Finite Group Databasebeta

← #36565010

Subgroups — #36565010

Order 22784.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151656C211yesmaximal, characteristic
2#26022876C211yesmaximal, characteristic
3#26022859C211yesmaximal, characteristic
4#26021023C211yesmaximal, characteristic, complemented
5#26021021C211yesmaximal, characteristic, complemented
6#26021014C211yesmaximal, characteristic, complemented
7#26021012C211yesmaximal, characteristic, complemented
8#26020453C211yesmaximal, characteristic
9#26020449C211yesmaximal, characteristic
10#26020445C211yesmaximal, characteristic
11#26020417C211yesmaximal, characteristic, complemented
12#26020407C211yesmaximal, characteristic, complemented
13#26020384C211yesmaximal, characteristic, complemented
14#26020381C211yesmaximal, characteristic, complemented
15C89×#1853C211yesmaximal, characteristic
16C2×#15667971C2211yescharacteristic
17C2×#15667971C2211yescharacteristic
18#19151656C418yescomplemented
19C2×#15667946C2211yescharacteristic
20C2×#15667945C2211yescharacteristic
21C2×#15667944C2211yescharacteristic
22C2×#15667943C2211yescharacteristic
23C2×#15667943C2211yescharacteristic
24C2×#15667943C2211yescharacteristic
25C2×#15667941C2211yescharacteristic
26C2×#15667941C2211yescharacteristic
27C2×C22⋊#10623494C2211yescharacteristic
28C2×#15667922C2211yescharacteristic
29C2×C8×Dic89C2211yescharacteristic
30#19151494—21no—
31#19151494—21no—
32#19151492—21no—
33#19151486—21no—
34#19151483—21no—
35#19151481—21no—
36#19151471—21no—
37#19151471—21no—
38#19151467C2211yescharacteristic
39#19151467C2211yescharacteristic
40#19151467C2211yescharacteristic
41#19151463C2211yescharacteristic
42#19151463C2211yescharacteristic
43#19151463C2211yescharacteristic
44#19151458C2211yescharacteristic
45C4⋊C4×Dic89C2211yescharacteristic
46C4⋊C4×Dic89C2211yescharacteristic
47#19151361C2211yescharacteristic
48#19151361C2211yescharacteristic
49#19151360C2211yescharacteristic
50#19151360C2211yescharacteristic
51#19151360C2211yescharacteristic
52#19151360C2211yescharacteristic
53C22×C4⋊C4×C89C2211yescharacteristic
54C2×#106×C89C2211yescharacteristic
55C2×#106×C89C2211yescharacteristic
56C2×C22⋊C8×C89C2211yescharacteristic
57#380×C89C2211yescharacteristic
58#340×C89C2211yescharacteristic
59#340×C89C2211yescharacteristic
60C22×#10623494C2311yescharacteristic
61C22×#10623494C2311yescharacteristic
62#15667971—41no—
63#15667971—41no—
64#15667971C2×C414yes—
65#15667971Q814yes—
66C2×C4⋊Dic89D411yescharacteristic
67C2×C4⋊Dic89D411yescharacteristic
68C2×C4⋊Dic89C2311yescharacteristic
69C2×C4⋊Dic89C2311yescharacteristic
70C2×C4⋊Dic89C2311yescharacteristic
71C2×C4⋊Dic89C2311yescharacteristic
72C2×#13083553—21no—
73C2×#13083553—21no—
74C2×#13083553—21no—
75C2×#13083553—21no—
76C2×#13083553—21no—
