Subgroups — C₉×#3231 #11767952

Order 1152.

489 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#533	C₂	1	1	yes	maximal, characteristic
2	C₉×#533	C₂	1	1	yes	maximal, characteristic
3	C₉×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#485	C₂	1	4	yes	maximal
8	C₉×#479	C₂	1	4	yes	maximal, complemented
9	C₉×#431	C₂	1	1	yes	maximal, characteristic
10	C₃×#3231	C₃	1	1	yes	maximal, characteristic
11	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₉×#125	—	2	1	no	—
16	C₉×#125	C₂²	1	2	yes	—
17	C₉×C₂²⋊Q₈	—	2	1	no	—
18	C₉×C₂²⋊Q₈	—	2	1	no	—
19	C₉×C₂²⋊Q₈	—	2	1	no	—
20	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₄×Q₈×C₉	—	2	1	no	—
22	C₄×Q₈×C₉	—	2	1	no	—
23	C₄×Q₈×C₉	—	2	1	no	—
24	C₄×Q₈×C₉	C₂²	1	2	yes	—
25	C₉×#117	—	2	1	no	—
26	C₉×#117	C₂²	1	2	yes	—
27	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
28	C₉×#107	C₂²	1	4	yes	—
29	C₉×#106	C₂²	1	4	yes	—
30	C₉×C₄⋊C₈	C₂²	1	2	yes	—
31	C₉×#103	C₂²	1	4	yes	—
32	C₉×#103	C₂²	1	4	yes	—
33	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
35	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
36	C₃×#533	C₂×C₃	1	1	yes	characteristic
37	C₃×#533	C₂×C₃	1	1	yes	characteristic
38	C₃×#500	C₂×C₃	1	1	yes	characteristic
39	C₃×#498	C₂×C₃	1	1	yes	characteristic
40	C₃×#487	C₂×C₃	1	1	yes	characteristic
41	C₃×#487	C₂×C₃	1	1	yes	characteristic
42	C₃×#485	C₂×C₃	1	4	yes	—
43	C₃×#479	C₂×C₃	1	4	yes	—
44	C₃×#431	C₂×C₃	1	1	yes	characteristic
45	C₂×Q₈×C₉	—	2	1	no	—
46	C₂×Q₈×C₉	—	2	1	no	—
47	C₂×Q₈×C₉	—	2	1	no	—
48	C₂×Q₈×C₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₉	C₂³	1	2	yes	—
53	C₂×C₈×C₉	C₂³	1	2	yes	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	—	2	1	no	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	—	2	1	no	—
59	C₉×C₄⋊C₄	—	2	1	no	—
60	C₉×C₄⋊C₄	—	2	1	no	—
61	C₉×C₄⋊C₄	—	2	2	no	—
62	C₉×C₄⋊C₄	—	2	2	no	—
63	C₉×C₄⋊C₄	C₂³	1	2	yes	—
64	C₉×C₄⋊C₄	C₂³	1	2	yes	—
65	C₉×C₄⋊C₄	C₂³	1	2	yes	—
66	C₉×C₂²⋊C₄	—	2	2	no	—
67	C₉×C₂²⋊C₄	—	2	2	no	—
68	C₄²×C₉	—	2	2	no	—
69	C₄²×C₉	—	2	2	no	—
70	C₄²×C₉	D₄	1	1	yes	characteristic
71	C₄²×C₉	D₄	1	1	yes	characteristic
72	C₄²×C₉	C₂³	1	1	yes	characteristic
73	C₄²×C₉	C₂³	1	1	yes	characteristic
74	#3231	C₉	1	1	yes	characteristic, direct product factor, Sylow
75	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
76	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
77	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
78	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
79	C₃×#125	—	2	1	no	—
80	C₃×#125	C₂²×C₃	1	2	yes	—
81	C₃×C₂²⋊Q₈	—	2	1	no	—
82	C₃×C₂²⋊Q₈	—	2	1	no	—
83	C₃×C₂²⋊Q₈	—	2	1	no	—
84	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
85	C₃×C₄×Q₈	—	2	1	no	—
86	C₃×C₄×Q₈	—	2	1	no	—
87	C₃×C₄×Q₈	—	2	1	no	—
88	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
89	C₃×#117	—	2	1	no	—
90	C₃×#117	C₂²×C₃	1	2	yes	—
91	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
92	C₃×#107	C₂²×C₃	1	4	yes	—
93	C₃×#106	C₂²×C₃	1	4	yes	—
94	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
95	C₃×#103	C₂²×C₃	1	4	yes	—
96	C₃×#103	C₂²×C₃	1	4	yes	—
97	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
98	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
99	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
100	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
101	Q₈×C₉	—	2	4	no	—
102	Q₈×C₉	—	4	2	no	—
103	Q₈×C₉	—	4	2	no	—
104	Q₈×C₉	—	4	2	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	2	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	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
118	C₈×C₉	—	2	2	no	—
119	C₈×C₉	—	2	2	no	—
120	#533	C₂×C₉	1	1	yes	characteristic
121	#533	C₂×C₉	1	1	yes	characteristic
122	#500	C₂×C₉	1	1	yes	characteristic, complemented
123	#498	C₂×C₉	1	1	yes	characteristic, complemented
124	#487	C₂×C₉	1	1	yes	characteristic, complemented
125	#487	C₂×C₉	1	1	yes	characteristic, complemented
126	#485	C₂×C₉	1	4	yes	—
127	#479	C₂×C₉	1	4	yes	complemented
128	#431	C₂×C₉	1	1	yes	characteristic
129	C₂×C₃×Q₈	—	2	1	no	—
130	C₂×C₃×Q₈	—	2	1	no	—
131	C₂×C₃×Q₈	—	2	1	no	—
132	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
133	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
137	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄⋊C₄	—	2	2	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	2	no	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
