Finite Group Databasebeta

← #36525323

Subgroups — #36525323

Order 22784.

680 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#26030258C211yesmaximal, characteristic
2#26029194C211yesmaximal, characteristic, complemented
3#26028988C211yesmaximal, characteristic
4#26019143C211yesmaximal, characteristic, complemented
5#26019140C211yesmaximal, characteristic, complemented
6#26019098C211yesmaximal, characteristic, complemented
7#26019091C211yesmaximal, characteristic, complemented
8#26019090C211yesmaximal, characteristic, complemented
9C8×#13083561C211yesmaximal, characteristic, complemented
10#26019051C211yesmaximal, characteristic, complemented
11#26019021C211yesmaximal, characteristic, complemented
12#26018705C211yesmaximal, characteristic, complemented
13D4⋊C8×D89C211yesmaximal, characteristic
14#26018531C211yesmaximal, characteristic, complemented
15C89×#1583C211yesmaximal, characteristic, complemented
16C2×M16×D89—21no—
17#19152152C2211yescharacteristic
18C4×D4×D89C2211yescharacteristic
19#19151930C411yescharacteristic
20#19151927C411yescharacteristic
21#19151887C411yescharacteristic
22#19151886C411yescharacteristic
23#19151802C411yescharacteristic
24#19151801C411yescharacteristic
25#19151750C411yescharacteristic
26#19151747C411yescharacteristic
27#19151729C2211yescharacteristic, complemented
28#19151728—21no—
29#19151728C2211yescharacteristic, complemented
30#19151727—21no—
31#19151722C2211yescharacteristic, complemented
32C4×#13083563C2211yescharacteristic
33C4×#13083561C2211yescharacteristic
34C4×#13083562C2211yescharacteristic
35#19151676C2211yescharacteristic
36#19151672—21no—
37#19151667—21no—
38C8×D356C2211yescharacteristic
39#19151663C2211yescharacteristic
40#19151649C2211yescharacteristic
41#19151623—21no—
42#19151620C2211yescharacteristic
43C4⋊C8×D89C2211yescharacteristic
44#19151560—21no—
45#19151552—21no—
46C4×#13083557C2211yescharacteristic
47C4×C8×D89C2211yescharacteristic
48#19151349C2211yescharacteristic, complemented
49#19151348C2211yescharacteristic, complemented
50#19151348C2211yescharacteristic, complemented
51#19151330C2211yescharacteristic, complemented
52Q8⋊#10623494C2211yescharacteristic, complemented
53#19151328C2211yescharacteristic, complemented
54#19151327C2211yescharacteristic
55D4⋊#10623494C2211yescharacteristic
56#19151325C2211yescharacteristic
57#19151320C2211yescharacteristic, complemented
58C8×#10623494C2211yescharacteristic, complemented
59C8×#10623494C2211yescharacteristic, complemented
60C8×Q8×C89C2211yescharacteristic, complemented
61C4×SD16×C89C2211yescharacteristic
62#436×C89C2211yescharacteristic
63#334×C89C2211yescharacteristic, complemented
64Q8⋊C8×C89C2211yescharacteristic, complemented
65D4⋊C8×C89C2211yescharacteristic
66#322×C89C2211yescharacteristic, complemented
67C2×D4×D89C2×C411yescharacteristic
68C22×C4×D89—21no—
69C2×#13083563C2×C411yescharacteristic
70C2×#13083562C2×C411yescharacteristic
71C2×#13083561C2×C411yescharacteristic
72C2×#13083558—21no—
73C2×#13083557—21no—
74C2×#13083557D411yescharacteristic
75C2×#13083557D411yescharacteristic
76M16×D89—44no—
77C2×C8×D89—21no—
