Finite Group Databasebeta

← #35412263

Subgroups — #35412263

Order 21248.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917249C211yesmaximal, characteristic
2#24916402C211yesmaximal, characteristic, complemented
3#24915890C211yesmaximal, characteristic
4#24908405C211yesmaximal, characteristic, complemented
5#24908237C211yesmaximal, characteristic, complemented
6#24908062C211yesmaximal, characteristic, complemented
7#24907741C211yesmaximal, characteristic
8#24907737C211yesmaximal, characteristic
9#24907482C211yesmaximal, characteristic
10#24907349C211yesmaximal, characteristic, complemented
11#24907341C211yesmaximal, characteristic, complemented
12#24907334C211yesmaximal, characteristic, complemented
13#24907306C211yesmaximal, characteristic
14#24907297C211yesmaximal, characteristic
15C83×#1922C211yesmaximal, 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#19127828—21no—
32C2×#14581728C2211yescharacteristic
33C2×#14581727C2211yescharacteristic
34#19127737C2211yescharacteristic
35#19127727C2211yescharacteristic
36#19127724C2211yescharacteristic
37C2×Dic83⋊C8C2211yescharacteristic
38C2×#14581720C2211yescharacteristic
39C2×#14581718C2211yescharacteristic
40#19127698C2211yescharacteristic
41#19127695C2211yescharacteristic
42#19127659C2211yescharacteristic
43#19127658—21no—
44#19127657C2211yescharacteristic
45#19127657C2211yescharacteristic
46#19127655—21no—
47M16×Dic83C2211yescharacteristic
48C4×#13050573—21no—
49C2×#14581702C2211yescharacteristic
50C2×#14581702C2211yescharacteristic
51#97×D83—21no—
52C2×C4×C83⋊C8C2211yescharacteristic
53C4×Dic83⋊C4C2211yescharacteristic
54#19127465C2211yescharacteristic
55#19127462C2211yescharacteristic
56#19127461C2211yescharacteristic
57#19127460C2211yescharacteristic
58#19127460C2211yescharacteristic
59#19127455C2211yescharacteristic
60#427×C83C2211yescharacteristic
61#424×C83C2211yescharacteristic
62#405×C83C2211yescharacteristic
63#341×C83C2211yescharacteristic
64#339×C83C2211yescharacteristic
65#339×C83C2211yescharacteristic
66#336×C83C2211yescharacteristic
67C2×#13050590C2311yescharacteristic
68#14581800—22no—
69#14581800Q814yescomplemented
70#14581798—22no—
71#14581798—22no—
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×Dic83⋊C4—21no—
97C2×C4×Dic83C2311yescharacteristic
98C2×C4×Dic83C2311yescharacteristic
99C42×D83—21no—
100#14581728—22no—
101#14581727—22no—
102#14581725C2×C411yescharacteristic
103#14581725C2×C411yescharacteristic
104#14581723C2×C411yescharacteristic
105#14581723C2×C411yescharacteristic
106Dic83⋊C8—21no—
107Dic83⋊C8—21no—
108Dic83⋊C8—21no—
109Dic83⋊C8—21no—
110Dic83⋊C8C2×C411yescharacteristic
111Dic83⋊C8C2×C411yescharacteristic
112Dic83⋊C8C2×C411yescharacteristic
113Dic83⋊C8C2×C411yescharacteristic
114#14581720—22no—
115#14581718—22no—
116#14581717C2×C411yescharacteristic
117#14581717C2×C411yescharacteristic
118#14581715C2×C411yescharacteristic
119#14581715C2×C411yescharacteristic
120#14581714C2×C411yescharacteristic
121#14581714C2×C411yescharacteristic
122#14581712C2×C411yescharacteristic
123#14581712C2×C411yescharacteristic
124#14581703—21no—
125#14581703—21no—
126#14581702—21no—
127#14581702—22no—
128#14581702—21no—
129#14581702D411yescharacteristic
130#14581702D411yescharacteristic
131#14581702D411yescharacteristic
132#14581702D411yescharacteristic
133C4×C83⋊C8—21no—
134C4×C83⋊C8—21no—
135C4×C83⋊C8—22no—
136C8×Dic83—21no—
