Subgroups — C₉×#3228 #11767949

Order 1152.

555 conjugacy classes of subgroups.

The subgroup lattice has 333 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×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#486	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#482	C₂	1	4	yes	maximal, complemented
8	C₉×#478	C₂	1	4	yes	maximal, complemented
9	C₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3228	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×#41	—	2	1	no	—
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×#127	C₂²	1	1	yes	characteristic
16	C₉×#125	C₂²	1	2	yes	—
17	C₉×#124	—	2	1	no	—
18	C₉×C₂²⋊Q₈	—	2	1	no	—
19	C₉×C₂²⋊Q₈	—	2	1	no	—
20	C₉×#121	—	2	1	no	—
21	C₉×#121	C₂²	1	2	yes	—
22	C₄×Q₈×C₉	—	2	1	no	—
23	C₄×Q₈×C₉	—	2	1	no	—
24	C₄×D₄×C₉	—	2	1	no	—
25	C₄×D₄×C₉	C₂²	1	2	yes	—
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×#102	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×#500	C₂×C₃	1	1	yes	characteristic
39	C₃×#498	C₂×C₃	1	1	yes	characteristic
40	C₃×#486	C₂×C₃	1	1	yes	characteristic
41	C₃×#486	C₂×C₃	1	1	yes	characteristic
42	C₃×#482	C₂×C₃	1	4	yes	—
43	C₃×#478	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	—	2	1	no	—
48	C₂×Q₈×C₉	—	2	1	no	—
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	C₂×C₈×C₉	C₂³	1	2	yes	—
58	C₂×C₈×C₉	C₂³	1	2	yes	—
59	C₉×C₄⋊C₄	—	2	1	no	—
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	—	2	2	no	—
62	C₉×C₄⋊C₄	—	2	1	no	—
63	C₉×C₄⋊C₄	—	2	1	no	—
64	C₉×C₄⋊C₄	C₂³	1	2	yes	—
65	C₉×C₄⋊C₄	C₂³	1	2	yes	—
66	C₉×C₄⋊C₄	C₂³	1	2	yes	—
67	C₉×C₂²⋊C₄	—	2	2	no	—
68	C₉×C₂²⋊C₄	—	2	2	no	—
69	C₉×C₂²⋊C₄	—	2	2	no	—
70	C₄²×C₉	—	2	2	no	—
71	C₄²×C₉	D₄	1	1	yes	characteristic
72	C₄²×C₉	D₄	1	1	yes	characteristic
73	C₄²×C₉	C₂³	1	1	yes	characteristic
74	C₄²×C₉	C₂³	1	1	yes	characteristic
75	#3228	C₉	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×C₃×#41	—	2	1	no	—
77	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
78	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
79	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
80	C₃×#127	C₂²×C₃	1	1	yes	characteristic
81	C₃×#125	C₂²×C₃	1	2	yes	—
82	C₃×#124	—	2	1	no	—
83	C₃×C₂²⋊Q₈	—	2	1	no	—
84	C₃×C₂²⋊Q₈	—	2	1	no	—
85	C₃×#121	—	2	1	no	—
86	C₃×#121	C₂²×C₃	1	2	yes	—
87	C₃×C₄×Q₈	—	2	1	no	—
88	C₃×C₄×Q₈	—	2	1	no	—
89	C₃×C₄×D₄	—	2	1	no	—
90	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
91	C₃×#117	C₂²×C₃	1	2	yes	—
92	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
93	C₃×#107	C₂²×C₃	1	4	yes	—
94	C₃×#106	C₂²×C₃	1	4	yes	—
95	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
96	C₃×#102	C₂²×C₃	1	4	yes	—
97	C₃×#102	C₂²×C₃	1	4	yes	—
98	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
99	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
100	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
101	C₂³×C₉	—	2	2	no	—
102	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₉	—	4	2	no	—
104	Q₈×C₉	—	4	2	no	—
105	Q₈×C₉	—	4	2	no	—
106	D₄×C₉	—	2	4	no	—
107	D₄×C₉	—	4	4	no	—
108	D₄×C₉	—	4	2	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	—	2	2	no	—
116	C₂×C₄×C₉	—	4	4	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	2	no	—
124	C₈×C₉	—	2	2	no	—
125	#533	C₂×C₉	1	1	yes	characteristic, complemented
126	#532	C₂×C₉	1	1	yes	characteristic
127	#500	C₂×C₉	1	1	yes	characteristic, complemented
128	#498	C₂×C₉	1	1	yes	characteristic, complemented
129	#486	C₂×C₉	1	1	yes	characteristic, complemented
130	#486	C₂×C₉	1	1	yes	characteristic, complemented
131	#482	C₂×C₉	1	4	yes	complemented
132	#478	C₂×C₉	1	4	yes	complemented
133	#431	C₂×C₉	1	1	yes	characteristic, complemented
134	C₃×#41	—	4	4	no	—
135	C₂×C₃×Q₈	—	2	1	no	—
136	C₂×C₃×Q₈	—	2	1	no	—
137	C₂×C₃×Q₈	—	2	1	no	—
138	C₂×C₃×D₄	—	2	2	no	—
139	C₂×C₃×D₄	—	2	1	no	—
140	C₂×C₃×D₄	—	2	1	no	—
141	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
142	C₂²×C₃×C₄	—	2	2	no	—
143	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
145	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
146	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
147	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	—	2	1	no	—
153	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
154	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
155	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
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	—	2	1	no	—
