Finite Group Databasebeta

← #36524261

Subgroups — #36524261

Order 22784.

662 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029334C211yesmaximal, characteristic, complemented
2#26029290C211yesmaximal, characteristic
3C4×SD16×D89C211yesmaximal, characteristic
4#26019156C211yesmaximal, characteristic, complemented
5#26019155C211yesmaximal, characteristic, complemented
6#26019153C211yesmaximal, characteristic, complemented
7#26019145C211yesmaximal, characteristic, complemented
8#26019142C211yesmaximal, characteristic, complemented
9#26019140C211yesmaximal, characteristic, complemented
10#26019139C211yesmaximal, characteristic, complemented
11#26018999C211yesmaximal, characteristic, complemented
12#26018645C211yesmaximal, characteristic, complemented
13#26018641C211yesmaximal, characteristic
14#26018537C211yesmaximal, characteristic, complemented
15C89×#1592C211yesmaximal, characteristic, complemented
16C2×SD16×D89C411yescharacteristic
17C4×Q8×D89C2211yescharacteristic
18C4×D4×D89C2211yescharacteristic
19#19151729C2211yescharacteristic, complemented
20#19151728C2211yescharacteristic, complemented
21#19151727—21no—
22#19151726—21no—
23#19151725C2211yescharacteristic, complemented
24#19151689C411yescharacteristic
25C4×#13083563C2211yescharacteristic
26SD16×Dic89C411yescharacteristic
27C4×#13083561C2211yescharacteristic
28#19151685C411yescharacteristic
29C4×#13083562C2211yescharacteristic
30#19151683C411yescharacteristic
31#19151677C2211yescharacteristic
32#19151676C2211yescharacteristic
33#19151675C2211yescharacteristic
34#19151674—21no—
35#19151673—21no—
36#19151654—21no—
37#19151653—21no—
38#19151650C2211yescharacteristic
39#19151650C2211yescharacteristic
40#19151649C2211yescharacteristic
41#19151649C2211yescharacteristic
42#106×D89C411yescharacteristic
43#103×D89C411yescharacteristic
44#102×D89C411yescharacteristic
45#19151561—21no—
46#19151560—21no—
47C4×C8×D89C2211yescharacteristic
48#19151351C2211yescharacteristic, complemented
49#19151348C2211yescharacteristic, complemented
50#19151348C2211yescharacteristic, complemented
51#19151330C2211yescharacteristic, complemented
52Q8⋊#10623494C2211yescharacteristic, complemented
53#19151328C2211yescharacteristic, complemented
54#19151327C2211yescharacteristic
55D4⋊#10623494C2211yescharacteristic
56#19151325C2211yescharacteristic
57#19151319C2211yescharacteristic, complemented
58C8×#10623494C2211yescharacteristic, complemented
59C8×#10623494C2211yescharacteristic, complemented
60#446×C89C2211yescharacteristic, complemented
61C4×SD16×C89C2211yescharacteristic
62#436×C89C2211yescharacteristic
63#335×C89C2211yescharacteristic, complemented
64Q8⋊C8×C89C2211yescharacteristic, complemented
65D4⋊C8×C89C2211yescharacteristic
66#322×C89C2211yescharacteristic, complemented
67C2×Q8×D89C2×C411yescharacteristic
68C2×D4×D89C2×C411yescharacteristic
69C22×C4×D89—21no—
70C2×#13083563C2×C411yescharacteristic
71C2×#13083562C2×C411yescharacteristic
72C2×#13083561C2×C411yescharacteristic
73SD16×D89—24no—