137C8×Dic83—21no—
138C83⋊#95—21no—
139C2×M16×C83C2311yescharacteristic
140C2×M16×C83C2311yescharacteristic
141C22×C8×C83C2311yescharacteristic
142C22×C8×C83C2311yescharacteristic
143#117×C83C2311yescharacteristic
144#117×C83C2311yescharacteristic
145C2×C42×C83C2311yescharacteristic
146#107×C83C2×C411yescharacteristic
147#107×C83C2×C411yescharacteristic
148#106×C83C2×C411yescharacteristic
149#106×C83C2×C411yescharacteristic
150C4⋊C8×C83—21no—
151C4⋊C8×C83—21no—
152#97×C83—21no—
153C4×C8×C83—21no—
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#13050574#4112yes—
173#13050573—24no—
174#13050573—24no—
175C2×C83⋊C8—21no—
176C2×C83⋊C8—21no—
177C2×C83⋊C8—21no—
178C2×C83⋊C8—21no—
179C2×C83⋊C8C22×C411yescharacteristic
180C2×C83⋊C8C22×C411yescharacteristic
181C2×C83⋊C8C2×D411yescharacteristic
182C2×C83⋊C8C2×D411yescharacteristic
183C2×C83⋊C8C2×Q812yes—
184C2×C83⋊C8#4112yes—
185C8×D83—24no—
186C8×D83—24no—
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—
197Dic83⋊C4—42no—
198C22⋊Dic83—21no—
199(C2×D83)⋊C4—21no—
200(C2×D83)⋊C4—21no—
201C4×Dic83—21no—
202C4×Dic83—21no—
203C4×Dic83—21no—
204C4×Dic83—21no—
205C4×Dic83—21no—
206C4×Dic83C22×C411yescharacteristic
207C4×Dic83C22×C411yescharacteristic
208C4×Dic83C2×D411yescharacteristic
209C4×Dic83C2×D411yescharacteristic
210C22×C4×C83—21no—
211C22×C4×C83C2411yescharacteristic
212M16×C83—21no—
213M16×C83—21no—
214M16×C83—21no—
215M16×C83C2×Q812yes—
216C2×C8×C83—21no—
217C2×C8×C83—21no—
218C2×C8×C83—21no—
219C2×C8×C83—21no—
220C2×C8×C83—21no—
221C2×C8×C83—21no—
222C2×C8×C83C22×C411yescharacteristic
223C2×C8×C83C22×C411yescharacteristic
224C2×C8×C83C2×D411yescharacteristic
225C2×C8×C83C2×D411yescharacteristic
226C4⋊C4×C83C22×C411yescharacteristic
227C4⋊C4×C83C22×C411yescharacteristic
228C4⋊C4×C83C22×C411yescharacteristic
229C4⋊C4×C83C22×C411yescharacteristic
230C22⋊C4×C83—21no—
231C22⋊C4×C83—21no—
232C42×C83—21no—
233C42×C83—21no—
234C42×C83—21no—
235C42×C83—21no—
236C23×C83C2×#4111yescharacteristic
237C22×D83—21no—
238C2×C4×C83—22no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—21no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83C23×C411yescharacteristic
246C2×C4×C83C22×D411yescharacteristic
247C2×C4×C83C22×Q811yescharacteristic
248C2×C4×C83C2×#4111yescharacteristic
249C2×C4×C83C2×#4111yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C22⋊D83—22no—
252C22⋊D83—22no—
253C2×Dic83—21no—
254C2×Dic83—21no—
255C2×Dic83—21no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83C4×D411yescharacteristic
262C2×Dic83C4×D411yescharacteristic
263C2×Dic83C4×D411yescharacteristic
264C2×Dic83C4×D411yescharacteristic
265D332—21no—
266D332C4×Q812yes—
267C4×D83—22no—
268C4×D83—22no—
269C4×D83—81no—
270C4×D83—81no—
271Dic166—21no—
272Dic166—81no—
273Dic166—81no—
274Dic166C4×Q812yes—
275C8×C83—21no—
276C8×C83—21no—
277C8×C83—22no—
278C8×C83—21no—
279C8×C83—21no—
280C8×C83—41no—
281C83⋊C8—21no—
282C83⋊C8—21no—
283C83⋊C8—22no—
284C83⋊C8—41no—
285C83⋊C8C22⋊Q812yes—
286C83⋊C8C22⋊Q812yes—
287C22×C83—21no—
288C22×C83—21no—
289C22×C83C2×C4×D411yescharacteristic
290C22×C83#52911yescharacteristic
291C22×C83#53311yescharacteristic
292C2×D83—22no—
293C2×D83—41no—
294C2×D83—41no—
295C4×C83—42no—
296C4×C83—41no—
297C4×C83—41no—
298C4×C83—41no—
299C4×C83—41no—
300C4×C83C2×C4×Q811yescharacteristic
301C4×C83C4×#4111yescharacteristic
302C4×C83C2×C22⋊Q811yescharacteristic
