Finite Group Databasebeta

← #36525243

Subgroups — #36525243

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030841C211yesmaximal, characteristic
2#26029403C211yesmaximal, characteristic, complemented
3#26019128C211yesmaximal, characteristic, complemented
4#26019116C211yesmaximal, characteristic, complemented
5#26019114C211yesmaximal, characteristic, complemented
6#26019109C211yesmaximal, characteristic, complemented
7#26019093C211yesmaximal, characteristic, complemented
8#26019088C211yesmaximal, characteristic, complemented
9#26019081C211yesmaximal, characteristic, complemented
10#26019079C211yesmaximal, characteristic, complemented
11#26019020C211yesmaximal, characteristic, complemented
12#26018710C211yesmaximal, characteristic, complemented
13#26018703C211yesmaximal, characteristic, complemented
14#26018542C211yesmaximal, characteristic, complemented
15C89×#2981C211yesmaximal, characteristic, complemented
16#19152132C2211yescharacteristic
17#19152004C411yescharacteristic
18#19151985C411yescharacteristic
19#19151962C411yescharacteristic
20#19151960C411yescharacteristic
21#19151849C411yescharacteristic
22#19151849C411yescharacteristic
23#19151842C411yescharacteristic
24#19151842C411yescharacteristic
25#19151728C2211yescharacteristic
26C8×Dic178C2211yescharacteristic
27C4×#13083562C2211yescharacteristic
28C4×#13083562C2211yescharacteristic
29C4×#13083559C2211yescharacteristic
30C4×#13083559C2211yescharacteristic
31#19151677C2211yescharacteristic
32#19151673—21no—
33#19151672—21no—
34C8×D356C2211yescharacteristic
35#19151658—21no—
36#19151623—21no—
37#19151622C2211yescharacteristic
38C4×#13083558C2211yescharacteristic
39C2×C4×#10623494C2211yescharacteristic
40#19151350C2211yescharacteristic
41#19151350C2211yescharacteristic
42#19151348C2211yescharacteristic
43#19151348C2211yescharacteristic
44#19151328C2211yescharacteristic
45#19151328C2211yescharacteristic
46D4⋊#10623494C2211yescharacteristic
47D4⋊#10623494C2211yescharacteristic
48D4⋊#10623494C2211yescharacteristic
49D4⋊#10623494C2211yescharacteristic
50#19151325C2211yescharacteristic
51#19151325C2211yescharacteristic
52#19151320C2211yescharacteristic
53#19151320C2211yescharacteristic
54#19151319C2211yescharacteristic
55#19151319C2211yescharacteristic
56#532×C89C2211yescharacteristic
57#436×C89C2211yescharacteristic
58#436×C89C2211yescharacteristic
59#435×C89—21no—
60#435×C89—21no—
61C8×D4×C89C2211yescharacteristic
62C8×D4×C89C2211yescharacteristic
63#432×C89C2211yescharacteristic
64#431×C89C2211yescharacteristic
65#408×C89—21no—
66#408×C89—21no—
67C2×#13083562C2×C411yescharacteristic
68C2×#13083562C2×C411yescharacteristic
69C2×#13083559C2×C411yescharacteristic
70C2×#13083559C2×C411yescharacteristic
71C2×#13083558—21no—
72C2×#13083558C2×C411yescharacteristic
73C2×#13083558C2×C411yescharacteristic
74C2×#13083557—21no—
75C22×#10623494—21no—
76C2×C8×D89—21no—
77#15668006—21no—
78#15668004C2×C411yescharacteristic
79#15668000C2×C411yescharacteristic