77C2×#13083553—21no—
78C2×C22⋊Dic89—21no—
79C2×C22⋊Dic89—21no—
80C2×C22⋊Dic89D411yescharacteristic
81C2×C22⋊Dic89D411yescharacteristic
82C2×C4×Dic89—21no—
83C2×C4×Dic89—21no—
84C2×C4×Dic89—21no—
85C2×C4×Dic89C2311yescharacteristic
86C2×C4×Dic89C2311yescharacteristic
87#15667946—41no—
88#15667945—41no—
89#15667944—41no—
90#15667943—41no—
91#15667943—41no—
92#15667943—41no—
93#15667941C2×C414yescomplemented
94#15667941C2×C414yescomplemented
95C22⋊#10623494C2×C414yes—
96#15667922C2×C414yescomplemented
97C8×Dic89C2×C414yescomplemented
98#15667919—21no—
99#15667919—21no—
100#15667919—21no—
101#15667919—21no—
102#15667919—21no—
103#15667919—21no—
104#15667919—21no—
105#15667919—21no—
106#15667919—21no—
107C23×C4×C89—21no—
108C23×C4×C89C2311yescharacteristic
109C22×C8×C89C2311yescharacteristic
110C22×C8×C89C2311yescharacteristic
111C2×C4⋊C4×C89—21no—
112C2×C4⋊C4×C89—41no—
113C2×C4⋊C4×C89—41no—
114C2×C4⋊C4×C89C2311yescharacteristic
115C2×C4⋊C4×C89C2311yescharacteristic
116C2×C4⋊C4×C89C2311yescharacteristic
117C2×C4⋊C4×C89C2311yescharacteristic
118C2×C22⋊C4×C89—21no—
119#106×C89—41no—
120#106×C89—41no—
121C22⋊C8×C89C2×C414yes—
122#95×C89—21no—
123C22×Dic89—21no—
124C22×Dic89—21no—
125C22×Dic89—21no—
126C22×Dic89—21no—
127C22×Dic89C2×D411yescharacteristic
128C22×Dic89C2×D411yescharacteristic
129C22×Dic89C2×D411yescharacteristic
130C22×Dic89C2×D411yescharacteristic
131C2×#10623494—21no—
132C2×#10623494—21no—
133C2×#10623494—21no—
134C2×#10623494—21no—
135C2×#10623494C22×C412yes—
136C2×#10623494C22×C412yes—
137C4⋊Dic89—21no—
138C4⋊Dic89—21no—
139C4⋊Dic89—21no—
140C4⋊Dic89—21no—
141C4⋊Dic89—21no—
142C4⋊Dic89—21no—
143C4⋊Dic89—21no—
144C4⋊Dic89—21no—
145C4⋊Dic89C4⋊C414yes—
146C4⋊Dic89C4⋊C414yes—
147#13083553—81no—
148#13083553—81no—
149#13083553—81no—
150#13083553—81no—
151#13083553—81no—
152#13083553—81no—
153C22⋊Dic89—42no—
154C22⋊Dic89—81no—
155C22⋊Dic89C4⋊C414yes—
156C22⋊Dic89C4⋊C414yes—
157C4×Dic89—42no—
158C4×Dic89—81no—
159C4×Dic89—81no—
160C4×Dic89C22×C412yes—
161C4×Dic89C22×C412yes—
162C4×Dic89C2×Q812yes—
163C4×Dic89C2×Q812yes—
164C24×C89C2×D411yescharacteristic
165C22×C4×C89—21no—
166C22×C4×C89—21no—
167C22×C4×C89—21no—
168C22×C4×C89—21no—
169C22×C4×C89—21no—
170C22×C4×C89—21no—
171C22×C4×C89—21no—
172C22×C4×C89—21no—
173C22×C4×C89—42no—
174C22×C4×C89—81no—
175C22×C4×C89—81no—
176C22×C4×C89C22×C412yes—
177C22×C4×C89C2×D411yescharacteristic
178C22×C4×C89C2×Q812yes—
179C22×C4×C89C2411yescharacteristic
180C2×C8×C89—21no—
181C2×C8×C89—21no—
182C2×C8×C89—21no—
183C2×C8×C89—21no—
184C2×C8×C89C22×C412yes—
185C2×C8×C89C22×C412yes—
186C4⋊C4×C89—21no—
187C4⋊C4×C89—21no—
188C4⋊C4×C89—21no—
189C4⋊C4×C89—21no—
190C4⋊C4×C89—21no—
191C4⋊C4×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—42no—
195C22⋊C4×C89—81no—