78C2×C8×D89—21no—
79C2×C8×D89—21no—
80C2×C8×D89—21no—
81#15668011C2×C411yescharacteristic
82#15668010C2×C411yescharacteristic
83#15668002—21no—
84#15667992C2×C411yescharacteristic
85#15667990—21no—
86#15667988C2×C411yescharacteristic
87C4×Dic178—21no—
88C4×Dic178C2311yescharacteristic
89C4×D356C2311yescharacteristic
90D4×Dic89C2×C411yescharacteristic
91#15667974C2×C411yescharacteristic
92C4×C22⋊D89—21no—
93#15667972—21no—
94#15667969—21no—
95C4⋊C4×D89C2×C411yescharacteristic
96C22⋊C4×D89—21no—
97C42×D89C2311yescharacteristic
98#15667944C2×C411yescharacteristic
99#15667943C2×C411yescharacteristic
100#15667943C2×C411yescharacteristic
101C4⋊#10623494—21no—
102C4⋊#10623494C2311yescharacteristic
103C4⋊#10623494C2311yescharacteristic
104#15667941—21no—
105#15667941—21no—
106#15667941—21no—
107#15667941—21no—
108#15667941C2×C411yescharacteristic
109#15667941C2×C411yescharacteristic
110#15667937C2×C411yescharacteristic
111#15667936C2×C411yescharacteristic
112#15667935C2×C411yescharacteristic
113#15667934C2×C411yescharacteristic
114#15667933C2×C411yescharacteristic
115#15667932C2×C411yescharacteristic
116C22⋊#10623494—21no—
117#15667924—21no—
118#15667924—21no—
119#15667924C2×C411yescharacteristic
120#15667924C2×C411yescharacteristic
121#15667923—21no—
122#15667922—21no—
123#15667922D411yescharacteristic
124#15667922D411yescharacteristic
125C4×#10623494C2311yescharacteristic
126C4×#10623494C2311yescharacteristic
127C4×#10623494C2311yescharacteristic
128C8×Dic89—21no—
129C8×Dic89—21no—
130C2×SD16×C89C2×C411yescharacteristic
131C2×M16×C89—21no—
132C4×Q8×C89C2311yescharacteristic
133C4×D4×C89C2311yescharacteristic
134#106×C89C2×C411yescharacteristic
135C4⋊C8×C89—21no—
136C4⋊C8×C89C2311yescharacteristic
137C4⋊C8×C89C2311yescharacteristic
138#103×C89C2×C411yescharacteristic
139#102×C89C2×C411yescharacteristic
140C22⋊C8×C89—21no—
141C4×C8×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145C23×D89—21no—
146C2×Dic178—21no—
147C2×Dic178C22×C411yescharacteristic
148C2×D356C22×C411yescharacteristic
149C2×C22⋊D89—21no—
150D4×D89—22no—
151D4×D89—41no—
152C22×Dic89—21no—
153C2×C4×D89—21no—
154C2×C4×D89—21no—
155C2×C4×D89—21no—
156C2×C4×D89—42no—
157C2×C4×D89—42no—
158C2×C4×D89C22×C411yescharacteristic
159C2×C4×D89C2×D411yescharacteristic
160C2×C4×D89#4111yescharacteristic
161#13083563—22no—
162#13083562—41no—
163#13083561—22no—
164#13083558—42no—
165#13083557—22no—
166#13083557—42no—
167#13083557SD1614yescomplemented
168C2×#10623494—21no—
169C2×#10623494—21no—
170C2×#10623494—21no—
171C2×#10623494C22×C411yescharacteristic
172C2×#10623494C22×C411yescharacteristic
173C2×#10623494C2×D411yescharacteristic
174C2×#10623494C2×D411yescharacteristic
175C8×D89—42no—
176C8×D89—42no—
177C8×D89—42no—
178C8×D89—42no—
179C4⋊Dic89—21no—
180C4⋊Dic89C22×C411yescharacteristic
181C4⋊Dic89C22×C411yescharacteristic
182#13083553—21no—
183#13083553—21no—
184#13083553—21no—
185#13083553—21no—
186#13083553—21no—
187#13083553—21no—
188C22⋊Dic89—21no—
189#13083551—21no—
190#13083551—21no—
191#13083551—21no—
192#13083551—21no—
193C4×Dic89—21no—
