Finite Group Databasebeta

← #35376167

Subgroups — #35376167

Order 21248.

654 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918019C211yesmaximal, characteristic
2#24916411C211yesmaximal, characteristic, complemented
3#24916202C211yesmaximal, characteristic
4#24906356C211yesmaximal, characteristic, complemented
5#24906353C211yesmaximal, characteristic, complemented
6#24906308C211yesmaximal, characteristic, complemented
7#24906307C211yesmaximal, characteristic, complemented
8#24906298C211yesmaximal, characteristic, complemented
9C8×#13050578C211yesmaximal, characteristic, complemented
10#24906260C211yesmaximal, characteristic, complemented
11#334×D83C211yesmaximal, characteristic, complemented
12#24905864C211yesmaximal, characteristic, complemented
13#24905860C211yesmaximal, characteristic
14#24905742C211yesmaximal, characteristic, complemented
15C83×#1582C211yesmaximal, characteristic, complemented
16C2×#14581798—21no—
17#19128255C2211yescharacteristic
18#19128248C2211yescharacteristic
19#19128098C411yescharacteristic
20#19128072C411yescharacteristic
21#19128050C411yescharacteristic
22#19128046C411yescharacteristic
23#19127933C411yescharacteristic
24#19127931C411yescharacteristic
25#19127929C411yescharacteristic
26#19127927C411yescharacteristic
27#19127829—21no—
28#19127828—21no—
29#19127828C2211yescharacteristic
30Q8×C83⋊C8C2211yescharacteristic
31C8×Dic166C2211yescharacteristic
32C4×#13050579C2211yescharacteristic
33C4×#13050577C2211yescharacteristic
34C4×#13050578C2211yescharacteristic
35#19127780C2211yescharacteristic
36#19127779C2211yescharacteristic
37#19127775—21no—
38#19127772—21no—
39#19127766C2211yescharacteristic
40#19127752C2211yescharacteristic
41#19127726—21no—
42C4⋊C8×D83C2211yescharacteristic
43C4⋊C8×D83C2211yescharacteristic
44#19127666—21no—
45#19127658—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127452C2211yescharacteristic
49#19127451C2211yescharacteristic
50#19127451C2211yescharacteristic
51#19127433C2211yescharacteristic
52Q8⋊(C83⋊C8)C2211yescharacteristic
53#19127431C2211yescharacteristic
54#19127430C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic
58C8×C83⋊C8C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60#446×C83C2211yescharacteristic
61C4×SD16×C83C2211yescharacteristic
62C8×D4×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic
67C2×#13050590—21no—
68C2×#13050579C2×C411yescharacteristic
69C2×#13050578C2×C411yescharacteristic
70C2×#13050577C2×C411yescharacteristic
71#14581798—44no—
72C2×#13050574—21no—
73C2×#13050573—21no—
74C2×#13050573D411yescharacteristic
75C2×#13050573D411yescharacteristic
76C2×C8×D83—21no—
77C2×C8×D83—21no—
78C2×C8×D83—21no—
79C2×C8×D83C2×C411yescharacteristic
80C2×C8×D83C2×C411yescharacteristic
81#14581791C2×C411yescharacteristic
82#14581791C2×C411yescharacteristic
