Finite Group Databasebeta

← #36525445

Subgroups — #36525445

Order 22784.

608 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#26030264C211yesmaximal, characteristic
2#26029270C211yesmaximal, characteristic, complemented
3#26019164C211yesmaximal, characteristic, complemented
4#26019152C211yesmaximal, characteristic, complemented
5#26019130C211yesmaximal, characteristic, complemented
6#26019128C211yesmaximal, characteristic, complemented
7#26019121C211yesmaximal, characteristic, complemented
8#26019116C211yesmaximal, characteristic, complemented
9#26019069C211yesmaximal, characteristic, complemented
10#26019064C211yesmaximal, characteristic, complemented
11#26019030C211yesmaximal, characteristic, complemented
12#26018712C211yesmaximal, characteristic, complemented
13#26018632C211yesmaximal, characteristic, complemented
14#26018539C211yesmaximal, characteristic, complemented
15C89×#2969C211yesmaximal, characteristic, complemented
16C4×#13083574C2211yescharacteristic
17#19151936C411yescharacteristic
18#19151919C411yescharacteristic
19#19151902C411yescharacteristic
20#19151895C411yescharacteristic
21#19151817C411yescharacteristic
22#19151810C411yescharacteristic
23#19151758C411yescharacteristic
24#19151741C411yescharacteristic
25C8×Dic178C2211yescharacteristic
26C8×Dic178C2211yescharacteristic
27C4×#13083564C2211yescharacteristic
28C4×#13083561C2211yescharacteristic
29C4×#13083562C2211yescharacteristic
30C4×#13083559C2211yescharacteristic
31#19151677C2211yescharacteristic
32#19151677C2211yescharacteristic
33#19151674—21no—
34#19151674—21no—
35#19151655C2211yescharacteristic
36#19151653—21no—
37#19151651C2211yescharacteristic
38#19151550—21no—
39C2×C4×#10623494C2211yescharacteristic
40#19151350C2211yescharacteristic
41#19151350C2211yescharacteristic
42#19151348C2211yescharacteristic
43#19151348C2211yescharacteristic
44Q8⋊#10623494C2211yescharacteristic
45Q8⋊#10623494C2211yescharacteristic
46#19151328C2211yescharacteristic
47#19151328C2211yescharacteristic
48D4⋊#10623494C2211yescharacteristic
49D4⋊#10623494C2211yescharacteristic
50#19151325C2211yescharacteristic
51#19151325C2211yescharacteristic
52#19151320C2211yescharacteristic
53#19151320C2211yescharacteristic
54#19151320C2211yescharacteristic
55#19151320C2211yescharacteristic
56#529×C89C2211yescharacteristic
57#446×C89C2211yescharacteristic
58C8×Q8×C89C2211yescharacteristic
59#436×C89—21no—
60#436×C89C2211yescharacteristic
61#435×C89—21no—
62C8×D4×C89C2211yescharacteristic
63#433×C89—21no—
64#431×C89C2211yescharacteristic
65#405×C89—21no—
66C4×M16×C89C2211yescharacteristic
67C2×#13083574C2×C411yescharacteristic
68C2×#13083564C2×C411yescharacteristic
69C2×#13083562C2×C411yescharacteristic
70C2×#13083561C2×C411yescharacteristic
71C2×#13083559C2×C411yescharacteristic
72C2×#13083557—21no—
73C2×#13083557—21no—
74C22×#10623494—21no—
75C4×Dic178C2×C411yescharacteristic
