Finite Group Databasebeta

← #36525999

Subgroups — #36525999

Order 22784.

640 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029269C211yesmaximal, characteristic
2Q8×#13083558C211yesmaximal, characteristic, complemented
3C4×#15668043C211yesmaximal, characteristic
4#26019157C211yesmaximal, characteristic, complemented
5#26019139C211yesmaximal, characteristic, complemented
6#26019127C211yesmaximal, characteristic, complemented
7#26019126C211yesmaximal, characteristic, complemented
8#26019105C211yesmaximal, characteristic, complemented
9#26019092C211yesmaximal, characteristic, complemented
10#26019068C211yesmaximal, characteristic, complemented
11#26019056C211yesmaximal, characteristic, complemented
12#26019020C211yesmaximal, characteristic, complemented
13#26018692C211yesmaximal, characteristic
14#26018555C211yesmaximal, characteristic, complemented
15C89×#1491C211yesmaximal, characteristic, complemented
16C2×#15668043C411yescharacteristic
17C4×#13083574C2211yescharacteristic
18#19151729—21no—
19#19151729—21no—
20#19151729C2211yescharacteristic
21#19151729C2211yescharacteristic
22Q8×#10623494C2211yescharacteristic
23Q8×#10623494C2211yescharacteristic
24#19151722C2211yescharacteristic
25#19151722C2211yescharacteristic
26#19151703C411yescharacteristic
27#19151703C411yescharacteristic
28#19151696C411yescharacteristic
29#19151696C411yescharacteristic
30C4×#13083564C2211yescharacteristic
31C4×#13083564C2211yescharacteristic
32C4×#13083561C2211yescharacteristic
33C4×#13083561C2211yescharacteristic
34#19151675C2211yescharacteristic
35#19151675C2211yescharacteristic
36#19151673—21no—
37#19151673—21no—
38#19151653—21no—
39#19151638C411yescharacteristic
40#19151622—21no—
41#19151622C2211yescharacteristic
42#19151607C411yescharacteristic
43#19151597C411yescharacteristic
44C4×#13083558—21no—
45C4×#13083558C2211yescharacteristic
46C4×#13083558C2211yescharacteristic
47#19151550—21no—
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#432×C89C2211yescharacteristic
62C4×M16×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×#13083558—21no—
74C2×#13083558C2×C411yescharacteristic
75C2×#13083558C2×C411yescharacteristic
76C2×#13083558D411yescharacteristic
77C2×#13083558D411yescharacteristic
78C2×#13083557—21no—
79C2×#13083557—21no—
80C4×Dic178C2×C411yescharacteristic
81C4×Dic178C2311yescharacteristic
82C4×D356C2×C411yescharacteristic
83C4×D356C2311yescharacteristic
84C4×C22⋊D89—21no—
85C4×C22⋊D89—21no—
86#15667971C2×C411yescharacteristic
87#15667969—21no—
88C42×D89—21no—
89#15667945C2×C411yescharacteristic
90#15667945C2×C411yescharacteristic
91#15667943C2×C411yescharacteristic
92#15667943C2×C411yescharacteristic
93C4⋊#10623494—21no—
94C4⋊#10623494—21no—
95C4⋊#10623494—21no—
96C4⋊#10623494—21no—
97C4⋊#10623494—21no—
98C4⋊#10623494C2311yescharacteristic
99C4⋊#10623494C2311yescharacteristic
100#15667941—21no—
101#15667941—21no—
102#15667936C2×C411yescharacteristic
103#15667936C2×C411yescharacteristic
104#15667936C2×C411yescharacteristic
105#15667936C2×C411yescharacteristic
106#15667935C2×C411yescharacteristic
107#15667935C2×C411yescharacteristic
108#15667932C2×C411yescharacteristic
109#15667932C2×C411yescharacteristic
110#15667924—21no—
111#15667924—21no—
112#15667923—21no—
113#15667923—21no—
