Subgroups — C₉×#3223 #11767944

Order 1152.

546 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#536	C₂	1	1	yes	maximal, characteristic
3	C₉×#497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#488	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#464	C₂	1	2	yes	maximal, complemented
8	C₉×#462	C₂	1	2	yes	maximal, complemented
9	C₉×#452	C₂	1	2	yes	maximal
10	C₉×#451	C₂	1	2	yes	maximal, complemented
11	C₉×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#3223	C₃	1	1	yes	maximal, characteristic
13	C₂×C₉×#41	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
15	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
18	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
19	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₉×#126	—	2	2	no	—
23	C₉×#125	—	2	2	no	—
24	C₉×#125	C₂²	1	1	yes	characteristic
25	C₉×#124	C₂²	1	1	yes	characteristic
26	C₉×#124	C₂²	1	1	yes	characteristic
27	C₉×#123	—	2	2	no	—
28	C₉×C₂²⋊Q₈	—	2	2	no	—
29	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
30	C₄×Q₈×C₉	C₂²	1	2	yes	—
31	C₉×#117	C₂²	1	1	yes	characteristic
32	C₉×C₄⋊C₈	C₂²	1	2	yes	—
33	C₉×#103	C₂²	1	2	yes	—
34	C₉×#103	C₂²	1	2	yes	—
35	C₉×#103	C₂²	1	2	yes	—
36	C₉×#102	C₂²	1	2	yes	—
37	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
38	C₉×#97	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
40	C₃×#541	C₂×C₃	1	1	yes	characteristic
41	C₃×#536	C₂×C₃	1	1	yes	characteristic
42	C₃×#497	C₂×C₃	1	1	yes	characteristic
43	C₃×#494	C₂×C₃	1	1	yes	characteristic
44	C₃×#491	C₂×C₃	1	1	yes	characteristic
45	C₃×#488	C₂×C₃	1	1	yes	characteristic
46	C₃×#464	C₂×C₃	1	2	yes	—
47	C₃×#462	C₂×C₃	1	2	yes	—
48	C₃×#452	C₂×C₃	1	2	yes	—
49	C₃×#451	C₂×C₃	1	2	yes	—
50	C₃×#433	C₂×C₃	1	1	yes	characteristic
51	C₉×#41	—	4	2	no	—
52	C₂×Q₈×C₉	—	2	1	no	—
53	C₂×Q₈×C₉	—	4	2	no	—
54	C₂×Q₈×C₉	C₂³	1	2	yes	—
55	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
56	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
57	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₉	—	2	1	no	—
59	C₂×D₄×C₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₉	—	2	1	no	—
61	C₂²×C₄×C₉	—	2	1	no	—
62	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
63	C₉×Q₁₆	—	4	2	no	—
64	C₉×Q₁₆	—	4	2	no	—
65	C₉×Q₁₆	—	4	2	no	—
66	C₉×SD₁₆	—	4	2	no	—
67	C₂×C₈×C₉	C₂³	1	2	yes	—
68	C₂×C₈×C₉	C₂³	1	2	yes	—
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	—	2	2	no	—
73	C₉×C₄⋊C₄	—	2	2	no	—
74	C₉×C₄⋊C₄	—	2	2	no	—
75	C₉×C₄⋊C₄	D₄	1	2	yes	—
76	C₉×C₄⋊C₄	C₂³	1	2	yes	—
77	C₉×C₂²⋊C₄	—	2	2	no	—
78	C₉×C₂²⋊C₄	—	2	2	no	—
79	C₉×C₂²⋊C₄	—	2	2	no	—
80	C₉×C₂²⋊C₄	D₄	1	2	yes	—
81	C₄²×C₉	—	2	2	no	—
82	C₄²×C₉	C₂³	1	1	yes	characteristic
83	C₄²×C₉	C₂³	1	1	yes	characteristic
84	#3223	C₉	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
86	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
88	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
89	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
90	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
91	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
92	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
93	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
94	C₃×#126	—	2	2	no	—
95	C₃×#125	—	2	2	no	—
96	C₃×#125	C₂²×C₃	1	1	yes	characteristic
97	C₃×#124	C₂²×C₃	1	1	yes	characteristic
98	C₃×#124	C₂²×C₃	1	1	yes	characteristic
99	C₃×#123	—	2	2	no	—
100	C₃×C₂²⋊Q₈	—	2	2	no	—
101	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
102	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
103	C₃×#117	C₂²×C₃	1	1	yes	characteristic
104	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
105	C₃×#103	C₂²×C₃	1	2	yes	—
106	C₃×#103	C₂²×C₃	1	2	yes	—
107	C₃×#103	C₂²×C₃	1	2	yes	—
108	C₃×#102	C₂²×C₃	1	2	yes	—
