Finite Group Databasebeta

← #36525837

Subgroups — #36525837

Order 22784.

598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030270C211yesmaximal, characteristic
2#26029414C211yesmaximal, characteristic, complemented
3#26019154C211yesmaximal, characteristic, complemented
4#26019135C211yesmaximal, characteristic, complemented
5#26019130C211yesmaximal, characteristic, complemented
6#26019128C211yesmaximal, characteristic, complemented
7#26019102C211yesmaximal, characteristic, complemented
8#26019079C211yesmaximal, characteristic, complemented
9#26019062C211yesmaximal, characteristic, complemented
10C8×#13083564C211yesmaximal, characteristic, complemented
11#26019028C211yesmaximal, characteristic, complemented
12#26018715C211yesmaximal, characteristic, complemented
13#26018639C211yesmaximal, characteristic, complemented
14#26018552C211yesmaximal, characteristic, complemented
15C89×#2991C211yesmaximal, characteristic, complemented
16#19152132C2211yescharacteristic
17#19151939C411yescharacteristic
18#19151922C411yescharacteristic
19#19151905C411yescharacteristic
20#19151898C411yescharacteristic
21#19151820C411yescharacteristic
22#19151813C411yescharacteristic
23#19151760C411yescharacteristic
24#19151743C411yescharacteristic
25#19151728C2211yescharacteristic
26C8×Dic178C2211yescharacteristic
27C4×#13083564C2211yescharacteristic
28C4×#13083561C2211yescharacteristic
29C4×#13083562C2211yescharacteristic
30C4×#13083559C2211yescharacteristic
31#19151677C2211yescharacteristic
32#19151666—21no—
33C8×D356C2211yescharacteristic
34#19151661—21no—
35#19151658—21no—
36#19151651C2211yescharacteristic
37#19151651C2211yescharacteristic
38#19151623—21no—
39C2×C4⋊#10623494C2211yescharacteristic
40#19151350C2211yescharacteristic
41#19151350C2211yescharacteristic
42#19151350C2211yescharacteristic
43#19151350C2211yescharacteristic
44Q8⋊#10623494C2211yescharacteristic
45Q8⋊#10623494C2211yescharacteristic
46#19151328C2211yescharacteristic
47#19151328C2211yescharacteristic
48D4⋊#10623494C2211yescharacteristic
49D4⋊#10623494C2211yescharacteristic
50#19151325C2211yescharacteristic
51#19151325C2211yescharacteristic
52#19151320C2211yescharacteristic
53#19151320C2211yescharacteristic
54#19151319C2211yescharacteristic
55C8×#10623494C2211yescharacteristic
56C4×#41×C89C2211yescharacteristic
57C8×Q8×C89C2211yescharacteristic
58C8×Q8×C89C2211yescharacteristic
59#436×C89—21no—
60#436×C89—21no—
61#436×C89C2211yescharacteristic
62#436×C89C2211yescharacteristic
63#431×C89—21no—
64#431×C89C2211yescharacteristic
65#423×C89—21no—
66#423×C89C2211yescharacteristic
67C2×#13083564C2×C411yescharacteristic
68C2×#13083562C2×C411yescharacteristic
69C2×#13083561C2×C411yescharacteristic
70C2×#13083559C2×C411yescharacteristic
71C2×#13083557—21no—
72C22×#10623494—21no—
73C2×C8×D89—21no—
74#15668006—21no—
75#15668004C2×C411yescharacteristic
76#15668000C2×C411yescharacteristic
77#15667995—21no—
78#15667991C2×C411yescharacteristic
79#15667986C2×C411yescharacteristic
80C4×Dic178C2311yescharacteristic
81C4×D356C2311yescharacteristic
82C4×C22⋊D89—21no—
83#15667969—21no—
84#15667945C2×C411yescharacteristic
85#15667945C2×C411yescharacteristic
86#15667943C2×C411yescharacteristic
87#15667943C2×C411yescharacteristic
88C4⋊#10623494—21no—
89C4⋊#10623494—21no—
90C4⋊#10623494C2×C411yescharacteristic
91C4⋊#10623494C2×C411yescharacteristic
92C4⋊#10623494C2311yescharacteristic
93C4⋊#10623494C2311yescharacteristic
94#15667941—21no—
95#15667941—21no—
96#15667936C2×C411yescharacteristic
97#15667936C2×C411yescharacteristic