80#15667995—21no—
81#15667991C2×C411yescharacteristic
82#15667986C2×C411yescharacteristic
83C4×Dic178C2311yescharacteristic
84C4×D356C2311yescharacteristic
85C4×C22⋊D89—21no—
86#15667969—21no—
87#15667945C2×C411yescharacteristic
88#15667945C2×C411yescharacteristic
89#15667943C2×C411yescharacteristic
90#15667943C2×C411yescharacteristic
91C4⋊#10623494—21no—
92C4⋊#10623494C2311yescharacteristic
93C4⋊#10623494C2311yescharacteristic
94#15667941—21no—
95#15667941—21no—
96#15667935C2×C411yescharacteristic
97#15667935C2×C411yescharacteristic
98#15667933C2×C411yescharacteristic
99#15667933C2×C411yescharacteristic
100#15667933C2×C411yescharacteristic
101#15667933C2×C411yescharacteristic
102#15667932C2×C411yescharacteristic
103#15667932C2×C411yescharacteristic
104#15667924—21no—
105#15667924—21no—
106#15667923C2×C411yescharacteristic
107#15667923C2×C411yescharacteristic
108#15667922—21no—
109C4×#10623494—21no—
110C4×#10623494C2311yescharacteristic
111C4×#10623494C2311yescharacteristic
112C4×#10623494C2311yescharacteristic
113C4×#10623494C2311yescharacteristic
114C8×Dic89—21no—
115C2×#41×C89—21no—
116C2×M16×C89—21no—
117C2×M16×C89—21no—
118C22×C8×C89—21no—
119C22×C8×C89—21no—
120C4⋊Q8×C89C2×C411yescharacteristic
121#127×C89C2×C411yescharacteristic
122#124×C89—21no—
123#121×C89—21no—
124#121×C89C2×C411yescharacteristic
125#121×C89C2×C411yescharacteristic
126C4×D4×C89—21no—
127C4×D4×C89C2311yescharacteristic
128C4×D4×C89C2311yescharacteristic
129C2×C42×C89C2311yescharacteristic
130C4⋊C8×C89D411yescharacteristic
131C4⋊C8×C89D411yescharacteristic
132C4⋊C8×C89C2311yescharacteristic
133C4⋊C8×C89C2311yescharacteristic
134C22⋊C8×C89—21no—
135C22⋊C8×C89—21no—
136C22⋊C8×C89—21no—
137C22⋊C8×C89—21no—
138C22⋊C8×C89—21no—
139C22⋊C8×C89D411yescharacteristic
140C22⋊C8×C89D411yescharacteristic
141#97×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C2×Dic178C22×C411yescharacteristic
145C2×D356C22×C411yescharacteristic
146C2×C22⋊D89—21no—
147C2×C4×D89—21no—
148#13083562—41no—
149#13083562—41no—
150#13083559—41no—
151#13083559—41no—
152#13083558—41no—
153#13083558—41no—
154#13083558—81no—
155#13083557—81no—
156C2×#10623494—21no—
157C2×#10623494—21no—
158C2×#10623494—21no—
159C2×#10623494—21no—
160C2×#10623494—42no—
161C2×#10623494C22×C411yescharacteristic
162C2×#10623494C22×C411yescharacteristic
163C2×#10623494C22×C411yescharacteristic
164C2×#10623494C22×C411yescharacteristic
165C8×D89—81no—
166C4⋊Dic89C22×C411yescharacteristic
167C4⋊Dic89C22×C411yescharacteristic
168#13083553—21no—
169#13083553—21no—
170#13083553—21no—
171#13083553—21no—
172C22⋊Dic89—21no—
173#13083551—21no—
174#13083551—21no—
175#13083551—21no—
176#13083551—21no—
177C4×Dic89—21no—
178#41×C89—81no—
179#41×C89—81no—
180C2×Q8×C89—21no—
181C2×D4×C89—21no—
182C2×D4×C89—21no—
183C2×D4×C89—21no—
184C2×D4×C89—21no—
185C2×D4×C89C22×C411yescharacteristic
186C2×D4×C89C22×C411yescharacteristic
187C22×C4×C89—21no—
188C22×C4×C89—21no—
