Subgroups — C₃×D₃×#499 #11874902

Order 1152.

763 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×#125	C₂	1	2	yes	maximal
2	C₃²×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5065	C₂	1	2	yes	maximal, complemented
4	C₃×D₃×#107	C₂	1	2	yes	maximal
5	C₃×#4878	C₂	1	2	yes	maximal, complemented
6	C₃×D₃×#106	C₂	1	2	yes	maximal
7	C₃×#4858	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
9	C₃×#4683	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×#499	—	3	1	no	maximal
11	D₃×#499	C₃	1	1	yes	maximal, characteristic, direct product factor
12	C₃²×#125	C₂²	1	2	yes	—
13	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
15	C₃×D₃×C₄⋊C₄	—	2	4	no	—
16	C₃×D₃×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×#890	C₂²	1	4	yes	—
18	C₃×#889	—	2	4	no	—
19	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
20	C₃×#870	C₂²	1	2	yes	—
21	C₃²×#107	C₂²	1	2	yes	—
22	C₃²×#106	C₂²	1	2	yes	—
23	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₃×#818	C₂²	1	2	yes	—
25	C₃×#817	C₂²	1	2	yes	—
26	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
27	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
28	C₃×#808	C₂²	1	4	yes	—
29	C₃×#807	C₂²	1	4	yes	—
30	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₃×#125	—	3	2	no	—
32	D₃×#125	C₂×C₃	1	2	yes	—
33	C₃×#499	—	2	1	no	—
34	C₃×#499	—	3	2	no	—
35	C₃×#499	—	3	2	no	—
36	C₃×#499	—	3	2	no	—
37	C₃×#499	—	3	1	no	—
38	C₃×#499	D₃	1	1	yes	characteristic, direct product factor
39	C₃×#499	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃×#107	—	3	2	no	—
41	C₂×C₃×#106	—	3	2	no	—
42	C₂×C₃×C₄×C₈	—	3	1	no	—
43	#5065	C₂×C₃	1	2	yes	complemented
44	D₃×#107	C₂×C₃	1	2	yes	—
45	#4878	C₂×C₃	1	2	yes	complemented
46	D₃×#106	C₂×C₃	1	2	yes	—
47	#4858	C₂×C₃	1	2	yes	complemented
48	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic
49	#4683	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×C₃×C₄×D₃	—	2	4	no	—
51	C₂×C₃×C₄×D₃	D₄	1	2	yes	—
52	C₂×C₃×C₄×D₃	C₂³	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
55	C₃²×C₄⋊C₄	—	2	4	no	—
56	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
57	C₃²×C₄²	C₂³	1	1	yes	characteristic
58	C₃×C₄⋊Dic₃	—	2	4	no	—
59	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
60	C₃×Dic₃⋊C₄	—	2	4	no	—
61	C₃×Dic₃⋊C₄	—	2	4	no	—
62	C₃×Dic₃⋊C₄	—	2	4	no	—
63	C₃×C₄×Dic₃	D₄	1	2	yes	—
64	C₃×C₄×Dic₃	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₃⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₃⋊C₈	C₂³	1	1	yes	characteristic
67	C₃×D₃×C₈	Q₈	1	4	yes	—
68	C₃×D₃×C₈	Q₈	1	4	yes	—
69	C₂×#499	—	3	1	no	Sylow
70	C₂²×C₃×C₈	—	3	1	no	—
71	C₂²×C₃×C₈	—	3	1	no	—
72	C₃×#125	—	2	2	no	—
73	C₃×#125	—	3	4	no	—
74	C₃×#125	—	3	2	no	—
75	C₃×#125	—	6	4	no	—
76	C₃×#125	C₂×D₃	1	2	yes	—
77	C₃×#125	C₂²×C₃	1	2	yes	—
78	C₂×C₃×C₄⋊C₄	—	3	4	no	—
79	C₂×C₃×C₄⋊C₄	—	6	4	no	—
80	C₂×C₃×C₄²	—	3	1	no	—
81	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
