Subgroups — C₉×#3214 #11767935

Order 1152.

651 conjugacy classes of subgroups.

The subgroup lattice has 354 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×#496	C₂	1	2	yes	maximal, complemented
4	C₉×#489	C₂	1	2	yes	maximal, complemented
5	C₉×#461	C₂	1	2	yes	maximal, complemented
6	C₉×#459	C₂	1	2	yes	maximal, complemented
7	C₉×#449	C₂	1	4	yes	maximal, complemented
8	C₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#3214	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	—	2	1	no	—
11	C₂×C₉×#41	C₂²	1	2	yes	—
12	C₂×C₉×SD₁₆	C₂²	1	4	yes	—
13	C₂×C₉×D₈	C₂²	1	4	yes	—
14	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₉×#127	C₂²	1	1	yes	characteristic
17	C₉×#127	C₂²	1	1	yes	characteristic
18	C₉×#124	—	2	1	no	—
19	C₉×#124	C₂²	1	2	yes	—
20	C₉×#121	—	2	1	no	—
21	C₉×#121	—	2	1	no	—
22	C₉×#121	—	2	1	no	—
23	C₉×#121	C₂²	1	2	yes	—
24	C₄×D₄×C₉	—	2	1	no	—
25	C₄×D₄×C₉	—	2	1	no	—
26	C₄×D₄×C₉	—	2	1	no	—
27	C₄×D₄×C₉	C₂²	1	2	yes	—
28	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
29	C₉×C₄⋊C₈	C₂²	1	2	yes	—
30	C₉×#103	C₂²	1	4	yes	—
31	C₉×#102	C₂²	1	4	yes	—
32	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₉×#97	C₂²	1	2	yes	—
34	C₃×#532	C₂×C₃	1	1	yes	characteristic
35	C₃×#532	C₂×C₃	1	1	yes	characteristic
36	C₃×#496	C₂×C₃	1	2	yes	—
37	C₃×#489	C₂×C₃	1	2	yes	—
38	C₃×#461	C₂×C₃	1	2	yes	—
39	C₃×#459	C₂×C₃	1	2	yes	—
40	C₃×#449	C₂×C₃	1	4	yes	—
41	C₃×#432	C₂×C₃	1	1	yes	characteristic
42	C₉×#41	—	4	4	no	—
43	C₉×#41	—	4	4	no	—
44	C₂×Q₈×C₉	—	2	1	no	—
45	C₂×Q₈×C₉	C₂³	1	2	yes	—
46	C₂×D₄×C₉	—	2	2	no	—
47	C₂×D₄×C₉	—	2	2	no	—
48	C₂×D₄×C₉	—	2	1	no	—
49	C₂×D₄×C₉	—	2	1	no	—
50	C₂×D₄×C₉	—	2	1	no	—
51	C₂×D₄×C₉	—	2	1	no	—
52	C₂×D₄×C₉	C₂³	1	2	yes	—
53	C₂×D₄×C₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₉	—	2	2	no	—
55	C₂²×C₄×C₉	—	2	2	no	—
56	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
59	C₉×SD₁₆	—	4	4	no	—
60	C₉×D₈	—	4	4	no	—
61	C₂×C₈×C₉	C₂³	1	4	yes	—
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	C₂³	1	2	yes	—
66	C₉×C₂²⋊C₄	—	2	2	no	—
67	C₉×C₂²⋊C₄	—	2	2	no	—
68	C₉×C₂²⋊C₄	—	2	2	no	—
69	C₉×C₂²⋊C₄	—	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	#3214	C₉	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×C₃×#41	—	2	1	no	—
76	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
77	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
78	C₂×C₃×D₈	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×#127	C₂²×C₃	1	1	yes	characteristic
82	C₃×#127	C₂²×C₃	1	1	yes	characteristic
83	C₃×#124	—	2	1	no	—
84	C₃×#124	C₂²×C₃	1	2	yes	—
85	C₃×#121	—	2	1	no	—
86	C₃×#121	—	2	1	no	—
87	C₃×#121	—	2	1	no	—
88	C₃×#121	C₂²×C₃	1	2	yes	—
89	C₃×C₄×D₄	—	2	1	no	—
90	C₃×C₄×D₄	—	2	1	no	—
91	C₃×C₄×D₄	—	2	1	no	—
92	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
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	—	2	2	no	—
101	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₉	—	2	4	no	—
103	Q₈×C₉	—	4	2	no	—
104	D₄×C₉	—	2	2	no	—
105	D₄×C₉	—	2	4	no	—
106	D₄×C₉	—	2	2	no	—
107	D₄×C₉	—	4	2	no	—
108	D₄×C₉	—	4	2	no	—
109	D₄×C₉	—	4	2	no	—
110	D₄×C₉	—	4	4	no	—
111	D₄×C₉	—	4	4	no	—
112	D₄×C₉	—	4	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	—	4	1	no	—
118	C₂×C₄×C₉	—	4	4	no	—
