Subgroups — C₉×#3210 #11767931

Order 1152.

591 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	C₉×#532	C₂	1	1	yes	maximal, characteristic
3	C₉×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#464	C₂	1	4	yes	maximal, complemented
8	C₉×#452	C₂	1	4	yes	maximal, complemented
9	C₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3210	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×#41	C₂²	1	2	yes	—
12	C₂×C₉×Q₁₆	C₂²	1	4	yes	—
13	C₂×C₉×SD₁₆	C₂²	1	4	yes	—
14	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₉×#127	C₂²	1	1	yes	characteristic
18	C₉×#125	—	2	1	no	—
19	C₉×#124	C₂²	1	2	yes	—
20	C₉×C₂²⋊Q₈	—	2	1	no	—
21	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₉×#121	—	2	1	no	—
23	C₉×#121	—	2	1	no	—
24	C₄×Q₈×C₉	—	2	1	no	—
25	C₄×Q₈×C₉	C₂²	1	2	yes	—
26	C₄×D₄×C₉	—	2	1	no	—
27	C₄×D₄×C₉	—	2	1	no	—
28	C₉×#117	—	2	1	no	—
29	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
30	C₉×C₄⋊C₈	C₂²	1	2	yes	—
31	C₉×#103	C₂²	1	4	yes	—
32	C₉×#102	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×#532	C₂×C₃	1	1	yes	characteristic
38	C₃×#494	C₂×C₃	1	1	yes	characteristic
39	C₃×#492	C₂×C₃	1	1	yes	characteristic
40	C₃×#487	C₂×C₃	1	1	yes	characteristic
41	C₃×#486	C₂×C₃	1	1	yes	characteristic
42	C₃×#464	C₂×C₃	1	4	yes	—
43	C₃×#452	C₂×C₃	1	4	yes	—
44	C₃×#431	C₂×C₃	1	1	yes	characteristic
45	C₉×#41	—	4	4	no	—
46	C₂×Q₈×C₉	—	2	1	no	—
47	C₂×Q₈×C₉	C₂³	1	2	yes	—
48	C₂×Q₈×C₉	C₂³	1	2	yes	—
49	C₂×D₄×C₉	—	2	2	no	—
50	C₂×D₄×C₉	—	2	1	no	—
51	C₂×D₄×C₉	—	2	1	no	—
52	C₂×D₄×C₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₉	—	2	2	no	—
54	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
57	C₉×Q₁₆	—	4	4	no	—
58	C₉×SD₁₆	—	4	4	no	—
59	C₂×C₈×C₉	C₂³	1	2	yes	—
60	C₂×C₈×C₉	C₂³	1	2	yes	—
61	C₉×C₄⋊C₄	—	2	1	no	—
62	C₉×C₄⋊C₄	—	2	1	no	—
63	C₉×C₄⋊C₄	—	2	1	no	—
64	C₉×C₄⋊C₄	—	2	1	no	—
65	C₉×C₄⋊C₄	—	2	1	no	—
66	C₉×C₄⋊C₄	—	2	2	no	—
67	C₉×C₄⋊C₄	—	2	2	no	—
68	C₉×C₄⋊C₄	C₂³	1	2	yes	—
69	C₉×C₂²⋊C₄	—	2	2	no	—
70	C₉×C₂²⋊C₄	—	2	2	no	—
71	C₉×C₂²⋊C₄	—	2	2	no	—
72	C₄²×C₉	—	2	2	no	—
73	C₄²×C₉	D₄	1	1	yes	characteristic
74	C₄²×C₉	D₄	1	1	yes	characteristic
75	C₄²×C₉	C₂³	1	1	yes	characteristic
76	C₄²×C₉	C₂³	1	1	yes	characteristic
77	#3210	C₉	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
79	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
80	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
81	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
82	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
83	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
84	C₃×#127	C₂²×C₃	1	1	yes	characteristic
85	C₃×#125	—	2	1	no	—
86	C₃×#124	C₂²×C₃	1	2	yes	—
87	C₃×C₂²⋊Q₈	—	2	1	no	—
88	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
89	C₃×#121	—	2	1	no	—
90	C₃×#121	—	2	1	no	—
91	C₃×C₄×Q₈	—	2	1	no	—
92	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
93	C₃×C₄×D₄	—	2	1	no	—
94	C₃×C₄×D₄	—	2	1	no	—
95	C₃×#117	—	2	1	no	—
96	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
97	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
98	C₃×#103	C₂²×C₃	1	4	yes	—
99	C₃×#102	C₂²×C₃	1	4	yes	—
100	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
101	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
102	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
103	C₂³×C₉	—	2	2	no	—
104	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
105	Q₈×C₉	—	2	4	no	—
106	Q₈×C₉	—	2	4	no	—
107	Q₈×C₉	—	4	2	no	—
108	D₄×C₉	—	2	4	no	—
109	D₄×C₉	—	4	4	no	—
