Subgroups — C₉×#4816 #13141893

Order 1728.

372 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#946	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#216	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
4	C₉×#897	C₂	1	1	yes	maximal, characteristic
5	C₃×C₉×#103	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#816	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#808	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4816	C₃	1	1	yes	maximal, characteristic
9	C₉×#470	—	3	1	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₉×#216	—	2	2	no	—
16	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
18	C₉×C₄⋊Dic₃	—	2	1	no	—
19	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
20	C₉×Dic₃⋊C₄	—	2	1	no	—
21	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₉×ΓL(1,𝔽₂₅)	—	2	2	no	—
23	C₃×#946	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#216	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
26	C₃×#897	C₂×C₃	1	1	yes	characteristic
27	C₃²×#103	C₂×C₃	1	1	yes	characteristic
28	C₃×#816	C₂×C₃	1	1	yes	characteristic
29	C₃×#808	C₂×C₃	1	1	yes	characteristic
30	C₂×C₉×Q₁₆	—	3	1	no	—
31	C₂×C₉×M₁₆	—	3	1	no	—
32	C₉×C₂²⋊Q₈	—	3	1	no	—
33	C₉×C₂²⋊Q₈	—	3	1	no	—
34	C₉×#106	—	3	1	no	—
35	C₉×#103	—	2	1	no	—
36	C₉×#103	—	3	1	no	—
37	C₉×#103	D₃	1	1	yes	characteristic, complemented
38	C₂²×D₃×C₉	D₄	1	1	yes	characteristic
39	C₃×Q₈×C₉	—	2	1	no	—
40	C₃×Q₈×C₉	—	2	1	no	—
41	C₂×C₃×C₄×C₉	—	2	1	no	—
42	C₂×C₃×C₄×C₉	—	2	1	no	—
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	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
47	C₄×D₃×C₉	—	2	1	no	—
48	C₄×D₃×C₉	—	2	1	no	—
49	C₉×Dic₆	—	2	1	no	—
50	C₉×Dic₆	—	2	1	no	—
51	C₃×C₈×C₉	—	2	1	no	—
52	C₉×C₃⋊C₈	D₄	1	2	yes	—
53	C₃×#470	—	3	1	no	—
54	#4816	C₉	1	1	yes	characteristic, direct product factor
55	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
60	C₃×#216	—	2	2	no	—
61	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
62	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
63	C₃×C₄⋊Dic₃	—	2	1	no	—
64	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
65	C₃×Dic₃⋊C₄	—	2	1	no	—
66	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
67	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
68	C₂×Q₈×C₉	—	2	1	no	—
69	C₂×Q₈×C₉	—	3	1	no	—
70	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
71	C₂²×C₄×C₉	—	3	1	no	—
72	C₉×Q₁₆	—	6	2	no	—
73	C₉×M₁₆	—	6	2	no	—
74	C₂×C₈×C₉	—	2	1	no	—
75	C₂×C₈×C₉	—	3	1	no	—
76	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
77	C₉×C₄⋊C₄	—	2	1	no	—
78	C₉×C₄⋊C₄	—	3	1	no	—
79	C₉×C₄⋊C₄	—	6	1	no	—
80	C₉×C₄⋊C₄	—	6	1	no	—
81	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
82	C₉×C₂²⋊C₄	—	6	1	no	—
83	C₉×C₂²⋊C₄	—	6	1	no	—
84	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
85	C₂×D₃×C₉	—	2	1	no	—
86	C₂×D₃×C₉	—	2	1	no	—
87	C₂×D₃×C₉	—	2	1	no	—
