Finite Group Databasebeta

← #36526041

Subgroups — #36526041

Order 22784.

626 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029377C211yesmaximal, characteristic
2#26029342C211yesmaximal, characteristic, complemented
3C4×#15668042C211yesmaximal, characteristic
4#26019158C211yesmaximal, characteristic, complemented
5#26019146C211yesmaximal, characteristic, complemented
6#26019126C211yesmaximal, characteristic, complemented
7#26019106C211yesmaximal, characteristic, complemented
8#26019105C211yesmaximal, characteristic, complemented
9#26019085C211yesmaximal, characteristic, complemented
10#26019061C211yesmaximal, characteristic, complemented
11#26019055C211yesmaximal, characteristic, complemented
12#26019020C211yesmaximal, characteristic, complemented
13#26018682C211yesmaximal, characteristic, complemented
14#26018547C211yesmaximal, characteristic, complemented
15C89×#1488C211yesmaximal, characteristic
16C2×#15668042C411yescharacteristic
17C2×C4×Dic178C2211yescharacteristic
18#19151729C2211yescharacteristic
19#19151729C2211yescharacteristic
20#19151726—21no—
21#19151726—21no—
22#19151725C2211yescharacteristic
23#19151725C2211yescharacteristic
24#19151722C2211yescharacteristic
25#19151722C2211yescharacteristic
26#19151703C411yescharacteristic
27#19151703C411yescharacteristic
28#19151698C411yescharacteristic
29#19151698C411yescharacteristic
30C4×#13083564C2211yescharacteristic
31C4×#13083564C2211yescharacteristic
32C4×#13083562C2211yescharacteristic
33C4×#13083562C2211yescharacteristic
34#19151670—21no—
35#19151670—21no—
36D4×#10623494C2211yescharacteristic
37D4×#10623494C2211yescharacteristic
38#19151655—21no—
39#19151652—21no—
40#19151651C2211yescharacteristic
41#19151648—21no—
42#19151641C411yescharacteristic
43#19151622C2211yescharacteristic
44C2×C4⋊#10623494—21no—
45#19151606C411yescharacteristic
46#19151599C411yescharacteristic
47C4×#13083558C2211yescharacteristic
48#19151350C2211yescharacteristic
49#19151350C2211yescharacteristic
50#19151348C2211yescharacteristic
51#19151348C2211yescharacteristic
52#19151330C2211yescharacteristic
53#19151330C2211yescharacteristic
54#19151330C2211yescharacteristic
55#19151330C2211yescharacteristic
56#19151320C2211yescharacteristic
57#19151320C2211yescharacteristic
58#19151319C2211yescharacteristic
59C8×#10623494C2211yescharacteristic
60C4×#41×C89C2211yescharacteristic
61#432×C89C2211yescharacteristic
62#431×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic
64Q8⋊C8×C89C2211yescharacteristic
65D4⋊C8×C89C2211yescharacteristic
66D4⋊C8×C89C2211yescharacteristic
67C22×Dic178C2×C411yescharacteristic
68#15668042—24no—
69C2×#13083564C2×C411yescharacteristic
70C2×#13083564C2×C411yescharacteristic
71C2×#13083562C2×C411yescharacteristic
72C2×#13083562C2×C411yescharacteristic
73C2×#13083558—21no—
74C2×#13083558C2×C411yescharacteristic
75C2×#13083558C2×C411yescharacteristic
76C22×#10623494—21no—
77C22×#10623494—21no—
78C4×Dic178—21no—
79C4×Dic178—21no—
80C4×Dic178C2×C411yescharacteristic
81C4×Dic178C2×C411yescharacteristic
82C4×Dic178C2311yescharacteristic
83C4×Dic178C2311yescharacteristic
84C2×C4⋊Dic89C2×C411yescharacteristic
85C2×#13083553—21no—
86C2×C4×Dic89—21no—
87#15667945C2×C411yescharacteristic
88#15667945C2×C411yescharacteristic
89#15667943C2×C411yescharacteristic
90#15667943C2×C411yescharacteristic
91C4⋊#10623494—21no—
92C4⋊#10623494—21no—
93C4⋊#10623494—21no—
94C4⋊#10623494D411yescharacteristic
95C4⋊#10623494D411yescharacteristic
96C4⋊#10623494C2311yescharacteristic
97C4⋊#10623494C2311yescharacteristic
98#15667941—21no—
