Finite Group Databasebeta

← D4×#66899 #14769205

Subgroups — D4×#66899 #14769205

Order 2816.

1530 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C11⋊C8C211yesmaximal, characteristic
2C22×#66899C212yesmaximal, complemented
3D4×C4⋊Dic11C211yesmaximal, characteristic
4#13069540C211yesmaximal, characteristic, complemented
5#13069419C211yesmaximal, characteristic, complemented
6#13069040C212yesmaximal, complemented
7#13068924C212yesmaximal, complemented
8#13068621C211yesmaximal, characteristic, complemented
9#13068551C212yesmaximal, complemented
10C4×#66899C211yesmaximal, characteristic, complemented
11C2×D4×Q8×C11C211yesmaximal, characteristic
12C23×Q8×C11C2212yes—
13D4×Q8×C11—24no—
14D4×Q8×C11C418yescomplemented
15C2×C11×C4⋊Q8—21no—
16C2×C11×C4⋊Q8C2211yescharacteristic
17C2×C11×C22⋊Q8—22no—
18C2×C11×C22⋊Q8C2212yes—
19C2×C4×Q8×C11C2211yescharacteristic
20C2×C4×D4×C11—21no—
21C2×C4×D4×C11C2211yescharacteristic
22C2×D4×Dic11—21no—
23C22×C4⋊Dic11C2212yes—
24C23×C11⋊C8C2212yes—
25C2×#66899—28no—
26C2×#66899C2211yescharacteristic
27C2×#66899C2212yes—
28C2×#66899C2212yes—
29C2×#66899C2212yes—
30C2×#66899C2212yes—
31C2×#66899C2212yes—
32C2×#66899C2211yescharacteristic
33#10622825C2212yes—
34C2×C22⋊(C11⋊C8)C2212yes—
35D4×C11⋊C8—24no—
36#10622598—21no—
37#10622563—22no—
38#10622548C2211yescharacteristic
39C4×C4⋊Dic11C2211yescharacteristic
40C2×C4⋊(C11⋊C8)C2211yescharacteristic
41C2×C4×C11⋊C8C2211yescharacteristic
42#10622167C2212yes—
43#10622167C2211yescharacteristic
44#10622167C2211yescharacteristic
45C22×Q8×C11—22no—
46C22×Q8×C11—24no—
47C22×Q8×C11—24no—
48C22×Q8×C11—24no—
49C22×Q8×C11—42no—
50C22×Q8×C11—42no—
51C22×Q8×C11C2×C418yescomplemented
52C22×Q8×C11C2312yes—
53C22×Q8×C11C2311yescharacteristic
54C22×D4×C11D411yescharacteristic
55C23×C4×C11—22no—
56C23×C4×C11C2312yes—
57C23×Dic11—22no—
58C11×C4⋊Q8—28no—
59C11×C4⋊Q8—22no—
60C11×C4⋊Q8C2×C414yescomplemented
61C11×C22⋊Q8—24no—
62C11×C22⋊Q8—216no—
63C11×C22⋊Q8C2×C418yescomplemented
64C4×Q8×C11—22no—
65C4×Q8×C11C2×C414yescomplemented
66C4×D4×C11—28no—
67C4×D4×C11C2×C414yes—
68C4×D4×C11D414yes—
69C2×C11×C4⋊C4—21no—
70C2×C11×C4⋊C4—21no—
71C2×C11×C4⋊C4—21no—
72C2×C11×C4⋊C4—22no—
73C2×C11×C4⋊C4D411yescharacteristic
74C2×C11×C4⋊C4C2311yescharacteristic
75C2×C11×C22⋊C4—22no—
76C2×C11×C22⋊C4D412yes—
77C2×C42×C11—21no—
78C2×C42×C11C2311yescharacteristic
79C2×C22⋊Dic11—22no—
80D4×Dic11—44no—
81C2×C4⋊Dic11—24no—
82C2×C4⋊Dic11—21no—
83C2×C4⋊Dic11—22no—
84C2×C4⋊Dic11—21no—
85C2×C4⋊Dic11—42no—
86C2×C4⋊Dic11C2311yescharacteristic
87C2×C4⋊Dic11C2312yes—
88C2×C4⋊Dic11C2311yescharacteristic
89C2×C4×Dic11—21no—
90C22×C11⋊C8—24no—
91C22×C11⋊C8—24no—
92C22×C11⋊C8—42no—
