Finite Group Databasebeta

← #36525788

Subgroups — #36525788

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030279C211yesmaximal, characteristic
2#26029339C211yesmaximal, characteristic, complemented
3#26019154C211yesmaximal, characteristic, complemented
4#26019140C211yesmaximal, characteristic, complemented
5#26019128C211yesmaximal, characteristic, complemented
6#26019109C211yesmaximal, characteristic, complemented
7#26019093C211yesmaximal, characteristic, complemented
8#26019088C211yesmaximal, characteristic, complemented
9#26019064C211yesmaximal, characteristic, complemented
10C8×#13083562C211yesmaximal, characteristic, complemented
11#26019030C211yesmaximal, characteristic, complemented
12#26018721C211yesmaximal, characteristic
13#26018646C211yesmaximal, characteristic, complemented
14#26018552C211yesmaximal, characteristic, complemented
15C89×#2979C211yesmaximal, characteristic
16C2×C4×Dic178C2211yescharacteristic
17#19151914C411yescharacteristic
18#19151914C411yescharacteristic
19#19151880C411yescharacteristic
20#19151880C411yescharacteristic
21#19151795C411yescharacteristic
22#19151795C411yescharacteristic
23#19151736C411yescharacteristic
24#19151736C411yescharacteristic
25#19151728C2211yescharacteristic
26#19151728C2211yescharacteristic
27#19151727—21no—
28#19151727—21no—
29C8×Dic178C2211yescharacteristic
30C8×Dic178C2211yescharacteristic
31C4×#13083562C2211yescharacteristic
32C4×#13083562C2211yescharacteristic
33C4×#13083562C2211yescharacteristic
34C4×#13083562C2211yescharacteristic
35#19151655C2211yescharacteristic
36#19151652—21no—
37#19151651C2211yescharacteristic
38#19151651C2211yescharacteristic
39#19151648—21no—
40#19151350C2211yescharacteristic
41#19151350C2211yescharacteristic
42#19151348C2211yescharacteristic
43#19151348C2211yescharacteristic
44#19151328C2211yescharacteristic
45#19151328C2211yescharacteristic
46#19151328C2211yescharacteristic
47#19151328C2211yescharacteristic
48D4⋊#10623494C2211yescharacteristic
49D4⋊#10623494C2211yescharacteristic
50D4⋊#10623494C2211yescharacteristic
51D4⋊#10623494C2211yescharacteristic
52#19151320C2211yescharacteristic
53#19151320C2211yescharacteristic
54#19151319C2211yescharacteristic
55C8×#10623494C2211yescharacteristic
56#532×C89C2211yescharacteristic
57#436×C89—21no—
58#436×C89C2211yescharacteristic
59#436×C89C2211yescharacteristic
60C8×D4×C89—21no—
61C8×D4×C89C2211yescharacteristic
62C8×D4×C89C2211yescharacteristic
63#431×C89C2211yescharacteristic
64#431×C89C2211yescharacteristic
65#408×C89—21no—
66#408×C89—21no—
67C22×Dic178C2×C411yescharacteristic
68C2×#13083562C2×C411yescharacteristic
69C2×#13083562C2×C411yescharacteristic
70C2×#13083562C2×C411yescharacteristic
71C2×#13083562C2×C411yescharacteristic
72C4×Dic178—21no—
73C4×Dic178—21no—
74C4×Dic178C2×C411yescharacteristic
75C4×Dic178C2×C411yescharacteristic
76C4×Dic178C2311yescharacteristic
77C4×Dic178C2311yescharacteristic
78C2×C4⋊Dic89C2×C411yescharacteristic
79C2×#13083553—21no—
80C2×C4×Dic89—21no—
81#15667943C2×C411yescharacteristic
82#15667943C2×C411yescharacteristic
83#15667943C2×C411yescharacteristic
84#15667943C2×C411yescharacteristic
85C4⋊#10623494—21no—
86C4⋊#10623494C2×C411yescharacteristic
87C4⋊#10623494C2×C411yescharacteristic
88C4⋊#10623494C2311yescharacteristic
89C4⋊#10623494C2311yescharacteristic
90#15667941—21no—
91#15667941—21no—
92#15667941—21no—
93#15667941—21no—
94#15667935C2×C411yescharacteristic
95#15667935C2×C411yescharacteristic
