Subgroups — C₉×#3221 #11767942

Order 1152.

582 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#536	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#496	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#492	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×#463	C₂	1	2	yes	maximal, complemented
8	C₉×#461	C₂	1	2	yes	maximal, complemented
9	C₉×#449	C₂	1	2	yes	maximal, complemented
10	C₉×#448	C₂	1	2	yes	maximal, complemented
11	C₉×#433	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#3221	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×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₉×D₈	C₂²	1	2	yes	—
19	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₉×#127	C₂²	1	1	yes	characteristic
22	C₉×#125	C₂²	1	1	yes	characteristic
23	C₉×#124	C₂²	1	1	yes	characteristic
24	C₉×#124	C₂²	1	1	yes	characteristic
25	C₉×#123	—	2	2	no	—
26	C₉×#123	—	2	2	no	—
27	C₉×C₂²⋊Q₈	—	2	2	no	—
28	C₉×#121	—	2	2	no	—
29	C₉×#121	C₂²	1	2	yes	—
30	C₄×D₄×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×#540	C₂×C₃	1	1	yes	characteristic
41	C₃×#536	C₂×C₃	1	1	yes	characteristic
42	C₃×#496	C₂×C₃	1	1	yes	characteristic
43	C₃×#492	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×#463	C₂×C₃	1	2	yes	—
47	C₃×#461	C₂×C₃	1	2	yes	—
48	C₃×#449	C₂×C₃	1	2	yes	—
49	C₃×#448	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	1	yes	characteristic
55	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
56	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₉	—	2	1	no	—
58	C₂×D₄×C₉	—	2	2	no	—
59	C₂×D₄×C₉	—	2	2	no	—
60	C₂×D₄×C₉	C₂³	1	2	yes	—
61	C₂×D₄×C₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₉	—	2	1	no	—
63	C₂²×C₄×C₉	—	2	1	no	—
64	C₂²×C₄×C₉	—	2	2	no	—
65	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
66	C₉×SD₁₆	—	4	2	no	—
67	C₉×SD₁₆	—	4	2	no	—
68	C₉×SD₁₆	—	4	2	no	—
69	C₉×D₈	—	4	2	no	—
70	C₂×C₈×C₉	C₂³	1	2	yes	—
71	C₂×C₈×C₉	C₂³	1	2	yes	—
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	—	2	2	no	—
81	C₉×C₂²⋊C₄	D₄	1	2	yes	—
82	C₄²×C₉	C₂³	1	1	yes	characteristic
83	C₄²×C₉	C₂³	1	1	yes	characteristic
84	#3221	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×SD₁₆	C₂²×C₃	1	2	yes	—
88	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
89	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
90	C₂×C₃×D₈	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×#127	C₂²×C₃	1	1	yes	characteristic
94	C₃×#125	C₂²×C₃	1	1	yes	characteristic
95	C₃×#124	C₂²×C₃	1	1	yes	characteristic
96	C₃×#124	C₂²×C₃	1	1	yes	characteristic
97	C₃×#123	—	2	2	no	—
98	C₃×#123	—	2	2	no	—
99	C₃×C₂²⋊Q₈	—	2	2	no	—
100	C₃×#121	—	2	2	no	—
101	C₃×#121	C₂²×C₃	1	2	yes	—
102	C₃×C₄×D₄	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	—	2	2	no	—
114	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₉	—	2	2	no	—
116	Q₈×C₉	—	2	2	no	—
117	Q₈×C₉	—	2	2	no	—
118	Q₈×C₉	—	4	2	no	—
119	D₄×C₉	—	2	2	no	—
120	D₄×C₉	—	2	2	no	—
121	D₄×C₉	—	2	2	no	—
122	D₄×C₉	—	4	2	no	—
123	D₄×C₉	—	8	2	no	—
124	D₄×C₉	—	8	2	no	—
125	C₂×C₄×C₉	—	2	2	no	—
126	C₂×C₄×C₉	—	2	2	no	—
127	C₂×C₄×C₉	—	2	1	no	—
128	C₂×C₄×C₉	—	2	1	no	—
129	C₂×C₄×C₉	—	2	1	no	—
130	C₂×C₄×C₉	—	4	2	no	—
131	C₂×C₄×C₉	—	4	2	no	—
132	C₂×C₄×C₉	—	8	2	no	—
133	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
134	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
135	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
137	C₈×C₉	—	2	2	no	—
138	C₈×C₉	—	2	2	no	—
139	#540	C₂×C₉	1	1	yes	characteristic, complemented
140	#536	C₂×C₉	1	1	yes	characteristic, complemented
141	#496	C₂×C₉	1	1	yes	characteristic, complemented
