Subgroups — C₄×D₄⋊(C₄₃⋊C₈) #24987189

Order 11008.

774 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
4	#19132436	C₂	1	2	yes	maximal, complemented
5	C₄²×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
8	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
9	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
10	#14582654	—	2	1	no	—
11	D₄⋊(C₄₃⋊C₈)	C₄	1	8	yes	direct product factor
12	C₂×C₄×D₄×C₄₃	C₄	1	2	yes	—
13	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₃	—	2	1	no	—
15	C₄×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊C₄×C₄₃	—	2	1	no	—
18	C₄×C₂²⋊C₄×C₄₃	—	2	2	no	—
19	C₄³×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₈	—	2	2	no	—
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	complemented
24	C₄×C₄₃⋊C₈	—	2	8	no	—
25	C₄×C₄₃⋊C₈	—	2	2	no	—
26	C₄×C₄₃⋊C₈	—	4	2	no	—
27	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	complemented
28	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₃	—	2	2	no	—
30	C₂³×C₄×C₄₃	—	2	1	no	—
31	C₄×D₄×C₄₃	—	2	4	no	—
32	C₄×D₄×C₄₃	—	2	2	no	—
33	C₄×D₄×C₄₃	C₈	1	8	yes	complemented
34	C₄×D₄×C₄₃	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
39	C₂×C₄²×C₄₃	—	2	2	no	—
40	C₂×C₄²×C₄₃	—	2	1	no	—
41	C₂×C₄²×C₄₃	—	4	2	no	—
42	C₂×C₄²×C₄₃	—	4	2	no	—
43	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₃	D₄	1	2	yes	—
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	#95×C₄₃	—	2	2	no	—
49	#95×C₄₃	—	2	1	no	—
50	C₂⁴×C₄₃	—	2	1	no	—
51	C₂×D₄×C₄₃	—	2	1	no	—
52	C₂×D₄×C₄₃	—	2	2	no	—
53	C₂×D₄×C₄₃	C₄²	1	2	yes	—
54	C₂×D₄×C₄₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₄₃	—	2	8	no	—
56	C₂²×C₄×C₄₃	—	2	4	no	—
57	C₂²×C₄×C₄₃	—	2	1	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	4	4	no	—
60	C₂²×C₄×C₄₃	—	4	4	no	—
61	C₂²×C₄×C₄₃	—	4	1	no	—
62	C₂²×C₄×C₄₃	—	4	1	no	—
63	C₂²×C₄×C₄₃	—	4	2	no	—
64	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₄₃	—	2	1	no	—
70	C₄⋊C₄×C₄₃	—	2	2	no	—
71	C₄⋊C₄×C₄₃	C₄²	1	2	yes	—
72	C₄⋊C₄×C₄₃	C₂×C₈	1	4	yes	—
73	C₂²⋊C₄×C₄₃	—	2	8	no	—
74	C₂²⋊C₄×C₄₃	—	2	4	no	—
75	C₄²×C₄₃	—	2	4	no	—
76	C₄²×C₄₃	—	2	2	no	—
77	C₄²×C₄₃	—	4	4	no	—
78	C₄²×C₄₃	—	4	1	no	—
79	C₄²×C₄₃	—	4	1	no	—
80	C₄²×C₄₃	C₂×C₈	1	4	yes	—
81	C₄²×C₄₃	M₁₆	1	4	yes	—
82	C₄²×C₄₃	D₈	1	2	yes	—
83	C₄²×C₄₃	SD₁₆	1	2	yes	—
84	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₄₃	#41	1	2	yes	—
86	C₂×C₄₃⋊C₈	—	2	8	no	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	4	2	no	—
90	C₂×C₄₃⋊C₈	—	4	2	no	—
91	C₂³×C₄₃	—	2	2	no	—
92	C₂³×C₄₃	—	2	4	no	—
93	C₂³×C₄₃	—	4	1	no	—
94	C₂³×C₄₃	—	4	1	no	—
95	C₂³×C₄₃	—	4	2	no	—
96	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	D₄×C₄₃	—	2	2	no	—
98	D₄×C₄₃	—	2	4	no	—
99	D₄×C₄₃	C₄×C₈	1	4	yes	complemented
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	2	4	no	—
105	C₂×C₄×C₄₃	—	4	2	no	—
106	C₂×C₄×C₄₃	—	4	1	no	—
107	C₂×C₄×C₄₃	—	4	1	no	—
108	C₂×C₄×C₄₃	—	4	2	no	—
109	C₂×C₄×C₄₃	—	4	16	no	—
110	C₂×C₄×C₄₃	—	4	4	no	—
111	C₂×C₄×C₄₃	—	4	4	no	—
112	C₂×C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