76C4×Dic178C2311yescharacteristic
77C4×D356C2×C411yescharacteristic
78C4×D356C2311yescharacteristic
79C4×C22⋊D89—21no—
80C4×C22⋊D89—21no—
81#15667971C2×C411yescharacteristic
82#15667969—21no—
83C42×D89—21no—
84#15667945C2×C411yescharacteristic
85#15667945C2×C411yescharacteristic
86#15667943C2×C411yescharacteristic
87#15667943C2×C411yescharacteristic
88C4⋊#10623494C2×C411yescharacteristic
89C4⋊#10623494C2×C411yescharacteristic
90C4⋊#10623494C2311yescharacteristic
91C4⋊#10623494C2311yescharacteristic
92#15667941—21no—
93#15667941—21no—
94#15667936C2×C411yescharacteristic
95#15667936C2×C411yescharacteristic
96#15667935C2×C411yescharacteristic
97#15667935C2×C411yescharacteristic
98#15667933C2×C411yescharacteristic
99#15667933C2×C411yescharacteristic
100#15667932C2×C411yescharacteristic
101#15667932C2×C411yescharacteristic
102C22⋊#10623494—21no—
103C22⋊#10623494C2×C411yescharacteristic
104C22⋊#10623494C2×C411yescharacteristic
105#15667924—21no—
106#15667924—21no—
107#15667923—21no—
108C4×#10623494—21no—
109C4×#10623494C2311yescharacteristic
110C4×#10623494C2311yescharacteristic
111C4×#10623494C2311yescharacteristic
112C4×#10623494C2311yescharacteristic
113C8×Dic89—21no—
114C8×Dic89—21no—
115C2×M16×C89—21no—
116C2×M16×C89D411yescharacteristic
117C2×M16×C89D411yescharacteristic
118C22×C8×C89—21no—
119#126×C89—21no—
120#125×C89C2×C411yescharacteristic
121#124×C89C2×C411yescharacteristic
122#123×C89—21no—
123C22⋊Q8×C89C2×C411yescharacteristic
124#121×C89C2×C411yescharacteristic
125C4×Q8×C89C2311yescharacteristic
126C4×D4×C89—21no—
127C4×D4×C89C2311yescharacteristic
128#117×C89—21no—
129C2×C42×C89C2311yescharacteristic
130C4⋊C8×C89—21no—
131C4⋊C8×C89—21no—
132C4⋊C8×C89C2311yescharacteristic
133C4⋊C8×C89C2311yescharacteristic
134C22⋊C8×C89—21no—
135C22⋊C8×C89—21no—
136C22⋊C8×C89—21no—
137#97×C89—21no—
138#97×C89D411yescharacteristic
139#97×C89D411yescharacteristic
140C4×C8×C89—21no—
141C4×C8×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144#13083574—22no—
145#13083574—41no—
146C2×Dic178C22×C411yescharacteristic
147C2×D356C22×C411yescharacteristic
148C2×C22⋊D89—21no—
149C2×C4×D89—21no—
150C2×C4×D89—21no—
151C2×C4×D89—21no—
152#13083564—22no—
153#13083562—22no—
154#13083561—41no—
155#13083559—41no—
156#13083557—42no—
157#13083557—42no—
158C2×#10623494—21no—
159C2×#10623494—21no—
160C2×#10623494—21no—
161C2×#10623494—21no—
162C2×#10623494—42no—
163C2×#10623494C22×C411yescharacteristic
164C2×#10623494C22×C411yescharacteristic
165C2×#10623494C22×C411yescharacteristic
166C2×#10623494C22×C411yescharacteristic
167C4⋊Dic89C22×C411yescharacteristic
168C4⋊Dic89C22×C411yescharacteristic
169#13083553—21no—
170#13083553—21no—
171C22⋊Dic89—21no—
172#13083551—21no—
173#13083551—21no—
174C4×Dic89—21no—
175C4×Dic89—21no—
176C4×Dic89—21no—
177C2×Q8×C89C22×C411yescharacteristic
178C2×D4×C89—21no—
179C2×D4×C89C22×C411yescharacteristic