110	D₄×C₉	—	4	2	no	—
111	D₄×C₉	—	4	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	—	2	2	no	—
116	C₂×C₄×C₉	—	2	2	no	—
117	C₂×C₄×C₉	—	2	2	no	—
118	C₂×C₄×C₉	—	4	4	no	—
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	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
125	C₈×C₉	—	2	2	no	—
126	C₈×C₉	—	2	2	no	—
127	#533	C₂×C₉	1	1	yes	characteristic, complemented
128	#532	C₂×C₉	1	1	yes	characteristic
129	#494	C₂×C₉	1	1	yes	characteristic, complemented
130	#492	C₂×C₉	1	1	yes	characteristic, complemented
131	#487	C₂×C₉	1	1	yes	characteristic, complemented
132	#486	C₂×C₉	1	1	yes	characteristic, complemented
133	#464	C₂×C₉	1	4	yes	complemented
134	#452	C₂×C₉	1	4	yes	complemented
135	#431	C₂×C₉	1	1	yes	characteristic, complemented
136	C₃×#41	—	4	4	no	—
137	C₂×C₃×Q₈	—	2	1	no	—
138	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
139	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
140	C₂×C₃×D₄	—	2	2	no	—
141	C₂×C₃×D₄	—	2	1	no	—
142	C₂×C₃×D₄	—	2	1	no	—
143	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
144	C₂²×C₃×C₄	—	2	2	no	—
145	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
146	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
148	C₃×Q₁₆	—	4	4	no	—
149	C₃×SD₁₆	—	4	4	no	—
150	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
151	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
152	C₃×C₄⋊C₄	—	2	1	no	—
153	C₃×C₄⋊C₄	—	2	1	no	—
154	C₃×C₄⋊C₄	—	2	1	no	—
155	C₃×C₄⋊C₄	—	2	1	no	—
156	C₃×C₄⋊C₄	—	2	1	no	—
157	C₃×C₄⋊C₄	—	2	2	no	—
158	C₃×C₄⋊C₄	—	2	2	no	—
159	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
160	C₃×C₂²⋊C₄	—	2	2	no	—
161	C₃×C₂²⋊C₄	—	2	2	no	—
162	C₃×C₂²⋊C₄	—	2	2	no	—
163	C₃×C₄²	—	2	2	no	—
164	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
165	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
166	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
167	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
168	C₂²×C₉	—	2	1	no	—
169	C₂²×C₉	—	2	1	no	—
170	C₂²×C₉	—	2	1	no	—
171	C₂²×C₉	—	4	2	no	—
172	C₂²×C₉	—	4	2	no	—
173	C₂²×C₉	—	4	2	no	—
174	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
175	C₄×C₉	—	2	1	no	—
176	C₄×C₉	—	2	1	no	—
177	C₄×C₉	—	2	1	no	—
178	C₄×C₉	—	4	2	no	—
179	C₄×C₉	—	4	2	no	—
180	C₄×C₉	—	4	2	no	—
181	C₄×C₉	#135	1	2	yes	—
182	C₄×C₉	#137	1	2	yes	—
183	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
184	C₂×#41	C₂²×C₉	1	2	yes	—
185	C₂×Q₁₆	C₂²×C₉	1	4	yes	—
186	C₂×SD₁₆	C₂²×C₉	1	4	yes	—
187	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
188	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
189	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
190	#127	C₂²×C₉	1	1	yes	characteristic
191	#125	—	2	1	no	—
192	#124	C₂²×C₉	1	2	yes	—
193	C₂²⋊Q₈	—	2	1	no	—
194	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
195	#121	—	2	1	no	—
196	#121	—	2	1	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	C₂²×C₉	1	2	yes	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	#117	—	2	1	no	—
202	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
203	C₄⋊C₈	C₂²×C₉	1	2	yes	—
204	#103	C₂²×C₉	1	4	yes	—
205	#102	C₂²×C₉	1	4	yes	—
206	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
207	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
208	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
209	C₂³×C₃	—	2	2	no	—
210	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
211	C₃×Q₈	—	2	4	no	—
212	C₃×Q₈	—	2	4	no	—
213	C₃×Q₈	—	4	2	no	—