96#15667935C2×C411yescharacteristic
97#15667935C2×C411yescharacteristic
98#15667933C2×C411yescharacteristic
99#15667933C2×C411yescharacteristic
100#15667933C2×C411yescharacteristic
101#15667933C2×C411yescharacteristic
102C22⋊#10623494—21no—
103C22⋊#10623494—21no—
104C22⋊#10623494C2×C411yescharacteristic
105C22⋊#10623494C2×C411yescharacteristic
106#15667923—21no—
107#15667922—21no—
108#15667922—21no—
109C4×#10623494C2311yescharacteristic
110C4×#10623494C2311yescharacteristic
111C4×#10623494C2311yescharacteristic
112C4×#10623494C2311yescharacteristic
113C8×Dic89—21no—
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
141C4×C8×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C2×Dic178—22no—
145C2×Dic178—21no—
146C2×Dic178—41no—
147C2×Dic178C22×C411yescharacteristic
148C2×Dic178C22×C411yescharacteristic
149C22×Dic89—21no—
150C22×Dic89—21no—
151#13083562—22no—
152#13083562—22no—
153#13083562—41no—
154#13083562—41no—
155C2×#10623494—21no—
156C2×#10623494—21no—
157C2×#10623494—21no—
158C2×#10623494—21no—
159C2×#10623494C22×C411yescharacteristic
160C2×#10623494C22×C411yescharacteristic
161C2×#10623494C22×C411yescharacteristic
162C2×#10623494C22×C411yescharacteristic
163C4⋊Dic89—21no—
164C4⋊Dic89C22×C411yescharacteristic
165C4⋊Dic89C22×C411yescharacteristic
166#13083553—21no—
167#13083553—21no—
168#13083553—21no—
169#13083553—21no—
170C4×Dic89—21no—
171C4×Dic89—21no—
172C4×Dic89—21no—
173C4×Dic89—21no—
174#41×C89—81no—
175#41×C89—81no—
176C2×Q8×C89—21no—
177C2×D4×C89—21no—
178C2×D4×C89—21no—
179C2×D4×C89—21no—
180C2×D4×C89—21no—
181C2×D4×C89C22×C411yescharacteristic
182C2×D4×C89C22×C411yescharacteristic
183C22×C4×C89—21no—
184C22×C4×C89—21no—
185C22×C4×C89C22×C411yescharacteristic
186C22×C4×C89C2×D411yescharacteristic
187C22×C4×C89#4111yescharacteristic
188M16×C89—81no—
189M16×C89—81no—
190C2×C8×C89—21no—
191C2×C8×C89—21no—
192C2×C8×C89—81no—
193C2×C8×C89—81no—
194C2×C8×C89C2×D411yescharacteristic
195C2×C8×C89C2×D411yescharacteristic
196C2×C8×C89C2×D411yescharacteristic
197C2×C8×C89C2×D411yescharacteristic
198C4⋊C4×C89—21no—
199C4⋊C4×C89C22×C411yescharacteristic
200C4⋊C4×C89C22×C411yescharacteristic
201C22⋊C4×C89—21no—
202C22⋊C4×C89—21no—
203C22⋊C4×C89—21no—
204C22⋊C4×C89—21no—
205C42×C89C22×C411yescharacteristic
206C42×C89C2×D411yescharacteristic
207C42×C89#4111yescharacteristic
208C42×C89C2411yescharacteristic
209C23×C89—21no—
210C23×C89—21no—
211C23×C89C4×D411yescharacteristic
212Q8×C89—81no—
213D4×C89—21no—
214D4×C89—21no—
215D4×C89—21no—
216D4×C89—21no—
217D4×C89—42no—
218D4×C89—81no—
219D4×C89—81no—
220D4×C89—81no—
221C2×C4×C89—21no—
222C2×C4×C89—21no—
223C2×C4×C89—21no—
224C2×C4×C89—21no—
225C2×C4×C89—21no—
226C2×C4×C89—21no—
227C2×C4×C89—41no—
228C2×C4×C89—41no—
229C2×C4×C89—41no—
230C2×C4×C89—41no—
231C2×C4×C89—41no—
232C2×C4×C89C4×D411yescharacteristic
233C2×C4×C89C4×D411yescharacteristic
234C2×C4×C89C4×D411yescharacteristic
235C2×C4×C89C23×C411yescharacteristic
236C2×C4×C89C22×D411yescharacteristic
237C2×C4×C89C2×#4111yescharacteristic
238C2×Dic89—21no—
239C2×Dic89—21no—
240C2×Dic89—21no—
241C2×Dic89—21no—
242C2×Dic89—42no—
243C2×Dic89—42no—
244Dic178—21no—
245Dic178—21no—
246Dic178—21no—
247Dic178—42no—
248Dic178#13112yes—
249C8×C89—21no—
250C8×C89—21no—
251C8×C89—21no—
252C8×C89—21no—
253C8×C89—41no—
254C8×C89—41no—
255#10623494—21no—
256#10623494—21no—
257#10623494—21no—
258#10623494—21no—
259#10623494—41no—
260#10623494—41no—
261#10623494—41no—
262#10623494—41no—
263C22×C89—21no—
