Finite Group Databasebeta

← #35374219

Subgroups — #35374219

Order 21248.

636 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916579C211yesmaximal, characteristic
2#24916545C211yesmaximal, characteristic, complemented
3C4×#14581824C211yesmaximal, characteristic
4#24906372C211yesmaximal, characteristic, complemented
5#24906370C211yesmaximal, characteristic, complemented
6#24906364C211yesmaximal, characteristic, complemented
7#24906363C211yesmaximal, characteristic, complemented
8#24906357C211yesmaximal, characteristic, complemented
9#24906355C211yesmaximal, characteristic, complemented
10#24906350C211yesmaximal, characteristic, complemented
11#24906214C211yesmaximal, characteristic, complemented
12#24905844C211yesmaximal, characteristic
13Q8⋊C8×D83C211yesmaximal, characteristic, complemented
14C8×#13050573C211yesmaximal, characteristic, complemented
15C83×#1591C211yesmaximal, characteristic, complemented
16C2×#14581824C411yescharacteristic
17C4×#13050591C2211yescharacteristic
18C4×Q8×D83C2211yescharacteristic
19#19127832C2211yescharacteristic
20#19127832C2211yescharacteristic
21#19127831C2211yescharacteristic
22#19127830—21no—
23#19127829—21no—
24#19127828C2211yescharacteristic
25#19127808C411yescharacteristic
26#19127805C411yescharacteristic
27#19127804C411yescharacteristic
28#19127800C411yescharacteristic
29C4×#13050580C2211yescharacteristic
30C4×#13050579C2211yescharacteristic
31C4×#13050577C2211yescharacteristic
32#19127780C2211yescharacteristic
33#19127778C2211yescharacteristic
34#19127774—21no—
35#19127771—21no—
36#19127757—21no—
37#19127756—21no—
38#19127756—21no—
39#19127753C2211yescharacteristic
40#19127752C2211yescharacteristic
41#19127742C411yescharacteristic
42#19127723—21no—
43C4⋊C8×D83C2211yescharacteristic
44#19127708C411yescharacteristic
45#19127701C411yescharacteristic
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127453C2211yescharacteristic
49#19127452C2211yescharacteristic
50#19127451C2211yescharacteristic
51#19127433C2211yescharacteristic
52Q8⋊(C83⋊C8)C2211yescharacteristic
53Q8⋊(C83⋊C8)C2211yescharacteristic
54#19127431C2211yescharacteristic
55#19127430C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic
58#19127422C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60#446×C83C2211yescharacteristic
61#446×C83C2211yescharacteristic
62C4×Q16×C83C2211yescharacteristic
63#335×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65Q8⋊C8×C83C2211yescharacteristic
66C82×C83C2211yescharacteristic
67C2×#13050591C2×C411yescharacteristic
68C2×Q8×D83C2×C411yescharacteristic
69#14581824—24no—
70C2×#13050580C2×C411yescharacteristic
71C2×#13050579C2×C411yescharacteristic
72C2×#13050577C2×C411yescharacteristic
73C2×#13050573—21no—
74C2×#13050573—21no—
75C2×#13050573C2×C411yescharacteristic
76C2×#13050573C2×C411yescharacteristic
77C2×C8×D83—21no—
78C2×C8×D83—21no—
