Subgroups — C₉×#5167 #13143346

Order 1728.

440 conjugacy classes of subgroups.

The subgroup lattice has 346 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
2	C₉×#944	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
5	C₂×C₉×#214	C₂	1	1	yes	maximal, characteristic
6	C₂×C₉×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#815	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5167	C₃	1	1	yes	maximal, characteristic
9	C₉×#497	—	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×Dic₆	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×#214	—	2	2	no	—
19	C₉×Dic₃⋊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×Dic₁₂	—	2	2	no	—
23	C₂×C₃²×SD₁₆	C₂×C₃	1	1	yes	characteristic
24	C₃×#944	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×#214	C₂×C₃	1	1	yes	characteristic
28	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
29	C₃×#815	C₂×C₃	1	1	yes	characteristic
30	C₂×C₉×Q₁₆	—	3	1	no	—
31	C₂×C₉×Q₁₆	—	3	1	no	—
32	C₂×C₉×SD₁₆	—	2	1	no	—
33	C₂×C₉×SD₁₆	—	3	1	no	—
34	C₂×C₉×SD₁₆	D₃	1	1	yes	characteristic
35	C₉×C₄⋊Q₈	—	3	1	no	—
36	C₉×#124	—	3	1	no	—
37	C₉×#97	—	3	1	no	—
38	C₂³×C₃×C₉	—	2	1	no	—
39	C₃×Q₈×C₉	—	2	1	no	—
40	C₃×Q₈×C₉	—	2	1	no	—
41	C₃×D₄×C₉	—	2	1	no	—
42	C₃×D₄×C₉	—	2	1	no	—
43	C₂×C₃×C₄×C₉	—	2	1	no	—
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂×C₉×Dic₃	—	2	1	no	—
46	C₂×C₉×Dic₃	—	2	1	no	—
47	C₂×C₉×Dic₃	D₄	1	2	yes	—
48	C₉×Dic₆	—	2	1	no	—
49	C₉×Dic₆	—	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×#497	—	3	1	no	—
55	#5167	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×Dic₆	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×#214	—	2	2	no	—
65	C₃×Dic₃⋊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×Dic₁₂	—	2	2	no	—
69	C₂×Q₈×C₉	—	2	1	no	—
70	C₂×Q₈×C₉	—	3	1	no	—
71	C₂×Q₈×C₉	—	3	1	no	—
72	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
73	C₂×D₄×C₉	—	2	1	no	—
74	C₂×D₄×C₉	C₂×D₃	1	1	yes	characteristic
75	C₉×Q₁₆	—	6	2	no	—
76	C₉×Q₁₆	—	6	2	no	—
77	C₉×SD₁₆	—	2	4	no	—
78	C₉×SD₁₆	—	2	2	no	—
79	C₉×SD₁₆	—	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×C₄⋊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	C₃×C₄×C₉	—	4	1	no	—
90	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
92	C₉×Dic₃	—	2	2	no	—
93	C₉×Dic₃	—	4	1	no	—
94	C₉×Dic₃	—	4	1	no	—
95	C₂×C₃×Q₁₆	—	3	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
100	C₂×C₃×SD₁₆	C₃×D₃	1	1	yes	characteristic
101	C₃×C₄⋊Q₈	—	3	1	no	—
102	C₃×#124	—	3	1	no	—
103	#944	C₂×C₉	1	1	yes	characteristic, complemented
104	C₂×#216	C₂×C₉	1	1	yes	characteristic, complemented
105	#936	C₂×C₉	1	1	yes	characteristic
106	C₂×#214	C₂×C₉	1	1	yes	characteristic
107	C₂×Dic₁₂	C₂×C₉	1	1	yes	characteristic, complemented
108	C₃×#97	—	3	1	no	—
109	#815	C₂×C₉	1	1	yes	characteristic, complemented
110	C₂³×C₃²	—	2	1	no	—
111	C₃²×Q₈	—	2	1	no	—
112	C₃²×Q₈	—	2	1	no	—
113	C₃²×D₄	—	2	1	no	—
114	C₃²×D₄	—	2	1	no	—
115	C₂×C₃²×C₄	—	2	1	no	—
116	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₃×Dic₃	—	2	1	no	—
118	C₂×C₃×Dic₃	—	2	1	no	—
119	C₂×C₃×Dic₃	C₃×D₄	1	2	yes	—
120	C₃×Dic₆	—	2	1	no	—
121	C₃×Dic₆	—	2	1	no	—
122	C₃×Dic₆	—	2	1	no	—
123	C₃×Dic₆	—	2	1	no	—
124	C₂³×C₉	—	2	1	no	—
125	C₂³×C₉	—	4	1	no	—
126	C₃²×C₈	C₃×D₄	1	2	yes	—
127	C₃×C₃⋊C₈	C₃×D₄	1	2	yes	—
128	Q₈×C₉	—	2	2	no	—
129	Q₈×C₉	—	2	1	no	—
130	Q₈×C₉	—	2	1	no	—
