Finite Group Databasebeta

← #35375346

Subgroups — #35375346

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#24917487C211yesmaximal, characteristic
2#24916482C211yesmaximal, characteristic, complemented
3#24916466C211yesmaximal, characteristic, complemented
4#24906350C211yesmaximal, characteristic, complemented
5#24906349C211yesmaximal, characteristic, complemented
6#24906314C211yesmaximal, characteristic, complemented
7#24906306C211yesmaximal, characteristic, complemented
8#24906306C211yesmaximal, characteristic, complemented
9#24906291C211yesmaximal, characteristic, complemented
10#24906260C211yesmaximal, characteristic, complemented
11#24906242C211yesmaximal, characteristic, complemented
12#24905922C211yesmaximal, characteristic, complemented
13#24905844C211yesmaximal, characteristic, complemented
14C8×#13050573C211yesmaximal, characteristic, complemented
15C8×Q16×C83C211yesmaximal, characteristic, complemented
16#19128238C2211yescharacteristic
17C4×#13050591C2211yescharacteristic
18#19128041C411yescharacteristic
19#19128028C411yescharacteristic
20#19128010C411yescharacteristic
21#19128004C411yescharacteristic
22#19127925C411yescharacteristic
23#19127919C411yescharacteristic
24#19127862C411yescharacteristic
25#19127849C411yescharacteristic
26#19127832C2211yescharacteristic
27Q8×C83⋊C8C2211yescharacteristic
28C4×#13050577C2211yescharacteristic
29C4×#13050577C2211yescharacteristic
30C4×D664C2211yescharacteristic
31#19127780—21no—
32#19127780—21no—
33#19127780C2211yescharacteristic
34#19127780C2211yescharacteristic
35#19127778—21no—
36#19127778C2211yescharacteristic
37#19127771—21no—
38#19127766C2211yescharacteristic
39#19127761—21no—
40#19127757—21no—
41#19127757C2211yescharacteristic
42#19127753C2211yescharacteristic
43#19127752C2211yescharacteristic
44#19127655—21no—
45C4×#13050573—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic
49#19127451C2211yescharacteristic
50#19127451C2211yescharacteristic
51Q8⋊(C83⋊C8)C2211yescharacteristic
52Q8⋊(C83⋊C8)C2211yescharacteristic
53#19127430C2211yescharacteristic
54#19127430C2211yescharacteristic
55#19127428C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic
58#19127422C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60C8×Q8×C83C2211yescharacteristic
61C8×Q8×C83C2211yescharacteristic
62C4×Q16×C83C2211yescharacteristic
63#335×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65Q8⋊C8×C83C2211yescharacteristic
66C82×C83C2211yescharacteristic
67C2×#13050591C2×C411yescharacteristic
68C2×#13050577C2×C411yescharacteristic
69C2×#13050577C2×C411yescharacteristic
70C2×D664C2×C411yescharacteristic
71C2×#13050573—21no—
72C2×#13050573—21no—
73C2×#13050573—21no—
74C2×#13050573D411yescharacteristic
75C2×#13050573D411yescharacteristic
76C2×C8×D83—21no—
77C2×C8×D83—21no—
78#14581787—21no—
79#14581783C2×C411yescharacteristic
80#14581781C2×C411yescharacteristic
81#14581777—21no—
82#14581768C2×C411yescharacteristic
83#14581767C2×C411yescharacteristic
84Q8×Dic83C2×C411yescharacteristic
85C4×D332—21no—
86C4×D332—21no—
87C4×D332C2311yescharacteristic
88C4×D332C2311yescharacteristic
89#14581754—21no—
90#14581754C2×C411yescharacteristic
91#14581750—21no—
