Subgroups — C₈×#14519799 #29649393

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11758222	C₂	1	2	yes	maximal, complemented
2	C₄×#14519799	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₂₄₄	C₂	1	2	yes	maximal, complemented
4	C₈×#11758190	C₂	1	4	yes	maximal, complemented
5	C₈×D₄₈₈	C₂	1	2	yes	maximal, complemented
6	#21444567	C₂	1	1	yes	maximal, characteristic, complemented
7	#21444202	C₂	1	2	yes	maximal, complemented
8	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14519799	C₄	1	1	yes	characteristic
10	C₄×#11758222	C₂²	1	2	yes	—
11	C₈×Dic₁₂₂	C₂²	1	4	yes	—
12	C₄×Dic₂₄₄	C₄	1	1	yes	characteristic
13	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
14	C₄×#11758190	C₄	1	2	yes	—
15	C₄×#11758190	C₂²	1	2	yes	—
16	C₄×D₄₈₈	C₄	1	1	yes	characteristic
17	C₄×D₄₈₈	C₂²	1	1	yes	characteristic
18	C₈×D₂₄₄	C₂²	1	4	yes	—
19	C₈×C₂²⋊D₆₁	—	2	4	no	—
20	#16839815	C₂²	1	2	yes	—
21	#16839813	—	2	2	no	—
22	#16839800	C₄	1	1	yes	characteristic
23	#16839759	C₄	1	2	yes	—
24	C₄×C₈×D₆₁	—	2	2	no	—
25	#16839515	C₂²	1	2	yes	—
26	#16839513	C₂²	1	2	yes	—
27	#16839494	C₂²	1	4	yes	—
28	#16839491	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₈²×C₆₁	C₂²	1	2	yes	—
33	C₈²×C₆₁	C₂²	1	2	yes	—
34	C₂×#11758222	C₂×C₄	1	2	yes	—
35	#14519799	C₈	1	8	yes	direct product factor
36	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
37	C₂×#11758190	C₂×C₄	1	2	yes	—
38	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₆₁	—	2	4	no	—
40	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₂₂	C₂³	1	2	yes	—
42	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
43	C₄×D₂₄₄	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₆₁	—	2	2	no	—
45	C₄×C₂²⋊D₆₁	—	2	2	no	—
46	#14519736	C₂×C₄	1	2	yes	—
47	#14519734	—	2	2	no	—
48	C₄²×D₆₁	—	2	2	no	—
49	#14519711	C₂×C₄	1	1	yes	characteristic
50	#14519711	C₂×C₄	1	1	yes	characteristic
51	#14519709	C₂×C₄	1	2	yes	—
52	C₄⋊#92418	C₂³	1	4	yes	—
53	#14519706	—	2	4	no	—
54	#14519702	C₂×C₄	1	2	yes	—
55	#14519702	C₂×C₄	1	2	yes	—
56	#14519699	C₂×C₄	1	2	yes	—
57	#14519699	C₂×C₄	1	2	yes	—
58	C₂²⋊#92418	—	2	2	no	—
59	#14519689	—	2	4	no	—
60	C₄×#92418	—	2	2	no	—
61	C₈×Dic₆₁	—	2	4	no	—
62	C₂²×C₈×C₆₁	—	2	1	no	—
63	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₆₁	D₄	1	2	yes	—
66	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₁	—	2	1	no	—
68	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
72	C₄×C₈×C₆₁	C₂³	1	2	yes	—
73	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₁	C₂³	1	2	yes	—
76	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₆₁	—	2	2	no	—
80	#11758222	—	2	4	no	—
81	#11758222	C₂×C₈	1	8	yes	complemented
82	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₆₁	—	2	2	no	—
84	C₂×C₄×D₆₁	—	2	2	no	—
85	C₂×C₄×D₆₁	—	2	2	no	—
86	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₆₁	—	2	4	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	—	2	2	no	—
90	C₂×C₈×C₆₁	—	2	4	no	—
91	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	complemented
92	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
98	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₁	#41	1	1	yes	characteristic
101	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
102	C₂²⋊Dic₆₁	—	2	2	no	—
103	#11758198	—	2	2	no	—
104	#11758198	—	2	2	no	—
105	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
106	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
107	#11758196	—	2	2	no	—
108	#11758196	—	2	2	no	—
109	C₄×Dic₆₁	—	2	2	no	—
110	C₄×Dic₆₁	—	2	2	no	—
111	C₄×Dic₆₁	—	2	2	no	—
112	C₂×#92418	—	2	4	no	—
113	Dic₂₄₄	C₂×C₈	1	4	yes	complemented
