Subgroups — C₂×Q₁₆×Dic₈₆ #25977046

Order 11008.

1830 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₈₆	C₂	1	2	yes	maximal
2	Q₁₆×Dic₈₆	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583304	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583303	C₂	1	2	yes	maximal
5	C₂×#14583194	C₂	1	2	yes	maximal
6	C₂×#14583192	C₂	1	4	yes	maximal
7	C₂×C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×Dic₄₃	C₂	1	2	yes	maximal
9	C₂×C₄×Q₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
10	Q₈×Dic₈₆	—	2	8	no	—
11	Q₈×Dic₈₆	C₂²	1	16	yes	—
12	C₂×#13063325	—	2	2	no	—
13	C₂×#13063325	C₂²	1	2	yes	—
14	C₂×#13063324	—	2	4	no	—
15	C₂×#13063324	C₂²	1	4	yes	—
16	C₂×Q₈×Dic₄₃	C₂²	1	4	yes	—
17	C₂×C₄×Dic₈₆	—	2	2	no	—
18	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
19	#14583304	C₂²	1	8	yes	—
20	#14583303	C₂²	1	16	yes	—
21	#14583194	C₂²	1	16	yes	—
22	#14583192	C₂²	1	32	yes	—
23	C₈×Dic₈₆	C₂²	1	8	yes	—
24	Q₁₆×Dic₄₃	C₂²	1	16	yes	—
25	C₂×#13063259	C₂²	1	2	yes	—
26	C₂×#13063259	C₂²	1	2	yes	—
27	C₂×#13063258	C₂²	1	4	yes	—
28	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
29	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
30	C₂×#13063249	C₂²	1	4	yes	—
31	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
32	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
34	C₄×Q₁₆×C₄₃	C₂²	1	8	yes	—
35	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#103×C₄₃	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₂²×Dic₈₆	—	2	2	no	—
39	C₂²×Dic₈₆	—	2	4	no	—
40	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
41	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
42	#13063325	—	2	4	no	—
43	#13063325	—	2	16	no	—
44	#13063325	C₂³	1	8	yes	—
45	#13063324	—	2	32	no	—
46	#13063324	—	2	8	no	—
47	#13063324	C₂³	1	16	yes	—
48	Q₈×Dic₄₃	—	2	8	no	—
49	Q₈×Dic₄₃	C₂³	1	16	yes	—
50	C₄×Dic₈₆	—	2	16	no	—
51	C₄×Dic₈₆	D₄	1	4	yes	—
52	C₄×Dic₈₆	C₂³	1	4	yes	—
53	C₂×C₄⋊Dic₄₃	—	2	2	no	—
54	C₂×C₄⋊Dic₄₃	—	2	4	no	—
55	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
56	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
57	C₂×Dic₄₃⋊C₄	—	2	4	no	—
58	C₂×Dic₄₃⋊C₄	D₄	1	2	yes	—
59	C₂×C₄×Dic₄₃	—	2	4	no	—
60	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
61	#13063259	C₂³	1	8	yes	—
62	#13063259	C₂³	1	8	yes	—
63	#13063258	C₂³	1	16	yes	—
64	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
65	Dic₄₃⋊C₈	C₂³	1	8	yes	—
66	#13063249	C₂³	1	16	yes	—
67	C₈×Dic₄₃	C₂³	1	8	yes	—
68	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₄₃	Q₈	1	8	yes	complemented
70	C₂×Q₁₆×C₄₃	C₂³	1	4	yes	—
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₄×Q₈×C₄₃	—	2	4	no	—
74	C₄×Q₈×C₄₃	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
76	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
77	C₂×C₄²×C₄₃	—	2	2	no	—
