Subgroups — Q₁₆×#6118 #16796344

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×#6118	C₂	1	2	yes	maximal
2	#13130475	C₂	1	2	yes	maximal, complemented
3	#13130447	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#1057	C₂	1	2	yes	maximal, complemented
5	#13130394	C₂	1	1	yes	maximal, characteristic
6	#13130165	C₂	1	2	yes	maximal
7	C₈×#6118	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₁₆×D₁₃	C₄	1	1	yes	characteristic
10	C₄×Q₈×D₁₃	C₂²	1	2	yes	—
11	Q₈×#1057	C₂²	1	4	yes	—
12	#11719080	—	2	4	no	—
13	#11719080	C₂²	1	4	yes	—
14	#11719052	—	2	2	no	—
15	#11719052	C₂²	1	2	yes	—
16	#11719051	C₂²	1	1	yes	characteristic
17	C₄×#6118	—	2	2	no	—
18	C₄×#6118	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
20	Q₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₄×#6107	C₄	1	2	yes	—
22	D₁₃×#107	C₄	1	1	yes	characteristic
23	#11718534	C₂²	1	1	yes	characteristic
24	D₁₃×#103	C₄	1	2	yes	—
25	#11718451	C₂²	1	2	yes	—
26	C₄×Dic₅₂	C₄	1	1	yes	characteristic
27	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
28	Q₈⋊#1057	C₂²	1	4	yes	—
29	#11718254	C₂²	1	2	yes	—
30	#11718254	C₂²	1	2	yes	—
31	#1057⋊C₈	C₂²	1	2	yes	—
32	C₈×#1057	C₂²	1	2	yes	—
33	#11718244	C₂²	1	4	yes	—
34	C₂×Q₈×D₁₃	C₂×C₄	1	2	yes	—
35	C₂×#6118	—	2	2	no	—
36	C₂×#6118	D₄	1	1	yes	characteristic
37	C₂×#6118	D₄	1	1	yes	characteristic
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
40	Q₈×Dic₁₃	C₂³	1	2	yes	—
41	C₂×#6107	C₂×C₄	1	2	yes	—
42	Q₁₆×D₁₃	—	2	4	no	—
43	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊C₄×D₁₃	—	2	2	no	—
47	C₄⋊C₄×D₁₃	C₂×C₄	1	2	yes	—
48	#88141	—	2	2	no	—
49	#88141	C₂³	1	2	yes	—
50	C₄²×D₁₃	—	2	2	no	—
51	C₄²×D₁₃	C₂³	1	1	yes	characteristic
52	C₄×Dic₂₆	—	2	2	no	—
53	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
54	#88111	—	2	1	no	—
55	#88110	—	2	4	no	—
56	#88110	D₄	1	2	yes	—
57	C₄⋊#1057	—	2	4	no	—
58	C₄⋊#1057	—	2	4	no	—
59	C₄⋊#1057	—	2	2	no	—
60	C₄⋊#1057	—	2	1	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	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
66	C₁₃×#103	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
68	#88075	C₂×C₄	1	2	yes	—
69	#88058	C₂×C₄	1	1	yes	characteristic
70	#88056	C₂×C₄	1	2	yes	—
71	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
72	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
73	#88050	C₂×C₄	1	2	yes	—
74	#88047	C₂×C₄	1	2	yes	—
75	C₄×#1055	C₂×C₄	1	1	yes	characteristic
76	M₁₆×Q₁₆	—	13	1	no	maximal, Sylow
77	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
78	Q₈×D₁₃	—	2	4	no	—
79	Q₈×D₁₃	—	4	2	no	—
80	C₂×C₄×D₁₃	—	2	2	no	—
81	C₂×C₄×D₁₃	—	2	2	no	—
82	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
85	C₂×Dic₂₆	—	2	2	no	—
86	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
87	C₂×#1057	—	2	4	no	—
88	C₂×#1057	—	2	2	no	—
89	C₂×#1057	C₂×D₄	1	2	yes	—
90	C₂×#1057	C₂×D₄	1	2	yes	—
91	#6118	—	2	2	no	—
92	#6118	—	4	4	no	—
93	#6118	Q₁₆	1	4	yes	direct product factor
94	C₁₃×Q₁₆	M₁₆	1	4	yes	complemented
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	#6107	—	2	4	no	—
102	C₄⋊Dic₁₃	—	2	2	no	—
103	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
104	#6101	—	2	2	no	—
105	#6101	—	2	2	no	—
106	#6101	—	2	2	no	—
107	C₄×Dic₁₃	—	2	2	no	—
108	C₄×Dic₁₃	—	2	2	no	—
109	C₄×Dic₁₃	—	2	2	no	—
110	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄×Dic₁₃	#41	1	1	yes	characteristic
112	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
113	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
114	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
115	Dic₅₂	—	2	2	no	—
116	C₈×D₁₃	—	2	2	no	—
117	C₈×D₁₃	—	4	1	no	—
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	C₂²×D₁₃	C₄×D₄	1	1	yes	characteristic
127	Q₈×C₁₃	—	2	2	no	—
128	Q₈×C₁₃	C₂×M₁₆	1	4	yes	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×Dic₁₃	—	2	2	no	—
135	C₂×Dic₁₃	—	2	2	no	—
136	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
137	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
139	C₄×D₁₃	—	2	1	no	—
140	C₄×D₁₃	—	2	1	no	—
141	C₄×D₁₃	—	4	1	no	—
142	C₄×D₁₃	—	4	4	no	—
143	C₄×D₁₃	—	8	2	no	—
144	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
145	C₄×D₁₃	#135	1	2	yes	—
146	Dic₂₆	—	2	2	no	—
