Subgroups — Q₈×#13063243 #25113169

Order 11008.

970 conjugacy classes of subgroups.

The subgroup lattice has 330 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	1	yes	maximal, characteristic
2	Q₈×#10621877	C₂	1	1	yes	maximal, characteristic
3	#19135584	C₂	1	3	yes	maximal
4	#19135583	C₂	1	3	yes	maximal
5	#19134588	C₂	1	3	yes	maximal
6	C₄×#13063243	C₂	1	3	yes	maximal
7	Q₈×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₈₆	—	2	1	no	—
9	Q₈×Dic₈₆	—	2	3	no	—
10	Q₈×Dic₈₆	—	2	1	no	—
11	Q₈×Dic₈₆	—	2	3	no	—
12	C₂×#13063325	C₂²	1	3	yes	—
13	C₂×#13063324	C₄	1	6	yes	—
14	C₂×Q₈×Dic₄₃	C₄	1	2	yes	—
15	C₂×C₄×Dic₈₆	C₂²	1	3	yes	—
16	#14583030	—	2	3	no	—
17	Q₈×C₄₃⋊C₈	—	2	1	no	—
18	#14582923	C₂²	1	3	yes	—
19	C₂×#13063243	C₂²	1	3	yes	—
20	C₂×#13063243	C₂²	1	1	yes	characteristic
21	C₄×#10621877	C₂²	1	3	yes	—
22	#14582667	C₂²	1	3	yes	—
23	#14582667	C₂²	1	6	yes	—
24	#14582667	C₂²	1	3	yes	—
25	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#446	—	2	3	no	—
27	C₈×Q₈×C₄₃	—	2	1	no	—
28	C₄₃×#423	C₂²	1	3	yes	—
29	C₄×M₁₆×C₄₃	C₂²	1	3	yes	—
30	C₂²×Dic₈₆	C₂×C₄	1	6	yes	—
31	C₂²×Dic₈₆	C₂³	1	1	yes	characteristic
32	C₂×#10621877	C₂³	1	3	yes	—
33	C₂×#10621877	C₂³	1	1	yes	characteristic
34	#13063325	—	2	3	no	—
35	#13063325	—	2	3	no	—
36	#13063325	—	2	3	no	—
37	#13063325	—	2	3	no	—
38	#13063324	—	2	12	no	—
39	#13063324	—	2	6	no	—
40	#13063324	—	2	6	no	—
41	Q₈×Dic₄₃	—	2	6	no	—
42	Q₈×Dic₄₃	—	2	2	no	—
43	C₄×Dic₈₆	—	2	3	no	—
44	C₄×Dic₈₆	—	2	3	no	—
45	C₄×Dic₈₆	—	2	3	no	—
46	C₄×Dic₈₆	—	2	3	no	—
47	C₂×C₄⋊Dic₄₃	C₂×C₄	1	6	yes	—
48	C₂×C₄⋊Dic₄₃	C₂³	1	3	yes	—
49	C₂×Dic₄₃⋊C₄	C₂×C₄	1	6	yes	—
50	C₂×C₄×Dic₄₃	C₂×C₄	1	6	yes	—
51	C₄⋊(C₄₃⋊C₈)	—	2	3	no	—
52	C₄⋊(C₄₃⋊C₈)	—	2	3	no	—
53	#13063243	—	2	6	no	—
54	#13063243	Q₈	1	4	yes	direct product factor
55	#13063236	—	2	3	no	—
56	C₄×C₄₃⋊C₈	—	2	3	no	—
57	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₂×M₁₆×C₄₃	C₂³	1	3	yes	—
59	C₂×M₁₆×C₄₃	C₂³	1	1	yes	characteristic
60	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
61	C₄×Q₈×C₄₃	D₄	1	3	yes	—
62	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
63	C₄×Q₈×C₄₃	D₄	1	3	yes	—
64	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	3	yes	—
65	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
66	C₄⋊C₈×C₄₃	—	2	3	no	—
67	C₄⋊C₈×C₄₃	—	2	3	no	—
68	#97×C₄₃	—	2	3	no	—
69	C₄×C₈×C₄₃	—	2	3	no	—
70	C₂×Q₈×C₄₃	—	2	4	no	—
71	C₂×Q₈×C₄₃	C₂²⋊C₄	1	3	yes	—
72	C₂×Q₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
77	C₂²×Dic₄₃	C₂²×C₄	1	6	yes	—
78	C₂×Dic₈₆	—	2	1	no	—
79	C₂×Dic₈₆	—	2	1	no	—
80	C₂×Dic₈₆	—	2	1	no	—
