Subgroups — Q₈×#13063245 #25113182

Order 11008.

1314 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#19135555	C₂	1	3	yes	maximal, complemented
4	#19135547	C₂	1	3	yes	maximal, complemented
5	#19134592	C₂	1	3	yes	maximal
6	C₄×#13063245	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×D₄₃	—	2	1	no	—
9	Q₈×D₁₇₂	—	2	4	no	—
10	Q₈×D₁₇₂	C₄	1	8	yes	complemented
11	C₂×#13063325	C₂²	1	3	yes	—
12	C₂×#13063305	—	2	3	no	—
13	C₂×C₄×D₁₇₂	C₂²	1	3	yes	—
14	Q₈×C₄₃⋊C₈	—	2	4	no	—
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	3	yes	—
16	C₂×#13063245	C₂²	1	3	yes	—
17	C₂×#13063245	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
19	#14582661	C₂²	1	3	yes	—
20	#14582661	C₂²	1	6	yes	—
21	#14582661	C₂²	1	3	yes	—
22	C₂×C₄×Q₈×C₄₃	—	2	1	no	—
23	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#389	—	2	3	no	—
25	C₄₃×#384	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄×C₄₃	C₂²	1	3	yes	—
27	C₂²×Dic₈₆	—	2	3	no	—
28	C₂×Q₈×D₄₃	—	2	8	no	—
29	C₂×Q₈×D₄₃	—	4	4	no	—
30	C₂×Q₈×D₄₃	—	4	4	no	—
31	C₂²×D₁₇₂	C₂³	1	1	yes	characteristic
32	C₂²×C₄×D₄₃	—	2	3	no	—
33	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
34	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
35	#13063325	—	2	6	no	—
36	#13063325	C₂×C₄	1	12	yes	—
37	#13063305	—	2	24	no	—
38	C₄×D₁₇₂	—	2	6	no	—
39	C₄×D₁₇₂	C₂×C₄	1	12	yes	—
40	C₂×C₄⋊Dic₄₃	—	2	3	no	—
41	C₂×C₄⋊Dic₄₃	C₂³	1	3	yes	—
42	C₂×(C₂×D₄₃)⋊C₄	—	2	3	no	—
43	C₄⋊(C₄₃⋊C₈)	—	2	6	no	—
44	#13063245	—	2	6	no	—
45	#13063245	Q₈	1	4	yes	direct product factor
46	C₄×C₄₃⋊C₈	—	2	6	no	—
47	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
48	C₄×Q₈×C₄₃	—	4	4	no	—
49	C₄×Q₈×C₄₃	C₂×C₄	1	4	yes	—
50	C₄×Q₈×C₄₃	D₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
52	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
53	C₂×C₄⋊C₄×C₄₃	D₄	1	3	yes	—
54	C₂×C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
55	C₂×C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	—	2	3	no	—
57	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
58	#95×C₄₃	—	2	3	no	—
59	C₂³×D₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	C₂²⋊C₄	1	4	yes	—
61	C₂×Q₈×C₄₃	D₈	1	4	yes	—
62	C₂×Q₈×C₄₃	SD₁₆	1	4	yes	—
63	C₂²×C₄×C₄₃	—	2	1	no	—
64	C₂²×C₄×C₄₃	—	2	3	no	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₃	—	2	3	no	—
69	Q₈×D₄₃	—	4	32	no	—
70	C₂×D₁₇₂	—	2	1	no	—
71	C₂×D₁₇₂	—	2	1	no	—
72	C₂×D₁₇₂	—	2	1	no	—
73	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
74	C₂×D₁₇₂	C₂×Q₈	1	2	yes	—
75	C₂×D₁₇₂	#41	1	2	yes	—
76	C₂×C₄×D₄₃	—	2	12	no	—
77	C₂×C₄×D₄₃	—	4	6	no	—
78	C₂×C₄×D₄₃	—	4	6	no	—
79	C₂×Dic₈₆	—	2	12	no	—
80	C₂×Dic₈₆	—	4	6	no	—
81	C₂×Dic₈₆	—	4	6	no	—
82	C₄⋊C₄×C₄₃	—	2	1	no	—
83	C₄⋊C₄×C₄₃	—	2	3	no	—
84	C₄⋊C₄×C₄₃	—	2	3	no	—
85	C₄⋊C₄×C₄₃	—	4	6	no	—
86	C₄⋊C₄×C₄₃	—	8	3	no	—
87	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	12	yes	—
