Subgroups — C₉×#4788 #13141895

Order 1728.

408 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#215	C₂	1	1	yes	maximal, characteristic
3	C₉×#893	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#103	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#821	C₂	1	1	yes	maximal, characteristic
6	C₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#808	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4788	C₃	1	1	yes	maximal, characteristic
9	C₄×C₉×SD₁₆	—	3	1	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃×C₉	—	2	1	no	—
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₉×#215	C₄	1	4	yes	complemented
16	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₉×C₄⋊Dic₃	—	2	1	no	—
18	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₄×C₉×Dic₃	—	2	1	no	—
20	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×Q₈×Dic₃	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#215	C₂×C₃	1	1	yes	characteristic
24	C₃×#893	C₂×C₃	1	1	yes	characteristic
25	C₃²×#103	C₂×C₃	1	1	yes	characteristic
26	C₃×#821	C₂×C₃	1	1	yes	characteristic
27	C₃×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
28	C₃×#808	C₂×C₃	1	1	yes	characteristic
29	C₂×C₉×SD₁₆	—	3	1	no	—
30	C₄×Q₈×C₉	—	3	1	no	—
31	C₄×D₄×C₉	—	3	1	no	—
32	C₉×#106	—	3	1	no	—
33	C₉×#103	—	2	1	no	—
34	C₉×#103	D₃	1	1	yes	characteristic, complemented
35	C₉×#102	—	3	1	no	—
36	C₄×C₈×C₉	—	3	1	no	—
37	C₂²×D₃×C₉	—	2	1	no	—
38	C₃×Q₈×C₉	—	2	1	no	—
39	C₃×Q₈×C₉	C₂×C₄	1	2	yes	complemented
40	C₂×C₃×C₄×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	—	2	1	no	—
42	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×C₉×Dic₃	—	2	1	no	—
44	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
45	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
46	C₉×D₁₂	—	2	1	no	—
47	C₉×D₁₂	C₂×C₄	1	2	yes	—
48	C₄×D₃×C₉	—	4	2	no	—
49	C₃×C₈×C₉	—	2	1	no	—
50	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	complemented
51	C₃×C₄×SD₁₆	—	3	1	no	—
52	#4788	C₉	1	1	yes	characteristic, direct product factor
53	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃×C₄×D₃	—	2	1	no	—
56	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
58	C₃×#215	C₃×C₄	1	4	yes	—
59	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
60	C₃×C₄⋊Dic₃	—	2	1	no	—
61	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
62	C₃×C₄×Dic₃	—	2	1	no	—
63	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×Q₈×C₉	—	2	1	no	—
66	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
67	C₂×D₄×C₉	—	3	1	no	—
68	C₂²×C₄×C₉	—	6	1	no	—
69	C₉×SD₁₆	—	3	4	no	—
70	C₂×C₈×C₉	—	2	1	no	—
71	C₂×C₈×C₉	—	3	1	no	—
72	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
73	C₉×C₄⋊C₄	—	2	1	no	—
74	C₉×C₄⋊C₄	—	3	1	no	—
75	C₉×C₄⋊C₄	—	6	1	no	—
76	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
77	C₉×C₂²⋊C₄	—	6	1	no	—
78	C₄²×C₉	—	3	1	no	—
79	C₄²×C₉	—	6	1	no	—
80	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
81	C₂×D₃×C₉	—	2	2	no	—
82	C₂×D₃×C₉	—	4	1	no	—
83	C₂×D₃×C₉	—	4	1	no	—
