Subgroups — #11790075

Order 1152.

569 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63630	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
5	#10589155	C₂	1	2	yes	maximal
6	#70469	—	3	1	no	maximal
7	#69485	—	3	1	no	maximal
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₄	1	2	yes	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
13	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
14	#63630	—	2	2	no	—
15	C₃×C₃⋊#95	C₄	1	1	yes	characteristic
16	C₃×C₃⋊#95	C₄	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
18	Dic₃⋊(C₃⋊C₈)	—	2	4	no	—
19	C₂×C₃×C₄⋊C₈	—	3	1	no	—
20	C₃×C₄×C₄⋊C₄	—	3	1	no	—
21	C₂×Dic₃⋊C₈	—	3	2	no	—
22	C₄×C₄⋊Dic₃	D₃	1	1	yes	characteristic
23	C₂×C₄⋊(C₃⋊C₈)	—	3	2	no	—
24	C₂×C₄⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
25	C₂×#803	—	3	1	no	—
26	C₂×#803	—	3	1	no	—
27	C₂×#803	—	6	1	no	—
28	#4550	—	3	2	no	—
29	#4524	—	3	2	no	—
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	C₃²×C₄²	D₄	1	2	yes	—
36	C₃²×C₄²	Q₈	1	2	yes	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
41	C₂×C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
42	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	2	yes	—
43	C₃×C₄⋊Dic₃	—	2	4	no	—
44	C₃×C₄×Dic₃	—	2	2	no	—
45	C₃×C₄×Dic₃	—	2	2	no	—
46	C₂×C₃×C₃⋊C₈	—	2	2	no	—
47	C₂×C₃×C₃⋊C₈	—	2	2	no	—
48	C₂×C₃×C₃⋊C₈	—	2	2	no	—
49	#1940	—	9	1	no	Sylow
50	C₂²×C₃×C₈	—	3	2	no	—
51	C₂×C₃×C₄⋊C₄	—	3	2	no	—
52	C₂×C₃×C₄²	—	2	1	no	—
53	C₂×C₃×C₄²	—	3	2	no	—
54	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₂×C₄⋊Dic₃	Dic₃	1	2	yes	—
57	C₂×C₄×Dic₃	C₂×D₃	1	2	yes	—
58	C₂²×C₃⋊C₈	—	3	1	no	—
59	C₂²×C₃⋊C₈	—	3	1	no	—
60	C₂²×C₃⋊C₈	—	3	1	no	—
61	C₂²×C₃⋊C₈	—	3	1	no	—
62	C₂²×C₃⋊C₈	—	6	1	no	—
63	C₂²×C₃⋊C₈	—	6	1	no	—
64	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
65	C₃×C₄⋊C₈	—	6	2	no	—
66	C₃×#95	—	3	1	no	—
67	C₃×#95	—	3	1	no	—
68	C₃⋊#95	Dic₃	1	1	yes	characteristic
69	C₃⋊#95	Dic₃	1	1	yes	characteristic
70	Dic₃⋊C₈	—	6	4	no	—
71	C₄⋊(C₃⋊C₈)	—	2	2	no	—
72	C₄⋊(C₃⋊C₈)	—	6	4	no	—
73	#803	—	6	4	no	—
74	#803	—	6	2	no	—
75	#803	—	6	2	no	—
76	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	—	2	2	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
80	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
81	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₃²×C₄	#41	1	1	yes	characteristic
86	C₂×C₃²×C₄	#41	1	1	yes	characteristic
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	—	2	2	no	—
92	C₂×C₃×Dic₃	M₁₆	1	4	yes	—
93	C₃⋊(C₃⋊C₈)	—	2	2	no	—
94	C₃⋊(C₃⋊C₈)	—	2	2	no	—
95	C₃×C₃⋊C₈	—	4	2	no	—
96	C₂×C₄⋊C₈	—	3	1	no	—
97	C₂×C₄⋊C₈	—	9	2	no	—
98	C₂×#97	—	9	1	no	—
99	C₄×C₄⋊C₄	—	3	1	no	—
100	#336	—	9	2	no	—
101	C₂²×C₃×C₄	—	2	1	no	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	2	1	no	—
104	C₂²×C₃×C₄	—	3	2	no	—
105	C₂²×C₃×C₄	—	3	2	no	—
106	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂²×Dic₃	C₂×Dic₃	1	2	yes	—
113	C₂²×Dic₃	C₂×Dic₃	1	2	yes	—
114	C₂×C₃×C₈	—	6	2	no	—
115	C₂×C₃×C₈	—	6	2	no	—
116	C₂×C₃×C₈	—	6	2	no	—
117	C₃×C₄⋊C₄	—	6	4	no	—
118	C₃×C₄²	—	2	2	no	—
119	C₃×C₄²	—	2	2	no	—
120	C₃×C₄²	—	6	2	no	—
121	C₃×C₄²	—	6	2	no	—
122	C₃×C₄²	Dic₆	1	2	yes	—
123	C₃×C₄²	Dic₆	1	2	yes	—
124	C₃×C₄²	D₁₂	1	2	yes	—
125	C₃×C₄²	#67	1	2	yes	—
126	C₄⋊Dic₃	—	2	4	no	—
127	C₄×Dic₃	—	2	2	no	—
128	C₄×Dic₃	—	2	2	no	—
