Subgroups — D₈×#681 #12070723

Order 1280.

794 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	D₄×#681	C₂	1	2	yes	maximal, complemented
3	#10602391	C₂	1	2	yes	maximal, complemented
4	#10601674	C₂	1	2	yes	maximal, complemented
5	#10601672	C₂	1	4	yes	maximal, complemented
6	D₈×#168	C₂	1	2	yes	maximal, complemented
7	#10600524	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#681	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×D₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
11	C₂²×#681	—	2	2	no	—
12	C₄×C₅×D₈	C₄	1	1	yes	characteristic
13	C₄×C₅×D₈	C₄	1	2	yes	—
14	C₄×C₅×D₈	C₄	1	1	yes	characteristic
15	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
16	C₄×C₅×D₈	C₂²	1	2	yes	—
17	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×#107	C₄	1	1	yes	characteristic
19	C₂×C₅×#102	C₄	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	#65496	—	2	2	no	—
22	#65301	C₂²	1	4	yes	complemented
23	#65300	—	2	4	no	—
24	D₄×#168	C₂²	1	4	yes	complemented
25	#65214	C₂²	1	1	yes	characteristic
26	#65213	C₂²	1	2	yes	—
27	C₄×#681	C₂²	1	1	yes	characteristic
28	D₄⋊#168	C₂²	1	4	yes	complemented
29	D₄⋊#168	C₂²	1	4	yes	complemented
30	#64678	C₂²	1	2	yes	complemented
31	#64678	C₂²	1	2	yes	complemented
32	#64674	C₂²	1	2	yes	complemented
33	C₈×#168	C₂²	1	2	yes	complemented
34	M₁₆×D₈	—	5	1	no	maximal, Sylow
35	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₅	—	2	2	no	—
37	C₂×C₅×D₈	—	2	4	no	—
38	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
39	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×D₄	—	2	2	no	—
44	C₄×C₅×D₄	—	2	2	no	—
45	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
46	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
47	C₄×C₅×D₄	C₂³	1	2	yes	—
48	C₄×C₅×D₄	C₂³	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×#681	—	2	2	no	—
53	C₂×#681	—	2	4	no	—
54	C₂×#681	—	4	8	no	—
55	C₂×#681	D₄	1	1	yes	characteristic
56	C₂×#681	D₄	1	1	yes	characteristic
57	C₂²×#168	—	2	4	no	—
58	C₅×#107	C₂×C₄	1	2	yes	—
59	C₅×#107	C₂×C₄	1	1	yes	characteristic
60	C₅×#107	C₂×C₄	1	1	yes	characteristic
61	C₅×#102	C₂×C₄	1	2	yes	—
62	C₅×#102	C₂×C₄	1	2	yes	—
63	C₅×#102	C₂×C₄	1	2	yes	—
64	C₅×#102	C₂×C₄	1	2	yes	—
65	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
69	C₂²⋊#168	—	2	1	no	—
70	C₂²⋊#168	—	2	4	no	—
71	C₂²⋊#168	—	2	4	no	—
72	C₄⋊#168	—	2	2	no	—
73	C₄⋊#168	—	2	1	no	—
74	C₄⋊#168	C₂³	1	4	yes	—
75	#3935	D₄	1	2	yes	—
76	C₄×#168	C₂³	1	2	yes	—
77	C₄×#168	C₂³	1	2	yes	—
78	C₂×C₄×D₈	D₅	1	1	yes	characteristic
79	D₄×M₁₆	—	5	2	no	—
80	#1589	—	5	2	no	—
81	#1581	—	5	4	no	—
82	C₈×D₈	—	5	2	no	—
83	#1567	—	5	1	no	—
84	#1486	—	5	2	no	—
85	C₈×M₁₆	—	5	1	no	—
86	C₂⁴×C₅	—	2	2	no	—
87	C₂×C₅×D₄	—	2	4	no	—
88	C₂×C₅×D₄	—	2	2	no	—
89	C₂×C₅×D₄	—	4	2	no	—
90	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
91	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	—	2	2	no	—
95	C₂²×C₄×C₅	—	2	2	no	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	—	4	4	no	—
98	C₂²×C₄×C₅	—	4	4	no	—
99	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	#41	1	1	yes	characteristic
102	C₅×D₈	—	2	4	no	—
103	C₅×D₈	—	2	2	no	—
104	C₅×D₈	M₁₆	1	4	yes	complemented
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	4	1	no	—
107	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
113	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
114	C₅×C₂²⋊C₄	—	2	2	no	—
115	C₅×C₂²⋊C₄	—	2	2	no	—
116	C₅×C₂²⋊C₄	—	2	2	no	—
117	C₅×C₂²⋊C₄	—	2	2	no	—
118	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅	#41	1	1	yes	characteristic
121	C₄²×C₅	C₂⁴	1	1	yes	characteristic
122	#681	—	2	2	no	—
123	#681	—	4	4	no	—
124	#681	—	4	8	no	—
125	#681	D₈	1	4	yes	direct product factor
