Subgroups — Q₈×C₁₁×#103 #14599713

Order 2816.

1086 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1990	C₂	1	3	yes	maximal
5	C₁₁×#1987	C₂	1	3	yes	maximal
6	C₁₁×#1803	C₂	1	3	yes	maximal
7	C₄×C₁₁×#103	C₂	1	3	yes	maximal
8	Q₈²×C₁₁	—	2	4	no	—
9	Q₈²×C₁₁	C₄	1	8	yes	complemented
10	C₂×C₁₁×C₄⋊Q₈	—	2	3	no	—
11	C₂×C₁₁×C₄⋊Q₈	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₁₁	—	2	1	no	—
13	C₂×C₄×Q₈×C₁₁	—	2	1	no	—
14	C₂×C₄×Q₈×C₁₁	C₂²	1	3	yes	—
15	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
16	C₈×Q₈×C₁₁	—	2	4	no	—
17	C₂×C₁₁×C₄⋊C₈	C₂²	1	3	yes	—
18	C₂×C₁₁×#103	C₂²	1	3	yes	—
19	C₂×C₁₁×#103	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
21	C₁₁×#389	—	2	3	no	—
22	C₁₁×#384	C₂²	1	3	yes	—
23	C₄×C₁₁×C₄⋊C₄	C₂²	1	3	yes	—
24	C₁₁×#340	C₂²	1	3	yes	—
25	C₁₁×#340	C₂²	1	6	yes	—
26	C₁₁×#340	C₂²	1	3	yes	—
27	C₂²×Q₈×C₁₁	—	2	3	no	—
28	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
29	C₂²×Q₈×C₁₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
31	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
32	C₁₁×C₄⋊Q₈	—	2	24	no	—
33	C₁₁×C₄⋊Q₈	—	2	6	no	—
34	C₁₁×C₄⋊Q₈	C₂×C₄	1	12	yes	—
35	C₄×Q₈×C₁₁	—	2	8	no	—
36	C₄×Q₈×C₁₁	—	2	6	no	—
37	C₄×Q₈×C₁₁	—	4	4	no	—
38	C₄×Q₈×C₁₁	C₂×C₄	1	12	yes	—
39	C₄×Q₈×C₁₁	C₂×C₄	1	4	yes	—
40	C₄×Q₈×C₁₁	D₄	1	4	yes	—
41	C₂×C₁₁×C₄⋊C₄	—	2	3	no	—
42	C₂×C₁₁×C₄⋊C₄	—	2	3	no	—
43	C₂×C₁₁×C₄⋊C₄	—	2	3	no	—
44	C₂×C₁₁×C₄⋊C₄	—	2	3	no	—
45	C₂×C₁₁×C₄⋊C₄	D₄	1	3	yes	—
46	C₂×C₁₁×C₄⋊C₄	C₂³	1	3	yes	—
47	C₂×C₁₁×C₄⋊C₄	C₂³	1	3	yes	—
48	C₂×C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₁	—	2	3	no	—
50	C₂×C₄²×C₁₁	—	2	3	no	—
51	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
52	C₁₁×C₄⋊C₈	—	2	6	no	—
53	C₁₁×#103	—	2	6	no	—
54	C₁₁×#103	Q₈	1	4	yes	direct product factor
55	C₄×C₈×C₁₁	—	2	6	no	—
56	C₁₁×#95	—	2	3	no	—
57	Q₈×#103	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂×Q₈×C₁₁	—	2	12	no	—
59	C₂×Q₈×C₁₁	—	2	1	no	—
60	C₂×Q₈×C₁₁	—	2	1	no	—
61	C₂×Q₈×C₁₁	—	2	1	no	—
62	C₂×Q₈×C₁₁	—	8	3	no	—
63	C₂×Q₈×C₁₁	—	8	3	no	—
64	C₂×Q₈×C₁₁	C₂²⋊C₄	1	4	yes	—
65	C₂×Q₈×C₁₁	SD₁₆	1	4	yes	—
66	C₂×Q₈×C₁₁	Q₁₆	1	4	yes	—
67	C₂×Q₈×C₁₁	C₂²×C₄	1	2	yes	—
68	C₂×Q₈×C₁₁	C₂×Q₈	1	2	yes	—
69	C₂×Q₈×C₁₁	#41	1	2	yes	—
70	C₂²×C₄×C₁₁	—	2	1	no	—
71	C₂²×C₄×C₁₁	—	2	1	no	—
72	C₂²×C₄×C₁₁	—	2	3	no	—
73	C₂²×C₄×C₁₁	—	2	3	no	—
74	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
75	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
76	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₁	—	2	1	no	—
78	C₂×C₈×C₁₁	—	2	3	no	—
79	C₂×C₈×C₁₁	—	2	3	no	—
80	C₂×C₈×C₁₁	—	2	1	no	—
81	C₂×C₈×C₁₁	—	2	3	no	—
