Subgroups — C₉×#3215 #11767936

Order 1152.

600 conjugacy classes of subgroups.

The subgroup lattice has 339 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, complemented
2	C₂×C₉×#124	C₂	1	1	yes	maximal, characteristic
3	C₉×#497	C₂	1	2	yes	maximal, complemented
4	C₉×#489	C₂	1	2	yes	maximal, complemented
5	C₉×#464	C₂	1	4	yes	maximal, complemented
6	C₉×#451	C₂	1	2	yes	maximal, complemented
7	C₉×#450	C₂	1	2	yes	maximal
8	C₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#3215	C₃	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×Q₁₆	C₂²	1	4	yes	complemented
13	C₂×C₉×SD₁₆	C₂²	1	4	yes	—
14	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₉×#125	—	2	1	no	—
17	C₉×#124	—	2	1	no	—
18	C₉×#124	—	2	1	no	—
19	C₉×#124	C₂²	1	2	yes	—
20	C₉×#124	C₂²	1	2	yes	—
21	C₉×C₂²⋊Q₈	—	2	1	no	—
22	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₄×Q₈×C₉	—	2	1	no	—
24	C₄×Q₈×C₉	C₂²	1	2	yes	complemented
25	C₉×#117	—	2	1	no	—
26	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
27	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
28	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
29	C₉×C₄⋊C₈	C₂²	1	2	yes	complemented
30	C₉×#103	C₂²	1	4	yes	complemented
31	C₉×#102	C₂²	1	4	yes	—
32	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₉×#97	C₂²	1	2	yes	complemented
34	C₃×#533	C₂×C₃	1	1	yes	characteristic
35	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
36	C₃×#497	C₂×C₃	1	2	yes	—
37	C₃×#489	C₂×C₃	1	2	yes	—
38	C₃×#464	C₂×C₃	1	4	yes	—
39	C₃×#451	C₂×C₃	1	2	yes	—
40	C₃×#450	C₂×C₃	1	2	yes	—
41	C₃×#432	C₂×C₃	1	1	yes	characteristic
42	C₂⁴×C₉	—	2	1	no	—
43	C₂×Q₈×C₉	—	2	1	no	—
44	C₂×Q₈×C₉	—	2	1	no	—
45	C₂×Q₈×C₉	—	4	2	no	—
46	C₂×Q₈×C₉	C₂³	1	2	yes	—
47	C₂×Q₈×C₉	C₂³	1	2	yes	—
48	C₂×D₄×C₉	—	2	1	no	—
49	C₂×D₄×C₉	—	4	2	no	—
50	C₂×D₄×C₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
55	C₉×Q₁₆	—	4	4	no	—
56	C₉×SD₁₆	—	4	4	no	—
57	C₂×C₈×C₉	C₂³	1	4	yes	—
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×C₂²⋊C₄	—	2	2	no	—
65	C₉×C₂²⋊C₄	—	2	2	no	—
66	C₉×C₂²⋊C₄	—	2	2	no	—
67	C₉×C₂²⋊C₄	—	2	2	no	—
68	C₉×C₂²⋊C₄	—	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	#3215	C₉	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
76	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
78	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
79	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
80	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
81	C₃×#125	—	2	1	no	—
82	C₃×#124	—	2	1	no	—
83	C₃×#124	—	2	1	no	—
84	C₃×#124	C₂²×C₃	1	2	yes	—
85	C₃×#124	C₂²×C₃	1	2	yes	—
86	C₃×C₂²⋊Q₈	—	2	1	no	—
87	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
88	C₃×C₄×Q₈	—	2	1	no	—
89	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
90	C₃×#117	—	2	1	no	—
91	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
92	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
93	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
94	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
95	C₃×#103	C₂²×C₃	1	4	yes	—
96	C₃×#102	C₂²×C₃	1	4	yes	—
97	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
98	C₃×#97	C₂²×C₃	1	2	yes	—
99	C₂³×C₉	—	2	2	no	—
100	C₂³×C₉	—	4	2	no	—
101	C₂³×C₉	—	4	2	no	—
102	C₂³×C₉	—	4	2	no	—
103	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
104	Q₈×C₉	—	2	4	no	—
105	Q₈×C₉	—	2	4	no	—
106	Q₈×C₉	—	4	2	no	—