83#14581790C2×C411yescharacteristic
84#14581788C2×C411yescharacteristic
85#14581782—21no—
86#14581778—21no—
87#14581772C2×C411yescharacteristic
88#14581770—21no—
89#14581768C2×C411yescharacteristic
90#14581767C2×C411yescharacteristic
91#14581765—21no—
92#14581763C2×C411yescharacteristic
93C4×Dic166—21no—
94C4×Dic166C2311yescharacteristic
95C4×D332C2311yescharacteristic
96C4×C22⋊D83—21no—
97#14581749—21no—
98C42×D83C2311yescharacteristic
99#14581724C2×C411yescharacteristic
100#14581723C2×C411yescharacteristic
101#14581723C2×C411yescharacteristic
102C4⋊(C83⋊C8)—21no—
103C4⋊(C83⋊C8)C2311yescharacteristic
104C4⋊(C83⋊C8)C2311yescharacteristic
105Dic83⋊C8—21no—
106Dic83⋊C8—21no—
107Dic83⋊C8—21no—
108Dic83⋊C8—21no—
109Dic83⋊C8—21no—
110#14581717C2×C411yescharacteristic
111#14581716C2×C411yescharacteristic
112#14581715C2×C411yescharacteristic
113#14581714C2×C411yescharacteristic
114#14581713C2×C411yescharacteristic
115#14581712C2×C411yescharacteristic
116C22⋊(C83⋊C8)—21no—
117(C2×D83)⋊C8—21no—
118(C2×D83)⋊C8—21no—
119(C2×D83)⋊C8—21no—
120#14581702—21no—
121#14581702D411yescharacteristic
122#14581702D411yescharacteristic
123C4×C83⋊C8—21no—
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C8×Dic83—21no—
128C8×Dic83C2×C411yescharacteristic
129C8×Dic83C2×C411yescharacteristic
130C2×SD16×C83C2×C411yescharacteristic
131C22×C8×C83—21no—
132C4×Q8×C83C2311yescharacteristic
133C4×D4×C83C2311yescharacteristic
134#106×C83C2×C411yescharacteristic
135C4⋊C8×C83—21no—
136C4⋊C8×C83C2311yescharacteristic
137C4⋊C8×C83C2311yescharacteristic
138#103×C83C2×C411yescharacteristic
139#102×C83C2×C411yescharacteristic
140C22⋊C8×C83—21no—
141#97×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145#13050590—42no—
146#13050590—42no—
147C2×Dic166—21no—
148C2×Dic166—21no—
149C2×Dic166C22×C411yescharacteristic
150C2×D332—21no—
151C2×D332C22×C411yescharacteristic
152C2×C22⋊D83—21no—
153C2×C22⋊D83—21no—
154C2×C22⋊D83—21no—
155C2×C4×D83—21no—
156C2×C4×D83C22×C411yescharacteristic
157C2×C4×D83C2×D411yescharacteristic
158C2×C4×D83#4111yescharacteristic
159#13050579—41no—
160#13050578—22no—
161#13050577—22no—
162#13050574—42no—
163#13050573—22no—
164#13050573—42no—
165#13050573SD1614yescomplemented
166C2×C83⋊C8—21no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8C22×C411yescharacteristic
170C2×C83⋊C8C22×C411yescharacteristic
171C2×C83⋊C8C2×D411yescharacteristic
172C2×C83⋊C8C2×D411yescharacteristic
173C8×D83—22no—
174C8×D83—41no—
175C8×D83—42no—
176C8×D83—42no—
177C8×D83—42no—
178C4⋊Dic83—21no—
179C4⋊Dic83C22×C411yescharacteristic
180C4⋊Dic83C22×C411yescharacteristic
181Dic83⋊C4—21no—
182Dic83⋊C4—21no—
183Dic83⋊C4—21no—
184Dic83⋊C4—21no—
185Dic83⋊C4—21no—
186Dic83⋊C4—21no—
187C22⋊Dic83—21no—
188(C2×D83)⋊C4—21no—
189(C2×D83)⋊C4—21no—
190(C2×D83)⋊C4—21no—
191(C2×D83)⋊C4—21no—
192C4×Dic83—21no—
