Finite Group Databasebeta

← #36525786

Subgroups — #36525786

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#26030283C211yesmaximal, characteristic
2#26029341C211yesmaximal, characteristic, complemented
3#26029318C211yesmaximal, characteristic
4#26019157C211yesmaximal, characteristic, complemented
5#26019139C211yesmaximal, characteristic, complemented
6#26019127C211yesmaximal, characteristic, complemented
7#26019112C211yesmaximal, characteristic, complemented
8#26019092C211yesmaximal, characteristic, complemented
9#26019091C211yesmaximal, characteristic, complemented
10#26019067C211yesmaximal, characteristic, complemented
11#26019049C211yesmaximal, characteristic, complemented
12#26019030C211yesmaximal, characteristic, complemented
13#26018722C211yesmaximal, characteristic
14#26018552C211yesmaximal, characteristic, complemented
15C89×#1479C211yesmaximal, characteristic, complemented
16#19152148C2211yescharacteristic
17#19151916C411yescharacteristic
18#19151916C411yescharacteristic
19#19151883C411yescharacteristic
20#19151883C411yescharacteristic
21#19151798C411yescharacteristic
22#19151798C411yescharacteristic
23#19151737C411yescharacteristic
24#19151737C411yescharacteristic
25#19151729—21no—
26#19151729—21no—
27#19151729C2211yescharacteristic
28#19151729C2211yescharacteristic
29Q8×#10623494C2211yescharacteristic
30Q8×#10623494C2211yescharacteristic
31C4×#13083561C2211yescharacteristic
32C4×#13083561C2211yescharacteristic
33C4×#13083561C2211yescharacteristic
34C4×#13083561C2211yescharacteristic
35#19151677—21no—
36#19151675C2211yescharacteristic
37#19151675C2211yescharacteristic
38C8×D356—21no—
39#19151663C2211yescharacteristic
40#19151663C2211yescharacteristic
41#19151655—21no—
42#19151655C2211yescharacteristic
43#19151651—21no—
44#19151651C2211yescharacteristic
45#19151651C2211yescharacteristic
46#19151563—21no—
47#19151563—21no—
48#19151350C2211yescharacteristic
49#19151350C2211yescharacteristic
50#19151348C2211yescharacteristic
51#19151348C2211yescharacteristic
52#19151327C2211yescharacteristic
53#19151327C2211yescharacteristic
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151320C2211yescharacteristic
57#19151320C2211yescharacteristic
58#19151319C2211yescharacteristic
59C8×#10623494C2211yescharacteristic
60C2×C4×Q8×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62#431×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic
64Q8⋊C8×C89C2211yescharacteristic
65Q8⋊C8×C89C2211yescharacteristic
66Q8⋊C8×C89C2211yescharacteristic
67C2×#13083574—21no—
68C2×#13083561C2×C411yescharacteristic
69C2×#13083561C2×C411yescharacteristic
70C2×#13083561C2×C411yescharacteristic
71C2×#13083561C2×C411yescharacteristic
72C2×#13083557—21no—
73C2×#13083557—21no—
74C2×C8×D89—21no—
75C2×C8×D89—21no—
76#15668012C2×C411yescharacteristic
77#15668009C2×C411yescharacteristic
78#15668000—21no—
79#15667986—21no—
80#15667986C2×C411yescharacteristic
81#15667986C2×C411yescharacteristic
82C4×D356—21no—
83C4×D356C2311yescharacteristic
84C4×D356C2311yescharacteristic
85#15667943C2×C411yescharacteristic
86#15667943C2×C411yescharacteristic
87#15667943C2×C411yescharacteristic
88#15667943C2×C411yescharacteristic
89C4⋊#10623494—21no—
90C4⋊#10623494—21no—
91C4⋊#10623494—21no—
92C4⋊#10623494—21no—
93C4⋊#10623494C2×C411yescharacteristic
94C4⋊#10623494C2×C411yescharacteristic
95C4⋊#10623494D411yescharacteristic
