Subgroups — #14907285

Order 2816.

530 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₁₁	C₂	1	1	yes	maximal, characteristic
2	#13070525	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070523	C₂	1	2	yes	maximal, complemented
4	#13070522	C₂	1	2	yes	maximal, complemented
5	C₁₆×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070518	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070370	C₂	1	1	yes	maximal, characteristic
8	#13070370	C₂	1	2	yes	maximal
9	#13070209	C₂	1	2	yes	maximal
10	C₄×#66862	C₂	1	1	yes	maximal, characteristic
11	C₁₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×D₁₁	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
14	Q₈×C₁₁⋊C₈	C₄	1	1	yes	characteristic
15	C₂×#66862	C₂²	1	1	yes	characteristic
16	C₂×#66862	C₂²	1	2	yes	—
17	C₂×#66862	C₂²	1	1	yes	characteristic
18	D₁₁×C₄⋊C₈	C₄	1	1	yes	characteristic
19	D₁₁×C₄⋊C₈	C₄	1	2	yes	—
20	#10622443	C₄	1	1	yes	characteristic
21	#10622443	C₄	1	2	yes	—
22	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₁₁	C₂²	1	2	yes	—
24	C₈×Dic₂₂	C₂²	1	1	yes	characteristic
25	C₈×Dic₂₂	C₂²	1	2	yes	—
26	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₁₁×#346	C₂²	1	2	yes	—
29	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
30	#10622140	C₂²	1	1	yes	characteristic
31	#10622140	C₂²	1	1	yes	characteristic
32	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
33	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
34	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
35	C₁₆×Dic₁₁	C₂²	1	2	yes	—
36	C₁₆×Dic₁₁	C₂²	1	2	yes	—
37	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
39	#10622099	C₂²	1	2	yes	—
40	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₁₁	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
43	Q₈×Dic₁₁	C₈	1	1	yes	characteristic
44	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×D₁₁	C₈	1	1	yes	characteristic
48	C₄⋊C₄×D₁₁	C₈	1	2	yes	—
49	#66940	C₈	1	2	yes	—
50	#66940	C₈	1	1	yes	characteristic
51	C₄²×D₁₁	C₂×C₄	1	1	yes	characteristic
52	C₄²×D₁₁	C₂×C₄	1	2	yes	—
53	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
58	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
59	C₁₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₁	C₂³	1	2	yes	—
62	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
63	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
64	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
65	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₁	C₂³	1	2	yes	—
68	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
69	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
71	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
72	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
73	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
74	C₄×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
75	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
76	#66862	—	2	2	no	—
77	#66862	—	2	4	no	—
78	#66862	Q₈	1	4	yes	complemented
79	Q₈×M₃₂	—	11	1	no	maximal, Sylow
80	C₂×Q₈×C₁₁	C₂×C₈	1	1	yes	characteristic
81	Q₈×D₁₁	—	2	4	no	—
82	C₂×C₄×D₁₁	C₂×C₈	1	2	yes	—
83	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₁	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
88	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
92	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
93	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
97	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
98	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
99	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
