Finite Group Databasebeta

← #35374419

Subgroups — #35374419

Order 21248.

712 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917508C211yesmaximal, characteristic
2#24916435C211yesmaximal, characteristic, complemented
3C8×D4×D83C211yesmaximal, characteristic, complemented
4#24906298C211yesmaximal, characteristic, complemented
5#24906296C211yesmaximal, characteristic, complemented
6#24906295C211yesmaximal, characteristic, complemented
7#24906293C211yesmaximal, characteristic, complemented
8#24906285C211yesmaximal, characteristic, complemented
9#24906277C211yesmaximal, characteristic, complemented
10#24906275C211yesmaximal, characteristic, complemented
11#24906245C211yesmaximal, characteristic, complemented
12#24905904C211yesmaximal, characteristic, complemented
13#24905873C211yesmaximal, characteristic, complemented
14#24905751C211yesmaximal, characteristic, complemented
15C83×#1580C211yesmaximal, characteristic, complemented
16C22×C8×D83—21no—
17#19128233C2211yescharacteristic
18C4×D4×D83C2211yescharacteristic
19#19128038C411yescharacteristic
20#19128018C411yescharacteristic
21#19128004C411yescharacteristic
22#19128003C411yescharacteristic
23#19127915C411yescharacteristic
24#19127914C411yescharacteristic
25#19127853C411yescharacteristic
26#19127833C411yescharacteristic
27#19127828C2211yescharacteristic, complemented
28C8×Dic166C2211yescharacteristic, complemented
29C4×#13050579C2211yescharacteristic
30C4×#13050578C2211yescharacteristic
31C4×#13050575C2211yescharacteristic
32#19127779C2211yescharacteristic, complemented
33#19127775—21no—
34C8×D332C2211yescharacteristic
35D4×C83⋊C8C2211yescharacteristic
36#19127766C2211yescharacteristic
37C8×C22⋊D83—21no—
38C8×C22⋊D83—21no—
39(C22⋊D83)⋊C8—21no—
40#19127759—21no—
41#19127757C2211yescharacteristic
42#19127752C2211yescharacteristic
43C4⋊C8×D83C2211yescharacteristic
44C22⋊C8×D83—21no—
45#19127656—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic, complemented
49#19127453C2211yescharacteristic, complemented
50#19127451C2211yescharacteristic, complemented
51#19127433C2211yescharacteristic, complemented
52#19127431C2211yescharacteristic, complemented
53#19127430C2211yescharacteristic
54D4⋊(C83⋊C8)C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic, complemented
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic, complemented
58#19127423C2211yescharacteristic, complemented
59#19127423C2211yescharacteristic, complemented
60C4×D8×C83C2211yescharacteristic
61C8×D4×C83C2211yescharacteristic
62C8×D4×C83C2211yescharacteristic, complemented
63#334×C83C2211yescharacteristic, complemented
64D4⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic, complemented
66#322×C83C2211yescharacteristic, complemented
67C2×D4×D83C2×C411yescharacteristic
68C22×C4×D83—21no—
69C2×#13050579C2×C411yescharacteristic
70C2×#13050578C2×C411yescharacteristic
71C2×#13050575C2×C411yescharacteristic
72C2×#13050574—21no—
73C2×#13050573—21no—
74C22×C83⋊C8—21no—
75C2×C8×D83—21no—
76C2×C8×D83—21no—
77C2×C8×D83—21no—
78C2×C8×D83—44no—
79C2×C8×D83D411yescharacteristic
80C2×C8×D83D411yescharacteristic
