Subgroups — C₉×#5168 #13143347

Order 1728.

458 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#947	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#216	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#936	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₉×#213	C₂	1	1	yes	maximal, characteristic
6	C₂×C₉×#204	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5168	C₃	1	1	yes	maximal, characteristic
9	C₉×#495	—	3	1	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
13	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
14	C₃×C₉×SD₁₆	—	2	2	no	—
15	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
16	C₉×#217	—	2	2	no	—
17	C₉×#216	—	2	2	no	—
18	C₉×#213	—	2	2	no	—
19	C₉×(C₂×D₃)⋊C₄	—	2	2	no	—
20	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₉×#204	—	2	2	no	—
23	C₂×C₃²×SD₁₆	C₂×C₃	1	1	yes	characteristic
24	C₃×#947	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×#216	C₂×C₃	1	1	yes	characteristic
26	C₃×#936	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×#213	C₂×C₃	1	1	yes	characteristic
28	C₂×C₃×#204	C₂×C₃	1	1	yes	characteristic
29	C₃×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
30	C₂×C₉×Q₁₆	—	3	1	no	—
31	C₂×C₉×SD₁₆	—	2	1	no	—
32	C₂×C₉×SD₁₆	—	3	1	no	—
33	C₂×C₉×SD₁₆	D₃	1	1	yes	characteristic, complemented
34	C₂×C₉×D₈	—	3	1	no	—
35	C₉×#124	—	3	1	no	—
36	C₉×#124	—	3	1	no	—
37	C₄×C₈×C₉	—	3	1	no	—
38	C₂³×C₃×C₉	—	2	1	no	—
39	C₂²×D₃×C₉	—	2	1	no	—
40	C₃×Q₈×C₉	—	2	1	no	—
41	C₃×Q₈×C₉	—	2	1	no	—
42	C₃×D₄×C₉	—	2	1	no	—
43	C₃×D₄×C₉	—	2	1	no	—
44	C₂×C₃×C₄×C₉	—	2	1	no	—
45	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₂×C₉×Dic₃	—	2	1	no	—
47	C₂×C₉×Dic₃	D₄	1	2	yes	—
48	C₉×D₁₂	—	2	1	no	—
49	C₉×D₁₂	—	2	1	no	—
50	C₉×Dic₆	—	2	1	no	—
51	C₉×Dic₆	—	2	1	no	—
52	C₃×C₈×C₉	D₄	1	2	yes	—
53	C₉×C₃⋊C₈	D₄	1	2	yes	—
54	C₃×#495	—	3	1	no	—
55	#5168	C₉	1	1	yes	characteristic, direct product factor
56	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
60	C₃²×SD₁₆	—	2	2	no	—
61	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₃×#217	—	2	2	no	—
63	C₃×#216	—	2	2	no	—
64	C₃×#213	—	2	2	no	—
65	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
66	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
68	C₃×#204	—	2	2	no	—
69	C₂×Q₈×C₉	—	2	1	no	—
70	C₂×Q₈×C₉	—	3	1	no	—
71	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
72	C₂×D₄×C₉	—	2	1	no	—
73	C₂×D₄×C₉	—	3	1	no	—
74	C₂×D₄×C₉	C₂×D₃	1	1	yes	characteristic
75	C₉×Q₁₆	—	6	2	no	—
76	C₉×SD₁₆	—	2	4	no	—
77	C₉×SD₁₆	—	2	2	no	—
78	C₉×SD₁₆	—	6	2	no	—
79	C₉×D₈	—	6	2	no	—
80	C₂×C₈×C₉	—	2	1	no	—
81	C₂×C₈×C₉	—	3	1	no	—
82	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
83	C₉×C₂²⋊C₄	—	6	2	no	—
84	C₉×C₂²⋊C₄	—	6	2	no	—
85	C₄²×C₉	—	3	1	no	—
86	C₂²×C₃×C₉	—	4	1	no	—
87	C₂²×C₃×C₉	—	4	2	no	—
88	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
89	C₂×D₃×C₉	—	4	1	no	—
90	C₂×D₃×C₉	—	4	2	no	—
91	C₃×C₄×C₉	—	4	1	no	—