74C2×#13083558—21no—
75C2×#13083557—21no—
76C2×#13083557—21no—
77C2×C8×D89C2×C411yescharacteristic
78C2×C8×D89C2×C411yescharacteristic
79Q8×Dic89C2×C411yescharacteristic
80C4×Dic178—21no—
81C4×Dic178C2311yescharacteristic
82#15667977—21no—
83#15667977C2×C411yescharacteristic
84C4×D356C2311yescharacteristic
85D4×Dic89C2×C411yescharacteristic
86#15667974C2×C411yescharacteristic
87C4×C22⋊D89—21no—
88#15667972—21no—
89C4⋊C4×D89—21no—
90C4⋊C4×D89C2×C411yescharacteristic
91C4⋊C4×D89C2×C411yescharacteristic
92C22⋊C4×D89—21no—
93C42×D89—21no—
94C42×D89C2311yescharacteristic
95#15667944C2×C411yescharacteristic
96#15667943C2×C411yescharacteristic
97#15667943C2×C411yescharacteristic
98C4⋊#10623494—21no—
99C4⋊#10623494C2311yescharacteristic
100C4⋊#10623494C2311yescharacteristic
101#15667941—21no—
102#15667941—21no—
103#15667941—21no—
104#15667941D411yescharacteristic
105#15667941D411yescharacteristic
106#15667937C2×C411yescharacteristic
107#15667936C2×C411yescharacteristic
108#15667935C2×C411yescharacteristic
109#15667934C2×C411yescharacteristic
110#15667933C2×C411yescharacteristic
111#15667932C2×C411yescharacteristic
112C22⋊#10623494—21no—
113#15667924—21no—
114#15667924—21no—
115#15667924—21no—
116#15667924—21no—
117#15667924—21no—
118#15667924D411yescharacteristic
119#15667924D411yescharacteristic
120#15667923—21no—
121#15667922—21no—
122#15667922—21no—
123C4×#10623494C2311yescharacteristic
124C4×#10623494C2311yescharacteristic
125C4×#10623494C2311yescharacteristic
126C8×Dic89—21no—
127C8×Dic89—21no—
128C8×Dic89C2×C411yescharacteristic
129C8×Dic89C2×C411yescharacteristic
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—
141#97×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145C23×D89—21no—
146C2×Dic178—21no—
147C2×Dic178C22×C411yescharacteristic
148Q8×D89—22no—
149Q8×D89—22no—
150C2×D356C22×C411yescharacteristic
151C2×C22⋊D89—21no—
152D4×D89—22no—
153D4×D89—22no—
154C22×Dic89—21no—
155C2×C4×D89—21no—
156C2×C4×D89—21no—
157C2×C4×D89—21no—
158C2×C4×D89—21no—
159C2×C4×D89—21no—
160C2×C4×D89—81no—
161C2×C4×D89—81no—
162C2×C4×D89C22×C411yescharacteristic
163C2×C4×D89C2×D411yescharacteristic
164C2×C4×D89#4111yescharacteristic
165#13083563—22no—
166#13083562—22no—
167#13083561—22no—
168#13083558—81no—
169#13083557—81no—
170#13083557—81no—
171C2×#10623494—21no—
172C2×#10623494—21no—
173C2×#10623494—21no—
174C2×#10623494C22×C411yescharacteristic
175C2×#10623494C22×C411yescharacteristic
176C2×#10623494C2×D411yescharacteristic
177C2×#10623494C2×D411yescharacteristic
178C8×D89—22no—
179C8×D89M1614yescomplemented
180C4⋊Dic89—21no—
181C4⋊Dic89C22×C411yescharacteristic
182C4⋊Dic89C22×C411yescharacteristic
183#13083553—21no—
184#13083553—21no—
185#13083553—21no—
186#13083553—21no—
187C22⋊Dic89—21no—
188#13083551—21no—
189#13083551—21no—
190C4×Dic89—21no—
191C4×Dic89—21no—
192C4×Dic89—21no—
193C4×Dic89C22×C411yescharacteristic
194C4×Dic89C2×D411yescharacteristic
195C4×Dic89#4111yescharacteristic
196C2×Q8×C89C22×C411yescharacteristic