81#14581785—21no—
82#14581783C2×C411yescharacteristic
83#14581779C2×C411yescharacteristic
84#14581774—21no—
85#14581772C2×C411yescharacteristic
86#14581763C2×C411yescharacteristic
87C4×Dic166C2311yescharacteristic
88C4×D332C2311yescharacteristic
89D4×Dic83C2×C411yescharacteristic
90#14581754C2×C411yescharacteristic
91C4×C22⋊D83—21no—
92C4×C22⋊D83—21no—
93(C22⋊D83)⋊C4—21no—
94#14581751—21no—
95C4⋊C4×D83C2×C411yescharacteristic
96C22⋊C4×D83—21no—
97C42×D83C2311yescharacteristic
98#14581725C2×C411yescharacteristic
99#14581724C2×C411yescharacteristic
100#14581723C2×C411yescharacteristic
101C4⋊(C83⋊C8)C2311yescharacteristic
102C4⋊(C83⋊C8)C2311yescharacteristic
103Dic83⋊C8—21no—
104Dic83⋊C8—21no—
105Dic83⋊C8—21no—
106Dic83⋊C8C2×C411yescharacteristic
107Dic83⋊C8C2×C411yescharacteristic
108#14581717C2×C411yescharacteristic
109#14581715C2×C411yescharacteristic
110#14581714C2×C411yescharacteristic
111#14581713C2×C411yescharacteristic
112#14581713C2×C411yescharacteristic
113#14581712C2×C411yescharacteristic
114C22⋊(C83⋊C8)—21no—
115C22⋊(C83⋊C8)—21no—
116(C2×D83)⋊C8—21no—
117(C2×D83)⋊C8—21no—
118(C2×D83)⋊C8—21no—
119(C2×D83)⋊C8C2×C411yescharacteristic
120(C2×D83)⋊C8C2×C411yescharacteristic
121#14581702—21no—
122#14581702—21no—
123#14581702—21no—
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C8×Dic83—21no—
128C8×Dic83D411yescharacteristic
129C8×Dic83D411yescharacteristic
130C2×D8×C83C2×C411yescharacteristic
131C22×C8×C83—21no—
132C22×C8×C83—21no—
133C4×D4×C83C2311yescharacteristic
134C4×D4×C83C2311yescharacteristic
135#107×C83C2×C411yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137C4⋊C8×C83C2311yescharacteristic
138#102×C83C2×C411yescharacteristic
139#102×C83C2×C411yescharacteristic
140C22⋊C8×C83—21no—
141C22⋊C8×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C23×D83—21no—
146C2×Dic166C22×C411yescharacteristic
147C2×D332C22×C411yescharacteristic
148C2×C22⋊D83—21no—
149C2×C22⋊D83—21no—
150D4×D83—22no—
151D4×D83—22no—
152C22×Dic83—21no—
153C2×C4×D83—21no—
154C2×C4×D83—21no—
155C2×C4×D83—21no—
156C2×C4×D83—42no—
157C2×C4×D83—42no—
158C2×C4×D83C22×C411yescharacteristic
159C2×C4×D83C2×D411yescharacteristic
160C2×C4×D83#4111yescharacteristic
161#13050579—22no—
162#13050578—22no—
163#13050575—22no—
164#13050574—42no—
165#13050573—42no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8—42no—
170C2×C83⋊C8C22×C411yescharacteristic
171C2×C83⋊C8C22×C411yescharacteristic
172C2×C83⋊C8C2×D411yescharacteristic
173C2×C83⋊C8C2×D411yescharacteristic
174C8×D83—22no—
175C8×D83—41no—
176C8×D83—42no—
177C8×D83—42no—
178C8×D83—42no—
179C4⋊Dic83—21no—
180C4⋊Dic83—21no—
181C4⋊Dic83C22×C411yescharacteristic
182C4⋊Dic83C22×C411yescharacteristic
183Dic83⋊C4—21no—
184Dic83⋊C4—21no—
185Dic83⋊C4—21no—
186C22⋊Dic83—21no—
187C22⋊Dic83—21no—
188C22⋊Dic83—21no—
189C22⋊Dic83—21no—
190(C2×D83)⋊C4—21no—
191(C2×D83)⋊C4—21no—
192(C2×D83)⋊C4—21no—
193C4×Dic83—21no—
194C4×Dic83C22×C411yescharacteristic
195C4×Dic83C2×D411yescharacteristic
196C4×Dic83#4111yescharacteristic
197C2×D4×C83C22×C411yescharacteristic
198C2×D4×C83C22×C411yescharacteristic
