Subgroups — Q₁₆×#11758213 #30613438

Order 15616.

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×#11758213	C₂	1	2	yes	maximal
2	#21461458	C₂	1	2	yes	maximal, complemented
3	#21461430	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#92420	C₂	1	2	yes	maximal, complemented
5	#21461377	C₂	1	1	yes	maximal, characteristic
6	#21461148	C₂	1	2	yes	maximal
7	C₈×#11758213	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
9	#16840812	—	2	4	no	—
10	#16840812	C₂²	1	4	yes	—
11	Q₈×#92420	C₂²	1	4	yes	—
12	#16840781	C₂²	1	1	yes	characteristic
13	#16840766	—	2	2	no	—
14	#16840766	C₂²	1	2	yes	—
15	C₄×#11758213	—	2	2	no	—
16	C₄×#11758213	C₂²	1	1	yes	characteristic
17	#16840625	C₂²	1	2	yes	—
18	#16840625	C₂²	1	2	yes	—
19	#16840612	C₂²	1	4	yes	—
20	Q₈⋊#92420	C₂²	1	4	yes	—
21	#92420⋊C₈	C₂²	1	2	yes	—
22	C₈×#92420	C₂²	1	2	yes	—
23	C₂×Q₁₆×D₆₁	C₄	1	1	yes	characteristic
24	C₄×Q₈×D₆₁	C₂²	1	2	yes	—
25	C₄×Dic₂₄₄	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
27	#16839856	C₂²	1	1	yes	characteristic
28	C₄×#11758202	C₄	1	2	yes	—
29	#16839854	C₂²	1	2	yes	—
30	#107×D₆₁	C₄	1	1	yes	characteristic
31	#103×D₆₁	C₄	1	2	yes	—
32	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
33	C₄×Q₁₆×C₆₁	C₄	1	1	yes	characteristic
34	C₂×#11758213	—	2	2	no	—
35	C₂×#11758213	D₄	1	1	yes	characteristic
36	C₂×#11758213	D₄	1	1	yes	characteristic
37	C₄⋊#92420	—	2	4	no	—
38	C₄⋊#92420	—	2	4	no	—
39	C₄⋊#92420	—	2	2	no	—
40	C₄⋊#92420	—	2	1	no	—
41	C₄⋊#92420	C₂³	1	4	yes	—
42	#14519834	—	2	1	no	—
43	#14519832	—	2	4	no	—
44	#14519832	D₄	1	2	yes	—
45	C₄×#92420	—	2	4	no	—
46	C₄×#92420	C₂³	1	2	yes	—
47	C₄×#92420	C₂³	1	2	yes	—
48	C₂×Q₈×D₆₁	C₂×C₄	1	2	yes	—
49	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
50	C₂×#11758202	C₂×C₄	1	2	yes	—
51	Q₁₆×D₆₁	—	2	4	no	—
52	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₆₁	C₂³	1	2	yes	—
55	C₄×Dic₁₂₂	—	2	2	no	—
56	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
57	#14519742	—	2	2	no	—
58	#14519742	C₂³	1	2	yes	—
59	C₄⋊C₄×D₆₁	—	2	2	no	—
60	C₄⋊C₄×D₆₁	C₂×C₄	1	2	yes	—
61	C₄²×D₆₁	—	2	2	no	—
62	C₄²×D₆₁	C₂³	1	1	yes	characteristic
63	#14519711	C₂×C₄	1	1	yes	characteristic
64	#14519710	C₂×C₄	1	2	yes	—
65	#14519702	C₂×C₄	1	2	yes	—
66	#14519701	C₂×C₄	1	2	yes	—
67	#14519700	C₂×C₄	1	2	yes	—
68	C₄×#92418	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
70	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
73	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
74	#103×C₆₁	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