81	C₂×Dic₈₆	—	2	1	no	—
82	C₂×Dic₈₆	—	2	6	no	—
83	C₂×Dic₈₆	—	2	6	no	—
84	C₂×Dic₈₆	—	8	6	no	—
85	C₂×Dic₈₆	C₂×Q₈	1	2	yes	—
86	C₂×Dic₈₆	#41	1	2	yes	—
87	M₁₆×C₄₃	—	2	1	no	—
88	M₁₆×C₄₃	—	2	3	no	—
89	M₁₆×C₄₃	—	2	3	no	—
90	M₁₆×C₄₃	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	2	3	no	—
93	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	6	yes	—
94	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	3	yes	—
95	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	3	yes	—
96	C₄²×C₄₃	C₂×D₄	1	3	yes	—
97	C₄²×C₄₃	C₂×D₄	1	3	yes	—
98	C₄²×C₄₃	C₂×D₄	1	3	yes	—
99	C₄²×C₄₃	C₂×D₄	1	3	yes	—
100	C₄⋊Dic₄₃	—	2	6	no	—
101	C₄⋊Dic₄₃	—	2	6	no	—
102	C₄⋊Dic₄₃	—	2	3	no	—
103	C₄⋊Dic₄₃	—	2	3	no	—
104	Dic₄₃⋊C₄	—	2	6	no	—
105	Dic₄₃⋊C₄	—	2	6	no	—
106	C₄×Dic₄₃	—	2	6	no	—
107	C₄×Dic₄₃	—	2	6	no	—
108	#10621877	—	2	1	no	—
109	#10621877	—	2	3	no	—
110	#10621877	—	2	3	no	—
111	#10621877	C₂×Q₈	1	2	yes	—
112	C₂×C₄₃⋊C₈	—	2	1	no	—
113	C₂×C₄₃⋊C₈	—	2	3	no	—
114	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
115	Q₈×C₄₃	—	2	6	no	—
116	Q₈×C₄₃	#101	1	4	yes	complemented
117	C₂×C₄×C₄₃	—	2	6	no	—
118	C₂×C₄×C₄₃	—	4	3	no	—
119	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	3	yes	—
120	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	3	yes	—
121	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	3	yes	—
122	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	3	yes	—
123	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	—	2	2	no	—
131	C₂×Dic₄₃	—	2	6	no	—
132	C₂×Dic₄₃	—	4	6	no	—
133	C₂×Dic₄₃	C₄×Q₈	1	4	yes	—
134	Dic₈₆	—	2	2	no	—
135	Dic₈₆	—	2	2	no	—
136	Dic₈₆	—	2	2	no	—
137	Dic₈₆	—	2	2	no	—
138	Dic₈₆	—	8	6	no	—
139	Dic₈₆	—	8	6	no	—
140	C₈×C₄₃	—	2	2	no	—
141	C₈×C₄₃	—	4	3	no	—
142	C₄₃⋊C₈	—	2	2	no	—
143	C₄₃⋊C₈	—	4	3	no	—
144	Q₈×#101	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₂²×C₄₃	#519	1	1	yes	characteristic
150	C₄×C₄₃	—	2	3	no	—
151	C₄×C₄₃	—	2	3	no	—
152	C₄×C₄₃	—	2	3	no	—
153	C₄×C₄₃	C₂×#101	1	6	yes	—
154	C₄×C₄₃	#548	1	1	yes	characteristic
155	C₄×C₄₃	#548	1	1	yes	characteristic
156	C₄×C₄₃	D₄×Q₈	1	1	yes	characteristic
157	C₄×C₄₃	D₄×Q₈	1	1	yes	characteristic
158	Dic₄₃	—	2	4	no	—
159	Dic₄₃	—	4	6	no	—
160	C₂×Q₈²	—	43	1	no	—
161	Q₈×M₁₆	—	43	1	no	—
162	Q₈×M₁₆	D₄₃	1	1	yes	characteristic
163	#1997	—	43	3	no	—
164	#1997	—	43	3	no	—
165	#1801	—	43	3	no	—
166	C₄×#101	—	43	3	no	—
167	C₂×C₄₃	—	2	1	no	—
168	C₂×C₄₃	—	2	1	no	—
169	C₂×C₄₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄₃	C₂²×#101	1	1	yes	characteristic
171	C₂×C₄₃	#2888	1	1	yes	characteristic
172	Q₈²	—	86	1	no	—