148	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
149	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
150	C₃×C₂²⋊C₄	—	2	2	no	—
151	C₃×C₂²⋊C₄	—	2	2	no	—
152	C₃×C₄²	—	2	2	no	—
153	C₃×C₄²	—	2	2	no	—
154	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
155	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
156	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
157	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
158	C₂²×C₉	—	2	1	no	—
159	C₂²×C₉	—	2	1	no	—
160	C₂²×C₉	—	2	1	no	—
161	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	—	2	1	no	—
165	C₄×C₉	—	4	2	no	—
166	C₄×C₉	—	4	2	no	—
167	C₄×C₉	—	4	2	no	—
168	C₄×C₉	—	4	2	no	—
169	C₄×C₉	#135	1	2	yes	—
170	C₄×C₉	#137	1	2	yes	—
171	C₄×C₉	2₋¹⁺⁴	1	2	yes	—
172	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
173	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
174	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
175	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
176	#125	—	2	1	no	—
177	#125	C₂²×C₉	1	2	yes	—
178	C₂²⋊Q₈	—	2	1	no	—
179	C₂²⋊Q₈	—	2	1	no	—
180	C₂²⋊Q₈	—	2	1	no	—
181	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	C₂²×C₉	1	2	yes	—
186	#117	—	2	1	no	—
187	#117	C₂²×C₉	1	2	yes	—
188	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
189	#107	C₂²×C₉	1	4	yes	—
190	#106	C₂²×C₉	1	4	yes	—
191	C₄⋊C₈	C₂²×C₉	1	2	yes	—
192	#103	C₂²×C₉	1	4	yes	—
193	#103	C₂²×C₉	1	4	yes	—
194	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
195	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
196	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
197	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
198	C₃×Q₈	—	2	4	no	—
199	C₃×Q₈	—	4	2	no	—
200	C₃×Q₈	—	4	2	no	—
201	C₃×Q₈	—	4	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	—	2	2	no	—
217	C₂×C₉	—	4	1	no	—
218	C₂×C₉	#536	1	1	yes	characteristic
219	C₂×C₉	C₂×#135	1	1	yes	characteristic
220	C₂×C₉	C₂×#137	1	1	yes	characteristic
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	2	1	no	—
223	C₂×Q₈	—	2	1	no	—
224	C₂×Q₈	C₂³×C₉	1	2	yes	—
225	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
226	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
227	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
228	C₂×C₈	C₂³×C₉	1	2	yes	—
229	C₂×C₈	C₂³×C₉	1	2	yes	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	C₂³×C₉	1	2	yes	—
240	C₄⋊C₄	C₂³×C₉	1	2	yes	—
241	C₄⋊C₄	C₂³×C₉	1	2	yes	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	D₄×C₉	1	1	yes	characteristic
247	C₄²	D₄×C₉	1	1	yes	characteristic
248	C₄²	C₂³×C₉	1	1	yes	characteristic
249	C₄²	C₂³×C₉	1	1	yes	characteristic
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	4	2	no	—
258	C₃×C₄	—	4	2	no	—
259	C₃×C₄	—	4	2	no	—
260	C₃×C₄	—	4	2	no	—
261	C₃×C₄	C₃×#135	1	2	yes	—
262	C₃×C₄	C₃×#137	1	2	yes	—
263	C₃×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
264	C₉	#3231	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
266	Q₈	—	2	4	no	—
267	Q₈	—	4	2	no	—
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
282	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₂×C₃	—	4	1	no	—
286	C₂×C₃	C₃×#536	1	1	yes	characteristic
287	C₂×C₃	C₂×C₃×#135	1	1	yes	characteristic
288	C₂×C₃	C₂×C₃×#137	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	4	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	C₉×#135	1	2	yes	—
301	C₄	C₉×#137	1	2	yes	—
302	C₄	C₉×2₋¹⁺⁴	1	2	yes	—
303	C₃	C₃×#3231	1	1	yes	characteristic
304	C₂	—	4	1	no	—
305	C₂	C₉×#536	1	1	yes	characteristic
306	C₂	C₂×C₉×#135	1	1	yes	characteristic
307	C₂	C₂×C₉×#137	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 ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₉×#500	C₂	semidirect
4. C₉×#498	C₂	semidirect
5. C₉×#487	C₂	semidirect
6. C₉×#487	C₂	semidirect
8. C₉×#479	C₂	semidirect
74. #3231	C₉	direct (N × H)
122. #500	C₂×C₉	semidirect
123. #498	C₂×C₉	semidirect
124. #487	C₂×C₉	semidirect
125. #487	C₂×C₉	semidirect
127. #479	C₂×C₉	semidirect
264. C₉	#3231	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×#533	2
C₉×#500	2
C₉×#498	2
C₉×#487	2
C₉×#485	2
C₉×#479	2
C₉×#431	2
C₃×#3231	3

Subgroups — C9×#3231 #11767952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3231 #11767952