Subgroups — C₃×#70465 #11784255

Order 1152.

661 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×Dic₃	C₂	1	2	yes	maximal
3	C₃×C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₃×#4550	C₂	1	2	yes	maximal
6	C₃×C₈×C₄⋊C₄	—	3	1	no	maximal
7	#70465	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₃	C₄	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃²×C₄⋊C₈	C₄	1	4	yes	complemented
15	C₃×C₃⋊#95	C₄	1	2	yes	—
16	C₃×C₈×Dic₃	—	2	4	no	—
17	C₃×C₄×C₃⋊C₈	C₄	1	4	yes	complemented
18	C₂×C₃×C₄⋊C₈	—	2	1	no	—
19	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₄×C₈	—	3	1	no	—
22	C₂×C₃×C₄×C₈	—	3	2	no	—
23	C₃×C₄×C₄⋊C₄	—	3	1	no	—
24	C₂×C₈×Dic₃	C₂×C₃	1	2	yes	—
25	C₄×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
26	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
27	C₃×#336	—	3	2	no	—
28	#4550	C₂×C₃	1	2	yes	—
29	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	C₂×C₄	1	4	yes	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	—	2	2	no	—
35	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
36	C₃²×C₄²	C₂×C₄	1	2	yes	—
37	C₃²×C₄²	C₂×C₄	1	2	yes	—
38	C₃×C₄⋊Dic₃	C₈	1	8	yes	complemented
39	C₃×C₄×Dic₃	—	2	2	no	—
40	C₃×C₄×Dic₃	—	2	2	no	—
41	C₂×C₃×C₃⋊C₈	—	2	1	no	—
42	C₂×C₃×C₃⋊C₈	—	2	1	no	—
43	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
46	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
47	C₈×C₄⋊C₄	—	3	1	no	Sylow
48	C₂²×C₃×C₈	—	2	2	no	—
49	C₂²×C₃×C₈	—	3	1	no	—
50	C₂²×C₃×C₈	—	3	1	no	—
51	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
52	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
53	C₂×C₃×C₄⋊C₄	—	3	2	no	—
54	C₂×C₃×C₄²	—	2	1	no	—
55	C₂×C₃×C₄²	—	3	2	no	—
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₄⋊Dic₃	C₃×C₄	1	2	yes	—
59	C₂×C₄×Dic₃	C₂²×C₃	1	2	yes	—
60	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
62	C₃×C₄⋊C₈	—	2	4	no	—
63	C₃×C₄⋊C₈	Dic₃	1	4	yes	complemented
64	C₃×C₄⋊C₈	C₃×C₄	1	4	yes	complemented
65	C₃×C₄×C₈	—	3	4	no	—
66	C₃×C₄×C₈	—	6	4	no	—
67	C₃×#95	—	3	2	no	—
68	C₃⋊#95	C₃×C₄	1	2	yes	—
69	C₈×Dic₃	—	2	4	no	—
70	C₄×C₃⋊C₈	C₃×C₄	1	4	yes	complemented
71	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	C₄²	1	2	yes	—
74	C₂×C₃²×C₄	C₄²	1	2	yes	—
75	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₃²×C₄	#41	1	1	yes	characteristic
81	C₂×C₃²×C₄	#41	1	1	yes	characteristic
82	C₂×C₃×Dic₃	—	2	4	no	—
83	C₂×C₃×Dic₃	—	2	4	no	—
84	C₂×C₃×Dic₃	C₂×C₈	1	8	yes	—
85	C₃²×C₈	—	2	4	no	—
86	C₃×C₃⋊C₈	—	2	2	no	—
87	C₃×C₃⋊C₈	C₄⋊C₄	1	4	yes	complemented
88	C₂×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
89	C₂×C₄×C₈	—	3	1	no	—
90	C₂×C₄×C₈	—	3	2	no	—
91	C₄×C₄⋊C₄	—	3	1	no	—
92	#336	—	3	2	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	—	2	1	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	3	4	no	—
97	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₂²×Dic₃	C₂×C₃×C₄	1	4	yes	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	2	4	no	—
109	C₂×C₃×C₈	—	3	2	no	—
110	C₂×C₃×C₈	—	3	2	no	—
111	C₂×C₃×C₈	—	3	2	no	—
112	C₂×C₃×C₈	—	3	2	no	—
113	C₂×C₃×C₈	—	4	2	no	—
114	C₂×C₃×C₈	—	4	2	no	—
115	C₂×C₃×C₈	—	6	1	no	—
116	C₂×C₃×C₈	—	6	1	no	—
117	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
118	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
119	C₃×C₄⋊C₄	—	3	8	no	—
120	C₃×C₄²	—	2	2	no	—
121	C₃×C₄²	—	2	2	no	—
122	C₃×C₄²	—	6	2	no	—
123	C₃×C₄²	—	6	2	no	—
124	C₃×C₄²	C₄×D₃	1	2	yes	—
125	C₃×C₄²	C₂×Dic₃	1	2	yes	—
126	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