199C22×C4×C83—21no—
200C22×C4×C83—21no—
201D8×C83—22no—
202C2×C8×C83—21no—
203C2×C8×C83—21no—
204C2×C8×C83—21no—
205C2×C8×C83—42no—
206C2×C8×C83—42no—
207C2×C8×C83C22×C411yescharacteristic
208C2×C8×C83C22×C411yescharacteristic
209C2×C8×C83C2×D411yescharacteristic
210C2×C8×C83C2×D411yescharacteristic
211C4⋊C4×C83C22×C411yescharacteristic
212C4⋊C4×C83C22×C411yescharacteristic
213C22⋊C4×C83—21no—
214C22⋊C4×C83—21no—
215C42×C83C2411yescharacteristic
216C23×C83—21no—
217C23×C83—21no—
218C22×D83—24no—
219C22×D83—21no—
220C22×D83—42no—
221C22×D83—42no—
222C22×D83C4×D411yescharacteristic
223D4×C83—21no—
224D4×C83—21no—
225D4×C83—21no—
226D4×C83#13112yes—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83C23×C411yescharacteristic
234C2×C4×C83C22×D411yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C22⋊D83—24no—
237C22⋊D83—42no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—42no—
242C2×Dic83C4×D411yescharacteristic
243C2×Dic83C4×D411yescharacteristic
244C2×Dic83C4×D411yescharacteristic
245D332—21no—
246D332#13112yes—
247C4×D83—22no—
248C4×D83—21no—
249C4×D83—21no—
250C4×D83—21no—
251C4×D83—21no—
252C4×D83—42no—
253C4×D83—42no—
254C4×D83—41no—
255C4×D83—41no—
256Dic166—21no—
257Dic166—21no—
258C8×C83—21no—
259C8×C83—21no—
260C8×C83—21no—
261C8×C83—21no—
262C8×C83—41no—
263C8×C83—41no—
264C8×C83—41no—
265C83⋊C8—21no—
266C83⋊C8—21no—
267C83⋊C8—21no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C83⋊C8AGL(1,ℤ/8)12yes—
272C22×C83—22no—
273C22×C83—41no—
274C22×C83—41no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83C2×C4×D411yescharacteristic
279C2×D83—21no—
280C2×D83—21no—
281C2×D83—22no—
282C2×D83—48no—
283C2×D83—41no—
284C2×D83—41no—
285C2×D83#44412yes—
286C4×C83—21no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83C2×#13111yescharacteristic
290C4×C83C2×#13111yescharacteristic
291C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C83#57511yescharacteristic
293Dic83—21no—
294Dic83—21no—
295Dic83—21no—
296Dic83—41no—
297Dic83—41no—
298Dic83—41no—
299#17914—831nomaximal, Sylow
300C2×C83—42no—
301C2×C83—81no—
302C2×C83#292911yescharacteristic
303C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C83C2×#44411yescharacteristic
305D83—22no—
306D83—42no—
307#3105—831no—
308#2966—831no—
309C2×C8×D4—831no—
310#1582—831no—
311#1582—831no—
312#1581—831no—
313#1581—831no—
314#1580—831no—
315#1580D8311yescharacteristic, complemented
316SD16⋊C8—831no—
317SD16⋊C8—831no—
318#1573—831no—
319#1490—831no—
320#1483—831no—
321#1451—831no—
322C83#1791411yescharacteristic, complemented, Sylow
323C23×C8—1661no—
324#529—831no—
325C2×C4×D4—831no—
326#484—831no—
327#481—831no—
328#478—831no—
329#478—831no—
330#463—831no—
331#463—831no—
332#450—831no—
333#447—831no—
334#446—831no—
335C8×Q8—831no—
336C4×SD16—831no—
337C4×SD16—831no—
338C4×D8—831no—
339C4×D8C2×D8311yescharacteristic
340#436—831no—
341#435—1661no—
342C8×D4—831no—
343C8×D4—831no—
344C8×D4—831no—
345C8×D4—1661no—
346C8×D4—1661no—
347C8×D4—1661no—
348C8×D4C2×D8311yescharacteristic
349C8×D4C2×D8311yescharacteristic, complemented
350#433—1661no—
351#432—831no—