129	C₂×C₃⋊C₈	—	2	2	no	—
130	C₂×C₃⋊C₈	—	2	2	no	—
131	C₂×C₃⋊C₈	—	2	2	no	—
132	C₂×C₃⋊C₈	—	3	2	no	—
133	C₂×C₃⋊C₈	—	3	2	no	—
134	C₂×C₃⋊C₈	—	3	2	no	—
135	C₂×C₃⋊C₈	—	3	2	no	—
136	C₂×C₃⋊C₈	—	6	2	no	—
137	C₂×C₃⋊C₈	—	6	2	no	—
138	C₂×C₃⋊C₈	—	6	2	no	—
139	C₂×C₃⋊C₈	—	6	2	no	—
140	C₂×C₃⋊C₈	—	6	2	no	—
141	C₂×C₃⋊C₈	—	6	2	no	—
142	C₂×C₃⋊C₈	—	6	1	no	—
143	C₂×C₃⋊C₈	—	6	1	no	—
144	C₂×C₃⋊C₈	—	6	1	no	—
145	C₂×C₃⋊C₈	—	6	1	no	—
146	C₂×C₃⋊C₈	—	6	1	no	—
147	C₂×C₃⋊C₈	—	6	1	no	—
148	C₂×C₃⋊C₈	—	6	1	no	—
149	C₂×C₃⋊C₈	—	6	1	no	—
150	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
153	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
154	C₂²×C₃²	#131	1	2	yes	—
155	C₃²×C₄	—	2	2	no	—
156	C₃²×C₄	—	2	2	no	—
157	C₃²×C₄	#121	1	1	yes	characteristic
158	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
159	C₃²×C₄	#125	1	1	yes	characteristic
160	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
161	C₃×Dic₃	—	2	4	no	—
162	C₃×Dic₃	—	4	2	no	—
163	C₂²×C₈	—	3	2	no	—
164	C₂²×C₈	—	9	1	no	—
165	C₂²×C₈	—	9	1	no	—
166	C₂×C₄⋊C₄	—	3	2	no	—
167	C₂×C₄²	—	3	2	no	—
168	C₂×C₄²	D₃²	1	1	yes	characteristic
169	C₄⋊C₈	—	6	2	no	—
170	C₄⋊C₈	—	18	4	no	—
171	#97	—	18	2	no	—
172	#95	—	3	1	no	—
173	#95	—	3	1	no	—
174	C₂³×C₃	—	2	1	no	—
175	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
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	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	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	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	3	4	no	—
190	C₂×C₃×C₄	—	4	2	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	—	6	2	no	—
193	C₂×C₃×C₄	—	6	2	no	—
194	C₂×C₃×C₄	—	6	2	no	—
195	C₂×C₃×C₄	—	6	2	no	—
196	C₂×C₃×C₄	—	6	2	no	—
197	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
198	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
199	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
200	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
201	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
204	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
206	C₂×C₃×C₄	#237	1	1	yes	characteristic
207	C₂×C₃×C₄	#237	1	1	yes	characteristic
208	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
209	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
210	C₂×C₃×C₄	#239	1	1	yes	characteristic
211	C₂×C₃×C₄	#239	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
213	C₂×Dic₃	—	2	2	no	—
214	C₂×Dic₃	—	2	2	no	—
215	C₂×Dic₃	—	2	2	no	—
216	C₂×Dic₃	—	2	2	no	—
217	C₂×Dic₃	—	2	2	no	—
218	C₂×Dic₃	#208	1	4	yes	—
219	C₃×C₈	—	12	2	no	—
220	C₃⋊C₈	—	4	2	no	—
221	C₃⋊C₈	—	6	4	no	—
222	C₃⋊C₈	—	6	4	no	—
223	C₃⋊C₈	—	6	2	no	—
224	C₃⋊C₈	—	6	2	no	—
225	C₃⋊C₈	—	6	2	no	—
226	C₃⋊C₈	—	6	2	no	—
227	C₂×C₃²	#389	1	1	yes	characteristic
228	C₂×C₃²	#423	1	1	yes	characteristic
229	C₂×C₃²	#424	1	1	yes	characteristic
230	C₂×C₃²	#435	1	2	yes	—
231	C₂×C₃²	#446	1	2	yes	—
232	C₂²×C₄	—	3	2	no	—
233	C₂²×C₄	—	3	2	no	—
234	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
235	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
237	C₂×C₈	—	6	2	no	—
238	C₂×C₈	—	6	2	no	—
239	C₂×C₈	—	6	2	no	—
240	C₂×C₈	—	9	2	no	—
241	C₂×C₈	—	9	2	no	—
242	C₂×C₈	—	18	1	no	—
243	C₂×C₈	—	18	1	no	—
244	C₂×C₈	—	18	1	no	—
245	C₂×C₈	—	18	1	no	—
246	C₄⋊C₄	—	6	4	no	—
247	C₄²	—	6	2	no	—