88	C₃×C₄×C₉	—	4	1	no	—
89	C₃×C₄×C₉	—	4	1	no	—
90	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
91	C₃×C₄×C₉	#41	1	1	yes	characteristic
92	C₉×Dic₃	—	2	1	no	—
93	C₉×Dic₃	—	4	1	no	—
94	C₉×Dic₃	—	4	1	no	—
95	C₂×C₃×Q₁₆	—	3	1	no	—
96	C₂×C₃×M₁₆	—	3	1	no	—
97	C₃×C₂²⋊Q₈	—	3	1	no	—
98	C₃×C₂²⋊Q₈	—	3	1	no	—
99	#946	C₂×C₉	1	1	yes	characteristic
100	C₂×#216	C₂×C₉	1	1	yes	characteristic, complemented
101	C₂×ΓL(1,𝔽₂₅)	C₂×C₉	1	1	yes	characteristic
102	#897	C₂×C₉	1	1	yes	characteristic
103	C₃×#106	—	3	1	no	—
104	C₃×#103	—	2	1	no	—
105	C₃×#103	—	3	1	no	—
106	C₃×#103	C₂×C₉	1	1	yes	characteristic, complemented
107	C₃×#103	C₃×D₃	1	1	yes	characteristic
108	#816	C₂×C₉	1	1	yes	characteristic, complemented
109	#808	C₂×C₉	1	1	yes	characteristic, complemented
110	C₂²×C₃×D₃	C₃×D₄	1	1	yes	characteristic
111	C₃²×Q₈	—	2	1	no	—
112	C₃²×Q₈	—	2	1	no	—
113	C₂×C₃²×C₄	—	2	1	no	—
114	C₂×C₃²×C₄	—	2	1	no	—
115	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
116	C₂×C₃×Dic₃	—	2	1	no	—
117	C₂×C₃×Dic₃	—	2	1	no	—
118	C₂×C₃×Dic₃	C₃×D₄	1	1	yes	characteristic
119	C₃×C₄×D₃	—	2	1	no	—
120	C₃×C₄×D₃	—	2	1	no	—
121	C₃×Dic₆	—	2	1	no	—
122	C₃×Dic₆	—	2	1	no	—
123	C₂³×C₉	—	3	1	no	—
124	C₃²×C₈	—	2	1	no	—
125	C₃×C₃⋊C₈	C₃×D₄	1	2	yes	—
126	Q₈×C₉	—	2	2	no	—
127	Q₈×C₉	—	2	1	no	—
128	Q₈×C₉	—	2	1	no	—
129	Q₈×C₉	—	4	1	no	—
130	Q₈×C₉	—	6	1	no	—
131	Q₈×C₉	—	6	1	no	—
132	C₂×C₄×C₉	—	2	1	no	—
133	C₂×C₄×C₉	—	2	1	no	—
134	C₂×C₄×C₉	—	2	1	no	—
135	C₂×C₄×C₉	—	3	1	no	—
136	C₂×C₄×C₉	—	4	1	no	—
137	C₂×C₄×C₉	—	4	1	no	—
138	C₂×C₄×C₉	—	6	1	no	—
139	C₂×C₄×C₉	—	6	1	no	—
140	C₂×C₄×C₉	—	6	1	no	—
141	C₂×C₄×C₉	—	6	1	no	—
142	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
143	C₈×C₉	—	2	1	no	—
144	C₈×C₉	—	3	2	no	—
145	C₈×C₉	—	4	1	no	—
146	#470	—	3	1	no	Sylow
147	C₂×C₃×C₉	#121	1	1	yes	characteristic
148	C₂×C₃×C₉	#137	1	1	yes	characteristic
149	C₂×C₃×C₉	#137	1	1	yes	characteristic
150	D₃×C₉	—	4	1	no	—
151	C₂×C₃×Q₈	—	2	1	no	—
152	C₂×C₃×Q₈	—	3	1	no	—
153	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
154	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
155	C₂²×C₃×C₄	—	3	1	no	—
156	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
157	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
158	C₃×Q₁₆	—	6	2	no	—
159	C₃×M₁₆	—	6	2	no	—
160	C₂×C₃×C₈	—	2	1	no	—
161	C₂×C₃×C₈	—	3	1	no	—
162	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
163	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
164	C₃×C₄⋊C₄	—	2	1	no	—
165	C₃×C₄⋊C₄	—	3	1	no	—
166	C₃×C₄⋊C₄	—	6	1	no	—
167	C₃×C₄⋊C₄	—	6	1	no	—
168	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
169	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
170	C₃×C₂²⋊C₄	—	6	1	no	—
171	C₃×C₂²⋊C₄	—	6	1	no	—
172	#216	—	2	2	no	—
173	(C₂×D₃)⋊C₄	—	2	1	no	—
174	(C₂×D₃)⋊C₄	—	2	1	no	—
175	C₄⋊Dic₃	—	2	1	no	—
176	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