92	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
94	C₉×Dic₃	—	2	2	no	—
95	C₉×Dic₃	—	4	1	no	—
96	C₂×C₃×Q₁₆	—	3	1	no	—
97	C₂×C₃×SD₁₆	—	2	1	no	—
98	C₂×C₃×SD₁₆	—	3	1	no	—
99	C₂×C₃×SD₁₆	C₂×C₉	1	1	yes	characteristic, complemented
100	C₂×C₃×SD₁₆	C₃×D₃	1	1	yes	characteristic
101	C₂×C₃×D₈	—	3	1	no	—
102	C₃×#124	—	3	1	no	—
103	C₃×#124	—	3	1	no	—
104	#947	C₂×C₉	1	1	yes	characteristic, complemented
105	C₂×#216	C₂×C₉	1	1	yes	characteristic, complemented
106	#936	C₂×C₉	1	1	yes	characteristic, complemented
107	C₂×#213	C₂×C₉	1	1	yes	characteristic
108	C₂×#204	C₂×C₉	1	1	yes	characteristic, complemented
109	C₃×C₄×C₈	—	3	1	no	—
110	C₈×Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
111	C₂³×C₃²	—	2	1	no	—
112	C₂²×C₃×D₃	—	2	1	no	—
113	C₃²×Q₈	—	2	1	no	—
114	C₃²×Q₈	—	2	1	no	—
115	C₃²×D₄	—	2	1	no	—
116	C₃²×D₄	—	2	1	no	—
117	C₂×C₃²×C₄	—	2	1	no	—
118	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
119	C₂×C₃×Dic₃	—	2	1	no	—
120	C₂×C₃×Dic₃	C₃×D₄	1	2	yes	—
121	C₃×D₁₂	—	2	1	no	—
122	C₃×D₁₂	—	2	1	no	—
123	C₃×Dic₆	—	2	1	no	—
124	C₃×Dic₆	—	2	1	no	—
125	C₂³×C₉	—	2	1	no	—
126	C₂³×C₉	—	4	1	no	—
127	C₂³×C₉	—	6	1	no	—
128	C₃²×C₈	C₃×D₄	1	2	yes	—
129	C₃×C₃⋊C₈	C₃×D₄	1	2	yes	—
130	Q₈×C₉	—	2	2	no	—
131	Q₈×C₉	—	2	1	no	—
132	Q₈×C₉	—	2	1	no	—
133	Q₈×C₉	—	4	1	no	—
134	Q₈×C₉	—	6	1	no	—
135	Q₈×C₉	—	6	1	no	—
136	D₄×C₉	—	2	2	no	—
137	D₄×C₉	—	2	1	no	—
138	D₄×C₉	—	2	1	no	—
139	D₄×C₉	—	4	1	no	—
140	D₄×C₉	—	6	1	no	—
141	D₄×C₉	—	6	1	no	—
142	C₂×C₄×C₉	—	2	1	no	—
143	C₂×C₄×C₉	—	2	1	no	—
144	C₂×C₄×C₉	—	3	2	no	—
145	C₂×C₄×C₉	—	4	1	no	—
146	C₂×C₄×C₉	—	6	1	no	—
147	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
148	C₈×C₉	—	2	2	no	—
149	C₈×C₉	—	3	2	no	—
150	C₈×C₉	#67	1	2	yes	—
151	#495	—	3	1	no	Sylow
152	C₂×C₃×C₉	—	8	1	no	—
153	C₂×C₃×C₉	#127	1	1	yes	characteristic
154	C₂×C₃×C₉	#135	1	2	yes	—
155	D₃×C₉	—	8	1	no	—
156	C₂×C₃×Q₈	—	2	1	no	—
157	C₂×C₃×Q₈	—	3	1	no	—
158	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
159	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
160	C₂×C₃×D₄	—	2	1	no	—
161	C₂×C₃×D₄	—	3	1	no	—
162	C₂×C₃×D₄	C₂²×C₉	1	1	yes	characteristic
163	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
164	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
165	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
166	C₃×Q₁₆	—	6	2	no	—
167	C₃×SD₁₆	—	2	4	no	—
168	C₃×SD₁₆	—	2	2	no	—
169	C₃×SD₁₆	—	2	2	no	—
170	C₃×SD₁₆	—	6	2	no	—
171	C₃×D₈	—	6	2	no	—
172	C₂×C₃×C₈	—	2	1	no	—
173	C₂×C₃×C₈	—	3	1	no	—
174	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
175	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
176	C₃×C₂²⋊C₄	—	6	2	no	—
177	C₃×C₂²⋊C₄	—	6	2	no	—
178	C₃×C₄²	—	3	1	no	—
179	#217	—	2	2	no	—
180	#216	—	2	2	no	—
181	#213	—	2	2	no	—
182	(C₂×D₃)⋊C₄	—	2	2	no	—
183	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
184	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
185	#204	—	2	2	no	—
186	C₂²×C₃²	—	4	1	no	—
187	C₂²×C₃²	—	4	2	no	—