114#15667923C2×C411yescharacteristic
115#15667923C2×C411yescharacteristic
116#15667923D411yescharacteristic
117#15667923D411yescharacteristic
118C4×#10623494—21no—
119C4×#10623494—21no—
120C4×#10623494C2311yescharacteristic
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C8×Dic89—21no—
125C8×Dic89—21no—
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×C89C2311yescharacteristic
139C4⋊C8×C89C2311yescharacteristic
140C22⋊C8×C89—21no—
141#97×C89—21no—
142#97×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#13083564M1614yescomplemented
155#13083561—22no—
156#13083561—22no—
157#13083558—22no—
158#13083558—22no—
159#13083558—41no—
160#13083558—41no—
161#13083558—42no—
162#13083557—42no—
163#13083557—42no—
164C2×#10623494—21no—
165C2×#10623494—21no—
166C2×#10623494C22×C411yescharacteristic
167C2×#10623494C22×C411yescharacteristic
168C2×#10623494C22×C411yescharacteristic
169C2×#10623494C22×C411yescharacteristic
170C2×#10623494C2×D411yescharacteristic
171C2×#10623494C2×D411yescharacteristic
172C2×#10623494C2×D411yescharacteristic
173C2×#10623494C2×D411yescharacteristic
174C4⋊Dic89C22×C411yescharacteristic
175C4⋊Dic89C22×C411yescharacteristic
176#13083553—21no—
177#13083553—21no—
178C22⋊Dic89—21no—
179#13083551—21no—
180#13083551—21no—
181C4×Dic89—21no—
182C4×Dic89—21no—
183C4×Dic89—21no—
184C2×Q8×C89—21no—
185C2×Q8×C89—22no—
186C2×Q8×C89—41no—
187C2×Q8×C89C22×C411yescharacteristic
188C2×Q8×C89C22×C411yescharacteristic
189C22×C4×C89—21no—
190C22×C4×C89—21no—
191C22×C4×C89C22×C411yescharacteristic
192C22×C4×C89C2×D411yescharacteristic
193C22×C4×C89#4111yescharacteristic
194M16×C89—42no—
195C2×C8×C89—21no—
196C2×C8×C89—21no—
197C2×C8×C89—21no—
198C2×C8×C89—21no—
199C4⋊C4×C89—21no—
200C4⋊C4×C89—21no—
201C4⋊C4×C89—21no—
202C4⋊C4×C89—21no—
203C4⋊C4×C89—21no—
204C4⋊C4×C89C22×C411yescharacteristic
205C4⋊C4×C89C22×C411yescharacteristic
206C42×C89—21no—
207C42×C89—21no—
208C42×C89—21no—
209C42×C89—21no—
210C42×C89C22×C411yescharacteristic
211C42×C89C2×D411yescharacteristic
212C42×C89#4111yescharacteristic
213C42×C89C2411yescharacteristic
214C23×C89C4×D411yescharacteristic
215C22×D89—21no—
216Q8×C89—21no—
217Q8×C89—21no—
218Q8×C89—21no—
219Q8×C89—42no—
220Q8×C89C2×M1612yes—
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—21no—
228C2×C4×C89—21no—
229C2×C4×C89—41no—
230C2×C4×C89—42no—
231C2×C4×C89—81no—
232C2×C4×C89C4×D411yescharacteristic
233C2×C4×C89C4×D411yescharacteristic
234C2×C4×C89C4×D411yescharacteristic
235C2×C4×C89C23×C411yescharacteristic
236C2×C4×C89C22×D411yescharacteristic
237C2×C4×C89C2×#4111yescharacteristic
238C22⋊D89—42no—
239C2×Dic89—21no—
240C2×Dic89—21no—
241C2×Dic89—21no—
242D356—21no—
243D356—21no—
244C4×D89—42no—
245C4×D89—41no—
246C4×D89—41no—
247C4×D89—81no—
248Dic178—21no—
249Dic178C2×M1612yes—
250C8×C89—22no—
251C8×C89—41no—
252C8×C89—41no—
253C8×C89—41no—
254#10623494—21no—
255#10623494—21no—
256#10623494—21no—
257#10623494—21no—
258#10623494—21no—
259#10623494—21no—
260#10623494—41no—
261#10623494—41no—
262#10623494C2×M1612yes—
263#10623494#13712yescomplemented
264C22×C89—21no—
265C22×C89—21no—
266C22×C89—21no—
267C22×C89C2×C4×D411yescharacteristic
268C2×D89—41no—
269C2×D89—41no—
270C2×D89—41no—
271C4×C89—21no—
272C4×C89—22no—
273C4×C89—21no—
274C4×C89—21no—
275C4×C89—21no—