93C22×C11⋊C8C2311yescharacteristic
94C22×C11⋊C8C2312yes—
95C22×C11⋊C8C2311yescharacteristic
96#66899—24no—
97#66899—22no—
98#66899—24no—
99#66899—24no—
100#66899—42no—
101#66899—42no—
102#66899D414yesdirect product factor
103C11⋊#95—21no—
104C11⋊#95—22no—
105C22⋊(C11⋊C8)—24no—
106C4⋊(C11⋊C8)—22no—
107C4×C11⋊C8—22no—
108D4×#103—111nomaximal, Sylow
109C24×C11C2×D412yes—
110C2×Q8×C11—28no—
111C2×Q8×C11—21no—
112C2×Q8×C11—22no—
113C2×Q8×C11—21no—
114C2×Q8×C11—21no—
115C2×Q8×C11—216no—
116C2×Q8×C11—22no—
117C2×Q8×C11—22no—
118C2×Q8×C11—22no—
119C2×Q8×C11—22no—
120C2×Q8×C11—44no—
121C2×Q8×C11—44no—
122C2×Q8×C11—48no—
123C2×Q8×C11—48no—
124C2×Q8×C11—48no—
125C2×Q8×C11C22×C414yes—
126C2×Q8×C11C22×C412yes—
127C2×Q8×C11C2×D412yes—
128C2×Q8×C11#4112yes—
129C2×D4×C11C22⋊C414yes—
130C2×D4×C11SD1614yes—
131C2×D4×C11Q1614yes—
132C22×C4×C11—21no—
133C22×C4×C11—22no—
134C22×C4×C11—22no—
135C22×C4×C11—22no—
136C22×C4×C11—22no—
137C22×C4×C11—28no—
138C22×C4×C11—21no—
139C22×C4×C11—22no—
140C22×C4×C11—44no—
141C22×C4×C11—44no—
142C22×C4×C11C22×C414yes—
143C22×C4×C11C2×D411yescharacteristic
144C22×C4×C11C2×D412yes—
145C22×C4×C11C2×D414yes—
146C22×C4×C11C2×D411yescharacteristic
147C22×C4×C11C2411yescharacteristic
148C22×Dic11—21no—
149C22×Dic11—21no—
150C22×Dic11—22no—
151C22×Dic11—44no—
152C22×Dic11—44no—
153C22×Dic11—82no—
154C11×C4⋊C4—24no—
155C11×C4⋊C4—24no—
156C11×C4⋊C4—24no—
157C11×C4⋊C4—28no—
158C11×C4⋊C4—21no—
159C11×C4⋊C4C22⋊C414yes—
160C11×C4⋊C4C22×C412yes—
161C11×C22⋊C4—28no—
162C11×C22⋊C4C22⋊C418yes—
163C42×C11—24no—
164C42×C11C22×C412yes—
165C42×C11C2×D412yes—
166C22⋊Dic11—44no—
167C4⋊Dic11—22no—
168C4⋊Dic11—21no—
169C4⋊Dic11—21no—
170C4⋊Dic11—21no—
171C4⋊Dic11—22no—
172C4⋊Dic11—44no—
173C4⋊Dic11—42no—
174C4⋊Dic11—81no—
175C4⋊Dic11C2×D412yes—
176C4×Dic11—42no—
177C2×C11⋊C8—21no—
178C2×C11⋊C8—21no—
179C2×C11⋊C8—22no—
180C2×C11⋊C8—22no—
181C2×C11⋊C8—21no—
182C2×C11⋊C8—21no—
183C2×C11⋊C8—21no—
184C2×C11⋊C8—22no—
185C2×C11⋊C8—48no—
186C2×C11⋊C8C2×D412yes—
187C2×D4×Q8D1111yescharacteristic
188C2×C8×D4—111no—
189C22×#103—112no—
190D4×C4⋊C4—111no—
191#1981—111no—
192#1956—111no—
193#1892—112no—
194#1872—112no—
195#1803—111no—
196#1787—112no—
197C4×#103—111no—
198C23×C11—22no—
199C23×C11—22no—
200C23×C11—22no—
201C23×C11—22no—
202C23×C11C2×C22⋊C414yes—
203C23×C11C2×SD1614yes—
204C23×C11C2×Q1614yes—
205C23×C11C22×D411yescharacteristic
206Q8×C11—22no—
207Q8×C11—22no—
208Q8×C11—22no—
209Q8×C11—24no—
210Q8×C11—24no—
211Q8×C11—416no—
212Q8×C11—42no—
213Q8×C11—42no—
214Q8×C11C4×D414yescomplemented
215D4×C11#103116yescomplemented
216C2×C4×C11—22no—
217C2×C4×C11—22no—
218C2×C4×C11—22no—
219C2×C4×C11—22no—
220C2×C4×C11—22no—
