Subgroups — C₉×#3207 #11767928

Order 1152.

657 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×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#493	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×#463	C₂	1	4	yes	maximal, complemented
8	C₉×#449	C₂	1	4	yes	maximal, complemented
9	C₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3207	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×#41	—	2	1	no	—
12	C₂×C₉×#41	C₂²	1	2	yes	—
13	C₂×C₉×SD₁₆	C₂²	1	4	yes	—
14	C₂×C₉×D₈	C₂²	1	4	yes	—
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×#127	C₂²	1	1	yes	characteristic
19	C₉×#124	—	2	1	no	—
20	C₉×#124	C₂²	1	2	yes	—
21	C₉×#121	—	2	1	no	—
22	C₉×#121	—	2	1	no	—
23	C₉×#121	—	2	1	no	—
24	C₉×#121	C₂²	1	2	yes	—
25	C₄×D₄×C₉	—	2	1	no	—
26	C₄×D₄×C₉	—	2	1	no	—
27	C₄×D₄×C₉	—	2	1	no	—
28	C₄×D₄×C₉	C₂²	1	2	yes	—
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×#532	C₂×C₃	1	1	yes	characteristic
37	C₃×#532	C₂×C₃	1	1	yes	characteristic
38	C₃×#493	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×#463	C₂×C₃	1	4	yes	—
43	C₃×#449	C₂×C₃	1	4	yes	—
44	C₃×#431	C₂×C₃	1	1	yes	characteristic
45	C₉×#41	—	4	4	no	—
46	C₉×#41	—	4	4	no	—
47	C₂×Q₈×C₉	—	2	1	no	—
48	C₂×Q₈×C₉	C₂³	1	2	yes	—
49	C₂×D₄×C₉	—	2	2	no	—
50	C₂×D₄×C₉	—	2	2	no	—
51	C₂×D₄×C₉	—	2	1	no	—
52	C₂×D₄×C₉	—	2	1	no	—
53	C₂×D₄×C₉	—	2	1	no	—
54	C₂×D₄×C₉	—	2	1	no	—
55	C₂×D₄×C₉	C₂³	1	2	yes	—
56	C₂×D₄×C₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₉	—	2	2	no	—
58	C₂²×C₄×C₉	—	2	2	no	—
59	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
62	C₉×SD₁₆	—	4	4	no	—
63	C₉×D₈	—	4	4	no	—
64	C₂×C₈×C₉	C₂³	1	2	yes	—
65	C₂×C₈×C₉	C₂³	1	2	yes	—
66	C₉×C₄⋊C₄	—	2	1	no	—
67	C₉×C₄⋊C₄	—	2	1	no	—
68	C₉×C₄⋊C₄	—	2	1	no	—
69	C₉×C₄⋊C₄	C₂³	1	2	yes	—
70	C₉×C₂²⋊C₄	—	2	2	no	—
71	C₉×C₂²⋊C₄	—	2	2	no	—
72	C₉×C₂²⋊C₄	—	2	2	no	—
73	C₉×C₂²⋊C₄	—	2	2	no	—
74	C₄²×C₉	D₄	1	1	yes	characteristic
75	C₄²×C₉	D₄	1	1	yes	characteristic
76	C₄²×C₉	C₂³	1	1	yes	characteristic
77	C₄²×C₉	C₂³	1	1	yes	characteristic
78	#3207	C₉	1	1	yes	characteristic, direct product factor, Sylow
79	C₂×C₃×#41	—	2	1	no	—
80	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
81	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
82	C₂×C₃×D₈	C₂²×C₃	1	4	yes	—
83	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
84	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
85	C₃×#127	C₂²×C₃	1	1	yes	characteristic
86	C₃×#127	C₂²×C₃	1	1	yes	characteristic
87	C₃×#124	—	2	1	no	—
88	C₃×#124	C₂²×C₃	1	2	yes	—
89	C₃×#121	—	2	1	no	—
90	C₃×#121	—	2	1	no	—
91	C₃×#121	—	2	1	no	—
92	C₃×#121	C₂²×C₃	1	2	yes	—
93	C₃×C₄×D₄	—	2	1	no	—
94	C₃×C₄×D₄	—	2	1	no	—
95	C₃×C₄×D₄	—	2	1	no	—
96	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
97	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
98	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
99	C₃×#103	C₂²×C₃	1	4	yes	—
100	C₃×#102	C₂²×C₃	1	4	yes	—
101	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
102	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
103	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
104	C₂³×C₉	—	2	2	no	—
105	C₂³×C₉	—	2	2	no	—
106	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₉	—	2	4	no	—
108	Q₈×C₉	—	4	2	no	—
109	D₄×C₉	—	2	4	no	—
110	D₄×C₉	—	2	4	no	—
111	D₄×C₉	—	4	4	no	—
112	D₄×C₉	—	4	4	no	—
113	D₄×C₉	—	4	2	no	—
114	D₄×C₉	—	4	2	no	—
115	D₄×C₉	—	4	2	no	—
116	D₄×C₉	—	4	2	no	—
117	C₂×C₄×C₉	—	2	2	no	—
118	C₂×C₄×C₉	—	2	2	no	—
119	C₂×C₄×C₉	—	2	2	no	—
120	C₂×C₄×C₉	—	2	2	no	—