107	Q₈×C₉	—	4	2	no	—
108	D₄×C₉	—	2	4	no	—
109	D₄×C₉	—	4	2	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	2	2	no	—
112	C₂×C₄×C₉	—	2	2	no	—
113	C₂×C₄×C₉	—	2	2	no	—
114	C₂×C₄×C₉	—	2	2	no	—
115	C₂×C₄×C₉	—	4	1	no	—
116	C₂×C₄×C₉	—	4	2	no	—
117	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
123	C₈×C₉	—	2	4	no	—
124	#533	C₂×C₉	1	1	yes	characteristic, complemented
125	C₂×#124	C₂×C₉	1	1	yes	characteristic
126	#497	C₂×C₉	1	2	yes	complemented
127	#489	C₂×C₉	1	2	yes	complemented
128	#464	C₂×C₉	1	4	yes	complemented
129	#451	C₂×C₉	1	2	yes	complemented
130	#450	C₂×C₉	1	2	yes	—
131	#432	C₂×C₉	1	1	yes	characteristic, complemented
132	C₂⁴×C₃	—	2	1	no	—
133	C₂×C₃×Q₈	—	2	1	no	—
134	C₂×C₃×Q₈	—	2	1	no	—
135	C₂×C₃×Q₈	—	4	2	no	—
136	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
137	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
138	C₂×C₃×D₄	—	2	1	no	—
139	C₂×C₃×D₄	—	4	2	no	—
140	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
141	C₂²×C₃×C₄	—	2	1	no	—
142	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
145	C₃×Q₁₆	—	4	4	no	—
146	C₃×SD₁₆	—	4	4	no	—
147	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	2	no	—
152	C₃×C₄⋊C₄	—	2	2	no	—
153	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
154	C₃×C₂²⋊C₄	—	2	2	no	—
155	C₃×C₂²⋊C₄	—	2	2	no	—
156	C₃×C₂²⋊C₄	—	2	2	no	—
157	C₃×C₂²⋊C₄	—	2	2	no	—
158	C₃×C₂²⋊C₄	—	2	2	no	—
159	C₃×C₄²	—	2	2	no	—
160	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
161	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
162	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
163	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
164	C₂²×C₉	—	2	1	no	—
165	C₂²×C₉	—	2	1	no	—
166	C₂²×C₉	—	4	2	no	—
167	C₂²×C₉	—	4	2	no	—
168	C₂²×C₉	—	4	2	no	—
169	C₂²×C₉	—	8	4	no	—
170	C₂²×C₉	#137	1	2	yes	—
171	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
172	C₄×C₉	—	2	1	no	—
173	C₄×C₉	—	2	1	no	—
174	C₄×C₉	—	2	1	no	—
175	C₄×C₉	—	2	1	no	—
176	C₄×C₉	—	2	1	no	—
177	C₄×C₉	—	4	2	no	—
178	C₄×C₉	—	4	2	no	—
179	C₄×C₉	—	4	2	no	—
180	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
181	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
182	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
183	C₂×Q₁₆	C₂²×C₉	1	4	yes	complemented
184	C₂×SD₁₆	C₂²×C₉	1	4	yes	—
185	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic, complemented
186	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic, complemented
187	#125	—	2	1	no	—
188	#124	—	2	1	no	—
189	#124	—	2	1	no	—
190	#124	C₂²×C₉	1	2	yes	—
191	#124	C₂²×C₉	1	2	yes	—
192	C₂²⋊Q₈	—	2	1	no	—
193	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
194	C₄×Q₈	—	2	1	no	—
195	C₄×Q₈	C₂²×C₉	1	2	yes	complemented
196	#117	—	2	1	no	—
197	C₂×C₂²⋊C₄	—	2	1	no	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
200	C₄⋊C₈	C₂²×C₉	1	2	yes	complemented
201	#103	C₂²×C₉	1	4	yes	complemented
202	#102	C₂²×C₉	1	4	yes	—
203	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
204	#97	C₂²×C₉	1	2	yes	complemented
205	C₂³×C₃	—	2	2	no	—
206	C₂³×C₃	—	4	2	no	—
207	C₂³×C₃	—	4	2	no	—
208	C₂³×C₃	—	4	2	no	—
209	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
210	C₃×Q₈	—	2	4	no	—
211	C₃×Q₈	—	2	4	no	—
212	C₃×Q₈	—	4	2	no	—
213	C₃×Q₈	—	4	2	no	—
214	C₃×D₄	—	2	4	no	—
215	C₃×D₄	—	4	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	4	1	no	—
222	C₂×C₃×C₄	—	4	2	no	—
