125	C₂²×C₃	1	2	yes	—
95	C₃×C₂²⋊Q₈	—	2	2	no	—
96	C₃×C₄×Q₈	—	2	1	no	—
97	C₃×C₄×Q₈	—	2	1	no	—
98	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
99	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
100	C₃×#117	—	2	2	no	—
101	C₃×#117	C₂²×C₃	1	1	yes	characteristic
102	C₃×#117	C₂²×C₃	1	1	yes	characteristic
103	C₃×#117	C₂²×C₃	1	1	yes	characteristic
104	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
105	C₃×#107	C₂²×C₃	1	1	yes	characteristic
106	C₃×#107	C₂²×C₃	1	1	yes	characteristic
107	C₃×#107	C₂²×C₃	1	2	yes	—
108	C₃×#106	C₂²×C₃	1	1	yes	characteristic
109	C₃×#106	C₂²×C₃	1	1	yes	characteristic
110	C₃×#106	C₂²×C₃	1	2	yes	—
111	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
112	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
113	C₃×#103	C₂²×C₃	1	2	yes	—
114	C₃×#103	C₂²×C₃	1	2	yes	—
115	C₃×#103	C₂²×C₃	1	2	yes	—
116	C₃×#103	C₂²×C₃	1	2	yes	—
117	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₉	—	2	4	no	—
119	Q₈×C₉	—	2	2	no	—
120	Q₈×C₉	—	8	2	no	—
121	Q₈×C₉	C₂×Q₈	1	4	yes	—
122	C₂×C₄×C₉	—	2	4	no	—
123	C₂×C₄×C₉	—	2	2	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	2	1	no	—
127	C₂×C₄×C₉	—	2	1	no	—
128	C₂×C₄×C₉	—	2	2	no	—
129	C₂×C₄×C₉	—	8	1	no	—
130	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
131	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
132	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
134	C₂×C₄×C₉	#41	1	2	yes	—
135	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
136	C₈×C₉	—	2	1	no	—
137	C₈×C₉	—	2	1	no	—
138	C₈×C₉	—	2	2	no	—
139	#544	C₂×C₉	1	1	yes	characteristic
140	#519	C₂×C₉	1	1	yes	characteristic
141	#479	C₂×C₉	1	2	yes	complemented
142	#479	C₂×C₉	1	2	yes	complemented
143	#477	C₂×C₉	1	2	yes	complemented
144	#475	C₂×C₉	1	2	yes	complemented
145	#428	C₂×C₉	1	1	yes	characteristic
146	#427	C₂×C₉	1	1	yes	characteristic
147	#424	C₂×C₉	1	1	yes	characteristic
148	#419	C₂×C₉	1	1	yes	characteristic
149	C₂×#103	C₂×C₉	1	1	yes	characteristic
150	C₂×C₃×Q₈	—	2	1	no	—
151	C₂×C₃×Q₈	—	2	2	no	—
152	C₂×C₃×Q₈	—	2	2	no	—
153	C₂×C₃×Q₈	C₃×Q₈	1	4	yes	—
154	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
155	C₂²×C₃×C₄	—	2	1	no	—
156	C₂²×C₃×C₄	—	2	1	no	—
157	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
158	C₃×M₁₆	—	2	2	no	—
159	C₂×C₃×C₈	—	2	2	no	—
160	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
161	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
162	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
163	C₃×C₄⋊C₄	—	2	1	no	—
164	C₃×C₄⋊C₄	—	2	2	no	—
165	C₃×C₄⋊C₄	—	2	2	no	—
166	C₃×C₄⋊C₄	—	2	2	no	—
167	C₃×C₄⋊C₄	—	2	2	no	—
168	C₃×C₄⋊C₄	—	2	2	no	—
169	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
170	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
171	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
172	C₃×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
173	C₃×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
174	C₃×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
175	C₃×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
176	C₃×C₂²⋊C₄	—	2	1	no	—
177	C₃×C₂²⋊C₄	—	2	1	no	—
178	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
179	C₃×C₄²	—	2	2	no	—
180	C₃×C₄²	—	2	2	no	—
181	C₃×C₄²	—	2	1	no	—
182	C₃×C₄²	—	2	1	no	—
183	C₃×C₄²	C₂³×C₃	1	2	yes	—
184	C₂²×C₉	—	2	1	no	—
185	C₂²×C₉	—	2	1	no	—
186	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
187	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
188	C₄×C₉	—	2	4	no	—
189	C₄×C₉	—	2	2	no	—
190	C₄×C₉	—	2	2	no	—
191	C₄×C₉	—	4	1	no	—
192	C₄×C₉	—	4	2	no	—
193	C₄×C₉	—	4	1	no	—
194	C₄×C₉	—	4	1	no	—
195	C₄×C₉	—	4	1	no	—
196	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
197	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
198	C₄×C₉	C₂×#41	1	1	yes	characteristic
199	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
200	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
201	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
204	#125	—	2	1	no	—
205	#125	C₂²×C₉	1	2	yes	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×Q₈	C₂²×C₉	1	2	yes	—
210	C₄×Q₈	C₂²×C₉	1	2	yes	—
211	#117	—	2	2	no	—
212	#117	C₂²×C₉	1	1	yes	characteristic
213	#117	C₂²×C₉	1	1	yes	characteristic
214	#117	C₂²×C₉	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
