Finite Group Databasebeta

← #35375140

Subgroups — #35375140

Order 21248.

658 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918019C211yesmaximal, characteristic
2#24916392C211yesmaximal, characteristic, complemented
3#24916172C211yesmaximal, characteristic
4#24906343C211yesmaximal, characteristic, complemented
5#24906341C211yesmaximal, characteristic, complemented
6#24906339C211yesmaximal, characteristic, complemented
7#24906323C211yesmaximal, characteristic, complemented
8#24906321C211yesmaximal, characteristic, complemented
9#24906319C211yesmaximal, characteristic, complemented
10#24906317C211yesmaximal, characteristic, complemented
11#24906220C211yesmaximal, characteristic, complemented
12#24905916C211yesmaximal, characteristic, complemented
13#24905912C211yesmaximal, characteristic
14#322×D83C211yesmaximal, characteristic, complemented
15C83×#1588C211yesmaximal, characteristic, complemented
16C2×#14581800—21no—
17#19128255C2211yescharacteristic
18#19128248C2211yescharacteristic
19#19128098C411yescharacteristic
20#19128072C411yescharacteristic
21#19128050C411yescharacteristic
22#19128046C411yescharacteristic
23#19127933C411yescharacteristic
24#19127931C411yescharacteristic
25#19127929C411yescharacteristic
26#19127927C411yescharacteristic
27#19127832C2211yescharacteristic
28#19127830—21no—
29#19127828C2211yescharacteristic
30C8×Dic166—21no—
31C8×Dic166C2211yescharacteristic
32C4×#13050579C2211yescharacteristic
33C4×#13050577C2211yescharacteristic
34C4×#13050578C2211yescharacteristic
35#19127780C2211yescharacteristic
36#19127777—21no—
37D4×C83⋊C8C2211yescharacteristic
38#19127766C2211yescharacteristic
39(C22⋊D83)⋊C8—21no—
40#19127726—21no—
41#19127723C2211yescharacteristic
42#19127723C2211yescharacteristic
43#19127666—21no—
44#19127655—21no—
45C4×C8×D83C2211yescharacteristic
46C4×C8×D83C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic
49#19127453C2211yescharacteristic
50#19127453C2211yescharacteristic
51#19127433C2211yescharacteristic
52Q8⋊(C83⋊C8)C2211yescharacteristic
53#19127431C2211yescharacteristic
54#19127430C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic
58#19127423C2211yescharacteristic
59#19127422C2211yescharacteristic
60C8×Q8×C83C2211yescharacteristic
61C4×SD16×C83C2211yescharacteristic
62#436×C83C2211yescharacteristic
63#335×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic
67C2×#13050590—21no—
68C2×#13050579C2×C411yescharacteristic
69C2×#13050578C2×C411yescharacteristic
70C2×#13050577C2×C411yescharacteristic
71#14581800—44no—
72C2×#13050573—21no—
73C2×#13050573—21no—
74C2×#13050573—21no—
75C22×C83⋊C8—21no—
76C2×C8×D83—21no—
77C2×C8×D83—21no—
78C2×C8×D83C2×C411yescharacteristic
79C2×C8×D83C2×C411yescharacteristic
80C2×C8×D83D411yescharacteristic
81C2×C8×D83D411yescharacteristic
82#14581791C2×C411yescharacteristic
83#14581791C2×C411yescharacteristic
84#14581790C2×C411yescharacteristic
85#14581788C2×C411yescharacteristic
86#14581782—21no—
87#14581778—21no—
88#14581772C2×C411yescharacteristic
89#14581770—21no—
90#14581768C2×C411yescharacteristic
91#14581767C2×C411yescharacteristic
92#14581765—21no—
93#14581763C2×C411yescharacteristic
94C4×Dic166—21no—
95C4×Dic166C2311yescharacteristic
96C4×D332C2311yescharacteristic
97C4×C22⋊D83—21no—
98#14581749—21no—
