Subgroups — Q₁₆×#13083558 #36525889

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13083558	C₂	1	2	yes	maximal
2	#26019149	C₂	1	2	yes	maximal, complemented
3	#26019103	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#10623494	C₂	1	2	yes	maximal, complemented
5	#26019011	C₂	1	1	yes	maximal, characteristic
6	#26018681	C₂	1	2	yes	maximal
7	C₈×#13083558	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19151729	—	2	4	no	—
10	#19151729	C₂²	1	4	yes	—
11	Q₈×#10623494	C₂²	1	4	yes	—
12	#19151651	C₂²	1	1	yes	characteristic
13	#19151622	—	2	2	no	—
14	#19151622	C₂²	1	2	yes	—
15	C₄×#13083558	—	2	2	no	—
16	C₄×#13083558	C₂²	1	1	yes	characteristic
17	#19151351	C₂²	1	2	yes	—
18	#19151351	C₂²	1	2	yes	—
19	Q₈⋊#10623494	C₂²	1	4	yes	—
20	Q₈⋊#10623494	C₂²	1	4	yes	—
21	#19151320	C₂²	1	2	yes	—
22	C₈×#10623494	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₈₉	C₄	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₈₉	C₄	1	2	yes	—
26	C₄×Q₁₆×C₈₉	C₄	1	1	yes	characteristic
27	C₄×Q₁₆×C₈₉	C₄	1	1	yes	characteristic
28	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
29	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#107×C₈₉	C₄	1	1	yes	characteristic
32	C₂×#103×C₈₉	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#13083558	—	2	2	no	—
35	C₂×#13083558	D₄	1	1	yes	characteristic
36	C₂×#13083558	D₄	1	1	yes	characteristic
37	C₄⋊#10623494	—	2	4	no	—
38	C₄⋊#10623494	—	2	4	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	C₄⋊#10623494	—	2	1	no	—
41	C₄⋊#10623494	C₂³	1	4	yes	—
42	C₂²⋊#10623494	—	2	1	no	—
43	#15667923	—	2	4	no	—
44	#15667923	D₄	1	2	yes	—
45	C₄×#10623494	—	2	4	no	—
46	C₄×#10623494	C₂³	1	2	yes	—
47	C₄×#10623494	C₂³	1	2	yes	—
48	C₂²×Q₈×C₈₉	C₂×C₄	1	2	yes	—
49	C₂×Q₁₆×C₈₉	—	2	4	no	—
50	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
51	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₈₉	—	2	2	no	—
56	C₄×Q₈×C₈₉	—	2	2	no	—
57	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
58	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
59	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
60	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
62	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₈₉	—	2	2	no	—
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	#107×C₈₉	C₂×C₄	1	2	yes	—
66	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
67	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
68	#103×C₈₉	C₂×C₄	1	2	yes	—
69	#103×C₈₉	C₂×C₄	1	2	yes	—
70	#103×C₈₉	C₂×C₄	1	2	yes	—
71	#103×C₈₉	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	#13083558	—	2	2	no	—
77	#13083558	—	4	4	no	—
78	#13083558	Q₁₆	1	4	yes	direct product factor
79	C₂×#10623494	—	2	4	no	—
80	C₂×#10623494	—	2	2	no	—
81	C₂×#10623494	C₂×D₄	1	2	yes	—
82	C₂×#10623494	C₂×D₄	1	2	yes	—
83	C₂×Q₈×C₈₉	—	2	2	no	—
84	C₂×Q₈×C₈₉	—	2	4	no	—
85	C₂×Q₈×C₈₉	—	4	2	no	—
86	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
87	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₈₉	—	2	2	no	—
89	C₂²×C₄×C₈₉	—	2	2	no	—
90	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
93	Q₁₆×C₈₉	—	2	4	no	—
94	Q₁₆×C₈₉	—	2	2	no	—
95	Q₁₆×C₈₉	M₁₆	1	4	yes	complemented
96	C₂×C₈×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	—	2	2	no	—
103	C₄⋊C₄×C₈₉	—	2	2	no	—
104	C₄⋊C₄×C₈₉	—	2	2	no	—
105	C₄⋊C₄×C₈₉	—	2	2	no	—
106	C₄⋊C₄×C₈₉	—	2	2	no	—
107	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
108	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
109	C₄²×C₈₉	—	2	2	no	—
110	C₄²×C₈₉	—	2	2	no	—
111	C₄²×C₈₉	—	2	2	no	—
112	C₄²×C₈₉	—	2	2	no	—
113	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	#41	1	1	yes	characteristic
116	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₈₉	—	2	2	no	—
119	Q₈×C₈₉	—	2	2	no	—
120	Q₈×C₈₉	—	2	2	no	—
121	Q₈×C₈₉	—	4	4	no	—
122	Q₈×C₈₉	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	4	4	no	—
131	C₂×C₄×C₈₉	—	8	2	no	—
132	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₈₉	#135	1	2	yes	—
137	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
144	#10623494	—	2	2	no	—
145	#10623494	—	4	2	no	—
146	#10623494	—	4	4	no	—
147	#10623494	C₂×Q₁₆	1	4	yes	complemented
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
151	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	2	4	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	4	2	no	—
157	C₄×C₈₉	—	4	2	no	—
158	C₄×C₈₉	—	4	2	no	—
159	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
160	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
161	C₄×C₈₉	#568	1	1	yes	characteristic
162	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
163	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
164	M₁₆×Q₁₆	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	2	2	no	—