193C4×Dic83C22×C411yescharacteristic
194C4×Dic83C2×D411yescharacteristic
195C4×Dic83#4111yescharacteristic
196C2×Q8×C83C22×C411yescharacteristic
197C2×D4×C83C22×C411yescharacteristic
198C22×C4×C83—21no—
199SD16×C83—41no—
200C2×C8×C83—21no—
201C2×C8×C83—21no—
202C2×C8×C83—21no—
203C2×C8×C83—42no—
204C2×C8×C83C22×C411yescharacteristic
205C2×C8×C83C22×C411yescharacteristic
206C2×C8×C83C2×D411yescharacteristic
207C2×C8×C83C2×D411yescharacteristic
208C4⋊C4×C83—21no—
209C4⋊C4×C83C22×C411yescharacteristic
210C4⋊C4×C83C22×C411yescharacteristic
211C22⋊C4×C83—21no—
212C42×C83—21no—
213C42×C83C2411yescharacteristic
214C23×C83—21no—
215C22×D83—21no—
216C22×D83C4×D411yescharacteristic
217Q8×C83—21no—
218Q8×C83—21no—
219D4×C83—21no—
220D4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83C23×C411yescharacteristic
227C2×C4×C83C22×D411yescharacteristic
228C2×C4×C83C2×#4111yescharacteristic
229C22⋊D83—42no—
230C22⋊D83—42no—
231C22⋊D83—81no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83C4×D411yescharacteristic
236C2×Dic83C4×D411yescharacteristic
237C2×Dic83C4×D411yescharacteristic
238D332—21no—
239D332—21no—
240D332—42no—
241C4×D83—21no—
242C4×D83—21no—
243C4×D83—41no—
244C4×D83—41no—
245C4×D83—41no—
246C4×D83C2×SD1612yes—
247C4×D83#13512yes—
248Dic166—21no—
249Dic166—21no—
250Dic166—42no—
251Dic166—81no—
252C8×C83—21no—
253C8×C83—21no—
254C8×C83—21no—
255C8×C83—41no—
256C8×C83—41no—
257C8×C83—41no—
258C8×C83C2×SD1612yes—
259C83⋊C8—21no—
260C83⋊C8—21no—
261C83⋊C8—41no—
262C83⋊C8—41no—
263C83⋊C8—41no—
264C83⋊C8#13112yescomplemented
265C83⋊C8C2×SD1612yes—
266C22×C83—41no—
267C22×C83—41no—
268C22×C83—41no—
269C22×C83C2×C4×D411yescharacteristic
270C2×D83—21no—
271C2×D83—21no—
272C2×D83—41no—
273C2×D83—41no—
274C2×D83—41no—
275C2×D83C4×SD1612yes—
276C4×C83—21no—
277C4×C83—41no—
278C4×C83—41no—
279C4×C83—41no—
280C4×C83C2×#13111yescharacteristic
281C4×C83#56811yescharacteristic
282C4×C83C22×SD1611yescharacteristic
283C4×C83C2×#13511yescharacteristic
284Dic83—21no—
285Dic83—21no—
286Dic83—41no—
287Dic83—41no—
288Dic83—41no—
289Dic83C4×SD1612yes—
290#18067—831nomaximal, Sylow
291C2×C83—81no—
292C2×C83#293511yescharacteristic
293C2×C83C2×C4×SD1611yescharacteristic
294C2×C83#295711yescharacteristic
295D83—22no—
296D83—81no—
297#3167—831no—
298#2962—831no—
299#2935—831no—
300#1590—831no—
301#1590—831no—
302#1583—831no—
303#1583—831no—
304#1582—831no—
305#1582D8311yescharacteristic, complemented
306C8×SD16—831no—
307C8×SD16—831no—
308C2×#334—831no—
309#1481—831no—
310#1480—831no—
311C8×M16—831no—
312C83#1806711yescharacteristic, complemented, Sylow
313C2×#131—1661no—
314#533—831no—
315#532—831no—
316#498—831no—
317#492—831no—
318#487—831no—
319#486—831no—
320#465—831no—
321#465—831no—
322#465—831no—
323#465—831no—
324#446—831no—
325#446—1661no—
326#446—1661no—
327#446C2×D8311yescharacteristic