100	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
101	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
103	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
104	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
107	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₈×D₁₁	—	2	2	no	—
111	C₈×D₁₁	—	2	4	no	—
112	C₈×D₁₁	C₂×Q₈	1	2	yes	—
113	C₈×D₁₁	#41	1	2	yes	—
114	C₁₁×C₁₆	—	2	1	no	—
115	C₁₁×C₁₆	—	2	2	no	—
116	C₁₁×C₁₆	C₂×Q₈	1	2	yes	complemented
117	C₁₁⋊C₁₆	—	2	1	no	—
118	C₁₁⋊C₁₆	—	2	2	no	—
119	C₁₁⋊C₁₆	C₂×Q₈	1	2	yes	complemented
120	C₂×C₈×Q₈	—	11	1	no	—
121	#2183	—	11	1	no	—
122	#2183	—	11	2	no	—
123	#2183	—	11	2	no	—
124	#2183	D₁₁	1	1	yes	characteristic, complemented
125	Q₈×C₁₆	—	11	1	no	—
126	Q₈×C₁₆	—	11	1	no	—
127	#2150	—	11	2	no	—
128	#2150	—	11	1	no	—
129	C₄×M₃₂	—	11	2	no	—
130	C₄×M₃₂	—	11	1	no	—
131	C₂²×D₁₁	C₂²×C₈	1	1	yes	characteristic
132	Q₈×C₁₁	—	2	2	no	—
133	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
138	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
139	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
140	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
141	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
143	C₄×D₁₁	—	2	2	no	—
144	C₄×D₁₁	—	2	4	no	—
145	C₄×D₁₁	—	4	1	no	—
146	C₄×D₁₁	—	4	2	no	—
147	C₄×D₁₁	C₄×Q₈	1	2	yes	—
148	C₄×D₁₁	#131	1	2	yes	—
149	Dic₂₂	—	2	4	no	—
150	Dic₂₂	M₃₂	1	4	yes	complemented
151	C₈×C₁₁	—	2	1	no	—
152	C₈×C₁₁	—	2	2	no	—
153	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
154	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
155	C₁₁⋊C₈	—	2	2	no	—
156	C₁₁⋊C₈	—	2	1	no	—
157	C₁₁⋊C₈	C₄×Q₈	1	2	yes	—
158	C₂×C₄×Q₈	—	11	1	no	—
159	C₂×M₃₂	—	11	1	no	—
160	C₂×M₃₂	—	11	2	no	—
161	C₂×M₃₂	—	11	1	no	—
162	C₈×Q₈	—	11	1	no	—
163	C₈×Q₈	—	11	2	no	—
164	C₈×Q₈	—	11	1	no	—
165	C₈×Q₈	—	11	1	no	—
166	C₈×Q₈	—	11	2	no	—
167	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
168	C₂×C₄⋊C₈	—	11	2	no	—
169	C₂×C₄⋊C₈	—	11	1	no	—
170	C₂×C₄×C₈	—	11	2	no	—
171	C₂×C₄×C₈	—	11	1	no	—
172	C₄⋊C₁₆	—	11	2	no	—
173	C₄⋊C₁₆	—	11	2	no	—
174	C₄⋊C₁₆	—	11	2	no	—
175	C₄⋊C₁₆	—	11	1	no	—
176	C₄⋊C₁₆	—	11	2	no	—
177	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
178	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
179	#346	—	11	1	no	—
180	#346	—	11	1	no	—
181	#346	—	11	2	no	—
182	#346	C₂×D₁₁	1	2	yes	—
183	C₄×C₁₆	—	11	2	no	—
184	C₄×C₁₆	—	11	2	no	—
185	C₄×C₁₆	—	11	1	no	—
186	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
187	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
188	C₂×D₁₁	—	2	1	no	—
189	C₂×D₁₁	—	2	1	no	—
190	C₂×D₁₁	C₈×Q₈	1	2	yes	—
191	C₄×C₁₁	—	2	2	no	—
192	C₄×C₁₁	—	2	1	no	—
193	C₄×C₁₁	—	2	2	no	—
194	C₄×C₁₁	C₂×M₃₂	1	2	yes	—
195	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
196	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
197	Dic₁₁	—	2	1	no	—
198	Dic₁₁	—	2	2	no	—
199	Dic₁₁	—	2	1	no	—
200	Dic₁₁	C₈×Q₈	1	2	yes	—
201	Dic₁₁	C₂×M₃₂	1	4	yes	—
202	C₂²×Q₈	—	11	1	no	—
203	C₂²×C₈	—	11	1	no	—
204	C₂²×C₈	—	11	2	no	—
205	C₂²×C₈	—	11	1	no	—
206	C₄×Q₈	—	11	1	no	—
207	C₄×Q₈	—	11	1	no	—
208	C₄×Q₈	—	11	2	no	—
209	C₄×Q₈	—	11	1	no	—
210	C₄×Q₈	—	11	2	no	—
211	C₄×Q₈	C₄×D₁₁	1	1	yes	characteristic
212	C₂×C₄⋊C₄	—	11	2	no	—
213	C₂×C₄⋊C₄	—	11	1	no	—
214	C₂×C₄²	—	11	2	no	—
215	C₂×C₄²	—	11	1	no	—
216	M₃₂	—	11	4	no	—
217	M₃₂	—	22	4	no	—
218	M₃₂	—	22	2	no	—
219	C₂×C₁₆	—	11	1	no	—
220	C₂×C₁₆	—	11	1	no	—
221	C₂×C₁₆	—	11	2	no	—
222	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
223	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