126	C₂×#168	—	2	4	no	—
127	C₂×#168	—	2	2	no	—
128	C₂×#168	—	4	8	no	—
129	C₂×#168	C₂×D₄	1	2	yes	—
130	C₂×#168	C₂×D₄	1	2	yes	—
131	C₂²×D₈	Dic₅	1	1	yes	characteristic
132	C₂²×M₁₆	—	10	2	no	—
133	C₂×C₄×D₄	C₂×D₅	1	2	yes	—
134	C₄×D₈	Dic₅	1	2	yes	—
135	C₄×D₈	Dic₅	1	1	yes	characteristic
136	C₄×D₈	Dic₅	1	1	yes	characteristic
137	C₄×D₈	C₂×D₅	1	1	yes	characteristic
138	C₄×D₈	C₂×D₅	1	2	yes	—
139	C₄×D₈	C₂×D₅	1	1	yes	characteristic
140	#436	—	5	4	no	—
141	#435	—	10	4	no	—
142	C₈×D₄	—	5	4	no	—
143	#431	—	5	1	no	—
144	C₂×#107	Dic₅	1	1	yes	characteristic
145	#423	—	5	2	no	—
146	C₂×#102	Dic₅	1	2	yes	—
147	#407	—	10	2	no	—
148	C₄×M₁₆	—	5	1	no	—
149	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
150	#335	—	5	2	no	—
151	#335	—	5	2	no	—
152	D₄⋊C₈	—	5	4	no	—
153	D₄⋊C₈	—	5	4	no	—
154	#322	—	5	2	no	—
155	C₈²	—	5	2	no	—
156	C₂³×C₅	—	2	2	no	—
157	C₂³×C₅	—	2	2	no	—
158	C₂³×C₅	—	4	4	no	—
159	C₂³×C₅	—	4	4	no	—
160	C₂³×C₅	—	8	2	no	—
161	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
162	C₅×D₄	—	2	2	no	—
163	C₅×D₄	—	2	2	no	—
164	C₅×D₄	—	2	2	no	—
165	C₅×D₄	—	4	4	no	—
166	C₅×D₄	C₂×M₁₆	1	4	yes	complemented
167	C₂×C₄×C₅	—	2	1	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	4	4	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	4	4	no	—
178	C₂×C₄×C₅	—	4	4	no	—
179	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
183	C₂×C₄×C₅	#135	1	2	yes	—
184	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
187	C₅×C₈	—	2	1	no	—
188	C₅×C₈	—	2	1	no	—
189	C₅×C₈	—	2	1	no	—
190	C₅×C₈	C₂×M₁₆	1	2	yes	complemented
191	#168	—	2	2	no	—
192	#168	—	4	2	no	—
193	#168	—	4	4	no	—
194	#168	C₂×D₈	1	4	yes	complemented
195	C₂²×D₄	C₂×Dic₅	1	2	yes	—
196	C₂³×C₄	—	2	2	no	—
197	C₂×D₈	—	2	4	no	—
198	C₂×D₈	C₂×Dic₅	1	2	yes	—
199	C₂×D₈	C₂×Dic₅	1	1	yes	characteristic
200	C₂×D₈	C₂×Dic₅	1	1	yes	characteristic
201	C₂×M₁₆	—	5	1	no	—
202	C₂×M₁₆	—	5	1	no	—
203	C₂×M₁₆	—	10	4	no	—
204	C₂×M₁₆	—	10	2	no	—
205	C₂×M₁₆	—	20	8	no	—
206	C₂²×C₈	—	10	4	no	—
207	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
208	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	C₂×Dic₅	1	2	yes	—
212	C₄×D₄	C₂×Dic₅	1	2	yes	—
213	C₄×D₄	C₂²×D₅	1	2	yes	—
214	C₄×D₄	C₂²×D₅	1	2	yes	—
215	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
218	#107	C₂×Dic₅	1	2	yes	—
219	#107	C₂×Dic₅	1	1	yes	characteristic
220	#107	C₂×Dic₅	1	1	yes	characteristic
221	C₄⋊C₈	—	5	4	no	—
222	C₄⋊C₈	—	10	2	no	—
223	C₄⋊C₈	—	10	1	no	—
224	#102	C₂×Dic₅	1	2	yes	—
225	#102	C₂×Dic₅	1	2	yes	—
226	#102	C₂×Dic₅	1	2	yes	—
227	#102	C₂×Dic₅	1	2	yes	—
228	C₂²⋊C₈	—	10	4	no	—
229	C₂²⋊C₈	—	10	4	no	—
230	C₂²⋊C₈	—	10	1	no	—
231	#97	—	5	2	no	—
232	C₄×C₈	—	5	2	no	—
233	C₄×C₈	—	5	2	no	—
234	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
235	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
238	C₂²×C₅	—	2	1	no	—
239	C₂²×C₅	—	2	1	no	—
240	C₂²×C₅	—	2	4	no	—
241	C₂²×C₅	—	4	2	no	—
242	C₂²×C₅	—	4	2	no	—
243	C₂²×C₅	—	4	2	no	—
244	C₂²×C₅	—	4	2	no	—
245	C₂²×C₅	—	4	2	no	—
246	C₂²×C₅	—	8	8	no	—
247	C₂²×C₅	C₄×D₈	1	2	yes	—
248	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
249	C₄×C₅	—	2	1	no	—
250	C₄×C₅	—	2	1	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	4	2	no	—
253	C₄×C₅	—	4	2	no	—
254	C₄×C₅	C₄×D₈	1	2	yes	—
255	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
256	C₄×C₅	#568	1	1	yes	characteristic
257	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
258	C₄×C₅	C₂×#135	1	1	yes	characteristic
259	C₂⁴	—	2	2	no	—
260	C₂×D₄	—	2	4	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	C₂²×Dic₅	1	2	yes	—
264	C₂×D₄	C₂²×Dic₅	1	2	yes	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	4	4	no	—
271	C₂²×C₄	—	4	4	no	—