79Q8×Dic83C2×C411yescharacteristic
80Q8×Dic83C2×C411yescharacteristic
81C4×Dic166—21no—
82C4×Dic166C2311yescharacteristic
83#14581757—21no—
84#14581757C2×C411yescharacteristic
85C4×D332—21no—
86C4×D332C2311yescharacteristic
87#14581754—21no—
88#14581754C2×C411yescharacteristic
89#14581750—21no—
90#14581750C2×C411yescharacteristic
91C4⋊C4×D83—21no—
92C4⋊C4×D83C2×C411yescharacteristic
93C42×D83—21no—
94C42×D83—21no—
95C42×D83C2311yescharacteristic
96#14581725C2×C411yescharacteristic
97#14581724C2×C411yescharacteristic
98#14581723C2×C411yescharacteristic
99C4⋊(C83⋊C8)—21no—
100C4⋊(C83⋊C8)—21no—
101C4⋊(C83⋊C8)C2311yescharacteristic
102C4⋊(C83⋊C8)C2311yescharacteristic
103Dic83⋊C8—21no—
104Dic83⋊C8—21no—
105Dic83⋊C8—21no—
106Dic83⋊C8D411yescharacteristic
107Dic83⋊C8D411yescharacteristic
108#14581717C2×C411yescharacteristic
109#14581716C2×C411yescharacteristic
110#14581716C2×C411yescharacteristic
111#14581715C2×C411yescharacteristic
112#14581714C2×C411yescharacteristic
113#14581712C2×C411yescharacteristic
114(C2×D83)⋊C8—21no—
115(C2×D83)⋊C8—21no—
116(C2×D83)⋊C8—21no—
117(C2×D83)⋊C8D411yescharacteristic
118(C2×D83)⋊C8D411yescharacteristic
119#14581703—21no—
120#14581703—21no—
121#14581702—21no—
122#14581702—21no—
123#14581702C2×C411yescharacteristic
124#14581702C2×C411yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C4×C83⋊C8C2311yescharacteristic
128C8×Dic83—21no—
129C8×Dic83—21no—
130C2×Q16×C83C2×C411yescharacteristic
131C4×Q8×C83C2311yescharacteristic
132C4×Q8×C83C2311yescharacteristic
133#107×C83C2×C411yescharacteristic
134C4⋊C8×C83—21no—
135C4⋊C8×C83—21no—
136C4⋊C8×C83C2311yescharacteristic
137C4⋊C8×C83C2311yescharacteristic
138#103×C83C2×C411yescharacteristic
139#103×C83C2×C411yescharacteristic
140#97×C83—21no—
141#97×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145#13050591—22no—
146#13050591—22no—
147C2×Dic166—21no—
148C2×Dic166C22×C411yescharacteristic
149Q8×D83—22no—
150Q8×D83—22no—
151C2×D332—21no—
152C2×D332C22×C411yescharacteristic
153C2×C4×D83—21no—
154C2×C4×D83—21no—
155C2×C4×D83—21no—
156C2×C4×D83—21no—
157C2×C4×D83—21no—
158C2×C4×D83C22×C411yescharacteristic
159C2×C4×D83C2×D411yescharacteristic
160C2×C4×D83#4111yescharacteristic
161#13050580—22no—
162#13050579—22no—
163#13050577—22no—
164#13050573—22no—
165#13050573—81no—
166#13050573—81no—
167#13050573M1614yescomplemented
168C2×C83⋊C8—21no—
169C2×C83⋊C8—21no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8C22×C411yescharacteristic
172C2×C83⋊C8C22×C411yescharacteristic
173C2×C83⋊C8C2×D411yescharacteristic
174C2×C83⋊C8C2×D411yescharacteristic
175C8×D83—42no—
176C8×D83—81no—
177C4⋊Dic83—21no—
178C4⋊Dic83—21no—
179C4⋊Dic83C22×C411yescharacteristic
180C4⋊Dic83C22×C411yescharacteristic
181Dic83⋊C4—21no—
182Dic83⋊C4—21no—
183Dic83⋊C4—21no—
184(C2×D83)⋊C4—21no—
185(C2×D83)⋊C4—21no—
186(C2×D83)⋊C4—21no—