99#15667941—21no—
100#15667941—21no—
101#15667941—21no—
102#15667936C2×C411yescharacteristic
103#15667936C2×C411yescharacteristic
104#15667935C2×C411yescharacteristic
105#15667935C2×C411yescharacteristic
106#15667935C2×C411yescharacteristic
107#15667935C2×C411yescharacteristic
108#15667933C2×C411yescharacteristic
109#15667933C2×C411yescharacteristic
110C22⋊#10623494—21no—
111C22⋊#10623494—21no—
112C22⋊#10623494D411yescharacteristic
113C22⋊#10623494D411yescharacteristic
114#15667923—21no—
115#15667923—21no—
116#15667923C2×C411yescharacteristic
117#15667923C2×C411yescharacteristic
118#15667922—21no—
119#15667922—21no—
120C4×#10623494C2311yescharacteristic
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C8×Dic89—21no—
125C8×Dic89—21no—
126C2×#41×C89C2×C411yescharacteristic
127C4×Q8×C89C2×C411yescharacteristic
128C4×Q8×C89C2311yescharacteristic
129C4×D4×C89—21no—
130C4×D4×C89—21no—
131C4×D4×C89C2×C411yescharacteristic
132C4×D4×C89C2311yescharacteristic
133#117×C89—21no—
134#117×C89C2×C411yescharacteristic
135C2×C42×C89—21no—
136C2×C42×C89C2311yescharacteristic
137C4⋊C8×C89—21no—
138C4⋊C8×C89C2311yescharacteristic
139C4⋊C8×C89C2311yescharacteristic
140C22⋊C8×C89—21no—
141C22⋊C8×C89—21no—
142#97×C89—21no—
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145C2×Dic178—22no—
146C2×Dic178—21no—
147C2×Dic178—41no—
148C2×Dic178C22×C411yescharacteristic
149C2×Dic178C22×C411yescharacteristic
150C22×Dic89—21no—
151C22×Dic89—21no—
152#13083564—22no—
153#13083564—22no—
154#13083562—22no—
155#13083562M1614yescomplemented
156#13083558—22no—
157#13083558—41no—
158#13083558—81no—
159C2×#10623494—21no—
160C2×#10623494—21no—
161C2×#10623494—81no—
162C2×#10623494—81no—
163C2×#10623494C22×C411yescharacteristic
164C2×#10623494C22×C411yescharacteristic
165C2×#10623494C22×C411yescharacteristic
166C2×#10623494C22×C411yescharacteristic
167C2×#10623494C2×D411yescharacteristic
168C2×#10623494C2×D411yescharacteristic
169C2×#10623494C2×D411yescharacteristic
170C2×#10623494C2×D411yescharacteristic
171C4⋊Dic89—21no—
172C4⋊Dic89C22×C411yescharacteristic
173C4⋊Dic89C22×C411yescharacteristic
174#13083553—21no—
175#13083553—21no—
176#13083553—21no—
177#13083553—21no—
178C4×Dic89—21no—
179C4×Dic89—21no—
180C4×Dic89—21no—
181C4×Dic89—21no—
182#41×C89—22no—
183#41×C89—41no—
184C2×Q8×C89C22×C411yescharacteristic
185C2×D4×C89—21no—
186C2×D4×C89C22×C411yescharacteristic
187C22×C4×C89—21no—
188C22×C4×C89—21no—
189C22×C4×C89—21no—
190C22×C4×C89C22×C411yescharacteristic
191C22×C4×C89C2×D411yescharacteristic
192C22×C4×C89#4111yescharacteristic
193C2×C8×C89—21no—
194C2×C8×C89—21no—
195C2×C8×C89—21no—
196C2×C8×C89—21no—
197C4⋊C4×C89—21no—
198C4⋊C4×C89—21no—
199C4⋊C4×C89C22×C411yescharacteristic
200C4⋊C4×C89C22×C411yescharacteristic
201C22⋊C4×C89—21no—
202C22⋊C4×C89—21no—
203C22⋊C4×C89—21no—
204C42×C89—21no—
205C42×C89—21no—
206C42×C89—21no—
207C42×C89C22×C411yescharacteristic
208C42×C89C2×D411yescharacteristic
209C42×C89#4111yescharacteristic
210C42×C89C2411yescharacteristic
211C23×C89—21no—
212C23×C89C4×D411yescharacteristic
213Q8×C89—21no—
214Q8×C89—21no—
215D4×C89—21no—
216D4×C89—42no—
217D4×C89C2×M1612yes—
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—42no—
227C2×C4×C89—42no—
228C2×C4×C89—41no—
229C2×C4×C89—41no—
230C2×C4×C89C4×D411yescharacteristic
231C2×C4×C89C4×D411yescharacteristic
232C2×C4×C89C4×D411yescharacteristic
233C2×C4×C89C23×C411yescharacteristic
234C2×C4×C89C22×D411yescharacteristic
235C2×C4×C89C2×#4111yescharacteristic
236C2×Dic89—21no—
237C2×Dic89—21no—
238C2×Dic89—21no—