88	C₄⋊C₄×C₄₃	C₂×Q₈	1	2	yes	complemented
89	C₄²×C₄₃	—	4	6	no	—
90	C₄²×C₄₃	C₂²×C₄	1	6	yes	—
91	C₄²×C₄₃	C₂×D₄	1	6	yes	—
92	(C₂×D₄₃)⋊C₄	—	2	12	no	—
93	C₄⋊Dic₄₃	—	2	12	no	—
94	C₄⋊Dic₄₃	—	2	3	no	—
95	C₄⋊Dic₄₃	C₂²×C₄	1	6	yes	—
96	C₂×C₄₃⋊C₈	—	2	1	no	—
97	C₂×C₄₃⋊C₈	—	2	1	no	—
98	C₂×C₄₃⋊C₈	—	2	3	no	—
99	C₂×C₄₃⋊C₈	—	2	3	no	—
100	C₂×C₄₃⋊C₈	—	2	3	no	—
101	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	complemented
102	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
103	C₂²×D₄₃	—	2	2	no	—
104	C₂²×D₄₃	—	2	2	no	—
105	C₂²×D₄₃	—	2	2	no	—
106	C₂²×D₄₃	—	4	1	no	—
107	C₂²×D₄₃	—	4	1	no	—
108	C₂²×D₄₃	—	4	1	no	—
109	C₂²×D₄₃	—	4	1	no	—
110	Q₈×C₄₃	#102	1	16	yes	complemented
111	C₂×C₄×C₄₃	—	2	3	no	—
112	C₂×C₄×C₄₃	—	2	3	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	4	1	no	—
115	C₂×C₄×C₄₃	—	4	1	no	—
116	C₂×C₄×C₄₃	—	4	3	no	—
117	C₂×C₄×C₄₃	—	4	3	no	—
118	C₂×C₄×C₄₃	—	4	3	no	—
119	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
120	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
121	C₂×C₄×C₄₃	C₂×D₈	1	6	yes	—
122	C₂×C₄×C₄₃	C₂×SD₁₆	1	6	yes	—
123	C₂×C₄×C₄₃	C₂³×C₄	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	6	no	—
130	C₂×Dic₄₃	—	4	3	no	—
131	C₂×Dic₄₃	—	4	3	no	—
132	D₁₇₂	—	2	2	no	—
133	D₁₇₂	—	2	2	no	—
134	D₁₇₂	—	2	2	no	—
135	D₁₇₂	C₄×Q₈	1	4	yes	complemented
136	C₄×D₄₃	—	4	24	no	—
137	Dic₈₆	—	4	24	no	—
138	C₄₃⋊C₈	—	2	2	no	—
139	C₄₃⋊C₈	—	4	3	no	—
140	Q₈×#102	—	43	1	no	maximal, Sylow
141	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₃	#548	1	1	yes	characteristic
143	C₂²×C₄₃	D₄×Q₈	1	1	yes	characteristic
144	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
145	C₂²×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
146	C₂²×C₄₃	#577	1	1	yes	characteristic
147	C₂²×C₄₃	#578	1	1	yes	characteristic
148	C₂×D₄₃	—	2	4	no	—
149	C₂×D₄₃	—	4	2	no	—
150	C₂×D₄₃	—	4	2	no	—
151	C₂×D₄₃	—	4	2	no	—
152	C₂×D₄₃	—	4	2	no	—
153	C₂×D₄₃	—	4	2	no	—
154	C₂×D₄₃	—	4	2	no	—
155	C₄×C₄₃	—	2	6	no	—
156	C₄×C₄₃	—	4	2	no	—
157	C₄×C₄₃	—	8	3	no	—
158	C₄×C₄₃	C₂×#102	1	12	yes	—
159	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
160	C₄×C₄₃	#519	1	1	yes	characteristic
161	C₄×C₄₃	#548	1	1	yes	characteristic
162	C₄×C₄₃	D₄×Q₈	1	1	yes	characteristic
163	Dic₄₃	—	4	6	no	—
164	C₂×D₄×Q₈	—	43	1	no	—
165	C₂×C₈×Q₈	—	43	1	no	—
166	Q₈×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
167	#1990	—	43	3	no	—
168	#1988	—	43	3	no	—
169	#1802	—	43	3	no	—
170	C₄×#102	—	43	3	no	—
171	C₂×C₄₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₄₃	C₂²×#102	1	1	yes	characteristic
173	C₂×C₄₃	#2898	1	1	yes	characteristic
174	C₂×C₄₃	#3358	1	1	yes	characteristic
175	C₂×C₄₃	#3359	1	1	yes	characteristic
176	C₂×C₄₃	Q₈×D₈	1	1	yes	characteristic
177	C₂×C₄₃	Q₈×SD₁₆	1	1	yes	characteristic
178	D₄₃	—	4	4	no	—
179	C₂³×Q₈	—	86	1	no	—
180	D₄×Q₈	—	43	8	no	—
181	D₄×Q₈	—	86	4	no	—
182	C₂×C₄⋊Q₈	—	43	3	no	—