127	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
128	C₄⋊Dic₃	C₃×C₈	1	8	yes	complemented
129	C₄×Dic₃	—	2	2	no	—
130	C₄×Dic₃	—	2	2	no	—
131	C₂×C₃⋊C₈	—	2	1	no	—
132	C₂×C₃⋊C₈	—	2	1	no	—
133	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
134	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
135	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
136	C₂×C₃⋊C₈	C₃×Q₈	1	2	yes	—
137	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
138	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
139	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
140	C₂²×C₃²	C₂²×C₈	1	2	yes	—
141	C₂²×C₃²	#131	1	2	yes	—
142	C₃²×C₄	—	2	2	no	—
143	C₃²×C₄	C₄×C₈	1	4	yes	complemented
144	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₃²×C₄	#117	1	1	yes	characteristic
146	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
147	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
148	C₃×Dic₃	—	2	8	no	—
149	C₂²×C₈	—	3	1	no	—
150	C₂²×C₈	—	3	1	no	—
151	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
152	C₂×C₄⋊C₄	—	3	2	no	—
153	C₂×C₄²	—	3	2	no	—
154	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
155	C₄⋊C₈	C₃×Dic₃	1	4	yes	complemented
156	C₄×C₈	—	3	4	no	—
157	C₄×C₈	—	6	4	no	—
158	#95	—	3	2	no	—
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	3	8	no	—
174	C₂×C₃×C₄	—	4	2	no	—
175	C₂×C₃×C₄	—	6	4	no	—
176	C₂×C₃×C₄	—	6	4	no	—
177	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
178	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
179	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
180	C₂×C₃×C₄	C₃×C₄²	1	2	yes	—
181	C₂×C₃×C₄	C₃×C₄²	1	2	yes	—
182	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
183	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
184	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	#237	1	1	yes	characteristic
186	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
187	C₂×C₃×C₄	#239	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
193	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
194	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
195	C₂×Dic₃	—	2	4	no	—
196	C₂×Dic₃	—	2	4	no	—
197	C₂×Dic₃	C₂×C₃×C₈	1	8	yes	—
198	C₃×C₈	—	2	4	no	—
199	C₃×C₈	—	2	4	no	—
200	C₃×C₈	—	3	4	no	—
201	C₃×C₈	—	4	4	no	—
202	C₃×C₈	—	6	2	no	—
203	C₃⋊C₈	—	2	2	no	—
204	C₃⋊C₈	C₃×C₄⋊C₄	1	4	yes	complemented
205	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
207	C₂×C₃²	#405	1	1	yes	characteristic
208	C₂×C₃²	C₈×D₄	1	2	yes	—
209	C₂×C₃²	C₈×Q₈	1	2	yes	—
210	C₂²×C₄	—	3	4	no	—
211	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
212	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	3	2	no	—
217	C₂×C₈	—	3	2	no	—
218	C₂×C₈	—	3	2	no	—
219	C₂×C₈	—	3	2	no	—
220	C₂×C₈	—	6	1	no	—
221	C₂×C₈	—	6	1	no	—
222	C₂×C₈	C₂×C₃×Dic₃	1	4	yes	—
223	C₄⋊C₄	—	3	8	no	—
224	C₄²	—	6	2	no	—
225	C₄²	—	6	2	no	—
226	C₄²	C₃×C₄×D₃	1	2	yes	—
227	C₄²	C₂×C₃×Dic₃	1	2	yes	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	#887	1	1	yes	characteristic
234	C₂²×C₃	#893	1	1	yes	characteristic
235	C₂²×C₃	C₂×D₃×C₈	1	2	yes	—
236	C₂²×C₃	#907	1	2	yes	—
237	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
238	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
239	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
240	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
241	C₂²×C₃	C₂²×C₃×C₈	1	2	yes	—
242	C₂²×C₃	C₃×#131	1	2	yes	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	4	no	—
