Subgroups — Q₈×C₂²⋊(C₄₃⋊C₈) #25113113

Order 11008.

1356 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₄₃⋊C₈	C₂	1	2	yes	maximal, complemented
2	#19135532	C₂	1	6	yes	maximal, complemented
3	#19134575	C₂	1	3	yes	maximal
4	C₄×C₂²⋊(C₄₃⋊C₈)	C₂	1	3	yes	maximal
5	C₂²×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₄₃⋊C₈	—	2	8	no	—
7	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	6	yes	—
8	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	3	yes	—
9	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₄₃⋊C₈	C₂²	1	6	yes	—
11	#14582654	C₂²	1	6	yes	—
12	#14582654	C₂²	1	6	yes	—
13	C₂³×Q₈×C₄₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₃	—	2	8	no	—
15	C₂×C₄×Q₈×C₄₃	C₄	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₄₃	C₄	1	3	yes	—
17	C₂×C₄×Q₈×C₄₃	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₃	C₄	1	3	yes	—
20	C₂²×C₄²×C₄₃	C₂²	1	3	yes	—
21	C₂²×C₄₃⋊C₈	C₂³	1	6	yes	—
22	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
23	C₄⋊(C₄₃⋊C₈)	—	2	12	no	—
24	C₂²⋊(C₄₃⋊C₈)	—	2	6	no	—
25	C₂²⋊(C₄₃⋊C₈)	Q₈	1	4	yes	direct product factor
26	C₄×C₄₃⋊C₈	—	2	12	no	—
27	C₂²×Q₈×C₄₃	—	2	4	no	—
28	C₂²×Q₈×C₄₃	—	2	4	no	—
29	C₂²×Q₈×C₄₃	C₈	1	8	yes	complemented
30	C₂²×Q₈×C₄₃	C₂×C₄	1	3	yes	—
31	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₃	C₂×C₄	1	3	yes	—
33	C₂³×C₄×C₄₃	C₂×C₄	1	3	yes	—
34	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₄₃	—	2	4	no	—
36	C₄×Q₈×C₄₃	—	2	12	no	—
37	C₄×Q₈×C₄₃	—	2	12	no	—
38	C₄×Q₈×C₄₃	D₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₄₃	—	2	12	no	—
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	3	yes	—
41	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	6	yes	—
42	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	3	yes	—
43	C₂×C₄²×C₄₃	—	2	12	no	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	3	yes	—
45	C₂×C₄²×C₄₃	C₂×C₄	1	3	yes	—
46	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
47	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
48	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×Q₈×C₄₃	—	2	32	no	—
50	C₂×Q₈×C₄₃	—	2	6	no	—
51	C₂×Q₈×C₄₃	—	2	6	no	—
52	C₂×Q₈×C₄₃	C₂²⋊C₄	1	4	yes	—
53	C₂×Q₈×C₄₃	C₂×C₈	1	12	yes	—
54	C₂×Q₈×C₄₃	C₂×C₈	1	4	yes	—
55	C₂×Q₈×C₄₃	M₁₆	1	4	yes	—
56	C₂²×C₄×C₄₃	—	2	6	no	—
57	C₂²×C₄×C₄₃	—	2	6	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	2	6	no	—
61	C₂²×C₄×C₄₃	—	2	6	no	—
62	C₂²×C₄×C₄₃	—	4	3	no	—
63	C₂²×C₄×C₄₃	C₂×C₈	1	12	yes	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
69	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₃	#41	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₄₃	—	2	12	no	—
