Finite Group Databasebeta

← #35426468

Subgroups — #35426468

Order 21248.

720 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917249C211yesmaximal, characteristic
2#24916409C211yesmaximal, characteristic, complemented
3#24916030C211yesmaximal, characteristic
4#24908438C211yesmaximal, characteristic, complemented
5#24908235C211yesmaximal, characteristic, complemented
6#24908210C211yesmaximal, characteristic, complemented
7#24908064C211yesmaximal, characteristic, complemented
8#24908059C211yesmaximal, characteristic, complemented
9#24907737C211yesmaximal, characteristic
10#24907735C211yesmaximal, characteristic
11C4×Dic83⋊C8C211yesmaximal, characteristic, complemented
12#24907300C211yesmaximal, characteristic
13#24907297C211yesmaximal, characteristic
14#24907267C211yesmaximal, characteristic
15C83×#1945C211yesmaximal, characteristic, complemented
16C2×#14581800C2211yescharacteristic
17C2×#14581798C2211yescharacteristic
18#19128254C2211yescharacteristic
19#19128169C2211yescharacteristic
20#19128011C411yescharacteristic, complemented
21#19128007C411yescharacteristic, complemented
22#19128004C411yescharacteristic, complemented
23#19128000C411yescharacteristic, complemented
24#19127995C411yescharacteristic, complemented
25#19127990C411yescharacteristic, complemented
26#19127985C411yescharacteristic, complemented
27#19127980C411yescharacteristic, complemented
28#19127830—21no—
29#19127830—21no—
30#19127828—21no—
31#19127825—21no—
32C2×#14581727C2211yescharacteristic
33#19127737C2211yescharacteristic
34#19127729C2211yescharacteristic
35#19127728C2211yescharacteristic
36#19127726—21no—
37#19127726—21no—
38#19127724C2211yescharacteristic
39#19127723—21no—
40C2×Dic83⋊C8C2211yescharacteristic
41C2×Dic83⋊C8C2211yescharacteristic
42C4⋊C8×D83—21no—
43C2×#14581720C2211yescharacteristic
44C2×#14581718C2211yescharacteristic
45#19127698C2211yescharacteristic
46#19127695C2211yescharacteristic
47#19127657C2211yescharacteristic
48#19127656C2211yescharacteristic
49M16×Dic83C2211yescharacteristic
50C2×C4×C83⋊C8C2211yescharacteristic
51C42×Dic83C2211yescharacteristic
52#19127466C2211yescharacteristic
53#19127465C2211yescharacteristic
54#19127462C2211yescharacteristic
55#19127461C2211yescharacteristic
56#19127460C2211yescharacteristic
57#19127460C2211yescharacteristic
58#19127455C2211yescharacteristic
59#19127455C2211yescharacteristic
60C2×#108×C83C2211yescharacteristic
61#427×C83C2211yescharacteristic
62#424×C83C2211yescharacteristic
63#424×C83C2211yescharacteristic
64C2×C4×C8×C83C2211yescharacteristic
65#339×C83C2211yescharacteristic
66#339×C83C2211yescharacteristic
67C2×#13050590C2311yescharacteristic
68#14581800—22no—
69#14581800—41no—
70#14581798—22no—
71#14581798—41no—
72C2×#13050574C2311yescharacteristic
73C2×#13050574C2311yescharacteristic
74C2×#13050573—21no—
75C2×#13050573—21no—
76C22×C83⋊C8C2311yescharacteristic
77C22×C83⋊C8C2311yescharacteristic
78C2×C8×D83—21no—
79C2×C8×D83—21no—
80#14581791C2×C411yescharacteristic
81#14581791C2×C411yescharacteristic
82#14581783C2×C411yescharacteristic
83#14581783C2×C411yescharacteristic
84#14581769—21no—
85#14581769—21no—
86C4×Dic166—21no—
