Subgroups — Q₁₆×#6122 #16796343

Order 3328.

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×#6122	C₂	1	2	yes	maximal
2	#13130476	C₂	1	2	yes	maximal, complemented
3	#13130449	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#1057	C₂	1	2	yes	maximal, complemented
5	#13130393	C₂	1	1	yes	maximal, characteristic
6	#13130164	C₂	1	2	yes	maximal
7	C₈×#6122	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
10	C₂×Q₈×Dic₁₃	C₂²	1	2	yes	—
11	#11719158	—	2	4	no	—
12	#11719158	C₂²	1	4	yes	—
13	Q₈×#1057	C₂²	1	4	yes	—
14	#11719129	—	2	2	no	—
15	#11719129	C₂²	1	2	yes	—
16	#11719125	C₂²	1	1	yes	characteristic
17	C₄×#6122	—	2	2	no	—
18	C₄×#6122	C₂²	1	1	yes	characteristic
19	C₂×#88075	C₄	1	2	yes	—
20	Q₁₆×Dic₁₃	C₄	1	1	yes	characteristic
21	Q₁₆×Dic₁₃	C₄	1	2	yes	—
22	Q₁₆×Dic₁₃	C₄	1	1	yes	characteristic
23	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
25	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×#88058	C₄	1	1	yes	characteristic
27	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
28	Q₈⋊#1057	C₂²	1	4	yes	—
29	Q₈⋊#1057	C₂²	1	4	yes	—
30	#11718256	C₂²	1	2	yes	—
31	#11718254	C₂²	1	2	yes	—
32	#1057⋊C₈	C₂²	1	2	yes	—
33	C₈×#1057	C₂²	1	2	yes	—
34	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×#6122	—	2	2	no	—
36	C₂×#6122	D₄	1	1	yes	characteristic
37	C₂×#6122	D₄	1	1	yes	characteristic
38	C₂×C₁₃×Q₁₆	—	2	4	no	—
39	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	Q₈×Dic₁₃	—	2	2	no	—
44	Q₈×Dic₁₃	—	2	2	no	—
45	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
46	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
47	Q₈×Dic₁₃	C₂³	1	2	yes	—
48	Q₈×Dic₁₃	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₁₃	—	2	2	no	—
50	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₁₃	—	2	2	no	—
52	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
53	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
54	C₂²⋊#1057	—	2	1	no	—
55	#88112	—	2	4	no	—
56	#88112	—	2	2	no	—
57	#88112	—	2	1	no	—
58	#88110	—	2	4	no	—
59	#88110	D₄	1	2	yes	—
60	C₄⋊#1057	—	2	4	no	—
61	C₄⋊#1057	C₂³	1	4	yes	—
62	C₄×#1057	—	2	4	no	—
63	C₄×#1057	C₂³	1	2	yes	—
64	C₄×#1057	C₂³	1	2	yes	—
65	#88075	C₂×C₄	1	2	yes	—
66	#88075	C₂×C₄	1	2	yes	—
67	#88075	C₂×C₄	1	2	yes	—
68	#88075	C₂×C₄	1	2	yes	—
69	#88058	C₂×C₄	1	2	yes	—
70	#88058	C₂×C₄	1	1	yes	characteristic
71	#88058	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
75	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
76	M₁₆×Q₁₆	—	13	1	no	maximal, Sylow
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	2	4	no	—
79	C₂×Q₈×C₁₃	—	4	2	no	—
80	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×Dic₁₃	—	2	2	no	—
85	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₁₃	#41	1	1	yes	characteristic
87	#6122	—	2	2	no	—
88	#6122	—	4	4	no	—
89	#6122	Q₁₆	1	4	yes	direct product factor
90	C₂×#1057	—	2	4	no	—
91	C₂×#1057	—	2	2	no	—
92	C₂×#1057	C₂×D₄	1	2	yes	—
93	C₂×#1057	C₂×D₄	1	2	yes	—
94	C₁₃×Q₁₆	—	2	4	no	—
95	C₁₃×Q₁₆	—	2	2	no	—
96	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₄⋊Dic₁₃	—	2	2	no	—
101	C₄⋊Dic₁₃	—	2	2	no	—
102	C₄⋊Dic₁₃	—	2	2	no	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