303C4×C83#52511yescharacteristic
304Dic83—21no—
305Dic83—21no—
306Dic83—21no—
307Dic83—21no—
308Dic83—22no—
309Dic83—41no—
310Dic83—41no—
311Dic83—81no—
312#21629—831nomaximal, Sylow
313C2×C83—21no—
314C2×C83—21no—
315C2×C83C4×C22⋊Q811yescharacteristic
316C2×C83#295611yescharacteristic
317C2×C83#295711yescharacteristic
318D83—42no—
319#3076—831no—
320#2961—831no—
321#2875—831no—
322#1986—831no—
323#1950—831no—
324#1922—831no—
325#1922D8311yescharacteristic, complemented
326#1866—831no—
327#1866—831no—
328#1806—831no—
329#1773—831no—
330#1772—831no—
331#1771—831no—
332#1765—831no—
333#1764—831no—
334C83#2162911yescharacteristic, complemented, Sylow
335C2×#131—831no—
336C2×#131—831no—
337#533—831no—
338C4×#41—831no—
339#479—831no—
340#479—831no—
341#478—831no—
342#478—831no—
343#477—831no—
344#476—831no—
345#475—831no—
346#474—831no—
347#446—1661no—
348#446—1661no—
349#446—1661no—
350#446—1661no—
351C2×#108—831no—
352C2×#108—831no—
353#427—831no—
354#427C2×D8311yescharacteristic
355#424—831no—
356#424C2×D8311yescharacteristic
357#423—831no—
358C2×C4⋊C8—831no—
359C2×#104—831no—
360C2×#104—831no—
361#416—831no—
362#416—831no—
363#405—831no—
364#405—831no—
365#405—831no—
366#405—1661no—
367#405—1661no—
368#405C2×D8311yescharacteristic
369C4×M16—831no—
370C4×M16—1661no—
371C2×#97—831no—
372C2×#97—831no—
373C2×#97—1661no—
374C2×C4×C8—831no—
375C4×C4⋊C4—831no—
376#341—831no—
377#341C2×D8311yescharacteristic
378#339—831no—
379#339—831no—
380#339—831no—
381#339—831no—
382#339C2×D8311yescharacteristic
383#339C2×D8311yescharacteristic
384#336—831no—
385#336C2×D8311yescharacteristic
386C2×#41—831no—
387#131—834no—
388#131—1662no—
389#131—1662no—
390#131—1662no—
391C2×M16—831no—
392C2×M16—831no—
393C2×M16—1661no—
394C2×M16—1661no—
395C2×M16C22×D8311yescharacteristic
396C2×M16C22×D8311yescharacteristic
397C22×C8—831no—
398C22×C8—831no—
399C22×C8—1661no—
400C22×C8—1661no—
401C22×C8C22×D8311yescharacteristic
402C22×C8C22×D8311yescharacteristic
403C4⋊Q8—831no—
404C4⋊Q8—831no—
405#125—831no—
406#125—831no—
407C22⋊Q8—1661no—
408C22⋊Q8—1661no—
409C4×Q8—831no—
410C4×Q8—831no—
411C4×Q8—1661no—
412C4×Q8—1661no—
413C4×D4—831no—
414C4×D4—831no—
415C4×D4—1661no—
416C4×D4—1661no—
417#117—831no—
418#117—1661no—
419#117C22×D8311yescharacteristic
420#117C22×D8311yescharacteristic
421C2×C4⋊C4—1661no—
422C2×C42—831no—
423C2×C42—831no—
424C2×C42—1661no—
425C2×C42C22×D8311yescharacteristic
426#108—1662no—
427#108—1662no—
428#107—831no—
429#107—831no—
430#107C4×D8311yescharacteristic
431#107C4×D8311yescharacteristic
432#106—831no—
433#106—831no—
434#106C4×D8311yescharacteristic
435#106C4×D8311yescharacteristic
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—831no—
439C4⋊C8—831no—
440C4⋊C8—831no—
441C4⋊C8—831no—
442C4⋊C8—1661no—
443C4⋊C8—1661no—
444C4⋊C8—1661no—
445C4⋊C8—1661no—
446#104—1662no—
447#104—1662no—
448#103—831no—
449#103—831no—
450#103—831no—
451#103—831no—
452#102—831no—
453#102—831no—
454#102—831no—
455#102—831no—
456#97—21no—
457#97—831no—
458#97—831no—
459#97—831no—
460#97—831no—
461#97—1661no—
462#97—1661no—
463#97—1661no—
464#97—1662no—
465#97—1661no—
466C4×C8—21no—
467C4×C8—1662no—
468C4×C8—1661no—
469C4×C8—1661no—
470C4×C8—1661no—
471C4×C8—1661no—
472#95—1661no—
473#41—832no—
474#41—832no—
475#41—1661no—
476#41—1661no—
477C2×Q8—831no—
478C2×Q8—1661no—
479C2×Q8—1661no—
480C2×D4—831no—
481C2×D4—1661no—
482C22×C4—21no—