78	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
79	#107×C₄₃	C₂³	1	4	yes	—
80	#103×C₄₃	C₂³	1	8	yes	—
81	C₄×C₈×C₄₃	C₂³	1	4	yes	—
82	C₂×Q₈×C₄₃	—	2	2	no	—
83	C₂×Q₈×C₄₃	—	2	4	no	—
84	C₂×Q₈×C₄₃	C₂×Q₈	1	8	yes	—
85	C₂×Q₈×C₄₃	C₂⁴	1	4	yes	—
86	C₂²×C₄×C₄₃	—	2	2	no	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₄₃	—	2	4	no	—
92	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
93	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
94	C₂×Dic₈₆	—	2	8	no	—
95	C₂×Dic₈₆	—	2	16	no	—
96	C₂×Dic₈₆	—	8	8	no	—
97	C₂×Dic₈₆	—	8	4	no	—
98	C₂×Dic₈₆	Q₁₆	1	8	yes	direct product factor
99	C₂×Dic₈₆	C₂×D₄	1	4	yes	—
100	Q₁₆×C₄₃	C₂×Q₈	1	16	yes	complemented
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
103	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
104	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
105	C₄⋊C₄×C₄₃	—	2	8	no	—
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	C₂⁴	1	4	yes	—
108	C₄²×C₄₃	—	2	8	no	—
109	C₄²×C₄₃	C₂×D₄	1	2	yes	—
110	C₄²×C₄₃	C₂⁴	1	2	yes	—
111	C₄⋊Dic₄₃	—	2	8	no	—
112	C₄⋊Dic₄₃	—	2	4	no	—
113	C₄⋊Dic₄₃	—	2	16	no	—
114	C₄⋊Dic₄₃	C₂×D₄	1	4	yes	—
115	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
116	Dic₄₃⋊C₄	—	2	16	no	—
117	Dic₄₃⋊C₄	C₂×D₄	1	8	yes	—
118	C₄×Dic₄₃	—	2	16	no	—
119	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
120	C₄×Dic₄₃	C₂⁴	1	4	yes	—
121	C₂×C₄₃⋊C₈	—	2	4	no	—
122	C₂×C₄₃⋊C₈	C₂⁴	1	4	yes	—
123	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₄₃	—	2	8	no	—
125	Q₈×C₄₃	—	2	4	no	—
126	Q₈×C₄₃	C₂²×Q₈	1	8	yes	—
127	C₂×C₄×C₄₃	—	2	8	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	4	no	—
130	C₂×C₄×C₄₃	—	2	4	no	—
131	C₂×C₄×C₄₃	—	4	4	no	—
132	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
133	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
134	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
137	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
138	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
139	C₂×Dic₄₃	—	2	4	no	—
140	C₂×Dic₄₃	—	2	8	no	—
141	C₂×Dic₄₃	—	4	8	no	—
142	C₂×Dic₄₃	C₂×Q₁₆	1	8	yes	—
143	C₂×Dic₄₃	C₂²×D₄	1	4	yes	—
144	C₂×Dic₄₃	C₂²×D₄	1	4	yes	—
145	Dic₈₆	—	8	16	no	—
146	Dic₈₆	—	8	8	no	—
147	Dic₈₆	C₂×Q₁₆	1	16	yes	direct product factor
148	C₈×C₄₃	—	2	2	no	—
149	C₈×C₄₃	C₂²×Q₈	1	4	yes	—
150	C₄₃⋊C₈	—	2	4	no	—
151	C₂×Q₈×Q₁₆	—	43	1	no	maximal, Sylow
152	C₂²×C₄₃	D₄×Q₈	1	2	yes	—
153	C₂²×C₄₃	C₂²×Q₁₆	1	2	yes	—
154	C₂²×C₄₃	#576	1	2	yes	—
155	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
156	C₄×C₄₃	—	2	8	no	—
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	4	4	no	—
159	C₄×C₄₃	D₄×Q₈	1	2	yes	—
160	C₄×C₄₃	C₂²×Q₁₆	1	4	yes	—
161	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
162	C₄×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
163	Dic₄₃	—	2	4	no	—
164	Dic₄₃	—	4	8	no	—
165	Dic₄₃	C₂²×Q₁₆	1	8	yes	—
166	C₂×Q₈²	—	43	2	no	—
