Subgroups — C₉×#5227 #13143593

Order 1728.

630 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#215	C₂	1	2	yes	maximal, complemented
3	C₉×#930	C₂	1	1	yes	maximal, characteristic
4	C₉×#835	C₂	1	2	yes	maximal, complemented
5	C₉×#830	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5227	C₃	1	1	yes	maximal, characteristic
7	C₉×#448	—	3	1	no	maximal
8	C₂×C₃×Q₈×C₉	—	2	4	no	—
9	C₂×C₃×Q₈×C₉	—	2	1	no	—
10	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	—	2	1	no	—
12	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₉×#67	—	2	1	no	—
14	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
15	C₉×#215	—	2	4	no	—
16	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
19	C₂²×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#215	C₂×C₃	1	2	yes	—
21	C₃×#930	C₂×C₃	1	1	yes	characteristic
22	C₂²×Q₈×C₉	—	2	1	no	—
23	C₂²×Q₈×C₉	D₃	1	1	yes	characteristic, complemented
24	C₃×#835	C₂×C₃	1	2	yes	—
25	C₃×#830	C₂×C₃	1	1	yes	characteristic
26	C₂×C₉×SD₁₆	—	3	2	no	—
27	C₉×#121	—	3	1	no	—
28	C₉×#103	—	3	2	no	—
29	C₉×C₂²⋊C₈	—	3	1	no	—
30	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
31	C₂²×D₃×C₉	—	2	1	no	—
32	C₃×Q₈×C₉	—	2	4	no	—
33	C₃×Q₈×C₉	—	2	2	no	—
34	C₃×Q₈×C₉	D₄	1	4	yes	complemented
35	C₂×C₃×C₄×C₉	—	2	4	no	—
36	C₂×C₃×C₄×C₉	—	2	2	no	—
37	C₂×C₃×C₄×C₉	—	2	2	no	—
38	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₉×#67	—	4	2	no	—
41	C₂×C₉×Dic₃	—	2	1	no	—
42	C₉×D₁₂	—	2	2	no	—
43	C₉×C₃⋊C₈	—	2	2	no	—
44	C₃×#448	—	3	1	no	—
45	#5227	C₉	1	1	yes	characteristic, direct product factor
46	C₂×C₃²×Q₈	—	2	4	no	—
47	C₂×C₃²×Q₈	—	2	1	no	—
48	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
49	C₂²×C₃²×C₄	—	2	1	no	—
50	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×#67	—	2	1	no	—
52	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
53	C₃×#215	—	2	4	no	—
54	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
55	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
57	C₂×Q₈×C₉	—	2	8	no	—
58	C₂×Q₈×C₉	—	2	2	no	—
59	C₂×Q₈×C₉	—	2	2	no	—
60	C₂×Q₈×C₉	—	2	4	no	—
61	C₂×Q₈×C₉	—	2	1	no	—
62	C₂×Q₈×C₉	C₂×D₃	1	2	yes	—
63	C₂×D₄×C₉	—	3	1	no	—
64	C₂×D₄×C₉	—	6	1	no	—
65	C₂²×C₄×C₉	—	2	2	no	—
66	C₂²×C₄×C₉	—	2	1	no	—
67	C₂²×C₄×C₉	—	2	1	no	—
68	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
69	C₉×SD₁₆	—	6	4	no	—
70	C₂×C₈×C₉	—	3	2	no	—
71	C₉×C₄⋊C₄	—	3	1	no	—
72	C₉×C₂²⋊C₄	—	6	1	no	—
73	C₂²×C₃×C₉	—	2	1	no	—
74	C₂²×C₃×C₉	—	2	1	no	—
75	C₂²×C₃×C₉	SD₁₆	1	2	yes	complemented
76	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
77	C₂×D₃×C₉	—	4	1	no	—
78	C₂×D₃×C₉	—	4	1	no	—
79	C₂×D₃×C₉	—	4	1	no	—
80	C₃×C₄×C₉	—	2	4	no	—
81	C₃×C₄×C₉	—	2	1	no	—
82	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
83	C₉×Dic₃	—	4	1	no	—
84	C₂²×C₃×Q₈	—	2	1	no	—