328C8×Q8—831no—
329C8×Q8—831no—
330C4×SD16—831no—
331C4×SD16—831no—
332C4×SD16—831no—
333C4×SD16C2×D8311yescharacteristic
334#436—831no—
335#436—831no—
336#435—1661no—
337#435—1661no—
338C8×D4—831no—
339C8×D4C2×D8311yescharacteristic
340#431—831no—
341#424—1661no—
342C2×C4⋊C8—831no—
343C2×C4⋊C8—831no—
344#408—1661no—
345#405—1661no—
346C4×M16—831no—
347C2×C4×C8—831no—
348#334—831no—
349#334—831no—
350#334—831no—
351#334C2×D8311yescharacteristic
352Q8⋊C8—831no—
353Q8⋊C8—831no—
354Q8⋊C8—831no—
355Q8⋊C8C2×D8311yescharacteristic
356D4⋊C8—831no—
357D4⋊C8—831no—
358D4⋊C8—831no—
359D4⋊C8C2×D8311yescharacteristic
360#322—831no—
361#322C2×D8311yescharacteristic
362C82—831no—
363C82—831no—
364C2×#41—1661no—
365C2×SD16—831no—
366C2×SD16—831no—
367C2×SD16—831no—
368C2×SD16C4×D8311yescharacteristic
369#131—3324no—
370C2×M16—831no—
371C2×M16—831no—
372C2×M16—1661no—
373C2×M16—1661no—
374C22×C8—21no—
375C22×C8—831no—
376C22×C8—831no—
377C22×C8—1661no—
378C22×C8—1661no—
379C22×C8—1661no—
380C4⋊Q8—831no—
381C4⋊Q8—831no—
382C4⋊Q8—831no—
383#127—831no—
384#125—1661no—
385#124—1661no—
386C22⋊Q8—831no—
387C22⋊Q8—831no—
388C22⋊Q8—1661no—
389#121—831no—
390#121—831no—
391#121—1661no—
392C4×Q8—831no—
393C4×Q8—1661no—
394C4×Q8C22×D8311yescharacteristic
395C4×D4—831no—
396C4×D4—1661no—
397C4×D4C22×D8311yescharacteristic
398#117—1661no—
399C2×C42—831no—
400#106—831no—
401#106—831no—
402#106—831no—
403#106C4×D8311yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—831no—
406C4⋊C8—831no—
407C4⋊C8—1661no—
408C4⋊C8—1661no—
409C4⋊C8—1661no—
410C4⋊C8—1661no—
411C4⋊C8—1661no—
412C4⋊C8—1661no—
413C4⋊C8C22×D8311yescharacteristic
414C4⋊C8C22×D8311yescharacteristic
415#103—831no—
416#103—831no—
417#103—831no—
418#103C4×D8311yescharacteristic
419#102—831no—
420#102—831no—
421#102—831no—
422#102C4×D8311yescharacteristic
423C22⋊C8—21no—
424C22⋊C8—1661no—
425C22⋊C8—1661no—
426C22⋊C8—1661no—
427C22⋊C8—1661no—
428#97—21no—
429#97—831no—
430#97—831no—
431#97—1661no—
432C4×C8—831no—
433C4×C8—831no—
434C4×C8—831no—
435C4×C8—831no—
436C4×C8—831no—
437C4×C8—1661no—
438C4×C8—1661no—
439C4×C8C22×D8311yescharacteristic
440C4×C8C22×D8311yescharacteristic
441C4×C8C22×D8311yescharacteristic
442#41—3322no—
443#41—3322no—
444C2×Q8—831no—
445C2×Q8—1661no—
446C2×Q8—1661no—
447C2×Q8C2×C4×D8311yescharacteristic
448C2×D4—831no—
449C2×D4—1661no—
450C2×D4—1661no—
451C2×D4—1661no—
452C2×D4—1661no—
453C2×D4C2×C4×D8311yescharacteristic
454C22×C4—21no—
455C22×C4—831no—
456C22×C4—831no—
457C22×C4—831no—
458C22×C4—1661no—
459SD16—41no—
460SD16—3321no—
461SD16—1662no—
462SD16—1662no—
463M16—3322no—
464M16—3322no—
465M16—834no—
466M16—1662no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—42no—
471C2×C8—3321no—
472C2×C8—3322no—
473C2×C8—3322no—
474C2×C8—3322no—
475C2×C8—831no—
476C2×C8—831no—
477C2×C8—831no—