223	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
229	C₃×C₈	—	2	4	no	—
230	C₂×C₉	—	2	2	no	—
231	C₂×C₉	—	8	2	no	—
232	C₂×C₉	#535	1	1	yes	characteristic
233	C₂×C₉	C₂×#137	1	1	yes	characteristic
234	C₂×C₉	#575	1	1	yes	characteristic
235	C₂⁴	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	C₂³×C₉	1	2	yes	—
240	C₂×Q₈	C₂³×C₉	1	2	yes	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	C₂³×C₉	1	2	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
246	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
247	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
248	Q₁₆	—	4	4	no	—
249	SD₁₆	—	4	4	no	—
250	C₂×C₈	C₂³×C₉	1	4	yes	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	C₂³×C₉	1	2	yes	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	2	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	D₄×C₉	1	1	yes	characteristic
264	C₄²	D₄×C₉	1	1	yes	characteristic
265	C₄²	C₂³×C₉	1	1	yes	characteristic
266	C₄²	C₂³×C₉	1	1	yes	characteristic
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	4	2	no	—
270	C₂²×C₃	—	4	2	no	—
271	C₂²×C₃	—	4	2	no	—
272	C₂²×C₃	—	8	4	no	—
273	C₂²×C₃	C₃×#137	1	2	yes	—
274	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	4	2	no	—
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	—	4	2	no	—
283	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
284	C₉	#3215	1	1	yes	characteristic, direct product factor, Sylow
285	C₂³	—	2	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
290	Q₈	—	2	4	no	—
291	Q₈	—	2	4	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	D₄	—	2	4	no	—
295	D₄	—	4	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
308	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
309	C₈	—	2	4	no	—
310	C₂×C₃	—	2	2	no	—
311	C₂×C₃	—	8	2	no	—
312	C₂×C₃	C₃×#535	1	1	yes	characteristic
313	C₂×C₃	C₂×C₃×#137	1	1	yes	characteristic
314	C₂×C₃	C₃×#575	1	1	yes	characteristic
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	8	4	no	—
321	C₂²	C₉×#137	1	2	yes	—
322	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
332	C₃	C₃×#3215	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	—	8	2	no	—
335	C₂	C₉×#535	1	1	yes	characteristic
336	C₂	C₂×C₉×#137	1	1	yes	characteristic
337	C₂	C₉×#575	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	C₂²×C₃×D₄	C₂²×C₃
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
1. C₉×#533	C₂	semidirect
3. C₉×#497	C₂	semidirect
4. C₉×#489	C₂	semidirect
5. C₉×#464	C₂	semidirect
6. C₉×#451	C₂	semidirect
8. C₉×#432	C₂	semidirect
12. C₂×C₉×Q₁₆	C₂²	semidirect
14. C₉×C₄⋊Q₈	C₂²	semidirect
15. C₉×C₄⋊Q₈	C₂²	semidirect
24. C₄×Q₈×C₉	C₂²	semidirect
29. C₉×C₄⋊C₈	C₂²	semidirect
30. C₉×#103	C₂²	semidirect
33. C₉×#97	C₂²	semidirect
74. #3215	C₉	direct (N × H)
124. #533	C₂×C₉	semidirect
126. #497	C₂×C₉	semidirect
127. #489	C₂×C₉	semidirect
128. #464	C₂×C₉	semidirect
129. #451	C₂×C₉	semidirect
131. #432	C₂×C₉	semidirect
183. C₂×Q₁₆	C₂²×C₉	semidirect
185. C₄⋊Q₈	C₂²×C₉	semidirect
186. C₄⋊Q₈	C₂²×C₉	semidirect
195. C₄×Q₈	C₂²×C₉	semidirect
200. C₄⋊C₈	C₂²×C₉	semidirect
201. #103	C₂²×C₉	semidirect
204. #97	C₂²×C₉	semidirect
284. C₉	#3215	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
C₂×C₉×#124	2
C₉×#497	2
C₉×#489	2
C₉×#464	2
C₉×#451	2
C₉×#450	2
C₉×#432	2
C₃×#3215	3

Subgroups — C9×#3215 #11767936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3215 #11767936