183	C₂×C₂²⋊Q₈	—	86	3	no	—
184	C₂×C₄×Q₈	—	2	1	no	—
185	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
186	C₂×C₄×D₄	—	43	3	no	—
187	C₈×Q₈	—	86	4	no	—
188	C₂×C₄⋊C₈	—	43	3	no	—
189	C₂×#102	—	43	1	no	—
190	C₂×#102	—	43	3	no	—
191	C₂×C₄×C₈	—	43	3	no	—
192	#389	—	2	3	no	—
193	#384	C₂×D₄₃	1	3	yes	—
194	C₄×C₄⋊C₄	C₂×D₄₃	1	3	yes	—
195	#340	—	43	6	no	—
196	#340	—	43	3	no	—
197	#340	—	43	3	no	—
198	C₄₃	Q₈×#102	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	—	86	8	no	—
200	C₂²×Q₈	—	86	3	no	—
201	C₂²×Q₈	—	172	4	no	—
202	C₂²×Q₈	—	172	4	no	—
203	C₂²×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
204	C₂²×D₄	—	43	1	no	—
205	C₂³×C₄	—	86	3	no	—
206	C₂²×C₈	—	43	1	no	—
207	C₂²×C₈	—	43	3	no	—
208	C₄⋊Q₈	—	43	12	no	—
209	C₄⋊Q₈	—	86	6	no	—
210	C₂²⋊Q₈	—	86	24	no	—
211	C₄×Q₈	—	4	4	no	—
212	C₄×Q₈	C₄×D₄₃	1	4	yes	—
213	C₄×Q₈	D₁₇₂	1	4	yes	—
214	C₄×D₄	—	43	12	no	—
215	C₄×D₄	—	86	6	no	—
216	C₂×C₄⋊C₄	—	2	3	no	—
217	C₂×C₄⋊C₄	—	2	3	no	—
218	C₂×C₄⋊C₄	—	43	3	no	—
219	C₂×C₄⋊C₄	—	86	3	no	—
220	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	3	yes	—
221	C₂×C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
222	C₂×C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	—	86	3	no	—
224	C₂×C₄²	—	2	3	no	—
225	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
226	C₄⋊C₈	—	86	6	no	—
227	#102	—	43	4	no	—
228	#102	—	86	6	no	—
229	C₄×C₈	—	86	6	no	—
230	#95	—	2	3	no	—
231	C₂⁴	—	86	1	no	—
232	C₂×Q₈	—	86	12	no	—
233	C₂×Q₈	—	172	32	no	—
234	C₂×Q₈	—	172	6	no	—
235	C₂×Q₈	—	172	6	no	—
236	C₂×Q₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
237	C₂×Q₈	#10621882	1	4	yes	—
238	C₂×Q₈	#10621884	1	4	yes	—
239	C₂×D₄	—	43	2	no	—
240	C₂×D₄	—	43	2	no	—
241	C₂×D₄	—	43	2	no	—
242	C₂×D₄	—	86	1	no	—
243	C₂×D₄	—	86	1	no	—
244	C₂×D₄	—	86	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	3	no	—
247	C₂²×C₄	—	86	3	no	—
248	C₂²×C₄	—	86	12	no	—
249	C₂²×C₄	—	172	6	no	—
250	C₂²×C₄	—	172	6	no	—
251	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
252	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
253	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
254	C₂×C₈	—	43	2	no	—
255	C₂×C₈	—	86	1	no	—
256	C₂×C₈	—	86	1	no	—
257	C₂×C₈	—	86	3	no	—
258	C₂×C₈	—	86	3	no	—
259	C₂×C₈	—	86	3	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	3	no	—
262	C₄⋊C₄	—	2	3	no	—
263	C₄⋊C₄	—	4	6	no	—
264	C₄⋊C₄	—	8	3	no	—
265	C₄⋊C₄	—	43	6	no	—
266	C₄⋊C₄	—	86	12	no	—
267	C₄⋊C₄	—	86	3	no	—
268	C₄⋊C₄	(C₂×D₄₃)⋊C₄	1	12	yes	—
269	C₄⋊C₄	Q₈×D₄₃	1	2	yes	complemented
270	C₂²⋊C₄	—	86	12	no	—
271	C₄²	—	4	6	no	—
272	C₄²	C₂×C₄×D₄₃	1	6	yes	—
273	C₄²	C₂×D₁₇₂	1	6	yes	—
274	C₂³	—	86	2	no	—
275	C₂³	—	86	2	no	—
276	C₂³	—	86	2	no	—
277	C₂³	—	172	1	no	—
278	C₂³	—	172	1	no	—
279	C₂³	—	172	1	no	—
280	C₂³	—	172	1	no	—
281	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
282	Q₈	—	172	24	no	—