92#14581750C2×C411yescharacteristic
93#14581749—21no—
94C42×D83—21no—
95C42×D83C2311yescharacteristic
96#14581726C2×C411yescharacteristic
97#14581723C2×C411yescharacteristic
98#14581723C2×C411yescharacteristic
99C4⋊(C83⋊C8)—21no—
100C4⋊(C83⋊C8)—21no—
101C4⋊(C83⋊C8)C2311yescharacteristic
102C4⋊(C83⋊C8)C2311yescharacteristic
103Dic83⋊C8C2×C411yescharacteristic
104Dic83⋊C8C2×C411yescharacteristic
105#14581716C2×C411yescharacteristic
106#14581716C2×C411yescharacteristic
107#14581714C2×C411yescharacteristic
108#14581714C2×C411yescharacteristic
109#14581712C2×C411yescharacteristic
110#14581712C2×C411yescharacteristic
111(C2×D83)⋊C8—21no—
112(C2×D83)⋊C8—21no—
113(C2×D83)⋊C8—21no—
114(C2×D83)⋊C8—21no—
115(C2×D83)⋊C8—21no—
116(C2×D83)⋊C8—21no—
117(C2×D83)⋊C8C2×C411yescharacteristic
118(C2×D83)⋊C8C2×C411yescharacteristic
119#14581703—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×Dic83—21no—
129C2×Q16×C83C2×C411yescharacteristic
130C4×Q8×C83C2311yescharacteristic
131C4×Q8×C83C2311yescharacteristic
132#107×C83C2×C411yescharacteristic
133C4⋊C8×C83—21no—
134C4⋊C8×C83—21no—
135C4⋊C8×C83C2311yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137#103×C83C2×C411yescharacteristic
138#103×C83C2×C411yescharacteristic
139C4×C8×C83—21no—
140C4×C8×C83—21no—
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144#13050591—22no—
145#13050591—41no—
146C2×D332—21no—
147C2×D332—21no—
148C2×D332C22×C411yescharacteristic
149C2×D332C22×C411yescharacteristic
150C2×C4×D83—21no—
151C2×C4×D83—21no—
152C2×C4×D83—21no—
153C2×C4×D83—21no—
154C2×C4×D83C22×C411yescharacteristic
155C2×C4×D83C2×D411yescharacteristic
156C2×C4×D83#4111yescharacteristic
157#13050577—22no—
158#13050577—41no—
159D664—41no—
160#13050573—22no—
161#13050573—22no—
162#13050573—42no—
163#13050573—42no—
164#13050573—42no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8C22×C411yescharacteristic
169C2×C83⋊C8C22×C411yescharacteristic
170C2×C83⋊C8C2×D411yescharacteristic
171C2×C83⋊C8C2×D411yescharacteristic
172C8×D83—42no—
173C8×D83—42no—
174C4⋊Dic83—21no—
175C4⋊Dic83—21no—
176C4⋊Dic83C22×C411yescharacteristic
177C4⋊Dic83C22×C411yescharacteristic
178Dic83⋊C4—21no—
179Dic83⋊C4—21no—
180(C2×D83)⋊C4—21no—
181(C2×D83)⋊C4—21no—
182(C2×D83)⋊C4—21no—
183(C2×D83)⋊C4—21no—
184(C2×D83)⋊C4—21no—
185(C2×D83)⋊C4—21no—
186(C2×D83)⋊C4—21no—
187(C2×D83)⋊C4—21no—
188C4×Dic83—21no—
189C4×Dic83—21no—
190C4×Dic83C22×C411yescharacteristic
191C4×Dic83C2×D411yescharacteristic
192C4×Dic83#4111yescharacteristic
193C2×Q8×C83C22×C411yescharacteristic
194C2×Q8×C83C22×C411yescharacteristic
195Q16×C83—22no—
196C2×C8×C83—21no—
197C2×C8×C83—21no—
198C2×C8×C83—21no—
199C2×C8×C83C22×C411yescharacteristic
200C2×C8×C83C22×C411yescharacteristic
201C2×C8×C83C2×D411yescharacteristic
202C2×C8×C83C2×D411yescharacteristic
203C4⋊C4×C83—21no—
204C4⋊C4×C83—21no—
205C4⋊C4×C83C22×C411yescharacteristic
206C4⋊C4×C83C22×C411yescharacteristic
207C42×C83—21no—
208C42×C83—21no—
209C42×C83C2411yescharacteristic
210C22×D83—21no—
211C22×D83—21no—
212C22×D83C4×D411yescharacteristic
213Q8×C83—21no—
214Q8×C83—21no—
215Q8×C83—21no—
216Q8×C83#13112yes—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83C23×C411yescharacteristic