194C4×Dic89C22×C411yescharacteristic
195C4×Dic89C2×D411yescharacteristic
196C4×Dic89#4111yescharacteristic
197C2×Q8×C89C22×C411yescharacteristic
198C2×D4×C89C22×C411yescharacteristic
199C22×C4×C89—21no—
200SD16×C89—41no—
201M16×C89—42no—
202C2×C8×C89—21no—
203C2×C8×C89—21no—
204C2×C8×C89—21no—
205C2×C8×C89C22×C411yescharacteristic
206C2×C8×C89C22×C411yescharacteristic
207C2×C8×C89C2×D411yescharacteristic
208C2×C8×C89C2×D411yescharacteristic
209C4⋊C4×C89—21no—
210C4⋊C4×C89C22×C411yescharacteristic
211C4⋊C4×C89C22×C411yescharacteristic
212C22⋊C4×C89—21no—
213C42×C89—21no—
214C42×C89C2411yescharacteristic
215C23×C89—21no—
216C22×D89—21no—
217C22×D89—42no—
218C22×D89—42no—
219C22×D89—81no—
220C22×D89C4×D411yescharacteristic
221Q8×C89—21no—
222Q8×C89—21no—
223D4×C89—21no—
224D4×C89—21no—
225C2×C4×C89—21no—
226C2×C4×C89—21no—
227C2×C4×C89—21no—
228C2×C4×C89—41no—
229C2×C4×C89—41no—
230C2×C4×C89C23×C411yescharacteristic
231C2×C4×C89C22×D411yescharacteristic
232C2×C4×C89C2×#4111yescharacteristic
233C22⋊D89—42no—
234C2×Dic89—21no—
235C2×Dic89—21no—
236C2×Dic89—21no—
237C2×Dic89—42no—
238C2×Dic89C4×D411yescharacteristic
239C2×Dic89C4×D411yescharacteristic
240C2×Dic89C4×D411yescharacteristic
241D356—21no—
242D356#13112yes—
243C4×D89—21no—
244C4×D89—21no—
245C4×D89—41no—
246C4×D89—42no—
247C4×D89—41no—
248C4×D89—41no—
249C4×D89—42no—
250C4×D89C2×SD1612yes—
251C4×D89#13512yes—
252Dic178—21no—
253Dic178—21no—
254Dic178—81no—
255C8×C89—21no—
256C8×C89—21no—
257C8×C89—21no—
258C8×C89—41no—
259C8×C89—41no—
260C8×C89—41no—
261C8×C89C2×SD1612yes—
262#10623494—22no—
263#10623494—21no—
264#10623494—21no—
265#10623494—21no—
266#10623494—41no—
267#10623494—41no—
268#10623494C2×SD1612yes—
269C22×C89—41no—
270C22×C89—41no—
271C22×C89—41no—
272C22×C89C2×C4×D411yescharacteristic
273C2×D89—21no—
274C2×D89—21no—
275C2×D89—22no—
276C2×D89—41no—
277C2×D89—41no—
278C2×D89—84no—
279C2×D89C4×SD1612yes—
280C4×C89—21no—
281C4×C89—41no—
282C4×C89—41no—
283C4×C89—41no—
284C4×C89C2×#13111yescharacteristic
285C4×C89#56811yescharacteristic
286C4×C89C22×SD1611yescharacteristic
287C4×C89C2×#13511yescharacteristic
288Dic89—21no—
289Dic89—21no—
290Dic89—41no—
291Dic89—41no—
292Dic89—41no—
293Dic89C4×SD1612yes—
294#17995—891nomaximal, Sylow
295C2×C89—81no—
296C2×C89#293311yescharacteristic
297C2×C89C2×C4×SD1611yescharacteristic
298C2×C89#295611yescharacteristic
299D89—22no—
300D89—42no—
301#3101—891no—
302#2962—891no—
303D4×M16—891no—
304#1590—891no—
305#1590—891no—
306#1583—891no—
307#1583D8911yescharacteristic, complemented
308#1582—891no—
309#1582—891no—
310C8×SD16—891no—
311C8×SD16—891no—
312#1570—891no—
313#1493—891no—
314C2×D4⋊C8—891no—
315C8×M16—891no—
316C89#1799511yescharacteristic, complemented, Sylow
317C22×M16—1781no—
318#533—891no—
319C2×C4×D4—891no—
320#483—891no—
321#483—891no—
322#476—891no—
323#476—891no—
324#461—891no—
325#461—891no—
326#450—891no—
327#450—891no—
328#446—891no—
329#446—891no—
330#446—1781no—
331#446—1781no—
332C8×Q8—891no—
333C8×Q8C2×D8911yescharacteristic, complemented
334C4×SD16—891no—
335C4×SD16—891no—
336C4×SD16—891no—
337C4×SD16C2×D8911yescharacteristic