98#15667935C2×C411yescharacteristic
99#15667935C2×C411yescharacteristic
100#15667933C2×C411yescharacteristic
101#15667933C2×C411yescharacteristic
102#15667932C2×C411yescharacteristic
103#15667932C2×C411yescharacteristic
104C22⋊#10623494—21no—
105C22⋊#10623494C2×C411yescharacteristic
106C22⋊#10623494C2×C411yescharacteristic
107#15667924—21no—
108#15667924—21no—
109#15667922—21no—
110C4×#10623494C2311yescharacteristic
111C4×#10623494C2311yescharacteristic
112C4×#10623494C2311yescharacteristic
113C4×#10623494C2311yescharacteristic
114C8×Dic89—21no—
115C2×#41×C89C2×C411yescharacteristic
116C2×M16×C89—21no—
117C2×M16×C89—21no—
118C2×M16×C89—21no—
119C4×Q8×C89C2×C411yescharacteristic
120C4×Q8×C89C2311yescharacteristic
121C4×D4×C89—21no—
122C4×D4×C89—21no—
123C4×D4×C89C2×C411yescharacteristic
124C4×D4×C89C2311yescharacteristic
125#117×C89—21no—
126#117×C89C2×C411yescharacteristic
127C2×C42×C89—21no—
128C2×C42×C89C2311yescharacteristic
129C4⋊C8×C89—21no—
130C4⋊C8×C89—21no—
131C4⋊C8×C89—21no—
132C4⋊C8×C89D411yescharacteristic
133C4⋊C8×C89D411yescharacteristic
134C4⋊C8×C89C2311yescharacteristic
135C4⋊C8×C89C2311yescharacteristic
136C22⋊C8×C89—21no—
137C22⋊C8×C89—21no—
138C22⋊C8×C89D411yescharacteristic
139C22⋊C8×C89D411yescharacteristic
140C4×C8×C89—21no—
141C4×C8×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C2×Dic178C22×C411yescharacteristic
145C2×D356C22×C411yescharacteristic
146C2×C22⋊D89—21no—
147C2×C4×D89—21no—
148#13083564—22no—
149#13083562—41no—
150#13083561—22no—
151#13083559—41no—
152#13083557—81no—
153C2×#10623494—21no—
154C2×#10623494—21no—
155C2×#10623494—21no—
156C2×#10623494—21no—
157C2×#10623494—81no—
158C2×#10623494C22×C411yescharacteristic
159C2×#10623494C22×C411yescharacteristic
160C2×#10623494C22×C411yescharacteristic
161C2×#10623494C22×C411yescharacteristic
162C8×D89—81no—
163C4⋊Dic89C22×C411yescharacteristic
164C4⋊Dic89C22×C411yescharacteristic
165#13083553—21no—
166#13083553—21no—
167#13083553—21no—
168#13083553—21no—
169C22⋊Dic89—21no—
170#13083551—21no—
171#13083551—21no—
172#13083551—21no—
173#13083551—21no—
174C4×Dic89—21no—
175#41×C89—22no—
176#41×C89—41no—
177C2×Q8×C89C22×C411yescharacteristic
178C2×D4×C89—21no—
179C2×D4×C89C22×C411yescharacteristic
180C22×C4×C89—21no—
181C22×C4×C89—21no—
182C22×C4×C89—21no—
183C22×C4×C89C22×C411yescharacteristic
184C22×C4×C89C2×D411yescharacteristic
185C22×C4×C89#4111yescharacteristic
186M16×C89—81no—
187M16×C89—81no—
188M16×C89—81no—
189C2×C8×C89—21no—
190C2×C8×C89—21no—
191C2×C8×C89C2×D411yescharacteristic
192C2×C8×C89C2×D411yescharacteristic
193C2×C8×C89C2×D411yescharacteristic
194C2×C8×C89C2×D411yescharacteristic
195C4⋊C4×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89C22×C411yescharacteristic
198C4⋊C4×C89C22×C411yescharacteristic
199C22⋊C4×C89—21no—
200C22⋊C4×C89—21no—
201C22⋊C4×C89—21no—
202C42×C89—21no—
203C42×C89—21no—
204C42×C89—21no—
205C42×C89C22×C411yescharacteristic
206C42×C89C2×D411yescharacteristic
207C42×C89#4111yescharacteristic
208C42×C89C2411yescharacteristic
209C23×C89—21no—
210C23×C89C4×D411yescharacteristic
211C22×D89—21no—
212Q8×C89—21no—
213Q8×C89#13112yes—
214D4×C89—21no—
215D4×C89—21no—
216D4×C89—42no—
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—41no—
225C2×C4×C89—42no—
226C2×C4×C89—42no—
227C2×C4×C89—41no—
228C2×C4×C89—41no—
229C2×C4×C89C4×D411yescharacteristic
230C2×C4×C89C4×D411yescharacteristic
231C2×C4×C89C4×D411yescharacteristic
232C2×C4×C89C23×C411yescharacteristic
233C2×C4×C89C22×D411yescharacteristic
234C2×C4×C89C2×#4111yescharacteristic
235C22⋊D89—42no—
236C2×Dic89—21no—