187C4×Dic83—21no—
188C4×Dic83—21no—
189C4×Dic83—21no—
190C4×Dic83—21no—
191C4×Dic83—21no—
192C4×Dic83C22×C411yescharacteristic
193C4×Dic83C2×D411yescharacteristic
194C4×Dic83#4111yescharacteristic
195C2×Q8×C83C22×C411yescharacteristic
196C2×Q8×C83C22×C411yescharacteristic
197Q16×C83—22no—
198C2×C8×C83—21no—
199C2×C8×C83—21no—
200C2×C8×C83—21no—
201C2×C8×C83C22×C411yescharacteristic
202C2×C8×C83C22×C411yescharacteristic
203C2×C8×C83C2×D411yescharacteristic
204C2×C8×C83C2×D411yescharacteristic
205C4⋊C4×C83—21no—
206C4⋊C4×C83—21no—
207C4⋊C4×C83C22×C411yescharacteristic
208C4⋊C4×C83C22×C411yescharacteristic
209C42×C83—21no—
210C42×C83—21no—
211C42×C83C2411yescharacteristic
212C22×D83—21no—
213C22×D83C4×D411yescharacteristic
214Q8×C83—21no—
215Q8×C83—21no—
216Q8×C83—21no—
217Q8×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83C23×C411yescharacteristic
223C2×C4×C83C22×D411yescharacteristic
224C2×C4×C83C2×#4111yescharacteristic
225C2×Dic83—21no—
226C2×Dic83—21no—
227C2×Dic83—21no—
228C2×Dic83C4×D411yescharacteristic
229C2×Dic83C4×D411yescharacteristic
230C2×Dic83C4×D411yescharacteristic
231D332—21no—
232D332—21no—
233D332—42no—
234C4×D83—21no—
235C4×D83—21no—
236C4×D83—41no—
237C4×D83—42no—
238C4×D83—41no—
239C4×D83—41no—
240C4×D83—42no—
241C4×D83—81no—
242C4×D83—81no—
243C4×D83C2×M1612yes—
244C4×D83#13112yes—
245Dic166—21no—
246Dic166—21no—
247Dic166—42no—
248C8×C83—22no—
249C8×C83—21no—
250C8×C83—21no—
251C8×C83—21no—
252C8×C83—41no—
253C8×C83—41no—
254C8×C83C2×M1612yes—
255C83⋊C8—21no—
256C83⋊C8—21no—
257C83⋊C8—21no—
258C83⋊C8—41no—
259C83⋊C8—41no—
260C83⋊C8—41no—
261C83⋊C8C2×M1612yes—
262C22×C83C2×C4×D411yescharacteristic
263C2×D83—21no—
264C2×D83—21no—
265C2×D83—41no—
266C2×D83—41no—
267C2×D83—41no—
268C2×D83#43612yes—
269C4×C83—21no—
270C4×C83—41no—
271C4×C83—41no—
272C4×C83—41no—
273C4×C83—41no—
274C4×C83C22×M1611yescharacteristic
275C4×C83C2×#13111yescharacteristic
276C4×C83#57711yescharacteristic
277C4×C83#57811yescharacteristic
278Dic83—21no—
279Dic83—21no—
280Dic83—41no—
281Dic83—41no—
282Dic83—41no—
283Dic83#43612yes—
284#17894—831nomaximal, Sylow
285C2×C83C2×#43611yescharacteristic
286C2×C83#294711yescharacteristic
287C2×C83#295611yescharacteristic
288D83—22no—
289D83—81no—
290#2989—831no—
291#2983—831no—
292C4×#137—831no—
293#1593—831no—
294#1593—831no—
295#1592—831no—
296#1592—831no—
297#1591—831no—
298#1591D8311yescharacteristic, complemented
299#1590—831no—
300#1590—831no—
301#1565—831no—
302#1476—831no—
303C2×Q8⋊C8—831no—
304C8×M16—831no—
305C83#1789411yescharacteristic, complemented, Sylow
306C2×#137—831no—
307C4×#41—831no—
308C2×C4×Q8—831no—
309#446—831no—
310#446—831no—
311#446—831no—
312#446—831no—
313#446—1661no—
314#446—1661no—
315#446C2×D8311yescharacteristic
316#446C2×D8311yescharacteristic
317#441—831no—
318#441—831no—
319#440—831no—
320#440—831no—
321C4×Q16—831no—