76	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
77	Q₈×D₆₁	—	2	4	no	—
78	Q₈×D₆₁	—	4	2	no	—
79	C₂×C₄×D₆₁	—	2	2	no	—
80	C₂×C₄×D₆₁	—	2	2	no	—
81	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
84	C₂×Dic₁₂₂	—	2	2	no	—
85	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
86	C₂×#92420	—	2	4	no	—
87	C₂×#92420	—	2	2	no	—
88	C₂×#92420	C₂×D₄	1	2	yes	—
89	C₂×#92420	C₂×D₄	1	2	yes	—
90	#11758213	—	2	2	no	—
91	#11758213	—	4	4	no	—
92	#11758213	Q₁₆	1	4	yes	direct product factor
93	Q₁₆×C₆₁	M₁₆	1	4	yes	complemented
94	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
98	C₄²×C₆₁	—	2	2	no	—
99	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	#11758202	—	2	4	no	—
101	C₄⋊Dic₆₁	—	2	2	no	—
102	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
103	#11758196	—	2	2	no	—
104	#11758196	—	2	2	no	—
105	#11758196	—	2	2	no	—
106	C₄×Dic₆₁	—	2	2	no	—
107	C₄×Dic₆₁	—	2	2	no	—
108	C₄×Dic₆₁	—	2	2	no	—
109	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₆₁	#41	1	1	yes	characteristic
111	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
112	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
113	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
114	Dic₂₄₄	—	2	2	no	—
115	C₈×D₆₁	—	2	2	no	—
116	C₈×D₆₁	—	4	1	no	—
117	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₆₁	—	2	2	no	—
119	Q₈×C₆₁	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₆₁	—	2	2	no	—
126	C₂×Dic₆₁	—	2	2	no	—
127	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
128	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
130	C₄×D₆₁	—	2	1	no	—
131	C₄×D₆₁	—	2	1	no	—
132	C₄×D₆₁	—	4	1	no	—
133	C₄×D₆₁	—	4	4	no	—
134	C₄×D₆₁	—	8	2	no	—
135	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
136	C₄×D₆₁	#135	1	2	yes	—
137	Dic₁₂₂	—	2	2	no	—
138	Dic₁₂₂	—	2	2	no	—
139	Dic₁₂₂	—	4	4	no	—
140	#92420	—	2	2	no	—
141	#92420	—	4	2	no	—
142	#92420	—	4	4	no	—
143	#92420	C₂×Q₁₆	1	4	yes	complemented
144	C₈×C₆₁	—	2	1	no	—
145	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
146	#92418	—	2	1	no	—
147	#92418	—	2	1	no	—
148	M₁₆×Q₁₆	—	61	1	no	maximal, Sylow
149	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×D₆₁	—	2	1	no	—
151	C₂×D₆₁	—	2	1	no	—
152	C₂×D₆₁	C₄×Q₁₆	1	2	yes	—
153	C₄×C₆₁	—	2	4	no	—
154	C₄×C₆₁	—	2	1	no	—
155	C₄×C₆₁	—	4	2	no	—
156	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
157	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
158	C₄×C₆₁	#568	1	1	yes	characteristic
159	Dic₆₁	—	2	1	no	—
160	Dic₆₁	—	2	1	no	—
161	Dic₆₁	—	4	2	no	—
162	Dic₆₁	—	4	2	no	—
163	Dic₆₁	C₂²×Q₁₆	1	1	yes	characteristic