131	Q₈×C₉	—	4	1	no	—
132	Q₈×C₉	—	6	1	no	—
133	Q₈×C₉	—	6	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	C₂×C₄×C₉	—	2	1	no	—
141	C₂×C₄×C₉	—	2	1	no	—
142	C₂×C₄×C₉	—	3	2	no	—
143	C₂×C₄×C₉	—	4	1	no	—
144	C₂×C₄×C₉	—	6	1	no	—
145	C₂×C₄×C₉	—	6	1	no	—
146	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
147	C₈×C₉	—	2	2	no	—
148	C₈×C₉	—	3	2	no	—
149	C₈×C₉	#67	1	2	yes	—
150	#497	—	3	1	no	Sylow
151	C₂×C₃×C₉	—	8	1	no	—
152	C₂×C₃×C₉	#127	1	1	yes	characteristic
153	C₂×C₃×C₉	#137	1	2	yes	—
154	C₂×C₃×Q₈	—	2	1	no	—
155	C₂×C₃×Q₈	—	3	1	no	—
156	C₂×C₃×Q₈	—	3	1	no	—
157	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
158	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
159	C₂×C₃×D₄	—	2	1	no	—
160	C₂×C₃×D₄	C₂²×C₉	1	1	yes	characteristic
161	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
162	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
163	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
164	C₃×Q₁₆	—	6	2	no	—
165	C₃×Q₁₆	—	6	2	no	—
166	C₃×SD₁₆	—	2	4	no	—
167	C₃×SD₁₆	—	2	2	no	—
168	C₃×SD₁₆	—	2	2	no	—
169	C₃×SD₁₆	—	6	2	no	—
170	C₂×C₃×C₈	—	2	1	no	—
171	C₂×C₃×C₈	—	3	1	no	—
172	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
173	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
174	C₃×C₄⋊C₄	—	6	2	no	—
175	C₃×C₂²⋊C₄	—	6	2	no	—
176	C₃×C₄²	—	3	1	no	—
177	#217	—	2	2	no	—
178	#216	—	2	2	no	—
179	#214	—	2	2	no	—
180	Dic₃⋊C₄	—	2	2	no	—
181	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
182	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
183	Dic₁₂	—	2	2	no	—
184	C₂²×C₃²	—	4	1	no	—
185	C₂²×C₃²	—	4	2	no	—
186	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
187	C₃²×C₄	—	4	1	no	—
188	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
189	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
190	C₃×Dic₃	—	2	2	no	—
191	C₃×Dic₃	—	4	1	no	—
192	C₃×Dic₃	—	4	1	no	—
193	C₂²×C₉	—	2	1	no	—
194	C₂²×C₉	—	4	2	no	—
195	C₂²×C₉	—	4	2	no	—
196	C₂²×C₉	—	4	1	no	—
197	C₂²×C₉	—	8	2	no	—
198	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
199	C₄×C₉	—	2	1	no	—
200	C₄×C₉	—	2	1	no	—
201	C₄×C₉	—	4	2	no	—
202	C₄×C₉	—	4	1	no	—
203	C₄×C₉	—	6	2	no	—
204	C₄×C₉	—	12	1	no	—
205	C₄×C₉	—	12	1	no	—
206	C₄×C₉	D₃×D₄	1	1	yes	characteristic
207	C₄×C₉	C₂×#67	1	1	yes	characteristic
208	C₂×Q₁₆	—	3	1	no	—
209	C₂×Q₁₆	—	3	1	no	—
210	C₂×SD₁₆	—	3	1	no	—
211	C₂×SD₁₆	D₃×C₉	1	1	yes	characteristic
212	C₄⋊Q₈	—	3	1	no	—
213	#124	—	3	1	no	—
214	#97	—	3	1	no	—
215	C₃×C₉	#497	1	1	yes	characteristic, complemented, Sylow
216	C₂³×C₃	—	2	1	no	—
217	C₂³×C₃	—	2	1	no	—
218	C₂³×C₃	—	4	1	no	—
219	C₃×Q₈	—	2	2	no	—
220	C₃×Q₈	—	2	1	no	—
221	C₃×Q₈	—	2	1	no	—
222	C₃×Q₈	—	2	1	no	—
223	C₃×Q₈	—	2	1	no	—
224	C₃×Q₈	—	4	1	no	—
225	C₃×Q₈	—	6	1	no	—
226	C₃×Q₈	—	6	1	no	—
227	C₃×Q₈	—	6	1	no	—
228	C₃×Q₈	—	6	1	no	—
229	C₃×D₄	—	2	2	no	—
230	C₃×D₄	—	2	1	no	—
231	C₃×D₄	—	2	1	no	—
232	C₃×D₄	—	2	1	no	—
233	C₃×D₄	—	2	1	no	—
234	C₃×D₄	—	4	1	no	—
235	C₂×C₃×C₄	—	2	1	no	—
236	C₂×C₃×C₄	—	2	1	no	—
237	C₂×C₃×C₄	—	2	1	no	—
238	C₂×C₃×C₄	—	3	2	no	—
239	C₂×C₃×C₄	—	4	1	no	—
240	C₂×C₃×C₄	—	6	1	no	—
241	C₂×C₃×C₄	—	6	1	no	—
242	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
244	C₂×Dic₃	—	2	1	no	—
245	C₂×Dic₃	—	2	1	no	—
246	C₂×Dic₃	D₄×C₉	1	2	yes	—
247	Dic₆	—	2	1	no	—
248	Dic₆	—	2	1	no	—
249	Dic₆	—	2	1	no	—
250	Dic₆	—	2	1	no	—
251	C₃×C₈	—	2	2	no	—