189C22×C4×C89C22×C411yescharacteristic
190C22×C4×C89C2×D411yescharacteristic
191C22×C4×C89#4111yescharacteristic
192M16×C89—81no—
193M16×C89—81no—
194C2×C8×C89—21no—
195C2×C8×C89—21no—
196C2×C8×C89—81no—
197C2×C8×C89—81no—
198C2×C8×C89C2×D411yescharacteristic
199C2×C8×C89C2×D411yescharacteristic
200C2×C8×C89C2×D411yescharacteristic
201C2×C8×C89C2×D411yescharacteristic
202C4⋊C4×C89—21no—
203C4⋊C4×C89C22×C411yescharacteristic
204C4⋊C4×C89C22×C411yescharacteristic
205C22⋊C4×C89—21no—
206C22⋊C4×C89—21no—
207C22⋊C4×C89—21no—
208C22⋊C4×C89—21no—
209C42×C89C22×C411yescharacteristic
210C42×C89C2×D411yescharacteristic
211C42×C89#4111yescharacteristic
212C42×C89C2411yescharacteristic
213C23×C89—21no—
214C23×C89—21no—
215C23×C89C4×D411yescharacteristic
216C22×D89—21no—
217Q8×C89—81no—
218D4×C89—21no—
219D4×C89—21no—
220D4×C89—21no—
221D4×C89—21no—
222D4×C89—81no—
223D4×C89—81no—
224D4×C89—81no—
225D4×C89—81no—
226C2×C4×C89—21no—
227C2×C4×C89—21no—
228C2×C4×C89—21no—
229C2×C4×C89—21no—
230C2×C4×C89—21no—
231C2×C4×C89—21no—
232C2×C4×C89—41no—
233C2×C4×C89—41no—
234C2×C4×C89—41no—
235C2×C4×C89—41no—
236C2×C4×C89—41no—
237C2×C4×C89C4×D411yescharacteristic
238C2×C4×C89C4×D411yescharacteristic
239C2×C4×C89C4×D411yescharacteristic
240C2×C4×C89C23×C411yescharacteristic
241C2×C4×C89C22×D411yescharacteristic
242C2×C4×C89C2×#4111yescharacteristic
243C22⋊D89—81no—
244C2×Dic89—21no—
245C2×Dic89—21no—
246C2×Dic89—21no—
247D356—21no—
248D356—21no—
249C4×D89—41no—
250C4×D89—41no—
251Dic178—21no—
252Dic178—21no—
253C8×C89—21no—
254C8×C89—21no—
255C8×C89—21no—
256C8×C89—21no—
257C8×C89—41no—
258C8×C89—41no—
259#10623494—22no—
260#10623494—21no—
261#10623494—21no—
262#10623494—21no—
263#10623494—21no—
264#10623494—41no—
265#10623494—41no—
266#10623494—41no—
267C22×C89—21no—
268C22×C89—21no—
269C22×C89—21no—
270C22×C89—41no—
271C22×C89—41no—
272C22×C89—41no—
273C22×C89—41no—
274C22×C89—41no—
275C22×C89—41no—
276C22×C89C2×C4×D411yescharacteristic
277C2×D89—41no—
278C2×D89—41no—
279C2×D89—41no—
280C4×C89—21no—
281C4×C89—21no—
282C4×C89—21no—
283C4×C89—21no—
284C4×C89—41no—
285C4×C89—41no—
286C4×C89#56811yescharacteristic
287C4×C89#56811yescharacteristic
288C4×C89#57611yescharacteristic
289C4×C89#57711yescharacteristic
290Dic89—41no—
291Dic89—41no—
292Dic89—41no—
293#17987—891nomaximal, Sylow
294C2×C89—41no—
295C2×C89—81no—
296C2×C89—81no—
297C2×C89#293211yescharacteristic
298C2×C89#294611yescharacteristic
299C2×C89#295611yescharacteristic
300D89—81no—
301#3171—891no—
302#2993—891no—
303#2981D8911yescharacteristic, complemented
304#1588—891no—
305#1586—891no—
306#1586—891no—
307#1585—891no—
308#1583—891no—
309#1582—891no—
310#1581—891no—
311#1581—891no—
312#1569—891no—
313#1494—891no—
314#1492—891no—
315#1451—891no—
316C89#1798711yescharacteristic, complemented, Sylow
317#532C2×D8911yescharacteristic
318#529—891no—
319#501—891no—
320#496—891no—
321#490—891no—
322#489—891no—
323#469—891no—
324#469—891no—
325#468—891no—
326#468—891no—
327#446—891no—