82	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
83	D₃×C₄⋊C₄	—	2	4	no	—
84	D₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
85	#890	C₂²×C₃	1	4	yes	—
86	#889	—	2	4	no	—
87	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
88	#870	C₂²×C₃	1	2	yes	—
89	C₃×#107	—	2	2	no	—
90	C₃×#107	—	3	2	no	—
91	C₃×#107	—	3	4	no	—
92	C₃×#107	C₂×D₃	1	2	yes	—
93	C₃×#107	C₂²×C₃	1	2	yes	—
94	C₃×#106	—	2	2	no	—
95	C₃×#106	—	3	2	no	—
96	C₃×#106	—	3	4	no	—
97	C₃×#106	C₂×D₃	1	2	yes	—
98	C₃×#106	C₂²×C₃	1	2	yes	—
99	C₃×C₄×C₈	—	2	1	no	—
100	C₃×C₄×C₈	—	3	1	no	—
101	C₃×C₄×C₈	—	3	1	no	—
102	C₃×C₄×C₈	—	3	1	no	—
103	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
104	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
105	#818	C₂²×C₃	1	2	yes	—
106	#817	C₂²×C₃	1	2	yes	—
107	C₈×Dic₃	C₂²×C₃	1	1	yes	characteristic
108	C₈×Dic₃	C₂²×C₃	1	1	yes	characteristic
109	#808	C₂²×C₃	1	4	yes	—
110	#807	C₂²×C₃	1	4	yes	—
111	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
112	C₂²×C₃×D₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₃²×C₄	—	2	4	no	—
114	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
115	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
116	C₂×C₃×Dic₃	—	2	4	no	—
117	C₂×C₃×Dic₃	C₂×D₄	1	2	yes	—
118	C₂×C₃×Dic₃	C₂×D₄	1	1	yes	characteristic
119	C₃×C₄×D₃	—	2	4	no	—
120	C₃×C₄×D₃	—	4	8	no	—
121	C₃×C₄×D₃	C₂×Q₈	1	2	yes	—
122	C₃×C₄×D₃	C₂×Q₈	1	2	yes	—
123	C₃²×C₈	C₂×Q₈	1	2	yes	—
124	C₃²×C₈	C₂×Q₈	1	2	yes	—
125	C₃×C₃⋊C₈	C₂×Q₈	1	2	yes	—
126	C₃×C₃⋊C₈	C₂×Q₈	1	2	yes	—
127	C₂×#125	—	3	2	no	—
128	#499	—	3	2	no	—
129	#499	—	3	2	no	—
130	#499	—	3	2	no	—
131	#499	—	3	1	no	—
132	#499	C₃×D₃	1	1	yes	characteristic, direct product factor
133	C₂×#107	—	3	2	no	—
134	C₂×#106	—	3	2	no	—
135	C₂×C₄×C₈	—	3	1	no	—
136	C₂²×C₃×C₄	—	3	1	no	—
137	C₂²×C₃×C₄	—	3	2	no	—
138	C₂²×C₃×C₄	—	6	4	no	—
139	C₂×C₄×D₃	—	2	4	no	—
140	C₂×C₄×D₃	C₃×D₄	1	2	yes	—
141	C₂×C₄×D₃	C₂³×C₃	1	1	yes	characteristic
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	—	3	1	no	—
145	C₂×C₃×C₈	—	3	1	no	—
146	C₂×C₃×C₈	—	3	4	no	—
147	C₂×C₃×C₈	—	3	4	no	—
148	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
150	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
151	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
152	C₃×C₄⋊C₄	—	2	4	no	—
153	C₃×C₄⋊C₄	—	2	4	no	—
154	C₃×C₄⋊C₄	—	2	4	no	—
155	C₃×C₄⋊C₄	—	3	4	no	—
156	C₃×C₄⋊C₄	—	4	4	no	—
157	C₃×C₄⋊C₄	—	6	4	no	—
158	C₃×C₄⋊C₄	—	6	4	no	—
159	C₃×C₄⋊C₄	—	6	4	no	—
160	C₃×C₄⋊C₄	—	6	4	no	—
161	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
162	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
163	C₃×C₄²	—	2	1	no	—
164	C₃×C₄²	—	3	2	no	—
165	C₃×C₄²	—	3	1	no	—
166	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
167	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
168	C₄⋊Dic₃	—	2	4	no	—
169	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
170	Dic₃⋊C₄	—	2	4	no	—
171	Dic₃⋊C₄	—	2	4	no	—
172	Dic₃⋊C₄	—	2	4	no	—