114	D₄₈₈	C₂×C₈	1	4	yes	complemented
115	#11758190	C₂×C₈	1	8	yes	complemented
116	C₈×D₆₁	—	4	8	no	—
117	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₆₁	—	2	2	no	—
119	C₂×C₄×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₆₁	#131	1	2	yes	—
126	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₆₁	—	2	8	no	—
130	C₂×Dic₆₁	—	2	2	no	—
131	C₂×Dic₆₁	—	2	2	no	—
132	C₂×Dic₆₁	—	2	2	no	—
133	D₂₄₄	—	2	2	no	—
134	D₂₄₄	C₂²×C₈	1	4	yes	—
135	C₄×D₆₁	—	2	8	no	—
136	C₄×D₆₁	—	4	4	no	—
137	C₄×D₆₁	—	4	4	no	—
138	Dic₁₂₂	—	2	2	no	—
139	Dic₁₂₂	C₂²×C₈	1	4	yes	—
140	C₈×C₆₁	—	2	4	no	—
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	—	2	1	no	—
143	C₈×C₆₁	—	2	2	no	—
144	C₈×C₆₁	C₂²×C₈	1	2	yes	—
145	C₈×C₆₁	C₂²×C₈	1	2	yes	—
146	C₈×C₆₁	#135	1	4	yes	complemented
147	#92418	—	4	4	no	—
148	C₈×#135	—	61	1	no	maximal, Sylow
149	C₂²×C₆₁	—	2	1	no	—
150	C₂²×C₆₁	—	2	1	no	—
151	C₂²×C₆₁	C₈×D₄	1	2	yes	—
152	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×D₆₁	—	2	4	no	—
154	C₂×D₆₁	—	4	2	no	—
155	C₂×D₆₁	—	4	2	no	—
156	C₄×C₆₁	—	2	1	no	—
157	C₄×C₆₁	—	2	1	no	—
158	C₄×C₆₁	C₈×D₄	1	2	yes	—
159	C₄×C₆₁	#443	1	2	yes	—
160	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
161	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
162	C₄×C₆₁	C₂×#135	1	2	yes	—
163	Dic₆₁	—	2	4	no	—
164	Dic₆₁	—	4	2	no	—
165	Dic₆₁	—	4	2	no	—
166	C₈×#41	—	61	2	no	—
167	C₄×#135	—	61	1	no	—
168	C₈×Q₁₆	—	61	2	no	—
169	C₈×SD₁₆	—	61	4	no	—
170	C₈×D₈	—	61	2	no	—
171	#1564	—	61	1	no	—
172	#1476	—	61	2	no	—
173	C₂×C₈²	D₆₁	1	1	yes	characteristic, complemented
174	C₂×C₆₁	—	2	2	no	—
175	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
176	C₂×C₆₁	C₄×#135	1	1	yes	characteristic
177	C₂×C₆₁	C₂×#443	1	1	yes	characteristic
178	D₆₁	—	4	4	no	—
179	C₂×#135	—	61	1	no	—
180	C₄×#41	—	61	2	no	—
181	C₈×Q₈	—	61	4	no	—
182	C₄×Q₁₆	—	61	1	no	—
183	C₄×Q₁₆	—	61	1	no	—
184	C₄×SD₁₆	—	61	2	no	—
185	C₄×SD₁₆	—	61	2	no	—
186	C₄×D₈	—	61	1	no	—
187	C₄×D₈	—	61	1	no	—
188	C₈×D₄	—	61	4	no	—
189	C₈×D₄	—	122	4	no	—
190	#431	—	61	2	no	—
191	#431	—	122	2	no	—
192	#427	—	61	1	no	—
193	#416	—	61	2	no	—
194	C₂×C₄×C₈	—	122	2	no	—
195	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
198	#335	—	61	2	no	—
199	#334	—	61	2	no	—
200	Q₈⋊C₈	—	61	4	no	—
201	D₄⋊C₈	—	61	4	no	—
202	C₈²	C₂×D₆₁	1	2	yes	—
203	C₈²	C₂×D₆₁	1	2	yes	—
204	C₆₁	C₈×#135	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	—	61	2	no	—
206	#135	—	61	8	no	—
207	C₂×Q₁₆	—	61	1	no	—
208	C₂×SD₁₆	—	61	2	no	—
209	C₂×D₈	—	61	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	122	4	no	—
212	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
213	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
214	C₂²×C₈	D₂₄₄	1	2	yes	—
215	C₄×Q₈	—	61	2	no	—
216	C₄×Q₈	—	61	2	no	—
217	C₄×D₄	—	61	2	no	—
218	C₄×D₄	—	61	2	no	—
219	C₄×D₄	—	122	2	no	—
220	C₄×D₄	—	122	2	no	—
221	#117	—	61	2	no	—
222	#117	—	122	2	no	—
223	C₂×C₄²	—	122	2	no	—
224	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
225	#107	—	61	1	no	—
226	#107	—	61	1	no	—
227	#106	—	61	2	no	—
228	C₄⋊C₈	—	61	4	no	—
229	C₄⋊C₈	—	122	4	no	—
230	#103	—	61	2	no	—
231	#103	—	61	2	no	—
232	#102	—	61	2	no	—
233	#102	—	61	2	no	—
234	C₂²⋊C₈	—	122	2	no	—
235	C₂²⋊C₈	—	122	4	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	122	4	no	—
238	C₄×C₈	—	122	2	no	—
239	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
240	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
241	C₄×C₈	D₂₄₄	1	1	yes	characteristic
242	C₄×C₈	D₂₄₄	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₆₁	1	2	yes	—
244	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₆₁	1	2	yes	—