84	C₃×C₄×C₉	—	2	2	no	—
85	C₃×C₄×C₉	—	4	1	no	—
86	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
87	C₃×C₄×C₉	#41	1	1	yes	characteristic
88	C₉×Dic₃	—	2	1	no	—
89	C₉×Dic₃	—	4	1	no	—
90	C₉×Dic₃	SD₁₆	1	2	yes	complemented
91	C₂×C₃×SD₁₆	—	3	1	no	—
92	C₃×C₄×Q₈	—	3	1	no	—
93	C₃×C₄×D₄	—	3	1	no	—
94	Q₈×Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
95	C₂×#215	C₂×C₉	1	1	yes	characteristic
96	#893	C₂×C₉	1	1	yes	characteristic
97	C₃×#106	—	3	1	no	—
98	C₃×#103	—	2	1	no	—
99	C₃×#103	C₂×C₉	1	1	yes	characteristic, complemented
100	C₃×#103	C₃×D₃	1	1	yes	characteristic
101	C₃×#102	—	3	1	no	—
102	C₃×C₄×C₈	—	3	1	no	—
103	#821	C₂×C₉	1	1	yes	characteristic
104	C₈×Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
105	#808	C₂×C₉	1	1	yes	characteristic, complemented
106	C₂²×C₃×D₃	—	2	1	no	—
107	C₃²×Q₈	—	2	1	no	—
108	C₃²×Q₈	C₂×C₃×C₄	1	2	yes	—
109	C₂×C₃²×C₄	—	2	1	no	—
110	C₂×C₃²×C₄	—	2	1	no	—
111	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂×C₃×Dic₃	—	2	1	no	—
113	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
114	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
115	C₃×D₁₂	—	2	1	no	—
116	C₃×D₁₂	C₂×C₃×C₄	1	2	yes	—
117	C₃×C₄×D₃	—	4	2	no	—
118	C₂³×C₉	—	6	1	no	—
119	C₃²×C₈	—	2	1	no	—
120	C₃×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
121	Q₈×C₉	—	2	1	no	—
122	Q₈×C₉	—	2	2	no	—
123	Q₈×C₉	—	4	1	no	—
124	Q₈×C₉	C₄×D₃	1	2	yes	complemented
125	D₄×C₉	—	3	2	no	—
126	D₄×C₉	—	6	1	no	—
127	C₂×C₄×C₉	—	2	1	no	—
128	C₂×C₄×C₉	—	2	1	no	—
129	C₂×C₄×C₉	—	2	1	no	—
130	C₂×C₄×C₉	—	3	1	no	—
131	C₂×C₄×C₉	—	3	1	no	—
132	C₂×C₄×C₉	—	4	1	no	—
133	C₂×C₄×C₉	—	4	1	no	—
134	C₂×C₄×C₉	—	6	1	no	—
135	C₂×C₄×C₉	—	12	2	no	—
136	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
137	C₈×C₉	—	2	1	no	—
138	C₈×C₉	—	3	2	no	—
139	C₈×C₉	—	4	1	no	—
140	C₄×SD₁₆	—	3	1	no	Sylow
141	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
142	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₃×C₉	#135	1	1	yes	characteristic
144	D₃×C₉	—	4	2	no	—
145	C₂×C₃×Q₈	—	2	1	no	—
146	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
147	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
148	C₂×C₃×D₄	—	3	1	no	—
149	C₂²×C₃×C₄	—	6	1	no	—
150	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
151	C₂×C₄×D₃	—	2	1	no	—
152	C₃×SD₁₆	—	3	4	no	—
153	C₂×C₃×C₈	—	2	1	no	—
154	C₂×C₃×C₈	—	3	1	no	—
155	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
156	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	2	1	no	—
158	C₃×C₄⋊C₄	—	3	1	no	—
159	C₃×C₄⋊C₄	—	6	1	no	—
160	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
161	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
162	C₃×C₂²⋊C₄	—	6	1	no	—
163	C₃×C₄²	—	3	1	no	—
164	C₃×C₄²	—	6	1	no	—
165	#215	C₄×C₉	1	4	yes	complemented
166	(C₂×D₃)⋊C₄	—	2	1	no	—
167	C₄⋊Dic₃	—	2	1	no	—
168	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
169	C₄×Dic₃	—	2	1	no	—
170	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
171	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