173	C₄×Dic₃	C₃×D₄	1	2	yes	—
174	C₄×Dic₃	C₂³×C₃	1	1	yes	characteristic
175	C₂×C₃⋊C₈	C₂³×C₃	1	1	yes	characteristic
176	C₂×C₃⋊C₈	C₂³×C₃	1	1	yes	characteristic
177	D₃×C₈	C₃×Q₈	1	4	yes	—
178	D₃×C₈	C₃×Q₈	1	4	yes	—
179	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₃×D₃	C₄⋊Q₈	1	2	yes	—
181	C₂×C₃×D₃	#135	1	4	yes	—
182	C₃²×C₄	—	2	2	no	—
183	C₃²×C₄	—	4	4	no	—
184	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
185	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
186	C₃×Dic₃	—	2	2	no	—
187	C₃×Dic₃	—	4	4	no	—
188	C₃×Dic₃	C₄⋊Q₈	1	2	yes	—
189	C₂²×C₈	—	3	1	no	—
190	C₂²×C₈	—	3	1	no	—
191	#125	—	3	4	no	—
192	#125	—	3	2	no	—
193	#125	—	6	4	no	—
194	#125	C₂×C₃×D₃	1	2	yes	—
195	C₂×C₄⋊C₄	—	3	4	no	—
196	C₂×C₄⋊C₄	—	6	4	no	—
197	C₂×C₄²	—	3	1	no	—
198	#107	—	3	2	no	—
199	#107	—	3	4	no	—
200	#107	C₂×C₃×D₃	1	2	yes	—
201	#106	—	3	2	no	—
202	#106	—	3	4	no	—
203	#106	C₂×C₃×D₃	1	2	yes	—
204	C₄×C₈	—	3	1	no	—
205	C₄×C₈	—	3	1	no	—
206	C₄×C₈	—	3	1	no	—
207	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
208	C₂³×C₃	—	3	1	no	—
209	C₂²×D₃	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	—	2	4	no	—
211	C₂×C₃×C₄	—	2	4	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	3	1	no	—
215	C₂×C₃×C₄	—	3	2	no	—
216	C₂×C₃×C₄	—	3	2	no	—
217	C₂×C₃×C₄	—	3	2	no	—
218	C₂×C₃×C₄	—	4	4	no	—
219	C₂×C₃×C₄	—	6	4	no	—
220	C₂×C₃×C₄	—	6	4	no	—
221	C₂×C₃×C₄	—	12	8	no	—
222	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
223	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
224	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
226	C₂×Dic₃	—	2	4	no	—
227	C₂×Dic₃	C₂×C₃×D₄	1	2	yes	—
228	C₂×Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
229	C₄×D₃	—	2	4	no	—
230	C₄×D₃	—	4	8	no	—
231	C₄×D₃	C₂×C₃×Q₈	1	2	yes	—
232	C₄×D₃	C₂×C₃×Q₈	1	2	yes	—
233	C₃×C₈	—	2	2	no	—
234	C₃×C₈	—	2	2	no	—
235	C₃×C₈	—	3	2	no	—
236	C₃×C₈	—	3	2	no	—
237	C₃×C₈	D₃×Q₈	1	2	yes	—
238	C₃×C₈	D₃×Q₈	1	2	yes	—
239	C₃×C₈	C₂×C₃×Q₈	1	2	yes	—
240	C₃×C₈	C₂×C₃×Q₈	1	2	yes	—
241	C₃⋊C₈	C₂×C₃×Q₈	1	2	yes	—
242	C₃⋊C₈	C₂×C₃×Q₈	1	2	yes	—
243	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
244	C₂×C₃²	C₂×#135	1	2	yes	—
245	C₃×D₃	#499	1	4	yes	direct product factor
246	C₂²×C₄	—	3	1	no	—
247	C₂²×C₄	—	3	2	no	—
248	C₂²×C₄	—	6	4	no	—
249	C₂×C₈	—	3	1	no	—
250	C₂×C₈	—	3	1	no	—
251	C₂×C₈	—	3	4	no	—
252	C₂×C₈	—	3	4	no	—
253	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
254	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	3	4	no	—
257	C₄⋊C₄	—	6	4	no	—
258	C₄⋊C₄	—	6	4	no	—
259	C₄⋊C₄	—	6	4	no	—
260	C₄⋊C₄	—	6	4	no	—
261	C₄⋊C₄	C₂²×C₃×D₃	1	4	yes	—
262	C₄²	—	3	1	no	—
263	C₄²	—	3	2	no	—
264	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	3	2	no	—
267	C₂²×C₃	—	3	4	no	—
268	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
269	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
270	C₂×D₃	C₃×C₄⋊Q₈	1	2	yes	—
271	C₂×D₃	C₃×#135	1	4	yes	—
