Finite Group Databasebeta

← #35375348

Subgroups — #35375348

Order 21248.

616 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917477C211yesmaximal, characteristic
2#24916479C211yesmaximal, characteristic, complemented
3#24916464C211yesmaximal, characteristic, complemented
4#24906351C211yesmaximal, characteristic, complemented
5#24906348C211yesmaximal, characteristic, complemented
6#24906315C211yesmaximal, characteristic, complemented
7#24906309C211yesmaximal, characteristic, complemented
8#24906309C211yesmaximal, characteristic, complemented
9#24906269C211yesmaximal, characteristic, complemented
10C8×#13050578C211yesmaximal, characteristic, complemented
11#24906242C211yesmaximal, characteristic, complemented
12#24905920C211yesmaximal, characteristic, complemented
13#24905842C211yesmaximal, characteristic, complemented
14#24905742C211yesmaximal, characteristic, complemented
15C83×#1581C211yesmaximal, characteristic, complemented
16#19128233C2211yescharacteristic
17C4×#13050589C2211yescharacteristic
18#19128044C411yescharacteristic
19#19128023C411yescharacteristic
20#19128007C411yescharacteristic
21#19127999C411yescharacteristic
22#19127922C411yescharacteristic
23#19127914C411yescharacteristic
24#19127864C411yescharacteristic
25#19127843C411yescharacteristic
26#19127831C2211yescharacteristic
27C8×Dic166C2211yescharacteristic
28#19127825C2211yescharacteristic
29#19127825C2211yescharacteristic
30C4×Dic332C2211yescharacteristic
31C4×#13050578C2211yescharacteristic
32C4×#13050578C2211yescharacteristic
33#19127779C2211yescharacteristic
34#19127779C2211yescharacteristic
35#19127777—21no—
36#19127776—21no—
37#19127776—21no—
38#19127772—21no—
39#19127759—21no—
40#19127757C2211yescharacteristic
41#19127755—21no—
42#19127753C2211yescharacteristic
43#19127752C2211yescharacteristic
44#19127656—21no—
45M16×Dic83—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic
49#19127451C2211yescharacteristic
50#19127451C2211yescharacteristic
51#19127433C2211yescharacteristic
52#19127433C2211yescharacteristic
53#19127431C2211yescharacteristic
54#19127431C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic
56D4⋊(C83⋊C8)C2211yescharacteristic
57#19127423C2211yescharacteristic
58C8×C83⋊C8C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60C4×D8×C83C2211yescharacteristic
61#436×C83C2211yescharacteristic
62C8×D4×C83C2211yescharacteristic
63#335×C83C2211yescharacteristic
64D4⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic
67C2×#13050589C2×C411yescharacteristic
68C2×Dic332C2×C411yescharacteristic
69C2×#13050578C2×C411yescharacteristic
70C2×#13050578C2×C411yescharacteristic
71C2×#13050574—21no—
72C2×#13050574—21no—
73C2×#13050573D411yescharacteristic
74C2×#13050573D411yescharacteristic
75C2×C8×D83—21no—
76#14581785—21no—
77#14581783C2×C411yescharacteristic