82	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
83	C₁₁×C₄⋊C₄	—	2	12	no	—
84	C₁₁×C₄⋊C₄	—	2	12	no	—
85	C₁₁×C₄⋊C₄	—	2	3	no	—
86	C₁₁×C₄⋊C₄	—	2	1	no	—
87	C₁₁×C₄⋊C₄	—	2	3	no	—
88	C₁₁×C₄⋊C₄	—	2	3	no	—
89	C₁₁×C₄⋊C₄	—	4	6	no	—
90	C₁₁×C₄⋊C₄	—	8	3	no	—
91	C₁₁×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
92	C₁₁×C₄⋊C₄	C₂²×C₄	1	6	yes	—
93	C₁₁×C₄⋊C₄	C₂×Q₈	1	2	yes	—
94	C₄²×C₁₁	—	2	12	no	—
95	C₄²×C₁₁	—	4	6	no	—
96	C₄²×C₁₁	C₂²×C₄	1	6	yes	—
97	C₄²×C₁₁	C₂×D₄	1	6	yes	—
98	C₂×Q₈²	C₂×C₁₁	1	1	yes	characteristic
99	C₂×C₈×Q₈	C₂×C₁₁	1	1	yes	characteristic
100	Q₈×C₄⋊C₄	C₂×C₁₁	1	1	yes	characteristic
101	#1990	C₂×C₁₁	1	3	yes	—
102	#1987	C₂×C₁₁	1	3	yes	—
103	#1803	C₂×C₁₁	1	3	yes	—
104	C₄×#103	C₂×C₁₁	1	3	yes	—
105	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₁₁	—	2	2	no	—
107	Q₈×C₁₁	—	2	2	no	—
108	Q₈×C₁₁	—	2	2	no	—
109	Q₈×C₁₁	—	8	12	no	—
110	Q₈×C₁₁	#103	1	16	yes	direct product factor
111	Q₈×C₁₁	C₄×Q₈	1	4	yes	complemented
112	C₂×C₄×C₁₁	—	2	2	no	—
113	C₂×C₄×C₁₁	—	2	2	no	—
114	C₂×C₄×C₁₁	—	2	2	no	—
115	C₂×C₄×C₁₁	—	2	3	no	—
116	C₂×C₄×C₁₁	—	2	3	no	—
117	C₂×C₄×C₁₁	—	2	6	no	—
118	C₂×C₄×C₁₁	—	2	2	no	—
119	C₂×C₄×C₁₁	—	4	1	no	—
120	C₂×C₄×C₁₁	—	4	3	no	—
121	C₂×C₄×C₁₁	—	4	1	no	—
122	C₂×C₄×C₁₁	—	4	3	no	—
123	C₂×C₄×C₁₁	—	4	3	no	—
124	C₂×C₄×C₁₁	—	4	3	no	—
125	C₂×C₄×C₁₁	—	4	3	no	—
126	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	6	yes	—
127	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	6	yes	—
128	C₂×C₄×C₁₁	C₂×SD₁₆	1	6	yes	—
129	C₂×C₄×C₁₁	C₂×Q₁₆	1	6	yes	—
130	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
136	C₈×C₁₁	—	2	2	no	—
137	C₈×C₁₁	—	4	3	no	—
138	Q₈²	—	2	4	no	—
139	Q₈²	C₄×C₁₁	1	8	yes	complemented
140	C₂×C₄⋊Q₈	—	2	3	no	—
141	C₂×C₄⋊Q₈	C₂²×C₁₁	1	3	yes	—
142	C₂×C₄×Q₈	—	2	1	no	—
143	C₂×C₄×Q₈	—	2	1	no	—
144	C₂×C₄×Q₈	C₂²×C₁₁	1	3	yes	—
145	C₂×C₄×Q₈	C₂²×C₁₁	1	1	yes	characteristic
146	C₈×Q₈	—	2	4	no	—
147	C₂×C₄⋊C₈	C₂²×C₁₁	1	3	yes	—
148	C₂×#103	C₂²×C₁₁	1	3	yes	—
149	C₂×#103	C₂²×C₁₁	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₁₁	1	3	yes	—
151	#389	—	2	3	no	—
152	#384	C₂²×C₁₁	1	3	yes	—
153	C₄×C₄⋊C₄	C₂²×C₁₁	1	3	yes	—
154	#340	C₂²×C₁₁	1	6	yes	—
155	#340	C₂²×C₁₁	1	3	yes	—
156	#340	C₂²×C₁₁	1	3	yes	—
157	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₁	#548	1	1	yes	characteristic
159	C₂²×C₁₁	D₄×Q₈	1	1	yes	characteristic
160	C₂²×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
161	C₂²×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
162	C₂²×C₁₁	#576	1	1	yes	characteristic
163	C₂²×C₁₁	#577	1	1	yes	characteristic
164	C₄×C₁₁	—	2	4	no	—