214	C₃×D₄	—	2	4	no	—
215	C₃×D₄	—	4	4	no	—
216	C₃×D₄	—	4	2	no	—
217	C₃×D₄	—	4	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	—	2	2	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	4	4	no	—
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	C₂×C₃×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
231	C₃×C₈	—	2	2	no	—
232	C₃×C₈	—	2	2	no	—
233	C₂×C₉	—	4	1	no	—
234	C₂×C₉	—	8	2	no	—
235	C₂×C₉	#535	1	1	yes	characteristic
236	C₂×C₉	C₂×#135	1	1	yes	characteristic
237	C₂×C₉	C₂×#137	1	1	yes	characteristic
238	#41	—	4	4	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	C₂³×C₉	1	2	yes	—
241	C₂×Q₈	C₂³×C₉	1	2	yes	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	C₂³×C₉	1	2	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
248	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
249	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
250	Q₁₆	—	4	4	no	—
251	SD₁₆	—	4	4	no	—
252	C₂×C₈	C₂³×C₉	1	2	yes	—
253	C₂×C₈	C₂³×C₉	1	2	yes	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	C₂³×C₉	1	2	yes	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	D₄×C₉	1	1	yes	characteristic
267	C₄²	D₄×C₉	1	1	yes	characteristic
268	C₄²	C₂³×C₉	1	1	yes	characteristic
269	C₄²	C₂³×C₉	1	1	yes	characteristic
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	4	2	no	—
274	C₂²×C₃	—	4	2	no	—
275	C₂²×C₃	—	4	2	no	—
276	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
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×#135	1	2	yes	—
284	C₃×C₄	C₃×#137	1	2	yes	—
285	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
286	C₉	#3210	1	1	yes	characteristic, direct product factor, Sylow
287	C₂³	—	2	2	no	—
288	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
289	Q₈	—	2	4	no	—
290	Q₈	—	2	4	no	—
291	Q₈	—	4	2	no	—
292	D₄	—	2	4	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	2	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	—	2	2	no	—
302	C₂×C₄	—	4	4	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	2	no	—
310	C₈	—	2	2	no	—
311	C₂×C₃	—	4	1	no	—
312	C₂×C₃	—	8	2	no	—
313	C₂×C₃	C₃×#535	1	1	yes	characteristic
314	C₂×C₃	C₂×C₃×#135	1	1	yes	characteristic
315	C₂×C₃	C₂×C₃×#137	1	1	yes	characteristic
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
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	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	2	no	—
329	C₄	C₉×#135	1	2	yes	—
330	C₄	C₉×#137	1	2	yes	—
331	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
332	C₃	C₃×#3210	1	1	yes	characteristic
333	C₂	—	4	1	no	—
334	C₂	—	8	2	no	—
335	C₂	C₉×#535	1	1	yes	characteristic
336	C₂	C₂×C₉×#135	1	1	yes	characteristic
337	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	C₃×#532	C₂×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×#494	C₂	semidirect
4. C₉×#492	C₂	semidirect
5. C₉×#487	C₂	semidirect
6. C₉×#486	C₂	semidirect
7. C₉×#464	C₂	semidirect
8. C₉×#452	C₂	semidirect
9. C₉×#431	C₂	semidirect
77. #3210	C₉	direct (N × H)
127. #533	C₂×C₉	semidirect
129. #494	C₂×C₉	semidirect
130. #492	C₂×C₉	semidirect
131. #487	C₂×C₉	semidirect
132. #486	C₂×C₉	semidirect
133. #464	C₂×C₉	semidirect
134. #452	C₂×C₉	semidirect
135. #431	C₂×C₉	semidirect
286. C₉	#3210	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×#532	2
C₉×#494	2
C₉×#492	2
C₉×#487	2
C₉×#486	2
C₉×#464	2
C₉×#452	2
C₉×#431	2
C₃×#3210	3

Subgroups — C9×#3210 #11767931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3210 #11767931