87C4×Dic166—21no—
88#14581757C2×C411yescharacteristic
89#14581757C2×C411yescharacteristic
90#14581754C2×C411yescharacteristic
91#14581754C2×C411yescharacteristic
92(C22⋊D83)⋊C4—21no—
93(C22⋊D83)⋊C4—21no—
94#14581751C2311yescharacteristic
95#14581749—21no—
96C2×C4×Dic83—21no—
97C2×C4×Dic83—21no—
98C2×C4×Dic83C2311yescharacteristic
99C2×C4×Dic83C2311yescharacteristic
100C42×D83—21no—
101#14581727—41no—
102#14581725C2×C411yescharacteristic
103#14581725C2×C411yescharacteristic
104#14581723C2×C411yescharacteristic
105#14581723C2×C411yescharacteristic
106C4⋊(C83⋊C8)—21no—
107C4⋊(C83⋊C8)—21no—
108Dic83⋊C8—21no—
109Dic83⋊C8—21no—
110Dic83⋊C8—21no—
111Dic83⋊C8—21no—
112Dic83⋊C8C2×C411yescharacteristic
113Dic83⋊C8C2×C411yescharacteristic
114Dic83⋊C8C2×C411yescharacteristic
115Dic83⋊C8C2×C411yescharacteristic
116Dic83⋊C8D411yescharacteristic
117Dic83⋊C8D411yescharacteristic
118Dic83⋊C8D411yescharacteristic
119Dic83⋊C8D411yescharacteristic
120#14581720Q814yescomplemented
121#14581718—22no—
122#14581717C2×C411yescharacteristic
123#14581717C2×C411yescharacteristic
124#14581715C2×C411yescharacteristic
125#14581715C2×C411yescharacteristic
126#14581714C2×C411yescharacteristic
127#14581714C2×C411yescharacteristic
128#14581712C2×C411yescharacteristic
129#14581712C2×C411yescharacteristic
130#14581703—21no—
131#14581702—21no—
132#14581702—21no—
133C4×C83⋊C8—21no—
134C4×C83⋊C8—41no—
135C8×Dic83—21no—
136C8×Dic83—21no—
137C2×M16×C83C2311yescharacteristic
138C2×M16×C83C2311yescharacteristic
139C22×C8×C83C2311yescharacteristic
140C22×C8×C83C2311yescharacteristic
141#117×C83C2311yescharacteristic
142#117×C83C2311yescharacteristic
143C2×C42×C83C2311yescharacteristic
144#108×C83—41no—
145#107×C83C2×C411yescharacteristic
146#107×C83C2×C411yescharacteristic
147#106×C83C2×C411yescharacteristic
148#106×C83C2×C411yescharacteristic
149C4⋊C8×C83—21no—
150C4⋊C8×C83—21no—
151C4⋊C8×C83—21no—
152C4⋊C8×C83—21no—
153C4×C8×C83—41no—
154#13050590—21no—
155#13050590—21no—
156#13050590C2×Q812yes—
157#13050590#4112yes—
158C2×Dic166—21no—
159C2×Dic166—21no—
160C2×Dic166C22×C411yescharacteristic
161C2×D332C22×C411yescharacteristic
162C2×C22⋊D83—21no—
163C22×Dic83—21no—
164C22×Dic83#4111yescharacteristic
165C22×Dic83#4111yescharacteristic
166C2×C4×D83—21no—
167C2×C4×D83—21no—
168C2×C4×D83—21no—
169#13050574—21no—
170#13050574—21no—
171#13050574—21no—
172#13050574C2×Q812yes—
173#13050573—42no—
174#13050573—42no—
175C2×C83⋊C8—21no—
176C2×C83⋊C8—21no—
177C2×C83⋊C8—21no—
178C2×C83⋊C8—21no—
179C2×C83⋊C8—21no—
180C2×C83⋊C8—21no—
181C2×C83⋊C8C22×C411yescharacteristic
182C2×C83⋊C8C22×C411yescharacteristic
183C2×C83⋊C8C2×D411yescharacteristic
184C2×C83⋊C8C2×D411yescharacteristic
185C8×D83—42no—
186C8×D83—42no—
187C4⋊Dic83—21no—
188C4⋊Dic83—21no—
189C4⋊Dic83C22×C411yescharacteristic
190C4⋊Dic83C22×C411yescharacteristic
191Dic83⋊C4—21no—
192Dic83⋊C4—21no—
193Dic83⋊C4—21no—
194Dic83⋊C4—21no—
195Dic83⋊C4—21no—
196Dic83⋊C4—21no—
197C22⋊Dic83—21no—
198(C2×D83)⋊C4—21no—
199(C2×D83)⋊C4—21no—
200C4×Dic83—24no—
201C4×Dic83—21no—
202C4×Dic83—21no—
203C4×Dic83—21no—
204C4×Dic83—21no—
205C4×Dic83—21no—
206C4×Dic83—42no—
207C4×Dic83C22×C411yescharacteristic