180C22×C4×C89—21no—
181C22×C4×C89C22×C411yescharacteristic
182C22×C4×C89C2×D411yescharacteristic
183C22×C4×C89#4111yescharacteristic
184M16×C89—41no—
185M16×C89—41no—
186M16×C89—42no—
187C2×C8×C89—21no—
188C2×C8×C89—21no—
189C2×C8×C89—42no—
190C2×C8×C89C2×D411yescharacteristic
191C2×C8×C89C2×D411yescharacteristic
192C2×C8×C89C2×D411yescharacteristic
193C2×C8×C89C2×D411yescharacteristic
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89—21no—
198C4⋊C4×C89C22×C411yescharacteristic
199C4⋊C4×C89C22×C411yescharacteristic
200C22⋊C4×C89—21no—
201C22⋊C4×C89—21no—
202C22⋊C4×C89—21no—
203C22⋊C4×C89—21no—
204C22⋊C4×C89—21no—
205C42×C89—21no—
206C42×C89C22×C411yescharacteristic
207C42×C89C2×D411yescharacteristic
208C42×C89#4111yescharacteristic
209C42×C89C2411yescharacteristic
210C23×C89—21no—
211C23×C89C4×D411yescharacteristic
212C22×D89—21no—
213Q8×C89—21no—
214Q8×C89—21no—
215D4×C89—21no—
216D4×C89—21no—
217D4×C89—81no—
218C2×C4×C89—21no—
219C2×C4×C89—21no—
220C2×C4×C89—21no—
221C2×C4×C89—21no—
222C2×C4×C89—21no—
223C2×C4×C89—21no—
224C2×C4×C89—21no—
225C2×C4×C89—41no—
226C2×C4×C89—41no—
227C2×C4×C89—41no—
228C2×C4×C89C4×D411yescharacteristic
229C2×C4×C89C4×D411yescharacteristic
230C2×C4×C89C4×D411yescharacteristic
231C2×C4×C89C23×C411yescharacteristic
232C2×C4×C89C22×D411yescharacteristic
233C2×C4×C89C2×#4111yescharacteristic
234C22⋊D89—42no—
235C2×Dic89—21no—
236C2×Dic89—21no—
237C2×Dic89—21no—
238D356—21no—
239D356—21no—
240C4×D89—42no—
241C4×D89—41no—
242C4×D89—41no—
243C4×D89—81no—
244Dic178—21no—
245Dic178#13112yes—
246C8×C89—21no—
247C8×C89—21no—
248C8×C89—21no—
249C8×C89—21no—
250C8×C89—41no—
251C8×C89—41no—
252#10623494—22no—
253#10623494—21no—
254#10623494—21no—
255#10623494—21no—
256#10623494—21no—
257#10623494—41no—
258#10623494—41no—
259#10623494—41no—
260C22×C89—21no—
261C22×C89—21no—
262C22×C89—21no—
263C22×C89—41no—
264C22×C89—41no—
265C22×C89—41no—
266C22×C89C2×C4×D411yescharacteristic
267C2×D89—41no—
268C2×D89—41no—
269C2×D89—41no—
270C4×C89—21no—
271C4×C89—21no—
272C4×C89—21no—
273C4×C89—21no—
274C4×C89—41no—
275C4×C89—41no—
276C4×C89—41no—
277C4×C89C2×#13111yescharacteristic
278C4×C89#56811yescharacteristic
279C4×C89#57511yescharacteristic
280C4×C89#57511yescharacteristic
281Dic89—22no—
282Dic89—41no—
283Dic89—41no—
284#18005—891nomaximal, Sylow
285C2×C89—41no—
286C2×C89—81no—
287C2×C89#293311yescharacteristic
288C2×C89#294311yescharacteristic
289C2×C89#295611yescharacteristic
290D89—81no—
291#3102—891no—
292#2971—891no—
293#2969D8911yescharacteristic, complemented
294#1594—891no—
295#1592—891no—
296#1588—891no—
297#1588—891no—
298#1587—891no—
299#1586—891no—
300#1579—891no—
301SD16⋊C8—891no—
302#1572—891no—
303#1494—891no—
304#1476—891no—
305#1451—891no—
306C89#1800511yescharacteristic, complemented, Sylow
307#529C2×D8911yescharacteristic
308C4×#41—891no—
309#485—891no—
310#482—891no—
311#479—891no—
