Finite Group Databasebeta

← #36526029

Subgroups — #36526029

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#26029378C211yesmaximal, characteristic
2#26029341C211yesmaximal, characteristic, complemented
3C4×#15668043C211yesmaximal, characteristic
4#26019158C211yesmaximal, characteristic, complemented
5#26019146C211yesmaximal, characteristic, complemented
6#26019120C211yesmaximal, characteristic, complemented
7#26019112C211yesmaximal, characteristic, complemented
8#26019099C211yesmaximal, characteristic, complemented
9#26019091C211yesmaximal, characteristic, complemented
10#26019067C211yesmaximal, characteristic, complemented
11#26019049C211yesmaximal, characteristic, complemented
12#26019017C211yesmaximal, characteristic, complemented
13#26018687C211yesmaximal, characteristic
14#26018547C211yesmaximal, characteristic, complemented
15C89×#1487C211yesmaximal, characteristic, complemented
16C2×#15668043C411yescharacteristic
17C4×#13083574C2211yescharacteristic
18#19151729—21no—
19#19151729—21no—
20#19151729C2211yescharacteristic
21#19151729C2211yescharacteristic
22#19151725C2211yescharacteristic
23#19151725C2211yescharacteristic
24Q8×#10623494C2211yescharacteristic
25Q8×#10623494C2211yescharacteristic
26#19151703C411yescharacteristic
27#19151703C411yescharacteristic
28#19151696C411yescharacteristic
29#19151696C411yescharacteristic
30C4×#13083564C2211yescharacteristic
31C4×#13083564C2211yescharacteristic
32C4×#13083561C2211yescharacteristic
33C4×#13083561C2211yescharacteristic
34#19151667—21no—
35#19151667—21no—
36#19151663C2211yescharacteristic
37#19151663C2211yescharacteristic
38#19151655—21no—
39#19151655C2211yescharacteristic
40#19151654—21no—
41#19151651—21no—
42#19151651C2211yescharacteristic
43#19151638C411yescharacteristic
44C4⋊C8×D89—21no—
45#19151607C411yescharacteristic
46#19151597C411yescharacteristic
47C4×#13083558C2211yescharacteristic
48#19151350C2211yescharacteristic
49#19151350C2211yescharacteristic
50#19151348C2211yescharacteristic
51#19151348C2211yescharacteristic
52#19151330C2211yescharacteristic
53#19151330C2211yescharacteristic
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151320C2211yescharacteristic
57#19151320C2211yescharacteristic
58#19151319C2211yescharacteristic
59C8×#10623494C2211yescharacteristic
60C2×C4×Q8×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62#423×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic
64Q8⋊C8×C89C2211yescharacteristic
65Q8⋊C8×C89C2211yescharacteristic
66Q8⋊C8×C89C2211yescharacteristic
67C2×#13083574C2×C411yescharacteristic
68#15668043—24no—
69C2×#13083564C2×C411yescharacteristic
70C2×#13083564C2×C411yescharacteristic
71C2×#13083561C2×C411yescharacteristic
72C2×#13083561C2×C411yescharacteristic
73C2×#13083558C2×C411yescharacteristic
74C2×#13083558C2×C411yescharacteristic
75C2×C8×D89—21no—
76C2×C8×D89—21no—
77C4×Dic178C2×C411yescharacteristic
78C4×Dic178C2311yescharacteristic
79C4×D356C2×C411yescharacteristic
80C4×D356C2311yescharacteristic
81C4×C22⋊D89—21no—
82C4×C22⋊D89—21no—
83#15667971C2×C411yescharacteristic
84#15667969—21no—
85C42×D89—21no—
86#15667945C2×C411yescharacteristic
87#15667945C2×C411yescharacteristic
88#15667943C2×C411yescharacteristic
89#15667943C2×C411yescharacteristic
90C4⋊#10623494—21no—
91C4⋊#10623494—21no—
92C4⋊#10623494—21no—
93C4⋊#10623494—21no—
94C4⋊#10623494D411yescharacteristic
95C4⋊#10623494D411yescharacteristic
96C4⋊#10623494C2311yescharacteristic
97C4⋊#10623494C2311yescharacteristic
98#15667941—21no—
99#15667941—21no—
100#15667936C2×C411yescharacteristic
101#15667936C2×C411yescharacteristic
102#15667936C2×C411yescharacteristic
103#15667936C2×C411yescharacteristic
104#15667935C2×C411yescharacteristic
105#15667935C2×C411yescharacteristic
106#15667932C2×C411yescharacteristic
107#15667932C2×C411yescharacteristic
108C22⋊#10623494—21no—
109C22⋊#10623494D411yescharacteristic
110C22⋊#10623494D411yescharacteristic
111#15667924—21no—
112#15667924—21no—
113#15667923—21no—
114#15667923—21no—
115#15667923—21no—