113	C₂×C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
114	C₂×C₄×C₄₃	#102	1	4	yes	—
115	C₂×C₄×C₄₃	#104	1	4	yes	—
116	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#135	1	2	yes	—
125	C₄₃⋊C₈	—	2	8	no	—
126	C₄₃⋊C₈	—	4	4	no	—
127	C₄×D₄⋊C₈	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	4	no	—
129	C₂²×C₄₃	—	4	4	no	—
130	C₂²×C₄₃	—	4	4	no	—
131	C₂²×C₄₃	—	4	2	no	—
132	C₂²×C₄₃	—	4	2	no	—
133	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₃	C₂×C₂²⋊C₈	1	2	yes	—
135	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
136	C₂²×C₄₃	#416	1	1	yes	characteristic
137	C₂²×C₄₃	C₂×#104	1	2	yes	—
138	C₄×C₄₃	—	2	4	no	—
139	C₄×C₄₃	—	2	2	no	—
140	C₄×C₄₃	—	4	4	no	—
141	C₄×C₄₃	—	4	2	no	—
142	C₄×C₄₃	D₄⋊C₈	1	8	yes	complemented
143	C₄×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
144	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
145	C₄×C₄₃	#431	1	2	yes	—
146	C₄×C₄₃	C₄×D₈	1	2	yes	—
147	C₄×C₄₃	C₄×SD₁₆	1	2	yes	—
148	C₄²×D₄	D₄₃	1	1	yes	characteristic
149	C₄×C₄⋊C₈	—	43	1	no	—
150	C₄²×C₈	—	43	1	no	—
151	C₂×D₄⋊C₈	—	43	2	no	—
152	#1279	—	43	2	no	—
153	C₂×C₄₃	—	4	4	no	—
154	C₂×C₄₃	C₂×D₄⋊C₈	1	2	yes	—
155	C₂×C₄₃	#1476	1	2	yes	—
156	C₂×C₄₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
157	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
158	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
159	C₂×C₄×D₄	Dic₄₃	1	2	yes	—
160	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
161	C₂²×C₄²	—	2	1	no	—
162	C₂×C₄⋊C₈	—	43	2	no	—
163	C₂×C₄×C₈	—	43	2	no	—
164	C₂×C₄×C₈	—	86	2	no	—
165	C₂×C₄×C₈	—	86	1	no	—
166	C₄×C₄⋊C₄	Dic₄₃	1	2	yes	—
167	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
168	C₄×C₂²⋊C₄	—	2	1	no	—
169	C₄×C₂²⋊C₄	—	2	2	no	—
170	C₄³	C₂×D₄₃	1	1	yes	characteristic
171	#336	—	86	1	no	—
172	D₄⋊C₈	—	43	8	no	—
173	C₄₃	C₄×D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
174	C₂²×D₄	C₂×Dic₄₃	1	1	yes	characteristic
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	—	2	1	no	—
177	C₂²×C₈	—	86	2	no	—
178	C₂²×C₈	—	86	2	no	—
179	C₄×D₄	—	2	4	no	—
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	C₄₃⋊C₈	1	8	yes	complemented
182	C₄×D₄	C₄×D₄₃	1	4	yes	—
183	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₂²⋊C₄	—	2	1	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	—	4	2	no	—
190	C₂×C₄²	—	4	2	no	—
191	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
194	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
196	C₄⋊C₈	—	43	4	no	—
197	C₄⋊C₈	—	86	2	no	—
198	C₄×C₈	—	43	4	no	—
199	C₄×C₈	—	86	8	no	—
200	C₄×C₈	—	86	2	no	—
201	C₄×C₈	—	172	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	1	no	—
204	C₂⁴	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	C₂×C₄₃⋊C₈	1	4	yes	—
208	C₂×D₄	C₄×Dic₄₃	1	2	yes	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
219	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
223	C₂×C₈	—	86	8	no	—
224	C₂×C₈	—	86	4	no	—
225	C₂×C₈	—	86	4	no	—
226	C₂×C₈	—	172	2	no	—
227	C₂×C₈	—	172	2	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₂×C₄₃⋊C₈	1	4	yes	—