106	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	—	2	2	no	—
109	C₄×Dic₁₃	—	2	2	no	—
110	C₄×Dic₁₃	—	2	2	no	—
111	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₁₃	#41	1	1	yes	characteristic
114	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
115	C₂×#1055	—	4	1	no	—
116	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
117	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
118	Q₈×M₁₆	—	13	2	no	—
119	C₂×C₄×Q₁₆	—	13	1	no	—
120	#1591	—	13	2	no	—
121	#1584	—	13	4	no	—
122	C₈×Q₁₆	—	13	2	no	—
123	#1567	—	13	1	no	—
124	#1487	—	13	2	no	—
125	C₈×M₁₆	—	13	1	no	—
126	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
127	Q₈×C₁₃	—	2	2	no	—
128	Q₈×C₁₃	—	2	2	no	—
129	Q₈×C₁₃	—	2	2	no	—
130	Q₈×C₁₃	—	4	4	no	—
131	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	4	4	no	—
137	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	—	4	1	no	—
142	C₂×Dic₁₃	—	8	2	no	—
143	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₁₃	C₂×Q₁₆	1	2	yes	—
146	C₂×Dic₁₃	#135	1	2	yes	—
147	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
149	#1057	—	2	2	no	—
150	#1057	—	4	2	no	—
151	#1057	—	4	4	no	—
152	#1057	C₂×Q₁₆	1	4	yes	complemented
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
155	#1055	—	2	1	no	—
156	#1055	—	2	1	no	—
157	C₂²×Q₁₆	#261	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	13	2	no	—
159	#446	—	13	4	no	—
160	#446	—	26	4	no	—
161	C₈×Q₈	—	13	4	no	—
162	C₄×Q₁₆	—	13	1	no	—
163	C₄×Q₁₆	—	13	2	no	—
164	C₄×Q₁₆	—	13	1	no	—
165	C₄×Q₁₆	—	13	2	no	—
166	C₄×Q₁₆	—	13	1	no	—
167	C₄×Q₁₆	—	13	1	no	—
168	#431	—	13	1	no	—
169	C₂×#107	—	13	1	no	—
170	#423	—	13	2	no	—
171	#423	—	26	2	no	—
172	C₂×#103	—	13	2	no	—
173	C₄×M₁₆	—	13	1	no	—
174	C₄×M₁₆	—	26	2	no	—
175	C₂×C₄×C₈	—	13	1	no	—
176	#335	—	13	2	no	—
177	#335	—	13	2	no	—
178	Q₈⋊C₈	—	13	4	no	—
179	Q₈⋊C₈	—	13	4	no	—
180	#322	—	13	2	no	—
181	C₈²	—	13	2	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
185	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	4	no	—
188	C₄×C₁₃	—	4	2	no	—
189	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
190	C₄×C₁₃	#568	1	1	yes	characteristic
191	Dic₁₃	—	2	1	no	—
192	Dic₁₃	—	2	1	no	—
193	Dic₁₃	—	4	2	no	—
194	Dic₁₃	—	4	2	no	—
195	Dic₁₃	C₄×Q₁₆	1	2	yes	—
196	Dic₁₃	C₂²×Q₁₆	1	1	yes	characteristic
197	Dic₁₃	C₂×#135	1	1	yes	characteristic
198	C₂²×Q₈	C₂×#261	1	2	yes	—
199	C₂×Q₁₆	—	2	4	no	—
200	C₂×Q₁₆	C₂×#261	1	2	yes	—
201	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
202	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
203	C₂×M₁₆	—	13	1	no	—
204	C₂×M₁₆	—	13	1	no	—
205	C₂×M₁₆	—	26	2	no	—
206	C₂²×C₈	—	13	1	no	—
207	C₂²×C₈	C₂×#261	1	1	yes	characteristic
208	C₄×Q₈	—	13	2	no	—
209	C₄×Q₈	—	13	2	no	—
210	C₄×Q₈	—	13	2	no	—
211	C₄×Q₈	—	13	2	no	—
212	C₄×Q₈	—	26	2	no	—
213	C₄×Q₈	—	26	2	no	—
214	C₂×C₄⋊C₄	—	13	2	no	—
215	C₂×C₄⋊C₄	—	26	2	no	—
216	C₂×C₄²	—	13	1	no	—
217	C₂×C₄²	—	26	2	no	—
218	#107	—	13	2	no	—
219	#107	—	13	1	no	—
220	#107	—	13	1	no	—
221	C₄⋊C₈	—	13	4	no	—
222	C₄⋊C₈	—	26	2	no	—
223	C₄⋊C₈	—	26	1	no	—
224	C₄⋊C₈	—	26	4	no	—
225	C₄⋊C₈	—	26	4	no	—
226	#103	—	13	2	no	—
227	#103	—	13	2	no	—
228	#103	—	13	2	no	—
229	#103	—	13	2	no	—
230	C₂²⋊C₈	—	26	1	no	—
231	#97	—	13	2	no	—
232	#97	—	26	4	no	—