78#14581779C2×C411yescharacteristic
79#14581774—21no—
80#14581772C2×C411yescharacteristic
81#14581763C2×C411yescharacteristic
82C4×Dic166C2311yescharacteristic
83C4×Dic166C2311yescharacteristic
84#14581757C2×C411yescharacteristic
85D4×Dic83C2×C411yescharacteristic
86C4×C22⋊D83—21no—
87C4×C22⋊D83—21no—
88(C22⋊D83)⋊C4—21no—
89#14581751—21no—
90#14581750C2×C411yescharacteristic
91#14581748—21no—
92C2×C4×Dic83—21no—
93C42×D83C2311yescharacteristic
94#14581726C2×C411yescharacteristic
95#14581723C2×C411yescharacteristic
96#14581723C2×C411yescharacteristic
97C4⋊(C83⋊C8)C2311yescharacteristic
98C4⋊(C83⋊C8)C2311yescharacteristic
99Dic83⋊C8—21no—
100Dic83⋊C8—21no—
101Dic83⋊C8—21no—
102Dic83⋊C8—21no—
103Dic83⋊C8C2×C411yescharacteristic
104Dic83⋊C8C2×C411yescharacteristic
105#14581717C2×C411yescharacteristic
106#14581717C2×C411yescharacteristic
107#14581715C2×C411yescharacteristic
108#14581715C2×C411yescharacteristic
109#14581713C2×C411yescharacteristic
110#14581713C2×C411yescharacteristic
111C22⋊(C83⋊C8)—21no—
112C22⋊(C83⋊C8)—21no—
113(C2×D83)⋊C8—21no—
114(C2×D83)⋊C8—21no—
115(C2×D83)⋊C8C2×C411yescharacteristic
116(C2×D83)⋊C8C2×C411yescharacteristic
117#14581702—21no—
118#14581702—21no—
119#14581702D411yescharacteristic
120#14581702D411yescharacteristic
121C4×C83⋊C8C2311yescharacteristic
122C4×C83⋊C8C2311yescharacteristic
123C4×C83⋊C8C2311yescharacteristic
124C8×Dic83—21no—
125C8×Dic83—21no—
126C8×Dic83—21no—
127C8×Dic83—21no—
128C8×Dic83—21no—
129C2×D8×C83C2×C411yescharacteristic
130C2×M16×C83—21no—
131C22×C8×C83—21no—
132C4×D4×C83C2311yescharacteristic
133C4×D4×C83C2311yescharacteristic
134#107×C83C2×C411yescharacteristic
135C4⋊C8×C83C2311yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137#102×C83C2×C411yescharacteristic
138#102×C83C2×C411yescharacteristic
139C22⋊C8×C83—21no—
140C22⋊C8×C83—21no—
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144#13050589—22no—
145#13050589—41no—
146C2×Dic166C22×C411yescharacteristic
147C2×Dic166C22×C411yescharacteristic
148C2×C22⋊D83—21no—
149C2×C22⋊D83—21no—
150C22×Dic83—21no—
151C22×Dic83—21no—
152C2×C4×D83C22×C411yescharacteristic
153C2×C4×D83C2×D411yescharacteristic
154C2×C4×D83#4111yescharacteristic
155Dic332—22no—
156#13050578—22no—
157#13050578—41no—
158#13050574—42no—
159#13050574—42no—
160#13050573—22no—
161#13050573—22no—
162C2×C83⋊C8—21no—
163C2×C83⋊C8—21no—
164C2×C83⋊C8—21no—
165C2×C83⋊C8C22×C411yescharacteristic
166C2×C83⋊C8C22×C411yescharacteristic
167C2×C83⋊C8C2×D411yescharacteristic
168C2×C83⋊C8C2×D411yescharacteristic
169C8×D83—42no—
170C4⋊Dic83—21no—
171C4⋊Dic83—21no—
172C4⋊Dic83C22×C411yescharacteristic
173C4⋊Dic83C22×C411yescharacteristic
174Dic83⋊C4—21no—
175Dic83⋊C4—21no—
176Dic83⋊C4—21no—
177Dic83⋊C4—21no—
178C22⋊Dic83—21no—
179C22⋊Dic83—21no—
180C22⋊Dic83—21no—
181C22⋊Dic83—21no—