119	C₂×C₄×C₉	—	4	4	no	—
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	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
126	C₈×C₉	—	2	4	no	—
127	#532	C₂×C₉	1	1	yes	characteristic, complemented
128	#532	C₂×C₉	1	1	yes	characteristic, complemented
129	#496	C₂×C₉	1	2	yes	complemented
130	#489	C₂×C₉	1	2	yes	complemented
131	#461	C₂×C₉	1	2	yes	complemented
132	#459	C₂×C₉	1	2	yes	complemented
133	#449	C₂×C₉	1	4	yes	complemented
134	#432	C₂×C₉	1	1	yes	characteristic, complemented
135	C₃×#41	—	4	4	no	—
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×D₄	—	2	2	no	—
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	—	2	1	no	—
144	C₂×C₃×D₄	—	2	1	no	—
145	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
146	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
147	C₂²×C₃×C₄	—	2	2	no	—
148	C₂²×C₃×C₄	—	2	2	no	—
149	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
151	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
152	C₃×SD₁₆	—	4	4	no	—
153	C₃×D₈	—	4	4	no	—
154	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
155	C₃×C₄⋊C₄	—	2	1	no	—
156	C₃×C₄⋊C₄	—	2	1	no	—
157	C₃×C₄⋊C₄	—	2	1	no	—
158	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
159	C₃×C₂²⋊C₄	—	2	2	no	—
160	C₃×C₂²⋊C₄	—	2	2	no	—
161	C₃×C₂²⋊C₄	—	2	2	no	—
162	C₃×C₂²⋊C₄	—	2	2	no	—
163	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
164	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
165	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
166	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
167	C₂²×C₉	—	2	1	no	—
168	C₂²×C₉	—	2	1	no	—
169	C₂²×C₉	—	2	1	no	—
170	C₂²×C₉	—	4	2	no	—
171	C₂²×C₉	—	4	2	no	—
172	C₂²×C₉	—	4	2	no	—
173	C₂²×C₉	—	4	2	no	—
174	C₂²×C₉	—	4	2	no	—
175	C₂²×C₉	—	4	2	no	—
176	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
177	C₄×C₉	—	2	1	no	—
178	C₄×C₉	—	2	1	no	—
179	C₄×C₉	—	2	1	no	—
180	C₄×C₉	—	2	1	no	—
181	C₄×C₉	—	4	2	no	—
182	C₄×C₉	—	4	2	no	—
183	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
184	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
185	C₂×#41	—	2	1	no	—
186	C₂×#41	C₂²×C₉	1	2	yes	—
187	C₂×SD₁₆	C₂²×C₉	1	4	yes	—
188	C₂×D₈	C₂²×C₉	1	4	yes	—
189	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
190	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
191	#127	C₂²×C₉	1	1	yes	characteristic
192	#127	C₂²×C₉	1	1	yes	characteristic
193	#124	—	2	1	no	—
194	#124	C₂²×C₉	1	2	yes	—
195	#121	—	2	1	no	—
196	#121	—	2	1	no	—
197	#121	—	2	1	no	—
198	#121	C₂²×C₉	1	2	yes	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	—	2	1	no	—
202	C₄×D₄	C₂²×C₉	1	2	yes	—
203	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
204	C₄⋊C₈	C₂²×C₉	1	2	yes	—
205	#103	C₂²×C₉	1	4	yes	—
206	#102	C₂²×C₉	1	4	yes	—
207	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
208	#97	C₂²×C₉	1	2	yes	—
209	C₂³×C₃	—	2	2	no	—
210	C₂³×C₃	—	2	2	no	—
211	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
212	C₃×Q₈	—	2	4	no	—
213	C₃×Q₈	—	4	2	no	—
214	C₃×D₄	—	2	2	no	—
215	C₃×D₄	—	2	4	no	—
216	C₃×D₄	—	2	2	no	—
217	C₃×D₄	—	4	2	no	—
218	C₃×D₄	—	4	2	no	—
219	C₃×D₄	—	4	2	no	—
220	C₃×D₄	—	4	4	no	—
221	C₃×D₄	—	4	4	no	—
222	C₃×D₄	—	4	2	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	2	2	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	4	1	no	—