165	C₄×C₁₁	—	2	6	no	—
166	C₄×C₁₁	—	4	2	no	—
167	C₄×C₁₁	—	4	6	no	—
168	C₄×C₁₁	—	8	3	no	—
169	C₄×C₁₁	C₂×#103	1	12	yes	—
170	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₄×C₁₁	#519	1	1	yes	characteristic
172	C₄×C₁₁	#548	1	1	yes	characteristic
173	C₄×C₁₁	D₄×Q₈	1	1	yes	characteristic
174	C₂²×Q₈	—	2	3	no	—
175	C₂²×Q₈	D₄×C₁₁	1	1	yes	characteristic
176	C₂²×Q₈	C₂³×C₁₁	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₁₁	1	3	yes	—
178	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	24	no	—
180	C₄⋊Q₈	—	2	6	no	—
181	C₄⋊Q₈	C₂×C₄×C₁₁	1	12	yes	—
182	C₄×Q₈	—	2	8	no	—
183	C₄×Q₈	—	2	6	no	—
184	C₄×Q₈	—	4	4	no	—
185	C₄×Q₈	C₂×C₄×C₁₁	1	12	yes	—
186	C₄×Q₈	C₂×C₄×C₁₁	1	4	yes	—
187	C₄×Q₈	D₄×C₁₁	1	4	yes	—
188	C₂×C₄⋊C₄	—	2	3	no	—
189	C₂×C₄⋊C₄	—	2	3	no	—
190	C₂×C₄⋊C₄	—	2	3	no	—
191	C₂×C₄⋊C₄	—	2	3	no	—
192	C₂×C₄⋊C₄	D₄×C₁₁	1	3	yes	—
193	C₂×C₄⋊C₄	C₂³×C₁₁	1	3	yes	—
194	C₂×C₄⋊C₄	C₂³×C₁₁	1	3	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₁	1	1	yes	characteristic
196	C₂×C₄²	—	2	3	no	—
197	C₂×C₄²	—	2	3	no	—
198	C₂×C₄²	C₂³×C₁₁	1	3	yes	—
199	C₄⋊C₈	—	2	6	no	—
200	#103	—	2	6	no	—
201	#103	Q₈×C₁₁	1	4	yes	direct product factor
202	C₄×C₈	—	2	6	no	—
203	#95	—	2	3	no	—
204	C₂×C₁₁	Q₈×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₁₁	C₂²×#103	1	1	yes	characteristic
206	C₂×C₁₁	#2897	1	1	yes	characteristic
207	C₂×C₁₁	#3362	1	1	yes	characteristic
208	C₂×C₁₁	#3363	1	1	yes	characteristic
209	C₂×C₁₁	Q₈×SD₁₆	1	1	yes	characteristic
210	C₂×C₁₁	Q₈×Q₁₆	1	1	yes	characteristic
211	C₂×Q₈	—	2	12	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	8	3	no	—
216	C₂×Q₈	—	8	3	no	—
217	C₂×Q₈	C₁₁×C₂²⋊C₄	1	4	yes	—
218	C₂×Q₈	C₁₁×SD₁₆	1	4	yes	—
219	C₂×Q₈	C₁₁×Q₁₆	1	4	yes	—
220	C₂×Q₈	C₂²×C₄×C₁₁	1	2	yes	—
221	C₂×Q₈	C₂×Q₈×C₁₁	1	2	yes	—
222	C₂×Q₈	C₁₁×#41	1	2	yes	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	3	no	—
226	C₂²×C₄	—	2	3	no	—
227	C₂²×C₄	C₂×D₄×C₁₁	1	3	yes	—
228	C₂²×C₄	C₂×D₄×C₁₁	1	3	yes	—
229	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	3	no	—
233	C₂×C₈	—	2	3	no	—
234	C₂×C₈	—	2	3	no	—
235	C₂×C₈	C₂×Q₈×C₁₁	1	2	yes	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	—	2	12	no	—
238	C₄⋊C₄	—	2	3	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	3	no	—
241	C₄⋊C₄	—	2	3	no	—
242	C₄⋊C₄	—	4	6	no	—
243	C₄⋊C₄	—	8	3	no	—
244	C₄⋊C₄	C₁₁×C₂²⋊C₄	1	12	yes	—
245	C₄⋊C₄	C₂²×C₄×C₁₁	1	6	yes	—
246	C₄⋊C₄	C₂×Q₈×C₁₁	1	2	yes	—
247	C₄²	—	2	12	no	—
248	C₄²	—	4	6	no	—
249	C₄²	C₂²×C₄×C₁₁	1	6	yes	—
250	C₄²	C₂×D₄×C₁₁	1	6	yes	—
251	C₁₁	Q₈×#103	1	1	yes	characteristic, direct product factor, Sylow