147	Dic₂₆	—	2	2	no	—
148	Dic₂₆	—	4	4	no	—
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	—	13	1	no	—
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	#261	1	1	yes	characteristic
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	C₂×C₄×D₄	1	1	yes	characteristic
183	C₂×D₁₃	—	2	1	no	—
184	C₂×D₁₃	—	2	1	no	—
185	C₂×D₁₃	C₄×Q₁₆	1	2	yes	—
186	C₄×C₁₃	—	2	4	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	4	2	no	—
189	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
190	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
191	C₄×C₁₃	#568	1	1	yes	characteristic
192	Dic₁₃	—	2	1	no	—
193	Dic₁₃	—	2	1	no	—
194	Dic₁₃	—	4	2	no	—
195	Dic₁₃	—	4	2	no	—
196	Dic₁₃	C₂²×Q₁₆	1	1	yes	characteristic
197	Dic₁₃	C₂×#135	1	1	yes	characteristic
198	C₂²×Q₈	—	13	2	no	—
199	C₂×Q₁₆	—	13	2	no	—
200	C₂×Q₁₆	—	13	1	no	—
201	C₂×Q₁₆	—	26	4	no	—
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	—	13	1	no	—
208	C₄×Q₈	—	13	2	no	—
209	C₄×Q₈	—	13	2	no	—
210	C₄×Q₈	—	13	2	no	—
211	C₄×Q₈	—	26	2	no	—
212	C₄×Q₈	—	26	2	no	—
213	C₄×Q₈	C₂×#261	1	2	yes	—
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	1	no	—
219	#107	—	13	2	no	—
220	#107	C₂×#261	1	1	yes	characteristic
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	C₂×#261	1	2	yes	—
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	2	no	—
237	C₄×C₈	—	13	1	no	—
238	C₄×C₈	—	26	4	no	—
239	C₄×C₈	C₂×#261	1	1	yes	characteristic
240	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
241	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
242	C₂×C₁₃	#2957	1	1	yes	characteristic
243	D₁₃	—	2	2	no	—
244	C₂×Q₈	—	13	2	no	—
245	C₂×Q₈	—	26	2	no	—
246	C₂×Q₈	—	26	4	no	—
247	C₂×Q₈	—	52	2	no	—
248	C₂×Q₈	C₂²×#261	1	2	yes	—
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	13	1	no	—
251	C₂²×C₄	—	13	1	no	—
252	C₂²×C₄	—	26	2	no	—
253	C₂²×C₄	—	26	2	no	—
254	Q₁₆	—	26	4	no	—
255	Q₁₆	—	26	2	no	—
256	Q₁₆	#6118	1	4	yes	direct product factor
257	M₁₆	—	13	4	no	—
258	M₁₆	—	26	2	no	—
259	M₁₆	—	52	4	no	—
260	C₂×C₈	—	13	2	no	—
261	C₂×C₈	—	13	2	no	—
262	C₂×C₈	—	13	1	no	—
263	C₂×C₈	—	13	1	no	—
264	C₂×C₈	—	26	2	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	—	2	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	C₂²×#261	1	2	yes	—
277	C₄²	—	2	2	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	C₂²×#261	1	1	yes	characteristic
285	C₁₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	13	1	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	26	2	no	—
289	Q₈	—	26	2	no	—
290	Q₈	—	52	4	no	—
291	Q₈	C₂×#6118	1	4	yes	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	13	1	no	—
295	C₂×C₄	—	13	2	no	—
296	C₂×C₄	—	13	2	no	—
297	C₂×C₄	—	13	1	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	26	1	no	—
300	C₂×C₄	—	26	1	no	—
301	C₂×C₄	—	26	2	no	—
302	C₂×C₄	—	26	2	no	—
303	C₂×C₄	—	52	1	no	—
304	C₂×C₄	—	52	4	no	—
305	C₂×C₄	—	104	2	no	—
306	C₂×C₄	D₄×#261	1	1	yes	characteristic
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×#6118	1	2	yes	—
317	C₂²	—	13	2	no	—
318	C₂²	—	26	1	no	—
319	C₂²	—	26	1	no	—
320	C₂²	C₂×D₄×#261	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	2	no	—
324	C₄	—	13	1	no	—
325	C₄	—	13	1	no	—
326	C₄	—	26	1	no	—
327	C₄	—	26	1	no	—
328	C₄	—	52	2	no	—
329	C₄	—	52	2	no	—
330	C₄	Q₁₆×#261	1	2	yes	—
331	C₄	C₂²×#6118	1	1	yes	characteristic
332	C₄	#11719629	1	1	yes	characteristic
333	C₂	—	26	2	no	—
334	C₂	D₄×#6118	1	1	yes	characteristic
335	C₂	C₂×Q₁₆×#261	1	1	yes	characteristic
336	C₂	#13131613	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×Q₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂²×D₁₃	C₄×D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₄×Q₁₆	#261
hypocenter	C₁₃	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13130475	C₂	semidirect
3. #13130447	C₂	semidirect
4. Q₁₆×#1057	C₂	semidirect
93. #6118	Q₁₆	direct (N × H)
94. C₁₃×Q₁₆	M₁₆	semidirect
152. #1057	C₂×Q₁₆	semidirect
256. Q₁₆	#6118	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×#6118	2
#13130475	2
#13130447	2
Q₁₆×#1057	2
#13130394	2
#13130165	2
C₈×#6118	2
C₄×Q₁₆×D₁₃	2
M₁₆×Q₁₆	13

Subgroups — Q16×#6118 #16796344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#6118 #16796344