166	C₂×C₈₉	D₄×M₁₆	1	1	yes	characteristic
167	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
168	C₂×C₈₉	#2957	1	1	yes	characteristic
169	Q₈×M₁₆	—	89	2	no	—
170	C₂×C₄×Q₁₆	D₈₉	1	1	yes	characteristic
171	#1591	—	89	2	no	—
172	#1584	—	89	4	no	—
173	C₈×Q₁₆	—	89	2	no	—
174	#1567	—	89	1	no	—
175	#1487	—	89	2	no	—
176	C₈×M₁₆	—	89	1	no	—
177	C₈₉	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	Dic₈₉	1	1	yes	characteristic
179	C₂×C₄×Q₈	C₂×D₈₉	1	2	yes	—
180	#446	—	89	4	no	—
181	#446	—	178	4	no	—
182	C₈×Q₈	—	89	4	no	—
183	C₄×Q₁₆	Dic₈₉	1	2	yes	—
184	C₄×Q₁₆	Dic₈₉	1	1	yes	characteristic
185	C₄×Q₁₆	Dic₈₉	1	1	yes	characteristic
186	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
187	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
188	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
189	#431	—	89	1	no	—
190	C₂×#107	Dic₈₉	1	1	yes	characteristic
191	#423	—	89	2	no	—
192	#423	—	178	2	no	—
193	C₂×#103	Dic₈₉	1	2	yes	—
194	C₄×M₁₆	—	89	1	no	—
195	C₄×M₁₆	—	178	2	no	—
196	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
197	#335	—	89	2	no	—
198	#335	—	89	2	no	—
199	Q₈⋊C₈	—	89	4	no	—
200	Q₈⋊C₈	—	89	4	no	—
201	#322	—	89	2	no	—
202	C₈²	—	89	2	no	—
203	C₂²×Q₈	C₂×Dic₈₉	1	2	yes	—
204	C₂×Q₁₆	—	2	4	no	—
205	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
206	C₂×Q₁₆	C₂×Dic₈₉	1	1	yes	characteristic
207	C₂×Q₁₆	C₂×Dic₈₉	1	1	yes	characteristic
208	C₂×M₁₆	—	89	1	no	—
209	C₂×M₁₆	—	89	1	no	—
210	C₂×M₁₆	—	178	2	no	—
211	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
212	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
216	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
217	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
218	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
223	#107	C₂×Dic₈₉	1	2	yes	—
224	#107	C₂×Dic₈₉	1	1	yes	characteristic
225	#107	C₂×Dic₈₉	1	1	yes	characteristic
226	C₄⋊C₈	—	89	4	no	—
227	C₄⋊C₈	—	178	4	no	—
228	C₄⋊C₈	—	178	4	no	—
229	C₄⋊C₈	—	178	2	no	—
230	C₄⋊C₈	—	178	1	no	—
231	#103	C₂×Dic₈₉	1	2	yes	—
232	#103	C₂×Dic₈₉	1	2	yes	—
233	#103	C₂×Dic₈₉	1	2	yes	—
234	#103	C₂×Dic₈₉	1	2	yes	—
235	C₂²⋊C₈	—	178	1	no	—
236	#97	—	89	2	no	—
237	#97	—	178	4	no	—
238	C₄×C₈	—	89	2	no	—
239	C₄×C₈	—	89	2	no	—
240	C₄×C₈	—	178	4	no	—
241	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
242	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
249	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	#13083573	1	1	yes	characteristic
255	Q₁₆	—	2	4	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	#13083558	1	4	yes	direct product factor
258	M₁₆	—	89	4	no	—
259	M₁₆	—	356	4	no	—
260	M₁₆	—	178	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	—	89	2	no	—
265	C₂×C₈	—	178	4	no	—
266	C₂×C₈	—	178	2	no	—
267	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
277	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
283	C₄²	D₄×D₈₉	1	1	yes	characteristic
284	C₄²	#13083573	1	1	yes	characteristic
285	C₄²	C₂³×D₈₉	1	1	yes	characteristic
286	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	4	no	—
291	Q₈	C₂×#13083558	1	4	yes	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
302	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
303	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
304	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
305	C₂×C₄	#15668035	1	2	yes	—
306	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	89	4	no	—
313	C₈	—	356	2	no	—
314	C₈	—	356	4	no	—
315	C₈	—	178	2	no	—
316	C₈	C₂×#13083558	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	Q₁₆×Dic₈₉	1	2	yes	—
320	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	4	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	Q₁₆×Dic₈₉	1	2	yes	—
329	C₄	C₂²×#13083558	1	1	yes	characteristic
330	C₄	#19152365	1	1	yes	characteristic
331	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
332	C₄	C₂×#15668035	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	D₄×#13083558	1	1	yes	characteristic
335	C₂	C₂×Q₁₆×Dic₈₉	1	1	yes	characteristic
336	C₂	#26029169	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₄×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₈₉
hypocenter	C₈₉	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #26019149	C₂	semidirect
3. #26019103	C₂	semidirect
4. Q₁₆×#10623494	C₂	semidirect
78. #13083558	Q₁₆	direct (N × H)
95. Q₁₆×C₈₉	M₁₆	semidirect
147. #10623494	C₂×Q₁₆	semidirect
177. C₈₉	M₁₆×Q₁₆	semidirect
257. Q₁₆	#13083558	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#13083558	2
#26019149	2
#26019103	2
Q₁₆×#10623494	2
#26019011	2
#26018681	2
C₈×#13083558	2
C₂×C₄×Q₁₆×C₈₉	2
M₁₆×Q₁₆	89

Subgroups — Q16×#13083558 #36525889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#13083558 #36525889