182(C2×D83)⋊C4—21no—
183(C2×D83)⋊C4—21no—
184C4×Dic83—21no—
185C4×Dic83—21no—
186C4×Dic83—21no—
187C4×Dic83—21no—
188C4×Dic83C22×C411yescharacteristic
189C4×Dic83C2×D411yescharacteristic
190C4×Dic83#4111yescharacteristic
191C2×D4×C83C22×C411yescharacteristic
192C2×D4×C83C22×C411yescharacteristic
193C22×C4×C83—21no—
194C22×C4×C83—21no—
195D8×C83—41no—
196M16×C83—42no—
197C2×C8×C83—21no—
198C2×C8×C83—21no—
199C2×C8×C83—21no—
200C2×C8×C83—42no—
201C2×C8×C83C22×C411yescharacteristic
202C2×C8×C83C22×C411yescharacteristic
203C2×C8×C83C2×D411yescharacteristic
204C2×C8×C83C2×D411yescharacteristic
205C4⋊C4×C83C22×C411yescharacteristic
206C4⋊C4×C83C22×C411yescharacteristic
207C22⋊C4×C83—21no—
208C22⋊C4×C83—21no—
209C42×C83C2411yescharacteristic
210C23×C83—21no—
211C23×C83—21no—
212C22×D83C4×D411yescharacteristic
213D4×C83—21no—
214D4×C83—21no—
215D4×C83—21no—
216D4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83C23×C411yescharacteristic
224C2×C4×C83C22×D411yescharacteristic
225C2×C4×C83C2×#4111yescharacteristic
226C22⋊D83—42no—
227C22⋊D83—81no—
228C2×Dic83—21no—
229C2×Dic83—21no—
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—42no—
233C2×Dic83—81no—
234C2×Dic83C4×D411yescharacteristic
235C2×Dic83C4×D411yescharacteristic
236C2×Dic83C4×D411yescharacteristic
237C4×D83—21no—
238C4×D83—21no—
239C4×D83—21no—
240C4×D83—21no—
241C4×D83—41no—
242Dic166—21no—
243Dic166—21no—
244Dic166—21no—
245Dic166#13112yes—
246C8×C83—21no—
247C8×C83—21no—
248C8×C83—21no—
249C8×C83—41no—
250C8×C83—41no—
251C8×C83—41no—
252C8×C83#13512yes—
253C83⋊C8—22no—
254C83⋊C8—21no—
255C83⋊C8—21no—
256C83⋊C8—21no—
257C83⋊C8—41no—
258C83⋊C8—41no—
259C83⋊C8#13512yes—
260C22×C83—41no—
261C22×C83—41no—
262C22×C83—41no—
263C22×C83—41no—
264C22×C83—41no—
265C22×C83—41no—
266C22×C83C2×C4×D411yescharacteristic
267C2×D83—21no—
268C2×D83—21no—
269C2×D83—21no—
270C4×C83—21no—
271C4×C83—41no—
272C4×C83—41no—
273C4×C83C2×#13111yescharacteristic
274C4×C83#56811yescharacteristic
275C4×C83C2×#13511yescharacteristic
276C4×C83C2×#13511yescharacteristic
277Dic83—22no—
278Dic83—21no—
279Dic83—21no—
280Dic83—21no—
281Dic83—41no—
282Dic83—41no—
283Dic83—41no—
284#17999—831nomaximal, Sylow
285C2×C83—81no—
286C2×C83—81no—
287C2×C83#293311yescharacteristic
288C2×C83C4×#13511yescharacteristic
289C2×C83#295611yescharacteristic
290D83—41no—
291#3102—831no—
292#2971—831no—
293#2969—831no—
294#1590—831no—
295#1589—831no—
296#1584—831no—
297#1583—831no—
298#1583—831no—
299#1581D8311yescharacteristic, complemented
300C8×Q16—831no—
301C8×SD16—831no—
302#1572—831no—
303#1494—831no—
304#1476—831no—
305C8×M16—831no—
306C83#1799911yescharacteristic, complemented, Sylow
307#529—831no—
308C4×#41—831no—
309#485—831no—
310#482—831no—
311#479—831no—
312#478—831no—
313#464—831no—
314#463—831no—
315#452—831no—
316#449—831no—
