Subgroups — C₉×#4710 #13141515

Order 1728.

400 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#204	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₄×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#821	C₂	1	1	yes	maximal, characteristic
6	C₉×#818	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#816	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4710	C₃	1	1	yes	maximal, characteristic
9	C₉×#440	—	3	1	no	maximal
10	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₃×C₉	—	2	1	no	—
12	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₉×Dic₃⋊C₄	—	2	1	no	—
20	C₄×C₉×Dic₃	—	2	1	no	—
21	C₉×#204	C₄	1	4	yes	complemented
22	C₂×C₃×#204	C₂×C₃	1	1	yes	characteristic
23	C₃×C₄×D₁₂	C₂×C₃	1	1	yes	characteristic
24	C₃×C₄×Dic₆	C₂×C₃	1	1	yes	characteristic
25	C₃²×#97	C₂×C₃	1	1	yes	characteristic
26	C₃×#821	C₂×C₃	1	1	yes	characteristic
27	C₃×#818	C₂×C₃	1	1	yes	characteristic
28	C₃×#816	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×#107	—	3	1	no	—
33	C₉×#103	—	3	1	no	—
34	C₉×#102	—	3	1	no	—
35	C₉×#97	—	2	1	no	—
36	C₉×#97	D₃	1	1	yes	characteristic, complemented
37	C₂²×D₃×C₉	—	2	1	no	—
38	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₉×Dic₃	—	2	1	no	—
42	C₂×C₉×Dic₃	—	2	1	no	—
43	C₂×C₉×Dic₃	—	2	1	no	—
44	C₉×D₁₂	—	2	1	no	—
45	C₉×D₁₂	C₂×C₄	1	2	yes	—
46	C₄×D₃×C₉	—	4	2	no	—
47	C₉×Dic₆	—	2	1	no	—
48	C₉×Dic₆	C₂×C₄	1	2	yes	complemented
49	C₃×C₈×C₉	—	2	1	no	—
50	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
51	C₃×#440	—	3	1	no	—
52	#4710	C₉	1	1	yes	characteristic, direct product factor
53	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×C₄×D₃	—	2	1	no	—
55	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
58	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
59	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
60	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
61	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
62	C₃×Dic₃⋊C₄	—	2	1	no	—
63	C₃×C₄×Dic₃	—	2	1	no	—
64	C₃×#204	C₃×C₄	1	4	yes	—
65	C₂×Q₈×C₉	—	3	1	no	—
66	C₂×D₄×C₉	—	3	1	no	—
67	C₂²×C₄×C₉	—	6	1	no	—
68	C₉×SD₁₆	—	3	4	no	—
69	C₂×C₈×C₉	—	2	1	no	—
70	C₂×C₈×C₉	—	2	1	no	—
71	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
72	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
73	C₉×C₄⋊C₄	—	3	1	no	—
74	C₉×C₄⋊C₄	—	3	1	no	—
75	C₉×C₄⋊C₄	—	6	1	no	—
76	C₉×C₂²⋊C₄	—	6	1	no	—
77	C₄²×C₉	—	2	1	no	—
78	C₄²×C₉	—	6	1	no	—
79	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
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	1	no	—
85	C₃×C₄×C₉	—	2	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	2	no	—
89	C₉×Dic₃	—	4	1	no	—
90	C₉×Dic₃	—	4	1	no	—
91	C₂×C₃×SD₁₆	—	3	1	no	—
92	C₃×C₄×Q₈	—	3	1	no	—
93	C₃×C₄×D₄	—	3	1	no	—
94	C₂×#204	C₂×C₉	1	1	yes	characteristic
95	C₄×D₁₂	C₂×C₉	1	1	yes	characteristic
96	C₄×Dic₆	C₂×C₉	1	1	yes	characteristic, complemented
97	C₃×#107	—	3	1	no	—
98	C₃×#103	—	3	1	no	—
99	C₃×#102	—	3	1	no	—
100	C₃×#97	—	2	1	no	—
101	C₃×#97	C₂×C₉	1	1	yes	characteristic, complemented
102	C₃×#97	C₃×D₃	1	1	yes	characteristic
103	#821	C₂×C₉	1	1	yes	characteristic
104	#818	C₂×C₉	1	1	yes	characteristic, complemented
105	#816	C₂×C₉	1	1	yes	characteristic, complemented
106	C₂²×C₃×D₃	—	2	1	no	—
107	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
108	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
109	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₃×Dic₃	—	2	1	no	—
111	C₂×C₃×Dic₃	—	2	1	no	—
112	C₂×C₃×Dic₃	—	2	1	no	—
113	C₃×D₁₂	—	2	1	no	—
114	C₃×D₁₂	C₂×C₃×C₄	1	2	yes	—
115	C₃×C₄×D₃	—	4	2	no	—
116	C₃×Dic₆	—	2	1	no	—
117	C₃×Dic₆	C₂×C₃×C₄	1	2	yes	—
118	C₂³×C₉	—	6	1	no	—
119	C₃²×C₈	—	2	1	no	—
120	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
121	Q₈×C₉	—	3	2	no	—
122	Q₈×C₉	—	6	1	no	—
123	D₄×C₉	—	3	2	no	—
124	D₄×C₉	—	6	1	no	—
125	C₂×C₄×C₉	—	2	1	no	—
126	C₂×C₄×C₉	—	2	1	no	—
127	C₂×C₄×C₉	—	2	1	no	—
128	C₂×C₄×C₉	—	6	1	no	—
129	C₂×C₄×C₉	—	6	1	no	—
130	C₂×C₄×C₉	—	6	1	no	—
131	C₂×C₄×C₉	—	12	2	no	—
132	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
133	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
134	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
135	C₈×C₉	—	2	2	no	—
136	C₈×C₉	—	2	1	no	—
137	C₈×C₉	—	2	2	no	—
138	C₈×C₉	C₄×D₃	1	2	yes	complemented
139	#440	—	3	1	no	Sylow
140	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₃×C₉	#137	1	1	yes	characteristic
143	D₃×C₉	—	4	2	no	—
144	C₂×C₃×Q₈	—	3	1	no	—
145	C₂×C₃×D₄	—	3	1	no	—
146	C₂²×C₃×C₄	—	6	1	no	—
147	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
148	C₂×C₄×D₃	—	2	1	no	—
149	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
150	C₃×SD₁₆	—	3	4	no	—
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	2	1	no	—
153	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
154	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
155	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
156	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	3	1	no	—
158	C₃×C₄⋊C₄	—	3	1	no	—
159	C₃×C₄⋊C₄	—	6	1	no	—
160	C₃×C₂²⋊C₄	—	6	1	no	—
161	C₃×C₄²	—	2	1	no	—
162	C₃×C₄²	—	6	1	no	—
163	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