483C22×C4—831no—
484C22×C4—831no—
485C22×C4—1661no—
486C22×C4—1661no—
487C22×C4—1661no—
488C22×C4—1661no—
489C22×C4C23×D8311yescharacteristic
490M16—21no—
491M16—21no—
492M16—21no—
493M16—832no—
494M16—1664no—
495M16—1664no—
496M16—1661no—
497M16—1661no—
498M16—1661no—
499M16Q8×D8312yes—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—831no—
507C2×C8—831no—
508C2×C8—832no—
509C2×C8—832no—
510C2×C8—831no—
511C2×C8—831no—
512C2×C8—1664no—
513C2×C8—1664no—
514C2×C8—1661no—
515C2×C8—1661no—
516C2×C8—1661no—
517C2×C8—1661no—
518C2×C8C2×C4×D8311yescharacteristic
519C2×C8C2×C4×D8311yescharacteristic
520C2×C8D4×D8311yescharacteristic
521C2×C8D4×D8311yescharacteristic
522C4⋊C4—3322no—
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—21no—
546C42—831no—
547C42—831no—
548C42—831no—
549C42—831no—
550C42—1661no—
551C42—1661no—
552C42—1661no—
553C42—1661no—
554C42—1661no—
555C23—1661no—
556C23#41×D8311yescharacteristic
557Q8—832no—
558Q8—6641no—
559Q8—6641no—
560Q8—1661no—
561D4—832no—
562D4—1662no—
563D4—1662no—
564D4—1661no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—3321no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—831no—
578C2×C4—831no—
579C2×C4—831no—
580C2×C4—831no—
581C2×C4—6641no—
582C2×C4—6641no—
583C2×C4—1661no—
584C2×C4—1661no—
585C2×C4—1661no—
586C2×C4—1662no—
587C2×C4—1662no—
588C2×C4C22×C4×D8311yescharacteristic
589C2×C4C2×D4×D8311yescharacteristic
590C2×C4C2×Q8×D8311yescharacteristic
591C2×C4#41×D8311yescharacteristic
592C2×C4#41×D8311yescharacteristic
593C2×C4#41×D8311yescharacteristic
594C8—21no—
595C8—21no—
596C8—22no—
597C8—21no—
598C8—21no—
599C8—41no—
600C8—3321no—
601C8—832no—
602C8—832no—
603C8—1661no—
604C8—1661no—
605C8—1662no—
606C22—21no—
607C22—21no—
608C22—3321no—
609C22—3321no—
610C22—1662no—
611C22C4×D4×D8311yescharacteristic
612C22#1912823411yescharacteristic
613C22#1912825711yescharacteristic
614C4—42no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—3321no—
620C4—3321no—
621C4—6641no—
622C4—1661no—
623C4—1661no—
624C4—1661no—
625C4—1661no—
626C4—1662no—
627C4C4×Q8×D8311yescharacteristic
628C4#1912816911yescharacteristic
629C4C22⋊Q8×D8311yescharacteristic
630C4#1912820911yescharacteristic
631C2—21no—
632C2—21no—
633C2—3322no—
634C2#2490999711yescharacteristic
635C2#2491636811yescharacteristic
636C2#2491637711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1922C2
socleC22×C83C2×C4×D4
prime radicalC2×#13050590C23
cosocleC22×C4×C83C24
hypercenter#1922D83
hypocenterC83#21629
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1922

Complements

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

NH ≅ G/Ntype
2. #24916402C2semidirect
4. #24908405C2semidirect
5. #24908237C2semidirect
6. #24908062C2semidirect
10. #24907349C2semidirect
11. #24907341C2semidirect
12. #24907334C2semidirect
15. C83×#1922C2semidirect
20. #19128011C4semidirect
21. #19128007C4semidirect
22. #19128004C4semidirect
23. #19128000C4semidirect
24. #19127995C4semidirect
25. #19127990C4semidirect
26. #19127985C4semidirect
27. #19127980C4semidirect
69. #14581800Q8semidirect
325. #1922D83semidirect
334. C83#21629semidirect

Maximal subgroups

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

subgroupindex
#249172492
#249164022
#249158902
#249084052
#249082372
#249080622
#249077412
#249077372
#249074822
#249073492
#249073412
#249073342
#249073062
#249072972
C83×#19222
#2162983