221C2×C4×C11—21no—
222C2×C4×C11—21no—
223C2×C4×C11—24no—
224C2×C4×C11—24no—
225C2×C4×C11—24no—
226C2×C4×C11—24no—
227C2×C4×C11—22no—
228C2×C4×C11—24no—
229C2×C4×C11—416no—
230C2×C4×C11—42no—
231C2×C4×C11—42no—
232C2×C4×C11—41no—
233C2×C4×C11—41no—
234C2×C4×C11C2×C22⋊C412yes—
235C2×C4×C11C2×C22⋊C414yes—
236C2×C4×C11C2×C22⋊C412yes—
237C2×C4×C11C2×SD1612yes—
238C2×C4×C11C2×Q1612yes—
239C2×C4×C11C23×C411yescharacteristic
240C2×C4×C11C22×D411yescharacteristic
241C2×C4×C11C22×D411yescharacteristic
242C2×C4×C11C2×#4111yescharacteristic
243C2×C4×C11H(5,𝔽2)11yescharacteristic
244C2×C4×C11H(5,𝔽2)11yescharacteristic
245C2×Dic11—22no—
246C2×Dic11—41no—
247C2×Dic11—41no—
248C2×Dic11—42no—
249C2×Dic11—41no—
250C2×Dic11—41no—
251C2×Dic11—42no—
252C2×Dic11—42no—
253C2×Dic11—41no—
254C2×Dic11—88no—
255C11⋊C8—22no—
256C11⋊C8—42no—
257C11⋊C8—41no—
258C23×Q8C2×D1112yes—
259C23×C8—112no—
260D4×Q8—24no—
261D4×Q8Dic1118yescomplemented
262C2×C4⋊Q8—21no—
263C2×C4⋊Q8C2×D1111yescharacteristic
264C2×C22⋊Q8—22no—
265C2×C22⋊Q8C2×D1112yes—
266C2×C4×Q8C2×D1111yescharacteristic
267C2×C4×D4—21no—
268C2×C4×D4—221no—
269C2×C4×D4C2×D1111yescharacteristic
270C22×C4⋊C4—112no—
271C8×D4—224no—
272C2×C4⋊C8—111no—
273C2×#103—111no—
274C2×#103—112no—
275C2×#103—112no—
276C2×#103—112no—
277C2×#103—111no—
278C2×#103—112no—
279C2×#103—112no—
280C2×#103—228no—
281C2×C22⋊C8—112no—
282C2×C4×C8—111no—
283#389—221no—
284#385—222no—
285#384—111no—
286#380—112no—
287C4×C4⋊C4—111no—
288#340—112no—
289#340—111no—
290#340—111no—
291C22×C11—24no—
292C22×C11—24no—
293C22×C11—24no—
294C22×C11—24no—
295C22×C11—24no—
296C22×C11—24no—
297C22×C11C2×#10318yescomplemented
298C22×C11C22×C22⋊C411yescharacteristic
299C22×C11D4211yescharacteristic
300C22×C11#54611yescharacteristic
301C22×C11C22×SD1611yescharacteristic
302C22×C11C22×Q1611yescharacteristic
303C22×C11#57711yescharacteristic
304C22×C11#57811yescharacteristic
305C4×C11—24no—
306C4×C11—24no—
307C4×C11—22no—
308C4×C11—44no—
309C4×C11—42no—
310C4×C11C2×#10314yescomplemented
311C4×C11C2×C4×D411yescharacteristic
312C4×C11#51811yescharacteristic
313C4×C11D4211yescharacteristic
314C4×C11#54611yescharacteristic
315Dic11—42no—
316Dic11—82no—
317Dic11—81no—
318C22×Q8—22no—
319C22×Q8—24no—
320C22×Q8—24no—
321C22×Q8—24no—
322C22×Q8—42no—
323C22×Q8—42no—
324C22×Q8C2×Dic1118yescomplemented
325C22×Q8C22×D1112yes—
326C22×Q8C22×D1111yescharacteristic
327C22×D4C22⋊D1111yescharacteristic
328C23×C4—22no—
329C23×C4—222no—
330C23×C4C22×D1112yes—
331C22×C8—111no—
332C22×C8—112no—
333C22×C8—111no—
334C22×C8—224no—
335C22×C8—224no—
336C22×C8—442no—
337C4⋊Q8—28no—
338C4⋊Q8—22no—
339C4⋊Q8C2×Dic1114yescomplemented
340C22⋊Q8—24no—