196C23×C89—21no—
197C23×C89—21no—
198C23×C89—21no—
199C23×C89—21no—
200C23×C89—21no—
201C23×C89—21no—
202C23×C89C2×C4⋊C412yes—
203C23×C89C22×D411yescharacteristic
204C2×C4×C89—21no—
205C2×C4×C89—21no—
206C2×C4×C89—22no—
207C2×C4×C89—22no—
208C2×C4×C89—22no—
209C2×C4×C89—22no—
210C2×C4×C89—41no—
211C2×C4×C89—41no—
212C2×C4×C89—41no—
213C2×C4×C89—41no—
214C2×C4×C89—41no—
215C2×C4×C89—41no—
216C2×C4×C89—41no—
217C2×C4×C89—41no—
218C2×C4×C89—41no—
219C2×C4×C89—41no—
220C2×C4×C89—41no—
221C2×C4×C89—41no—
222C2×C4×C89—84no—
223C2×C4×C89C2×C4⋊C412yes—
224C2×C4×C89C23×C411yescharacteristic
225C2×C4×C89C22×Q811yescharacteristic
226C2×C4×C89H(5,𝔽2)11yescharacteristic
227C2×C4×C89H(5,𝔽2)11yescharacteristic
228C2×C4×C892−1+411yescharacteristic
229C2×C4×C892−1+411yescharacteristic
230C2×Dic89—21no—
231C2×Dic89—21no—
232C2×Dic89—21no—
233C2×Dic89—21no—
234C2×Dic89—21no—
235C2×Dic89—21no—
236C2×Dic89—21no—
237C2×Dic89—21no—
238C2×Dic89—41no—
239C2×Dic89—41no—
240C2×Dic89—41no—
241C2×Dic89—41no—
242C2×Dic89—41no—
243C2×Dic89—41no—
244C2×Dic89—41no—
245C2×Dic89—41no—
246C2×Dic89—41no—
247C2×Dic89—41no—
248C2×Dic89—41no—
249C2×Dic89—41no—
250C2×Dic89C2×C4⋊C412yes—
251C2×Dic89C2×C4⋊C412yes—
252C2×Dic89C2×C4⋊C412yes—
253C2×Dic89C2×C4⋊C412yes—
254C8×C89—22no—
255C8×C89—22no—
256#10623494—22no—
257#10623494—22no—
258C22×C89—22no—
259C22×C89—22no—
260C22×C89—22no—
261C22×C89—41no—
262C22×C89—41no—
263C22×C89—41no—
264C22×C89—41no—
265C22×C89C22×C4⋊C411yescharacteristic
266C22×C89#53511yescharacteristic
267C22×C89#53611yescharacteristic
268C22×C89#57711yescharacteristic
269C22×C89#57711yescharacteristic
270C22×C89#57711yescharacteristic
271C22×C89#57711yescharacteristic
272C4×C89—22no—
273C4×C89—81no—
274C4×C89—81no—
275C4×C89—81no—
276C4×C89—81no—
277C4×C89#52011yescharacteristic
278C4×C89#52011yescharacteristic
279C4×C89#54411yescharacteristic
280C4×C89#54411yescharacteristic
281Dic89—22no—
282Dic89—22no—
283Dic89—22no—
284Dic89—22no—
285Dic89—81no—
286Dic89—81no—
287Dic89—81no—
288Dic89—81no—
289#21990—891nomaximal, Sylow
290C2×C89—42no—
291C2×C89#231711yescharacteristic
292C2×C89#291211yescharacteristic
293C2×C89#291211yescharacteristic
294C2×C89#322111yescharacteristic
295C2×C89#322211yescharacteristic
296C2×C89#324411yescharacteristic
297C2×C89#324411yescharacteristic
298C2×#433—891no—
299#2317—891no—
300#2314—891no—
301#1949—891no—
302#1949—891no—
303#1948—891no—
304#1948—891no—
305#1854—891no—
306#1854—891no—
307#1853—891no—
308#1853D8911yescharacteristic
309#1847—891no—
310#1844—891no—
311#1836—891no—
312#1835—891no—
313C89#2199011yescharacteristic, complemented, Sylow
314C2×#117—891no—
315C2×#117—891no—
316C22×C4⋊C4C2×D8911yescharacteristic
317#433—898no—
318C2×#107—891no—
319C2×#107—891no—
320C2×#106—891no—
321C2×#106—891no—