216	#107	C₂²×C₉	1	1	yes	characteristic
217	#107	C₂²×C₉	1	1	yes	characteristic
218	#107	C₂²×C₉	1	2	yes	—
219	#106	C₂²×C₉	1	1	yes	characteristic
220	#106	C₂²×C₉	1	1	yes	characteristic
221	#106	C₂²×C₉	1	2	yes	—
222	C₄⋊C₈	C₂²×C₉	1	2	yes	—
223	C₄⋊C₈	C₂²×C₉	1	2	yes	—
224	#103	C₂²×C₉	1	2	yes	—
225	#103	C₂²×C₉	1	2	yes	—
226	#103	C₂²×C₉	1	2	yes	—
227	#103	C₂²×C₉	1	2	yes	—
228	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
229	C₃×Q₈	—	2	4	no	—
230	C₃×Q₈	—	2	2	no	—
231	C₃×Q₈	—	8	2	no	—
232	C₃×Q₈	C₂×C₃×Q₈	1	4	yes	—
233	C₂×C₃×C₄	—	2	4	no	—
234	C₂×C₃×C₄	—	2	2	no	—
235	C₂×C₃×C₄	—	2	1	no	—
236	C₂×C₃×C₄	—	2	1	no	—
237	C₂×C₃×C₄	—	2	1	no	—
238	C₂×C₃×C₄	—	2	1	no	—
239	C₂×C₃×C₄	—	2	2	no	—
240	C₂×C₃×C₄	—	8	1	no	—
241	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
242	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
243	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
245	C₂×C₃×C₄	C₃×#41	1	2	yes	—
246	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
247	C₃×C₈	—	2	1	no	—
248	C₃×C₈	—	2	1	no	—
249	C₃×C₈	—	2	2	no	—
250	C₂×C₉	—	2	2	no	—
251	C₂×C₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₉	#577	1	1	yes	characteristic
253	C₂×C₉	#578	1	1	yes	characteristic
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	Q₈×C₉	1	4	yes	complemented
258	C₂×Q₈	C₂³×C₉	1	2	yes	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
262	M₁₆	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
265	C₂×C₈	C₂³×C₉	1	2	yes	—
266	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
274	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂³×C₉	1	2	yes	—
276	C₄⋊C₄	C₂³×C₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂³×C₉	1	1	yes	characteristic
278	C₄⋊C₄	C₂³×C₉	1	1	yes	characteristic
279	C₄⋊C₄	C₂³×C₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	D₄×C₉	1	2	yes	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	C₂³×C₉	1	2	yes	—
288	C₂²×C₃	—	2	1	no	—
289	C₂²×C₃	—	2	1	no	—
290	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
291	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	2	no	—
294	C₃×C₄	—	2	4	no	—
295	C₃×C₄	—	4	2	no	—
296	C₃×C₄	—	4	1	no	—
297	C₃×C₄	—	4	1	no	—
298	C₃×C₄	—	4	1	no	—
299	C₃×C₄	—	4	1	no	—
300	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
301	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
302	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
303	C₉	#3082	1	1	yes	characteristic, direct product factor, Sylow
304	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
305	Q₈	—	2	4	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	8	2	no	—
308	Q₈	C₂×Q₈×C₉	1	4	yes	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	8	1	no	—
317	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
318	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
319	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
320	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
321	C₂×C₄	C₉×#41	1	2	yes	—
322	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	2	no	—
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
328	C₂×C₃	C₃×#577	1	1	yes	characteristic
329	C₂×C₃	C₃×#578	1	1	yes	characteristic
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	C₉×C₂²⋊Q₈	1	2	yes	—
333	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
334	C₄	—	2	4	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
343	C₄	C₂²×Q₈×C₉	1	1	yes	characteristic
344	C₄	C₂×C₉×#41	1	1	yes	characteristic
345	C₃	C₃×#3082	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	C₂×C₉×C₂²⋊Q₈	1	1	yes	characteristic
348	C₂	C₉×#577	1	1	yes	characteristic
349	C₂	C₉×#578	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₉×#479	C₂	semidirect
4. C₉×#479	C₂	semidirect
5. C₉×#477	C₂	semidirect
6. C₉×#475	C₂	semidirect
56. C₂×Q₈×C₉	Q₈	semidirect
87. #3082	C₉	direct (N × H)
141. #479	C₂×C₉	semidirect
142. #479	C₂×C₉	semidirect
143. #477	C₂×C₉	semidirect
144. #475	C₂×C₉	semidirect
257. C₂×Q₈	Q₈×C₉	semidirect
303. C₉	#3082	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#544	2
C₉×#519	2
C₉×#479	2
C₉×#477	2
C₉×#475	2
C₉×#428	2
C₉×#427	2
C₉×#424	2
C₉×#419	2
C₂×C₉×#103	2
C₃×#3082	3

Subgroups — C9×#3082 #11767803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3082 #11767803