248	C₄²	—	6	2	no	—
249	C₄²	#626	1	2	yes	—
250	C₄²	#627	1	2	yes	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	#887	1	1	yes	characteristic
257	C₂²×C₃	#893	1	1	yes	characteristic
258	C₂²×C₃	#901	1	2	yes	—
259	C₂²×C₃	D₃×M₁₆	1	2	yes	—
260	C₂²×C₃	C₂×#208	1	2	yes	—
261	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
262	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
263	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
264	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
265	C₂²×C₃	#948	1	2	yes	—
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	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	4	2	no	—
275	C₃×C₄	—	4	2	no	—
276	C₃×C₄	—	6	4	no	—
277	C₃×C₄	—	12	2	no	—
278	C₃×C₄	#888	1	1	yes	characteristic
279	C₃×C₄	#888	1	1	yes	characteristic
280	C₃×C₄	#890	1	1	yes	characteristic
281	C₃×C₄	#890	1	1	yes	characteristic
282	C₃×C₄	#895	1	1	yes	characteristic
283	C₃×C₄	#897	1	1	yes	characteristic
284	C₃×C₄	#938	1	1	yes	characteristic
285	C₃×C₄	#946	1	1	yes	characteristic
286	Dic₃	—	2	4	no	—
287	Dic₃	—	4	2	no	—
288	C₃²	#1940	1	1	yes	characteristic, complemented, Sylow
289	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	3	4	no	—
293	C₂×C₄	—	6	2	no	—
294	C₂×C₄	—	6	2	no	—
295	C₂×C₄	—	6	2	no	—
296	C₂×C₄	—	6	2	no	—
297	C₂×C₄	—	6	2	no	—
298	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
299	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
300	C₂×C₄	#3796	1	1	yes	characteristic
301	C₂×C₄	#3796	1	1	yes	characteristic
302	C₂×C₄	#3797	1	1	yes	characteristic
303	C₂×C₄	#3802	1	1	yes	characteristic
304	C₂×C₄	C₂×#626	1	1	yes	characteristic
305	C₂×C₄	C₂×#627	1	1	yes	characteristic
306	C₈	—	12	2	no	—
307	C₈	—	18	2	no	—
308	C₈	—	18	2	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	#4828	1	2	yes	—
315	C₂×C₃	#4836	1	2	yes	—
316	C₂×C₃	#4924	1	1	yes	characteristic
317	C₂×C₃	#4972	1	1	yes	characteristic
318	C₂×C₃	#4984	1	1	yes	characteristic
319	C₂×C₃	#4999	1	1	yes	characteristic
320	C₂×C₃	#5011	1	2	yes	—
321	C₂×C₃	#5024	1	2	yes	—
322	C₂×C₃	#5102	1	1	yes	characteristic
323	C₂×C₃	#5118	1	1	yes	characteristic
324	C₂²	D₃×#208	1	2	yes	—
325	C₂²	#63814	1	2	yes	—
326	C₂²	#63843	1	1	yes	characteristic
327	C₂²	#63898	1	1	yes	characteristic
328	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	6	4	no	—
332	C₄	—	12	2	no	—
333	C₄	#63871	1	1	yes	characteristic
334	C₄	#63872	1	1	yes	characteristic
335	C₄	#63916	1	1	yes	characteristic
336	C₄	#63919	1	1	yes	characteristic
337	C₃	—	2	1	no	—
338	C₃	#69485	1	1	yes	characteristic, complemented
339	C₃	#70469	1	1	yes	characteristic, complemented
340	C₂	#10590252	1	2	yes	—
341	C₂	#10590330	1	2	yes	—
342	C₂	#10591115	1	1	yes	characteristic
343	C₂	#10591116	1	1	yes	characteristic
344	C₂	#10591357	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃²
commutator, socle, prime radical	C₂³×C₃²	C₂²×C₄
Fitting	C₂×C₃²×C₄²	C₂²
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1940
LCS	G ▷ C₂³×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
288. C₃²	#1940	semidirect
338. C₃	#69485	semidirect
339. C₃	#70469	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63630	2
C₃×C₄×C₄⋊Dic₃	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₂×Dic₃⋊(C₃⋊C₈)	2
#10589155	2
#70469	3
#69485	3