164	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
165	(C₂×D₃)⋊C₄	—	2	1	no	—
166	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
167	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
168	Dic₃⋊C₄	—	2	1	no	—
169	C₄×Dic₃	—	2	1	no	—
170	#204	C₄×C₉	1	4	yes	complemented
171	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂×C₃×D₃	—	2	2	no	—
173	C₂×C₃×D₃	—	4	1	no	—
174	C₂×C₃×D₃	—	4	1	no	—
175	C₃²×C₄	—	2	1	no	—
176	C₃²×C₄	—	2	1	no	—
177	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₃²×C₄	C₃×#41	1	1	yes	characteristic
179	C₃×Dic₃	—	2	2	no	—
180	C₃×Dic₃	—	4	1	no	—
181	C₃×Dic₃	—	4	1	no	—
182	C₂²×C₉	—	2	1	no	—
183	C₂²×C₉	—	6	2	no	—
184	C₂²×C₉	—	12	1	no	—
185	C₂²×C₉	—	12	1	no	—
186	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
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	1	no	—
193	C₄×C₉	—	6	2	no	—
194	C₄×C₉	—	12	1	no	—
195	C₄×C₉	—	12	1	no	—
196	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
197	C₄×C₉	#235	1	1	yes	characteristic
198	C₂×SD₁₆	—	3	1	no	—
199	C₄×Q₈	—	3	1	no	—
200	C₄×D₄	—	3	1	no	—
201	#107	—	3	1	no	—
202	#103	—	3	1	no	—
203	#102	—	3	1	no	—
204	#97	D₃×C₉	1	1	yes	characteristic, complemented
205	C₃×C₉	#440	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	6	1	no	—
207	C₂²×D₃	—	2	1	no	—
208	C₃×Q₈	—	3	2	no	—
209	C₃×Q₈	—	6	1	no	—
210	C₃×D₄	—	3	2	no	—
211	C₃×D₄	—	6	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	6	1	no	—
216	C₂×C₃×C₄	—	6	1	no	—
217	C₂×C₃×C₄	—	6	1	no	—
218	C₂×C₃×C₄	—	12	2	no	—
219	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
220	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
222	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
223	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
225	C₂×Dic₃	—	2	1	no	—
226	C₂×Dic₃	—	2	1	no	—
227	C₂×Dic₃	—	2	1	no	—
228	D₁₂	—	2	1	no	—
229	D₁₂	C₂×C₄×C₉	1	2	yes	—
230	C₄×D₃	—	4	2	no	—
231	Dic₆	—	2	1	no	—
232	Dic₆	C₂×C₄×C₉	1	2	yes	complemented
233	C₃×C₈	—	2	2	no	—
234	C₃×C₈	—	2	1	no	—
235	C₃×C₈	—	2	1	no	—
236	C₃×C₈	—	2	2	no	—
237	C₃×C₈	C₂×C₄×C₉	1	2	yes	complemented
238	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
239	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
240	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
241	C₂×C₃²	C₃×#137	1	1	yes	characteristic
242	C₃×D₃	—	4	2	no	—
243	C₂×C₉	—	2	1	no	—
244	C₂×C₉	—	2	1	no	—
245	C₂×C₉	—	2	1	no	—
246	C₂×C₉	—	12	2	no	—
247	C₂×C₉	C₄×D₁₂	1	1	yes	characteristic
248	C₂×C₉	#908	1	1	yes	characteristic
249	C₂×C₉	#909	1	1	yes	characteristic
250	C₂×Q₈	—	3	1	no	—
251	C₂×D₄	—	3	1	no	—
252	C₂²×C₄	—	6	1	no	—
253	SD₁₆	—	3	4	no	—
254	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
255	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
256	C₄⋊C₄	—	3	1	no	—
257	C₄⋊C₄	—	3	1	no	—
258	C₄⋊C₄	—	6	1	no	—
259	C₂²⋊C₄	—	6	1	no	—
260	C₄²	—	6	1	no	—
261	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	6	2	no	—
264	C₂²×C₃	—	12	1	no	—
265	C₂²×C₃	—	12	1	no	—
266	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
267	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
268	C₂×D₃	—	2	2	no	—
269	C₂×D₃	—	4	1	no	—
270	C₂×D₃	—	4	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	4	1	no	—
278	C₃×C₄	—	4	1	no	—
279	C₃×C₄	—	6	2	no	—
280	C₃×C₄	—	12	1	no	—
281	C₃×C₄	—	12	1	no	—
282	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
283	C₃×C₄	C₉×#41	1	1	yes	characteristic
284	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
285	C₃×C₄	C₃×#235	1	1	yes	characteristic
286	Dic₃	—	2	2	no	—
287	Dic₃	—	4	1	no	—
288	Dic₃	—	4	1	no	—
289	C₃²	C₃×#440	1	1	yes	characteristic
290	C₉	—	2	1	no	—
291	C₉	#4710	1	1	yes	characteristic, direct product factor
292	C₂³	—	6	1	no	—
293	Q₈	—	3	2	no	—
294	Q₈	—	6	1	no	—
295	D₄	—	3	2	no	—
296	D₄	—	6	1	no	—
297	C₂×C₄	—	6	1	no	—
298	C₂×C₄	—	6	1	no	—
299	C₂×C₄	—	6	1	no	—
300	C₂×C₄	—	12	2	no	—
301	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
302	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	C₄×D₃×C₉	1	2	yes	complemented
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	12	2	no	—
310	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
311	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
312	C₂×C₃	C₉×#137	1	1	yes	characteristic
313	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
314	C₂×C₃	C₃×#908	1	1	yes	characteristic
315	C₂×C₃	C₃×#909	1	1	yes	characteristic
316	D₃	—	4	2	no	—
317	C₂²	—	6	2	no	—
318	C₂²	—	12	1	no	—
319	C₂²	—	12	1	no	—
320	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	6	2	no	—
324	C₄	—	12	1	no	—
325	C₄	—	12	1	no	—
326	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
327	C₄	C₉×#235	1	1	yes	characteristic
328	C₃	—	2	1	no	—
329	C₃	C₉×#440	1	1	yes	characteristic, complemented
330	C₃	C₃×#4710	1	1	yes	characteristic
331	C₂	—	12	2	no	—
332	C₂	C₄×C₉×D₁₂	1	1	yes	characteristic
333	C₂	C₉×#908	1	1	yes	characteristic
334	C₂	C₉×#909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#97	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₁₂	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#97	D₃
hypocenter	C₃	C₉×#440
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×#97