85	C₂²×C₃×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
86	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
87	C₂×C₃×SD₁₆	—	3	2	no	—
88	C₃×#121	—	3	1	no	—
89	C₂×#215	C₂×C₉	1	2	yes	complemented
90	#930	C₂×C₉	1	1	yes	characteristic
91	C₃×#103	—	3	2	no	—
92	C₃×C₂²⋊C₈	—	3	1	no	—
93	#835	C₂×C₉	1	2	yes	complemented
94	#830	C₂×C₉	1	1	yes	characteristic, complemented
95	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
96	C₂²×C₃×D₃	—	2	1	no	—
97	C₃²×Q₈	—	2	4	no	—
98	C₃²×Q₈	—	2	2	no	—
99	C₃²×Q₈	C₃×D₄	1	4	yes	—
100	C₂×C₃²×C₄	—	2	4	no	—
101	C₂×C₃²×C₄	—	2	2	no	—
102	C₂×C₃²×C₄	—	2	2	no	—
103	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₃×#67	—	4	2	no	—
106	C₂×C₃×Dic₃	—	2	1	no	—
107	C₃×D₁₂	—	2	2	no	—
108	C₂³×C₉	—	2	1	no	—
109	C₂³×C₉	—	6	1	no	—
110	C₂³×C₉	#67	1	1	yes	characteristic
111	C₃×C₃⋊C₈	—	2	2	no	—
112	Q₈×C₉	—	2	8	no	—
113	Q₈×C₉	—	2	4	no	—
114	Q₈×C₉	—	2	4	no	—
115	Q₈×C₉	—	2	2	no	—
116	Q₈×C₉	—	2	4	no	—
117	Q₈×C₉	#67	1	4	yes	complemented
118	D₄×C₉	—	6	2	no	—
119	D₄×C₉	—	12	2	no	—
120	C₂×C₄×C₉	—	2	8	no	—
121	C₂×C₄×C₉	—	2	4	no	—
122	C₂×C₄×C₉	—	2	4	no	—
123	C₂×C₄×C₉	—	2	4	no	—
124	C₂×C₄×C₉	—	2	2	no	—
125	C₂×C₄×C₉	—	2	2	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	#67	1	1	yes	characteristic
130	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
131	C₈×C₉	—	6	2	no	—
132	#448	—	3	1	no	Sylow
133	C₂×C₃×C₉	—	2	2	no	—
134	C₂×C₃×C₉	#120	1	1	yes	characteristic
135	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₃×C₉	#137	1	1	yes	characteristic
137	D₃×C₉	—	8	1	no	—
138	C₂×C₃×Q₈	—	2	8	no	—
139	C₂×C₃×Q₈	—	2	2	no	—
140	C₂×C₃×Q₈	—	2	4	no	—
141	C₂×C₃×Q₈	—	2	4	no	—
142	C₂×C₃×Q₈	—	2	1	no	—
143	C₂×C₃×Q₈	—	2	1	no	—
144	C₂×C₃×Q₈	—	2	2	no	—
145	C₂×C₃×Q₈	C₂²×C₉	1	2	yes	—
146	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
147	C₂×C₃×D₄	—	3	1	no	—
148	C₂×C₃×D₄	—	6	1	no	—
149	C₂²×C₃×C₄	—	2	2	no	—
150	C₂²×C₃×C₄	—	2	1	no	—
151	C₂²×C₃×C₄	—	2	1	no	—
152	C₂²×C₃×C₄	—	2	1	no	—
153	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
154	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
155	C₂×#67	—	2	1	no	—
156	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
157	C₃×SD₁₆	—	6	4	no	—
158	C₂×C₃×C₈	—	3	2	no	—
159	C₃×C₄⋊C₄	—	3	1	no	—
160	C₃×C₂²⋊C₄	—	6	1	no	—
161	#215	—	2	4	no	—
162	(C₂×D₃)⋊C₄	—	2	1	no	—
163	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
164	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
165	C₂²×C₃²	—	2	1	no	—
166	C₂²×C₃²	—	2	1	no	—
167	C₂²×C₃²	C₃×SD₁₆	1	2	yes	—
168	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×C₃×D₃	—	4	1	no	—
170	C₂×C₃×D₃	—	4	1	no	—
171	C₂×C₃×D₃	—	4	1	no	—
172	C₃²×C₄	—	2	4	no	—
173	C₃²×C₄	—	2	1	no	—
174	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