264C22×C89—21no—
265C22×C89—21no—
266C22×C89—41no—
267C22×C89—41no—
268C22×C89—41no—
269C22×C89—41no—
270C22×C89—41no—
271C22×C89—41no—
272C22×C89C2×C4×D411yescharacteristic
273C4×C89—21no—
274C4×C89—21no—
275C4×C89—21no—
276C4×C89—21no—
277C4×C89—41no—
278C4×C89—41no—
279C4×C89C2×#13111yescharacteristic
280C4×C89#56811yescharacteristic
281C4×C89#57711yescharacteristic
282C4×C89#57711yescharacteristic
283Dic89—22no—
284Dic89—41no—
285Dic89—41no—
286Dic89—41no—
287#18030—891nomaximal, Sylow
288C2×C89—41no—
289C2×C89—81no—
290C2×C89—81no—
291C2×C89#293311yescharacteristic
292C2×C89#294911yescharacteristic
293C2×C89#295511yescharacteristic
294#3103—891no—
295#2984—891no—
296#2979D8911yescharacteristic
297#1592—891no—
298#1590—891no—
299#1588—891no—
300#1585—891no—
301#1583—891no—
302#1582—891no—
303SD16⋊C8—891no—
304C8×SD16—891no—
305#1572—891no—
306#1496—891no—
307#1479—891no—
308#1454—891no—
309C89#1803011yescharacteristic, complemented, Sylow
310#532C2×D8911yescharacteristic
311C2×C4×Q8—891no—
312#481—891no—
313#481—891no—
314#475—891no—
315#475—891no—
316#460—891no—
317#460—891no—
318#448—891no—
319#448—891no—
320#446—891no—
321#446—891no—
322#446—1781no—
323#446—1781no—
324C8×Q8—891no—
325C8×Q8—891no—
326C4×SD16—891no—
327C4×SD16—891no—
328C4×SD16—891no—
329C4×SD16—891no—
330#436—21no—
331#436C2×D8911yescharacteristic
332#436C2×D8911yescharacteristic
333C8×D4—21no—
334C8×D4C2×D8911yescharacteristic
335C8×D4C2×D8911yescharacteristic
336#432—891no—
337#432—1781no—
338#431—891no—
339#431—891no—
340#431—1781no—
341#431C2×D8911yescharacteristic
342#431C2×D8911yescharacteristic
343#408—21no—
344#408—21no—
345#335—891no—
346#335—891no—
347#334—891no—
348#334—891no—
349Q8⋊C8—891no—
350Q8⋊C8—891no—
351Q8⋊C8—891no—
352Q8⋊C8—891no—
353D4⋊C8—891no—
354D4⋊C8—891no—
355D4⋊C8—891no—
356D4⋊C8—891no—
357#322—891no—
358#322—891no—
359#322—891no—
360C82—891no—
361C2×#41—21no—
362C22×Q8—891no—
363C2×SD16—891no—
364C2×SD16—891no—
365C2×SD16—891no—
366C2×SD16—891no—
367C2×M16—21no—
368C2×M16—21no—
369C22×C8—21no—
370C22×C8—21no—
371C4⋊Q8C4×D8911yescharacteristic
372#127C4×D8911yescharacteristic
373#124—21no—
374#121—21no—
375#121C4×D8911yescharacteristic
376#121C4×D8911yescharacteristic
377C4×Q8—891no—
378C4×Q8—891no—
379C4×Q8—891no—
380C4×Q8—891no—
381C4×Q8—1781no—
382C4×Q8—1781no—
383C4×D4—21no—
384C4×D4C22×D8911yescharacteristic
385C4×D4C22×D8911yescharacteristic
386C2×C4⋊C4—891no—
387C2×C4⋊C4—1781no—
388C2×C42—1781no—
389C2×C42C22×D8911yescharacteristic
390#106—891no—
391#106—891no—
392#106—891no—
393#106—891no—
394C4⋊C8—891no—
395C4⋊C8—891no—
396C4⋊C8—891no—
397C4⋊C8—891no—
398C4⋊C8—1781no—
399C4⋊C8—1781no—
400C4⋊C8—1781no—
401C4⋊C8—1781no—
402C4⋊C8—1781no—
403C4⋊C8C22⋊D8911yescharacteristic
404C4⋊C8C22⋊D8911yescharacteristic
405C4⋊C8C22×D8911yescharacteristic
406C4⋊C8C22×D8911yescharacteristic
407#103—891no—
408#103—891no—
409#103—891no—
410#103—891no—
411#102—891no—
412#102—891no—
413#102—891no—
414#102—891no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8—21no—
419C22⋊C8—21no—
420C22⋊C8—891no—
421C22⋊C8—891no—
422C22⋊C8—1781no—
423C22⋊C8—1781no—
424C22⋊C8C22⋊D8911yescharacteristic
425C22⋊C8C22⋊D8911yescharacteristic
426#97—1781no—
427#97—1781no—
428#97—1781no—
429C4×C8—21no—
430C4×C8—891no—
431C4×C8—891no—
432C4×C8—891no—
433C4×C8—891no—
434C4×C8—1781no—
435C4×C8—1781no—
436C4×C8C22×D8911yescharacteristic
437C4×C8C22×D8911yescharacteristic
438#41—81no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—891no—