322C2×#106—891no—
323C2×#106—891no—
324C2×#106C2×D8911yescharacteristic
325C2×#106C2×D8911yescharacteristic
326C2×C4⋊C8—891no—
327C2×C4⋊C8—891no—
328C2×C22⋊C8—891no—
329C2×C22⋊C8C2×D8911yescharacteristic
330C2×#97—891no—
331C2×C4×C8—891no—
332#389—1781no—
333#389—1781no—
334#388—1781no—
335#388—1781no—
336#387—1781no—
337#387—1781no—
338#385—1781no—
339#385—1781no—
340#384—891no—
341#384—891no—
342#384—891no—
343#382—891no—
344#382—891no—
345#382—891no—
346#380—891no—
347#380C2×D8911yescharacteristic
348C4×C4⋊C4—891no—
349C4×C4⋊C4—891no—
350#340—891no—
351#340—891no—
352#340—891no—
353#340—891no—
354#340—891no—
355#340—891no—
356#340C2×D8911yescharacteristic
357#340C2×D8911yescharacteristic
358C23×C4—21no—
359C23×C4C22×D8911yescharacteristic
360C22×C8—891no—
361C22×C8—891no—
362C22×C8C22×D8911yescharacteristic
363C22×C8C22×D8911yescharacteristic
364#117—894no—
365#117—894no—
366#117—3561no—
367#117—3561no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4—41no—
371C2×C4⋊C4—891no—
372C2×C4⋊C4—891no—
373C2×C4⋊C4—891no—
374C2×C4⋊C4—891no—
375C2×C4⋊C4—891no—
376C2×C4⋊C4—891no—
377C2×C4⋊C4—1781no—
378C2×C4⋊C4—1781no—
379C2×C4⋊C4—1781no—
380C2×C4⋊C4—1781no—
381C2×C4⋊C4—1781no—
382C2×C4⋊C4—1781no—
383C2×C4⋊C4C22×D8911yescharacteristic
384C2×C4⋊C4C22×D8911yescharacteristic
385C2×C4⋊C4C22×D8911yescharacteristic
386C2×C4⋊C4C22×D8911yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—891no—
389C2×C22⋊C4—891no—
390C2×C22⋊C4—1781no—
391C2×C22⋊C4—1781no—
392C2×C42—891no—
393C2×C42—891no—
394C2×C42—1781no—
395C2×C42—1781no—
396C2×C42—1781no—
397#107—3561no—
398#107—3561no—
399#106—41no—
400#106—41no—
401#106—3561no—
402#106—3561no—
403#106—3561no—
404#106—3561no—
405C4⋊C8—894no—
406C4⋊C8—894no—
407C22⋊C8—894no—
408C22⋊C8C4×D8914yes—
409#97—894no—
410C4×C8—894no—
411#95—21no—
412#95—1781no—
413#95—1781no—
414#95—1781no—
415#95—1781no—
416#95—1781no—
417#95—1781no—
418#95—1781no—
419#95—1781no—
420#95—1781no—
421C24D4×D8911yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—891no—
434C22×C4—891no—
435C22×C4—891no—
436C22×C4—891no—
437C22×C4—1781no—
438C22×C4—1781no—
439C22×C4—1781no—
440C22×C4—1781no—
441C22×C4C2×C4×D8912yes—
442C22×C4D4×D8911yescharacteristic
443C22×C4Q8×D8912yes—
444C22×C4C23×D8911yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—892no—
450C2×C8—892no—
451C2×C8—1781no—
452C2×C8—1781no—
453C2×C8—1781no—
454C2×C8—1781no—
455C2×C8C2×C4×D8912yes—
456C2×C8C2×C4×D8912yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—894no—
467C4⋊C4—894no—
468C4⋊C4—1781no—
469C4⋊C4—1781no—
470C4⋊C4—1781no—
471C4⋊C4—1781no—
472C4⋊C4—1781no—
473C4⋊C4—1781no—
474C4⋊C4—1781no—
475C4⋊C4—1781no—
476C4⋊C4—7121no—
477C4⋊C4—7121no—
478C4⋊C4—7121no—
479C4⋊C4—7121no—
480C4⋊C4—7121no—
481C4⋊C4—7121no—
482C22⋊C4—81no—
