Finite Group Databasebeta

← #35416704

Subgroups — #35416704

Order 21248.

648 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917247C211yesmaximal, characteristic
2#24916396C211yesmaximal, characteristic, complemented
3#24916103C211yesmaximal, characteristic
4#24908405C211yesmaximal, characteristic, complemented
5#24908237C211yesmaximal, characteristic, complemented
6#24908211C211yesmaximal, characteristic, complemented
7#24908208C211yesmaximal, characteristic, complemented
8#24908170C211yesmaximal, characteristic
9#24908114C211yesmaximal, characteristic
10#24908113C211yesmaximal, characteristic
11#24908061C211yesmaximal, characteristic, complemented
12#24908060C211yesmaximal, characteristic, complemented
13#24907481C211yesmaximal, characteristic
14C8×Dic83⋊C4C211yesmaximal, characteristic, complemented
15C83×#1937C211yesmaximal, characteristic
16#19128254C2211yescharacteristic
17#41×Dic83C2211yescharacteristic
18#19128010C411yescharacteristic, complemented
19#19128006C411yescharacteristic, complemented
20#19128003C411yescharacteristic, complemented
21#19127999C411yescharacteristic, complemented
22#19127994C411yescharacteristic, complemented
23#19127989C411yescharacteristic, complemented
24#19127984C411yescharacteristic, complemented
25#19127979C411yescharacteristic, complemented
26#19127830—21no—
27#19127830—21no—
28#19127825—21no—
29#19127825—21no—
30#19127759—21no—
31#19127759—21no—
32#19127751—21no—
33#19127751—21no—
34C2×#14581728C2211yescharacteristic
35C2×#14581727C2211yescharacteristic
36#19127739C2211yescharacteristic
37#19127737C2211yescharacteristic
38#19127729C2211yescharacteristic
39#19127727C2211yescharacteristic
40#19127727C2211yescharacteristic
41#19127724C2211yescharacteristic
42C2×Dic83⋊C8C2211yescharacteristic
43C2×#14581719C2211yescharacteristic
44#19127697C2211yescharacteristic
45#19127668C2211yescharacteristic
46#19127659C2211yescharacteristic
47#19127657C2211yescharacteristic
48M16×Dic83C2211yescharacteristic
49C2×C8×Dic83C2211yescharacteristic
50C2×C8×Dic83C2211yescharacteristic
51C4×Dic83⋊C4C2211yescharacteristic
52#19127465C2211yescharacteristic
53#19127462C2211yescharacteristic
54#19127462C2211yescharacteristic
55#19127461C2211yescharacteristic
56#19127461C2211yescharacteristic
57#19127460C2211yescharacteristic
58#19127456C2211yescharacteristic
59#19127455C2211yescharacteristic
60#424×C83C2211yescharacteristic
61C2×#104×C83C2211yescharacteristic
62#416×C83C2211yescharacteristic
63#408×C83C2211yescharacteristic
64C2×C4×C8×C83C2211yescharacteristic
65#341×C83C2211yescharacteristic
66#339×C83C2211yescharacteristic
67C2×#13050574C2311yescharacteristic
68C2×#13050574C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71#14581791C2×C411yescharacteristic
72#14581791C2×C411yescharacteristic
73#14581783C2×C411yescharacteristic
74#14581783C2×C411yescharacteristic
75#14581769—21no—
76#14581769—21no—
77Q8×Dic83C2×C411yescharacteristic
78Q8×Dic83C2×C411yescharacteristic
79C4×Dic166—21no—
80C4×Dic166—21no—
