Subgroups — C₉×#4979 #13142571

Order 1728.

638 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×Dic₃⋊C₄	C₂	1	2	yes	maximal
4	C₉×C₃⋊#95	C₂	1	2	yes	maximal
5	C₉×C₃⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₃×#4979	C₃	1	1	yes	maximal, characteristic
7	C₉×#395	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
13	C₂×C₉×Dic₆	—	2	2	no	—
14	C₂×C₉×Dic₆	—	2	4	no	—
15	C₃×C₉×C₄⋊C₄	—	2	2	no	—
16	C₉×Dic₃⋊C₄	—	2	4	no	—
17	C₂²×C₃×Dic₆	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×Dic₃⋊C₄	C₂×C₃	1	2	yes	—
20	C₂²×Q₈×C₉	—	3	1	no	—
21	C₃×C₃⋊#95	C₂×C₃	1	2	yes	—
22	C₃×C₃⋊#95	C₂×C₃	1	1	yes	characteristic
23	C₂×C₉×C₄⋊C₄	—	2	1	no	—
24	C₂×C₉×C₄⋊C₄	—	3	2	no	—
25	C₂×C₉×C₄⋊C₄	D₃	1	1	yes	characteristic
26	C₉×#95	—	3	1	no	—
27	C₉×#95	—	3	2	no	—
28	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	—	2	1	no	—
33	C₂×C₃×C₄×C₉	—	2	2	no	—
34	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
35	C₂×C₃×C₄×C₉	Q₈	1	2	yes	complemented
36	C₂×C₉×Dic₃	—	2	2	no	—
37	C₂×C₉×Dic₃	—	2	2	no	—
38	C₂×C₉×Dic₃	—	2	2	no	—
39	C₂×C₉×Dic₃	—	2	2	no	—
40	C₂×C₉×Dic₃	D₄	1	4	yes	—
41	C₂×C₉×Dic₃	Q₈	1	4	yes	complemented
42	C₉×Dic₆	—	2	8	no	—
43	C₃×#395	—	3	1	no	—
44	#4979	C₉	1	1	yes	characteristic, direct product factor
45	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
49	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
50	C₂×C₃×Dic₆	—	2	2	no	—
51	C₂×C₃×Dic₆	—	2	4	no	—
52	C₃²×C₄⋊C₄	—	2	2	no	—
53	C₃×Dic₃⋊C₄	—	2	4	no	—
54	C₂×Q₈×C₉	—	6	4	no	—
55	C₂×Q₈×C₉	—	6	2	no	—
56	C₂²×C₄×C₉	—	2	1	no	—
57	C₂²×C₄×C₉	—	2	1	no	—
58	C₂²×C₄×C₉	—	2	1	no	—
59	C₂²×C₄×C₉	—	3	2	no	—
60	C₂²×C₄×C₉	—	3	2	no	—
61	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
62	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
64	C₉×C₄⋊C₄	—	2	4	no	—
65	C₉×C₄⋊C₄	—	2	2	no	—
66	C₉×C₄⋊C₄	—	6	4	no	—
67	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
68	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
71	C₂²×C₃×C₉	#41	1	1	yes	characteristic
72	C₃×C₄×C₉	—	2	2	no	—
73	C₃×C₄×C₉	—	2	2	no	—
74	C₃×C₄×C₉	—	4	1	no	—
75	C₉×Dic₃	—	2	4	no	—
76	C₉×Dic₃	—	2	4	no	—
77	C₂²×C₃×Q₈	—	3	1	no	—
78	C₂²×Dic₆	C₂×C₉	1	1	yes	characteristic
79	C₂×C₃×C₄⋊C₄	—	2	1	no	—
80	C₂×C₃×C₄⋊C₄	—	3	2	no	—
81	C₂×C₃×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
82	C₂×C₃×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
83	C₂×Dic₃⋊C₄	C₂×C₉	1	2	yes	—
84	C₃×#95	—	3	1	no	—
85	C₃×#95	—	3	2	no	—
86	C₃⋊#95	C₂×C₉	1	2	yes	—
87	C₃⋊#95	C₂×C₉	1	1	yes	characteristic
88	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
89	C₂×C₃²×C₄	—	2	1	no	—
90	C₂×C₃²×C₄	—	2	1	no	—
91	C₂×C₃²×C₄	—	2	1	no	—
92	C₂×C₃²×C₄	—	2	2	no	—
93	C₂×C₃²×C₄	—	2	2	no	—
94	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
95	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
96	C₂×C₃×Dic₃	—	2	2	no	—
97	C₂×C₃×Dic₃	—	2	2	no	—
98	C₂×C₃×Dic₃	—	2	2	no	—
99	C₂×C₃×Dic₃	—	2	2	no	—
100	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
101	C₂×C₃×Dic₃	C₃×Q₈	1	4	yes	—
102	C₃×Dic₆	—	2	8	no	—
103	C₂³×C₉	—	2	1	no	—
104	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
105	Q₈×C₉	—	6	8	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	1	no	—
111	C₂×C₄×C₉	—	2	2	no	—