352#431—831no—
353C2×C4⋊C8—831no—
354C2×C22⋊C8—1661no—
355#405—1661no—
356C4×M16—831no—
357C2×C4×C8—831no—
358#335—831no—
359#335—831no—
360#334—831no—
361#334C2×D8311yescharacteristic, complemented
362Q8⋊C8—831no—
363Q8⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8—831no—
366D4⋊C8—831no—
367D4⋊C8—831no—
368D4⋊C8C2×D8311yescharacteristic
369D4⋊C8C2×D8311yescharacteristic, complemented
370#322—831no—
371#322—831no—
372#322—831no—
373#322C2×D8311yescharacteristic, complemented
374C22×D4—831no—
375C23×C4—1661no—
376C2×SD16—831no—
377C2×SD16—831no—
378C2×D8—831no—
379C2×D8C4×D8311yescharacteristic
380C2×M16—1661no—
381C2×M16—1661no—
382C22×C8—21no—
383C22×C8—21no—
384C22×C8—3324no—
385C22×C8—831no—
386C22×C8—831no—
387C22×C8—1661no—
388C22×C8—1661no—
389C22×C8—1661no—
390C22×C8—1661no—
391#126—1661no—
392#125—831no—
393#124—831no—
394#123—1661no—
395C22⋊Q8—831no—
396#121—831no—
397C4×Q8—831no—
398C4×D4—831no—
399C4×D4—831no—
400C4×D4—831no—
401C4×D4—1661no—
402C4×D4—1661no—
403C4×D4—1661no—
404C4×D4C22×D8311yescharacteristic
405C4×D4C22×D8311yescharacteristic
406#117—1661no—
407C2×C4⋊C4—831no—
408C2×C22⋊C4—1661no—
409C2×C42—831no—
410#107—831no—
411#107C4×D8311yescharacteristic
412#106—831no—
413#106—831no—
414C4⋊C8—831no—
415C4⋊C8—831no—
416C4⋊C8—831no—
417C4⋊C8—831no—
418C4⋊C8—1661no—
419C4⋊C8—1661no—
420C4⋊C8—1661no—
421C4⋊C8C22×D8311yescharacteristic
422C4⋊C8C22×D8311yescharacteristic
423#103—831no—
424#103—831no—
425#102—831no—
426#102—831no—
427#102—831no—
428#102—831no—
429#102C4×D8311yescharacteristic
430#102C4×D8311yescharacteristic
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8—831no—
434C22⋊C8—831no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438C22⋊C8—1661no—
439C22⋊C8—1661no—
440#97—1661no—
441#97—1661no—
442#97—1661no—
443C4×C8—831no—
444C4×C8—831no—
445C4×C8—831no—
446C4×C8—831no—
447C4×C8—831no—
448C4×C8—1661no—
449C4×C8C22×D8311yescharacteristic
450C4×C8C22×D8311yescharacteristic
451C4×C8C22×D8311yescharacteristic
452C24—1661no—
453C2×Q8—831no—
454C2×D4—831no—
455C2×D4—1661no—
456C2×D4—1662no—
457C2×D4—1661no—
458C2×D4—1662no—
459C2×D4C2×C4×D8311yescharacteristic
460C2×D4C2×C4×D8311yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—3322no—
464C22×C4—3322no—
465C22×C4—831no—
466C22×C4—831no—
467C22×C4—831no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1661no—
471C22×C4—1661no—
472SD16—1662no—
473SD16—1662no—
474D8—22no—
475D8—1662no—
476M16—3322no—
477M16—3322no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—3322no—
484C2×C8—3321no—
485C2×C8—3322no—
486C2×C8—3322no—
487C2×C8—3322no—
488C2×C8—831no—
489C2×C8—831no—
490C2×C8—831no—
491C2×C8—831no—
492C2×C8—1662no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8—1661no—
496C2×C8C2×C4×D8311yescharacteristic
497C2×C8C2×C4×D8311yescharacteristic
498C2×C8D4×D8311yescharacteristic
499C2×C8D4×D8311yescharacteristic
500C4⋊C4—831no—
501C4⋊C4—831no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4—1661no—
507C4⋊C4C2×C4×D8311yescharacteristic
508C4⋊C4C2×C4×D8311yescharacteristic
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—1661no—