317#446—831no—
318C8×Q8—831no—
319C8×Q8—831no—
320C8×Q8—831no—
321C4×Q16—831no—
322C4×SD16—831no—
323C4×SD16—831no—
324C4×D8C2×D8311yescharacteristic
325#436—831no—
326#436—831no—
327#436—1661no—
328#436—1661no—
329#436—1661no—
330#436C2×D8311yescharacteristic
331#435—1661no—
332C8×D4C2×D8311yescharacteristic
333#433—1661no—
334#432—831no—
335#432—1661no—
336#431—831no—
337#431—831no—
338#405—1661no—
339C4×M16—831no—
340C4×M16—1661no—
341C2×C4×C8—831no—
342#335—831no—
343#335C2×D8311yescharacteristic
344#334—831no—
345#334—831no—
346Q8⋊C8—831no—
347Q8⋊C8—831no—
348Q8⋊C8—831no—
349Q8⋊C8—831no—
350D4⋊C8—831no—
351D4⋊C8—831no—
352D4⋊C8C2×D8311yescharacteristic
353D4⋊C8C2×D8311yescharacteristic
354#322—831no—
355#322C2×D8311yescharacteristic
356C82—831no—
357C82—831no—
358C2×#41—831no—
359C2×Q16—831no—
360C2×SD16—831no—
361C2×SD16—831no—
362C2×D8C4×D8311yescharacteristic
363C2×M16—21no—
364C2×M16—831no—
365C2×M16—831no—
366C2×M16—1661no—
367C2×M16—1661no—
368C22×C8—21no—
369C22×C8—1661no—
370#126—1661no—
371#125—831no—
372#124—831no—
373#123—1661no—
374C22⋊Q8—831no—
375#121—831no—
376C4×Q8—831no—
377C4×Q8—831no—
378C4×Q8—831no—
379C4×D4—831no—
380C4×D4—1661no—
381C4×D4—1661no—
382C4×D4—1661no—
383C4×D4C22×D8311yescharacteristic
384C4×D4C22×D8311yescharacteristic
385#117—831no—
386#117—1661no—
387#117—1661no—
388C2×C42—831no—
389C2×C42—1661no—
390#107—831no—
391#107C4×D8311yescharacteristic
392#106—831no—
393#106—831no—
394C4⋊C8—831no—
395C4⋊C8—831no—
396C4⋊C8—831no—
397C4⋊C8—831no—
398C4⋊C8—1661no—
399C4⋊C8—1661no—
400C4⋊C8—1661no—
401C4⋊C8—1661no—
402C4⋊C8C22×D8311yescharacteristic
403C4⋊C8C22×D8311yescharacteristic
404#103—831no—
405#103—831no—
406#103—831no—
407#103—831no—
408#102—831no—
409#102—831no—
410#102C4×D8311yescharacteristic
411#102C4×D8311yescharacteristic
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—831no—
415C22⋊C8—831no—
416C22⋊C8—1661no—
417C22⋊C8—1661no—
418C22⋊C8—1661no—
419C22⋊C8—1661no—
420#97—831no—
421#97—831no—
422#97—1661no—
423#97—1661no—
424C4×C8—831no—
425C4×C8—831no—
426C4×C8—831no—
427C4×C8—1661no—
428C4×C8—1661no—
429C4×C8—1661no—
430C4×C8—1661no—
431C4×C8—1661no—
432C4×C8C22×D8311yescharacteristic
433C4×C8C22×D8311yescharacteristic
434C4×C8C22×D8311yescharacteristic
435#41—3321no—
436#41—1662no—
437C2×Q8—831no—
438C2×Q8—831no—
439C2×D4—1661no—
440C2×D4—1661no—
441C2×D4C2×C4×D8311yescharacteristic
442C2×D4C2×C4×D8311yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—831no—
446C22×C4—831no—
447C22×C4—831no—
448C22×C4—1661no—
449C22×C4—1661no—
450Q16—1662no—
451SD16—3321no—
452SD16—1662no—
453D8—41no—
454M16—42no—
455M16—3322no—
456M16—3322no—
457M16—1662no—
458M16—1662no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—42no—
463C2×C8—3322no—
464C2×C8—831no—
465C2×C8—831no—
466C2×C8—831no—