197C2×D4×C89C22×C411yescharacteristic
198C22×C4×C89—21no—
199SD16×C89—22no—
200M16×C89—81no—
201C2×C8×C89—21no—
202C2×C8×C89—21no—
203C2×C8×C89—21no—
204C2×C8×C89C22×C411yescharacteristic
205C2×C8×C89C22×C411yescharacteristic
206C2×C8×C89C2×D411yescharacteristic
207C2×C8×C89C2×D411yescharacteristic
208C4⋊C4×C89—21no—
209C4⋊C4×C89C22×C411yescharacteristic
210C4⋊C4×C89C22×C411yescharacteristic
211C22⋊C4×C89—21no—
212C42×C89—21no—
213C42×C89C2411yescharacteristic
214C23×C89—21no—
215C22×D89—21no—
216C22×D89—42no—
217C22×D89—81no—
218C22×D89—81no—
219C22×D89C4×D411yescharacteristic
220Q8×C89—21no—
221Q8×C89—21no—
222D4×C89—21no—
223D4×C89—21no—
224C2×C4×C89—21no—
225C2×C4×C89—21no—
226C2×C4×C89—21no—
227C2×C4×C89—41no—
228C2×C4×C89—41no—
229C2×C4×C89C23×C411yescharacteristic
230C2×C4×C89C22×D411yescharacteristic
231C2×C4×C89C2×#4111yescharacteristic
232C22⋊D89—42no—
233C2×Dic89—21no—
234C2×Dic89—21no—
235C2×Dic89—21no—
236C2×Dic89—81no—
237C2×Dic89C4×D411yescharacteristic
238C2×Dic89C4×D411yescharacteristic
239C2×Dic89C4×D411yescharacteristic
240D356—21no—
241D356—21no—
242C4×D89—21no—
243C4×D89—21no—
244C4×D89—41no—
245C4×D89—42no—
246C4×D89—41no—
247C4×D89—41no—
248C4×D89—81no—
249C4×D89—81no—
250C4×D89—81no—
251C4×D89C2×M1612yes—
252C4×D89#13112yes—
253Dic178—21no—
254Dic178—21no—
255Dic178—42no—
256C8×C89—21no—
257C8×C89—21no—
258C8×C89—21no—
259C8×C89—41no—
260C8×C89—41no—
261C8×C89—41no—
262C8×C89C2×M1612yes—
263#10623494—21no—
264#10623494—21no—
265#10623494—21no—
266#10623494—41no—
267#10623494—41no—
268#10623494—41no—
269#10623494C2×M1612yes—
270C22×C89—41no—
271C22×C89—41no—
272C22×C89—41no—
273C22×C89C2×C4×D411yescharacteristic
274C2×D89—21no—
275C2×D89—21no—
276C2×D89—41no—
277C2×D89—41no—
278C2×D89—41no—
279C2×D89—84no—
280C2×D89#43612yes—
281C4×C89—21no—
282C4×C89—41no—
283C4×C89—41no—
284C4×C89—41no—
285C4×C89C22×M1611yescharacteristic
286C4×C89C2×#13111yescharacteristic
287C4×C89#57711yescharacteristic
288C4×C89#57711yescharacteristic
289Dic89—21no—
290Dic89—21no—
291Dic89—41no—
292Dic89—41no—
293Dic89—41no—
294Dic89#43612yes—
295#17896—891nomaximal, Sylow
296C2×C89—81no—
297C2×C89C2×#43611yescharacteristic
298C2×C89#294911yescharacteristic
299C2×C89#295711yescharacteristic
300D89—22no—
301D89—81no—
302#2983—891no—
303#2975—891no—
304C2×C4×SD16—891no—
305#1592—891no—
306#1592—891no—
307#1592—891no—
308#1592D8911yescharacteristic, complemented
309#1590—891no—
310#1590—891no—
311#1590—891no—
312#1590—891no—
313#1564—891no—
314#1479—891no—
315#1478—891no—
316#1450—891no—
317C89#1789611yescharacteristic, complemented, Sylow
318C22×SD16—891no—
319C2×C4×Q8—891no—
320C2×C4×D4—891no—
321#446—891no—
322#446—891no—
323#446—891no—
324#446—1781no—
325#446—1781no—
326#446C2×D8911yescharacteristic, complemented
327C4×SD16—891no—
328C4×SD16—891no—
329C4×SD16—891no—
330C4×SD16—891no—
331C4×SD16—891no—
332C4×SD16—891no—
333C4×SD16—891no—
334C4×SD16C2×D8911yescharacteristic
335#436—891no—
336#436—891no—
337#436—891no—
338#436—1781no—
339#436—1781no—