112	C₂×C₄×C₉	—	2	2	no	—
113	C₂×C₄×C₉	—	2	1	no	—
114	C₂×C₄×C₉	—	2	2	no	—
115	C₂×C₄×C₉	—	2	2	no	—
116	C₂×C₄×C₉	—	3	4	no	—
117	C₂×C₄×C₉	—	3	4	no	—
118	C₂×C₄×C₉	—	4	2	no	—
119	C₂×C₄×C₉	—	4	2	no	—
120	C₂×C₄×C₉	—	6	2	no	—
121	C₂×C₄×C₉	—	6	2	no	—
122	C₂×C₄×C₉	—	6	2	no	—
123	C₂×C₄×C₉	—	6	2	no	—
124	C₂×C₄×C₉	Dic₆	1	2	yes	complemented
125	C₂×C₄×C₉	#67	1	2	yes	—
126	#395	—	3	1	no	Sylow
127	C₂×C₃×C₉	#120	1	1	yes	characteristic
128	C₂×C₃×C₉	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₃×C₉	C₄⋊Q₈	1	2	yes	—
131	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₃×Q₈	—	6	4	no	—
133	C₂×C₃×Q₈	—	6	2	no	—
134	C₂²×C₃×C₄	—	2	1	no	—
135	C₂²×C₃×C₄	—	2	1	no	—
136	C₂²×C₃×C₄	—	2	1	no	—
137	C₂²×C₃×C₄	—	3	2	no	—
138	C₂²×C₃×C₄	—	3	2	no	—
139	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
145	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
146	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
147	C₂×Dic₆	—	2	2	no	—
148	C₂×Dic₆	—	2	4	no	—
149	C₃×C₄⋊C₄	—	2	4	no	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	—	2	2	no	—
152	C₃×C₄⋊C₄	—	6	4	no	—
153	Dic₃⋊C₄	—	2	4	no	—
154	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
155	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
157	C₂²×C₃²	C₂×C₃×Q₈	1	2	yes	—
158	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
159	C₃²×C₄	—	2	2	no	—
160	C₃²×C₄	—	2	2	no	—
161	C₃²×C₄	—	4	1	no	—
162	C₃×Dic₃	—	2	4	no	—
163	C₃×Dic₃	—	2	4	no	—
164	C₂²×C₉	—	2	1	no	—
165	C₂²×C₉	—	2	1	no	—
166	C₂²×C₉	—	2	2	no	—
167	C₂²×C₉	—	2	2	no	—
168	C₂²×C₉	—	2	1	no	—
169	C₂²×C₉	C₂×Dic₆	1	1	yes	characteristic
170	C₂²×C₉	#235	1	1	yes	characteristic
171	C₂²×C₉	D₃×D₄	1	2	yes	—
172	C₂²×C₉	D₃×Q₈	1	2	yes	—
173	C₂²×C₉	C₂×#67	1	1	yes	characteristic
174	C₄×C₉	—	2	4	no	—
175	C₄×C₉	—	2	2	no	—
176	C₄×C₉	—	2	2	no	—
177	C₄×C₉	—	4	2	no	—
178	C₄×C₉	—	4	2	no	—
179	C₄×C₉	—	4	1	no	—
180	C₄×C₉	—	6	4	no	—
181	C₄×C₉	—	6	4	no	—
182	C₂²×Q₈	—	3	1	no	—
183	C₂×C₄⋊C₄	—	3	2	no	—
184	C₂×C₄⋊C₄	D₃×C₉	1	1	yes	characteristic
185	#95	—	3	1	no	—
186	#95	—	3	2	no	—
187	C₃×C₉	#395	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₃	—	2	1	no	—
189	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
190	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
191	C₃×Q₈	—	6	8	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	3	4	no	—
208	C₂×C₃×C₄	—	3	4	no	—
209	C₂×C₃×C₄	—	4	2	no	—
210	C₂×C₃×C₄	—	4	2	no	—
211	C₂×C₃×C₄	—	6	2	no	—
212	C₂×C₃×C₄	—	6	2	no	—
213	C₂×C₃×C₄	—	6	2	no	—
214	C₂×C₃×C₄	—	6	2	no	—
215	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
216	C₂×C₃×C₄	Q₈×C₉	1	2	yes	complemented
217	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
218	C₂×C₃×C₄	C₃×#67	1	2	yes	—
219	C₂×Dic₃	—	2	2	no	—
220	C₂×Dic₃	—	2	2	no	—
221	C₂×Dic₃	—	2	2	no	—
222	C₂×Dic₃	—	2	2	no	—
223	C₂×Dic₃	D₄×C₉	1	4	yes	—
224	C₂×Dic₃	Q₈×C₉	1	4	yes	complemented
225	Dic₆	—	2	8	no	—
226	C₂×C₃²	C₃×#120	1	1	yes	characteristic
227	C₂×C₃²	C₃×C₂²⋊Q₈	1	2	yes	—
228	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
229	C₂×C₃²	C₃×C₄⋊Q₈	1	2	yes	—
230	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
231	C₂×C₉	—	2	2	no	—
232	C₂×C₉	—	2	2	no	—
233	C₂×C₉	—	2	1	no	—
234	C₂×C₉	—	2	1	no	—
235	C₂×C₉	—	2	1	no	—