188	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
189	C₂×C₃×D₃	—	4	1	no	—
190	C₂×C₃×D₃	—	4	2	no	—
191	C₃²×C₄	—	4	1	no	—
192	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
193	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
194	C₃×Dic₃	—	2	2	no	—
195	C₃×Dic₃	—	4	1	no	—
196	C₂²×C₉	—	2	1	no	—
197	C₂²×C₉	—	4	2	no	—
198	C₂²×C₉	—	4	2	no	—
199	C₂²×C₉	—	4	1	no	—
200	C₂²×C₉	—	8	2	no	—
201	C₂²×C₉	—	12	2	no	—
202	C₂²×C₉	—	12	1	no	—
203	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
204	C₄×C₉	—	2	1	no	—
205	C₄×C₉	—	2	1	no	—
206	C₄×C₉	—	4	2	no	—
207	C₄×C₉	—	4	1	no	—
208	C₄×C₉	—	6	2	no	—
209	C₄×C₉	—	12	1	no	—
210	C₄×C₉	D₃×D₄	1	1	yes	characteristic
211	C₄×C₉	C₂×#67	1	1	yes	characteristic
212	C₂×Q₁₆	—	3	1	no	—
213	C₂×SD₁₆	—	3	1	no	—
214	C₂×SD₁₆	D₃×C₉	1	1	yes	characteristic, complemented
215	C₂×D₈	—	3	1	no	—
216	#124	—	3	1	no	—
217	#124	—	3	1	no	—
218	C₄×C₈	—	3	1	no	—
219	C₃×C₉	#495	1	1	yes	characteristic, complemented, Sylow
220	C₂³×C₃	—	2	1	no	—
221	C₂³×C₃	—	2	1	no	—
222	C₂³×C₃	—	4	1	no	—
223	C₂³×C₃	—	6	1	no	—
224	C₂²×D₃	—	2	1	no	—
225	C₃×Q₈	—	2	2	no	—
226	C₃×Q₈	—	2	1	no	—
227	C₃×Q₈	—	2	1	no	—
228	C₃×Q₈	—	2	1	no	—
229	C₃×Q₈	—	2	1	no	—
230	C₃×Q₈	—	4	1	no	—
231	C₃×Q₈	—	6	1	no	—
232	C₃×Q₈	—	6	1	no	—
233	C₃×D₄	—	2	2	no	—
234	C₃×D₄	—	2	1	no	—
235	C₃×D₄	—	2	1	no	—
236	C₃×D₄	—	2	1	no	—
237	C₃×D₄	—	2	1	no	—
238	C₃×D₄	—	4	1	no	—
239	C₃×D₄	—	6	1	no	—
240	C₃×D₄	—	6	1	no	—
241	C₂×C₃×C₄	—	2	1	no	—
242	C₂×C₃×C₄	—	2	1	no	—
243	C₂×C₃×C₄	—	2	1	no	—
244	C₂×C₃×C₄	—	3	2	no	—
245	C₂×C₃×C₄	—	4	1	no	—
246	C₂×C₃×C₄	—	6	1	no	—
247	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
248	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
249	C₂×Dic₃	—	2	1	no	—
250	C₂×Dic₃	D₄×C₉	1	2	yes	—
251	D₁₂	—	2	1	no	—
252	D₁₂	—	2	1	no	—
253	Dic₆	—	2	1	no	—
254	Dic₆	—	2	1	no	—
255	C₃×C₈	—	2	2	no	—
256	C₃×C₈	—	3	2	no	—
257	C₃×C₈	D₄×C₉	1	2	yes	—
258	C₃×C₈	C₃×#67	1	2	yes	—
259	C₃⋊C₈	D₄×C₉	1	2	yes	—
260	C₂×C₃²	—	8	1	no	—
261	C₂×C₃²	C₃×#127	1	1	yes	characteristic
262	C₂×C₃²	C₃×#135	1	2	yes	—
263	C₃×D₃	—	8	1	no	—
264	C₂×C₉	—	2	2	no	—
265	C₂×C₉	—	2	1	no	—
266	C₂×C₉	—	8	2	no	—
267	C₂×C₉	—	8	1	no	—
268	C₂×C₉	—	24	1	no	—
269	C₂×C₉	#916	1	2	yes	—
270	C₂×C₉	#940	1	1	yes	characteristic
271	C₂×Q₈	—	3	1	no	—
272	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
273	C₂×D₄	—	3	1	no	—
274	C₂×D₄	C₂×D₃×C₉	1	1	yes	characteristic
275	Q₁₆	—	6	2	no	—
276	SD₁₆	—	2	2	no	—
277	SD₁₆	—	6	2	no	—
278	D₈	—	6	2	no	—
279	C₂×C₈	—	3	1	no	—
280	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
281	C₂²⋊C₄	—	6	2	no	—
282	C₂²⋊C₄	—	6	2	no	—
283	C₄²	—	3	1	no	—
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	—	4	2	no	—
286	C₂²×C₃	—	4	2	no	—
287	C₂²×C₃	—	4	1	no	—
288	C₂²×C₃	—	4	2	no	—
289	C₂²×C₃	—	4	1	no	—
290	C₂²×C₃	—	8	2	no	—
291	C₂²×C₃	—	12	2	no	—
292	C₂²×C₃	—	12	1	no	—
293	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
