Subgroups — C₃×#70469 #11790056

Order 1152.

565 conjugacy classes of subgroups.

The subgroup lattice has 341 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×Dic₃⋊C₈	C₂	1	2	yes	maximal
3	C₃×C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic
5	C₃×#4550	C₂	1	2	yes	maximal
6	C₃×#1940	—	3	1	no	maximal
7	#70469	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	—	2	2	no	—
15	C₃×C₃⋊#95	C₄	1	1	yes	characteristic
16	C₃×C₃⋊#95	C₄	1	1	yes	characteristic
17	C₃×Dic₃⋊C₈	—	2	4	no	—
18	C₃×#803	—	2	2	no	—
19	C₂×C₃×C₄⋊C₈	—	2	1	no	—
20	C₂×C₃×C₄⋊C₈	—	3	2	no	—
21	C₂×C₃×C₄⋊C₈	D₃	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#97	—	3	1	no	—
24	C₃×C₄×C₄⋊C₄	—	3	1	no	—
25	C₂×Dic₃⋊C₈	C₂×C₃	1	2	yes	—
26	C₄×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
27	C₂×#803	C₂×C₃	1	1	yes	characteristic
28	C₃×#336	—	3	2	no	—
29	#4550	C₂×C₃	1	2	yes	—
30	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
34	C₂²×C₃×Dic₃	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	—	2	2	no	—
36	C₂×C₃²×C₈	—	2	2	no	—
37	C₂×C₃²×C₈	—	2	2	no	—
38	C₃²×C₄²	D₄	1	2	yes	—
39	C₃²×C₄²	Q₈	1	2	yes	—
40	C₃×C₄⋊Dic₃	—	2	4	no	—
41	C₃×C₄×Dic₃	—	2	2	no	—
42	C₃×C₄×Dic₃	—	2	2	no	—
43	C₂×C₃×C₃⋊C₈	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃×C₃⋊C₈	—	2	1	no	—
46	C₂×C₃×C₃⋊C₈	—	2	1	no	—
47	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
48	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
49	#1940	—	3	1	no	Sylow
50	C₂²×C₃×C₈	—	2	2	no	—
51	C₂²×C₃×C₈	—	3	1	no	—
52	C₂²×C₃×C₈	—	3	1	no	—
53	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
54	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
55	C₂×C₃×C₄⋊C₄	—	3	2	no	—
56	C₂×C₃×C₄²	—	2	1	no	—
57	C₂×C₃×C₄²	—	3	2	no	—
58	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄⋊Dic₃	C₃×C₄	1	2	yes	—
61	C₂×C₄×Dic₃	C₂²×C₃	1	2	yes	—
62	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₃×C₄⋊C₈	—	2	4	no	—
65	C₃×C₄⋊C₈	—	2	2	no	—
66	C₃×C₄⋊C₈	—	2	2	no	—
67	C₃×C₄⋊C₈	—	6	4	no	—
68	C₃×#97	—	6	2	no	—
69	C₃×#95	—	3	1	no	—
70	C₃×#95	—	3	1	no	—
71	C₃⋊#95	C₃×C₄	1	1	yes	characteristic
72	C₃⋊#95	C₃×C₄	1	1	yes	characteristic
73	Dic₃⋊C₈	—	2	4	no	—
74	#803	—	2	2	no	—
75	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	—	2	2	no	—
77	C₂×C₃²×C₄	—	2	2	no	—
78	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
79	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
80	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₃²×C₄	#41	1	1	yes	characteristic
85	C₂×C₃²×C₄	#41	1	1	yes	characteristic
86	C₂×C₃×Dic₃	—	2	2	no	—
87	C₂×C₃×Dic₃	—	2	2	no	—
88	C₂×C₃×Dic₃	—	2	2	no	—
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₂×C₃×Dic₃	—	2	2	no	—
91	C₂×C₃×Dic₃	M₁₆	1	4	yes	—
92	C₃²×C₈	—	4	2	no	—
93	C₃×C₃⋊C₈	—	2	2	no	—
94	C₃×C₃⋊C₈	—	2	2	no	—
95	C₂×C₄⋊C₈	—	3	2	no	—
96	C₂×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
97	C₂×#97	—	3	1	no	—
98	C₄×C₄⋊C₄	—	3	1	no	—
99	#336	—	3	2	no	—
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	—	2	1	no	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	3	2	no	—
104	C₂²×C₃×C₄	—	3	2	no	—
105	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
111	C₂²×Dic₃	C₂×C₃×C₄	1	2	yes	—
112	C₂²×Dic₃	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₃×C₈	—	2	4	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	—	3	2	no	—
121	C₂×C₃×C₈	—	3	2	no	—