208C4×Dic83C22×C411yescharacteristic
209C4×Dic83C2×D411yescharacteristic
210C4×Dic83C2×D411yescharacteristic
211C22×C4×C83—21no—
212C22×C4×C83C2411yescharacteristic
213M16×C83—21no—
214M16×C83—21no—
215M16×C83—21no—
216M16×C83—21no—
217C2×C8×C83—21no—
218C2×C8×C83—21no—
219C2×C8×C83—21no—
220C2×C8×C83—21no—
221C2×C8×C83—21no—
222C2×C8×C83—21no—
223C2×C8×C83C22×C411yescharacteristic
224C2×C8×C83C22×C411yescharacteristic
225C2×C8×C83C2×D411yescharacteristic
226C2×C8×C83C2×D411yescharacteristic
227C4⋊C4×C83C22×C411yescharacteristic
228C4⋊C4×C83C22×C411yescharacteristic
229C4⋊C4×C83C22×C411yescharacteristic
230C4⋊C4×C83C22×C411yescharacteristic
231C22⋊C4×C83—21no—
232C22⋊C4×C83—21no—
233C42×C83—21no—
234C42×C83—21no—
235C42×C83—21no—
236C42×C83C2×Q812yes—
237C23×C83C2×#4111yescharacteristic
238C22×D83—21no—
239C2×C4×C83—22no—
240C2×C4×C83—22no—
241C2×C4×C83—22no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—21no—
246C2×C4×C83C23×C411yescharacteristic
247C2×C4×C83C22×D411yescharacteristic
248C2×C4×C83C22×Q811yescharacteristic
249C2×C4×C83C2×#4111yescharacteristic
250C2×C4×C83H(5,𝔽2)11yescharacteristic
251C2×C4×C832−1+411yescharacteristic
252C22⋊D83—22no—
253C22⋊D83—22no—
254C2×Dic83—21no—
255C2×Dic83—22no—
256C2×Dic83—22no—
257C2×Dic83—21no—
258C2×Dic83—21no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83C4×D411yescharacteristic
263C2×Dic83C4×D411yescharacteristic
264C2×Dic83C4×D411yescharacteristic
265C2×Dic83C4×D411yescharacteristic
266D332—21no—
267D332C4×Q812yes—
268C4×D83—22no—
269C4×D83—22no—
270C4×D83—42no—
271C4×D83—81no—
272Dic166—21no—
273Dic166—42no—
274Dic166—81no—
275Dic166C4×Q812yes—
276C8×C83—21no—
277C8×C83—21no—
278C8×C83—21no—
279C8×C83—21no—
280C8×C83—41no—
281C8×C83—41no—
282C83⋊C8—21no—
283C83⋊C8—21no—
284C83⋊C8—21no—
285C83⋊C8—21no—
286C83⋊C8—22no—
287C83⋊C8—41no—
288C22×C83—21no—
289C22×C83—21no—
290C22×C83C2×C4×D411yescharacteristic
291C22×C83#55211yescharacteristic
292C22×C83#55411yescharacteristic
293C2×D83—22no—
294C2×D83—41no—
295C2×D83—41no—
296C4×C83—24no—
297C4×C83—41no—
298C4×C83—41no—
299C4×C83—41no—
300C4×C83—41no—
301C4×C83C2×C4×Q811yescharacteristic
302C4×C83#52011yescharacteristic
303C4×C83#54611yescharacteristic
304C4×C83D4×Q811yescharacteristic
305Dic83—22no—
306Dic83—21no—
307Dic83—21no—
308Dic83—21no—
309Dic83—42no—
310Dic83—41no—
311Dic83—41no—
312Dic83#44312yes—
313#22982—831nomaximal, Sylow
314C2×C83—21no—
315C2×C83—21no—
316C2×C83#236011yescharacteristic
317C2×C83C2×#44311yescharacteristic
318C2×C83#295611yescharacteristic
319D83—42no—
320#3076—831no—
321#2962—831no—
322#2906—831no—
323#1994—831no—
324#1950—831no—
325#1945—831no—
326#1945D8311yescharacteristic, complemented
327#1922—831no—
328#1922—831no—
329#1866—831no—
330#1866—831no—
331C4×C4⋊C8—831no—
332#1764—831no—
333#1764—831no—
334C4×#104—831no—
335C83#2298211yescharacteristic, complemented, Sylow
336C2×#131—831no—
337C2×#131—831no—
338#533—831no—
339C4×#41—831no—
340#479—831no—
341#479—831no—
342#478—831no—
343#478—831no—
344#477—831no—
345#476—831no—
346#475—831no—
347#474—831no—
348#446—1661no—
349#446—1661no—
350#446—1661no—
351C8×Q8—1661no—