116#15667923C2×C411yescharacteristic
117#15667923C2×C411yescharacteristic
118#15667922—21no—
119#15667922—21no—
120C4×#10623494—21no—
121C4×#10623494—21no—
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C4×#10623494C2311yescharacteristic
125C4×#10623494C2311yescharacteristic
126C22×Q8×C89C2×C411yescharacteristic
127C2×M16×C89—21no—
128C4×Q8×C89—21no—
129C4×Q8×C89—21no—
130C4×Q8×C89C2×C411yescharacteristic
131C4×Q8×C89C2×C411yescharacteristic
132C4×Q8×C89C2311yescharacteristic
133C4×Q8×C89C2311yescharacteristic
134C2×C4⋊C4×C89—21no—
135C2×C4⋊C4×C89C2×C411yescharacteristic
136C2×C42×C89—21no—
137C2×C42×C89C2311yescharacteristic
138C4⋊C8×C89—21no—
139C4⋊C8×C89—21no—
140C4⋊C8×C89C2311yescharacteristic
141C4⋊C8×C89C2311yescharacteristic
142C22⋊C8×C89—21no—
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145#13083574—22no—
146#13083574—41no—
147C2×Dic178C22×C411yescharacteristic
148C2×D356C22×C411yescharacteristic
149C2×C22⋊D89—21no—
150C2×C4×D89—21no—
151C2×C4×D89—21no—
152C2×C4×D89—21no—
153#13083564—22no—
154#13083564—22no—
155#13083561—22no—
156#13083561M1614yescomplemented
157#13083558—22no—
158#13083558—41no—
159C2×#10623494—21no—
160C2×#10623494—21no—
161C2×#10623494C22×C411yescharacteristic
162C2×#10623494C22×C411yescharacteristic
163C2×#10623494C22×C411yescharacteristic
164C2×#10623494C22×C411yescharacteristic
165C2×#10623494C2×D411yescharacteristic
166C2×#10623494C2×D411yescharacteristic
167C2×#10623494C2×D411yescharacteristic
168C2×#10623494C2×D411yescharacteristic
169C8×D89—81no—
170C8×D89—81no—
171C4⋊Dic89C22×C411yescharacteristic
172C4⋊Dic89C22×C411yescharacteristic
173#13083553—21no—
174#13083553—21no—
175C22⋊Dic89—21no—
176#13083551—21no—
177#13083551—21no—
178C4×Dic89—21no—
179C4×Dic89—21no—
180C4×Dic89—21no—
181C2×Q8×C89—21no—
182C2×Q8×C89—22no—
183C2×Q8×C89—41no—
184C2×Q8×C89C22×C411yescharacteristic
185C2×Q8×C89C22×C411yescharacteristic
186C22×C4×C89—21no—
187C22×C4×C89—21no—
188C22×C4×C89C22×C411yescharacteristic
189C22×C4×C89C2×D411yescharacteristic
190C22×C4×C89#4111yescharacteristic
191M16×C89—81no—
192C2×C8×C89—21no—
193C2×C8×C89—21no—
194C2×C8×C89—21no—
195C2×C8×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89—21no—
198C4⋊C4×C89—21no—
199C4⋊C4×C89—21no—
200C4⋊C4×C89—21no—
201C4⋊C4×C89C22×C411yescharacteristic
202C4⋊C4×C89C22×C411yescharacteristic
203C42×C89—21no—
204C42×C89—21no—
205C42×C89—21no—
206C42×C89—21no—
207C42×C89C22×C411yescharacteristic
208C42×C89C2×D411yescharacteristic
209C42×C89#4111yescharacteristic
210C42×C89C2411yescharacteristic
211C23×C89C4×D411yescharacteristic
212C22×D89—21no—
213Q8×C89—21no—
214Q8×C89—21no—
215Q8×C89—21no—
216Q8×C89—42no—
217Q8×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—21no—
226C2×C4×C89—41no—
227C2×C4×C89—42no—
228C2×C4×C89—42no—
229C2×C4×C89C4×D411yescharacteristic
230C2×C4×C89C4×D411yescharacteristic
231C2×C4×C89C4×D411yescharacteristic
232C2×C4×C89C23×C411yescharacteristic
233C2×C4×C89C22×D411yescharacteristic
234C2×C4×C89C2×#4111yescharacteristic
235C22⋊D89—42no—
236C2×Dic89—21no—
237C2×Dic89—21no—
238C2×Dic89—21no—
239D356—21no—
240D356C2×M1612yes—
241C4×D89—42no—
242C4×D89—42no—
243C4×D89—41no—
244C4×D89—41no—
245Dic178—21no—
246Dic178—21no—
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—21no—
263C22×C89—21no—
264C22×C89C2×C4×D411yescharacteristic
265C2×D89—22no—
266C2×D89—41no—
267C2×D89—41no—
268C4×C89—21no—
269C4×C89—22no—
270C4×C89—21no—
271C4×C89—21no—
272C4×C89—21no—
273C4×C89—41no—
274C4×C89—41no—
275C4×C89—41no—
276C4×C89C22×M1611yescharacteristic
277C4×C89#56811yescharacteristic
278C4×C89#57711yescharacteristic
279C4×C89#57811yescharacteristic
280Dic89—41no—
281Dic89—41no—
282Dic89—41no—
283#18051—891nomaximal, Sylow
284C2×C89—41no—
285C2×C89D4×M1611yescharacteristic
286C2×C89#294711yescharacteristic
287C2×C89#295511yescharacteristic
288D89—42no—
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#1488—891no—
302#1487D8911yescharacteristic, complemented
303#1453—891no—
304C89#1805111yescharacteristic, complemented, Sylow