177	Dic₃⋊C₄	—	2	1	no	—
178	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
179	ΓL(1,𝔽₂₅)	—	2	2	no	—
180	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×C₃×D₃	—	2	1	no	—
182	C₂×C₃×D₃	—	2	1	no	—
183	C₂×C₃×D₃	—	2	1	no	—
184	C₃²×C₄	—	4	1	no	—
185	C₃²×C₄	—	4	1	no	—
186	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
187	C₃²×C₄	C₃×#41	1	1	yes	characteristic
188	C₃×Dic₃	—	2	1	no	—
189	C₃×Dic₃	—	4	1	no	—
190	C₃×Dic₃	—	4	1	no	—
191	C₂²×C₉	—	2	1	no	—
192	C₂²×C₉	—	6	1	no	—
193	C₂²×C₉	—	6	1	no	—
194	C₂²×C₉	—	6	1	no	—
195	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
196	C₄×C₉	—	2	1	no	—
197	C₄×C₉	—	2	1	no	—
198	C₄×C₉	—	4	2	no	—
199	C₄×C₉	—	4	1	no	—
200	C₄×C₉	—	4	1	no	—
201	C₄×C₉	—	6	1	no	—
202	C₄×C₉	—	8	1	no	—
203	C₄×C₉	—	12	1	no	—
204	C₄×C₉	—	12	1	no	—
205	C₄×C₉	#235	1	1	yes	characteristic
206	C₄×C₉	D₃×D₄	1	1	yes	characteristic
207	C₂×Q₁₆	—	3	1	no	—
208	C₂×M₁₆	—	3	1	no	—
209	C₂²⋊Q₈	—	3	1	no	—
210	C₂²⋊Q₈	—	3	1	no	—
211	#106	—	3	1	no	—
212	#103	—	3	1	no	—
213	#103	D₃×C₉	1	1	yes	characteristic, complemented
214	C₃×C₉	#470	1	1	yes	characteristic, complemented, Sylow
215	C₂³×C₃	—	3	1	no	—
216	C₂²×D₃	D₄×C₉	1	1	yes	characteristic
217	C₃×Q₈	—	2	2	no	—
218	C₃×Q₈	—	2	1	no	—
219	C₃×Q₈	—	2	1	no	—
220	C₃×Q₈	—	2	1	no	—
221	C₃×Q₈	—	2	1	no	—
222	C₃×Q₈	—	4	1	no	—
223	C₃×Q₈	—	6	1	no	—
224	C₃×Q₈	—	6	1	no	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	—	2	1	no	—
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	3	1	no	—
231	C₂×C₃×C₄	—	4	1	no	—
232	C₂×C₃×C₄	—	4	1	no	—
233	C₂×C₃×C₄	—	6	1	no	—
234	C₂×C₃×C₄	—	6	1	no	—
235	C₂×C₃×C₄	—	6	1	no	—
236	C₂×C₃×C₄	—	6	1	no	—
237	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
239	C₂×Dic₃	—	2	1	no	—
240	C₂×Dic₃	—	2	1	no	—
241	C₂×Dic₃	D₄×C₉	1	1	yes	characteristic
242	C₄×D₃	—	2	1	no	—
243	C₄×D₃	—	2	1	no	—
244	Dic₆	—	2	1	no	—
245	Dic₆	—	2	1	no	—
246	C₃×C₈	—	2	1	no	—
247	C₃×C₈	—	2	1	no	—
248	C₃×C₈	—	3	2	no	—
249	C₃×C₈	—	4	1	no	—
250	C₃⋊C₈	D₄×C₉	1	2	yes	—
251	C₂×C₃²	C₃×#121	1	1	yes	characteristic
252	C₂×C₃²	C₃×#137	1	1	yes	characteristic
253	C₂×C₃²	C₃×#137	1	1	yes	characteristic
254	C₃×D₃	—	4	1	no	—
255	C₂×C₉	—	2	1	no	—
256	C₂×C₉	—	2	1	no	—
257	C₂×C₉	—	2	1	no	—
258	C₂×C₉	—	12	1	no	—
259	C₂×C₉	#884	1	1	yes	characteristic
260	C₂×C₉	#915	1	1	yes	characteristic
261	C₂×C₉	#918	1	1	yes	characteristic
262	C₂×Q₈	—	3	1	no	—
263	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
264	C₂²×C₄	—	3	1	no	—
265	Q₁₆	—	6	2	no	—
266	M₁₆	—	6	2	no	—
267	C₂×C₈	—	3	1	no	—
268	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
269	C₄⋊C₄	—	3	1	no	—
270	C₄⋊C₄	—	6	1	no	—
271	C₄⋊C₄	—	6	1	no	—
272	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
273	C₂²⋊C₄	—	6	1	no	—
274	C₂²⋊C₄	—	6	1	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	6	1	no	—
277	C₂²×C₃	—	6	1	no	—
278	C₂²×C₃	—	6	1	no	—