352C2×#108—831no—
353C2×#108C2×D8311yescharacteristic
354#427—831no—
355#427C2×D8311yescharacteristic
356#424—831no—
357#424—831no—
358#424—1661no—
359#424—1661no—
360#424C2×D8311yescharacteristic
361#424C2×D8311yescharacteristic
362#423—831no—
363#423—1661no—
364C2×C4⋊C8—831no—
365C2×C4⋊C8—831no—
366C2×C4⋊C8—1661no—
367C2×#104—831no—
368C2×#104—831no—
369#416—831no—
370#416—831no—
371#405—831no—
372#405—831no—
373C4×M16—831no—
374C2×C4×C8—831no—
375C2×C4×C8C2×D8311yescharacteristic
376C43—831no—
377#341—831no—
378#341—831no—
379#339—831no—
380#339—831no—
381#339—831no—
382#339—831no—
383#339C2×D8311yescharacteristic
384#339C2×D8311yescharacteristic
385#336—831no—
386#336—831no—
387C2×#41—831no—
388#131—3321no—
389#131—3321no—
390#131—1662no—
391#131—1662no—
392C2×M16—831no—
393C2×M16—831no—
394C2×M16—1661no—
395C2×M16—1661no—
396C2×M16C22×D8311yescharacteristic
397C2×M16C22×D8311yescharacteristic
398C22×C8—831no—
399C22×C8—831no—
400C22×C8—1661no—
401C22×C8—1661no—
402C22×C8C22×D8311yescharacteristic
403C22×C8C22×D8311yescharacteristic
404C4⋊Q8—831no—
405C4⋊Q8—831no—
406#125—831no—
407#125—831no—
408C22⋊Q8—1661no—
409C22⋊Q8—1661no—
410C4×Q8—831no—
411C4×Q8—831no—
412C4×Q8—1661no—
413C4×Q8—1661no—
414C4×D4—831no—
415C4×D4—831no—
416C4×D4—1661no—
417C4×D4—1661no—
418#117—831no—
419#117—1661no—
420#117C22×D8311yescharacteristic
421#117C22×D8311yescharacteristic
422C2×C42—831no—
423C2×C42—831no—
424C2×C42—1661no—
425C2×C42—1661no—
426C2×C42—1661no—
427C2×C42C22×D8311yescharacteristic
428#108—41no—
429#108—3321no—
430#107—831no—
431#107—831no—
432#107C4×D8311yescharacteristic
433#107C4×D8311yescharacteristic
434#106—831no—
435#106—831no—
436#106C4×D8311yescharacteristic
437#106C4×D8311yescharacteristic
438C4⋊C8—21no—
439C4⋊C8—21no—
440C4⋊C8—21no—
441C4⋊C8—21no—
442C4⋊C8—831no—
443C4⋊C8—831no—
444C4⋊C8—831no—
445C4⋊C8—831no—
446C4⋊C8—831no—
447C4⋊C8—831no—
448C4⋊C8—831no—
449C4⋊C8—831no—
450C4⋊C8—1661no—
451C4⋊C8—1661no—
452C4⋊C8—1661no—
453C4⋊C8—1661no—
454C4⋊C8—1661no—
455C4⋊C8—1661no—
456#104—834no—
457#104—1662no—
458#103—831no—
459#103—831no—
460#103—831no—
461#103—831no—
462#102—831no—
463#102—831no—
464#102—831no—
465#102—831no—
466#97—1661no—
467#97—1661no—
468#97—1661no—
469C4×C8—41no—
470C4×C8—3321no—
471C4×C8—1661no—
472C4×C8—1661no—
473C4×C8—1661no—
474#41—832no—
475#41—832no—
476#41—1661no—
477#41—1661no—
478C2×Q8—831no—
479C2×Q8—1661no—
480C2×Q8—1661no—
481C2×D4—831no—
482C2×D4—1661no—
483C22×C4—21no—
484C22×C4—831no—
485C22×C4—831no—
486C22×C4—1661no—
487C22×C4—1661no—
488C22×C4—1661no—
489C22×C4—1661no—
490C22×C4C23×D8311yescharacteristic
491M16—21no—
492M16—21no—
493M16—21no—
494M16—21no—
495M16—3322no—
496M16—3322no—
497M16—832no—
498M16—1661no—
499M16—1661no—
500M16—1661no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—3322no—
508C2×C8—3322no—
509C2×C8—831no—
510C2×C8—831no—
511C2×C8—831no—
512C2×C8—831no—
513C2×C8—1661no—
514C2×C8—1661no—
515C2×C8—1661no—
516C2×C8—1661no—
517C2×C8—1661no—
518C2×C8—1661no—
519C2×C8C2×C4×D8311yescharacteristic
520C2×C8C2×C4×D8311yescharacteristic
521C2×C8D4×D8311yescharacteristic
522C2×C8D4×D8311yescharacteristic