81D4×Dic83—21no—
82D4×Dic83—21no—
83D4×Dic83C2×C411yescharacteristic
84D4×Dic83C2×C411yescharacteristic
85#14581751—21no—
86#14581751C2311yescharacteristic
87#14581751C2311yescharacteristic
88C2×Dic83⋊C4—21no—
89C2×C4×Dic83—21no—
90C2×C4×Dic83C2311yescharacteristic
91C2×C4×Dic83C2311yescharacteristic
92#14581728—41no—
93#14581727—41no—
94#14581726C2×C411yescharacteristic
95#14581726C2×C411yescharacteristic
96#14581725C2×C411yescharacteristic
97#14581725C2×C411yescharacteristic
98#14581724C2×C411yescharacteristic
99#14581724C2×C411yescharacteristic
100#14581723C2×C411yescharacteristic
101#14581723C2×C411yescharacteristic
102Dic83⋊C8—21no—
103Dic83⋊C8—21no—
104Dic83⋊C8—21no—
105Dic83⋊C8—21no—
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#14581719—41no—
115#14581716C2×C411yescharacteristic
116#14581716C2×C411yescharacteristic
117#14581713C2×C411yescharacteristic
118#14581713C2×C411yescharacteristic
119C22⋊(C83⋊C8)—21no—
120C22⋊(C83⋊C8)—21no—
121#14581703—21no—
122#14581702—21no—
123#14581702—21no—
124C4×C83⋊C8—21no—
125C8×Dic83—21no—
126C8×Dic83—21no—
127C8×Dic83—41no—
128C8×Dic83D411yescharacteristic
129C8×Dic83D411yescharacteristic
130C8×Dic83D411yescharacteristic
131C8×Dic83D411yescharacteristic
132C83⋊#95—21no—
133C2×#41×C83C2311yescharacteristic
134C2×M16×C83C2311yescharacteristic
135C2×M16×C83C2311yescharacteristic
136C22×C8×C83C2311yescharacteristic
137C22×C8×C83C2311yescharacteristic
138#117×C83C2311yescharacteristic
139C2×C42×C83C2311yescharacteristic
140C4⋊C8×C83—21no—
141C4⋊C8×C83—21no—
142#104×C83—41no—
143#103×C83C2×C411yescharacteristic
144#103×C83C2×C411yescharacteristic
145#102×C83C2×C411yescharacteristic
146#102×C83C2×C411yescharacteristic
147C22⋊C8×C83—21no—
148C22⋊C8×C83—21no—
149C4×C8×C83—41no—
150C2×Dic166—21no—
151C2×Dic166—21no—
152C22×Dic83—21no—
153C22×Dic83—21no—
154C22×Dic83—21no—
155C22×Dic83#4111yescharacteristic
156C22×Dic83#4111yescharacteristic
157#13050574—21no—
158#13050574—21no—
159#13050574—21no—
160#13050574—21no—
161C2×C83⋊C8—21no—
162C2×C83⋊C8—21no—
163C2×C83⋊C8—21no—
164C2×C83⋊C8—21no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8C22×C411yescharacteristic
168C2×C83⋊C8C22×C411yescharacteristic
169C2×C83⋊C8C2×D411yescharacteristic
170C2×C83⋊C8C2×D411yescharacteristic
171C4⋊Dic83—21no—
172C4⋊Dic83—21no—
173C4⋊Dic83—21no—
174C4⋊Dic83—21no—
175C4⋊Dic83C22×C411yescharacteristic
176C4⋊Dic83C22×C411yescharacteristic
177C4⋊Dic83C22×C411yescharacteristic
178C4⋊Dic83C22×C411yescharacteristic
179Dic83⋊C4—24no—
180Dic83⋊C4—21no—
181Dic83⋊C4—21no—
182Dic83⋊C4—21no—
183Dic83⋊C4—21no—
184C22⋊Dic83—21no—
185C22⋊Dic83—21no—
186C22⋊Dic83—21no—
187C22⋊Dic83—21no—
188C4×Dic83—21no—
189C4×Dic83—21no—
190C4×Dic83—21no—
191C4×Dic83—21no—
192C4×Dic83—21no—
193C4×Dic83—21no—
194C4×Dic83C22×C411yescharacteristic
195C4×Dic83C22×C411yescharacteristic
196C4×Dic83C2×D411yescharacteristic
197C4×Dic83C2×D411yescharacteristic
198#41×C83—21no—
199#41×C83—21no—
200#41×C83—21no—
201#41×C83—21no—