340#436C2×D8911yescharacteristic
341#432—1781no—
342#432—1781no—
343#431—891no—
344#431—891no—
345#431—891no—
346#431—891no—
347C2×#106—891no—
348C2×#103—891no—
349C2×#102—891no—
350#407—1781no—
351#407—1781no—
352C2×C4×C8—891no—
353#335—891no—
354#335C2×D8911yescharacteristic, complemented
355#334—891no—
356#334—891no—
357Q8⋊C8—891no—
358Q8⋊C8—891no—
359Q8⋊C8—891no—
360Q8⋊C8C2×D8911yescharacteristic, complemented
361D4⋊C8—891no—
362D4⋊C8—891no—
363D4⋊C8—891no—
364D4⋊C8C2×D8911yescharacteristic
365#322—891no—
366#322C2×D8911yescharacteristic, complemented
367C82—891no—
368C82—891no—
369C22×Q8—891no—
370C22×D4—891no—
371C23×C4—1781no—
372C2×SD16—891no—
373C2×SD16—891no—
374C2×SD16—891no—
375C2×SD16—1784no—
376C2×SD16C4×D8911yescharacteristic
377C2×M16—21no—
378C2×M16—1781no—
379C2×M16—1781no—
380C2×M16—1781no—
381C22×C8—891no—
382C22×C8—891no—
383C4×Q8—891no—
384C4×Q8—891no—
385C4×Q8—891no—
386C4×Q8—1781no—
387C4×Q8—1781no—
388C4×Q8C22×D8911yescharacteristic
389C4×D4—891no—
390C4×D4—891no—
391C4×D4—891no—
392C4×D4—1781no—
393C4×D4—1781no—
394C4×D4C22×D8911yescharacteristic
395C2×C4⋊C4—891no—
396C2×C4⋊C4—891no—
397C2×C4⋊C4—1781no—
398C2×C22⋊C4—1781no—
399C2×C42—891no—
400C2×C42—1781no—
401#106—891no—
402#106—891no—
403#106—891no—
404#106C4×D8911yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—891no—
407C4⋊C8—891no—
408C4⋊C8—891no—
409C4⋊C8—891no—
410C4⋊C8—1781no—
411C4⋊C8—1781no—
412C4⋊C8—1781no—
413C4⋊C8—1781no—
414C4⋊C8C22×D8911yescharacteristic
415C4⋊C8C22×D8911yescharacteristic
416#103—891no—
417#103—891no—
418#103—891no—
419#103C4×D8911yescharacteristic
420#102—891no—
421#102—891no—
422#102—891no—
423#102C4×D8911yescharacteristic
424C22⋊C8—21no—
425C22⋊C8—891no—
426C22⋊C8—891no—
427C22⋊C8—1781no—
428C22⋊C8—1781no—
429C22⋊C8—1781no—
430C22⋊C8—1781no—
431C22⋊C8—1781no—
432C22⋊C8—1781no—
433#97—21no—
434#97—1781no—
435#97—1781no—
436#97—1781no—
437C4×C8—891no—
438C4×C8—891no—
439C4×C8—891no—
440C4×C8—891no—
441C4×C8—891no—
442C4×C8—1781no—
443C4×C8—1781no—
444C4×C8C22×D8911yescharacteristic
445C4×C8C22×D8911yescharacteristic
446C4×C8C22×D8911yescharacteristic
447C24—1781no—
448C2×Q8—891no—
449C2×Q8—1781no—
450C2×Q8—1782no—
451C2×Q8—1782no—
452C2×Q8C2×C4×D8911yescharacteristic
453C2×D4—891no—
454C2×D4—1782no—
455C2×D4—1782no—
456C2×D4—1781no—
457C2×D4C2×C4×D8911yescharacteristic
458C22×C4—21no—
459C22×C4—891no—
460C22×C4—891no—
461C22×C4—891no—
462C22×C4—1781no—
463C22×C4—1781no—
464C22×C4—1781no—
465C22×C4—1781no—
466C22×C4—1781no—
467C22×C4—1781no—
468C22×C4—7121no—
469C22×C4—7121no—
470SD16—22no—
471SD16—1782no—
472SD16—1782no—
473SD16—1782no—
474M16—81no—
475M16—7121no—
476M16—7121no—
477M16—7121no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—891no—
482C2×C8—894no—
483C2×C8—891no—
484C2×C8—891no—
485C2×C8—891no—
486C2×C8—1782no—
487C2×C8—1781no—
488C2×C8—1781no—
489C2×C8—1781no—
490C2×C8C2×C4×D8911yescharacteristic
491C2×C8C2×C4×D8911yescharacteristic
492C2×C8D4×D8911yescharacteristic