99C42×D83C2311yescharacteristic
100#14581724C2×C411yescharacteristic
101#14581723C2×C411yescharacteristic
102#14581723C2×C411yescharacteristic
103C4⋊(C83⋊C8)—21no—
104C4⋊(C83⋊C8)C2311yescharacteristic
105C4⋊(C83⋊C8)C2311yescharacteristic
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#14581703—21no—
120#14581702—21no—
121C4×C83⋊C8C2311yescharacteristic
122C4×C83⋊C8C2311yescharacteristic
123C4×C83⋊C8C2311yescharacteristic
124C8×Dic83—21no—
125C8×Dic83—21no—
126C8×Dic83C2×C411yescharacteristic
127C8×Dic83C2×C411yescharacteristic
128C8×Dic83D411yescharacteristic
129C8×Dic83D411yescharacteristic
130C2×SD16×C83C2×C411yescharacteristic
131C2×M16×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—
141C4×C8×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—41no—
161#13050577—41no—
162#13050573—42no—
163#13050573—42no—
164#13050573—42no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—42no—
169C2×C83⋊C8C22×C411yescharacteristic
170C2×C83⋊C8C22×C411yescharacteristic
171C2×C83⋊C8C2×D411yescharacteristic
172C2×C83⋊C8C2×D411yescharacteristic
173C8×D83—24no—
174C8×D83—22no—
175C8×D83—22no—
176C8×D83—41no—
177C8×D83—41no—
178C8×D83—42no—
179C4⋊Dic83—21no—
180C4⋊Dic83C22×C411yescharacteristic
181C4⋊Dic83C22×C411yescharacteristic
182Dic83⋊C4—21no—
183Dic83⋊C4—21no—
184Dic83⋊C4—21no—
185Dic83⋊C4—21no—
186Dic83⋊C4—21no—
187Dic83⋊C4—21no—
188C22⋊Dic83—21no—
189(C2×D83)⋊C4—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×Q8×C83C22×C411yescharacteristic
198C2×D4×C83C22×C411yescharacteristic
199C22×C4×C83—21no—
200SD16×C83—41no—
201M16×C83—42no—
202C2×C8×C83—21no—
203C2×C8×C83—21no—
204C2×C8×C83—21no—
205C2×C8×C83C22×C411yescharacteristic
206C2×C8×C83C22×C411yescharacteristic
207C2×C8×C83C2×D411yescharacteristic
208C2×C8×C83C2×D411yescharacteristic
209C4⋊C4×C83—21no—
210C4⋊C4×C83C22×C411yescharacteristic
211C4⋊C4×C83C22×C411yescharacteristic
212C22⋊C4×C83—21no—
213C42×C83—21no—
214C42×C83C2411yescharacteristic
215C23×C83—21no—
216C22×D83—21no—
217C22×D83C4×D411yescharacteristic
218Q8×C83—21no—
219Q8×C83—21no—
220D4×C83—21no—
221D4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83C23×C411yescharacteristic
228C2×C4×C83C22×D411yescharacteristic
229C2×C4×C83C2×#4111yescharacteristic
230C22⋊D83—42no—
231C22⋊D83—42no—
232C22⋊D83—42no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83C4×D411yescharacteristic
237C2×Dic83C4×D411yescharacteristic
238C2×Dic83C4×D411yescharacteristic
239D332—21no—
240D332—21no—
241D332—42no—
242C4×D83—21no—
243C4×D83—21no—
244C4×D83—22no—
245C4×D83—41no—
246C4×D83—41no—
247C4×D83AGL(1,ℤ/8)12yes—
248C4×D83#13712yes—
249Dic166—21no—
250Dic166—21no—
251Dic166—42no—
252Dic166—42no—
253C8×C83—22no—
254C8×C83—21no—
255C8×C83—21no—
256C8×C83—21no—
257C8×C83—21no—
258C8×C83—41no—
259C8×C83—41no—
260C83⋊C8—22no—
261C83⋊C8—21no—
262C83⋊C8—21no—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—41no—
266C83⋊C8—41no—
267C22×C83—41no—
268C22×C83—41no—
269C22×C83—41no—
270C22×C83C2×C4×D411yescharacteristic
271C2×D83—21no—
272C2×D83—21no—
273C2×D83—41no—
274C2×D83—41no—
275C2×D83—41no—
276C2×D83#44012yes—
277C4×C83—21no—
278C4×C83—41no—
279C4×C83—41no—
280C4×C83—41no—
281C4×C83#56811yescharacteristic
282C4×C83#56811yescharacteristic
283C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