312#478—891no—
313#464—891no—
314#463—891no—
315#452—891no—
316#449—891no—
317#446C2×D8911yescharacteristic
318C8×Q8—891no—
319C8×Q8—891no—
320C8×Q8C2×D8911yescharacteristic
321C4×Q16—891no—
322C4×SD16—891no—
323C4×SD16—891no—
324C4×D8—891no—
325#436—21no—
326#436—891no—
327#436—891no—
328#436—1781no—
329#436—1781no—
330#436C2×D8911yescharacteristic
331#435—21no—
332C8×D4C2×D8911yescharacteristic
333#433—21no—
334#432—891no—
335#432—1781no—
336#431—891no—
337#431C2×D8911yescharacteristic
338#405—21no—
339C4×M16—1781no—
340C4×M16C2×D8911yescharacteristic
341C2×C4×C8—891no—
342#335—891no—
343#335—891no—
344#334—891no—
345#334—891no—
346Q8⋊C8—891no—
347Q8⋊C8—891no—
348Q8⋊C8—891no—
349Q8⋊C8—891no—
350D4⋊C8—891no—
351D4⋊C8—891no—
352D4⋊C8—891no—
353D4⋊C8—891no—
354#322—891no—
355#322—891no—
356#322—891no—
357#322—891no—
358C2×#41—891no—
359C2×Q16—891no—
360C2×SD16—891no—
361C2×SD16—891no—
362C2×D8—891no—
363C2×M16—21no—
364C2×M16—1781no—
365C2×M16—1781no—
366C2×M16C22⋊D8911yescharacteristic
367C2×M16C22⋊D8911yescharacteristic
368C22×C8—21no—
369C22×C8—1781no—
370#126—21no—
371#125C4×D8911yescharacteristic
372#124C4×D8911yescharacteristic
373#123—21no—
374C22⋊Q8C4×D8911yescharacteristic
375#121C4×D8911yescharacteristic
376C4×Q8—891no—
377C4×Q8—891no—
378C4×Q8C22×D8911yescharacteristic
379C4×D4—21no—
380C4×D4—891no—
381C4×D4—891no—
382C4×D4—1781no—
383C4×D4—1781no—
384C4×D4C22×D8911yescharacteristic
385#117—21no—
386#117—891no—
387#117—1781no—
388C2×C42—1781no—
389C2×C42C22×D8911yescharacteristic
390#107—891no—
391#107—891no—
392#106—891no—
393#106—891no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—891no—
397C4⋊C8—891no—
398C4⋊C8—891no—
399C4⋊C8—891no—
400C4⋊C8—1781no—
401C4⋊C8—1781no—
402C4⋊C8C22×D8911yescharacteristic
403C4⋊C8C22×D8911yescharacteristic
404#103—891no—
405#103—891no—
406#103—891no—
407#103—891no—
408#102—891no—
409#102—891no—
410#102—891no—
411#102—891no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415C22⋊C8—891no—
416C22⋊C8—891no—
417C22⋊C8—1781no—
418C22⋊C8—1781no—
419C22⋊C8—1781no—
420#97—21no—
421#97—1781no—
422#97C22⋊D8911yescharacteristic
423#97C22⋊D8911yescharacteristic
424C4×C8—21no—
425C4×C8—21no—
426C4×C8—891no—
427C4×C8—891no—
428C4×C8—891no—
429C4×C8—891no—
430C4×C8—1781no—
431C4×C8—1781no—
432C4×C8—1781no—
433C4×C8C22×D8911yescharacteristic
434C4×C8C22×D8911yescharacteristic
435#41—3561no—
436#41—1782no—
437C2×Q8—891no—
438C2×Q8C2×C4×D8911yescharacteristic
439C2×D4—21no—
440C2×D4—891no—
441C2×D4—1781no—
442C2×D4C2×C4×D8911yescharacteristic
443C22×C4—21no—
444C22×C4—1781no—
445C22×C4—1781no—
446C22×C4—1781no—
447C22×C4C2×C4×D8911yescharacteristic
448C22×C4C2×C22⋊D8911yescharacteristic
449C22×C4#1308357411yescharacteristic
450Q16—1782no—
451SD16—3561no—
452SD16—1782no—
453D8—3561no—
454M16—41no—
455M16—41no—
456M16—42no—
457M16—3562no—
458M16—3562no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—42no—