222C2×C4×C83C22×D411yescharacteristic
223C2×C4×C83C2×#4111yescharacteristic
224C2×Dic83—21no—
225C2×Dic83—21no—
226C2×Dic83C4×D411yescharacteristic
227C2×Dic83C4×D411yescharacteristic
228C2×Dic83C4×D411yescharacteristic
229D332—21no—
230D332—21no—
231D332—21no—
232D332—21no—
233D332—42no—
234D332—81no—
235C4×D83—21no—
236C4×D83—21no—
237C4×D83—21no—
238C4×D83—21no—
239C4×D83—41no—
240C4×D83—42no—
241C4×D83—41no—
242C4×D83—41no—
243C4×D83—41no—
244C4×D83—41no—
245C4×D83—81no—
246C8×C83—22no—
247C8×C83—21no—
248C8×C83—21no—
249C8×C83—21no—
250C8×C83—41no—
251C8×C83—41no—
252C8×C83#13512yes—
253C83⋊C8—21no—
254C83⋊C8—21no—
255C83⋊C8—21no—
256C83⋊C8—41no—
257C83⋊C8—41no—
258C83⋊C8—41no—
259C83⋊C8#13512yes—
260C22×C83C2×C4×D411yescharacteristic
261C2×D83—21no—
262C2×D83—21no—
263C2×D83—21no—
264C2×D83—41no—
265C2×D83—41no—
266C2×D83—41no—
267C2×D83—41no—
268C2×D83—41no—
269C2×D83—41no—
270C4×C83—22no—
271C4×C83—21no—
272C4×C83—41no—
273C4×C83—41no—
274C4×C83—41no—
275C4×C83C2×#13111yescharacteristic
276C4×C83#56811yescharacteristic
277C4×C83C2×#13511yescharacteristic
278C4×C83C2×#13511yescharacteristic
279Dic83—21no—
280Dic83—21no—
281Dic83—21no—
282Dic83—41no—
283Dic83—41no—
284#17999—831nomaximal, Sylow
285C2×C83#293311yescharacteristic
286C2×C83C4×#13511yescharacteristic
287C2×C83#295611yescharacteristic
288D83—41no—
289D83—81no—
290D83—81no—
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#1581—831no—
300C8×Q16D8311yescharacteristic, complemented
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×Q8C2×D8311yescharacteristic
320C8×Q8C2×D8311yescharacteristic
321C4×Q16C2×D8311yescharacteristic
322C4×SD16—831no—
323C4×SD16—831no—
324C4×D8—831no—
325#436—831no—
326#436—831no—
327#436—831no—
328#436—1661no—
329#436—1661no—
330#436—1661no—
331#435—1661no—
332C8×D4—831no—
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⋊C8C2×D8311yescharacteristic
349Q8⋊C8C2×D8311yescharacteristic
350D4⋊C8—831no—
351D4⋊C8—831no—
352D4⋊C8—831no—
353D4⋊C8—831no—
354#322—831no—
355#322—831no—
356C82—831no—
357C82C2×D8311yescharacteristic
358C2×#41—831no—
359C2×Q16C4×D8311yescharacteristic
360C2×SD16—831no—
361C2×SD16—831no—
362C2×D8—831no—
363C2×M16—831no—
364C2×M16—831no—
365C2×M16—1661no—
366C2×M16—1661no—
367C2×M16—1661no—
368C22×C8—1661no—
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×Q8C22×D8311yescharacteristic
378C4×Q8C22×D8311yescharacteristic
379C4×D4—831no—
380C4×D4—831no—
381C4×D4—831no—
382C4×D4—1661no—
383C4×D4—1661no—
384C4×D4—1661no—
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—21no—
395C4⋊C8—21no—
396C4⋊C8—831no—
397C4⋊C8—831no—
398C4⋊C8—831no—
399C4⋊C8—831no—
400C4⋊C8—1661no—
401C4⋊C8—1661no—
402C4⋊C8C22×D8311yescharacteristic
403C4⋊C8C22×D8311yescharacteristic
404#103—831no—
405#103—831no—
406#103C4×D8311yescharacteristic
407#103C4×D8311yescharacteristic
408#102—831no—
409#102—831no—
410#102—831no—
411#102—831no—
412C22⋊C8—831no—
413C22⋊C8—831no—
414C22⋊C8—1661no—
415C22⋊C8—1661no—
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—21no—
425C4×C8—21no—
426C4×C8—831no—
427C4×C8—831no—
428C4×C8—831no—
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×Q8C2×C4×D8311yescharacteristic