276C4×C89—41no—
277C4×C89—41no—
278C4×C89—41no—
279C4×C89C22×M1611yescharacteristic
280C4×C89#56811yescharacteristic
281C4×C89C2×#13711yescharacteristic
282C4×C89#57511yescharacteristic
283Dic89—22no—
284Dic89—41no—
285Dic89—41no—
286#18049—891nomaximal, Sylow
287C2×C89—41no—
288C2×C89D4×M1611yescharacteristic
289C2×C89C4×#13711yescharacteristic
290C2×C89#295611yescharacteristic
291D89—81no—
292#2971—891no—
293Q8×M16—891no—
294C4×#137—891no—
295#1592—891no—
296#1590—891no—
297#1588—891no—
298#1587—891no—
299#1584—891no—
300#1583—891no—
301#1579—891no—
302C8×Q16—891no—
303#1569—891no—
304#1491D8911yescharacteristic, complemented
305#1489—891no—
306#1455—891no—
307C89#1804911yescharacteristic, complemented, Sylow
308C2×#137—891no—
309C4×#41—891no—
310C2×C4×Q8C2×D8911yescharacteristic
311#446—891no—
312#446—891no—
313#446—1781no—
314#446—1781no—
315C8×Q8—891no—
316C8×Q8—891no—
317C8×Q8—891no—
318C8×Q8—891no—
319#441—891no—
320#441—891no—
321#440—891no—
322#440—891no—
323C4×Q16—891no—
324C4×Q16—891no—
325C4×SD16—891no—
326C4×SD16—891no—
327#436—891no—
328#436—891no—
329#436—1781no—
330#436—1781no—
331#432—1781no—
332#432C2×D8911yescharacteristic
333#428—891no—
334#423—891no—
335#423—1781no—
336#419—891no—
337#417—891no—
338C4×M16—891no—
339C4×M16—891no—
340C4×M16—1781no—
341C4×M16—1781no—
342C4×M16C2×D8911yescharacteristic
343#335—891no—
344#335—891no—
345#334—891no—
346#334—891no—
347Q8⋊C8—891no—
348Q8⋊C8—891no—
349Q8⋊C8C2×D8911yescharacteristic
350Q8⋊C8C2×D8911yescharacteristic
351Q8⋊C8C2×D8911yescharacteristic
352Q8⋊C8C2×D8911yescharacteristic
353D4⋊C8—891no—
354D4⋊C8—891no—
355#322—891no—
356#322—891no—
357#322—891no—
358C82—891no—
359C2×#41—891no—
360C22×Q8C4×D8911yescharacteristic
361#137—1784no—
362C2×Q16—891no—
363C2×Q16—891no—
364C2×SD16—891no—
365C2×SD16—891no—
366C2×M16—21no—
367C2×M16—891no—
368C2×M16—891no—
369C2×M16—891no—
370C2×M16—891no—
371C2×M16—1781no—
372C2×M16—1781no—
373C2×M16—1781no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—891no—
377C4×Q8—891no—
378C4×Q8C4×D8911yescharacteristic
379C4×Q8C4×D8911yescharacteristic
380C4×Q8C22×D8911yescharacteristic
381C4×Q8C22×D8911yescharacteristic
382C4×D4—891no—
383C4×D4—891no—
384C4×D4—1781no—
385C4×D4—1781no—
386#117—891no—
387#117—1781no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4C4×D8911yescharacteristic
390C2×C42—21no—
391C2×C42—1781no—
392C2×C42C22×D8911yescharacteristic
393#107—891no—
394#107—891no—
395#106—891no—
396#106—891no—
397C4⋊C8—891no—
398C4⋊C8—891no—
399C4⋊C8—1781no—
400C4⋊C8—1781no—
401C4⋊C8—1781no—
402C4⋊C8—1781no—
403C4⋊C8—1781no—
404C4⋊C8—1781no—
405C4⋊C8—1781no—
406C4⋊C8C22×D8911yescharacteristic
407C4⋊C8C22×D8911yescharacteristic
408#103—891no—
409#103—891no—
410#103—891no—
411#103—891no—
412#103—891no—
413#103—891no—
414#102—891no—
415#102—891no—
416C22⋊C8—21no—
417C22⋊C8—1781no—
418C22⋊C8—1781no—
419#97—21no—
420#97—21no—
421#97—891no—
422#97—891no—
423#97—891no—
424#97—891no—
425#97—1781no—
426#97—1781no—
427C4×C8—891no—
428C4×C8—891no—
429C4×C8—891no—
430C4×C8—891no—
431C4×C8—1781no—
432C4×C8—1781no—
433C4×C8—1781no—
434C4×C8—1781no—
435C4×C8C22×D8911yescharacteristic
436C4×C8C22×D8911yescharacteristic
437#41—3561no—
438#41—1782no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—41no—
