Subgroups — C₉×#5009 #13142720

Order 1728.

400 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	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#832	C₂	1	1	yes	maximal, characteristic
4	C₉×#832	C₂	1	1	yes	maximal, characteristic
5	C₉×#808	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#807	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5009	C₃	1	1	yes	maximal, characteristic
9	C₉×#478	—	3	1	no	maximal
10	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₉	—	2	1	no	—
12	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₉×Dic₃⋊C₄	—	2	1	no	—
18	C₉×Dic₃⋊C₄	—	2	1	no	—
19	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
22	C₃×#870	C₂×C₃	1	1	yes	characteristic
23	C₃×#832	C₂×C₃	1	1	yes	characteristic
24	C₃×#832	C₂×C₃	1	1	yes	characteristic
25	C₃×#808	C₂×C₃	1	1	yes	characteristic
26	C₃×#807	C₂×C₃	1	1	yes	characteristic
27	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
28	C₉×#125	—	3	1	no	—
29	C₄×D₄×C₉	—	2	1	no	—
30	C₄×D₄×C₉	D₃	1	1	yes	characteristic
31	C₉×#107	—	3	1	no	—
32	C₉×#106	—	3	1	no	—
33	C₉×C₄⋊C₈	—	3	1	no	—
34	C₉×#102	—	3	1	no	—
35	C₉×#102	—	3	1	no	—
36	C₂³×C₃×C₉	—	2	1	no	—
37	C₃×D₄×C₉	—	2	1	no	—
38	C₃×D₄×C₉	Q₈	1	2	yes	—
39	C₂×C₃×C₄×C₉	—	2	1	no	—
40	C₂×C₃×C₄×C₉	—	4	2	no	—
41	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₉×Dic₃	—	2	1	no	—
45	C₂×C₉×Dic₃	—	2	1	no	—
46	C₉×C₃⋊C₈	—	2	1	no	—
47	C₉×C₃⋊C₈	—	2	1	no	—
48	C₃×#478	—	3	1	no	—
49	#5009	C₉	1	1	yes	characteristic, direct product factor
50	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
51	C₂²×C₃²×C₄	—	2	1	no	—
52	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
53	C₃²×C₂²⋊C₄	—	2	1	no	—
54	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
56	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
57	C₃×Dic₃⋊C₄	—	2	1	no	—
58	C₃×Dic₃⋊C₄	—	2	1	no	—
59	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂×D₄×C₉	—	2	1	no	—
62	C₂×D₄×C₉	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₄×C₉	—	2	2	no	—
64	C₂²×C₄×C₉	—	2	1	no	—
65	C₂×C₈×C₉	—	3	1	no	—
66	C₂×C₈×C₉	—	3	1	no	—
67	C₉×C₄⋊C₄	—	2	1	no	—
68	C₉×C₄⋊C₄	—	3	1	no	—
69	C₉×C₄⋊C₄	—	3	1	no	—
70	C₉×C₄⋊C₄	—	6	1	no	—
71	C₉×C₄⋊C₄	—	6	1	no	—
72	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
73	C₉×C₂²⋊C₄	—	2	2	no	—
74	C₉×C₂²⋊C₄	—	2	1	no	—
75	C₄²×C₉	—	2	1	no	—
76	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
77	C₂²×C₃×C₉	—	2	2	no	—
78	C₂²×C₃×C₉	—	4	1	no	—
79	C₂²×C₃×C₉	—	4	1	no	—
80	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
81	C₃×C₄×C₉	—	2	1	no	—
82	C₃×C₄×C₉	—	2	1	no	—
83	C₃×C₄×C₉	—	4	1	no	—
84	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
85	C₃×C₄×C₉	#41	1	1	yes	characteristic
86	C₉×Dic₃	—	4	1	no	—
87	C₉×Dic₃	—	4	1	no	—
88	C₃×#125	—	3	1	no	—
89	C₃×C₄×D₄	—	2	1	no	—
90	C₃×C₄×D₄	C₂×C₉	1	1	yes	characteristic
91	C₃×C₄×D₄	C₃×D₃	1	1	yes	characteristic
92	#870	C₂×C₉	1	1	yes	characteristic, complemented
93	C₃×#107	—	3	1	no	—
94	C₃×#106	—	3	1	no	—
95	C₃×C₄⋊C₈	—	3	1	no	—
96	C₃×#102	—	3	1	no	—
97	C₃×#102	—	3	1	no	—
98	#832	C₂×C₉	1	1	yes	characteristic
99	#832	C₂×C₉	1	1	yes	characteristic
100	#808	C₂×C₉	1	1	yes	characteristic, complemented
101	#807	C₂×C₉	1	1	yes	characteristic, complemented
102	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic, complemented
103	C₂³×C₃²	—	2	1	no	—
104	C₃²×D₄	—	2	1	no	—
105	C₃²×D₄	C₃×Q₈	1	2	yes	—
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	—	4	2	no	—
108	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
109	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
111	C₂×C₃×Dic₃	—	2	1	no	—
112	C₂×C₃×Dic₃	—	2	1	no	—
113	C₂³×C₉	—	2	2	no	—
114	C₂³×C₉	—	2	1	no	—
115	C₃×C₃⋊C₈	—	2	1	no	—
116	C₃×C₃⋊C₈	—	2	1	no	—