96C4⋊#10623494D411yescharacteristic
97C4⋊#10623494C2311yescharacteristic
98C4⋊#10623494C2311yescharacteristic
99#15667936C2×C411yescharacteristic
100#15667936C2×C411yescharacteristic
101#15667936C2×C411yescharacteristic
102#15667936C2×C411yescharacteristic
103#15667932C2×C411yescharacteristic
104#15667932C2×C411yescharacteristic
105#15667932C2×C411yescharacteristic
106#15667932C2×C411yescharacteristic
107C22⋊#10623494—21no—
108C22⋊#10623494C2×C411yescharacteristic
109C22⋊#10623494C2×C411yescharacteristic
110C22⋊#10623494D411yescharacteristic
111C22⋊#10623494D411yescharacteristic
112#15667924—21no—
113#15667924—21no—
114#15667924—21no—
115#15667924—21no—
116#15667923—21no—
117#15667923—21no—
118#15667923—21no—
119C4×#10623494—21no—
120C4×#10623494—21no—
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C4×#10623494C2311yescharacteristic
125C22×Q8×C89C2×C411yescharacteristic
126C4×Q8×C89—21no—
127C4×Q8×C89—21no—
128C4×Q8×C89C2×C411yescharacteristic
129C4×Q8×C89C2×C411yescharacteristic
130C4×Q8×C89C2311yescharacteristic
131C4×Q8×C89C2311yescharacteristic
132C2×C4⋊C4×C89—21no—
133C2×C4⋊C4×C89C2×C411yescharacteristic
134C2×C42×C89—21no—
135C2×C42×C89C2311yescharacteristic
136C4⋊C8×C89—21no—
137C4⋊C8×C89C2311yescharacteristic
138C4⋊C8×C89C2311yescharacteristic
139C22⋊C8×C89—21no—
140C22⋊C8×C89—21no—
141C4×C8×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144#13083574—81no—
145#13083574—81no—
146C2×Dic178—21no—
147C2×D356—21no—
148C2×D356—21no—
149C2×D356C22×C411yescharacteristic
150C2×D356C22×C411yescharacteristic
151C2×C22⋊D89—21no—
152C2×C22⋊D89—21no—
153C2×C4×D89—21no—
154C2×C4×D89—21no—
155#13083561—22no—
156#13083561—22no—
157#13083561—41no—
158#13083561—41no—
159#13083557—81no—
160#13083557—81no—
161C2×#10623494—21no—
162C2×#10623494—21no—
163C2×#10623494C22×C411yescharacteristic
164C2×#10623494C22×C411yescharacteristic
165C2×#10623494C22×C411yescharacteristic
166C2×#10623494C22×C411yescharacteristic
167C2×#10623494C2×D411yescharacteristic
168C2×#10623494C2×D411yescharacteristic
169C2×#10623494C2×D411yescharacteristic
170C2×#10623494C2×D411yescharacteristic
171C8×D89—81no—
172C8×D89—81no—
173C4⋊Dic89—21no—
174C4⋊Dic89C22×C411yescharacteristic
175C4⋊Dic89C22×C411yescharacteristic
176#13083551—21no—
177#13083551—21no—
178#13083551—21no—
179#13083551—21no—
180C2×Q8×C89—21no—
181C2×Q8×C89—22no—
182C2×Q8×C89—41no—
183C2×Q8×C89C22×C411yescharacteristic
184C2×Q8×C89C22×C411yescharacteristic
185C22×C4×C89—21no—
186C22×C4×C89—21no—
187C22×C4×C89C22×C411yescharacteristic
188C22×C4×C89C2×D411yescharacteristic
189C22×C4×C89#4111yescharacteristic
190C2×C8×C89—21no—
191C2×C8×C89—21no—
192C2×C8×C89—21no—
193C2×C8×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89—21no—
198C4⋊C4×C89—21no—
199C4⋊C4×C89C22×C411yescharacteristic
200C4⋊C4×C89C22×C411yescharacteristic
201C42×C89—21no—
202C42×C89—21no—
203C42×C89—21no—
204C42×C89—21no—
205C42×C89C22×C411yescharacteristic
206C42×C89C2×D411yescharacteristic
207C42×C89#4111yescharacteristic
208C42×C89C2411yescharacteristic
209C23×C89C4×D411yescharacteristic
210C22×D89—21no—
211C22×D89—21no—
212Q8×C89—21no—
213Q8×C89—21no—
214Q8×C89—21no—
215Q8×C89—42no—
216Q8×C89#13112yes—
217C2×C4×C89—21no—
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×C89C4×D411yescharacteristic
229C2×C4×C89C4×D411yescharacteristic