272	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
273	C₂²×C₄	#711	1	1	yes	characteristic
274	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
275	D₈	—	2	4	no	—
276	D₈	—	2	2	no	—
277	D₈	#681	1	4	yes	direct product factor
278	M₁₆	—	5	4	no	—
279	M₁₆	—	10	2	no	—
280	M₁₆	—	20	8	no	—
281	M₁₆	—	20	4	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	5	2	no	—
285	C₂×C₈	—	5	2	no	—
286	C₂×C₈	—	10	4	no	—
287	C₂×C₈	—	10	2	no	—
288	C₂×C₈	—	20	8	no	—
289	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
290	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
291	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
292	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
295	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₄²	D₄×D₅	1	1	yes	characteristic
301	C₄²	#711	1	1	yes	characteristic
302	C₄²	C₂²×Dic₅	1	1	yes	characteristic
303	C₄²	C₂³×D₅	1	1	yes	characteristic
304	C₂×C₅	—	2	2	no	—
305	C₂×C₅	—	4	4	no	—
306	C₂×C₅	—	8	2	no	—
307	C₂×C₅	D₄×M₁₆	1	1	yes	characteristic
308	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
309	C₂×C₅	#2957	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	—	4	4	no	—
313	C₂³	—	4	4	no	—
314	C₂³	—	8	2	no	—
315	C₂³	D₄×Dic₅	1	1	yes	characteristic
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	4	4	no	—
320	D₄	C₂×#681	1	4	yes	complemented
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	D₅×D₈	1	2	yes	—
334	C₂×C₄	#4058	1	2	yes	—
335	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
336	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
337	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
338	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
339	C₂×C₄	C₂×#711	1	1	yes	characteristic
340	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	5	4	no	—
345	C₈	—	10	2	no	—
346	C₈	—	20	2	no	—
347	C₈	—	20	4	no	—
348	C₈	C₂×#681	1	2	yes	complemented
349	C₅	M₁₆×D₈	1	1	yes	characteristic, complemented, Sylow
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	4	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	8	8	no	—
359	C₂²	D₈×Dic₅	1	2	yes	—
360	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	4	2	no	—
365	C₄	—	4	2	no	—
366	C₄	D₈×Dic₅	1	2	yes	—
367	C₄	C₂×D₅×D₈	1	1	yes	characteristic
368	C₄	C₂×#4058	1	1	yes	characteristic
369	C₄	C₂²×#681	1	1	yes	characteristic
370	C₄	#66153	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	4	4	no	—
373	C₂	—	8	2	no	—
374	C₂	D₄×#681	1	1	yes	characteristic
375	C₂	C₂×D₈×Dic₅	1	1	yes	characteristic
376	C₂	#10615641	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₄×C₅×D₈	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂²×C₅×D₈	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×C₄×D₈	D₅
hypocenter	C₅	M₁₆×D₈
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. D₄×#681	C₂	semidirect
3. #10602391	C₂	semidirect
4. #10601674	C₂	semidirect
5. #10601672	C₂	semidirect
6. D₈×#168	C₂	semidirect
7. #10600524	C₂	semidirect
8. C₈×#681	C₂	semidirect
22. #65301	C₂²	semidirect
24. D₄×#168	C₂²	semidirect
28. D₄⋊#168	C₂²	semidirect
29. D₄⋊#168	C₂²	semidirect
30. #64678	C₂²	semidirect
31. #64678	C₂²	semidirect
32. #64674	C₂²	semidirect
33. C₈×#168	C₂²	semidirect
104. C₅×D₈	M₁₆	semidirect
125. #681	D₈	direct (N × H)
166. C₅×D₄	C₂×M₁₆	semidirect
190. C₅×C₈	C₂×M₁₆	semidirect
194. #168	C₂×D₈	semidirect
277. D₈	#681	direct (N × H)
320. D₄	C₂×#681	semidirect
348. C₈	C₂×#681	semidirect
349. C₅	M₁₆×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×D₈	2
D₄×#681	2
#10602391	2
#10601674	2
#10601672	2
D₈×#168	2
#10600524	2
C₈×#681	2
M₁₆×D₈	5

Subgroups — D8×#681 #12070723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×#681 #12070723