279	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
280	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
281	C₂×D₃	—	2	1	no	—
282	C₂×D₃	—	2	1	no	—
283	C₂×D₃	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	4	2	no	—
287	C₃×C₄	—	4	1	no	—
288	C₃×C₄	—	4	1	no	—
289	C₃×C₄	—	4	1	no	—
290	C₃×C₄	—	4	1	no	—
291	C₃×C₄	—	6	1	no	—
292	C₃×C₄	—	8	1	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	C₉×#41	1	1	yes	characteristic
297	C₃×C₄	C₃×#235	1	1	yes	characteristic
298	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
299	Dic₃	—	2	1	no	—
300	Dic₃	—	4	1	no	—
301	Dic₃	—	4	1	no	—
302	C₃²	C₃×#470	1	1	yes	characteristic
303	C₉	—	2	1	no	—
304	C₉	#4816	1	1	yes	characteristic, direct product factor
305	C₂³	—	3	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	6	1	no	—
309	Q₈	—	6	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	3	1	no	—
313	C₂×C₄	—	6	1	no	—
314	C₂×C₄	—	6	1	no	—
315	C₂×C₄	—	6	1	no	—
316	C₂×C₄	—	6	1	no	—
317	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	3	2	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	12	1	no	—
324	C₂×C₃	C₉×#121	1	1	yes	characteristic
325	C₂×C₃	C₉×#137	1	1	yes	characteristic
326	C₂×C₃	C₉×#137	1	1	yes	characteristic
327	C₂×C₃	C₃×#884	1	1	yes	characteristic
328	C₂×C₃	C₃×#915	1	1	yes	characteristic
329	C₂×C₃	C₃×#918	1	1	yes	characteristic
330	D₃	—	4	1	no	—
331	C₂²	—	6	1	no	—
332	C₂²	—	6	1	no	—
333	C₂²	—	6	1	no	—
334	C₂²	D₃×D₄×C₉	1	1	yes	characteristic
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	6	1	no	—
338	C₄	—	12	1	no	—
339	C₄	—	12	1	no	—
340	C₄	C₉×#235	1	1	yes	characteristic
341	C₄	D₃×D₄×C₉	1	1	yes	characteristic
342	C₃	—	2	1	no	—
343	C₃	C₉×#470	1	1	yes	characteristic, complemented
344	C₃	C₃×#4816	1	1	yes	characteristic
345	C₂	—	12	1	no	—
346	C₂	C₉×#884	1	1	yes	characteristic
347	C₂	C₉×#915	1	1	yes	characteristic
348	C₂	C₉×#918	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	C₃×C₉×#103	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂²×C₃×D₃	C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#103	D₃
hypocenter	C₃	C₉×#470
LCS	G ▷ C₂×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
2. C₂×C₉×#216	C₂	semidirect
5. C₃×C₉×#103	C₂	semidirect
6. C₉×#816	C₂	semidirect
7. C₉×#808	C₂	semidirect
37. C₉×#103	D₃	semidirect
54. #4816	C₉	direct (N × H)
100. C₂×#216	C₂×C₉	semidirect
106. C₃×#103	C₂×C₉	semidirect
108. #816	C₂×C₉	semidirect
109. #808	C₂×C₉	semidirect
213. #103	D₃×C₉	semidirect
214. C₃×C₉	#470	semidirect
304. C₉	#4816	direct (N × H)
343. C₃	C₉×#470	semidirect

Maximal subgroups

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

subgroup	index
C₉×#946	2
C₂×C₉×#216	2
C₂×C₉×ΓL(1,𝔽₂₅)	2
C₉×#897	2
C₃×C₉×#103	2
C₉×#816	2
C₉×#808	2
C₃×#4816	3
C₉×#470	3

Subgroups — C9×#4816 #13141893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4816 #13141893