109	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
110	C₃×#97	C₂²×C₃	1	1	yes	characteristic
111	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
112	C₂³×C₉	—	2	1	no	—
113	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₉	—	2	2	no	—
115	Q₈×C₉	—	2	2	no	—
116	Q₈×C₉	—	2	2	no	—
117	Q₈×C₉	—	2	2	no	—
118	Q₈×C₉	—	2	2	no	—
119	Q₈×C₉	—	4	2	no	—
120	D₄×C₉	—	2	2	no	—
121	D₄×C₉	—	4	2	no	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	—	2	2	no	—
124	C₂×C₄×C₉	—	2	2	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	—	4	2	no	—
129	C₂×C₄×C₉	—	4	2	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	C₂⁴	1	1	yes	characteristic
134	C₈×C₉	—	2	2	no	—
135	C₈×C₉	—	2	2	no	—
136	#541	C₂×C₉	1	1	yes	characteristic, complemented
137	#536	C₂×C₉	1	1	yes	characteristic
138	#497	C₂×C₉	1	1	yes	characteristic, complemented
139	#494	C₂×C₉	1	1	yes	characteristic, complemented
140	#491	C₂×C₉	1	1	yes	characteristic, complemented
141	#488	C₂×C₉	1	1	yes	characteristic, complemented
142	#464	C₂×C₉	1	2	yes	complemented
143	#462	C₂×C₉	1	2	yes	complemented
144	#452	C₂×C₉	1	2	yes	—
145	#451	C₂×C₉	1	2	yes	complemented
146	#433	C₂×C₉	1	1	yes	characteristic, complemented
147	C₃×#41	—	4	2	no	—
148	C₂×C₃×Q₈	—	2	1	no	—
149	C₂×C₃×Q₈	—	4	2	no	—
150	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
151	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
152	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
153	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
154	C₂×C₃×D₄	—	2	1	no	—
155	C₂×C₃×D₄	C₂³×C₃	1	1	yes	characteristic
156	C₂²×C₃×C₄	—	2	1	no	—
157	C₂²×C₃×C₄	—	2	1	no	—
158	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
159	C₃×Q₁₆	—	4	2	no	—
160	C₃×Q₁₆	—	4	2	no	—
161	C₃×Q₁₆	—	4	2	no	—
162	C₃×SD₁₆	—	4	2	no	—
163	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
164	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
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	—	2	2	no	—
170	C₃×C₄⋊C₄	—	2	2	no	—
171	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
172	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
173	C₃×C₂²⋊C₄	—	2	2	no	—
174	C₃×C₂²⋊C₄	—	2	2	no	—
175	C₃×C₂²⋊C₄	—	2	2	no	—
176	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
177	C₃×C₄²	—	2	2	no	—
178	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
179	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
180	C₂²×C₉	—	2	1	no	—
181	C₂²×C₉	—	2	2	no	—
182	C₂²×C₉	—	4	1	no	—
183	C₂²×C₉	—	4	2	no	—
184	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	2	2	no	—
187	C₄×C₉	—	2	2	no	—
188	C₄×C₉	—	4	1	no	—
189	C₄×C₉	—	4	1	no	—
190	C₄×C₉	—	4	1	no	—
191	C₄×C₉	—	4	2	no	—
192	C₄×C₉	—	4	2	no	—
193	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
194	C₂×#41	C₂²×C₉	1	1	yes	characteristic
195	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
196	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
197	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
198	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
199	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
200	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
201	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
202	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
203	#126	—	2	2	no	—
204	#125	—	2	2	no	—
205	#125	C₂²×C₉	1	1	yes	characteristic
206	#124	C₂²×C₉	1	1	yes	characteristic
207	#124	C₂²×C₉	1	1	yes	characteristic
208	#123	—	2	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
211	C₄×Q₈	C₂²×C₉	1	2	yes	—
212	#117	C₂²×C₉	1	1	yes	characteristic
213	C₄⋊C₈	C₂²×C₉	1	2	yes	—
214	#103	C₂²×C₉	1	2	yes	—
215	#103	C₂²×C₉	1	2	yes	—
216	#103	C₂²×C₉	1	2	yes	—
217	#102	C₂²×C₉	1	2	yes	—
218	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
219	#97	C₂²×C₉	1	1	yes	characteristic