164	Dic₆₁	C₂×#135	1	1	yes	characteristic
165	Q₈×M₁₆	—	61	2	no	—
166	C₂×C₄×Q₁₆	—	61	1	no	—
167	#1591	—	61	2	no	—
168	#1584	—	61	4	no	—
169	C₈×Q₁₆	—	61	2	no	—
170	#1567	—	61	1	no	—
171	#1487	—	61	2	no	—
172	C₈×M₁₆	—	61	1	no	—
173	C₂×C₆₁	D₄×M₁₆	1	1	yes	characteristic
174	C₂×C₆₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
175	C₂×C₆₁	#2957	1	1	yes	characteristic
176	D₆₁	—	2	2	no	—
177	C₂²×Q₁₆	—	61	1	no	—
178	C₂×C₄×Q₈	—	61	2	no	—
179	#446	—	61	4	no	—
180	#446	—	122	4	no	—
181	C₈×Q₈	—	61	4	no	—
182	C₄×Q₁₆	—	61	1	no	—
183	C₄×Q₁₆	—	61	2	no	—
184	C₄×Q₁₆	—	61	1	no	—
185	C₄×Q₁₆	—	61	2	no	—
186	C₄×Q₁₆	—	61	1	no	—
187	C₄×Q₁₆	#6923	1	1	yes	characteristic
188	#431	—	61	1	no	—
189	C₂×#107	—	61	1	no	—
190	#423	—	61	2	no	—
191	#423	—	122	2	no	—
192	C₂×#103	—	61	2	no	—
193	C₄×M₁₆	—	61	1	no	—
194	C₄×M₁₆	—	122	2	no	—
195	C₂×C₄×C₈	—	61	1	no	—
196	#335	—	61	2	no	—
197	#335	—	61	2	no	—
198	Q₈⋊C₈	—	61	4	no	—
199	Q₈⋊C₈	—	61	4	no	—
200	#322	—	61	2	no	—
201	C₈²	—	61	2	no	—
202	C₆₁	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	61	2	no	—
204	C₂×Q₁₆	—	61	2	no	—
205	C₂×Q₁₆	—	61	1	no	—
206	C₂×Q₁₆	—	122	4	no	—
207	C₂×Q₁₆	C₂×#6923	1	1	yes	characteristic
208	C₂×M₁₆	—	61	1	no	—
209	C₂×M₁₆	—	61	1	no	—
210	C₂×M₁₆	—	122	2	no	—
211	C₂²×C₈	—	61	1	no	—
212	C₂²×C₈	—	61	1	no	—
213	C₄×Q₈	—	61	2	no	—
214	C₄×Q₈	—	61	2	no	—
215	C₄×Q₈	—	61	2	no	—
216	C₄×Q₈	—	122	2	no	—
217	C₄×Q₈	—	122	2	no	—
218	C₄×Q₈	C₂×#6923	1	2	yes	—
219	C₂×C₄⋊C₄	—	61	2	no	—
220	C₂×C₄⋊C₄	—	122	2	no	—
221	C₂×C₄²	—	61	1	no	—
222	C₂×C₄²	—	122	2	no	—
223	#107	—	61	1	no	—
224	#107	—	61	2	no	—
225	#107	C₂×#6923	1	1	yes	characteristic
226	C₄⋊C₈	—	61	4	no	—
227	C₄⋊C₈	—	122	2	no	—
228	C₄⋊C₈	—	122	1	no	—
229	C₄⋊C₈	—	122	4	no	—
230	C₄⋊C₈	—	122	4	no	—
231	#103	—	61	2	no	—
232	#103	—	61	2	no	—
233	#103	—	61	2	no	—
234	#103	C₂×#6923	1	2	yes	—
235	C₂²⋊C₈	—	122	1	no	—
236	#97	—	61	2	no	—
237	#97	—	122	4	no	—
238	C₄×C₈	—	61	2	no	—
239	C₄×C₈	—	61	1	no	—
240	C₄×C₈	—	61	1	no	—
241	C₄×C₈	—	61	2	no	—
242	C₄×C₈	—	61	1	no	—
243	C₄×C₈	—	122	4	no	—
244	C₄×C₈	C₂×#6923	1	1	yes	characteristic
245	C₂×Q₈	—	61	2	no	—
246	C₂×Q₈	—	122	2	no	—
247	C₂×Q₈	—	122	4	no	—
248	C₂×Q₈	—	244	2	no	—
249	C₂×Q₈	C₂²×#6923	1	2	yes	—
250	C₂²×C₄	—	61	1	no	—
251	C₂²×C₄	—	61	1	no	—
252	C₂²×C₄	—	61	1	no	—
253	C₂²×C₄	—	122	2	no	—
254	C₂²×C₄	—	122	2	no	—
255	Q₁₆	—	122	4	no	—
256	Q₁₆	—	122	2	no	—
257	Q₁₆	#11758213	1	4	yes	direct product factor
258	M₁₆	—	61	4	no	—
259	M₁₆	—	122	2	no	—
260	M₁₆	—	244	4	no	—
261	C₂×C₈	—	61	2	no	—
262	C₂×C₈	—	61	2	no	—
263	C₂×C₈	—	61	1	no	—
264	C₂×C₈	—	61	1	no	—