175	C₃×Dic₃	—	4	1	no	—
176	C₂²×C₉	—	2	2	no	—
177	C₂²×C₉	—	2	2	no	—
178	C₂²×C₉	—	2	1	no	—
179	C₂²×C₉	—	2	1	no	—
180	C₂²×C₉	—	2	2	no	—
181	C₂²×C₉	—	2	1	no	—
182	C₂²×C₉	—	12	1	no	—
183	C₂²×C₉	—	12	1	no	—
184	C₂²×C₉	—	12	1	no	—
185	C₂²×C₉	#215	1	2	yes	complemented
186	C₂²×C₉	C₂×#67	1	1	yes	characteristic
187	C₄×C₉	—	2	8	no	—
188	C₄×C₉	—	2	2	no	—
189	C₄×C₉	—	2	4	no	—
190	C₄×C₉	—	2	1	no	—
191	C₄×C₉	—	2	2	no	—
192	C₄×C₉	—	12	1	no	—
193	C₄×C₉	C₂×#67	1	2	yes	—
194	C₂²×Q₈	D₃×C₉	1	1	yes	characteristic, complemented
195	C₂×SD₁₆	—	3	2	no	—
196	#121	—	3	1	no	—
197	#103	—	3	2	no	—
198	C₂²⋊C₈	—	3	1	no	—
199	C₃×C₉	#448	1	1	yes	characteristic, complemented, Sylow
200	C₂³×C₃	—	2	1	no	—
201	C₂³×C₃	—	6	1	no	—
202	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
203	C₂³×C₃	C₃×#67	1	1	yes	characteristic
204	C₂²×D₃	—	2	1	no	—
205	C₃×Q₈	—	2	4	no	—
206	C₃×Q₈	—	2	8	no	—
207	C₃×Q₈	—	2	4	no	—
208	C₃×Q₈	—	2	4	no	—
209	C₃×Q₈	—	2	2	no	—
210	C₃×Q₈	—	2	2	no	—
211	C₃×Q₈	—	2	4	no	—
212	C₃×Q₈	D₄×C₉	1	4	yes	complemented
213	C₃×Q₈	C₃×#67	1	4	yes	—
214	C₃×D₄	—	6	2	no	—
215	C₃×D₄	—	12	2	no	—
216	C₂×C₃×C₄	—	2	4	no	—
217	C₂×C₃×C₄	—	2	8	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	2	2	no	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	—	6	1	no	—
228	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂³×C₉	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	#67	—	4	2	no	—
233	C₂×Dic₃	—	2	1	no	—
234	D₁₂	—	2	2	no	—
235	C₃×C₈	—	6	2	no	—
236	C₃⋊C₈	—	2	2	no	—
237	C₂×C₃²	—	2	2	no	—
238	C₂×C₃²	C₃×#120	1	1	yes	characteristic
239	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
240	C₂×C₃²	C₃×#137	1	1	yes	characteristic
241	C₃×D₃	—	8	1	no	—
242	C₂×C₉	—	2	4	no	—
243	C₂×C₉	—	2	2	no	—
244	C₂×C₉	—	2	1	no	—
245	C₂×C₉	—	2	1	no	—
246	C₂×C₉	—	2	1	no	—
247	C₂×C₉	—	24	1	no	—
248	C₂×C₉	C₂×#215	1	1	yes	characteristic
249	C₂×C₉	#942	1	1	yes	characteristic
250	C₂×C₉	#953	1	1	yes	characteristic
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	C₂×D₃×C₉	1	2	yes	—
254	C₂×D₄	—	3	1	no	—
255	C₂×D₄	—	6	1	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
258	SD₁₆	—	6	4	no	—
259	C₂×C₈	—	3	2	no	—
260	C₄⋊C₄	—	3	1	no	—
261	C₂²⋊C₄	—	6	1	no	—
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	2	2	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	2	no	—
270	C₂²×C₃	—	12	1	no	—
271	C₂²×C₃	—	12	1	no	—
272	C₂²×C₃	—	12	1	no	—
273	C₂²×C₃	C₉×SD₁₆	1	2	yes	complemented
274	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
275	C₂²×C₃	C₃×#215	1	2	yes	—
276	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
277	C₂×D₃	—	4	1	no	—
278	C₂×D₃	—	4	1	no	—
279	C₂×D₃	—	4	1	no	—
280	C₃×C₄	—	2	8	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	2	4	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	12	1	no	—