220	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
221	C₂³×C₃	—	2	1	no	—
222	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
223	C₃×Q₈	—	2	2	no	—
224	C₃×Q₈	—	2	2	no	—
225	C₃×Q₈	—	2	2	no	—
226	C₃×Q₈	—	2	2	no	—
227	C₃×Q₈	—	2	2	no	—
228	C₃×Q₈	—	4	2	no	—
229	C₃×D₄	—	2	2	no	—
230	C₃×D₄	—	4	2	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	—	2	2	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	—	2	1	no	—
236	C₂×C₃×C₄	—	2	1	no	—
237	C₂×C₃×C₄	—	4	2	no	—
238	C₂×C₃×C₄	—	4	2	no	—
239	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
240	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
241	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
243	C₃×C₈	—	2	2	no	—
244	C₃×C₈	—	2	2	no	—
245	C₂×C₉	—	4	1	no	—
246	C₂×C₉	—	8	1	no	—
247	C₂×C₉	#535	1	1	yes	characteristic
248	C₂×C₉	#578	1	2	yes	—
249	#41	—	4	2	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
253	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
254	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
255	C₂×Q₈	C₂³×C₉	1	2	yes	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	C₂³×C₉	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
261	Q₁₆	—	4	2	no	—
262	Q₁₆	—	4	2	no	—
263	Q₁₆	—	4	2	no	—
264	SD₁₆	—	4	2	no	—
265	C₂×C₈	C₂³×C₉	1	2	yes	—
266	C₂×C₈	C₂³×C₉	1	2	yes	—
267	C₄⋊C₄	—	2	2	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	2	yes	—
274	C₄⋊C₄	C₂³×C₉	1	2	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	D₄×C₉	1	2	yes	—
279	C₄²	—	2	2	no	—
280	C₄²	C₂³×C₉	1	1	yes	characteristic
281	C₄²	C₂³×C₉	1	1	yes	characteristic
282	C₂²×C₃	—	2	2	no	—
283	C₂²×C₃	—	2	1	no	—
284	C₂²×C₃	—	4	2	no	—
285	C₂²×C₃	—	4	1	no	—
286	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	4	2	no	—
291	C₃×C₄	—	4	1	no	—
292	C₃×C₄	—	4	1	no	—
293	C₃×C₄	—	4	1	no	—
294	C₃×C₄	—	4	2	no	—
295	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
296	C₉	#3223	1	1	yes	characteristic, direct product factor, Sylow
297	C₂³	—	2	1	no	—
298	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
299	Q₈	—	2	2	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	2	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	4	2	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
316	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
317	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
318	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	2	no	—
321	C₂×C₃	—	4	1	no	—
322	C₂×C₃	—	8	1	no	—
323	C₂×C₃	C₃×#535	1	1	yes	characteristic
324	C₂×C₃	C₃×#578	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	2	no	—
329	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	2	no	—
337	C₄	—	4	2	no	—
338	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
339	C₃	C₃×#3223	1	1	yes	characteristic
340	C₂	—	4	1	no	—
341	C₂	—	8	1	no	—
342	C₂	C₉×#535	1	1	yes	characteristic
343	C₂	C₉×#578	1	2	yes	—

	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	C₂×C₃×#41	C₂²×C₃
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
1. C₉×#541	C₂	semidirect
3. C₉×#497	C₂	semidirect
4. C₉×#494	C₂	semidirect
5. C₉×#491	C₂	semidirect
6. C₉×#488	C₂	semidirect
7. C₉×#464	C₂	semidirect
8. C₉×#462	C₂	semidirect
10. C₉×#451	C₂	semidirect
11. C₉×#433	C₂	semidirect
84. #3223	C₉	direct (N × H)
136. #541	C₂×C₉	semidirect
138. #497	C₂×C₉	semidirect
139. #494	C₂×C₉	semidirect
140. #491	C₂×C₉	semidirect
141. #488	C₂×C₉	semidirect
142. #464	C₂×C₉	semidirect
143. #462	C₂×C₉	semidirect
145. #451	C₂×C₉	semidirect
146. #433	C₂×C₉	semidirect
296. C₉	#3223	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×#541	2
C₉×#536	2
C₉×#497	2
C₉×#494	2
C₉×#491	2
C₉×#488	2
C₉×#464	2
C₉×#462	2
C₉×#452	2
C₉×#451	2
C₉×#433	2
C₃×#3223	3

Subgroups — C9×#3223 #11767944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3223 #11767944