284C4×C83C2×#13711yescharacteristic
285Dic83—21no—
286Dic83—41no—
287Dic83—41no—
288Dic83—41no—
289Dic83#44012yes—
290Dic83#44312yes—
291#17979—831nomaximal, Sylow
292C2×C83—81no—
293C2×C83#293211yescharacteristic
294C2×C83C2×#44011yescharacteristic
295C2×C83C2×#44311yescharacteristic
296D83—22no—
297D83—81no—
298#3167—831no—
299#2960—831no—
300#2931—831no—
301#1588—831no—
302#1588—831no—
303#1588—831no—
304#1588D8311yescharacteristic, complemented
305#1585—831no—
306#1585—831no—
307#1585—831no—
308#1585—831no—
309#1567—831no—
310#1493—831no—
311#1492—831no—
312C2×#322—831no—
313C83#1797911yescharacteristic, complemented, Sylow
314C2×#131—1661no—
315#533—831no—
316#532—831no—
317#498—831no—
318#492—831no—
319#487—831no—
320#486—831no—
321#465—831no—
322#465—831no—
323#465—831no—
324#465—831no—
325#446—831no—
326#446—831no—
327#446—1661no—
328C8×Q8—831no—
329C8×Q8—1661no—
330C8×Q8C2×D8311yescharacteristic
331C4×SD16—831no—
332C4×SD16—831no—
333C4×SD16—831no—
334C4×SD16C2×D8311yescharacteristic
335#436—831no—
336#436—1661no—
337#436C2×D8311yescharacteristic
338C8×D4—831no—
339C8×D4—831no—
340C8×D4—1661no—
341#424—1661no—
342#423—831no—
343#423—831no—
344#408—1661no—
345#405—1661no—
346C2×C4×C8—831no—
347C2×C4×C8—831no—
348C2×C4×C8—831no—
349#335—831no—
350#335—831no—
351#335—831no—
352#335C2×D8311yescharacteristic
353Q8⋊C8—831no—
354Q8⋊C8—831no—
355Q8⋊C8—831no—
356Q8⋊C8C2×D8311yescharacteristic
357D4⋊C8—831no—
358D4⋊C8—831no—
359D4⋊C8—831no—
360D4⋊C8C2×D8311yescharacteristic
361#322—831no—
362#322—831no—
363#322—831no—
364#322C2×D8311yescharacteristic
365C2×#41—1661no—
366C2×SD16—831no—
367C2×SD16—831no—
368C2×SD16—831no—
369C2×SD16C4×D8311yescharacteristic
370#131—3324no—
371C2×M16—21no—
372C2×M16—1661no—
373C2×M16—1661no—
374C2×M16—1661no—
375C22×C8—831no—
376C22×C8—831no—
377C22×C8—831no—
378C22×C8—831no—
379C22×C8—1661no—
380C22×C8—1661no—
381C22×C8—1661no—
382C4⋊Q8—831no—
383C4⋊Q8—831no—
384C4⋊Q8—831no—
385#127—831no—
386#125—1661no—
387#124—1661no—
388C22⋊Q8—831no—
389C22⋊Q8—831no—
390C22⋊Q8—1661no—
391#121—831no—
392#121—831no—
393#121—1661no—
394C4×Q8—831no—
395C4×Q8—1661no—
396C4×Q8C22×D8311yescharacteristic
397C4×D4—831no—
398C4×D4—1661no—
399C4×D4C22×D8311yescharacteristic
400#117—1661no—
401C2×C42—831no—
402#106—831no—
403#106—831no—
404#106—831no—
405#106C4×D8311yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—831no—
408C4⋊C8—831no—
409C4⋊C8—1661no—
410C4⋊C8—1661no—
411C4⋊C8—1661no—
412C4⋊C8—1661no—
413C4⋊C8—1661no—
414C4⋊C8C22×D8311yescharacteristic
415C4⋊C8C22×D8311yescharacteristic
416#103—831no—
417#103—831no—
418#103—831no—
419#103C4×D8311yescharacteristic
420#102—831no—
421#102—831no—
422#102—831no—
423#102C4×D8311yescharacteristic
424C22⋊C8—21no—
425C22⋊C8—1661no—
426C22⋊C8—1661no—
427C22⋊C8—1661no—
428#97—1661no—
429#97—1661no—
430C4×C8—21no—
431C4×C8—831no—
432C4×C8—831no—
433C4×C8—831no—
434C4×C8—831no—
435C4×C8—831no—
436C4×C8—831no—
437C4×C8—831no—
438C4×C8—1661no—
439C4×C8—1661no—
440C4×C8C22×D8311yescharacteristic
441C4×C8C22×D8311yescharacteristic
442C4×C8C22×D8311yescharacteristic
443#41—3322no—
444#41—3322no—
445C2×Q8—831no—
446C2×Q8—1661no—
447C2×Q8—1661no—
448C2×Q8C2×C4×D8311yescharacteristic
449C2×D4—831no—
450C2×D4—1661no—
451C2×D4—1661no—
452C2×D4—1661no—
453C2×D4—1661no—