442C2×Q8—891no—
443C2×Q8—3561no—
444C2×Q8—1782no—
445C2×Q8—1781no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×C4×D8911yescharacteristic
451C2×D4C2×C4×D8911yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—1781no—
455C22×C4—1781no—
456C22×C4C2×C4×D8911yescharacteristic
457C22×C4C2×C22⋊D8911yescharacteristic
458C22×C4#1308357411yescharacteristic
459SD16—3561no—
460SD16—3561no—
461SD16—1782no—
462SD16—1782no—
463M16—81no—
464M16—81no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—81no—
468C2×C8—81no—
469C2×C8—891no—
470C2×C8—891no—
471C2×C8—891no—
472C2×C8—891no—
473C2×C8—1781no—
474C2×C8—1781no—
475C2×C8—1781no—
476C2×C8—1781no—
477C2×C8C2×C22⋊D8911yescharacteristic
478C2×C8C2×C22⋊D8911yescharacteristic
479C2×C8C2×C22⋊D8911yescharacteristic
480C2×C8C2×C22⋊D8911yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—891no—
483C4⋊C4—891no—
484C4⋊C4—1781no—
485C4⋊C4—1781no—
486C4⋊C4—1781no—
487C4⋊C4—1781no—
488C4⋊C4—1781no—
489C4⋊C4C2×C4×D8911yescharacteristic
490C4⋊C4C2×C4×D8911yescharacteristic
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C42—1781no—
496C42—1781no—
497C42—1781no—
498C42—1781no—
499C42C2×C4×D8911yescharacteristic
500C42C2×C22⋊D8911yescharacteristic
501C42#1308357411yescharacteristic
502C42C23×D8911yescharacteristic
503C23—21no—
504C23—21no—
505C23C4×C22⋊D8911yescharacteristic
506Q8—81no—
507Q8—892no—
508Q8—3562no—
509Q8—1781no—
510Q8—1781no—
511Q8—1781no—
512D4—21no—
513D4—21no—
514D4—21no—
515D4—21no—
516D4—42no—
517D4—81no—
518D4—81no—
519D4—81no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—3562no—
532C2×C4—3562no—
533C2×C4—1781no—
534C2×C4—1781no—
535C2×C4—1781no—
536C2×C4—1781no—
537C2×C4C4×C22⋊D8911yescharacteristic
538C2×C4C4×C22⋊D8911yescharacteristic
539C2×C4C4×C22⋊D8911yescharacteristic
540C2×C4C22×C4×D8911yescharacteristic
541C2×C4C22×C22⋊D8911yescharacteristic
542C2×C4C2×#1308357411yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8—21no—
547C8—41no—
548C8—41no—
549C8—3561no—
550C8—3561no—
551C8—3561no—
552C8—3561no—
553C8—1781no—
554C8—1781no—
555C8—1781no—
556C8—1781no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22C2×C4×C22⋊D8911yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—41no—
572C4—41no—
573C4—3561no—
574C4—3561no—
575C4—3561no—
576C4—1782no—
577C4#131×D8911yescharacteristic
578C4#1915236311yescharacteristic
579C4#1915240911yescharacteristic
580C4#1915240911yescharacteristic
581C2—41no—
582C2—81no—
583C2—81no—
584C2#2602897411yescharacteristic
585C2#2602910911yescharacteristic
586C2#2602915311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2979C2
socleC22×C89C2×C4×D4
prime radical#532×C89C22
cosocleC42×C89C24
hypercenter#2979D89
hypocenterC89#18030
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2979

Complements

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

NH ≅ G/Ntype
2. #26029339C2semidirect
3. #26019154C2semidirect
4. #26019140C2semidirect
5. #26019128C2semidirect
6. #26019109C2semidirect
7. #26019093C2semidirect
8. #26019088C2semidirect
9. #26019064C2semidirect
10. C8×#13083562C2semidirect
11. #26019030C2semidirect
13. #26018646C2semidirect
14. #26018552C2semidirect
309. C89#18030semidirect

Maximal subgroups

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

subgroupindex
#260302792
#260293392
#260191542
#260191402
#260191282
#260191092
#260190932
#260190882
#260190642
C8×#130835622
#260190302
#260187212
#260186462
#260185522
C89×#29792
#1803089