247	#41	—	61	8	no	—
248	#41	—	122	4	no	—
249	C₂×Q₈	—	61	2	no	—
250	C₂×D₄	—	61	2	no	—
251	C₂×D₄	—	122	2	no	—
252	C₂²×C₄	—	122	2	no	—
253	C₂²×C₄	—	122	2	no	—
254	C₂²×C₄	—	122	2	no	—
255	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
257	C₂²×C₄	#11758222	1	1	yes	characteristic
258	Q₁₆	—	61	4	no	—
259	SD₁₆	—	61	8	no	—
260	D₈	—	61	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	122	4	no	—
266	C₂×C₈	—	244	8	no	—
267	C₂×C₈	C₈×D₆₁	1	4	yes	complemented
268	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
272	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
273	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
274	C₄⋊C₄	—	61	2	no	—
275	C₄⋊C₄	—	61	2	no	—
276	C₄⋊C₄	—	122	2	no	—
277	C₄⋊C₄	—	122	2	no	—
278	C₂²⋊C₄	—	122	2	no	—
279	C₂²⋊C₄	—	122	2	no	—
280	C₂²⋊C₄	—	122	2	no	—
281	C₄²	—	122	2	no	—
282	C₄²	—	122	2	no	—
283	C₄²	—	122	2	no	—
284	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
285	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
286	C₄²	#11758222	1	1	yes	characteristic
287	C₄²	C₂³×D₆₁	1	1	yes	characteristic
288	C₂³	—	122	2	no	—
289	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
290	Q₈	—	61	4	no	—
291	Q₈	—	122	2	no	—
292	D₄	—	61	4	no	—
293	D₄	—	122	8	no	—
294	D₄	—	122	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	122	2	no	—
298	C₂×C₄	—	122	8	no	—
299	C₂×C₄	—	122	2	no	—
300	C₂×C₄	—	122	2	no	—
301	C₂×C₄	—	244	4	no	—
302	C₂×C₄	—	244	4	no	—
303	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
304	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
305	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
306	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
307	C₂×C₄	#14519783	1	2	yes	—
308	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
310	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	244	4	no	—
316	C₈	C₂×C₈×D₆₁	1	2	yes	—
317	C₈	C₂×C₈×D₆₁	1	2	yes	—
318	C₈	#14519799	1	4	yes	direct product factor
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	122	4	no	—
322	C₂²	—	244	2	no	—
323	C₂²	—	244	2	no	—
324	C₂²	C₈×D₂₄₄	1	2	yes	—
325	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	122	4	no	—
329	C₄	—	244	2	no	—
330	C₄	—	244	2	no	—
331	C₄	C₈×D₂₄₄	1	2	yes	—
332	C₄	#16839880	1	2	yes	—
333	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
334	C₄	C₂×#14519783	1	1	yes	characteristic
335	C₄	C₂×#14519799	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	244	4	no	—
338	C₂	C₂×C₈×D₂₄₄	1	1	yes	characteristic
339	C₂	C₄×#14519799	1	1	yes	characteristic
340	C₂	C₂×#16839880	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	D₂₄₄
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#14519799	C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	C₈×#135
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. C₈×#11758222	C₂	semidirect
3. C₈×Dic₂₄₄	C₂	semidirect
4. C₈×#11758190	C₂	semidirect
5. C₈×D₄₈₈	C₂	semidirect
6. #21444567	C₂	semidirect
7. #21444202	C₂	semidirect
8. C₂×C₈²×C₆₁	C₂	semidirect
35. #14519799	C₈	direct (N × H)
81. #11758222	C₂×C₈	semidirect
91. C₂×C₈×C₆₁	C₂×C₈	semidirect
113. Dic₂₄₄	C₂×C₈	semidirect
114. D₄₈₈	C₂×C₈	semidirect
115. #11758190	C₂×C₈	semidirect
146. C₈×C₆₁	#135	semidirect
173. C₂×C₈²	D₆₁	semidirect
204. C₆₁	C₈×#135	semidirect
267. C₂×C₈	C₈×D₆₁	semidirect
318. C₈	#14519799	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#11758222	2
C₄×#14519799	2
C₈×Dic₂₄₄	2
C₈×#11758190	2
C₈×D₄₈₈	2
#21444567	2
#21444202	2
C₂×C₈²×C₆₁	2
C₈×#135	61

Subgroups — C8×#14519799 #29649393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14519799 #29649393