117	D₄×C₉	—	2	2	no	—
118	D₄×C₉	—	2	1	no	—
119	D₄×C₉	—	2	2	no	—
120	D₄×C₉	Dic₆	1	2	yes	—
121	C₂×C₄×C₉	—	2	2	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	4	4	no	—
127	C₂×C₄×C₉	—	4	2	no	—
128	C₂×C₄×C₉	—	6	1	no	—
129	C₂×C₄×C₉	—	6	1	no	—
130	C₂×C₄×C₉	#67	1	1	yes	characteristic
131	C₂×C₄×C₉	#67	1	1	yes	characteristic
132	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
133	C₈×C₉	—	6	1	no	—
134	C₈×C₉	—	6	1	no	—
135	#478	—	3	1	no	Sylow
136	C₂×C₃×C₉	—	4	2	no	—
137	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₃×C₉	#135	1	1	yes	characteristic
139	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₃×D₄	—	2	1	no	—
141	C₂×C₃×D₄	C₂²×C₉	1	1	yes	characteristic
142	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
143	C₂²×C₃×C₄	—	2	2	no	—
144	C₂²×C₃×C₄	—	2	1	no	—
145	C₂²×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₈	—	3	1	no	—
147	C₂×C₃×C₈	—	3	1	no	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	3	1	no	—
150	C₃×C₄⋊C₄	—	3	1	no	—
151	C₃×C₄⋊C₄	—	6	1	no	—
152	C₃×C₄⋊C₄	—	6	1	no	—
153	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
154	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
155	C₃×C₂²⋊C₄	—	2	2	no	—
156	C₃×C₂²⋊C₄	—	2	1	no	—
157	C₃×C₂²⋊C₄	—	2	1	no	—
158	C₃×C₄²	—	2	1	no	—
159	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
160	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
161	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
162	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
163	Dic₃⋊C₄	—	2	1	no	—
164	Dic₃⋊C₄	—	2	1	no	—
165	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
166	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
167	C₂²×C₃²	—	2	2	no	—
168	C₂²×C₃²	—	4	1	no	—
169	C₂²×C₃²	—	4	1	no	—
170	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
171	C₃²×C₄	—	2	1	no	—
172	C₃²×C₄	—	2	1	no	—
173	C₃²×C₄	—	4	1	no	—
174	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
175	C₃²×C₄	C₃×#41	1	1	yes	characteristic
176	C₃×Dic₃	—	4	1	no	—
177	C₃×Dic₃	—	4	1	no	—
178	C₂²×C₉	—	2	4	no	—
179	C₂²×C₉	—	2	2	no	—
180	C₂²×C₉	—	2	1	no	—
181	C₂²×C₉	—	4	2	no	—
182	C₂²×C₉	—	4	2	no	—
183	C₂²×C₉	—	4	1	no	—
184	C₂²×C₉	—	4	1	no	—
185	C₂²×C₉	C₂×#67	1	1	yes	characteristic
186	C₄×C₉	—	2	2	no	—
187	C₄×C₉	—	2	1	no	—
188	C₄×C₉	—	2	1	no	—
189	C₄×C₉	—	2	1	no	—
190	C₄×C₉	—	2	1	no	—
191	C₄×C₉	—	4	1	no	—
192	C₄×C₉	—	4	2	no	—
193	C₄×C₉	—	4	1	no	—
194	C₄×C₉	—	12	1	no	—
195	C₄×C₉	—	12	1	no	—
196	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
197	C₄×C₉	#235	1	1	yes	characteristic
198	#125	—	3	1	no	—
199	C₄×D₄	D₃×C₉	1	1	yes	characteristic
200	#107	—	3	1	no	—
201	#106	—	3	1	no	—
202	C₄⋊C₈	—	3	1	no	—
203	#102	—	3	1	no	—
204	#102	—	3	1	no	—
205	C₃×C₉	#478	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	2	2	no	—
207	C₂³×C₃	—	2	1	no	—
208	C₂³×C₃	—	2	1	no	—
209	C₃×D₄	—	2	2	no	—
210	C₃×D₄	—	2	1	no	—
211	C₃×D₄	—	2	1	no	—
212	C₃×D₄	—	2	2	no	—
213	C₃×D₄	Q₈×C₉	1	2	yes	—
214	C₃×D₄	C₃×Dic₆	1	2	yes	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	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	—	4	2	no	—
222	C₂×C₃×C₄	—	4	4	no	—
223	C₂×C₃×C₄	—	4	2	no	—
224	C₂×C₃×C₄	—	6	1	no	—
225	C₂×C₃×C₄	—	6	1	no	—
226	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
227	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
229	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
230	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
232	C₂×Dic₃	—	2	1	no	—
233	C₂×Dic₃	—	2	1	no	—
234	C₃×C₈	—	6	1	no	—
235	C₃×C₈	—	6	1	no	—
236	C₃⋊C₈	—	2	1	no	—
237	C₃⋊C₈	—	2	1	no	—
238	C₂×C₃²	—	4	2	no	—
239	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
240	C₂×C₃²	C₃×#135	1	1	yes	characteristic