173	Q₈²	—	86	3	no	—
174	Q₈²	—	86	1	no	—
175	Q₈²	—	86	3	no	—
176	C₂×C₄⋊Q₈	—	43	6	no	—
177	C₂×C₄⋊Q₈	—	43	3	no	—
178	C₂×C₄×Q₈	—	43	2	no	—
179	C₂×C₄×Q₈	—	43	3	no	—
180	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
181	#446	—	2	3	no	—
182	#446	—	86	3	no	—
183	C₈×Q₈	—	2	1	no	—
184	C₈×Q₈	—	86	1	no	—
185	#423	—	43	3	no	—
186	#423	C₂×D₄₃	1	3	yes	—
187	C₂×#101	—	43	1	no	—
188	C₂×#101	—	43	3	no	—
189	C₄×M₁₆	—	43	3	no	—
190	C₄×M₁₆	C₂×D₄₃	1	3	yes	—
191	#343	—	43	6	no	—
192	#343	—	43	3	no	—
193	#343	—	43	3	no	—
194	C₄₃	Q₈×#101	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	43	6	no	—
196	C₂²×Q₈	—	43	1	no	—
197	C₂²×Q₈	C₄×D₄₃	1	1	yes	characteristic
198	C₂×M₁₆	—	43	1	no	—
199	C₂×M₁₆	—	43	3	no	—
200	C₂×M₁₆	C₂²×D₄₃	1	3	yes	—
201	C₂×M₁₆	C₂²×D₄₃	1	1	yes	characteristic
202	C₄⋊Q₈	—	86	12	no	—
203	C₄⋊Q₈	—	86	6	no	—
204	C₄⋊Q₈	—	86	6	no	—
205	C₄⋊Q₈	—	86	3	no	—
206	C₄⋊Q₈	—	86	3	no	—
207	C₄⋊Q₈	—	86	3	no	—
208	C₄⋊Q₈	—	86	3	no	—
209	C₄×Q₈	—	86	6	no	—
210	C₄×Q₈	—	86	2	no	—
211	C₄×Q₈	—	86	3	no	—
212	C₄×Q₈	—	86	3	no	—
213	C₄×Q₈	—	86	3	no	—
214	C₄×Q₈	—	86	3	no	—
215	C₄×Q₈	D₁₇₂	1	1	yes	characteristic
216	C₄×Q₈	D₁₇₂	1	3	yes	—
217	C₄×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
218	C₄×Q₈	C₂²⋊D₄₃	1	3	yes	—
219	C₂×C₄⋊C₄	—	43	6	no	—
220	C₂×C₄⋊C₄	—	43	6	no	—
221	C₂×C₄⋊C₄	—	43	3	no	—
222	C₂×C₄⋊C₄	C₄×D₄₃	1	3	yes	—
223	C₂×C₄²	—	43	6	no	—
224	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
225	C₄⋊C₈	—	2	3	no	—
226	C₄⋊C₈	—	2	3	no	—
227	C₄⋊C₈	—	86	3	no	—
228	C₄⋊C₈	—	86	3	no	—
229	#101	—	43	4	no	—
230	#101	—	86	6	no	—
231	#97	—	2	3	no	—
232	#97	—	86	3	no	—
233	C₄×C₈	—	2	3	no	—
234	C₄×C₈	—	86	3	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	43	2	no	—
237	C₂×Q₈	—	43	2	no	—
238	C₂×Q₈	—	86	6	no	—
239	C₂×Q₈	—	86	6	no	—
240	C₂×Q₈	—	86	1	no	—
241	C₂×Q₈	—	86	1	no	—
242	C₂×Q₈	—	86	1	no	—
243	C₂×Q₈	—	86	1	no	—
244	C₂×Q₈	—	344	6	no	—
245	C₂×Q₈	(C₂×D₄₃)⋊C₄	1	3	yes	—
246	C₂×Q₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	—	43	2	no	—
248	C₂²×C₄	—	43	6	no	—
249	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
250	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
251	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
252	M₁₆	—	2	1	no	—
253	M₁₆	—	2	3	no	—
254	M₁₆	—	2	3	no	—
255	M₁₆	—	43	2	no	—
256	M₁₆	—	86	1	no	—
257	M₁₆	—	86	3	no	—
258	M₁₆	—	86	3	no	—
259	M₁₆	Q₈×D₄₃	1	2	yes	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	3	no	—
262	C₂×C₈	—	86	1	no	—
263	C₂×C₈	—	86	3	no	—
264	C₄⋊C₄	—	86	6	no	—
265	C₄⋊C₄	—	86	6	no	—
266	C₄⋊C₄	—	86	6	no	—
267	C₄⋊C₄	—	86	6	no	—