167	C₂²×C₉	—	4	2	no	—
168	C₂²×C₉	—	4	2	no	—
169	C₂²×C₉	—	4	2	no	—
170	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
171	C₄×C₉	—	2	1	no	—
172	C₄×C₉	—	2	1	no	—
173	C₄×C₉	—	2	1	no	—
174	C₄×C₉	—	4	2	no	—
175	C₄×C₉	—	4	2	no	—
176	C₄×C₉	—	4	2	no	—
177	C₄×C₉	#135	1	2	yes	—
178	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
179	C₄×C₉	2₋¹⁺⁴	1	2	yes	—
180	C₂×#41	—	2	1	no	—
181	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
182	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
183	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
184	#127	C₂²×C₉	1	1	yes	characteristic
185	#125	C₂²×C₉	1	2	yes	—
186	#124	—	2	1	no	—
187	C₂²⋊Q₈	—	2	1	no	—
188	C₂²⋊Q₈	—	2	1	no	—
189	#121	—	2	1	no	—
190	#121	C₂²×C₉	1	2	yes	—
191	C₄×Q₈	—	2	1	no	—
192	C₄×Q₈	—	2	1	no	—
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	C₂²×C₉	1	2	yes	—
195	#117	C₂²×C₉	1	2	yes	—
196	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
197	#107	C₂²×C₉	1	4	yes	—
198	#106	C₂²×C₉	1	4	yes	—
199	C₄⋊C₈	C₂²×C₉	1	2	yes	—
200	#102	C₂²×C₉	1	4	yes	—
201	#102	C₂²×C₉	1	4	yes	—
202	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
203	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
204	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
205	C₂³×C₃	—	2	2	no	—
206	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
207	C₃×Q₈	—	4	2	no	—
208	C₃×Q₈	—	4	2	no	—
209	C₃×Q₈	—	4	2	no	—
210	C₃×D₄	—	2	4	no	—
211	C₃×D₄	—	4	4	no	—
212	C₃×D₄	—	4	2	no	—
213	C₃×D₄	—	4	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	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	—	4	4	no	—
221	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
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	C₂⁴×C₃	1	1	yes	characteristic
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	2	2	no	—
229	C₂×C₉	—	4	1	no	—
230	C₂×C₉	—	8	2	no	—
231	C₂×C₉	#536	1	1	yes	characteristic
232	C₂×C₉	C₂×#135	1	1	yes	characteristic
233	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
234	#41	—	4	4	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂³×C₉	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
244	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
245	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
246	C₂×C₈	C₂³×C₉	1	2	yes	—
247	C₂×C₈	C₂³×C₉	1	2	yes	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	C₂³×C₉	1	2	yes	—
254	C₄⋊C₄	C₂³×C₉	1	2	yes	—
255	C₄⋊C₄	C₂³×C₉	1	2	yes	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	D₄×C₉	1	1	yes	characteristic
261	C₄²	D₄×C₉	1	1	yes	characteristic
262	C₄²	C₂³×C₉	1	1	yes	characteristic
263	C₄²	C₂³×C₉	1	1	yes	characteristic
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	4	2	no	—
268	C₂²×C₃	—	4	2	no	—
269	C₂²×C₃	—	4	2	no	—
270	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	4	2	no	—
275	C₃×C₄	—	4	2	no	—
276	C₃×C₄	—	4	2	no	—
277	C₃×C₄	C₃×#135	1	2	yes	—
278	C₃×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
279	C₃×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
280	C₉	#3228	1	1	yes	characteristic, direct product factor, Sylow
281	C₂³	—	2	2	no	—
282	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
302	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	2	2	no	—
305	C₂×C₃	—	4	1	no	—
306	C₂×C₃	—	8	2	no	—
307	C₂×C₃	C₃×#536	1	1	yes	characteristic
308	C₂×C₃	C₂×C₃×#135	1	1	yes	characteristic
309	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	C₉×#135	1	2	yes	—
324	C₄	C₉×AGL(1,ℤ/8)	1	2	yes	—
325	C₄	C₉×2₋¹⁺⁴	1	2	yes	—
326	C₃	C₃×#3228	1	1	yes	characteristic
327	C₂	—	4	1	no	—
328	C₂	—	8	2	no	—
329	C₂	C₉×#536	1	1	yes	characteristic
330	C₂	C₂×C₉×#135	1	1	yes	characteristic
331	C₂	C₂×C₉×AGL(1,ℤ/8)	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×#500	C₂	semidirect
4. C₉×#498	C₂	semidirect
5. C₉×#486	C₂	semidirect
6. C₉×#486	C₂	semidirect
7. C₉×#482	C₂	semidirect
8. C₉×#478	C₂	semidirect
9. C₉×#431	C₂	semidirect
75. #3228	C₉	direct (N × H)
125. #533	C₂×C₉	semidirect
127. #500	C₂×C₉	semidirect
128. #498	C₂×C₉	semidirect
129. #486	C₂×C₉	semidirect
130. #486	C₂×C₉	semidirect
131. #482	C₂×C₉	semidirect
132. #478	C₂×C₉	semidirect
133. #431	C₂×C₉	semidirect
280. C₉	#3228	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×#500	2
C₉×#498	2
C₉×#486	2
C₉×#482	2
C₉×#478	2
C₉×#431	2
C₃×#3228	3

Subgroups — C9×#3228 #11767949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3228 #11767949