252	C₃×C₈	—	3	2	no	—
253	C₃×C₈	D₄×C₉	1	2	yes	—
254	C₃×C₈	C₃×#67	1	2	yes	—
255	C₃⋊C₈	D₄×C₉	1	2	yes	—
256	C₂×C₃²	—	8	1	no	—
257	C₂×C₃²	C₃×#127	1	1	yes	characteristic
258	C₂×C₃²	C₃×#137	1	2	yes	—
259	C₂×C₉	—	2	2	no	—
260	C₂×C₉	—	2	1	no	—
261	C₂×C₉	—	8	2	no	—
262	C₂×C₉	—	8	1	no	—
263	C₂×C₉	#915	1	2	yes	—
264	C₂×C₉	#940	1	1	yes	characteristic
265	C₂×Q₈	—	3	1	no	—
266	C₂×Q₈	—	3	1	no	—
267	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
268	C₂×D₄	C₂×D₃×C₉	1	1	yes	characteristic
269	Q₁₆	—	6	2	no	—
270	Q₁₆	—	6	2	no	—
271	SD₁₆	—	2	2	no	—
272	SD₁₆	—	6	2	no	—
273	C₂×C₈	—	3	1	no	—
274	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
275	C₄⋊C₄	—	6	2	no	—
276	C₂²⋊C₄	—	6	2	no	—
277	C₄²	—	3	1	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	4	2	no	—
280	C₂²×C₃	—	4	2	no	—
281	C₂²×C₃	—	4	1	no	—
282	C₂²×C₃	—	4	2	no	—
283	C₂²×C₃	—	4	1	no	—
284	C₂²×C₃	—	8	2	no	—
285	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
286	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	—	4	1	no	—
291	C₃×C₄	—	4	1	no	—
292	C₃×C₄	—	6	2	no	—
293	C₃×C₄	—	12	1	no	—
294	C₃×C₄	—	12	1	no	—
295	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
296	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
297	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
298	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
299	Dic₃	—	2	2	no	—
300	Dic₃	—	4	1	no	—
301	Dic₃	—	4	1	no	—
302	C₃²	C₃×#497	1	1	yes	characteristic
303	C₉	—	2	1	no	—
304	C₉	#5167	1	1	yes	characteristic, direct product factor
305	C₂³	—	2	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	6	1	no	—
309	Q₈	—	6	1	no	—
310	Q₈	—	6	1	no	—
311	Q₈	—	6	1	no	—
312	D₄	—	2	1	no	—
313	D₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	3	2	no	—
316	C₂×C₄	—	6	1	no	—
317	C₂×C₄	—	6	1	no	—
318	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
319	C₈	—	3	2	no	—
320	C₈	C₉×#67	1	2	yes	—
321	C₂×C₃	—	2	2	no	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	8	2	no	—
324	C₂×C₃	—	8	1	no	—
325	C₂×C₃	—	8	1	no	—
326	C₂×C₃	C₉×#127	1	1	yes	characteristic
327	C₂×C₃	C₉×#137	1	2	yes	—
328	C₂×C₃	C₃×#915	1	2	yes	—
329	C₂×C₃	C₃×#940	1	1	yes	characteristic
330	C₂²	—	4	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
333	C₄	—	4	1	no	—
334	C₄	—	6	2	no	—
335	C₄	—	12	1	no	—
336	C₄	—	12	1	no	—
337	C₄	D₃×D₄×C₉	1	1	yes	characteristic
338	C₄	C₂×C₉×#67	1	1	yes	characteristic
339	C₃	—	2	1	no	—
340	C₃	C₉×#497	1	1	yes	characteristic, complemented
341	C₃	C₃×#5167	1	1	yes	characteristic
342	C₂	—	8	1	no	—
343	C₂	C₉×#915	1	2	yes	—
344	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×C₃²×D₄	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₉×SD₁₆	D₃
hypocenter	C₃	C₉×#497
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
2. C₉×#944	C₂	semidirect
3. C₂×C₉×#216	C₂	semidirect
6. C₂×C₉×Dic₁₂	C₂	semidirect
7. C₉×#815	C₂	semidirect
55. #5167	C₉	direct (N × H)
103. #944	C₂×C₉	semidirect
104. C₂×#216	C₂×C₉	semidirect
107. C₂×Dic₁₂	C₂×C₉	semidirect
109. #815	C₂×C₉	semidirect
215. C₃×C₉	#497	semidirect
304. C₉	#5167	direct (N × H)
340. C₃	C₉×#497	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×#944	2
C₂×C₉×#216	2
C₉×#936	2
C₂×C₉×#214	2
C₂×C₉×Dic₁₂	2
C₉×#815	2
C₃×#5167	3
C₉×#497	3

Subgroups — C9×#5167 #13143346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5167 #13143346