167	Q₈×Q₁₆	—	43	16	no	—
168	C₂×#500	—	43	2	no	—
169	C₂×#500	—	43	1	no	—
170	C₂×#477	—	43	4	no	—
171	C₂×#477	—	43	2	no	—
172	C₂×C₈×Q₈	—	43	1	no	—
173	C₂×C₄×Q₁₆	—	43	2	no	—
174	C₂×C₄×Q₁₆	D₄₃	1	1	yes	characteristic
175	C₂×C₄₃	Q₈×Q₁₆	1	4	yes	complemented
176	C₂×C₄₃	C₂×D₄×Q₈	1	1	yes	characteristic
177	C₂×C₄₃	C₂³×Q₁₆	1	1	yes	characteristic
178	C₂×C₄₃	C₂×#576	1	1	yes	characteristic
179	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
180	Q₈²	—	43	16	no	—
181	Q₈²	—	86	8	no	—
182	C₂×C₄⋊Q₈	—	43	4	no	—
183	C₂×C₄⋊Q₈	—	43	2	no	—
184	C₂×C₄⋊Q₈	—	86	2	no	—
185	C₂×C₄⋊Q₈	—	86	4	no	—
186	C₂×C₄×Q₈	—	43	1	no	—
187	C₂×C₄×Q₈	—	43	4	no	—
188	C₂×C₄×Q₈	—	86	2	no	—
189	C₂×C₄×Q₈	C₂×D₄₃	1	2	yes	—
190	#500	—	43	16	no	—
191	#500	—	43	8	no	—
192	#477	—	43	16	no	—
193	#477	—	43	32	no	—
194	C₈×Q₈	—	43	8	no	—
195	C₄×Q₁₆	—	43	16	no	—
196	C₄×Q₁₆	C₂×D₄₃	1	8	yes	—
197	C₂×#107	—	43	2	no	—
198	C₂×#107	—	43	2	no	—
199	C₂×#107	—	43	4	no	—
200	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
201	C₂×C₄⋊C₈	—	43	2	no	—
202	C₂×C₄⋊C₈	—	43	1	no	—
203	C₂×#103	—	43	4	no	—
204	C₂×#103	C₂×D₄₃	1	2	yes	—
205	C₂×C₄×C₈	—	43	2	no	—
206	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
207	C₄₃	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₈	—	43	1	no	—
209	C₂²×Q₈	—	86	4	no	—
210	C₂²×Q₈	—	86	2	no	—
211	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
212	C₂×Q₁₆	Dic₈₆	1	8	yes	direct product factor
213	C₂×Q₁₆	C₂²×D₄₃	1	4	yes	—
214	C₂²×C₈	—	43	2	no	—
215	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	43	16	no	—
218	C₄⋊Q₈	—	43	8	no	—
219	C₄⋊Q₈	—	86	16	no	—
220	C₄⋊Q₈	—	86	8	no	—
221	C₄⋊Q₈	—	86	32	no	—
222	C₄⋊Q₈	—	86	4	no	—
223	C₄×Q₈	—	2	4	no	—
224	C₄×Q₈	—	43	4	no	—
225	C₄×Q₈	—	43	4	no	—
226	C₄×Q₈	—	43	16	no	—
227	C₄×Q₈	—	86	16	no	—
228	C₄×Q₈	—	86	8	no	—
229	C₄×Q₈	C₂²×D₄₃	1	8	yes	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	43	2	no	—
232	C₂×C₄⋊C₄	—	43	1	no	—
233	C₂×C₄⋊C₄	—	43	4	no	—
234	C₂×C₄⋊C₄	—	86	4	no	—
235	C₂×C₄⋊C₄	—	86	4	no	—
236	C₂×C₄⋊C₄	—	86	2	no	—
237	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
238	C₂×C₄²	—	2	2	no	—
239	C₂×C₄²	—	43	2	no	—
240	C₂×C₄²	—	86	4	no	—
241	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
242	#107	—	43	8	no	—
243	#107	—	43	8	no	—
244	#107	—	43	16	no	—
245	#107	C₂²×D₄₃	1	4	yes	—
246	C₄⋊C₈	—	43	8	no	—
247	C₄⋊C₈	—	43	4	no	—
248	#103	—	43	16	no	—
249	#103	C₂²×D₄₃	1	8	yes	—
250	C₄×C₈	—	43	8	no	—
251	C₄×C₈	C₂²×D₄₃	1	4	yes	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	43	8	no	—
255	C₂×Q₈	—	43	4	no	—
256	C₂×Q₈	—	86	16	no	—
257	C₂×Q₈	—	86	8	no	—
258	C₂×Q₈	—	344	4	no	—
259	C₂×Q₈	—	344	8	no	—
260	C₂×Q₈	C₂×Dic₈₆	1	8	yes	—
261	C₂×Q₈	C₂³×D₄₃	1	4	yes	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	43	2	no	—
265	C₂²×C₄	—	43	2	no	—
266	C₂²×C₄	—	86	4	no	—
267	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