172	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×C₃×D₃	—	2	2	no	—
174	C₂×C₃×D₃	—	4	1	no	—
175	C₂×C₃×D₃	—	4	1	no	—
176	C₃²×C₄	—	2	2	no	—
177	C₃²×C₄	—	4	1	no	—
178	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
179	C₃²×C₄	C₃×#41	1	1	yes	characteristic
180	C₃×Dic₃	—	2	1	no	—
181	C₃×Dic₃	—	4	1	no	—
182	C₃×Dic₃	C₃×SD₁₆	1	2	yes	—
183	C₂²×C₉	—	2	1	no	—
184	C₂²×C₉	—	6	2	no	—
185	C₂²×C₉	—	12	1	no	—
186	C₂²×C₉	—	12	1	no	—
187	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
188	C₄×C₉	—	2	2	no	—
189	C₄×C₉	—	2	1	no	—
190	C₄×C₉	—	2	1	no	—
191	C₄×C₉	—	3	2	no	—
192	C₄×C₉	—	4	2	no	—
193	C₄×C₉	—	4	1	no	—
194	C₄×C₉	—	6	1	no	—
195	C₄×C₉	—	8	1	no	—
196	C₄×C₉	—	12	1	no	—
197	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
198	C₄×C₉	#237	1	1	yes	characteristic
199	C₂×SD₁₆	—	3	1	no	—
200	C₄×Q₈	—	3	1	no	—
201	C₄×D₄	—	3	1	no	—
202	#106	—	3	1	no	—
203	#103	D₃×C₉	1	1	yes	characteristic, complemented
204	#102	—	3	1	no	—
205	C₄×C₈	—	3	1	no	—
206	C₃×C₉	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
207	C₂³×C₃	—	6	1	no	—
208	C₂²×D₃	—	2	1	no	—
209	C₃×Q₈	—	2	1	no	—
210	C₃×Q₈	—	2	1	no	—
211	C₃×Q₈	—	2	2	no	—
212	C₃×Q₈	—	4	1	no	—
213	C₃×Q₈	C₂×C₄×C₉	1	2	yes	complemented
214	C₃×Q₈	C₃×C₄×D₃	1	2	yes	—
215	C₃×D₄	—	3	2	no	—
216	C₃×D₄	—	6	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	3	1	no	—
223	C₂×C₃×C₄	—	3	1	no	—
224	C₂×C₃×C₄	—	4	1	no	—
225	C₂×C₃×C₄	—	4	1	no	—
226	C₂×C₃×C₄	—	6	1	no	—
227	C₂×C₃×C₄	—	12	2	no	—
228	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
230	C₂×Dic₃	—	2	1	no	—
231	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
232	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
233	D₁₂	—	2	1	no	—
234	D₁₂	C₂×C₄×C₉	1	2	yes	—
235	C₄×D₃	—	4	2	no	—
236	C₃×C₈	—	2	1	no	—
237	C₃×C₈	—	2	1	no	—
238	C₃×C₈	—	3	2	no	—
239	C₃×C₈	—	4	1	no	—
240	C₃⋊C₈	C₂×C₄×C₉	1	2	yes	complemented
241	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
242	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
243	C₂×C₃²	C₃×#135	1	1	yes	characteristic
244	C₃×D₃	—	4	2	no	—
245	C₂×C₉	—	2	1	no	—
246	C₂×C₉	—	2	1	no	—
247	C₂×C₉	—	2	1	no	—
248	C₂×C₉	—	12	2	no	—
249	C₂×C₉	#67⋊C₄	1	1	yes	characteristic
250	C₂×C₉	D₃×SD₁₆	1	1	yes	characteristic
251	C₂×C₉	#919	1	1	yes	characteristic
252	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
253	C₂×D₄	—	3	1	no	—
254	C₂²×C₄	—	6	1	no	—
255	SD₁₆	—	3	4	no	—
256	C₂×C₈	—	3	1	no	—
257	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
258	C₄⋊C₄	—	3	1	no	—
259	C₄⋊C₄	—	6	1	no	—
260	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	6	1	no	—
262	C₄²	—	3	1	no	—
263	C₄²	—	6	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	6	2	no	—
266	C₂²×C₃	—	12	1	no	—
267	C₂²×C₃	—	12	1	no	—
268	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
269	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
270	C₂×D₃	—	2	2	no	—
271	C₂×D₃	—	4	1	no	—
272	C₂×D₃	—	4	1	no	—
273	C₃×C₄	—	2	2	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	3	2	no	—
278	C₃×C₄	—	4	2	no	—