230C2×C4×C89C4×D411yescharacteristic
231C2×C4×C89C23×C411yescharacteristic
232C2×C4×C89C22×D411yescharacteristic
233C2×C4×C89C2×#4111yescharacteristic
234C22⋊D89—81no—
235C22⋊D89—81no—
236C2×Dic89—21no—
237C2×Dic89—21no—
238D356—21no—
239D356—21no—
240D356—21no—
241D356—21no—
242D356—42no—
243D356—81no—
244C4×D89—41no—
245C4×D89—41no—
246C4×D89—41no—
247C4×D89—41no—
248Dic178—81no—
249C8×C89—41no—
250C8×C89—41no—
251C8×C89—41no—
252C8×C89—41no—
253#10623494—21no—
254#10623494—21no—
255#10623494—21no—
256#10623494—21no—
257#10623494—21no—
258#10623494—21no—
259#10623494—21no—
260#10623494—21no—
261#10623494—41no—
262#10623494—41no—
263C22×C89—21no—
264C22×C89—21no—
265C22×C89—21no—
266C22×C89C2×C4×D411yescharacteristic
267C2×D89—41no—
268C2×D89—41no—
269C2×D89—41no—
270C2×D89—41no—
271C2×D89—41no—
272C2×D89—41no—
273C4×C89—21no—
274C4×C89—22no—
275C4×C89—21no—
276C4×C89—21no—
277C4×C89—21no—
278C4×C89—41no—
279C4×C89—41no—
280C4×C89—41no—
281C4×C89C2×#13111yescharacteristic
282C4×C89#56811yescharacteristic
283C4×C89#57711yescharacteristic
284C4×C89#57711yescharacteristic
285Dic89—41no—
286Dic89—41no—
287#18030—891nomaximal, Sylow
288C2×C89—41no—
289C2×C89#293311yescharacteristic
290C2×C89#294911yescharacteristic
291C2×C89#295511yescharacteristic
292D89—81no—
293D89—81no—
294#3103—891no—
295#2984—891no—
296#2979—891no—
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#1479D8911yescharacteristic, complemented
308#1454—891no—
309C89#1803011yescharacteristic, complemented, Sylow
310#532—891no—
311C2×C4×Q8C2×D8911yescharacteristic
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—891no—
331#436—891no—
332#436—1781no—
333C8×D4—891no—
334C8×D4—891no—
335C8×D4—1781no—
336#432—891no—
337#432—1781no—
338#431—891no—
339#431—891no—
340#431—1781no—
341#431C2×D8911yescharacteristic
342#431C2×D8911yescharacteristic
343#408—1781no—
344#408—1781no—
345#335—891no—
346#335—891no—
347#334—891no—
348#334—891no—
349Q8⋊C8C2×D8911yescharacteristic
350Q8⋊C8C2×D8911yescharacteristic
351Q8⋊C8C2×D8911yescharacteristic
352Q8⋊C8C2×D8911yescharacteristic
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—1781no—
362C22×Q8C4×D8911yescharacteristic
363C2×SD16—891no—
364C2×SD16—891no—
365C2×SD16—891no—
366C2×SD16—891no—
367C2×M16—1781no—
368C2×M16—1781no—
369C22×C8—1781no—
370C22×C8—1781no—
371C4⋊Q8—891no—
372#127—891no—
373#124—1781no—
374#121—891no—
375#121—891no—
376#121—1781no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8C4×D8911yescharacteristic
380C4×Q8C4×D8911yescharacteristic
381C4×Q8C22×D8911yescharacteristic
382C4×Q8C22×D8911yescharacteristic
383C4×D4—891no—
384C4×D4—891no—
385C4×D4—1781no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4C4×D8911yescharacteristic
388C2×C42—21no—
389C2×C42C22×D8911yescharacteristic
390#106—891no—
391#106—891no—
392#106—891no—
393#106—891no—
394C4⋊C8—21no—
395C4⋊C8—891no—
396C4⋊C8—891no—
397C4⋊C8—891no—
398C4⋊C8—891no—
399C4⋊C8—891no—
400C4⋊C8—891no—
401C4⋊C8—1781no—
402C4⋊C8—1781no—
403C4⋊C8—1781no—
404C4⋊C8—1781no—
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—891no—
418C22⋊C8—891no—
419C22⋊C8—891no—
420C22⋊C8—891no—
421C22⋊C8—1781no—
422C22⋊C8—1781no—
423C22⋊C8—1781no—
424C22⋊C8—1781no—
425C22⋊C8—1781no—
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—7121no—
439#41—7121no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—1781no—