288	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
289	C₃×C₄	C₂×C₃×#67	1	2	yes	—
290	Dic₃	—	4	1	no	—
291	C₃²	C₃×#448	1	1	yes	characteristic
292	C₉	—	2	1	no	—
293	C₉	#5227	1	1	yes	characteristic, direct product factor
294	C₂³	—	6	1	no	—
295	C₂³	C₉×#67	1	1	yes	characteristic
296	Q₈	—	2	4	no	—
297	Q₈	—	2	2	no	—
298	Q₈	C₉×#67	1	4	yes	complemented
299	D₄	—	6	2	no	—
300	D₄	—	12	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	6	1	no	—
305	C₂×C₄	C₉×#67	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
307	C₈	—	6	2	no	—
308	C₂×C₃	—	2	4	no	—
309	C₂×C₃	—	2	2	no	—
310	C₂×C₃	—	2	2	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	2	1	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	24	1	no	—
315	C₂×C₃	C₉×#120	1	1	yes	characteristic
316	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
317	C₂×C₃	C₉×#137	1	1	yes	characteristic
318	C₂×C₃	C₂×C₃×#215	1	1	yes	characteristic
319	C₂×C₃	C₃×#942	1	1	yes	characteristic
320	C₂×C₃	C₃×#953	1	1	yes	characteristic
321	D₃	—	8	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	12	1	no	—
325	C₂²	—	12	1	no	—
326	C₂²	—	12	1	no	—
327	C₂²	C₉×#215	1	2	yes	complemented
328	C₂²	C₂×C₉×#67	1	1	yes	characteristic
329	C₄	—	2	4	no	—
330	C₄	—	2	1	no	—
331	C₄	—	12	1	no	—
332	C₄	C₂×C₉×#67	1	2	yes	—
333	C₃	—	2	1	no	—
334	C₃	C₉×#448	1	1	yes	characteristic, complemented
335	C₃	C₃×#5227	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	24	1	no	—
338	C₂	C₂×C₉×#215	1	1	yes	characteristic
339	C₂	C₉×#942	1	1	yes	characteristic
340	C₂	C₉×#953	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	C₂²×C₃×Q₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#930	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂²×Q₈×C₉	D₃
hypocenter	C₃	C₉×#448
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×Q₈×C₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₈×C₉	C₂	semidirect
2. C₂×C₉×#215	C₂	semidirect
4. C₉×#835	C₂	semidirect
5. C₉×#830	C₂	semidirect
23. C₂²×Q₈×C₉	D₃	semidirect
34. C₃×Q₈×C₉	D₄	semidirect
45. #5227	C₉	direct (N × H)
75. C₂²×C₃×C₉	SD₁₆	semidirect
85. C₂²×C₃×Q₈	C₂×C₉	semidirect
89. C₂×#215	C₂×C₉	semidirect
93. #835	C₂×C₉	semidirect
94. #830	C₂×C₉	semidirect
117. Q₈×C₉	#67	semidirect
185. C₂²×C₉	#215	semidirect
194. C₂²×Q₈	D₃×C₉	semidirect
199. C₃×C₉	#448	semidirect
212. C₃×Q₈	D₄×C₉	semidirect
273. C₂²×C₃	C₉×SD₁₆	semidirect
293. C₉	#5227	direct (N × H)
298. Q₈	C₉×#67	semidirect
327. C₂²	C₉×#215	semidirect
334. C₃	C₉×#448	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈×C₉	2
C₂×C₉×#215	2
C₉×#930	2
C₉×#835	2
C₉×#830	2
C₃×#5227	3
C₉×#448	3

Subgroups — C9×#5227 #13143593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5227 #13143593