142	#492	C₂×C₉	1	1	yes	characteristic, complemented
143	#491	C₂×C₉	1	1	yes	characteristic, complemented
144	#488	C₂×C₉	1	1	yes	characteristic, complemented
145	#463	C₂×C₉	1	2	yes	complemented
146	#461	C₂×C₉	1	2	yes	complemented
147	#449	C₂×C₉	1	2	yes	complemented
148	#448	C₂×C₉	1	2	yes	complemented
149	#433	C₂×C₉	1	1	yes	characteristic, complemented
150	C₃×#41	—	4	2	no	—
151	C₂×C₃×Q₈	—	2	1	no	—
152	C₂×C₃×Q₈	—	4	2	no	—
153	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
154	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
155	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
156	C₂×C₃×D₄	—	2	1	no	—
157	C₂×C₃×D₄	—	2	2	no	—
158	C₂×C₃×D₄	—	2	2	no	—
159	C₂×C₃×D₄	C₂³×C₃	1	1	yes	characteristic
160	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
161	C₂²×C₃×C₄	—	2	1	no	—
162	C₂²×C₃×C₄	—	2	1	no	—
163	C₂²×C₃×C₄	—	2	2	no	—
164	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
165	C₃×SD₁₆	—	4	2	no	—
166	C₃×SD₁₆	—	4	2	no	—
167	C₃×SD₁₆	—	4	2	no	—
168	C₃×D₈	—	4	2	no	—
169	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
170	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
171	C₃×C₄⋊C₄	—	2	2	no	—
172	C₃×C₄⋊C₄	—	2	2	no	—
173	C₃×C₄⋊C₄	—	2	2	no	—
174	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
175	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
176	C₃×C₂²⋊C₄	—	2	2	no	—
177	C₃×C₂²⋊C₄	—	2	2	no	—
178	C₃×C₂²⋊C₄	—	2	2	no	—
179	C₃×C₂²⋊C₄	—	2	2	no	—
180	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
181	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
182	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
183	C₂²×C₉	—	2	1	no	—
184	C₂²×C₉	—	2	2	no	—
185	C₂²×C₉	—	4	1	no	—
186	C₂²×C₉	—	4	2	no	—
187	C₂²×C₉	—	4	2	no	—
188	C₂²×C₉	—	4	2	no	—
189	C₂²×C₉	—	4	2	no	—
190	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	2	2	no	—
193	C₄×C₉	—	2	2	no	—
194	C₄×C₉	—	4	1	no	—
195	C₄×C₉	—	4	1	no	—
196	C₄×C₉	—	4	1	no	—
197	C₄×C₉	—	4	2	no	—
198	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
199	C₂×#41	C₂²×C₉	1	1	yes	characteristic
200	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
201	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
202	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
203	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
204	C₂×D₈	C₂²×C₉	1	2	yes	—
205	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
206	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
207	#127	C₂²×C₉	1	1	yes	characteristic
208	#125	C₂²×C₉	1	1	yes	characteristic
209	#124	C₂²×C₉	1	1	yes	characteristic
210	#124	C₂²×C₉	1	1	yes	characteristic
211	#123	—	2	2	no	—
212	#123	—	2	2	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	#121	—	2	2	no	—
215	#121	C₂²×C₉	1	2	yes	—
216	C₄×D₄	C₂²×C₉	1	2	yes	—
217	#117	C₂²×C₉	1	1	yes	characteristic
218	C₄⋊C₈	C₂²×C₉	1	2	yes	—
219	#103	C₂²×C₉	1	2	yes	—
220	#103	C₂²×C₉	1	2	yes	—
221	#103	C₂²×C₉	1	2	yes	—
222	#102	C₂²×C₉	1	2	yes	—
223	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
224	#97	C₂²×C₉	1	1	yes	characteristic
225	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
226	C₂³×C₃	—	2	1	no	—
227	C₂³×C₃	—	2	2	no	—
228	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
229	C₃×Q₈	—	2	2	no	—
230	C₃×Q₈	—	2	2	no	—
231	C₃×Q₈	—	2	2	no	—
232	C₃×Q₈	—	4	2	no	—
233	C₃×D₄	—	2	2	no	—
234	C₃×D₄	—	2	2	no	—
235	C₃×D₄	—	2	2	no	—
236	C₃×D₄	—	4	2	no	—
237	C₃×D₄	—	8	2	no	—
238	C₃×D₄	—	8	2	no	—
239	C₂×C₃×C₄	—	2	2	no	—
240	C₂×C₃×C₄	—	2	2	no	—
241	C₂×C₃×C₄	—	2	1	no	—
242	C₂×C₃×C₄	—	2	1	no	—
243	C₂×C₃×C₄	—	2	1	no	—
244	C₂×C₃×C₄	—	4	2	no	—
245	C₂×C₃×C₄	—	4	2	no	—