237C2×Dic89—21no—
238C2×Dic89—21no—
239D356—21no—
240D356—21no—
241C4×D89—41no—
242C4×D89—41no—
243Dic178—21no—
244Dic178—21no—
245C8×C89—21no—
246C8×C89—21no—
247C8×C89—21no—
248C8×C89—21no—
249C8×C89—41no—
250C8×C89—41no—
251#10623494—21no—
252#10623494—21no—
253#10623494—21no—
254#10623494—21no—
255#10623494—41no—
256#10623494—41no—
257#10623494—41no—
258#10623494—41no—
259C22×C89—21no—
260C22×C89—21no—
261C22×C89—21no—
262C22×C89—41no—
263C22×C89—41no—
264C22×C89—41no—
265C22×C89C2×C4×D411yescharacteristic
266C2×D89—41no—
267C2×D89—41no—
268C2×D89—41no—
269C4×C89—21no—
270C4×C89—22no—
271C4×C89—21no—
272C4×C89—21no—
273C4×C89—21no—
274C4×C89—41no—
275C4×C89—41no—
276C4×C89C2×#13111yescharacteristic
277C4×C89#56811yescharacteristic
278C4×C89#57611yescharacteristic
279C4×C89#57811yescharacteristic
280Dic89—41no—
281Dic89—41no—
282Dic89—41no—
283#18033—891nomaximal, Sylow
284C2×C89—41no—
285C2×C89—81no—
286C2×C89#293311yescharacteristic
287C2×C89#295111yescharacteristic
288C2×C89#295511yescharacteristic
289D89—81no—
290#3102—891no—
291#2994—891no—
292#2991D8911yescharacteristic, complemented
293#1592—891no—
294#1589—891no—
295#1588—891no—
296#1588—891no—
297#1584—891no—
298#1581—891no—
299SD16⋊C8—891no—
300C8×Q16—891no—
301#1571—891no—
302#1495—891no—
303#1477—891no—
304#1454—891no—
305C89#1803311yescharacteristic, complemented, Sylow
306#529—891no—
307C4×#41C2×D8911yescharacteristic
308#485—891no—
309#482—891no—
310#479—891no—
311#478—891no—
312#464—891no—
313#463—891no—
314#452—891no—
315#449—891no—
316#446—891no—
317C8×Q8—891no—
318C8×Q8C2×D8911yescharacteristic
319C8×Q8C2×D8911yescharacteristic
320C4×Q16—891no—
321C4×SD16—891no—
322C4×SD16—891no—
323C4×D8—891no—
324#436—21no—
325#436—21no—
326#436—891no—
327#436C2×D8911yescharacteristic
328#436C2×D8911yescharacteristic
329#435—1781no—
330C8×D4—891no—
331C8×D4—1781no—
332#433—1781no—
333#431—21no—
334#431—891no—
335#431—891no—
336#431C2×D8911yescharacteristic
337#424—1781no—
338#423—21no—
339#423C2×D8911yescharacteristic
340C2×C4⋊C8—891no—
341#335—891no—
342#335—891no—
343#335—891no—
344#335—891no—
345Q8⋊C8—891no—
346Q8⋊C8—891no—
347Q8⋊C8—891no—
348Q8⋊C8—891no—
349D4⋊C8—891no—
350D4⋊C8—891no—
351D4⋊C8—891no—
352D4⋊C8—891no—
353#322—891no—
354#322—891no—
355#322—891no—
356C82—891no—
357C2×#41C4×D8911yescharacteristic
358C2×Q16—891no—
359C2×SD16—891no—
360C2×SD16—891no—
361C2×D8—891no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—1781no—
366C22×C8—1781no—
367C22×C8—1781no—
368#126—1781no—
369#125—891no—
370#124—891no—
371#123—1781no—
372C22⋊Q8—891no—
373#121—891no—
374C4×Q8—891no—
375C4×Q8C4×D8911yescharacteristic
376C4×Q8C22×D8911yescharacteristic
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—891no—
380C4×D4—1781no—
381C4×D4C4×D8911yescharacteristic
382C4×D4C22×D8911yescharacteristic
383#117—21no—
384#117—1781no—
385#117C4×D8911yescharacteristic
386C2×C42—21no—
387C2×C42C22×D8911yescharacteristic
388#107—891no—
389#107—891no—
390#106—891no—
391#106—891no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—891no—
396C4⋊C8—891no—
397C4⋊C8—891no—
398C4⋊C8—891no—
399C4⋊C8—1781no—
400C4⋊C8—1781no—
401C4⋊C8—1781no—
402C4⋊C8—1781no—
403C4⋊C8C22⋊D8911yescharacteristic
404C4⋊C8C22⋊D8911yescharacteristic
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—1781no—
420C22⋊C8—1781no—
421C22⋊C8—1781no—
422C22⋊C8C22⋊D8911yescharacteristic
423C22⋊C8C22⋊D8911yescharacteristic
424#97—1781no—
425C4×C8—21no—
426C4×C8—21no—
427C4×C8—891no—
428C4×C8—891no—
429C4×C8—891no—
430C4×C8—891no—
431C4×C8—1781no—