478C2×C8—831no—
479C2×C8—1662no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8C2×C4×D8311yescharacteristic
484C2×C8C2×C4×D8311yescharacteristic
485C2×C8D4×D8311yescharacteristic
486C2×C8D4×D8311yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—831no—
489C4⋊C4—831no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4C2×C4×D8311yescharacteristic
498C4⋊C4C2×C4×D8311yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—1661no—
501C22⋊C4—1661no—
502C22⋊C4—1661no—
503C22⋊C4—1661no—
504C22⋊C4—1661no—
505C42—21no—
506C42—831no—
507C42—831no—
508C42—831no—
509C42—1661no—
510C42C23×D8311yescharacteristic
511C23—21no—
512C23—831no—
513C23—1661no—
514Q8—21no—
515Q8—21no—
516Q8—3322no—
517Q8—6641no—
518Q8—1661no—
519Q8—1661no—
520D4—21no—
521D4—21no—
522D4—3322no—
523D4—3322no—
524D4—3322no—
525D4—6641no—
526D4—1661no—
527D4—1661no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—3321no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—831no—
537C2×C4—832no—
538C2×C4—832no—
539C2×C4—831no—
540C2×C4—831no—
541C2×C4—1661no—
542C2×C4—1661no—
543C2×C4—1661no—
544C2×C4—1661no—
545C2×C4—1661no—
546C2×C4C22×C4×D8311yescharacteristic
547C2×C4C2×D4×D8311yescharacteristic
548C2×C4#41×D8311yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—3321no—
556C8—3321no—
557C8—3321no—
558C8—832no—
559C8—832no—
560C8—1661no—
561C8—1661no—
562C8SD16×D8312yes—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—3321no—
567C22—3321no—
568C22—3321no—
569C22—832no—
570C22—1661no—
571C22—1661no—
572C22C4×D4×D8311yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—3321no—
578C4—3321no—
579C4—3321no—
580C4—832no—
581C4—1661no—
582C4—1661no—
583C4#131×D8311yescharacteristic
584C4#1912846711yescharacteristic
585C4C2×SD16×D8311yescharacteristic
586C4#135×D8311yescharacteristic
587C2—81no—
588C2—6641no—
589C2—1662no—
590C2#2491620411yescharacteristic
591C2C4×SD16×D8311yescharacteristic
592C2#2491637711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1582C2
socleC22×C83C2×C4×D4
prime radical#19128248C22
cosocleC42×C83C24
hypercenter#1582D83
hypocenterC83#18067
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
2. #24916411C2semidirect
4. #24906356C2semidirect
5. #24906353C2semidirect
6. #24906308C2semidirect
7. #24906307C2semidirect
8. #24906298C2semidirect
9. C8×#13050578C2semidirect
10. #24906260C2semidirect
11. #334×D83C2semidirect
12. #24905864C2semidirect
14. #24905742C2semidirect
15. C83×#1582C2semidirect
165. #13050573SD16semidirect
264. C83⋊C8#131semidirect
305. #1582D83semidirect
312. C83#18067semidirect

Maximal subgroups

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

subgroupindex
#249180192
#249164112
#249162022
#249063562
#249063532
#249063082
#249063072
#249062982
C8×#130505782
#249062602
#334×D832
#249058642
#249058602
#249057422
C83×#15822
#1806783