338#436—891no—
339#436C2×D8911yescharacteristic
340#435—1781no—
341#435—1781no—
342C8×D4—891no—
343C8×D4—891no—
344#431—891no—
345#424—1781no—
346#423—891no—
347C2×C4⋊C8—891no—
348#407—1781no—
349#405—1781no—
350C4×M16—891no—
351C2×C4×C8—891no—
352#334—891no—
353#334—891no—
354#334—891no—
355#334C2×D8911yescharacteristic, complemented
356Q8⋊C8—891no—
357Q8⋊C8—891no—
358Q8⋊C8—891no—
359Q8⋊C8C2×D8911yescharacteristic, complemented
360D4⋊C8—891no—
361D4⋊C8—891no—
362D4⋊C8—891no—
363D4⋊C8C2×D8911yescharacteristic
364#322—891no—
365#322C2×D8911yescharacteristic, complemented
366C82—891no—
367C82—891no—
368C22×D4—891no—
369C23×C4—1781no—
370C2×SD16—891no—
371C2×SD16—891no—
372C2×SD16—891no—
373C2×SD16C4×D8911yescharacteristic
374C2×M16—21no—
375C2×M16—891no—
376C2×M16—891no—
377C2×M16—3564no—
378C2×M16—1781no—
379C2×M16—1781no—
380C22×C8—1781no—
381C22×C8—1781no—
382C22×C8—1781no—
383C22×C8—1781no—
384C4⋊Q8—891no—
385C4⋊Q8—891no—
386#125—1781no—
387C22⋊Q8—891no—
388C22⋊Q8—891no—
389C22⋊Q8—1781no—
390C4×Q8—891no—
391C4×Q8—1781no—
392C4×Q8C22×D8911yescharacteristic
393C4×D4—891no—
394C4×D4—891no—
395C4×D4—891no—
396C4×D4—1781no—
397C4×D4—1781no—
398C4×D4C22×D8911yescharacteristic
399#117—1781no—
400C2×C4⋊C4—891no—
401C2×C22⋊C4—1781no—
402C2×C42—891no—
403#106—891no—
404#106—891no—
405#106—891no—
406#106C4×D8911yescharacteristic
407C4⋊C8—21no—
408C4⋊C8—891no—
409C4⋊C8—891no—
410C4⋊C8—891no—
411C4⋊C8—891no—
412C4⋊C8—1781no—
413C4⋊C8—1781no—
414C4⋊C8—1781no—
415C4⋊C8—1781no—
416C4⋊C8—1781no—
417C4⋊C8C22×D8911yescharacteristic
418C4⋊C8C22×D8911yescharacteristic
419#103—891no—
420#103—891no—
421#103—891no—
422#103C4×D8911yescharacteristic
423#102—891no—
424#102—891no—
425#102—891no—
426#102C4×D8911yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—891no—
429C22⋊C8—891no—
430C22⋊C8—1781no—
431C22⋊C8—1781no—
432C22⋊C8—1781no—
433#97—891no—
434#97—891no—
435#97—1781no—
436#97—1781no—
437C4×C8—21no—
438C4×C8—891no—
439C4×C8—891no—
440C4×C8—891no—
441C4×C8—1781no—
442C4×C8—1781no—
443C4×C8C22×D8911yescharacteristic
444C4×C8C22×D8911yescharacteristic
445C4×C8C22×D8911yescharacteristic
446C24—1781no—
447C2×Q8—891no—
448C2×Q8—1781no—
449C2×Q8C2×C4×D8911yescharacteristic
450C2×D4—891no—
451C2×D4—3561no—
452C2×D4—1781no—
453C2×D4—1782no—
454C2×D4C2×C4×D8911yescharacteristic
455C22×C4—21no—
456C22×C4—891no—
457C22×C4—891no—
458C22×C4—891no—
459C22×C4—3562no—
460C22×C4—3562no—
461C22×C4—1781no—
462C22×C4—1781no—
463C22×C4—1781no—
464C22×C4—1781no—
465SD16—41no—
466SD16—3561no—
467SD16—1782no—
468SD16—1782no—
469M16—42no—
470M16—894no—
471M16—3562no—
472M16—3562no—
473M16—1782no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—891no—
478C2×C8—891no—
479C2×C8—891no—
480C2×C8—891no—
481C2×C8—3562no—
482C2×C8—3562no—
483C2×C8—3562no—
484C2×C8—3562no—
485C2×C8—1781no—
486C2×C8—1781no—
487C2×C8—1781no—
488C2×C8C2×C4×D8911yescharacteristic
489C2×C8C2×C4×D8911yescharacteristic
490C2×C8D4×D8911yescharacteristic
491C2×C8D4×D8911yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—891no—
494C4⋊C4—891no—
495C4⋊C4—1781no—
496C4⋊C4—1781no—