Complements

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

N	H ≅ G/N	type
3. C₄×C₉×Dic₆	C₂	semidirect
4. C₃×C₉×#97	C₂	semidirect
6. C₉×#818	C₂	semidirect
7. C₉×#816	C₂	semidirect
21. C₉×#204	C₄	semidirect
36. C₉×#97	D₃	semidirect
48. C₉×Dic₆	C₂×C₄	semidirect
50. C₃×C₈×C₉	C₂×C₄	semidirect
52. #4710	C₉	direct (N × H)
96. C₄×Dic₆	C₂×C₉	semidirect
101. C₃×#97	C₂×C₉	semidirect
104. #818	C₂×C₉	semidirect
105. #816	C₂×C₉	semidirect
138. C₈×C₉	C₄×D₃	semidirect
170. #204	C₄×C₉	semidirect
204. #97	D₃×C₉	semidirect
205. C₃×C₉	#440	semidirect
232. Dic₆	C₂×C₄×C₉	semidirect
237. C₃×C₈	C₂×C₄×C₉	semidirect
291. C₉	#4710	direct (N × H)
305. C₈	C₄×D₃×C₉	semidirect
329. C₃	C₉×#440	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#204	2
C₄×C₉×D₁₂	2
C₄×C₉×Dic₆	2
C₃×C₉×#97	2
C₉×#821	2
C₉×#818	2
C₉×#816	2
C₃×#4710	3
C₉×#440	3

Subgroups — C9×#4710 #13141515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4710 #13141515