236	C₂×C₉	#888	1	2	yes	—
237	C₂×C₉	#896	1	2	yes	—
238	C₂×C₉	#924	1	1	yes	characteristic
239	C₂×C₉	#937	1	1	yes	characteristic
240	C₂×C₉	#944	1	1	yes	characteristic
241	C₂×Q₈	—	6	4	no	—
242	C₂×Q₈	—	6	2	no	—
243	C₂²×C₄	—	3	2	no	—
244	C₂²×C₄	—	3	2	no	—
245	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	6	4	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	C₂×D₄×C₉	1	2	yes	—
256	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
257	C₂²×C₃	C₂×Q₈×C₉	1	1	yes	characteristic
258	C₂²×C₃	C₂×Q₈×C₉	1	2	yes	—
259	C₂²×C₃	C₉×#41	1	1	yes	characteristic
260	C₂²×C₃	C₂×C₃×Dic₆	1	1	yes	characteristic
261	C₂²×C₃	C₃×#235	1	1	yes	characteristic
262	C₂²×C₃	C₃×D₃×D₄	1	2	yes	—
263	C₂²×C₃	C₃×D₃×Q₈	1	2	yes	—
264	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
265	C₃×C₄	—	2	4	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	4	2	no	—
271	C₃×C₄	—	4	2	no	—
272	C₃×C₄	—	4	1	no	—
273	C₃×C₄	—	4	1	no	—
274	C₃×C₄	—	6	4	no	—
275	C₃×C₄	—	6	4	no	—
276	Dic₃	—	2	4	no	—
277	Dic₃	—	2	4	no	—
278	C₃²	C₃×#395	1	1	yes	characteristic
279	C₉	—	2	1	no	—
280	C₉	#4979	1	1	yes	characteristic, direct product factor
281	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
282	Q₈	—	6	8	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	3	4	no	—
289	C₂×C₄	—	3	4	no	—
290	C₂×C₄	—	6	2	no	—
291	C₂×C₄	—	6	2	no	—
292	C₂×C₄	—	6	2	no	—
293	C₂×C₄	—	6	2	no	—
294	C₂×C₄	C₉×Dic₆	1	2	yes	complemented
295	C₂×C₄	C₉×#67	1	2	yes	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	2	2	no	—
300	C₂×C₃	—	2	2	no	—
301	C₂×C₃	C₉×#120	1	1	yes	characteristic
302	C₂×C₃	C₉×C₂²⋊Q₈	1	1	yes	characteristic
303	C₂×C₃	C₉×C₂²⋊Q₈	1	2	yes	—
304	C₂×C₃	C₉×C₄⋊Q₈	1	1	yes	characteristic
305	C₂×C₃	C₉×C₄⋊Q₈	1	2	yes	—
306	C₂×C₃	C₃×#888	1	2	yes	—
307	C₂×C₃	C₃×#896	1	2	yes	—
308	C₂×C₃	C₃×#924	1	1	yes	characteristic
309	C₂×C₃	C₃×#937	1	1	yes	characteristic
310	C₂×C₃	C₃×#944	1	1	yes	characteristic
311	C₂²	C₂×C₉×Dic₆	1	1	yes	characteristic
312	C₂²	C₉×#235	1	1	yes	characteristic
313	C₂²	D₃×D₄×C₉	1	2	yes	—
314	C₂²	D₃×Q₈×C₉	1	2	yes	—
315	C₂²	C₂×C₉×#67	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	1	no	—
319	C₄	—	6	4	no	—
320	C₄	—	6	4	no	—
321	C₃	—	2	1	no	—
322	C₃	C₉×#395	1	1	yes	characteristic, complemented
323	C₃	C₃×#4979	1	1	yes	characteristic
324	C₂	C₉×#888	1	2	yes	—
325	C₂	C₉×#896	1	2	yes	—
326	C₂	C₉×#924	1	1	yes	characteristic
327	C₂	C₉×#937	1	1	yes	characteristic
328	C₂	C₉×#944	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂³×C₃	C₂³×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₂×C₉×C₄⋊C₄	D₃
hypocenter	C₃	C₉×#395
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₉×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
35. C₂×C₃×C₄×C₉	Q₈	semidirect
41. C₂×C₉×Dic₃	Q₈	semidirect
44. #4979	C₉	direct (N × H)
124. C₂×C₄×C₉	Dic₆	semidirect
187. C₃×C₉	#395	semidirect
216. C₂×C₃×C₄	Q₈×C₉	semidirect
224. C₂×Dic₃	Q₈×C₉	semidirect
280. C₉	#4979	direct (N × H)
294. C₂×C₄	C₉×Dic₆	semidirect
322. C₃	C₉×#395	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×Dic₆	2
C₂×C₃×C₉×C₄⋊C₄	2
C₂×C₉×Dic₃⋊C₄	2
C₉×C₃⋊#95	2
C₃×#4979	3
C₉×#395	3

Subgroups — C9×#4979 #13142571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4979 #13142571