512C22⋊C4—1661no—
513C22⋊C4—1661no—
514C22⋊C4—1661no—
515C22⋊C4—1661no—
516C22⋊C4—1661no—
517C22⋊C4—1661no—
518C42—831no—
519C42—831no—
520C42—831no—
521C42—1661no—
522C42C23×D8311yescharacteristic
523C23—21no—
524C23—21no—
525C23—3322no—
526C23—3322no—
527C23—831no—
528C23—1664no—
529C23—1661no—
530Q8—1661no—
531Q8—1661no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—3322no—
536D4—832no—
537D4—1664no—
538D4—1661no—
539D4#1458179812yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—3322no—
547C2×C4—3322no—
548C2×C4—3322no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—831no—
552C2×C4—831no—
553C2×C4—831no—
554C2×C4—1662no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4—1661no—
559C2×C4—1661no—
560C2×C4—1661no—
561C2×C4—1661no—
562C2×C4C22×C4×D8311yescharacteristic
563C2×C4C2×D4×D8311yescharacteristic
564C2×C4#41×D8311yescharacteristic
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—3321no—
573C8—3321no—
574C8—3321no—
575C8—832no—
576C8—1661no—
577C8—1661no—
578C8—1661no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—3328no—
586C22—3321no—
587C22—3321no—
588C22—832no—
589C22—1662no—
590C22—1661no—
591C22—1661no—
592C22C4×D4×D8311yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—3321no—
597C4—3321no—
598C4—3321no—
599C4—1661no—
600C4—1661no—
601C4—1661no—
602C4#131×D8311yescharacteristic
603C4C2×#1458179811yescharacteristic
604C4AGL(1,ℤ/8)×D8311yescharacteristic
605C4#1912849711yescharacteristic
606C2—42no—
607C2—81no—
608C2—3322no—
609C2—1662no—
610C2#2491615211yescharacteristic
611C2#2491627411yescharacteristic
612C2#444×D8311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1580C2
socleC22×C83C2×C4×D4
prime radical#19127833C4
cosocleC42×C83C24
hypercenter#1580D83
hypocenterC83#17914
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1580

Complements

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

NH ≅ G/Ntype
2. #24916435C2semidirect
3. C8×D4×D83C2semidirect
4. #24906298C2semidirect
5. #24906296C2semidirect
6. #24906295C2semidirect
7. #24906293C2semidirect
8. #24906285C2semidirect
9. #24906277C2semidirect
10. #24906275C2semidirect
11. #24906245C2semidirect
12. #24905904C2semidirect
13. #24905873C2semidirect
14. #24905751C2semidirect
15. C83×#1580C2semidirect
27. #19127828C22semidirect
28. C8×Dic166C22semidirect
32. #19127779C22semidirect
48. #19127454C22semidirect
49. #19127453C22semidirect
50. #19127451C22semidirect
51. #19127433C22semidirect
52. #19127431C22semidirect
55. D4⋊(C83⋊C8)C22semidirect
57. #19127423C22semidirect
58. #19127423C22semidirect
59. #19127423C22semidirect
62. C8×D4×C83C22semidirect
63. #334×C83C22semidirect
65. D4⋊C8×C83C22semidirect
66. #322×C83C22semidirect
315. #1580D83semidirect
322. C83#17914semidirect
349. C8×D4C2×D83semidirect
361. #334C2×D83semidirect
369. D4⋊C8C2×D83semidirect
373. #322C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249175082
#249164352
C8×D4×D832
#249062982
#249062962
#249062952
#249062932
#249062852
#249062772
#249062752
#249062452
#249059042
#249058732
#249057512
C83×#15802
#1791483