523C4⋊C4—831no—
524C4⋊C4—831no—
525C4⋊C4—1661no—
526C4⋊C4—1661no—
527C4⋊C4—1661no—
528C4⋊C4—1661no—
529C4⋊C4—1661no—
530C4⋊C4—1661no—
531C4⋊C4—1661no—
532C4⋊C4—1661no—
533C4⋊C4C2×C4×D8311yescharacteristic
534C4⋊C4C2×C4×D8311yescharacteristic
535C4⋊C4C2×C4×D8311yescharacteristic
536C4⋊C4C2×C4×D8311yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—1661no—
540C22⋊C4—1661no—
541C22⋊C4—1661no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—3322no—
546C42—831no—
547C42—831no—
548C42—831no—
549C42—831no—
550C42—1661no—
551C42—1661no—
552C42—1664no—
553C42—1661no—
554C42—1661no—
555C42—1661no—
556C42Q8×D8312yes—
557C23—1661no—
558C23#41×D8311yescharacteristic
559Q8—3322no—
560Q8—832no—
561Q8—6641no—
562Q8—1661no—
563D4—832no—
564D4—1662no—
565D4—1662no—
566D4—1661no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—3322no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—831no—
579C2×C4—831no—
580C2×C4—831no—
581C2×C4—831no—
582C2×C4—6641no—
583C2×C4—1662no—
584C2×C4—1661no—
585C2×C4—1661no—
586C2×C4—1662no—
587C2×C4—1661no—
588C2×C4—1662no—
589C2×C4—1662no—
590C2×C4C22×C4×D8311yescharacteristic
591C2×C4C2×D4×D8311yescharacteristic
592C2×C4C2×Q8×D8311yescharacteristic
593C2×C4#41×D8311yescharacteristic
594C2×C4#1458183711yescharacteristic
595C2×C4#1458184011yescharacteristic
596C8—21no—
597C8—21no—
598C8—21no—
599C8—21no—
600C8—41no—
601C8—41no—
602C8—3321no—
603C8—1661no—
604C8—1661no—
605C8—1662no—
606C8—1661no—
607C8—1661no—
608C22—21no—
609C22—21no—
610C22—3321no—
611C22—3321no—
612C22—1662no—
613C22C4×D4×D8311yescharacteristic
614C22#1912838311yescharacteristic
615C22#1912840511yescharacteristic
616C4—24no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—3322no—
622C4—3321no—
623C4—3321no—
624C4—832no—
625C4—1661no—
626C4—1662no—
627C4—1661no—
628C4—1661no—
629C4C4×Q8×D8311yescharacteristic
630C4#1912818211yescharacteristic
631C4#1912834011yescharacteristic
632C4#1912836611yescharacteristic
633C2—21no—
634C2—21no—
635C2—3322no—
636C2#2491046511yescharacteristic
637C2#443×D8311yescharacteristic
638C2#2491636811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1945C2
socleC22×C83C2×C4×D4
prime radicalC2×#13050590C23
cosocleC22×C4×C83C24
hypercenter#1945D83
hypocenterC83#22982
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1945

Complements

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

NH ≅ G/Ntype
2. #24916409C2semidirect
4. #24908438C2semidirect
5. #24908235C2semidirect
6. #24908210C2semidirect
7. #24908064C2semidirect
8. #24908059C2semidirect
11. C4×Dic83⋊C8C2semidirect
15. C83×#1945C2semidirect
20. #19128011C4semidirect
21. #19128007C4semidirect
22. #19128004C4semidirect
23. #19128000C4semidirect
24. #19127995C4semidirect
25. #19127990C4semidirect
26. #19127985C4semidirect
27. #19127980C4semidirect
120. #14581720Q8semidirect
326. #1945D83semidirect
335. C83#22982semidirect

Maximal subgroups

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

subgroupindex
#249172492
#249164092
#249160302
#249084382
#249082352
#249082102
#249080642
#249080592
#249077372
#249077352
C4×Dic83⋊C82
#249073002
#249072972
#249072672
C83×#19452
#2298283