233	C₄×C₈	—	13	2	no	—
234	C₄×C₈	—	13	1	no	—
235	C₄×C₈	—	13	1	no	—
236	C₄×C₈	—	13	1	no	—
237	C₄×C₈	—	13	1	no	—
238	C₄×C₈	—	13	2	no	—
239	C₄×C₈	—	26	4	no	—
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
242	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
243	C₂×C₁₃	#2957	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	C₂²×#261	1	2	yes	—
248	C₂×Q₈	C₂²×#261	1	2	yes	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	13	1	no	—
251	C₂²×C₄	—	13	1	no	—
252	C₂²×C₄	—	26	2	no	—
253	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
254	Q₁₆	—	2	4	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	#6122	1	4	yes	direct product factor
257	M₁₆	—	13	4	no	—
258	M₁₆	—	26	2	no	—
259	M₁₆	—	52	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	13	2	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	13	1	no	—
264	C₂×C₈	—	13	1	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	—	26	4	no	—
267	C₂×C₈	—	52	1	no	—
268	C₂×C₈	C₂²×#261	1	1	yes	characteristic
269	C₂×C₈	C₂²×#261	1	1	yes	characteristic
270	C₄⋊C₄	—	13	2	no	—
271	C₄⋊C₄	—	13	2	no	—
272	C₄⋊C₄	—	26	2	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	26	2	no	—
275	C₄⋊C₄	—	26	2	no	—
276	C₄⋊C₄	—	26	2	no	—
277	C₄²	—	13	1	no	—
278	C₄²	—	13	1	no	—
279	C₄²	—	13	1	no	—
280	C₄²	—	13	1	no	—
281	C₄²	—	26	2	no	—
282	C₄²	—	26	2	no	—
283	C₄²	—	26	2	no	—
284	C₄²	—	26	2	no	—
285	C₁₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
286	C₂³	D₄×#261	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×#6122	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	—	4	4	no	—
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	13	1	no	—
301	C₂×C₄	—	13	1	no	—
302	C₂×C₄	—	13	1	no	—
303	C₂×C₄	—	26	2	no	—
304	C₂×C₄	—	26	2	no	—
305	C₂×C₄	—	52	1	no	—
306	C₂×C₄	—	104	2	no	—
307	C₂×C₄	D₄×#261	1	1	yes	characteristic
308	C₂×C₄	C₂³×#261	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	13	4	no	—
311	C₈	—	26	2	no	—
312	C₈	—	26	1	no	—
313	C₈	—	26	1	no	—
314	C₈	—	52	4	no	—
315	C₈	—	52	2	no	—
316	C₈	C₂×#6122	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	Q₁₆×#261	1	2	yes	—
320	C₂²	C₂×D₄×#261	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	13	1	no	—
325	C₄	—	13	1	no	—
326	C₄	—	13	2	no	—
327	C₄	—	26	1	no	—
328	C₄	—	26	1	no	—
329	C₄	—	52	2	no	—
330	C₄	—	52	2	no	—
331	C₄	#11719621	1	1	yes	characteristic
332	C₄	C₂²×#6122	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	D₄×#6122	1	1	yes	characteristic
335	C₂	C₂×Q₁₆×#261	1	1	yes	characteristic
336	C₂	#13131616	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₁₃×Q₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂²×Q₁₆	#261
hypocenter	C₁₃	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13130476	C₂	semidirect
3. #13130449	C₂	semidirect
4. Q₁₆×#1057	C₂	semidirect
89. #6122	Q₁₆	direct (N × H)
96. C₁₃×Q₁₆	M₁₆	semidirect
152. #1057	C₂×Q₁₆	semidirect
256. Q₁₆	#6122	direct (N × H)
285. C₁₃	M₁₆×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6122	2
#13130476	2
#13130449	2
Q₁₆×#1057	2
#13130393	2
#13130164	2
C₈×#6122	2
C₂×Q₁₆×Dic₁₃	2
M₁₆×Q₁₆	13

Subgroups — Q16×#6122 #16796343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#6122 #16796343