224	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
225	C₄⋊C₈	—	11	2	no	—
226	C₄⋊C₈	—	11	2	no	—
227	C₄⋊C₈	—	11	2	no	—
228	C₄⋊C₈	—	11	1	no	—
229	C₄⋊C₈	—	11	2	no	—
230	C₄⋊C₈	C₄×D₁₁	1	2	yes	—
231	C₄⋊C₈	C₄×D₁₁	1	1	yes	characteristic
232	C₄×C₈	—	11	2	no	—
233	C₄×C₈	—	11	1	no	—
234	C₄×C₈	—	11	2	no	—
235	C₄×C₈	—	11	1	no	—
236	C₄×C₈	—	11	2	no	—
237	C₄×C₈	—	11	1	no	—
238	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₁	1	2	yes	—
240	C₂×C₁₁	C₂×C₈×Q₈	1	1	yes	characteristic
241	C₂×C₁₁	C₂²×M₃₂	1	1	yes	characteristic
242	C₂×C₁₁	#3407	1	1	yes	characteristic
243	D₁₁	—	2	2	no	—
244	C₂×Q₈	—	11	1	no	—
245	C₂×Q₈	—	11	2	no	—
246	C₂×Q₈	—	22	4	no	—
247	C₂×Q₈	C₈×D₁₁	1	1	yes	characteristic
248	C₂²×C₄	—	11	1	no	—
249	C₂²×C₄	—	11	2	no	—
250	C₂²×C₄	—	11	1	no	—
251	C₂²×C₄	—	11	2	no	—
252	C₂²×C₄	—	11	1	no	—
253	C₂×C₈	—	11	1	no	—
254	C₂×C₈	—	11	2	no	—
255	C₂×C₈	—	11	2	no	—
256	C₂×C₈	—	11	2	no	—
257	C₂×C₈	—	11	1	no	—
258	C₂×C₈	—	22	2	no	—
259	C₂×C₈	—	22	4	no	—
260	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
262	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
263	C₄⋊C₄	—	11	2	no	—
264	C₄⋊C₄	—	11	2	no	—
265	C₄⋊C₄	—	11	2	no	—
266	C₄⋊C₄	—	11	2	no	—
267	C₄⋊C₄	—	11	1	no	—
268	C₄⋊C₄	C₈×D₁₁	1	2	yes	—
269	C₄⋊C₄	C₈×D₁₁	1	1	yes	characteristic
270	C₄²	—	11	2	no	—
271	C₄²	—	11	2	no	—
272	C₄²	—	11	1	no	—
273	C₄²	—	11	1	no	—
274	C₄²	—	11	2	no	—
275	C₄²	—	11	1	no	—
276	C₄²	C₂×C₄×D₁₁	1	2	yes	—
277	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	—	11	2	no	—
281	C₁₆	—	22	1	no	—
282	C₁₆	—	22	2	no	—
283	C₁₆	Q₈×D₁₁	1	2	yes	complemented
284	C₁₁	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
285	C₂³	—	11	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	11	4	no	—
288	Q₈	—	22	4	no	—
289	C₂×C₄	—	11	1	no	—
290	C₂×C₄	—	11	2	no	—
291	C₂×C₄	—	11	2	no	—
292	C₂×C₄	—	11	2	no	—
293	C₂×C₄	—	11	2	no	—
294	C₂×C₄	—	11	1	no	—
295	C₂×C₄	—	11	1	no	—
296	C₂×C₄	—	22	4	no	—
297	C₂×C₄	—	22	2	no	—
298	C₂×C₄	—	44	2	no	—
299	C₂×C₄	—	44	1	no	—
300	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
301	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
302	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
303	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	—	11	2	no	—
308	C₈	—	22	2	no	—
309	C₈	—	22	1	no	—
310	C₈	C₂×Q₈×D₁₁	1	1	yes	characteristic
311	C₈	#41×D₁₁	1	1	yes	characteristic
312	C₂²	—	11	2	no	—
313	C₂²	—	22	1	no	—
314	C₂²	—	22	1	no	—
315	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	11	2	no	—
320	C₄	—	11	4	no	—
321	C₄	—	22	1	no	—
322	C₄	—	22	2	no	—
323	C₄	—	22	1	no	—
324	C₄	D₁₁×M₃₂	1	2	yes	—
325	C₄	C₄×Q₈×D₁₁	1	1	yes	characteristic
326	C₄	D₁₁×#131	1	1	yes	characteristic
327	C₂	—	22	2	no	—
328	C₂	C₈×Q₈×D₁₁	1	1	yes	characteristic
329	C₂	C₂×D₁₁×M₃₂	1	1	yes	characteristic
330	C₂	#13081717	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2183	C₂
prime radical	C₂²×D₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2183	D₁₁
hypocenter	C₁₁	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #13070525	C₂	semidirect
3. #13070523	C₂	semidirect
4. #13070522	C₂	semidirect
5. C₁₆×Dic₂₂	C₂	semidirect
6. #13070518	C₂	semidirect
11. C₁₁×#2183	C₂	semidirect
78. #66862	Q₈	semidirect
116. C₁₁×C₁₆	C₂×Q₈	semidirect
119. C₁₁⋊C₁₆	C₂×Q₈	semidirect
124. #2183	D₁₁	semidirect
150. Dic₂₂	M₃₂	semidirect
283. C₁₆	Q₈×D₁₁	semidirect
284. C₁₁	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₁₁	2
#13070525	2
#13070523	2
#13070522	2
C₁₆×Dic₂₂	2
#13070518	2
#13070370	2
#13070209	2
C₄×#66862	2
C₁₁×#2183	2
Q₈×M₃₂	11