121	C₂×C₄×C₉	—	2	2	no	—
122	C₂×C₄×C₉	—	4	4	no	—
123	C₂×C₄×C₉	—	4	4	no	—
124	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
130	C₈×C₉	—	2	2	no	—
131	C₈×C₉	—	2	2	no	—
132	#532	C₂×C₉	1	1	yes	characteristic, complemented
133	#532	C₂×C₉	1	1	yes	characteristic, complemented
134	#493	C₂×C₉	1	1	yes	characteristic, complemented
135	#492	C₂×C₉	1	1	yes	characteristic, complemented
136	#487	C₂×C₉	1	1	yes	characteristic, complemented
137	#486	C₂×C₉	1	1	yes	characteristic, complemented
138	#463	C₂×C₉	1	4	yes	complemented
139	#449	C₂×C₉	1	4	yes	complemented
140	#431	C₂×C₉	1	1	yes	characteristic, complemented
141	C₃×#41	—	4	4	no	—
142	C₃×#41	—	4	4	no	—
143	C₂×C₃×Q₈	—	2	1	no	—
144	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
145	C₂×C₃×D₄	—	2	2	no	—
146	C₂×C₃×D₄	—	2	2	no	—
147	C₂×C₃×D₄	—	2	1	no	—
148	C₂×C₃×D₄	—	2	1	no	—
149	C₂×C₃×D₄	—	2	1	no	—
150	C₂×C₃×D₄	—	2	1	no	—
151	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
152	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
153	C₂²×C₃×C₄	—	2	2	no	—
154	C₂²×C₃×C₄	—	2	2	no	—
155	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
156	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
158	C₃×SD₁₆	—	4	4	no	—
159	C₃×D₈	—	4	4	no	—
160	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
161	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
162	C₃×C₄⋊C₄	—	2	1	no	—
163	C₃×C₄⋊C₄	—	2	1	no	—
164	C₃×C₄⋊C₄	—	2	1	no	—
165	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
166	C₃×C₂²⋊C₄	—	2	2	no	—
167	C₃×C₂²⋊C₄	—	2	2	no	—
168	C₃×C₂²⋊C₄	—	2	2	no	—
169	C₃×C₂²⋊C₄	—	2	2	no	—
170	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
171	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
172	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
173	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
174	C₂²×C₉	—	2	1	no	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	2	1	no	—
177	C₂²×C₉	—	4	2	no	—
178	C₂²×C₉	—	4	2	no	—
179	C₂²×C₉	—	4	2	no	—
180	C₂²×C₉	—	4	2	no	—
181	C₂²×C₉	—	4	2	no	—
182	C₂²×C₉	—	4	2	no	—
183	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
184	C₄×C₉	—	2	1	no	—
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	2	1	no	—
187	C₄×C₉	—	4	2	no	—
188	C₄×C₉	—	4	2	no	—
189	C₄×C₉	#135	1	2	yes	—
190	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
191	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
192	C₂×#41	—	2	1	no	—
193	C₂×#41	C₂²×C₉	1	2	yes	—
194	C₂×SD₁₆	C₂²×C₉	1	4	yes	—
195	C₂×D₈	C₂²×C₉	1	4	yes	—
196	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
197	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
198	#127	C₂²×C₉	1	1	yes	characteristic
199	#127	C₂²×C₉	1	1	yes	characteristic
200	#124	—	2	1	no	—
201	#124	C₂²×C₉	1	2	yes	—
202	#121	—	2	1	no	—
203	#121	—	2	1	no	—
204	#121	—	2	1	no	—
205	#121	C₂²×C₉	1	2	yes	—
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	C₂²×C₉	1	2	yes	—
210	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
211	C₄⋊C₈	C₂²×C₉	1	2	yes	—
212	#103	C₂²×C₉	1	4	yes	—
213	#102	C₂²×C₉	1	4	yes	—
214	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
215	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
216	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
217	C₂³×C₃	—	2	2	no	—
218	C₂³×C₃	—	2	2	no	—
219	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
220	C₃×Q₈	—	2	4	no	—
221	C₃×Q₈	—	4	2	no	—
222	C₃×D₄	—	2	4	no	—
223	C₃×D₄	—	2	4	no	—
224	C₃×D₄	—	4	4	no	—
225	C₃×D₄	—	4	4	no	—
226	C₃×D₄	—	4	2	no	—