454C2×D4C2×C4×D8311yescharacteristic
455C22×C4—21no—
456C22×C4—831no—
457C22×C4—831no—
458C22×C4—831no—
459C22×C4—1661no—
460SD16—41no—
461SD16—3321no—
462SD16—3321no—
463SD16—3321no—
464M16—42no—
465M16—3322no—
466M16—3322no—
467M16—3322no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—3321no—
472C2×C8—3321no—
473C2×C8—3322no—
474C2×C8—3322no—
475C2×C8—831no—
476C2×C8—831no—
477C2×C8—831no—
478C2×C8—831no—
479C2×C8—1662no—
480C2×C8—1662no—
481C2×C8—1664no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8C2×C4×D8311yescharacteristic
486C2×C8C2×C4×D8311yescharacteristic
487C2×C8D4×D8311yescharacteristic
488C2×C8D4×D8311yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—831no—
491C4⋊C4—831no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4C2×C4×D8311yescharacteristic
500C4⋊C4C2×C4×D8311yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—1661no—
503C22⋊C4—1661no—
504C22⋊C4—1661no—
505C22⋊C4—1661no—
506C22⋊C4—1661no—
507C42—21no—
508C42—831no—
509C42—831no—
510C42—831no—
511C42—1661no—
512C42C23×D8311yescharacteristic
513C23—21no—
514C23—831no—
515C23—1661no—
516Q8—21no—
517Q8—21no—
518Q8—3322no—
519Q8—3322no—
520Q8—1661no—
521Q8—1661no—
522D4—21no—
523D4—21no—
524D4—3322no—
525D4—3322no—
526D4—3322no—
527D4—3322no—
528D4—1661no—
529D4—1661no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—831no—
538C2×C4—831no—
539C2×C4—832no—
540C2×C4—832no—
541C2×C4—831no—
542C2×C4—1662no—
543C2×C4—1661no—
544C2×C4—1661no—
545C2×C4—1661no—
546C2×C4—1661no—
547C2×C4—1661no—
548C2×C4C22×C4×D8311yescharacteristic
549C2×C4C2×D4×D8311yescharacteristic
550C2×C4#41×D8311yescharacteristic
551C8—22no—
552C8—21no—
553C8—21no—
554C8—21no—
555C8—21no—
556C8—41no—
557C8—41no—
558C8—3321no—
559C8—3321no—
560C8—1662no—
561C8—1661no—
562C8—1661no—
563C8—1661no—
564C8—1661no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—3321no—
569C22—3321no—
570C22—3321no—
571C22—832no—
572C22—1661no—
573C22—1661no—
574C22C4×D4×D8311yescharacteristic
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—3321no—
580C4—3321no—
581C4—3321no—
582C4—832no—
583C4—832no—
584C4—1661no—
585C4#1912846611yescharacteristic
586C4#1912846711yescharacteristic
587C4AGL(1,ℤ/8)×D8311yescharacteristic
588C4#137×D8311yescharacteristic
589C2—81no—
590C2—6641no—
591C2—1662no—
592C2#2491617711yescharacteristic
593C2#440×D8311yescharacteristic
594C2#443×D8311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1588C2
socleC22×C83C2×C4×D4
prime radical#19128248C22
cosocleC42×C83C24
hypercenter#1588D83
hypocenterC83#17979
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1588

Complements

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

NH ≅ G/Ntype
2. #24916392C2semidirect
4. #24906343C2semidirect
5. #24906341C2semidirect
6. #24906339C2semidirect
7. #24906323C2semidirect
8. #24906321C2semidirect
9. #24906319C2semidirect
10. #24906317C2semidirect
11. #24906220C2semidirect
12. #24905916C2semidirect
14. #322×D83C2semidirect
15. C83×#1588C2semidirect
304. #1588D83semidirect
313. C83#17979semidirect

Maximal subgroups

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

subgroupindex
#249180192
#249163922
#249161722
#249063432
#249063412
#249063392
#249063232
#249063212
#249063192
#249063172
#249062202
#249059162
#249059122
#322×D832
C83×#15882
#1797983