483C22⋊C4—894no—
484C22⋊C4—894no—
485C22⋊C4—3562no—
486C22⋊C4—7121no—
487C42—892no—
488C42—892no—
489C42—892no—
490C42—892no—
491C42—3562no—
492C42—7121no—
493C42—7121no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C4⋊C4×D8912yes—
501C23C2×D4×D8911yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—84no—
521C2×C4—892no—
522C2×C4—892no—
523C2×C4—892no—
524C2×C4—892no—
525C2×C4—3561no—
526C2×C4—3561no—
527C2×C4—3561no—
528C2×C4—3561no—
529C2×C4—3561no—
530C2×C4—3561no—
531C2×C4—3561no—
532C2×C4—3561no—
533C2×C4—3561no—
534C2×C4—3561no—
535C2×C4—3561no—
536C2×C4—3561no—
537C2×C4—1781no—
538C2×C4—1781no—
539C2×C4—1781no—
540C2×C4—1781no—
541C2×C4—1781no—
542C2×C4—1781no—
543C2×C4—1781no—
544C2×C4—1781no—
545C2×C4C4⋊C4×D8912yes—
546C2×C4C22×C4×D8911yescharacteristic
547C2×C4C2×Q8×D8911yescharacteristic
548C2×C4#1566805711yescharacteristic
549C2×C4#1566805711yescharacteristic
550C2×C4#1566805911yescharacteristic
551C2×C4#1566805911yescharacteristic
552C8—22no—
553C8—22no—
554C8—1782no—
555C8—1782no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C2×C4⋊C4×D8911yescharacteristic
564C22#1915216311yescharacteristic
565C22#1915217011yescharacteristic
566C22#1915240411yescharacteristic
567C22#1915240411yescharacteristic
568C22#1915240711yescharacteristic
569C22#1915240811yescharacteristic
570C4—22no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—1782no—
576C4—1782no—
577C4—1782no—
578C4—1782no—
579C4—7121no—
580C4—7121no—
581C4—7121no—
582C4—7121no—
583C4#1915207711yescharacteristic
584C4#1915207711yescharacteristic
585C4#1915222711yescharacteristic
586C4#1915222711yescharacteristic
587C2—42no—
588C2#2602287111yescharacteristic
589C2#2602884311yescharacteristic
590C2#2602884611yescharacteristic
591C2#2603130911yescharacteristic
592C2#2603132011yescharacteristic
593C2#2603154611yescharacteristic
594C2#2603155411yescharacteristic
subgroupG/H
centerC23C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC89×#1853C2
socleC23×C89C22×D4
prime radicalC24×C89C2×D4
cosocleC22×C4×C89C24
hypercenter#1853D89
hypocenterC89#21990
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #26021023C2semidirect
5. #26021021C2semidirect
6. #26021014C2semidirect
7. #26021012C2semidirect
11. #26020417C2semidirect
12. #26020407C2semidirect
13. #26020384C2semidirect
14. #26020381C2semidirect
18. #19151656C4semidirect
93. #15667941C2×C4semidirect
94. #15667941C2×C4semidirect
96. #15667922C2×C4semidirect
97. C8×Dic89C2×C4semidirect
313. C89#21990semidirect

Maximal subgroups

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

subgroupindex
C2×#191516562
#260228762
#260228592
#260210232
#260210212
#260210142
#260210122
#260204532
#260204492
#260204452
#260204172
#260204072
#260203842
#260203812
C89×#18532
#2199089