444C2×Q8C2×C4×D8911yescharacteristic
445C2×Q8C2×C4×D8911yescharacteristic
446C2×D4—891no—
447C2×D4—891no—
448C2×D4—1781no—
449C2×D4—1781no—
450C2×D4—1781no—
451C2×D4—1781no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—1781no—
455C22×C4—1781no—
456C22×C4C2×C4×D8911yescharacteristic
457C22×C4D4×D8911yescharacteristic
458C22×C4#1308357311yescharacteristic
459SD16—3561no—
460SD16—3561no—
461SD16—1782no—
462SD16—1782no—
463M16—7121no—
464M16—7121no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—891no—
470C2×C8—891no—
471C2×C8—891no—
472C2×C8—891no—
473C2×C8—891no—
474C2×C8—891no—
475C2×C8—891no—
476C2×C8—891no—
477C2×C8—1781no—
478C2×C8—1781no—
479C2×C8—7121no—
480C2×C8—7121no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—891no—
487C4⋊C4—891no—
488C4⋊C4—1781no—
489C4⋊C4C2×C4×D8911yescharacteristic
490C4⋊C4C2×C4×D8911yescharacteristic
491C22⋊C4—1781no—
492C22⋊C4—1781no—
493C22⋊C4—1781no—
494C22⋊C4—1781no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42C2×C4×D8911yescharacteristic
500C42D4×D8911yescharacteristic
501C42#1308357311yescharacteristic
502C42C23×D8911yescharacteristic
503C23—1781no—
504C23—1781no—
505C23#1566797211yescharacteristic
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—42no—
510Q8—7121no—
511Q8#1566801812yes—
512D4—3562no—
513D4—1781no—
514D4—1781no—
515D4—1781no—
516D4—1781no—
517D4—7121no—
518D4—7121no—
519D4—7121no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—3561no—
532C2×C4—3561no—
533C2×C4—3561no—
534C2×C4—3561no—
535C2×C4—1781no—
536C2×C4—1781no—
537C2×C4#1566797211yescharacteristic
538C2×C4#1566797211yescharacteristic
539C2×C4#1566797211yescharacteristic
540C2×C4C22×C4×D8911yescharacteristic
541C2×C4C2×D4×D8911yescharacteristic
542C2×C4C2×#1308357311yescharacteristic
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—3561no—
548C8—3561no—
549C8—1781no—
550C8—1781no—
551C8—1781no—
552C8—1781no—
553C8—1781no—
554C8—1781no—
555C8—1781no—
556C8—1781no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—3561no—
561C22—3561no—
562C22—3561no—
563C22—3561no—
564C22—3561no—
565C22—3561no—
566C22C2×#1566797211yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—22no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—3561no—
576C4—3561no—
577C4C2×#1566801811yescharacteristic
578C4#1915236411yescharacteristic
579C4#1915240711yescharacteristic
580C4#1915240711yescharacteristic
581C2—41no—
582C2—7121no—
583C2—7121no—
584C2#2602897311yescharacteristic
585C2#2602910711yescharacteristic
586C2#2602915111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1479C2
socleC22×C89C2×C4×D4
prime radical#19152148C22
cosocleC42×C89C24
hypercenter#1479D89
hypocenterC89#18030
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
2. #26029341C2semidirect
4. #26019157C2semidirect
5. #26019139C2semidirect
6. #26019127C2semidirect
7. #26019112C2semidirect
8. #26019092C2semidirect
9. #26019091C2semidirect
10. #26019067C2semidirect
11. #26019049C2semidirect
12. #26019030C2semidirect
14. #26018552C2semidirect
15. C89×#1479C2semidirect
307. #1479D89semidirect
309. C89#18030semidirect

Maximal subgroups

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

subgroupindex
#260302832
#260293412
#260293182
#260191572
#260191392
#260191272
#260191122
#260190922
#260190912
#260190672
#260190492
#260190302
#260187222
#260185522
C89×#14792
#1803089