294	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
295	C₂×D₃	—	4	1	no	—
296	C₂×D₃	—	4	2	no	—
297	C₃×C₄	—	2	1	no	—
298	C₃×C₄	—	2	1	no	—
299	C₃×C₄	—	4	2	no	—
300	C₃×C₄	—	4	1	no	—
301	C₃×C₄	—	4	1	no	—
302	C₃×C₄	—	6	2	no	—
303	C₃×C₄	—	12	1	no	—
304	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
305	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
306	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
307	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
308	Dic₃	—	2	2	no	—
309	Dic₃	—	4	1	no	—
310	C₃²	C₃×#495	1	1	yes	characteristic
311	C₉	—	2	1	no	—
312	C₉	#5168	1	1	yes	characteristic, direct product factor
313	C₂³	—	2	1	no	—
314	C₂³	—	6	1	no	—
315	Q₈	—	2	1	no	—
316	Q₈	—	2	1	no	—
317	Q₈	—	6	1	no	—
318	Q₈	—	6	1	no	—
319	D₄	—	2	1	no	—
320	D₄	—	2	1	no	—
321	D₄	—	6	1	no	—
322	D₄	—	6	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	3	2	no	—
325	C₂×C₄	—	6	1	no	—
326	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
327	C₈	—	3	2	no	—
328	C₈	C₉×#67	1	2	yes	—
329	C₂×C₃	—	2	2	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	8	2	no	—
332	C₂×C₃	—	8	1	no	—
333	C₂×C₃	—	8	1	no	—
334	C₂×C₃	—	24	1	no	—
335	C₂×C₃	C₉×#127	1	1	yes	characteristic
336	C₂×C₃	C₉×#135	1	2	yes	—
337	C₂×C₃	C₃×#916	1	2	yes	—
338	C₂×C₃	C₃×#940	1	1	yes	characteristic
339	D₃	—	8	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	12	2	no	—
343	C₂²	—	12	1	no	—
344	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
345	C₄	—	4	1	no	—
346	C₄	—	6	2	no	—
347	C₄	—	12	1	no	—
348	C₄	D₃×D₄×C₉	1	1	yes	characteristic
349	C₄	C₂×C₉×#67	1	1	yes	characteristic
350	C₃	—	2	1	no	—
351	C₃	C₉×#495	1	1	yes	characteristic, complemented
352	C₃	C₃×#5168	1	1	yes	characteristic
353	C₂	—	8	1	no	—
354	C₂	—	24	1	no	—
355	C₂	C₉×#916	1	2	yes	—
356	C₂	C₉×#940	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	D₃×D₄
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×SD₁₆	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×#213	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₉×SD₁₆	D₃
hypocenter	C₃	C₉×#495
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄×C₉ ◁ C₂×C₉×SD₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×SD₁₆	C₂	semidirect
2. C₉×#947	C₂	semidirect
3. C₂×C₉×#216	C₂	semidirect
4. C₉×#936	C₂	semidirect
6. C₂×C₉×#204	C₂	semidirect
7. C₈×C₉×Dic₃	C₂	semidirect
33. C₂×C₉×SD₁₆	D₃	semidirect
55. #5168	C₉	direct (N × H)
99. C₂×C₃×SD₁₆	C₂×C₉	semidirect
104. #947	C₂×C₉	semidirect
105. C₂×#216	C₂×C₉	semidirect
106. #936	C₂×C₉	semidirect
108. C₂×#204	C₂×C₉	semidirect
110. C₈×Dic₃	C₂×C₉	semidirect
214. C₂×SD₁₆	D₃×C₉	semidirect
219. C₃×C₉	#495	semidirect
312. C₉	#5168	direct (N × H)
351. C₃	C₉×#495	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×SD₁₆	2
C₉×#947	2
C₂×C₉×#216	2
C₉×#936	2
C₂×C₉×#213	2
C₂×C₉×#204	2
C₈×C₉×Dic₃	2
C₃×#5168	3
C₉×#495	3

Subgroups — C9×#5168 #13143347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5168 #13143347