250	C₃×C₄	—	4	2	no	—
251	C₃×C₄	—	6	8	no	—
252	C₃×C₄	C₈×Dic₃	1	4	yes	complemented
253	C₃×C₄	C₃×C₄×C₈	1	4	yes	complemented
254	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
255	C₃×C₄	#892	1	1	yes	characteristic
256	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
257	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
258	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
259	C₃×C₄	C₃×#117	1	1	yes	characteristic
260	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
261	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
262	Dic₃	—	2	8	no	—
263	C₃²	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
264	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	3	8	no	—
267	C₂×C₄	—	6	4	no	—
268	C₂×C₄	—	6	4	no	—
269	C₂×C₄	C₃×D₃×C₈	1	4	yes	—
270	C₂×C₄	C₃×C₄×Dic₃	1	2	yes	—
271	C₂×C₄	C₃×C₄×Dic₃	1	2	yes	—
272	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
273	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
274	C₂×C₄	C₃×#237	1	1	yes	characteristic
275	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
276	C₂×C₄	C₃×#239	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	—	3	4	no	—
280	C₈	—	6	2	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	2	no	—
283	C₂×C₃	—	2	2	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	#4830	1	2	yes	—
287	C₂×C₃	#4834	1	2	yes	—
288	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
289	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
290	C₂×C₃	#5122	1	1	yes	characteristic
291	C₂×C₃	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
292	C₂×C₃	C₂×C₃×C₄×C₈	1	1	yes	characteristic
293	C₂×C₃	C₃×#405	1	1	yes	characteristic
294	C₂×C₃	C₃×C₈×D₄	1	2	yes	—
295	C₂×C₃	C₃×C₈×Q₈	1	2	yes	—
296	C₂²	C₃×#887	1	1	yes	characteristic
297	C₂²	C₃×#893	1	1	yes	characteristic
298	C₂²	C₂×C₃×D₃×C₈	1	2	yes	—
299	C₂²	C₃×#907	1	2	yes	—
300	C₂²	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	6	8	no	—
303	C₄	C₃×C₈×Dic₃	1	4	yes	complemented
304	C₄	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
305	C₄	C₃×#892	1	1	yes	characteristic
306	C₄	C₃×D₄×Dic₃	1	1	yes	characteristic
307	C₄	C₃×Q₈×Dic₃	1	1	yes	characteristic
308	C₃	—	2	1	no	—
309	C₃	#70465	1	1	yes	characteristic, direct product factor
310	C₃	C₃×C₈×C₄⋊C₄	1	1	yes	characteristic, complemented
311	C₂	C₃×#4830	1	2	yes	—
312	C₂	C₃×#4834	1	2	yes	—
313	C₂	C₃×Dic₃×C₄⋊C₄	1	1	yes	characteristic
314	C₂	C₂×C₃×C₈×Dic₃	1	1	yes	characteristic
315	C₂	C₃×#5122	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂×C₃	C₂×C₃×C₄×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊C₈	D₃
hypocenter	C₃	C₃×C₈×C₄⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
7. #70465	C₃	direct (N × H)
14. C₃²×C₄⋊C₈	C₄	semidirect
17. C₃×C₄×C₃⋊C₈	C₄	semidirect
38. C₃×C₄⋊Dic₃	C₈	semidirect
63. C₃×C₄⋊C₈	Dic₃	semidirect
64. C₃×C₄⋊C₈	C₃×C₄	semidirect
70. C₄×C₃⋊C₈	C₃×C₄	semidirect
87. C₃×C₃⋊C₈	C₄⋊C₄	semidirect
128. C₄⋊Dic₃	C₃×C₈	semidirect
143. C₃²×C₄	C₄×C₈	semidirect
155. C₄⋊C₈	C₃×Dic₃	semidirect
204. C₃⋊C₈	C₃×C₄⋊C₄	semidirect
252. C₃×C₄	C₈×Dic₃	semidirect
253. C₃×C₄	C₃×C₄×C₈	semidirect
263. C₃²	C₈×C₄⋊C₄	semidirect
303. C₄	C₃×C₈×Dic₃	semidirect
309. C₃	#70465	direct (N × H)
310. C₃	C₃×C₈×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄⋊C₈	2
C₂×C₃×C₈×Dic₃	2
C₃×C₄×C₄⋊Dic₃	2
C₂×C₃×C₄×C₃⋊C₈	2
C₃×#4550	2
C₃×C₈×C₄⋊C₄	3
#70465	3

Subgroups — C3×#70465 #11784255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70465 #11784255