328C8×Q8—891no—
329C4×SD16—891no—
330C4×SD16—891no—
331C4×D8—891no—
332C4×D8—891no—
333#436—891no—
334#436—1781no—
335#436C2×D8911yescharacteristic
336#436C2×D8911yescharacteristic
337#435—21no—
338#435—21no—
339#435—1781no—
340C8×D4—891no—
341C8×D4C2×D8911yescharacteristic
342C8×D4C2×D8911yescharacteristic
343#433—1781no—
344#432C2×D8911yescharacteristic
345#431C2×D8911yescharacteristic
346#424—1781no—
347#423—891no—
348#408—21no—
349#408—21no—
350C4×M16—891no—
351C2×C4×C8—891no—
352#335—891no—
353#335—891no—
354#334—891no—
355#334—891no—
356Q8⋊C8—891no—
357Q8⋊C8—891no—
358D4⋊C8—891no—
359D4⋊C8—891no—
360D4⋊C8—891no—
361D4⋊C8—891no—
362D4⋊C8—891no—
363D4⋊C8—891no—
364#322—891no—
365#322—891no—
366#322—891no—
367#322—891no—
368C2×#41—21no—
369C2×SD16—891no—
370C2×SD16—891no—
371C2×D8—891no—
372C2×D8—891no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—891no—
376C2×M16—891no—
377C2×M16—1781no—
378C2×M16—1781no—
379C22×C8—21no—
380C22×C8—21no—
381C22×C8—1781no—
382C22×C8—1781no—
383C4⋊Q8C4×D8911yescharacteristic
384#127C4×D8911yescharacteristic
385#126—1781no—
386#125—891no—
387#124—21no—
388#124—891no—
389#123—1781no—
390C22⋊Q8—891no—
391#121—21no—
392#121—891no—
393#121C4×D8911yescharacteristic
394#121C4×D8911yescharacteristic
395C4×Q8—891no—
396C4×D4—21no—
397C4×D4—891no—
398C4×D4—1781no—
399C4×D4C22×D8911yescharacteristic
400C4×D4C22×D8911yescharacteristic
401#117—1781no—
402C2×C42C22×D8911yescharacteristic
403#107—891no—
404#107—891no—
405#106—891no—
406#106—891no—
407C4⋊C8—891no—
408C4⋊C8—891no—
409C4⋊C8—1781no—
410C4⋊C8—1781no—
411C4⋊C8—1781no—
412C4⋊C8C22⋊D8911yescharacteristic
413C4⋊C8C22⋊D8911yescharacteristic
414C4⋊C8C22×D8911yescharacteristic
415C4⋊C8C22×D8911yescharacteristic
416#103—891no—
417#103—891no—
418#102—891no—
419#102—891no—
420#102—891no—
421#102—891no—
422#102—891no—
423#102—891no—
424C22⋊C8—21no—
425C22⋊C8—21no—
426C22⋊C8—21no—
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—1781no—
430C22⋊C8—1781no—
431C22⋊C8C22⋊D8911yescharacteristic
432C22⋊C8C22⋊D8911yescharacteristic
433#97—21no—
434#97—891no—
435#97—891no—
436#97—1781no—
437C4×C8—891no—
438C4×C8—891no—
439C4×C8—891no—
440C4×C8—891no—
441C4×C8—1781no—
442C4×C8—1781no—
443C4×C8C22×D8911yescharacteristic
444C4×C8C22×D8911yescharacteristic
445#41—81no—
446#41—81no—
447C2×Q8—21no—
448C2×Q8—891no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—891no—
454C2×D4—1781no—
455C2×D4C2×C4×D8911yescharacteristic
456C2×D4C2×C4×D8911yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—1781no—
460C22×C4C2×C4×D8911yescharacteristic
461C22×C4C2×C22⋊D8911yescharacteristic
462C22×C4#1308357411yescharacteristic
463SD16—3561no—
464SD16—3561no—
465D8—3561no—
466D8—3561no—
467M16—81no—
468M16—81no—
469M16—3561no—
470M16—3561no—
471M16—7121no—
472M16—7121no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—81no—
476C2×C8—81no—
477C2×C8—891no—
478C2×C8—891no—
479C2×C8—891no—
480C2×C8—891no—
481C2×C8—3562no—