202C2×Q8×C83C22×C411yescharacteristic
203C2×D4×C83—21no—
204C2×D4×C83C22×C411yescharacteristic
205C22×C4×C83—21no—
206C22×C4×C83—21no—
207C22×C4×C83C2411yescharacteristic
208M16×C83—21no—
209M16×C83—21no—
210M16×C83—21no—
211M16×C83—21no—
212C2×C8×C83—21no—
213C2×C8×C83—21no—
214C2×C8×C83—21no—
215C2×C8×C83—21no—
216C2×C8×C83—21no—
217C2×C8×C83—21no—
218C2×C8×C83C22×C411yescharacteristic
219C2×C8×C83C22×C411yescharacteristic
220C2×C8×C83C2×D411yescharacteristic
221C2×C8×C83C2×D411yescharacteristic
222C4⋊C4×C83C22×C411yescharacteristic
223C4⋊C4×C83C22×C411yescharacteristic
224C22⋊C4×C83—21no—
225C42×C83—21no—
226C42×C83—21no—
227C42×C83—21no—
228C23×C83—21no—
229C23×C83C2×#4111yescharacteristic
230Q8×C83—21no—
231Q8×C83—21no—
232D4×C83—21no—
233D4×C83—21no—
234D4×C83—41no—
235D4×C83—41no—
236C2×C4×C83—22no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83C23×C411yescharacteristic
245C2×C4×C83C22×D411yescharacteristic
246C2×C4×C83H(5,𝔽2)11yescharacteristic
247C2×C4×C832−1+411yescharacteristic
248C2×C4×C832−1+411yescharacteristic
249C2×C4×C832−1+411yescharacteristic
250C2×Dic83—22no—
251C2×Dic83—22no—
252C2×Dic83—21no—
253C2×Dic83—21no—
254C2×Dic83—21no—
255C2×Dic83—21no—
256C2×Dic83—41no—
257C2×Dic83—42no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—42no—
261C2×Dic83C4×D411yescharacteristic
262C2×Dic83C4×D411yescharacteristic
263C2×Dic83C4×D411yescharacteristic
264C2×Dic83C4×D411yescharacteristic
265Dic166—42no—
266Dic166—42no—
267C8×C83—21no—
268C8×C83—21no—
269C8×C83—21no—
270C8×C83—21no—
271C8×C83—41no—
272C8×C83—41no—
273C83⋊C8—21no—
274C83⋊C8—21no—
275C83⋊C8—21no—
276C83⋊C8—21no—
277C83⋊C8—41no—
278C83⋊C8—41no—
279C22×C83—21no—
280C22×C83—21no—
281C22×C83—41no—
282C22×C83—41no—
283C22×C83—41no—
284C22×C83C2×C4×D411yescharacteristic
285C22×C83#54111yescharacteristic
286C22×C83#54211yescharacteristic
287C4×C83—42no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83#51911yescharacteristic
292C4×C83#52011yescharacteristic
293C4×C83#53611yescharacteristic
294C4×C83#53711yescharacteristic
295Dic83—21no—
296Dic83—21no—
297Dic83—21no—
298Dic83—42no—
299Dic83—41no—
300Dic83—41no—
301Dic83—41no—
302Dic83—41no—
303Dic83#44312yes—
304#22130—831nomaximal, Sylow
305C2×C83—21no—
306C2×C83—21no—
307C2×C83—81no—
308C2×C83#232311yescharacteristic
309C2×C83C2×#44311yescharacteristic
310C2×C83#295511yescharacteristic
311#3076—831no—
312#2960—831no—
313#2922—831no—
314#1986—831no—
315#1950—831no—
316#1945—831no—
317#1945—831no—
318#1937—831no—
319#1937D8311yescharacteristic
320#1929—831no—
321#1929—831no—
322#1922—831no—
323#1922—831no—
324#1806—831no—
325C8×C4⋊C4—831no—
326C83#2213011yescharacteristic, complemented, Sylow
327#533—831no—
328C4×#41—831no—
329#479—831no—
330#479—831no—
331#478—831no—
332#478—831no—
333#477—831no—
334#476—831no—
335#475—831no—
336#474—831no—
337#446—1661no—
338#446—1661no—
339C8×Q8—1661no—
340C8×Q8—1661no—
341#433—1661no—
342#433—1661no—
343#431—1661no—
344#431—1661no—
345C2×#108—831no—
346C2×#108—831no—