122	C₂×C₃×C₈	—	4	2	no	—
123	C₂×C₃×C₈	—	4	2	no	—
124	C₂×C₃×C₈	—	6	1	no	—
125	C₂×C₃×C₈	—	6	1	no	—
126	C₂×C₃×C₈	—	6	1	no	—
127	C₂×C₃×C₈	—	6	1	no	—
128	C₃×C₄⋊C₄	—	6	4	no	—
129	C₃×C₄²	—	2	2	no	—
130	C₃×C₄²	—	2	2	no	—
131	C₃×C₄²	—	6	2	no	—
132	C₃×C₄²	—	6	2	no	—
133	C₃×C₄²	Dic₆	1	2	yes	—
134	C₃×C₄²	#67	1	2	yes	—
135	C₃×C₄²	C₃×D₄	1	2	yes	—
136	C₃×C₄²	C₃×Q₈	1	2	yes	—
137	C₄⋊Dic₃	—	2	4	no	—
138	C₄×Dic₃	—	2	2	no	—
139	C₄×Dic₃	—	2	2	no	—
140	C₂×C₃⋊C₈	—	2	1	no	—
141	C₂×C₃⋊C₈	—	2	1	no	—
142	C₂×C₃⋊C₈	—	2	1	no	—
143	C₂×C₃⋊C₈	—	2	1	no	—
144	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
145	C₂×C₃⋊C₈	C₃×Q₈	1	2	yes	—
146	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
148	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
149	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
150	C₂²×C₃²	#131	1	2	yes	—
151	C₃²×C₄	—	2	2	no	—
152	C₃²×C₄	—	2	2	no	—
153	C₃²×C₄	#121	1	1	yes	characteristic
154	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
155	C₃²×C₄	#125	1	1	yes	characteristic
156	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
157	C₃×Dic₃	—	2	4	no	—
158	C₃×Dic₃	—	4	2	no	—
159	C₂²×C₈	—	3	1	no	—
160	C₂²×C₈	—	3	1	no	—
161	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
162	C₂×C₄⋊C₄	—	3	2	no	—
163	C₂×C₄²	—	3	2	no	—
164	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	6	4	no	—
167	#97	—	6	2	no	—
168	#95	—	3	1	no	—
169	#95	—	3	1	no	—
170	C₂³×C₃	—	2	1	no	—
171	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	3	4	no	—
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	—	6	2	no	—
189	C₂×C₃×C₄	—	6	2	no	—
190	C₂×C₃×C₄	—	6	2	no	—
191	C₂×C₃×C₄	—	6	2	no	—
192	C₂×C₃×C₄	—	6	2	no	—
193	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
194	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
195	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
196	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
197	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
198	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	#237	1	1	yes	characteristic
200	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
201	C₂×C₃×C₄	#239	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
207	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
208	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
209	C₂×Dic₃	—	2	2	no	—
210	C₂×Dic₃	—	2	2	no	—
211	C₂×Dic₃	—	2	2	no	—
212	C₂×Dic₃	—	2	2	no	—
213	C₂×Dic₃	—	2	2	no	—
214	C₂×Dic₃	C₃×M₁₆	1	4	yes	—
215	C₃×C₈	—	4	4	no	—
216	C₃×C₈	—	4	2	no	—
217	C₃×C₈	—	4	2	no	—
218	C₃×C₈	—	6	2	no	—
219	C₃×C₈	—	6	2	no	—
220	C₃⋊C₈	—	2	2	no	—
221	C₃⋊C₈	—	2	2	no	—
222	C₂×C₃²	#389	1	1	yes	characteristic
223	C₂×C₃²	#423	1	1	yes	characteristic
224	C₂×C₃²	#424	1	1	yes	characteristic
225	C₂×C₃²	#435	1	2	yes	—
226	C₂×C₃²	#446	1	2	yes	—
227	C₂²×C₄	—	3	2	no	—
228	C₂²×C₄	—	3	2	no	—
229	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
230	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	3	2	no	—
236	C₂×C₈	—	3	2	no	—
237	C₂×C₈	—	6	1	no	—
238	C₂×C₈	—	6	1	no	—
239	C₂×C₈	—	6	1	no	—
240	C₂×C₈	—	6	1	no	—
241	C₄⋊C₄	—	6	4	no	—
242	C₄²	—	6	2	no	—
243	C₄²	—	6	2	no	—
244	C₄²	C₃×Dic₆	1	2	yes	—
245	C₄²	C₃×#67	1	2	yes	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	2	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	#887	1	1	yes	characteristic
252	C₂²×C₃	#893	1	1	yes	characteristic