341C22⋊Q8—216no—
342C22⋊Q8C2×Dic1118yescomplemented
343C4×Q8—22no—
344C4×Q8C2×Dic1114yescomplemented
345C4×D4—28no—
346C4×D4—444no—
347C4×D4C2×Dic1114yes—
348C4×D4C22⋊D1114yes—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—111no—
354C2×C4⋊C4—112no—
355C2×C4⋊C4—111no—
356C2×C4⋊C4—224no—
357C2×C4⋊C4—222no—
358C2×C4⋊C4—221no—
359C2×C4⋊C4—221no—
360C2×C4⋊C4—442no—
361C2×C4⋊C4C22⋊D1111yescharacteristic
362C2×C4⋊C4C22×D1111yescharacteristic
363C2×C22⋊C4—22no—
364C2×C22⋊C4—222no—
365C2×C22⋊C4C22⋊D1112yes—
366C2×C42—21no—
367C2×C42—221no—
368C2×C42C22×D1111yescharacteristic
369C4⋊C8—222no—
370#103—114no—
371#103—224no—
372#103—224no—
373#103—224no—
374#103—222no—
375#103—442no—
376#103—442no—
377C22⋊C8—224no—
378C4×C8—222no—
379#95—221no—
380#95—222no—
381C2×C11—28no—
382C2×C11D4×C22⋊C411yescharacteristic
383C2×C11C22×#10311yescharacteristic
384C2×C11#289811yescharacteristic
385C2×C11D4×SD1611yescharacteristic
386C2×C11D4×Q1611yescharacteristic
387C2×C11#329811yescharacteristic
388C2×C11#329911yescharacteristic
389C24C2×C22⋊D1112yes—
390C2×Q8—28no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—216no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8D4×D1112yes—
406C2×Q8#434112yes—
407C2×Q8C22×Dic1114yes—
408C2×Q8C22×Dic1112yes—
409C2×D4#432514yes—
410C2×D4#432614yes—
411C2×D4C22⋊Dic1114yes—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—28no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—221no—
423C22×C4—221no—
424C22×C4—222no—
425C22×C4—444no—
426C22×C4—444no—
427C22×C4—882no—
428C22×C4C22×Dic1114yes—
429C22×C4C2×C22⋊D1111yescharacteristic
430C22×C4C2×C22⋊D1112yes—
431C22×C4C2×C22⋊D1114yes—
432C22×C4C2×C22⋊D1111yescharacteristic
433C22×C4C23×D1111yescharacteristic
434C2×C8—112no—
435C2×C8—221no—
436C2×C8—221no—
437C2×C8—222no—
438C2×C8—221no—
439C2×C8—221no—
440C2×C8—222no—
441C2×C8—221no—
442C2×C8—222no—
443C2×C8—448no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—21no—
449C4⋊C4—112no—
450C4⋊C4—221no—
451C4⋊C4—222no—
452C4⋊C4—221no—
453C4⋊C4—221no—
454C4⋊C4—222no—
455C4⋊C4—444no—
456C4⋊C4—442no—
457C4⋊C4—881no—
458C4⋊C4C22⋊Dic1114yes—
459C4⋊C4C22×Dic1112yes—
460C22⋊C4—28no—
461C22⋊C4—444no—
462C22⋊C4C22⋊Dic1118yes—
463C42—24no—
464C42—442no—
465C42C22×Dic1112yes—
466C42C2×C22⋊D1112yes—
467C11D4×#10311yescharacteristic, complemented, Sylow
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23C2×#432514yes—
473C23C2×#432614yes—
474C23C2×C22⋊Dic1114yes—
475C23C22×C22⋊D1111yescharacteristic
476Q8—22no—
477Q8—22no—
478Q8—22no—
479Q8—24no—
480Q8—24no—
481Q8—416no—
482Q8—42no—
483Q8—42no—
484Q8D4×Dic1114yescomplemented
485D4#66899116yesdirect product factor