493C2×C8D4×D8911yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—891no—
496C4⋊C4—891no—
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—
508C42—21no—
509C42—891no—
510C42—891no—
511C42—891no—
512C42—1781no—
513C42—1781no—
514C42—1781no—
515C42C23×D8911yescharacteristic
516C23—21no—
517C23—891no—
518C23—3562no—
519C23—1781no—
520C23—7121no—
521C23—7121no—
522Q8—21no—
523Q8—21no—
524Q8—3562no—
525Q8—1781no—
526Q8—1781no—
527D4—21no—
528D4—21no—
529D4—3562no—
530D4—1781no—
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—3561no—
545C2×C4—3561no—
546C2×C4—1781no—
547C2×C4—1781no—
548C2×C4—1781no—
549C2×C4—1781no—
550C2×C4—1781no—
551C2×C4—7121no—
552C2×C4—7121no—
553C2×C4—7121no—
554C2×C4—7121no—
555C2×C4C22×C4×D8911yescharacteristic
556C2×C4C2×D4×D8911yescharacteristic
557C2×C4#41×D8911yescharacteristic
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—892no—
565C8—3561no—
566C8—3561no—
567C8—3561no—
568C8—1781no—
569C8—1781no—
570C8—1781no—
571C8M16×D8912yes—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—892no—
576C22—3561no—
577C22—3561no—
578C22—3561no—
579C22—1781no—
580C22—1781no—
581C22—7124no—
582C22C4×D4×D8911yescharacteristic
583C4—21no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—892no—
588C4—3561no—
589C4—3561no—
590C4—3561no—
591C4—1781no—
592C4—1781no—
593C4C2×M16×D8911yescharacteristic
594C4#131×D8911yescharacteristic
595C4#1915240711yescharacteristic
596C4#1915240811yescharacteristic
597C2—81no—
598C2—1782no—
599C2—7121no—
600C2#436×D8911yescharacteristic
601C2#2602910811yescharacteristic
602C2#2602916611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1592C2
socleC22×C89C2×C4×D4
prime radicalC2×D4×D89C2×C4
cosocleC42×C89C24
hypercenter#1592D89
hypocenterC89#17896
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1592

Complements

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

NH ≅ G/Ntype
1. #26029334C2semidirect
4. #26019156C2semidirect
5. #26019155C2semidirect
6. #26019153C2semidirect
7. #26019145C2semidirect
8. #26019142C2semidirect
9. #26019140C2semidirect
10. #26019139C2semidirect
11. #26018999C2semidirect
12. #26018645C2semidirect
14. #26018537C2semidirect
15. C89×#1592C2semidirect
19. #19151729C22semidirect
20. #19151728C22semidirect
23. #19151725C22semidirect
48. #19151351C22semidirect
49. #19151348C22semidirect
50. #19151348C22semidirect
51. #19151330C22semidirect
52. Q8⋊#10623494C22semidirect
53. #19151328C22semidirect
57. #19151319C22semidirect
58. C8×#10623494C22semidirect
59. C8×#10623494C22semidirect
60. #446×C89C22semidirect
63. #335×C89C22semidirect
64. Q8⋊C8×C89C22semidirect
66. #322×C89C22semidirect
179. C8×D89M16semidirect
308. #1592D89semidirect
317. C89#17896semidirect
326. #446C2×D89semidirect
354. #335C2×D89semidirect
360. Q8⋊C8C2×D89semidirect
366. #322C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260293342
#260292902
C4×SD16×D892
#260191562
#260191552
#260191532
#260191452
#260191422
#260191402
#260191392
#260189992
#260186452
#260186412
#260185372
C89×#15922
#1789689