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	2	2	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	3	2	no	—
277	C₃×C₄	—	4	2	no	—
278	C₃×C₄	—	4	4	no	—
279	C₃×C₄	—	4	4	no	—
280	C₃×C₄	—	6	2	no	—
281	C₃×C₄	—	8	4	no	—
282	C₃×C₄	—	12	4	no	—
283	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
284	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
285	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
286	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
287	Dic₃	—	2	2	no	—
288	Dic₃	—	4	4	no	—
289	Dic₃	C₃×C₄⋊Q₈	1	2	yes	—
290	C₃²	C₂×#499	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	3	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	2	no	—
295	C₂×C₄	—	3	2	no	—
296	C₂×C₄	—	3	2	no	—
297	C₂×C₄	—	6	4	no	—
298	C₂×C₄	—	6	4	no	—
299	C₂×C₄	—	12	8	no	—
300	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
301	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
302	C₈	—	3	2	no	—
303	C₈	—	3	2	no	—
304	C₈	C₃×D₃×Q₈	1	2	yes	—
305	C₈	C₃×D₃×Q₈	1	2	yes	—
306	C₂×C₃	—	2	2	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	3	4	no	—
309	C₂×C₃	D₃×C₄⋊Q₈	1	1	yes	characteristic
310	C₂×C₃	D₃×#135	1	2	yes	—
311	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
312	C₂×C₃	C₂×C₃×#135	1	2	yes	—
313	D₃	C₃×#499	1	4	yes	direct product factor
314	C₂²	—	3	4	no	—
315	C₂²	—	3	2	no	—
316	C₂²	C₂×C₃×D₃×D₄	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	3	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	6	2	no	—
321	C₄	—	12	4	no	—
322	C₄	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
323	C₄	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
324	C₃	—	2	1	no	—
325	C₃	D₃×#499	1	1	yes	characteristic, direct product factor
326	C₃	C₂×C₃×#499	1	1	yes	characteristic, complemented
327	C₂	—	3	4	no	—
328	C₂	C₃×D₃×C₄⋊Q₈	1	1	yes	characteristic
329	C₂	C₃×D₃×#135	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#499	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂²×C₃×D₃	C₂×D₄
hypercenter	C₃×#499	D₃
hypocenter	C₃	C₂×C₃×#499
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#499

Complements

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

N	H ≅ G/N	type
2. C₃²×#499	C₂	semidirect
3. C₃×#5065	C₂	semidirect
5. C₃×#4878	C₂	semidirect
7. C₃×#4858	C₂	semidirect
9. C₃×#4683	C₂	semidirect
11. D₃×#499	C₃	direct (N × H)
38. C₃×#499	D₃	direct (N × H)
39. C₃×#499	C₂×C₃	semidirect
43. #5065	C₂×C₃	semidirect
45. #4878	C₂×C₃	semidirect
47. #4858	C₂×C₃	semidirect
49. #4683	C₂×C₃	semidirect
132. #499	C₃×D₃	direct (N × H)
245. C₃×D₃	#499	direct (N × H)
290. C₃²	C₂×#499	semidirect
313. D₃	C₃×#499	direct (N × H)
325. C₃	D₃×#499	direct (N × H)
326. C₃	C₂×C₃×#499	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×#125	2
C₃²×#499	2
C₃×#5065	2
C₃×D₃×#107	2
C₃×#4878	2
C₃×D₃×#106	2
C₃×#4858	2
C₃×C₄×D₃×C₈	2
C₃×#4683	2
C₂×C₃×#499	3
D₃×#499	3

Subgroups — C3×D3×#499 #11874902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×#499 #11874902