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—416no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—222no—
505C2×C4—441no—
506C2×C4—441no—
507C2×C4—442no—
508C2×C4—441no—
509C2×C4—441no—
510C2×C4—442no—
511C2×C4—442no—
512C2×C4—441no—
513C2×C4—888no—
514C2×C4C2×#432512yes—
515C2×C4C2×#432612yes—
516C2×C4C2×C22⋊Dic1112yes—
517C2×C4C2×C22⋊Dic1114yes—
518C2×C4C2×C22⋊Dic1112yes—
519C2×C4C2×D4×D1111yescharacteristic
520C2×C4C2×#434111yescharacteristic
521C2×C4#6703711yescharacteristic
522C2×C4#6703711yescharacteristic
523C2×C4C23×Dic1111yescharacteristic
524C2×C4C22×C22⋊D1111yescharacteristic
525C8—222no—
526C8—442no—
527C8—441no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22C2×#6689918yescomplemented
535C22D4×C22⋊D1111yescharacteristic
536C22#1062333711yescharacteristic
537C22C22×#432511yescharacteristic
538C22C22×#432611yescharacteristic
539C22#1062336811yescharacteristic
540C22#1062336911yescharacteristic
541C22C22×C22⋊Dic1111yescharacteristic
542C4—24no—
543C4—24no—
544C4—22no—
545C4—44no—
546C4—42no—
547C4—442no—
548C4—881no—
549C4—882no—
550C4C2×#6689914yescomplemented
551C4C2×D4×Dic1111yescharacteristic
552C4#1062332911yescharacteristic
553C4D4×C22⋊D1111yescharacteristic
554C4#1062333711yescharacteristic
555C2—28no—
556C2D4×C22⋊Dic1111yescharacteristic
557C2C22×#6689911yescharacteristic
558C2#1307714911yescharacteristic
559C2D4×#432511yescharacteristic
560C2D4×#432611yescharacteristic
561C2#1308048111yescharacteristic
562C2#1308049211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC2×D4×Q8×C11C2
socleC23×C11C22×D4
prime radicalC22×D4×C11D4
cosocleC22×C4×C11C24
hypercenterC2×D4×Q8D11
hypocenterC11D4×#103
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C22×#66899C2semidirect
4. #13069540C2semidirect
5. #13069419C2semidirect
6. #13069040C2semidirect
7. #13068924C2semidirect
8. #13068621C2semidirect
9. #13068551C2semidirect
10. C4×#66899C2semidirect
14. D4×Q8×C11C4semidirect
51. C22×Q8×C11C2×C4semidirect
60. C11×C4⋊Q8C2×C4semidirect
63. C11×C22⋊Q8C2×C4semidirect
65. C4×Q8×C11C2×C4semidirect
102. #66899D4direct (N × H)
214. Q8×C11C4×D4semidirect
215. D4×C11#103semidirect
261. D4×Q8Dic11semidirect
297. C22×C11C2×#103semidirect
310. C4×C11C2×#103semidirect
324. C22×Q8C2×Dic11semidirect
339. C4⋊Q8C2×Dic11semidirect
342. C22⋊Q8C2×Dic11semidirect
344. C4×Q8C2×Dic11semidirect
467. C11D4×#103semidirect
484. Q8D4×Dic11semidirect
485. D4#66899direct (N × H)
534. C22C2×#66899semidirect
550. C4C2×#66899semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C11⋊C82
C22×#668992
D4×C4⋊Dic112
#130695402
#130694192
#130690402
#130689242
#130686212
#130685512
C4×#668992
C2×D4×Q8×C112
D4×#10311