253	C₂²×C₃	#901	1	2	yes	—
254	C₂²×C₃	D₃×M₁₆	1	2	yes	—
255	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
256	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
257	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
258	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
259	C₂²×C₃	C₂×C₃×M₁₆	1	2	yes	—
260	C₂²×C₃	C₃×#131	1	2	yes	—
261	C₃×C₄	—	2	4	no	—
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	4	2	no	—
271	C₃×C₄	—	6	4	no	—
272	C₃×C₄	—	12	2	no	—
273	C₃×C₄	#888	1	1	yes	characteristic
274	C₃×C₄	#890	1	1	yes	characteristic
275	C₃×C₄	#938	1	1	yes	characteristic
276	C₃×C₄	#946	1	1	yes	characteristic
277	C₃×C₄	C₃×#121	1	1	yes	characteristic
278	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
279	C₃×C₄	C₃×#125	1	1	yes	characteristic
280	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
281	Dic₃	—	2	4	no	—
282	Dic₃	—	4	2	no	—
283	C₃²	#1940	1	1	yes	characteristic, complemented, Sylow
284	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	3	4	no	—
288	C₂×C₄	—	6	2	no	—
289	C₂×C₄	—	6	2	no	—
290	C₂×C₄	—	6	2	no	—
291	C₂×C₄	—	6	2	no	—
292	C₂×C₄	—	6	2	no	—
293	C₂×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
294	C₂×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
295	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
296	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
297	C₂×C₄	C₃×#237	1	1	yes	characteristic
298	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
299	C₂×C₄	C₃×#239	1	1	yes	characteristic
300	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
301	C₈	—	4	2	no	—
302	C₈	—	6	2	no	—
303	C₈	—	6	2	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	2	2	no	—
306	C₂×C₃	—	2	2	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	#4828	1	2	yes	—
310	C₂×C₃	#4836	1	2	yes	—
311	C₂×C₃	#4972	1	1	yes	characteristic
312	C₂×C₃	#5102	1	1	yes	characteristic
313	C₂×C₃	#5118	1	1	yes	characteristic
314	C₂×C₃	C₃×#389	1	1	yes	characteristic
315	C₂×C₃	C₃×#423	1	1	yes	characteristic
316	C₂×C₃	C₃×#424	1	1	yes	characteristic
317	C₂×C₃	C₃×#435	1	2	yes	—
318	C₂×C₃	C₃×#446	1	2	yes	—
319	C₂²	C₃×#887	1	1	yes	characteristic
320	C₂²	C₃×#893	1	1	yes	characteristic
321	C₂²	C₃×#901	1	2	yes	—
322	C₂²	C₃×D₃×M₁₆	1	2	yes	—
323	C₂²	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	6	4	no	—
327	C₄	—	12	2	no	—
328	C₄	C₃×#888	1	1	yes	characteristic
329	C₄	C₃×#890	1	1	yes	characteristic
330	C₄	C₃×#938	1	1	yes	characteristic
331	C₄	C₃×#946	1	1	yes	characteristic
332	C₃	—	2	1	no	—
333	C₃	#70469	1	1	yes	characteristic, direct product factor
334	C₃	C₃×#1940	1	1	yes	characteristic, complemented
335	C₂	C₃×#4828	1	2	yes	—
336	C₂	C₃×#4836	1	2	yes	—
337	C₂	C₃×#4972	1	1	yes	characteristic
338	C₂	C₃×#5102	1	1	yes	characteristic
339	C₂	C₃×#5118	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₃
commutator	C₂³×C₃	C₂²×C₃×C₄
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×#1940
LCS	G ▷ C₂³×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. #70469	C₃	direct (N × H)
283. C₃²	#1940	semidirect
333. C₃	#70469	direct (N × H)
334. C₃	C₃×#1940	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×Dic₃⋊C₈	2
C₃×C₄×C₄⋊Dic₃	2
C₂×C₃×#803	2
C₃×#4550	2
C₃×#1940	3
#70469	3

Subgroups — C3×#70469 #11790056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70469 #11790056