268	C₄⋊C₄	—	86	3	no	—
269	C₄⋊C₄	—	86	3	no	—
270	C₄⋊C₄	(C₂×D₄₃)⋊C₄	1	6	yes	—
271	C₄⋊C₄	(C₂×D₄₃)⋊C₄	1	3	yes	—
272	C₄⋊C₄	(C₂×D₄₃)⋊C₄	1	3	yes	—
273	C₄²	—	86	6	no	—
274	C₄²	—	86	6	no	—
275	C₄²	C₂×D₁₇₂	1	3	yes	—
276	C₄²	C₂×D₁₇₂	1	3	yes	—
277	C₄²	C₂×C₂²⋊D₄₃	1	3	yes	—
278	C₄²	C₂×C₂²⋊D₄₃	1	3	yes	—
279	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
280	Q₈	—	2	6	no	—
281	Q₈	—	86	2	no	—
282	Q₈	—	86	2	no	—
283	Q₈	—	86	2	no	—
284	Q₈	—	86	2	no	—
285	Q₈	—	344	6	no	—
286	Q₈	—	344	6	no	—
287	Q₈	#13063243	1	4	yes	direct product factor
288	C₂×C₄	—	2	6	no	—
289	C₂×C₄	—	4	3	no	—
290	C₂×C₄	—	43	4	no	—
291	C₂×C₄	—	86	2	no	—
292	C₂×C₄	—	86	2	no	—
293	C₂×C₄	—	86	6	no	—
294	C₂×C₄	—	172	6	no	—
295	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	3	yes	—
296	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	3	yes	—
297	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	3	yes	—
298	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	3	yes	—
299	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
302	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
303	C₂×C₄	#13063372	1	1	yes	characteristic
304	C₂×C₄	#13063373	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	4	3	no	—
307	C₈	—	86	2	no	—
308	C₈	—	172	3	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	C₂²×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
312	C₂²	C₄×Q₈×D₄₃	1	1	yes	characteristic
313	C₂²	#14583375	1	1	yes	characteristic
314	C₄	—	2	3	no	—
315	C₄	—	2	3	no	—
316	C₄	—	2	3	no	—
317	C₄	—	86	4	no	—
318	C₄	—	172	6	no	—
319	C₄	C₂×#13063243	1	6	yes	—
320	C₄	#14583556	1	1	yes	characteristic
321	C₄	#14583561	1	1	yes	characteristic
322	C₄	Q₈×C₂²⋊D₄₃	1	1	yes	characteristic
323	C₄	Q₈×D₁₇₂	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	Q₈×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
327	C₂	C₂²×#13063243	1	1	yes	characteristic
328	C₂	#19143073	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×(C₂×D₄₃)⋊C₄
commutator, prime radical	C₂³×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	Q₈×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	Q₈×M₁₆	D₄₃
hypocenter	C₄₃	Q₈×#101
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
54. #13063243	Q₈	direct (N × H)
116. Q₈×C₄₃	#101	semidirect
194. C₄₃	Q₈×#101	semidirect
287. Q₈	#13063243	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
Q₈×#10621877	2
#19135584	2
#19135583	2
#19134588	2
C₄×#13063243	2
Q₈×M₁₆×C₄₃	2
Q₈×#101	43

Subgroups — Q8×#13063243 #25113169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13063243 #25113169