347#427—831no—
348#427—831no—
349#424—831no—
350#424—831no—
351#424—831no—
352#424C2×D8311yescharacteristic
353#423—831no—
354C2×C4⋊C8—831no—
355C2×#104—831no—
356C2×#104C2×D8311yescharacteristic
357#416—831no—
358#416C2×D8311yescharacteristic
359#408—831no—
360#408C2×D8311yescharacteristic
361#405—831no—
362#405—831no—
363C4×M16—831no—
364C2×C4×C8—831no—
365C2×C4×C8—831no—
366C2×C4×C8C2×D8311yescharacteristic
367C4×C4⋊C4—831no—
368#341—831no—
369#341C2×D8311yescharacteristic
370#339—831no—
371#339—831no—
372#339—831no—
373#339—831no—
374#339—831no—
375#339C2×D8311yescharacteristic
376#336—831no—
377#336—831no—
378C2×#41C22×D8311yescharacteristic
379C2×M16—831no—
380C2×M16—831no—
381C2×M16C22×D8311yescharacteristic
382C2×M16C22×D8311yescharacteristic
383C22×C8—831no—
384C22×C8—831no—
385C22×C8C22×D8311yescharacteristic
386C22×C8C22×D8311yescharacteristic
387C4⋊Q8—831no—
388C4⋊Q8—831no—
389#125—831no—
390#125—831no—
391C22⋊Q8—1661no—
392C22⋊Q8—1661no—
393C4×Q8—831no—
394C4×Q8—831no—
395C4×Q8—1661no—
396C4×Q8—1661no—
397C4×D4—831no—
398C4×D4—831no—
399C4×D4—1661no—
400C4×D4—1661no—
401#117—831no—
402#117—831no—
403#117—1661no—
404#117C22×D8311yescharacteristic
405C2×C4⋊C4—1661no—
406C2×C42—831no—
407C2×C42—831no—
408C2×C42—1661no—
409C2×C42C22×D8311yescharacteristic
410#108—3321no—
411#108—3321no—
412#107—831no—
413#107—831no—
414#107—831no—
415#107—831no—
416#106—831no—
417#106—831no—
418#106—831no—
419#106—831no—
420C4⋊C8—21no—
421C4⋊C8—21no—
422C4⋊C8—831no—
423C4⋊C8—831no—
424C4⋊C8—831no—
425C4⋊C8—831no—
426C4⋊C8—1661no—
427C4⋊C8—1661no—
428C4⋊C8—1661no—
429C4⋊C8—1661no—
430C4⋊C8—1661no—
431C4⋊C8—1661no—
432C4⋊C8—1661no—
433C4⋊C8—1661no—
434#104—41no—
435#104—3321no—
436#103—831no—
437#103—831no—
438#103C4×D8311yescharacteristic
439#103C4×D8311yescharacteristic
440#102—831no—
441#102—831no—
442#102C4×D8311yescharacteristic
443#102C4×D8311yescharacteristic
444C22⋊C8—21no—
445C22⋊C8—21no—
446C22⋊C8—1661no—
447C22⋊C8—1661no—
448#97—1661no—
449#97—1661no—
450#97—1661no—
451C4×C8—41no—
452C4×C8—3321no—
453C4×C8—831no—
454C4×C8—831no—
455C4×C8—831no—
456C4×C8—831no—
457C4×C8—1661no—
458C4×C8—1661no—
459C4×C8—1661no—
460#95—1661no—
461#41—21no—
462#41—21no—
463#41—21no—
464#41—21no—
465C2×Q8—1661no—
466C2×Q8—1661no—
467C2×Q8C2×C4×D8311yescharacteristic
468C2×D4—21no—
469C2×D4C2×C4×D8311yescharacteristic
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—831no—
473C22×C4—831no—
474C22×C4—1661no—
475C22×C4—1661no—
476C22×C4—1661no—
477C22×C4C23×D8311yescharacteristic
478M16—21no—
479M16—21no—
480M16—21no—
481M16—21no—
482M16—1661no—
483M16—1661no—
484M16—1661no—
485M16—1661no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—831no—
493C2×C8—831no—
494C2×C8—831no—
495C2×C8—831no—
496C2×C8—1661no—
497C2×C8—1661no—
498C2×C8—1661no—
499C2×C8—1661no—
500C2×C8—1661no—
501C2×C8—1661no—
502C2×C8C2×C4×D8311yescharacteristic
503C2×C8C2×C4×D8311yescharacteristic
504C2×C8D4×D8311yescharacteristic
505C2×C8D4×D8311yescharacteristic
506C4⋊C4—831no—