497C4⋊C4—1781no—
498C4⋊C4—1781no—
499C4⋊C4—1781no—
500C4⋊C4—1781no—
501C4⋊C4—1781no—
502C4⋊C4C2×C4×D8911yescharacteristic
503C4⋊C4C2×C4×D8911yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1781no—
506C22⋊C4—1781no—
507C22⋊C4—1781no—
508C22⋊C4—1781no—
509C22⋊C4—1781no—
510C42—21no—
511C42—891no—
512C42—891no—
513C42—891no—
514C42—1781no—
515C42C23×D8911yescharacteristic
516C23—21no—
517C23—891no—
518C23—3562no—
519C23—3562no—
520C23—1781no—
521C23—7121no—
522Q8—21no—
523Q8—21no—
524Q8—1781no—
525Q8—1781no—
526Q8—7121no—
527D4—21no—
528D4—21no—
529D4—892no—
530D4—3562no—
531D4—1781no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—891no—
538C2×C4—892no—
539C2×C4—892no—
540C2×C4—891no—
541C2×C4—891no—
542C2×C4—3561no—
543C2×C4—3562no—
544C2×C4—3562no—
545C2×C4—3562no—
546C2×C4—3561no—
547C2×C4—3561no—
548C2×C4—1781no—
549C2×C4—1781no—
550C2×C4—1781no—
551C2×C4—1781no—
552C2×C4—1781no—
553C2×C4C22×C4×D8911yescharacteristic
554C2×C4C2×D4×D8911yescharacteristic
555C2×C4#41×D8911yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—892no—
563C8—3561no—
564C8—3561no—
565C8—1782no—
566C8—1781no—
567C8—1781no—
568C8—1781no—
569C8SD16×D8912yes—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—892no—
574C22—3561no—
575C22—3561no—
576C22—1782no—
577C22—1781no—
578C22—1781no—
579C22—7124no—
580C22C4×D4×D8911yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—892no—
586C4—3561no—
587C4—3561no—
588C4—3561no—
589C4—1781no—
590C4—1781no—
591C4#131×D8911yescharacteristic
592C4#1915236411yescharacteristic
593C4C2×SD16×D8911yescharacteristic
594C4#135×D8911yescharacteristic
595C2—81no—
596C2—3562no—
597C2—1782no—
598C2#2602897511yescharacteristic
599C2C4×SD16×D8911yescharacteristic
600C2#2602916011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1583C2
socleC22×C89C2×C4×D4
prime radicalC2×D4×D89C2×C4
cosocleC42×C89C24
hypercenter#1583D89
hypocenterC89#17995
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
2. #26029194C2semidirect
4. #26019143C2semidirect
5. #26019140C2semidirect
6. #26019098C2semidirect
7. #26019091C2semidirect
8. #26019090C2semidirect
9. C8×#13083561C2semidirect
10. #26019051C2semidirect
11. #26019021C2semidirect
12. #26018705C2semidirect
14. #26018531C2semidirect
15. C89×#1583C2semidirect
27. #19151729C22semidirect
29. #19151728C22semidirect
31. #19151722C22semidirect
48. #19151349C22semidirect
49. #19151348C22semidirect
50. #19151348C22semidirect
51. #19151330C22semidirect
52. Q8⋊#10623494C22semidirect
53. #19151328C22semidirect
57. #19151320C22semidirect
58. C8×#10623494C22semidirect
59. C8×#10623494C22semidirect
60. C8×Q8×C89C22semidirect
63. #334×C89C22semidirect
64. Q8⋊C8×C89C22semidirect
66. #322×C89C22semidirect
167. #13083557SD16semidirect
307. #1583D89semidirect
316. C89#17995semidirect
333. C8×Q8C2×D89semidirect
355. #334C2×D89semidirect
359. Q8⋊C8C2×D89semidirect
365. #322C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260302582
#260291942
#260289882
#260191432
#260191402
#260190982
#260190912
#260190902
C8×#130835612
#260190512
#260190212
#260187052
D4⋊C8×D892
#260185312
C89×#15832
#1799589