265	C₂×C₈	—	122	2	no	—
266	C₂×C₈	—	122	2	no	—
267	C₂×C₈	—	122	4	no	—
268	C₂×C₈	—	244	1	no	—
269	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
270	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	61	2	no	—
273	C₄⋊C₄	—	122	2	no	—
274	C₄⋊C₄	—	122	2	no	—
275	C₄⋊C₄	—	122	2	no	—
276	C₄⋊C₄	—	122	2	no	—
277	C₄⋊C₄	C₂²×#6923	1	2	yes	—
278	C₄²	—	2	2	no	—
279	C₄²	—	61	1	no	—
280	C₄²	—	61	1	no	—
281	C₄²	—	61	1	no	—
282	C₄²	—	122	2	no	—
283	C₄²	—	122	2	no	—
284	C₄²	—	122	2	no	—
285	C₄²	C₂²×#6923	1	1	yes	characteristic
286	C₂³	—	61	1	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	122	2	no	—
289	Q₈	—	122	2	no	—
290	Q₈	—	244	4	no	—
291	Q₈	C₂×#11758213	1	4	yes	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	61	1	no	—
295	C₂×C₄	—	61	2	no	—
296	C₂×C₄	—	61	2	no	—
297	C₂×C₄	—	61	1	no	—
298	C₂×C₄	—	61	1	no	—
299	C₂×C₄	—	122	1	no	—
300	C₂×C₄	—	122	1	no	—
301	C₂×C₄	—	122	2	no	—
302	C₂×C₄	—	122	2	no	—
303	C₂×C₄	—	488	2	no	—
304	C₂×C₄	—	244	1	no	—
305	C₂×C₄	—	244	4	no	—
306	C₂×C₄	D₄×#6923	1	1	yes	characteristic
307	C₂×C₄	D₄×#6923	1	1	yes	characteristic
308	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	61	4	no	—
311	C₈	—	122	2	no	—
312	C₈	—	122	1	no	—
313	C₈	—	122	1	no	—
314	C₈	—	244	4	no	—
315	C₈	—	244	2	no	—
316	C₈	C₂×#11758213	1	2	yes	—
317	C₂²	—	61	2	no	—
318	C₂²	—	122	1	no	—
319	C₂²	—	122	1	no	—
320	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	2	no	—
324	C₄	—	61	1	no	—
325	C₄	—	61	1	no	—
326	C₄	—	122	1	no	—
327	C₄	—	122	1	no	—
328	C₄	—	244	2	no	—
329	C₄	—	244	2	no	—
330	C₄	Q₁₆×#6923	1	2	yes	—
331	C₄	C₂²×#11758213	1	1	yes	characteristic
332	C₄	#16840841	1	1	yes	characteristic
333	C₂	—	122	2	no	—
334	C₂	D₄×#11758213	1	1	yes	characteristic
335	C₂	C₂×Q₁₆×#6923	1	1	yes	characteristic
336	C₂	#21462596	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₄×Q₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂²×D₆₁	C₄×D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₄×Q₁₆	#6923
hypocenter	C₆₁	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #21461458	C₂	semidirect
3. #21461430	C₂	semidirect
4. Q₁₆×#92420	C₂	semidirect
92. #11758213	Q₁₆	direct (N × H)
93. Q₁₆×C₆₁	M₁₆	semidirect
143. #92420	C₂×Q₁₆	semidirect
202. C₆₁	M₁₆×Q₁₆	semidirect
257. Q₁₆	#11758213	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×#11758213	2
#21461458	2
#21461430	2
Q₁₆×#92420	2
#21461377	2
#21461148	2
C₈×#11758213	2
C₄×Q₁₆×D₆₁	2
M₁₆×Q₁₆	61

Subgroups — Q16×#11758213 #30613438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#11758213 #30613438