73	C₄⋊C₄×C₄₃	—	2	6	no	—
74	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	12	yes	—
75	C₄²×C₄₃	—	2	6	no	—
76	C₄²×C₄₃	—	2	6	no	—
77	C₄²×C₄₃	—	2	6	no	—
78	C₄²×C₄₃	C₂×D₄	1	12	yes	—
79	C₂×C₄₃⋊C₈	—	2	2	no	—
80	C₂×C₄₃⋊C₈	—	2	2	no	—
81	C₂×C₄₃⋊C₈	—	2	6	no	—
82	C₂×C₄₃⋊C₈	—	2	6	no	—
83	C₂×C₄₃⋊C₈	—	2	6	no	—
84	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	complemented
85	C₂³×C₄₃	—	2	1	no	—
86	C₂³×C₄₃	—	2	1	no	—
87	C₂³×C₄₃	—	2	1	no	—
88	C₂³×C₄₃	—	2	1	no	—
89	C₂³×C₄₃	C₄×Q₈	1	2	yes	—
90	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
91	C₂³×C₄₃	#131	1	2	yes	—
92	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₄₃	—	2	24	no	—
94	Q₈×C₄₃	C₂²⋊C₈	1	16	yes	complemented
95	C₂×C₄×C₄₃	—	2	1	no	—
96	C₂×C₄×C₄₃	—	2	3	no	—
97	C₂×C₄×C₄₃	—	2	1	no	—
98	C₂×C₄×C₄₃	—	2	3	no	—
99	C₂×C₄×C₄₃	—	2	3	no	—
100	C₂×C₄×C₄₃	—	2	3	no	—
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	2	24	no	—
104	C₂×C₄×C₄₃	—	2	3	no	—
105	C₂×C₄×C₄₃	—	2	3	no	—
106	C₂×C₄×C₄₃	—	2	3	no	—
107	C₂×C₄×C₄₃	—	4	12	no	—
108	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
109	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
110	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	C₂²×C₈	1	6	yes	—
112	C₂×C₄×C₄₃	C₂²×C₈	1	6	yes	—
113	C₂×C₄×C₄₃	C₂×M₁₆	1	6	yes	—
114	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
115	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
118	C₄₃⋊C₈	—	2	4	no	—
119	C₄₃⋊C₈	—	4	6	no	—
120	Q₈×C₂²⋊C₈	—	43	1	no	maximal, Sylow
121	C₂²×C₄₃	—	2	2	no	—
122	C₂²×C₄₃	—	2	2	no	—
123	C₂²×C₄₃	—	2	2	no	—
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	—	2	2	no	—
127	C₂²×C₄₃	C₈×Q₈	1	4	yes	complemented
128	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₂²×C₄₃	#519	1	1	yes	characteristic
131	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
132	C₂²×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
133	C₂²×C₄₃	C₂×#131	1	1	yes	characteristic
134	C₂²×C₄₃	#568	1	1	yes	characteristic
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	—	2	6	no	—
137	C₄×C₄₃	—	2	6	no	—
138	C₄×C₄₃	—	4	3	no	—
139	C₄×C₄₃	C₂×C₂²⋊C₈	1	12	yes	—
140	C₄×C₄₃	#548	1	2	yes	—
141	C₄×C₄₃	D₄×Q₈	1	2	yes	—
142	C₂²×C₄×Q₈	D₄₃	1	1	yes	characteristic
143	C₂×C₈×Q₈	—	43	2	no	—
144	#1984	—	43	6	no	—
145	#1797	—	43	3	no	—
146	C₄×C₂²⋊C₈	—	43	3	no	—
147	C₂×C₄₃	—	2	4	no	—
148	C₂×C₄₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
150	C₂×C₄₃	#2880	1	1	yes	characteristic
151	C₂×C₄₃	C₂×C₈×Q₈	1	1	yes	characteristic
152	C₂×C₄₃	Q₈×M₁₆	1	1	yes	characteristic
153	C₂×C₄₃	#2982	1	1	yes	characteristic