241	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	—	2	1	no	—
244	C₂×C₉	—	2	1	no	—
245	C₂×C₉	—	4	4	no	—
246	C₂×C₉	—	4	2	no	—
247	C₂×C₉	#924	1	1	yes	characteristic
248	C₂×C₉	#932	1	1	yes	characteristic
249	C₂×C₉	#950	1	1	yes	characteristic
250	C₂×D₄	C₂×D₃×C₉	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	C₂×C₈	—	3	1	no	—
253	C₂×C₈	—	3	1	no	—
254	C₄⋊C₄	—	3	1	no	—
255	C₄⋊C₄	—	3	1	no	—
256	C₄⋊C₄	—	6	1	no	—
257	C₄⋊C₄	—	6	1	no	—
258	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
259	C₂²⋊C₄	—	2	1	no	—
260	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
261	C₂²×C₃	—	2	4	no	—
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	2	2	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	4	2	no	—
266	C₂²×C₃	—	4	2	no	—
267	C₂²×C₃	—	4	1	no	—
268	C₂²×C₃	—	4	1	no	—
269	C₂²×C₃	—	4	1	no	—
270	C₂²×C₃	—	4	1	no	—
271	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
272	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
273	C₃×C₄	—	2	2	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	4	1	no	—
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	—	4	1	no	—
283	C₃×C₄	—	4	1	no	—
284	C₃×C₄	—	12	1	no	—
285	C₃×C₄	—	12	1	no	—
286	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
287	C₃×C₄	C₉×#41	1	1	yes	characteristic
288	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
289	C₃×C₄	C₃×#235	1	1	yes	characteristic
290	Dic₃	—	4	1	no	—
291	Dic₃	—	4	1	no	—
292	C₃²	C₃×#478	1	1	yes	characteristic
293	C₉	—	2	1	no	—
294	C₉	#5009	1	1	yes	characteristic, direct product factor
295	C₂³	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	C₉×Dic₆	1	2	yes	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	6	1	no	—
301	C₂×C₄	—	6	1	no	—
302	C₂×C₄	C₉×#67	1	1	yes	characteristic
303	C₂×C₄	C₉×#67	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
305	C₈	—	6	1	no	—
306	C₈	—	6	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	4	4	no	—
311	C₂×C₃	—	4	2	no	—
312	C₂×C₃	—	4	2	no	—
313	C₂×C₃	C₉×C₂²⋊Q₈	1	1	yes	characteristic
314	C₂×C₃	C₉×#135	1	1	yes	characteristic
315	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
316	C₂×C₃	C₃×#924	1	1	yes	characteristic
317	C₂×C₃	C₃×#932	1	1	yes	characteristic
318	C₂×C₃	C₃×#950	1	1	yes	characteristic
319	C₂²	—	2	2	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	C₂×C₉×#67	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	12	1	no	—
327	C₄	—	12	1	no	—
328	C₄	C₂×C₉×Dic₆	1	1	yes	characteristic
329	C₄	C₉×#235	1	1	yes	characteristic
330	C₃	—	2	1	no	—
331	C₃	C₉×#478	1	1	yes	characteristic, complemented
332	C₃	C₃×#5009	1	1	yes	characteristic
333	C₂	—	4	2	no	—
334	C₂	C₉×#924	1	1	yes	characteristic
335	C₂	C₉×#932	1	1	yes	characteristic
336	C₂	C₉×#950	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×D₄×C₉	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃²×D₄	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×D₄×C₉	D₃
hypocenter	C₃	C₉×#478
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×D₄×C₉

Complements

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

N	H ≅ G/N	type
2. C₉×#870	C₂	semidirect
5. C₉×#808	C₂	semidirect
6. C₉×#807	C₂	semidirect
7. C₉×C₄⋊(C₃⋊C₈)	C₂	semidirect
49. #5009	C₉	direct (N × H)
92. #870	C₂×C₉	semidirect
100. #808	C₂×C₉	semidirect
101. #807	C₂×C₉	semidirect
102. C₄⋊(C₃⋊C₈)	C₂×C₉	semidirect
205. C₃×C₉	#478	semidirect
294. C₉	#5009	direct (N × H)
331. C₃	C₉×#478	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄×C₉	2
C₉×#870	2
C₉×#832	2
C₉×#808	2
C₉×#807	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×#5009	3
C₉×#478	3

Subgroups — C9×#5009 #13142720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5009 #13142720