322C4×Q16C2×D8311yescharacteristic
323C4×SD16—831no—
324C4×SD16—831no—
325#436—831no—
326#436—831no—
327#435—1661no—
328#435—1661no—
329#432—1661no—
330#432—1661no—
331#432—1661no—
332#431—831no—
333#431—831no—
334#428—831no—
335#423—1661no—
336C2×C4⋊C8—831no—
337#419—831no—
338#417—831no—
339C4×M16—831no—
340C2×C4×C8—831no—
341#335—831no—
342#335C2×D8311yescharacteristic
343#334—831no—
344#334—831no—
345Q8⋊C8—831no—
346Q8⋊C8—831no—
347Q8⋊C8—831no—
348Q8⋊C8—831no—
349Q8⋊C8C2×D8311yescharacteristic
350Q8⋊C8C2×D8311yescharacteristic
351D4⋊C8—831no—
352D4⋊C8—831no—
353#322—831no—
354#322—831no—
355C82—831no—
356C82C2×D8311yescharacteristic
357C2×#41—831no—
358C22×Q8—831no—
359#137—1664no—
360C2×Q16—831no—
361C2×Q16C4×D8311yescharacteristic
362C2×SD16—831no—
363C2×SD16—831no—
364C2×M16—831no—
365C2×M16—831no—
366C2×M16—1661no—
367C2×M16—1661no—
368C22×C8—1661no—
369C22×C8—1661no—
370C4×Q8—831no—
371C4×Q8—831no—
372C4×Q8—831no—
373C4×Q8—831no—
374C4×Q8—1661no—
375C4×Q8—1661no—
376C4×Q8C22×D8311yescharacteristic
377C4×Q8C22×D8311yescharacteristic
378C4×D4—831no—
379C4×D4—831no—
380C4×D4—1661no—
381C4×D4—1661no—
382#117—831no—
383#117—1661no—
384C2×C4⋊C4—831no—
385C2×C4⋊C4—1661no—
386C2×C42—831no—
387C2×C42—1661no—
388C2×C42—1661no—
389#107—831no—
390#107C4×D8311yescharacteristic
391#106—831no—
392#106—831no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—831no—
396C4⋊C8—831no—
397C4⋊C8—831no—
398C4⋊C8—831no—
399C4⋊C8—1661no—
400C4⋊C8—1661no—
401C4⋊C8—1661no—
402C4⋊C8—1661no—
403C4⋊C8—1661no—
404C4⋊C8C22×D8311yescharacteristic
405C4⋊C8C22×D8311yescharacteristic
406#103—831no—
407#103—831no—
408#103—831no—
409#103—831no—
410#103C4×D8311yescharacteristic
411#103C4×D8311yescharacteristic
412#102—831no—
413#102—831no—
414C22⋊C8—831no—
415C22⋊C8—831no—
416C22⋊C8—1661no—
417C22⋊C8—1661no—
418C22⋊C8—1661no—
419#97—21no—
420#97—21no—
421#97—831no—
422#97—831no—
423#97—1661no—
424#97—1661no—
425#97—1661no—
426#97—1661no—
427C4×C8—831no—
428C4×C8—831no—
429C4×C8—831no—
430C4×C8—1661no—
431C4×C8—1661no—
432C4×C8C22×D8311yescharacteristic
433C4×C8C22×D8311yescharacteristic
434C4×C8C22×D8311yescharacteristic
435#41—1662no—
436#41—1662no—
437C2×Q8—831no—
438C2×Q8—1661no—
439C2×Q8—1662no—
440C2×Q8—1662no—
441C2×Q8C2×C4×D8311yescharacteristic
442C2×Q8C2×C4×D8311yescharacteristic
443C2×D4—831no—
444C2×D4—1661no—
445C22×C4—831no—
446C22×C4—831no—
447C22×C4—831no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4—1661no—
451C22×C4—1661no—
452C22×C4—1661no—
453Q16—22no—
454Q16—1662no—
455SD16—1662no—
456SD16—1662no—
457M16—834no—
458M16—6641no—
459M16—6641no—
460M16—1662no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—3322no—
465C2×C8—831no—
466C2×C8—831no—
467C2×C8—831no—
468C2×C8—831no—
469C2×C8—6641no—
470C2×C8—1661no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8C2×C4×D8311yescharacteristic