442C2×Q8—891no—
443C2×Q8C2×C4×D8911yescharacteristic
444C2×Q8C2×C4×D8911yescharacteristic
445C2×D4—891no—
446C2×D4—1781no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—1781no—
450C22×C4—1781no—
451C22×C4—1781no—
452C22×C4C2×C4×D8911yescharacteristic
453C22×C4D4×D8911yescharacteristic
454C22×C4#1308357311yescharacteristic
455Q16—894no—
456Q16—1782no—
457SD16—1782no—
458SD16—1782no—
459M16—42no—
460M16—3562no—
461M16—3562no—
462M16—3562no—
463M16—3561no—
464M16—3561no—
465M16—1782no—
466M16—1782no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—891no—
472C2×C8—891no—
473C2×C8—891no—
474C2×C8—891no—
475C2×C8—891no—
476C2×C8—891no—
477C2×C8—891no—
478C2×C8—891no—
479C2×C8—1781no—
480C2×C8—1781no—
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⋊C4—1781no—
490C4⋊C4C2×C4×D8911yescharacteristic
491C4⋊C4C2×C4×D8911yescharacteristic
492C22⋊C4—1781no—
493C22⋊C4—1781no—
494C22⋊C4—1781no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—1781no—
500C42—1781no—
501C42—1781no—
502C42C2×C4×D8911yescharacteristic
503C42D4×D8911yescharacteristic
504C42#1308357311yescharacteristic
505C42C23×D8911yescharacteristic
506C23—1781no—
507C23#1566797211yescharacteristic
508Q8—21no—
509Q8—21no—
510Q8—21no—
511Q8—42no—
512Q8—892no—
513Q8—1781no—
514Q8M16×D8912yes—
515D4—3562no—
516D4—1781no—
517D4—1781no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—3562no—
530C2×C4—3561no—
531C2×C4—3561no—
532C2×C4—1781no—
533C2×C4—1781no—
534C2×C4—1781no—
535C2×C4—7121no—
536C2×C4#1566797211yescharacteristic
537C2×C4#1566797211yescharacteristic
538C2×C4#1566797211yescharacteristic
539C2×C4C22×C4×D8911yescharacteristic
540C2×C4C2×D4×D8911yescharacteristic
541C2×C4C2×#1308357311yescharacteristic
542C8—22no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—892no—
547C8—892no—
548C8—3561no—
549C8—3561no—
550C8—1781no—
551C8—1781no—
552C8—1781no—
553C8—1781no—
554C8—1781no—
555C8—1781no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—3561no—
560C22—3561no—
561C22—3561no—
562C22C2×#1566797211yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—3561no—
572C4—3561no—
573C4—1782no—
574C4C2×M16×D8911yescharacteristic
575C4#1915236411yescharacteristic
576C4#137×D8911yescharacteristic
577C4#1915239711yescharacteristic
578C2—41no—
579C2—7121no—
580C2#2602898411yescharacteristic
581C2#2602906711yescharacteristic
582C2#2602915811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1491C2
socleC22×C89C2×C4×D4
prime radicalC2×#13083574C2×C4
cosocleC42×C89C24
hypercenter#1491D89
hypocenterC89#18049
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
2. Q8×#13083558C2semidirect
4. #26019157C2semidirect
5. #26019139C2semidirect
6. #26019127C2semidirect
7. #26019126C2semidirect
8. #26019105C2semidirect
9. #26019092C2semidirect
10. #26019068C2semidirect
11. #26019056C2semidirect
12. #26019020C2semidirect
14. #26018555C2semidirect
15. C89×#1491C2semidirect
154. #13083564M16semidirect
263. #10623494#137semidirect
304. #1491D89semidirect
307. C89#18049semidirect

Maximal subgroups

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

subgroupindex
#260292692
Q8×#130835582
C4×#156680432
#260191572
#260191392
#260191272
#260191262
#260191052
#260190922
#260190682
#260190562
#260190202
#260186922
#260185552
C89×#14912
#1804989