154	C₂×C₄₃	#2984	1	1	yes	characteristic
155	C₂³×Q₈	Dic₄₃	1	1	yes	characteristic
156	C₂×C₄×Q₈	—	2	8	no	—
157	C₂×C₄×Q₈	Dic₄₃	1	1	yes	characteristic
158	C₂×C₄×Q₈	Dic₄₃	1	3	yes	—
159	C₂×C₄×Q₈	C₂×D₄₃	1	3	yes	—
160	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
161	C₂²×C₄⋊C₄	Dic₄₃	1	3	yes	—
162	C₂²×C₄²	C₂×D₄₃	1	3	yes	—
163	C₈×Q₈	—	86	8	no	—
164	C₂×C₄⋊C₈	—	43	6	no	—
165	C₂×C₂²⋊C₈	—	43	1	no	—
166	C₂×C₂²⋊C₈	—	43	3	no	—
167	C₂×C₄×C₈	—	43	6	no	—
168	#336	—	43	6	no	—
169	#336	—	43	6	no	—
170	C₄₃	Q₈×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₈	—	2	4	no	—
172	C₂²×Q₈	—	2	4	no	—
173	C₂²×Q₈	C₄₃⋊C₈	1	8	yes	complemented
174	C₂²×Q₈	C₂×Dic₄₃	1	3	yes	—
175	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
176	C₂³×C₄	C₂×Dic₄₃	1	3	yes	—
177	C₂³×C₄	C₂×Dic₄₃	1	3	yes	—
178	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
179	C₂²×C₈	—	43	2	no	—
180	C₂²×C₈	—	43	6	no	—
181	C₄×Q₈	—	2	4	no	—
182	C₄×Q₈	—	2	12	no	—
183	C₄×Q₈	—	2	12	no	—
184	C₄×Q₈	C₂²⋊D₄₃	1	8	yes	—
185	C₂×C₄⋊C₄	—	2	12	no	—
186	C₂×C₄⋊C₄	C₂×Dic₄₃	1	3	yes	—
187	C₂×C₄⋊C₄	C₂×Dic₄₃	1	6	yes	—
188	C₂×C₄⋊C₄	C₂×Dic₄₃	1	3	yes	—
189	C₂×C₄²	—	2	12	no	—
190	C₂×C₄²	C₂×Dic₄₃	1	3	yes	—
191	C₂×C₄²	C₂×Dic₄₃	1	3	yes	—
192	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
193	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
194	C₄⋊C₈	—	86	12	no	—
195	C₂²⋊C₈	—	43	4	no	—
196	C₂²⋊C₈	—	86	6	no	—
197	C₄×C₈	—	86	12	no	—
198	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
199	C₂×Q₈	—	2	32	no	—
200	C₂×Q₈	—	2	6	no	—
201	C₂×Q₈	—	2	6	no	—
202	C₂×Q₈	C₂×C₄₃⋊C₈	1	12	yes	—
203	C₂×Q₈	C₂×C₄₃⋊C₈	1	4	yes	—
204	C₂×Q₈	#10621877	1	4	yes	—
205	C₂×Q₈	C₂²⋊Dic₄₃	1	4	yes	—
206	C₂²×C₄	—	2	6	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	6	no	—
211	C₂²×C₄	—	2	6	no	—
212	C₂²×C₄	—	4	3	no	—
213	C₂²×C₄	C₂×C₄₃⋊C₈	1	12	yes	—
214	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
215	C₂²×C₄	#10621902	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₄₃	1	3	yes	—
217	C₂²×C₄	C₂²×Dic₄₃	1	3	yes	—
218	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₄₃	1	3	yes	—
220	C₂²×C₄	C₂²×Dic₄₃	1	3	yes	—
221	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
222	C₂×C₈	—	43	4	no	—
223	C₂×C₈	—	86	2	no	—
224	C₂×C₈	—	86	2	no	—
225	C₂×C₈	—	86	6	no	—
226	C₂×C₈	—	86	6	no	—
227	C₂×C₈	—	86	6	no	—
228	C₄⋊C₄	—	2	12	no	—
229	C₄⋊C₄	—	2	6	no	—
230	C₄⋊C₄	C₂²⋊Dic₄₃	1	12	yes	—
231	C₄²	—	2	6	no	—
232	C₄²	—	2	6	no	—
233	C₄²	—	2	6	no	—
234	C₄²	C₂×C₂²⋊D₄₃	1	12	yes	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	Q₈×Dic₄₃	1	2	yes	—
240	C₂³	C₂²×C₄₃⋊C₈	1	2	yes	—