305C2×#137—891no—
306C4×#41—891no—
307C2×C4×Q8C2×D8911yescharacteristic
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—891no—
329#432—1781no—
330#432—1781no—
331#431—891no—
332#431—1781no—
333#431C2×D8911yescharacteristic
334#428—891no—
335#423C2×D8911yescharacteristic
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⋊C8C2×D8911yescharacteristic
347Q8⋊C8C2×D8911yescharacteristic
348Q8⋊C8C2×D8911yescharacteristic
349Q8⋊C8C2×D8911yescharacteristic
350D4⋊C8—891no—
351D4⋊C8—891no—
352#322—891no—
353#322—891no—
354#322—891no—
355C82—891no—
356C2×#41—891no—
357C22×Q8C4×D8911yescharacteristic
358#137—1784no—
359C2×Q16—891no—
360C2×Q16—891no—
361C2×SD16—891no—
362C2×SD16—891no—
363C2×M16—21no—
364C2×M16—891no—
365C2×M16—891no—
366C22×C8—1781no—
367C22×C8—1781no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—891no—
371C4×Q8—891no—
372C4×Q8C4×D8911yescharacteristic
373C4×Q8C4×D8911yescharacteristic
374C4×Q8C22×D8911yescharacteristic
375C4×Q8C22×D8911yescharacteristic
376C4×D4—891no—
377C4×D4—891no—
378C4×D4—1781no—
379C4×D4—1781no—
380#117—891no—
381#117—1781no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4C4×D8911yescharacteristic
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—21no—
393C4⋊C8—891no—
394C4⋊C8—891no—
395C4⋊C8—891no—
396C4⋊C8—891no—
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—891no—
415C22⋊C8—891no—
416C22⋊C8—1781no—
417C22⋊C8—1781no—
418C22⋊C8—1781no—
419#97—891no—
420#97—891no—
421#97—1781no—
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—3561no—
435#41—1782no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—41no—
439C2×Q8—891no—
440C2×Q8C2×C4×D8911yescharacteristic
441C2×Q8C2×C4×D8911yescharacteristic
442C2×D4—891no—
443C2×D4—1781no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1781no—
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—81no—
457M16—3561no—
458M16—1782no—
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—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—891no—
481C4⋊C4—891no—
482C4⋊C4—1781no—
483C4⋊C4—1781no—
484C4⋊C4C2×C4×D8911yescharacteristic
485C4⋊C4C2×C4×D8911yescharacteristic
486C22⋊C4—1781no—
487C22⋊C4—1781no—
488C22⋊C4—1781no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—1781no—
494C42—1781no—
495C42—1781no—
496C42C2×C4×D8911yescharacteristic
497C42D4×D8911yescharacteristic
498C42#1308357311yescharacteristic
499C42C23×D8911yescharacteristic
500C23—1781no—
501C23#1566797211yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—42no—
506Q8—1781no—
507Q8—1781no—
508Q8M16×D8912yes—
509D4—892no—
510D4—3562no—
511D4—1781no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—3562no—
524C2×C4—3562no—
525C2×C4—3561no—
526C2×C4—3561no—
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—3561no—
554C22—3561no—
555C22—1782no—
556C22C2×#1566797211yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—3561no—
566C4—3561no—
567C4—3561no—
568C4C2×M16×D8911yescharacteristic
569C4#1915236411yescharacteristic
570C4#1915240411yescharacteristic
571C4#1915241111yescharacteristic
572C2—41no—
573C2—3562no—
574C2#2602898411yescharacteristic
575C2#2602908911yescharacteristic
576C2#2602915111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1487C2
socleC22×C89C2×C4×D4
prime radicalC2×#13083574C2×C4
cosocleC42×C89C24
hypercenter#1487D89
hypocenterC89#18051
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
2. #26029341C2semidirect
4. #26019158C2semidirect
5. #26019146C2semidirect
6. #26019120C2semidirect
7. #26019112C2semidirect
8. #26019099C2semidirect
9. #26019091C2semidirect
10. #26019067C2semidirect
11. #26019049C2semidirect
12. #26019017C2semidirect
14. #26018547C2semidirect
15. C89×#1487C2semidirect
156. #13083561M16semidirect
302. #1487D89semidirect
304. C89#18051semidirect

Maximal subgroups

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

subgroupindex
#260293782
#260293412
C4×#156680432
#260191582
#260191462
#260191202
#260191122
#260190992
#260190912
#260190672
#260190492
#260190172
#260186872
#260185472
C89×#14872
#1805189