432C4×C8C22×D8911yescharacteristic
433C4×C8C22×D8911yescharacteristic
434#41—22no—
435#41—41no—
436C2×Q8—891no—
437C2×Q8C2×C4×D8911yescharacteristic
438C2×D4—21no—
439C2×D4—891no—
440C2×D4—1781no—
441C2×D4C2×C4×D8911yescharacteristic
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—1781no—
446C22×C4C2×C4×D8911yescharacteristic
447C22×C4C2×C22⋊D8911yescharacteristic
448C22×C4#1308357411yescharacteristic
449Q16—1782no—
450SD16—3561no—
451SD16—1782no—
452D8—3561no—
453M16—81no—
454M16—81no—
455M16—81no—
456M16—7121no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—891no—
460C2×C8—891no—
461C2×C8—891no—
462C2×C8—891no—
463C2×C8—1781no—
464C2×C8—1781no—
465C2×C8—1781no—
466C2×C8—1781no—
467C2×C8—7121no—
468C2×C8—7121no—
469C2×C8C2×C22⋊D8911yescharacteristic
470C2×C8C2×C22⋊D8911yescharacteristic
471C2×C8C2×C22⋊D8911yescharacteristic
472C2×C8C2×C22⋊D8911yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—891no—
476C4⋊C4—891no—
477C4⋊C4—1781no—
478C4⋊C4—1781no—
479C4⋊C4—1781no—
480C4⋊C4—1781no—
481C4⋊C4C2×C4×D8911yescharacteristic
482C4⋊C4C2×C4×D8911yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—1781no—
487C22⋊C4—1781no—
488C22⋊C4—1781no—
489C22⋊C4—1781no—
490C22⋊C4—1781no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—1781no—
495C42C2×C4×D8911yescharacteristic
496C42C2×C22⋊D8911yescharacteristic
497C42#1308357411yescharacteristic
498C42C23×D8911yescharacteristic
499C23—21no—
500C23—1781no—
501C23C4×C22⋊D8911yescharacteristic
502Q8—21no—
503Q8—1781no—
504Q8—1781no—
505Q8#1566802012yes—
506D4—21no—
507D4—21no—
508D4—42no—
509D4—3562no—
510D4—1781no—
511D4—1781no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—3561no—
525C2×C4—3561no—
526C2×C4—1781no—
527C2×C4—1781no—
528C2×C4—1781no—
529C2×C4C4×C22⋊D8911yescharacteristic
530C2×C4C4×C22⋊D8911yescharacteristic
531C2×C4C4×C22⋊D8911yescharacteristic
532C2×C4C22×C4×D8911yescharacteristic
533C2×C4C22×C22⋊D8911yescharacteristic
534C2×C4C2×#1308357411yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—3561no—
542C8—3561no—
543C8—3561no—
544C8—3561no—
545C8—1781no—
546C8—1781no—
547C8—1781no—
548C8—1781no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—3561no—
556C22—3561no—
557C22—3561no—
558C22C2×C4×C22⋊D8911yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—3561no—
567C4—3561no—
568C4—3561no—
569C4C2×#1566802011yescharacteristic
570C4#1915236411yescharacteristic
571C4#1915240111yescharacteristic
572C4#1915241311yescharacteristic
573C2—41no—
574C2—81no—
575C2—7121no—
576C2#2602897711yescharacteristic
577C2#2602912511yescharacteristic
578C2#2602915311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2991C2
socleC22×C89C2×C4×D4
prime radical#19151743C4
cosocleC42×C89C24
hypercenter#2991D89
hypocenterC89#18033
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2991

Complements

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

NH ≅ G/Ntype
2. #26029414C2semidirect
3. #26019154C2semidirect
4. #26019135C2semidirect
5. #26019130C2semidirect
6. #26019128C2semidirect
7. #26019102C2semidirect
8. #26019079C2semidirect
9. #26019062C2semidirect
10. C8×#13083564C2semidirect
11. #26019028C2semidirect
12. #26018715C2semidirect
13. #26018639C2semidirect
14. #26018552C2semidirect
15. C89×#2991C2semidirect
292. #2991D89semidirect
305. C89#18033semidirect

Maximal subgroups

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

subgroupindex
#260302702
#260294142
#260191542
#260191352
#260191302
#260191282
#260191022
#260190792
#260190622
C8×#130835642
#260190282
#260187152
#260186392
#260185522
C89×#29912
#1803389