283	Q₈	#13063245	1	16	yes	direct product factor
284	D₄	—	43	4	no	—
285	D₄	—	86	2	no	—
286	D₄	—	86	2	no	—
287	D₄	—	86	2	no	—
288	C₂×C₄	—	2	3	no	—
289	C₂×C₄	—	2	3	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	3	no	—
294	C₂×C₄	—	4	3	no	—
295	C₂×C₄	—	4	3	no	—
296	C₂×C₄	—	86	6	no	—
297	C₂×C₄	—	172	3	no	—
298	C₂×C₄	—	172	3	no	—
299	C₂×C₄	—	172	24	no	—
300	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
301	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
302	C₂×C₄	C₂×#10621882	1	6	yes	—
303	C₂×C₄	C₂×#10621884	1	6	yes	—
304	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
306	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
307	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
308	C₂×C₄	#13063372	1	1	yes	characteristic
309	C₂×C₄	#13063373	1	1	yes	characteristic
310	C₈	—	86	2	no	—
311	C₈	—	172	3	no	—
312	C₂²	—	86	4	no	—
313	C₂²	—	172	2	no	—
314	C₂²	—	172	2	no	—
315	C₂²	—	172	2	no	—
316	C₂²	—	172	2	no	—
317	C₂²	—	172	2	no	—
318	C₂²	—	172	2	no	—
319	C₂²	C₂²×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
320	C₂²	#14583556	1	1	yes	characteristic
321	C₂²	Q₈×C₂²⋊D₄₃	1	1	yes	characteristic
322	C₂²	C₂²×#10621882	1	1	yes	characteristic
323	C₂²	C₂²×#10621884	1	1	yes	characteristic
324	C₂²	#14583710	1	1	yes	characteristic
325	C₂²	#14583714	1	1	yes	characteristic
326	C₄	—	2	6	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	3	no	—
329	C₄	—	172	6	no	—
330	C₄	C₂×#13063245	1	12	yes	—
331	C₄	C₄×Q₈×D₄₃	1	1	yes	characteristic
332	C₄	#14583375	1	1	yes	characteristic
333	C₄	#14583561	1	1	yes	characteristic
334	C₄	Q₈×D₁₇₂	1	1	yes	characteristic
335	C₂	—	172	4	no	—
336	C₂	Q₈×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
337	C₂	C₂²×#13063245	1	1	yes	characteristic
338	C₂	#19143125	1	1	yes	characteristic
339	C₂	#19147226	1	1	yes	characteristic
340	C₂	#19147233	1	1	yes	characteristic
341	C₂	Q₈×#10621882	1	1	yes	characteristic
342	C₂	Q₈×#10621884	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	Q₈×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₁₇₂	C₂³
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₄₃
hypocenter	C₄₃	Q₈×#102
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×C₄₃⋊C₈	C₂	semidirect
3. #19135555	C₂	semidirect
4. #19135547	C₂	semidirect
7. Q₈×C₄⋊C₄×C₄₃	C₂	semidirect
10. Q₈×D₁₇₂	C₄	semidirect
45. #13063245	Q₈	direct (N × H)
88. C₄⋊C₄×C₄₃	C₂×Q₈	semidirect
101. C₂×C₄₃⋊C₈	C₂×Q₈	semidirect
110. Q₈×C₄₃	#102	semidirect
135. D₁₇₂	C₄×Q₈	semidirect
166. Q₈×C₄⋊C₄	D₄₃	semidirect
198. C₄₃	Q₈×#102	semidirect
269. C₄⋊C₄	Q₈×D₄₃	semidirect
283. Q₈	#13063245	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×D₁₇₂	2
C₂×Q₈×C₄₃⋊C₈	2
#19135555	2
#19135547	2
#19134592	2
C₄×#13063245	2
Q₈×C₄⋊C₄×C₄₃	2
Q₈×#102	43

Subgroups — Q8×#13063245 #25113182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13063245 #25113182