227	C₃×D₄	—	4	2	no	—
228	C₃×D₄	—	4	2	no	—
229	C₃×D₄	—	4	2	no	—
230	C₂×C₃×C₄	—	2	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	2	no	—
235	C₂×C₃×C₄	—	4	4	no	—
236	C₂×C₃×C₄	—	4	4	no	—
237	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
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	2	no	—
247	C₂×C₉	—	8	2	no	—
248	C₂×C₉	#535	1	1	yes	characteristic
249	C₂×C₉	C₂×#135	1	1	yes	characteristic
250	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
251	#41	—	4	4	no	—
252	#41	—	4	4	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	C₂³×C₉	1	2	yes	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	C₂³×C₉	1	2	yes	—
262	C₂×D₄	C₂³×C₉	1	2	yes	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
266	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
267	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
268	SD₁₆	—	4	4	no	—
269	D₈	—	4	4	no	—
270	C₂×C₈	C₂³×C₉	1	2	yes	—
271	C₂×C₈	C₂³×C₉	1	2	yes	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	C₂³×C₉	1	2	yes	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₄²	D₄×C₉	1	1	yes	characteristic
281	C₄²	D₄×C₉	1	1	yes	characteristic
282	C₄²	C₂³×C₉	1	1	yes	characteristic
283	C₄²	C₂³×C₉	1	1	yes	characteristic
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	—	2	1	no	—
286	C₂²×C₃	—	2	1	no	—
287	C₂²×C₃	—	4	2	no	—
288	C₂²×C₃	—	4	2	no	—
289	C₂²×C₃	—	4	2	no	—
290	C₂²×C₃	—	4	2	no	—
291	C₂²×C₃	—	4	2	no	—
292	C₂²×C₃	—	4	2	no	—
293	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	—	2	1	no	—
297	C₃×C₄	—	4	2	no	—
298	C₃×C₄	—	4	2	no	—
299	C₃×C₄	C₃×#135	1	2	yes	—
300	C₃×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
301	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
302	C₉	#3207	1	1	yes	characteristic, direct product factor, Sylow
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
306	Q₈	—	2	4	no	—
307	Q₈	—	4	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	2	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
326	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
328	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	2	no	—
331	C₂×C₃	—	4	1	no	—
332	C₂×C₃	—	8	2	no	—
333	C₂×C₃	—	8	2	no	—
334	C₂×C₃	C₃×#535	1	1	yes	characteristic
335	C₂×C₃	C₂×C₃×#135	1	1	yes	characteristic
336	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	C₉×#135	1	2	yes	—
353	C₄	C₉×AGL(1,ℤ/8)	1	2	yes	—
354	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
355	C₃	C₃×#3207	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	8	2	no	—
358	C₂	—	8	2	no	—
359	C₂	C₉×#535	1	1	yes	characteristic
360	C₂	C₂×C₉×#135	1	1	yes	characteristic
361	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×#3207	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×#532	C₂	semidirect
2. C₉×#532	C₂	semidirect
3. C₉×#493	C₂	semidirect
4. C₉×#492	C₂	semidirect
5. C₉×#487	C₂	semidirect
6. C₉×#486	C₂	semidirect
7. C₉×#463	C₂	semidirect
8. C₉×#449	C₂	semidirect
9. C₉×#431	C₂	semidirect
78. #3207	C₉	direct (N × H)
132. #532	C₂×C₉	semidirect
133. #532	C₂×C₉	semidirect
134. #493	C₂×C₉	semidirect
135. #492	C₂×C₉	semidirect
136. #487	C₂×C₉	semidirect
137. #486	C₂×C₉	semidirect
138. #463	C₂×C₉	semidirect
139. #449	C₂×C₉	semidirect
140. #431	C₂×C₉	semidirect
302. C₉	#3207	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×#532	2
C₉×#493	2
C₉×#492	2
C₉×#487	2
C₉×#486	2
C₉×#463	2
C₉×#449	2
C₉×#431	2
C₃×#3207	3

Subgroups — C9×#3207 #11767928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3207 #11767928