482C2×C8—1781no—
483C2×C8—1781no—
484C2×C8—1781no—
485C2×C8—1781no—
486C2×C8—7121no—
487C2×C8C2×C22⋊D8911yescharacteristic
488C2×C8C2×C22⋊D8911yescharacteristic
489C2×C8C2×C22⋊D8911yescharacteristic
490C2×C8C2×C22⋊D8911yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—891no—
493C4⋊C4—891no—
494C4⋊C4—1781no—
495C4⋊C4—1781no—
496C4⋊C4—1781no—
497C4⋊C4—1781no—
498C4⋊C4C2×C4×D8911yescharacteristic
499C4⋊C4C2×C4×D8911yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—1781no—
505C22⋊C4—1781no—
506C22⋊C4—1781no—
507C22⋊C4—1781no—
508C22⋊C4—1781no—
509C42—1781no—
510C42C2×C4×D8911yescharacteristic
511C42C2×C22⋊D8911yescharacteristic
512C42#1308357411yescharacteristic
513C42C23×D8911yescharacteristic
514C23—21no—
515C23—21no—
516C23—1781no—
517C23C4×C22⋊D8911yescharacteristic
518Q8—81no—
519Q8—1781no—
520Q8—1781no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—81no—
526D4—81no—
527D4—81no—
528D4—81no—
529D4—1781no—
530D4—1781no—
531D4—7121no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—3561no—
544C2×C4—3561no—
545C2×C4—1781no—
546C2×C4—1781no—
547C2×C4—1781no—
548C2×C4C4×C22⋊D8911yescharacteristic
549C2×C4C4×C22⋊D8911yescharacteristic
550C2×C4C4×C22⋊D8911yescharacteristic
551C2×C4C22×C4×D8911yescharacteristic
552C2×C4C22×C22⋊D8911yescharacteristic
553C2×C4C2×#1308357411yescharacteristic
554C8—21no—
555C8—21no—
556C8—21no—
557C8—21no—
558C8—41no—
559C8—41no—
560C8—3561no—
561C8—3561no—
562C8—3561no—
563C8—1782no—
564C8—1781no—
565C8—1781no—
566C8—1781no—
567C8—1781no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—3561no—
578C22—3561no—
579C22—3561no—
580C22C2×C4×C22⋊D8911yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—3561no—
588C4—3561no—
589C4—3561no—
590C4#1915236311yescharacteristic
591C4#1915236311yescharacteristic
592C4#1915240111yescharacteristic
593C4#1915240911yescharacteristic
594C2—41no—
595C2—81no—
596C2—81no—
597C2—7121no—
598C2#2602896811yescharacteristic
599C2#2602908411yescharacteristic
600C2#2602915911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2981C2
socleC22×C89C2×C4×D4
prime radical#26030841C2
cosocleC42×C89C24
hypercenter#2981D89
hypocenterC89#17987
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2981

Complements

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

NH ≅ G/Ntype
2. #26029403C2semidirect
3. #26019128C2semidirect
4. #26019116C2semidirect
5. #26019114C2semidirect
6. #26019109C2semidirect
7. #26019093C2semidirect
8. #26019088C2semidirect
9. #26019081C2semidirect
10. #26019079C2semidirect
11. #26019020C2semidirect
12. #26018710C2semidirect
13. #26018703C2semidirect
14. #26018542C2semidirect
15. C89×#2981C2semidirect
303. #2981D89semidirect
316. C89#17987semidirect

Maximal subgroups

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

subgroupindex
#260308412
#260294032
#260191282
#260191162
#260191142
#260191092
#260190932
#260190882
#260190812
#260190792
#260190202
#260187102
#260187032
#260185422
C89×#29812
#1798789