438C2×Q8C2×C4×D8311yescharacteristic
439C2×D4—831no—
440C2×D4—831no—
441C2×D4—1661no—
442C2×D4—1661no—
443C22×C4—831no—
444C22×C4—831no—
445C22×C4—831no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450Q16—22no—
451SD16—3321no—
452SD16—1662no—
453D8—3321no—
454M16—3322no—
455M16—3322no—
456M16—3322no—
457M16—1662no—
458M16—1662no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—3322no—
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—21no—
476C4⋊C4—21no—
477C4⋊C4—831no—
478C4⋊C4—831no—
479C4⋊C4—1661no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C4⋊C4—1661no—
483C4⋊C4C2×C4×D8311yescharacteristic
484C4⋊C4C2×C4×D8311yescharacteristic
485C22⋊C4—1661no—
486C22⋊C4—1661no—
487C22⋊C4—1661no—
488C22⋊C4—1661no—
489C22⋊C4—1661no—
490C22⋊C4—1661no—
491C22⋊C4—1661no—
492C22⋊C4—1661no—
493C42—21no—
494C42—21no—
495C42—831no—
496C42—831no—
497C42—831no—
498C42—1661no—
499C42—1661no—
500C42C23×D8311yescharacteristic
501C23—831no—
502C23—1661no—
503C23—1661no—
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8#1458179812yes—
508D4—3322no—
509D4—6641no—
510D4—1661no—
511D4—1661no—
512D4—1661no—
513D4—1661no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—3322no—
519C2×C4—3321no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—3321no—
523C2×C4—3321no—
524C2×C4—831no—
525C2×C4—831no—
526C2×C4—831no—
527C2×C4—6641no—
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—22no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—41no—
542C8—41no—
543C8—3321no—
544C8—3321no—
545C8—3321no—
546C8—832no—
547C8—1661no—
548C8—1661no—
549C8—1661no—
550C8#1458181512yes—
551C22—3321no—
552C22—3321no—
553C22—3321no—
554C22—3321no—
555C22—3321no—
556C22—3321no—
557C22—1661no—
558C22—1661no—
559C22—1661no—
560C22C4×D4×D8311yescharacteristic
561C4—21no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—3321no—
567C4—3321no—
568C4—1661no—
569C4—1661no—
570C4—1661no—
571C4C2×#1458179811yescharacteristic
572C4#1912846611yescharacteristic
573C4#135×D8311yescharacteristic
574C4C2×#1458181511yescharacteristic
575C2—3321no—
576C2—6641no—
577C2—6641no—
578C2#2491618311yescharacteristic
579C2C4×#1458181511yescharacteristic
580C2#2491636811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC8×Q16×C83C2
socleC22×C83C2×C4×D4
prime radical#19127849C4
cosocleC42×C83C24
hypercenterC8×Q16D83
hypocenterC83#17999
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×Q16

Complements

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

NH ≅ G/Ntype
2. #24916482C2semidirect
3. #24916466C2semidirect
4. #24906350C2semidirect
5. #24906349C2semidirect
6. #24906314C2semidirect
7. #24906306C2semidirect
8. #24906306C2semidirect
9. #24906291C2semidirect
10. #24906260C2semidirect
11. #24906242C2semidirect
12. #24905922C2semidirect
13. #24905844C2semidirect
14. C8×#13050573C2semidirect
15. C8×Q16×C83C2semidirect
300. C8×Q16D83semidirect
306. C83#17999semidirect

Maximal subgroups

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

subgroupindex
#249174872
#249164822
#249164662
#249063502
#249063492
#249063142
#249063062
#249062912
#249062602
#249062422
#249059222
#249058442
C8×#130505732
C8×Q16×C832
#1799983