474C2×C8C2×C4×D8311yescharacteristic
475C2×C8D4×D8311yescharacteristic
476C2×C8D4×D8311yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—831no—
480C4⋊C4—831no—
481C4⋊C4—1661no—
482C4⋊C4—1661no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C4⋊C4—1661no—
486C4⋊C4C2×C4×D8311yescharacteristic
487C4⋊C4C2×C4×D8311yescharacteristic
488C22⋊C4—1661no—
489C22⋊C4—1661no—
490C22⋊C4—1661no—
491C42—21no—
492C42—21no—
493C42—831no—
494C42—831no—
495C42—831no—
496C42—1661no—
497C42—1661no—
498C42—1661no—
499C42—1661no—
500C42—1661no—
501C42C23×D8311yescharacteristic
502C23—831no—
503C23—1661no—
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8—21no—
508Q8—3322no—
509Q8—1661no—
510Q8—1661no—
511D4—3322no—
512D4—1661no—
513D4—1661no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—3321no—
519C2×C4—3322no—
520C2×C4—3322no—
521C2×C4—3321no—
522C2×C4—3321no—
523C2×C4—831no—
524C2×C4—832no—
525C2×C4—832no—
526C2×C4—831no—
527C2×C4—831no—
528C2×C4—6641no—
529C2×C4—6641no—
530C2×C4—1661no—
531C2×C4—1661no—
532C2×C4—1661no—
533C2×C4—1661no—
534C2×C4—1661no—
535C2×C4C22×C4×D8311yescharacteristic
536C2×C4C2×D4×D8311yescharacteristic
537C2×C4#41×D8311yescharacteristic
538C8—22no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—3321no—
545C8—3321no—
546C8—3321no—
547C8—832no—
548C8—1661no—
549C8—1661no—
550C8—1661no—
551C8M16×D8312yes—
552C22—3321no—
553C22—3321no—
554C22—3321no—
555C22—832no—
556C22—1661no—
557C22—1661no—
558C22C4×D4×D8311yescharacteristic
559C4—21no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—3321no—
565C4—3321no—
566C4—3321no—
567C4—832no—
568C4—1661no—
569C4—1661no—
570C4C2×M16×D8311yescharacteristic
571C4#131×D8311yescharacteristic
572C4#1912850811yescharacteristic
573C4#1912851411yescharacteristic
574C2—6641no—
575C2—1662no—
576C2#436×D8311yescharacteristic
577C2#2491630111yescharacteristic
578C2#2491636811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1591C2
socleC22×C83C2×C4×D4
prime radicalC2×#13050591C2×C4
cosocleC42×C83C24
hypercenter#1591D83
hypocenterC83#17894
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1591

Complements

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

NH ≅ G/Ntype
2. #24916545C2semidirect
4. #24906372C2semidirect
5. #24906370C2semidirect
6. #24906364C2semidirect
7. #24906363C2semidirect
8. #24906357C2semidirect
9. #24906355C2semidirect
10. #24906350C2semidirect
11. #24906214C2semidirect
13. Q8⋊C8×D83C2semidirect
14. C8×#13050573C2semidirect
15. C83×#1591C2semidirect
167. #13050573M16semidirect
298. #1591D83semidirect
305. C83#17894semidirect

Maximal subgroups

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

subgroupindex
#249165792
#249165452
C4×#145818242
#249063722
#249063702
#249063642
#249063632
#249063572
#249063552
#249063502
#249062142
#249058442
Q8⋊C8×D832
C8×#130505732
C83×#15912
#1789483