507C4⋊C4—831no—
508C4⋊C4—831no—
509C4⋊C4—831no—
510C4⋊C4—1661no—
511C4⋊C4—1661no—
512C4⋊C4—1661no—
513C4⋊C4—1661no—
514C4⋊C4—1661no—
515C4⋊C4—1661no—
516C4⋊C4—1661no—
517C4⋊C4—1661no—
518C4⋊C4—1664no—
519C4⋊C4C2×C4×D8311yescharacteristic
520C4⋊C4C2×C4×D8311yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—1661no—
523C22⋊C4—1661no—
524C22⋊C4—1661no—
525C22⋊C4—1661no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—831no—
530C42—831no—
531C42—831no—
532C42—831no—
533C42—1661no—
534C42—1661no—
535C42—1661no—
536C42—1661no—
537C42—1661no—
538C42—1661no—
539C23—21no—
540C23#41×D8311yescharacteristic
541Q8—21no—
542Q8—21no—
543Q8—3322no—
544Q8—3322no—
545D4—21no—
546D4—21no—
547D4—41no—
548D4—41no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3322no—
560C2×C4—3322no—
561C2×C4—3321no—
562C2×C4—831no—
563C2×C4—831no—
564C2×C4—831no—
565C2×C4—831no—
566C2×C4—1662no—
567C2×C4—1661no—
568C2×C4—1661no—
569C2×C4—1661no—
570C2×C4—1661no—
571C2×C4—1662no—
572C2×C4C22×C4×D8311yescharacteristic
573C2×C4C2×D4×D8311yescharacteristic
574C2×C4#1458183711yescharacteristic
575C2×C4#1458183911yescharacteristic
576C2×C4#1458183911yescharacteristic
577C2×C4#1458183911yescharacteristic
578C8—21no—
579C8—21no—
580C8—21no—
581C8—21no—
582C8—41no—
583C8—41no—
584C8—3321no—
585C8—3321no—
586C8—1661no—
587C8—1661no—
588C8—1661no—
589C8—1661no—
590C22—21no—
591C22—21no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C4×D4×D8311yescharacteristic
596C22#1912830811yescharacteristic
597C22#1912831911yescharacteristic
598C4—42no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—3322no—
603C4—3321no—
604C4—3321no—
605C4—3321no—
606C4—3321no—
607C4—832no—
608C4—1661no—
609C4—1661no—
610C4—1661no—
611C4#1912817811yescharacteristic
612C4#1912818011yescharacteristic
613C4#1912827111yescharacteristic
614C4#1912827811yescharacteristic
615C2—21no—
616C2—21no—
617C2—81no—
618C2#2491013811yescharacteristic
619C2#443×D8311yescharacteristic
620C2#2491636111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1937C2
socleC22×C83C2×C4×D4
prime radicalC2×#41×C83C23
cosocleC22×C4×C83C24
hypercenter#1937D83
hypocenterC83#22130
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1937

Complements

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

NH ≅ G/Ntype
2. #24916396C2semidirect
4. #24908405C2semidirect
5. #24908237C2semidirect
6. #24908211C2semidirect
7. #24908208C2semidirect
11. #24908061C2semidirect
12. #24908060C2semidirect
14. C8×Dic83⋊C4C2semidirect
18. #19128010C4semidirect
19. #19128006C4semidirect
20. #19128003C4semidirect
21. #19127999C4semidirect
22. #19127994C4semidirect
23. #19127989C4semidirect
24. #19127984C4semidirect
25. #19127979C4semidirect
326. C83#22130semidirect

Maximal subgroups

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

subgroupindex
#249172472
#249163962
#249161032
#249084052
#249082372
#249082112
#249082082
#249081702
#249081142
#249081132
#249080612
#249080602
#249074812
C8×Dic83⋊C42
C83×#19372
#2213083