279	C₃×C₄	—	4	1	no	—
280	C₃×C₄	—	4	1	no	—
281	C₃×C₄	—	6	1	no	—
282	C₃×C₄	—	8	1	no	—
283	C₃×C₄	—	12	1	no	—
284	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
285	C₃×C₄	C₉×#41	1	1	yes	characteristic
286	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
287	C₃×C₄	C₃×#237	1	1	yes	characteristic
288	Dic₃	—	2	1	no	—
289	Dic₃	—	4	1	no	—
290	Dic₃	C₉×SD₁₆	1	2	yes	complemented
291	C₃²	C₃×C₄×SD₁₆	1	1	yes	characteristic
292	C₉	—	2	1	no	—
293	C₉	#4788	1	1	yes	characteristic, direct product factor
294	C₂³	—	6	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	C₄×D₃×C₉	1	2	yes	complemented
297	D₄	—	3	2	no	—
298	D₄	—	6	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	3	1	no	—
302	C₂×C₄	—	3	1	no	—
303	C₂×C₄	—	6	1	no	—
304	C₂×C₄	—	12	2	no	—
305	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	3	2	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	12	2	no	—
312	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
313	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
314	C₂×C₃	C₉×#135	1	1	yes	characteristic
315	C₂×C₃	C₃×#67⋊C₄	1	1	yes	characteristic
316	C₂×C₃	C₃×D₃×SD₁₆	1	1	yes	characteristic
317	C₂×C₃	C₃×#919	1	1	yes	characteristic
318	D₃	—	4	2	no	—
319	C₂²	—	6	2	no	—
320	C₂²	—	12	1	no	—
321	C₂²	—	12	1	no	—
322	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	3	2	no	—
325	C₄	—	4	1	no	—
326	C₄	—	6	1	no	—
327	C₄	—	12	1	no	—
328	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
329	C₄	C₉×#237	1	1	yes	characteristic
330	C₃	—	2	1	no	—
331	C₃	C₄×C₉×SD₁₆	1	1	yes	characteristic, complemented
332	C₃	C₃×#4788	1	1	yes	characteristic
333	C₂	—	12	2	no	—
334	C₂	C₉×#67⋊C₄	1	1	yes	characteristic
335	C₂	D₃×C₉×SD₁₆	1	1	yes	characteristic
336	C₂	C₉×#919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#103	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₁₂	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#103	D₃
hypocenter	C₃	C₄×C₉×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#103

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. Q₈×C₉×Dic₃	C₂	semidirect
4. C₃×C₉×#103	C₂	semidirect
6. C₈×C₉×Dic₃	C₂	semidirect
7. C₉×#808	C₂	semidirect
15. C₉×#215	C₄	semidirect
34. C₉×#103	D₃	semidirect
39. C₃×Q₈×C₉	C₂×C₄	semidirect
50. C₉×C₃⋊C₈	C₂×C₄	semidirect
52. #4788	C₉	direct (N × H)
90. C₉×Dic₃	SD₁₆	semidirect
94. Q₈×Dic₃	C₂×C₉	semidirect
99. C₃×#103	C₂×C₉	semidirect
104. C₈×Dic₃	C₂×C₉	semidirect
105. #808	C₂×C₉	semidirect
124. Q₈×C₉	C₄×D₃	semidirect
165. #215	C₄×C₉	semidirect
203. #103	D₃×C₉	semidirect
206. C₃×C₉	C₄×SD₁₆	semidirect
213. C₃×Q₈	C₂×C₄×C₉	semidirect
240. C₃⋊C₈	C₂×C₄×C₉	semidirect
290. Dic₃	C₉×SD₁₆	semidirect
293. C₉	#4788	direct (N × H)
296. Q₈	C₄×D₃×C₉	semidirect
331. C₃	C₄×C₉×SD₁₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×C₉×Dic₃	2
C₂×C₉×#215	2
C₉×#893	2
C₃×C₉×#103	2
C₉×#821	2
C₈×C₉×Dic₃	2
C₉×#808	2
C₃×#4788	3
C₄×C₉×SD₁₆	3

Subgroups — C9×#4788 #13141895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4788 #13141895