239C2×Dic89—21no—
240C2×Dic89—42no—
241C2×Dic89—42no—
242Dic178—21no—
243Dic178—21no—
244Dic178—21no—
245Dic178—42no—
246Dic178C2×M1612yes—
247C8×C89—41no—
248C8×C89—41no—
249C8×C89—41no—
250C8×C89—41no—
251#10623494—21no—
252#10623494—21no—
253#10623494—21no—
254#10623494—21no—
255#10623494—21no—
256#10623494—21no—
257#10623494—21no—
258#10623494—41no—
259#10623494—41no—
260#10623494C2×M1612yes—
261C22×C89—21no—
262C22×C89—22no—
263C22×C89—21no—
264C22×C89—21no—
265C22×C89—41no—
266C22×C89—41no—
267C22×C89C2×C4×D411yescharacteristic
268C4×C89—21no—
269C4×C89—21no—
270C4×C89—21no—
271C4×C89—21no—
272C4×C89—41no—
273C4×C89—41no—
274C4×C89—41no—
275C4×C89C22×M1611yescharacteristic
276C4×C89#56811yescharacteristic
277C4×C89#57711yescharacteristic
278C4×C89#57811yescharacteristic
279Dic89—22no—
280Dic89—41no—
281Dic89—41no—
282Dic89—41no—
283#18051—891nomaximal, Sylow
284C2×C89—41no—
285C2×C89—42no—
286C2×C89D4×M1611yescharacteristic
287C2×C89#294711yescharacteristic
288C2×C89#295511yescharacteristic
289#2990—891no—
290#2984—891no—
291C4×#137—891no—
292#1593—891no—
293#1591—891no—
294#1587—891no—
295#1585—891no—
296#1584—891no—
297#1582—891no—
298SD16⋊C8—891no—
299C8×SD16—891no—
300#1569—891no—
301#1488D8911yescharacteristic
302#1487—891no—
303#1453—891no—
304C89#1805111yescharacteristic, complemented, Sylow
305C2×#137—891no—
306C4×#41C2×D8911yescharacteristic
307C2×C4×Q8—891no—
308#446—891no—
309#446—891no—
310#446—891no—
311#446—891no—
312#446—1781no—
313#446—1781no—
314C8×Q8—891no—
315C8×Q8—891no—
316#441—891no—
317#441—891no—
318#440—891no—
319#440—891no—
320C4×Q16—891no—
321C4×Q16—891no—
322C4×SD16—891no—
323C4×SD16—891no—
324#435—1781no—
325#435—1781no—
326C8×D4—891no—
327C8×D4—891no—
328#432—1781no—
329#432—1781no—
330#432C2×D8911yescharacteristic
331#431—891no—
332#431—1781no—
333#431C2×D8911yescharacteristic
334#428—891no—
335#423—891no—
336C2×C4⋊C8—1781no—
337#419—891no—
338#417—891no—
339C4×M16—891no—
340#335—891no—
341#335—891no—
342#334—891no—
343#334—891no—
344Q8⋊C8—891no—
345Q8⋊C8—891no—
346Q8⋊C8—891no—
347Q8⋊C8—891no—
348Q8⋊C8C2×D8911yescharacteristic
349Q8⋊C8C2×D8911yescharacteristic
350D4⋊C8C2×D8911yescharacteristic
351D4⋊C8C2×D8911yescharacteristic
352#322—891no—
353#322—891no—
354#322—891no—
355C82—891no—
356C2×#41C4×D8911yescharacteristic
357C22×Q8—891no—
358#137—1784no—
359C2×Q16—891no—
360C2×Q16—891no—
361C2×SD16—891no—
362C2×SD16—891no—
363C2×M16—891no—
364C2×M16—891no—
365C2×M16—1781no—
366C22×C8—1781no—
367C22×C8—1781no—
368C4×Q8—891no—
369C4×Q8—891no—
370C4×Q8—891no—
371C4×Q8—891no—
372C4×Q8—1781no—
373C4×Q8—1781no—
374C4×Q8C4×D8911yescharacteristic
375C4×Q8C22×D8911yescharacteristic
376C4×D4—21no—
377C4×D4—21no—
378C4×D4C4×D8911yescharacteristic
379C4×D4C22×D8911yescharacteristic
380#117—21no—
381#117C4×D8911yescharacteristic
382C2×C4⋊C4—891no—
383C2×C4⋊C4—1781no—
384C2×C42—21no—
385C2×C42—1781no—
386C2×C42C22×D8911yescharacteristic
387#107—891no—
388#107—891no—
389#106—891no—
390#106—891no—
391C4⋊C8—21no—
392C4⋊C8—891no—
393C4⋊C8—891no—
394C4⋊C8—891no—
395C4⋊C8—891no—
396C4⋊C8—1781no—
397C4⋊C8—1781no—
398C4⋊C8—1781no—
399C4⋊C8—1781no—
400C4⋊C8—1781no—
401C4⋊C8—1781no—
402C4⋊C8—1781no—
403C4⋊C8C22×D8911yescharacteristic
404C4⋊C8C22×D8911yescharacteristic
405#103—891no—
406#103—891no—
407#103—891no—
408#103—891no—
409#103—891no—
410#103—891no—
411#102—891no—
412#102—891no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415C22⋊C8—891no—
416C22⋊C8—891no—
417C22⋊C8—1781no—
418C22⋊C8—1781no—
419#97—21no—
420#97—891no—
421#97—891no—