462C2×C8—891no—
463C2×C8—891no—
464C2×C8—891no—
465C2×C8—891no—
466C2×C8—3562no—
467C2×C8—1781no—
468C2×C8—1781no—
469C2×C8—1781no—
470C2×C8—1781no—
471C2×C8C2×C22⋊D8911yescharacteristic
472C2×C8C2×C22⋊D8911yescharacteristic
473C2×C8C2×C22⋊D8911yescharacteristic
474C2×C8C2×C22⋊D8911yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—891no—
480C4⋊C4—891no—
481C4⋊C4—1781no—
482C4⋊C4—1781no—
483C4⋊C4C2×C4×D8911yescharacteristic
484C4⋊C4C2×C4×D8911yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—1781no—
491C22⋊C4—1781no—
492C22⋊C4—1781no—
493C42—21no—
494C42—1781no—
495C42—1781no—
496C42—1781no—
497C42C2×C4×D8911yescharacteristic
498C42C2×C22⋊D8911yescharacteristic
499C42#1308357411yescharacteristic
500C42C23×D8911yescharacteristic
501C23—21no—
502C23—1781no—
503C23C4×C22⋊D8911yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—892no—
507Q8—1781no—
508D4—21no—
509D4—21no—
510D4—81no—
511D4—3562no—
512D4—1781no—
513D4—1781no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—3562no—
525C2×C4—3561no—
526C2×C4—3561no—
527C2×C4—1781no—
528C2×C4—1781no—
529C2×C4—1781no—
530C2×C4—7121no—
531C2×C4C4×C22⋊D8911yescharacteristic
532C2×C4C4×C22⋊D8911yescharacteristic
533C2×C4C4×C22⋊D8911yescharacteristic
534C2×C4C22×C4×D8911yescharacteristic
535C2×C4C22×C22⋊D8911yescharacteristic
536C2×C4C2×#1308357411yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—41no—
542C8—41no—
543C8—3561no—
544C8—3561no—
545C8—3561no—
546C8—1782no—
547C8—1781no—
548C8—1781no—
549C8—1781no—
550C8—1781no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—3561no—
558C22—3561no—
559C22—3561no—
560C22C2×C4×C22⋊D8911yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—3561no—
569C4—3561no—
570C4—1782no—
571C4#131×D8911yescharacteristic
572C4#1915236311yescharacteristic
573C4#1915239611yescharacteristic
574C4#1915239611yescharacteristic
575C2—41no—
576C2—81no—
577C2—7121no—
578C2#2602897411yescharacteristic
579C2#2602905011yescharacteristic
580C2#2602915911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2969C2
socleC22×C89C2×C4×D4
prime radical#19151741C4
cosocleC42×C89C24
hypercenter#2969D89
hypocenterC89#18005
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. #26029270C2semidirect
3. #26019164C2semidirect
4. #26019152C2semidirect
5. #26019130C2semidirect
6. #26019128C2semidirect
7. #26019121C2semidirect
8. #26019116C2semidirect
9. #26019069C2semidirect
10. #26019064C2semidirect
11. #26019030C2semidirect
12. #26018712C2semidirect
13. #26018632C2semidirect
14. #26018539C2semidirect
15. C89×#2969C2semidirect
293. #2969D89semidirect
306. C89#18005semidirect

Maximal subgroups

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

subgroupindex
#260302642
#260292702
#260191642
#260191522
#260191302
#260191282
#260191212
#260191162
#260190692
#260190642
#260190302
#260187122
#260186322
#260185392
C89×#29692
#1800589