246	C₂×C₃×C₄	—	8	2	no	—
247	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
248	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
249	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
251	C₃×C₈	—	2	2	no	—
252	C₃×C₈	—	2	2	no	—
253	C₂×C₉	—	4	1	no	—
254	C₂×C₉	—	8	1	no	—
255	C₂×C₉	—	8	2	no	—
256	C₂×C₉	#535	1	1	yes	characteristic
257	C₂×C₉	#577	1	2	yes	—
258	#41	—	4	2	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
262	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
263	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	C₂³×C₉	1	1	yes	characteristic
268	C₂×D₄	C₂³×C₉	1	2	yes	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
273	SD₁₆	—	4	2	no	—
274	SD₁₆	—	4	2	no	—
275	SD₁₆	—	4	2	no	—
276	D₈	—	4	2	no	—
277	C₂×C₈	C₂³×C₉	1	2	yes	—
278	C₂×C₈	C₂³×C₉	1	2	yes	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	D₄×C₉	1	2	yes	—
283	C₄⋊C₄	C₂³×C₉	1	2	yes	—
284	C₂²⋊C₄	—	2	2	no	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	D₄×C₉	1	2	yes	—
289	C₄²	C₂³×C₉	1	1	yes	characteristic
290	C₄²	C₂³×C₉	1	1	yes	characteristic
291	C₂²×C₃	—	2	2	no	—
292	C₂²×C₃	—	2	1	no	—
293	C₂²×C₃	—	4	2	no	—
294	C₂²×C₃	—	4	2	no	—
295	C₂²×C₃	—	4	2	no	—
296	C₂²×C₃	—	4	1	no	—
297	C₂²×C₃	—	4	2	no	—
298	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
299	C₃×C₄	—	2	2	no	—
300	C₃×C₄	—	2	2	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	4	2	no	—
303	C₃×C₄	—	4	1	no	—
304	C₃×C₄	—	4	1	no	—
305	C₃×C₄	—	4	1	no	—
306	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
307	C₉	#3221	1	1	yes	characteristic, direct product factor, Sylow
308	C₂³	—	2	1	no	—
309	C₂³	—	2	2	no	—
310	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
311	Q₈	—	2	2	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	4	2	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	8	2	no	—
320	D₄	—	8	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
330	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
331	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
332	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	2	2	no	—
335	C₂×C₃	—	4	1	no	—
336	C₂×C₃	—	8	2	no	—
337	C₂×C₃	—	8	1	no	—
338	C₂×C₃	C₃×#535	1	1	yes	characteristic
339	C₂×C₃	C₃×#577	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	2	no	—
347	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	2	no	—
355	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
356	C₃	C₃×#3221	1	1	yes	characteristic
357	C₂	—	4	1	no	—
358	C₂	—	8	1	no	—
359	C₂	—	8	2	no	—
360	C₂	C₉×#535	1	1	yes	characteristic
361	C₂	C₉×#577	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	C₃×#3221	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×#540	C₂	semidirect
2. C₉×#536	C₂	semidirect
3. C₉×#496	C₂	semidirect
4. C₉×#492	C₂	semidirect
5. C₉×#491	C₂	semidirect
6. C₉×#488	C₂	semidirect
7. C₉×#463	C₂	semidirect
8. C₉×#461	C₂	semidirect
9. C₉×#449	C₂	semidirect
10. C₉×#448	C₂	semidirect
11. C₉×#433	C₂	semidirect
84. #3221	C₉	direct (N × H)
139. #540	C₂×C₉	semidirect
140. #536	C₂×C₉	semidirect
141. #496	C₂×C₉	semidirect
142. #492	C₂×C₉	semidirect
143. #491	C₂×C₉	semidirect
144. #488	C₂×C₉	semidirect
145. #463	C₂×C₉	semidirect
146. #461	C₂×C₉	semidirect
147. #449	C₂×C₉	semidirect
148. #448	C₂×C₉	semidirect
149. #433	C₂×C₉	semidirect
307. C₉	#3221	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×#540	2
C₉×#536	2
C₉×#496	2
C₉×#492	2
C₉×#491	2
C₉×#488	2
C₉×#463	2
C₉×#461	2
C₉×#449	2
C₉×#448	2
C₉×#433	2
C₃×#3221	3

Subgroups — C9×#3221 #11767942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3221 #11767942