228	C₂×C₃×C₄	—	4	4	no	—
229	C₂×C₃×C₄	—	4	4	no	—
230	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
236	C₃×C₈	—	2	4	no	—
237	C₂×C₉	—	4	1	no	—
238	C₂×C₉	—	8	2	no	—
239	C₂×C₉	—	8	2	no	—
240	C₂×C₉	#535	1	1	yes	characteristic
241	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂×C₉	#575	1	1	yes	characteristic
243	#41	—	4	4	no	—
244	#41	—	4	4	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	C₂³×C₉	1	2	yes	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	C₂³×C₉	1	2	yes	—
254	C₂×D₄	C₂³×C₉	1	2	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
258	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
259	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
260	SD₁₆	—	4	4	no	—
261	D₈	—	4	4	no	—
262	C₂×C₈	C₂³×C₉	1	4	yes	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	C₂³×C₉	1	2	yes	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₄²	D₄×C₉	1	1	yes	characteristic
272	C₄²	D₄×C₉	1	1	yes	characteristic
273	C₄²	C₂³×C₉	1	1	yes	characteristic
274	C₄²	C₂³×C₉	1	1	yes	characteristic
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	4	2	no	—
279	C₂²×C₃	—	4	2	no	—
280	C₂²×C₃	—	4	2	no	—
281	C₂²×C₃	—	4	2	no	—
282	C₂²×C₃	—	4	2	no	—
283	C₂²×C₃	—	4	2	no	—
284	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	—	4	2	no	—
291	C₃×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
292	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
293	C₉	#3214	1	1	yes	characteristic, direct product factor, Sylow
294	C₂³	—	2	2	no	—
295	C₂³	—	2	2	no	—
296	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
297	Q₈	—	2	4	no	—
298	Q₈	—	4	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	2	4	no	—
301	D₄	—	2	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
316	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
320	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
321	C₈	—	2	4	no	—
322	C₂×C₃	—	4	1	no	—
323	C₂×C₃	—	8	2	no	—
324	C₂×C₃	—	8	2	no	—
325	C₂×C₃	C₃×#535	1	1	yes	characteristic
326	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₂×C₃	C₃×#575	1	1	yes	characteristic
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	C₉×AGL(1,ℤ/8)	1	2	yes	—
345	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
346	C₃	C₃×#3214	1	1	yes	characteristic
347	C₂	—	4	1	no	—
348	C₂	—	8	2	no	—
349	C₂	—	8	2	no	—
350	C₂	C₉×#535	1	1	yes	characteristic
351	C₂	C₂×C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
352	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	C₃×#3214	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×#496	C₂	semidirect
4. C₉×#489	C₂	semidirect
5. C₉×#461	C₂	semidirect
6. C₉×#459	C₂	semidirect
7. C₉×#449	C₂	semidirect
8. C₉×#432	C₂	semidirect
74. #3214	C₉	direct (N × H)
127. #532	C₂×C₉	semidirect
128. #532	C₂×C₉	semidirect
129. #496	C₂×C₉	semidirect
130. #489	C₂×C₉	semidirect
131. #461	C₂×C₉	semidirect
132. #459	C₂×C₉	semidirect
133. #449	C₂×C₉	semidirect
134. #432	C₂×C₉	semidirect
293. C₉	#3214	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×#496	2
C₉×#489	2
C₉×#461	2
C₉×#459	2
C₉×#449	2
C₉×#432	2
C₃×#3214	3

Subgroups — C9×#3214 #11767935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3214 #11767935