268	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
269	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
270	Q₁₆	C₂×Dic₈₆	1	16	yes	direct product factor
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	43	4	no	—
273	C₂×C₈	—	86	4	no	—
274	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
275	C₂×C₈	C₂³×D₄₃	1	2	yes	—
276	C₂×C₈	C₂³×D₄₃	1	2	yes	—
277	C₄⋊C₄	—	2	8	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	43	8	no	—
280	C₄⋊C₄	—	43	4	no	—
281	C₄⋊C₄	—	43	8	no	—
282	C₄⋊C₄	—	86	16	no	—
283	C₄⋊C₄	—	86	16	no	—
284	C₄⋊C₄	—	86	4	no	—
285	C₄⋊C₄	—	86	8	no	—
286	C₄⋊C₄	C₂³×D₄₃	1	4	yes	—
287	C₄²	—	2	8	no	—
288	C₄²	—	43	4	no	—
289	C₄²	—	43	4	no	—
290	C₄²	—	86	16	no	—
291	C₄²	D₄×D₄₃	1	2	yes	—
292	C₄²	C₂³×D₄₃	1	2	yes	—
293	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
294	Q₈	—	2	8	no	—
295	Q₈	—	2	4	no	—
296	Q₈	—	43	16	no	—
297	Q₈	—	344	8	no	—
298	Q₈	—	344	16	no	—
299	Q₈	C₂²×Dic₈₆	1	8	yes	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	43	8	no	—
306	C₂×C₄	—	43	4	no	—
307	C₂×C₄	—	43	4	no	—
308	C₂×C₄	—	86	4	no	—
309	C₂×C₄	—	86	8	no	—
310	C₂×C₄	—	172	8	no	—
311	C₂×C₄	Q₁₆×D₄₃	1	4	yes	—
312	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
313	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
315	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
316	C₂×C₄	#13063372	1	2	yes	—
317	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	86	4	no	—
320	C₈	C₂²×Dic₈₆	1	4	yes	—
321	C₂²	D₄×Dic₈₆	1	2	yes	—
322	C₂²	C₂×Q₁₆×D₄₃	1	2	yes	—
323	C₂²	#14583701	1	2	yes	—
324	C₂²	C₂²×D₄×D₄₃	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	2	8	no	—
327	C₄	—	4	4	no	—
328	C₄	—	43	8	no	—
329	C₄	—	86	4	no	—
330	C₄	—	172	8	no	—
331	C₄	D₄×Dic₈₆	1	2	yes	—
332	C₄	C₂×Q₁₆×D₄₃	1	4	yes	—
333	C₄	C₂³×Dic₈₆	1	1	yes	characteristic
334	C₄	C₂×#13063372	1	1	yes	characteristic
335	C₂	Q₁₆×Dic₈₆	1	4	yes	direct product factor
336	C₂	C₂×D₄×Dic₈₆	1	1	yes	characteristic
337	C₂	C₂²×Q₁₆×D₄₃	1	1	yes	characteristic
338	C₂	C₂×#14583701	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₄×Q₁₆×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₂×C₄×Q₁₆	D₄₃
hypocenter	C₄₃	C₂×Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₈₆	C₂	direct (N × H)
69. C₂×Q₁₆×C₄₃	Q₈	semidirect
98. C₂×Dic₈₆	Q₁₆	direct (N × H)
100. Q₁₆×C₄₃	C₂×Q₈	semidirect
147. Dic₈₆	C₂×Q₁₆	direct (N × H)
175. C₂×C₄₃	Q₈×Q₁₆	semidirect
207. C₄₃	C₂×Q₈×Q₁₆	semidirect
212. C₂×Q₁₆	Dic₈₆	direct (N × H)
270. Q₁₆	C₂×Dic₈₆	direct (N × H)
335. C₂	Q₁₆×Dic₈₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₈₆	2
Q₁₆×Dic₈₆	2
C₂×#14583304	2
C₂×#14583303	2
C₂×#14583194	2
C₂×#14583192	2
C₂×C₈×Dic₈₆	2
C₂×Q₁₆×Dic₄₃	2
C₂×C₄×Q₁₆×C₄₃	2
C₂×Q₈×Q₁₆	43

Subgroups — C2×Q16×Dic86 #25977046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₁₆×Dic₈₆ #25977046