252	C₂³	C₂²×D₄×C₁₁	1	1	yes	characteristic
253	Q₈	—	2	2	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	2	2	no	—
256	Q₈	—	8	12	no	—
257	Q₈	C₁₁×#103	1	16	yes	direct product factor
258	Q₈	C₄×Q₈×C₁₁	1	4	yes	complemented
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	3	no	—
263	C₂×C₄	—	2	3	no	—
264	C₂×C₄	—	2	6	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	3	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	3	no	—
270	C₂×C₄	—	4	3	no	—
271	C₂×C₄	—	4	3	no	—
272	C₂×C₄	—	4	3	no	—
273	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	6	yes	—
274	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	6	yes	—
275	C₂×C₄	C₂×C₁₁×SD₁₆	1	6	yes	—
276	C₂×C₄	C₂×C₁₁×Q₁₆	1	6	yes	—
277	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
279	C₂×C₄	C₂²×Q₈×C₁₁	1	1	yes	characteristic
280	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
281	C₂×C₄	C₁₁×2₋¹⁺⁴	1	1	yes	characteristic
282	C₂×C₄	C₁₁×2₋¹⁺⁴	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	3	no	—
285	C₂²	C₂²×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
286	C₂²	C₁₁×#548	1	1	yes	characteristic
287	C₂²	D₄×Q₈×C₁₁	1	1	yes	characteristic
288	C₂²	C₂²×C₁₁×SD₁₆	1	1	yes	characteristic
289	C₂²	C₂²×C₁₁×Q₁₆	1	1	yes	characteristic
290	C₂²	C₁₁×#576	1	1	yes	characteristic
291	C₂²	C₁₁×#577	1	1	yes	characteristic
292	C₄	—	2	6	no	—
293	C₄	—	2	4	no	—
294	C₄	—	4	6	no	—
295	C₄	—	4	2	no	—
296	C₄	—	8	3	no	—
297	C₄	C₂×C₁₁×#103	1	12	yes	—
298	C₄	C₂×C₄×Q₈×C₁₁	1	1	yes	characteristic
299	C₄	C₁₁×#519	1	1	yes	characteristic
300	C₄	C₁₁×#548	1	1	yes	characteristic
301	C₄	D₄×Q₈×C₁₁	1	1	yes	characteristic
302	C₂	Q₈×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
303	C₂	C₂²×C₁₁×#103	1	1	yes	characteristic
304	C₂	C₁₁×#2897	1	1	yes	characteristic
305	C₂	C₁₁×#3362	1	1	yes	characteristic
306	C₂	C₁₁×#3363	1	1	yes	characteristic
307	C₂	Q₈×C₁₁×SD₁₆	1	1	yes	characteristic
308	C₂	Q₈×C₁₁×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ C₂×C₄×Q₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. Q₈²×C₁₁	C₄	semidirect
54. C₁₁×#103	Q₈	direct (N × H)
57. Q₈×#103	C₁₁	direct (N × H)
110. Q₈×C₁₁	#103	direct (N × H)
111. Q₈×C₁₁	C₄×Q₈	semidirect
139. Q₈²	C₄×C₁₁	semidirect
201. #103	Q₈×C₁₁	direct (N × H)
251. C₁₁	Q₈×#103	direct (N × H)
257. Q₈	C₁₁×#103	direct (N × H)
258. Q₈	C₄×Q₈×C₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₁₁	2
C₂×C₈×Q₈×C₁₁	2
Q₈×C₁₁×C₄⋊C₄	2
C₁₁×#1990	2
C₁₁×#1987	2
C₁₁×#1803	2
C₄×C₁₁×#103	2
Q₈×#103	11

Subgroups — Q8×C11×#103 #14599713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₁₁×#103 #14599713