241	C₂³	#13063332	1	2	yes	—
242	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
243	Q₈	—	2	24	no	—
244	Q₈	C₂²⋊(C₄₃⋊C₈)	1	16	yes	direct product factor
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	3	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	3	no	—
249	C₂×C₄	—	2	3	no	—
250	C₂×C₄	—	2	3	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	24	no	—
254	C₂×C₄	—	2	3	no	—
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	4	12	no	—
258	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	6	yes	—
259	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	6	yes	—
260	C₂×C₄	Q₈×Dic₄₃	1	2	yes	—
261	C₂×C₄	C₂²×C₄₃⋊C₈	1	6	yes	—
262	C₂×C₄	C₂²×C₄₃⋊C₈	1	6	yes	—
263	C₂×C₄	C₂×#10621877	1	6	yes	—
264	C₂×C₄	C₂²×C₂²⋊D₄₃	1	2	yes	—
265	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
267	C₂×C₄	#13063373	1	2	yes	—
268	C₈	—	86	4	no	—
269	C₈	—	172	6	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	Q₈×C₄₃⋊C₈	1	4	yes	complemented
277	C₂²	C₂²×C₂²⋊Dic₄₃	1	1	yes	characteristic
278	C₂²	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
279	C₂²	#14583377	1	1	yes	characteristic
280	C₂²	C₂³×C₄₃⋊C₈	1	1	yes	characteristic
281	C₂²	C₂²×#10621877	1	1	yes	characteristic
282	C₂²	C₂×#13063332	1	1	yes	characteristic
283	C₂²	#14583666	1	1	yes	characteristic
284	C₄	—	2	6	no	—
285	C₄	—	2	6	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	3	no	—
288	C₄	C₂×C₂²⋊(C₄₃⋊C₈)	1	12	yes	—
289	C₄	#14583560	1	2	yes	—
290	C₄	Q₈×C₂²⋊D₄₃	1	2	yes	—
291	C₂	—	2	4	no	—
292	C₂	Q₈×C₂²⋊Dic₄₃	1	1	yes	characteristic
293	C₂	C₂²×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
294	C₂	#19143044	1	1	yes	characteristic
295	C₂	C₂×Q₈×C₄₃⋊C₈	1	1	yes	characteristic
296	C₂	Q₈×#10621877	1	1	yes	characteristic
297	C₂	#19143676	1	1	yes	characteristic
298	C₂	#19143684	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₈
Fitting	C₂²×C₄×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₄₃
hypocenter	C₄₃	Q₈×C₂²⋊C₈
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₄₃⋊C₈	C₂	semidirect
2. #19135532	C₂	semidirect
25. C₂²⋊(C₄₃⋊C₈)	Q₈	direct (N × H)
29. C₂²×Q₈×C₄₃	C₈	semidirect
84. C₂×C₄₃⋊C₈	C₂×Q₈	semidirect
94. Q₈×C₄₃	C₂²⋊C₈	semidirect
127. C₂²×C₄₃	C₈×Q₈	semidirect
170. C₄₃	Q₈×C₂²⋊C₈	semidirect
173. C₂²×Q₈	C₄₃⋊C₈	semidirect
244. Q₈	C₂²⋊(C₄₃⋊C₈)	direct (N × H)
276. C₂²	Q₈×C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₄₃⋊C₈	2
#19135532	2
#19134575	2
C₄×C₂²⋊(C₄₃⋊C₈)	2
C₂²×C₄×Q₈×C₄₃	2
Q₈×C₂²⋊C₈	43

Subgroups — Q8×C22⋊(C43⋊C8) #25113113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂²⋊(C₄₃⋊C₈) #25113113