467C2×C8—831no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8C2×C4×D8311yescharacteristic
472C2×C8C2×C4×D8311yescharacteristic
473C2×C8D4×D8311yescharacteristic
474C2×C8D4×D8311yescharacteristic
475C4⋊C4—831no—
476C4⋊C4—831no—
477C4⋊C4—1661no—
478C4⋊C4—1661no—
479C4⋊C4—1661no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C4⋊C4—1661no—
483C4⋊C4C2×C4×D8311yescharacteristic
484C4⋊C4C2×C4×D8311yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—1661no—
488C22⋊C4—1661no—
489C22⋊C4—1661no—
490C22⋊C4—1661no—
491C22⋊C4—1661no—
492C22⋊C4—1661no—
493C42—831no—
494C42—831no—
495C42—831no—
496C42—1661no—
497C42—1661no—
498C42—1661no—
499C42—1661no—
500C42C23×D8311yescharacteristic
501C23—21no—
502C23—21no—
503C23—831no—
504Q8—832no—
505Q8—1661no—
506Q8—1661no—
507Q8—1661no—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—21no—
512D4—3322no—
513D4—6641no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—3321no—
521C2×C4—3322no—
522C2×C4—831no—
523C2×C4—831no—
524C2×C4—831no—
525C2×C4—6641no—
526C2×C4—1661no—
527C2×C4—1661no—
528C2×C4—1661no—
529C2×C4—1661no—
530C2×C4—1661no—
531C2×C4—1661no—
532C2×C4—1661no—
533C2×C4—1661no—
534C2×C4C22×C4×D8311yescharacteristic
535C2×C4C2×D4×D8311yescharacteristic
536C2×C4#41×D8311yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—3321no—
544C8—3321no—
545C8—832no—
546C8—1662no—
547C8—1661no—
548C8—1661no—
549C8—1661no—
550C8#1458181112yes—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—1661no—
558C22—1661no—
559C22—1661no—
560C22C4×D4×D8311yescharacteristic
561C4—21no—
562C4—41no—
563C4—41no—
564C4—3321no—
565C4—3321no—
566C4—3321no—
567C4—1661no—
568C4—1661no—
569C4—1661no—
570C4—1662no—
571C4#131×D8311yescharacteristic
572C4#1912846711yescharacteristic
573C4#135×D8311yescharacteristic
574C4C2×#1458181111yescharacteristic
575C2—81no—
576C2—81no—
577C2—3321no—
578C2#2491618611yescharacteristic
579C2C4×#1458181111yescharacteristic
580C2#2491637111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1581C2
socleC22×C83C2×C4×D4
prime radical#19127843C4
cosocleC42×C83C24
hypercenter#1581D83
hypocenterC83#17999
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #24916479C2semidirect
3. #24916464C2semidirect
4. #24906351C2semidirect
5. #24906348C2semidirect
6. #24906315C2semidirect
7. #24906309C2semidirect
8. #24906309C2semidirect
9. #24906269C2semidirect
10. C8×#13050578C2semidirect
11. #24906242C2semidirect
12. #24905920C2semidirect
13. #24905842C2semidirect
14. #24905742C2semidirect
15. C83×#1581C2semidirect
299. #1581D83semidirect
306. C83#17999semidirect

Maximal subgroups

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

subgroupindex
#249174772
#249164792
#249164642
#249063512
#249063482
#249063152
#249063092
#249062692
C8×#130505782
#249062422
#249059202
#249058422
#249057422
C83×#15812
#1799983