422#97—1781no—
423#97—1781no—
424#97—1781no—
425#97—1781no—
426C4×C8—891no—
427C4×C8—891no—
428C4×C8—891no—
429C4×C8—891no—
430C4×C8—1781no—
431C4×C8—1781no—
432C4×C8C22×D8911yescharacteristic
433C4×C8C22×D8911yescharacteristic
434#41—22no—
435#41—41no—
436C2×Q8—891no—
437C2×Q8—891no—
438C2×Q8—3561no—
439C2×Q8—1782no—
440C2×Q8—1781no—
441C2×Q8C2×C4×D8911yescharacteristic
442C2×D4—21no—
443C2×D4C2×C4×D8911yescharacteristic
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—1781no—
448C22×C4—1781no—
449C22×C4C2×C4×D8911yescharacteristic
450C22×C4D4×D8911yescharacteristic
451C22×C4#1308357311yescharacteristic
452Q16—1782no—
453Q16—1782no—
454SD16—894no—
455SD16—1782no—
456M16—3561no—
457M16—1782no—
458M16—7121no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—891no—
464C2×C8—891no—
465C2×C8—891no—
466C2×C8—891no—
467C2×C8—891no—
468C2×C8—891no—
469C2×C8—891no—
470C2×C8—891no—
471C2×C8—1781no—
472C2×C8—1781no—
473C2×C8—7121no—
474C2×C8—7121no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—891no—
478C4⋊C4—891no—
479C4⋊C4—1781no—
480C4⋊C4—1781no—
481C4⋊C4—1781no—
482C4⋊C4—1781no—
483C4⋊C4—1781no—
484C4⋊C4C2×C4×D8911yescharacteristic
485C4⋊C4C2×C4×D8911yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—1781no—
493C42—1781no—
494C42—1781no—
495C42—1781no—
496C42C2×C4×D8911yescharacteristic
497C42D4×D8911yescharacteristic
498C42#1308357311yescharacteristic
499C42C23×D8911yescharacteristic
500C23—21no—
501C23#1566797211yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—892no—
505Q8—3562no—
506Q8—1781no—
507Q8—1781no—
508Q8—1781no—
509D4—21no—
510D4—42no—
511D4M16×D8912yes—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—3562no—
525C2×C4—3562no—
526C2×C4—1781no—
527C2×C4—1781no—
528C2×C4—1781no—
529C2×C4—1781no—
530C2×C4#1566797211yescharacteristic
531C2×C4#1566797211yescharacteristic
532C2×C4#1566797211yescharacteristic
533C2×C4C22×C4×D8911yescharacteristic
534C2×C4C2×D4×D8911yescharacteristic
535C2×C4C2×#1308357311yescharacteristic
536C8—41no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—892no—
541C8—3561no—
542C8—3561no—
543C8—1781no—
544C8—1781no—
545C8—1781no—
546C8—1781no—
547C8—1781no—
548C8—1781no—
549C8—1781no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22C2×#1566797211yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—3561no—
565C4—3561no—
566C4—3561no—
567C4—1782no—
568C4C2×M16×D8911yescharacteristic
569C4#1915236411yescharacteristic
570C4#1915240511yescharacteristic
571C4#1915241211yescharacteristic
572C2—41no—
573C2—42no—
574C2#2602898411yescharacteristic
575C2#2602909811yescharacteristic
576C2#2602915111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1488C2
socleC22×C89C2×C4×D4
prime radicalC2×#41×C89C2×C4
cosocleC42×C89C24
hypercenter#1488D89
hypocenterC89#18051
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1488

Complements

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

NH ≅ G/Ntype
2. #26029342C2semidirect
4. #26019158C2semidirect
5. #26019146C2semidirect
6. #26019126C2semidirect
7. #26019106C2semidirect
8. #26019105C2semidirect
9. #26019085C2semidirect
10. #26019061C2semidirect
11. #26019055C2semidirect
12. #26019020C2semidirect
13. #26018682C2semidirect
14. #26018547C2semidirect
155. #13083562M16semidirect
304. C89#18051semidirect

Maximal subgroups

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

subgroupindex
#260293772
#260293422
C4×#156680422
#260191582
#260191462
#260191262
#260191062
#260191052
#260190852
#260190612
#260190552
#260190202
#260186822
#260185472
C89×#14882
#1805189