231	C₄⋊C₄	C₄×Dic₄₃	1	2	yes	—
232	C₂²⋊C₄	—	2	8	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	4	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂×C₄₃⋊C₈	1	4	yes	—
240	C₄²	#10621877	1	4	yes	—
241	C₄²	#10621882	1	2	yes	—
242	C₄²	#10621883	1	2	yes	—
243	C₄²	C₂×C₄×D₄₃	1	2	yes	—
244	C₄²	#10621898	1	2	yes	—
245	C₂³	—	2	4	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	4	1	no	—
248	C₂³	—	4	1	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
251	D₄	—	2	2	no	—
252	D₄	—	2	4	no	—
253	D₄	C₄×C₄₃⋊C₈	1	4	yes	complemented
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	4	yes	—
267	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	4	yes	—
268	C₂×C₄	#13063246	1	4	yes	—
269	C₂×C₄	#13063252	1	4	yes	—
270	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
271	C₂×C₄	#13063284	1	1	yes	characteristic
272	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
273	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
274	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
275	C₂×C₄	C₂×#10621877	1	2	yes	—
276	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
277	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
278	C₂×C₄	#13063345	1	2	yes	—
279	C₈	—	86	8	no	—
280	C₈	—	172	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	C₄×C₂²⋊Dic₄₃	1	1	yes	characteristic
287	C₂²	C₂×C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
288	C₂²	C₂×#13063246	1	1	yes	characteristic
289	C₂²	#14582895	1	1	yes	characteristic
290	C₂²	C₂×#13063252	1	2	yes	—
291	C₄	—	2	4	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	4	2	no	—
295	C₄	D₄⋊(C₄₃⋊C₈)	1	8	yes	direct product factor
296	C₄	C₂×C₄×C₄₃⋊C₈	1	1	yes	characteristic
297	C₄	C₄×#10621877	1	1	yes	characteristic
298	C₄	#14582952	1	2	yes	—
299	C₄	C₄×#10621882	1	2	yes	—
300	C₄	C₄×#10621883	1	2	yes	—
301	C₂	—	4	4	no	—
302	C₂	C₂×D₄⋊(C₄₃⋊C₈)	1	2	yes	—
303	C₂	#19132977	1	2	yes	—
304	C₂	C₄×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
305	C₂	C₄×#13063252	1	1	yes	characteristic
306	C₂	C₄×#13063246	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄²×D₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₃	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄²×D₄	D₄₃
hypocenter	C₄₃	C₄×D₄⋊C₈
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
1. C₄×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
2. C₄²×C₄₃⋊C₈	C₂	semidirect
4. #19132436	C₂	semidirect
11. D₄⋊(C₄₃⋊C₈)	C₄	direct (N × H)
23. C₄⋊(C₄₃⋊C₈)	C₂×C₄	semidirect
27. C₄×C₄₃⋊C₈	C₂×C₄	semidirect
33. C₄×D₄×C₄₃	C₈	semidirect
99. D₄×C₄₃	C₄×C₈	semidirect
142. C₄×C₄₃	D₄⋊C₈	semidirect
173. C₄₃	C₄×D₄⋊C₈	semidirect
181. C₄×D₄	C₄₃⋊C₈	semidirect
253. D₄	C₄×C₄₃⋊C₈	semidirect
295. C₄	D₄⋊(C₄₃⋊C₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₃⋊C₈)	2
C₄²×C₄₃⋊C₈	2
C₂×D₄⋊(C₄₃⋊C₈)	2
#19132436	2
C₄²×D₄×C₄₃	2
C₄×D₄⋊C₈	43

Subgroups — C4×D4⋊(C43⋊C8) #24987189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₄⋊(C₄₃⋊C₈) #24987189