Finite Group Databasebeta

← #36525254

Subgroups — #36525254

Order 22784.

566 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030859C211yesmaximal, characteristic
2#26029346C211yesmaximal, characteristic, complemented
3#26019128C211yesmaximal, characteristic, complemented
4#26019123C211yesmaximal, characteristic, complemented
5#26019121C211yesmaximal, characteristic, complemented
6#26019109C211yesmaximal, characteristic, complemented
7#26019102C211yesmaximal, characteristic, complemented
8#26019100C211yesmaximal, characteristic, complemented
9#26019093C211yesmaximal, characteristic, complemented
10#26019088C211yesmaximal, characteristic, complemented
11#26019017C211yesmaximal, characteristic, complemented
12#26018712C211yesmaximal, characteristic
13#26018706C211yesmaximal, characteristic, complemented
14#26018542C211yesmaximal, characteristic, complemented
15C89×#2993C211yesmaximal, characteristic
16#19152155C2211yescharacteristic
17#19152008C411yescharacteristic
18#19151993C411yescharacteristic
19#19151968C411yescharacteristic
20#19151957C411yescharacteristic
21#19151854C411yescharacteristic
22#19151854C411yescharacteristic
23#19151842C411yescharacteristic
24#19151842C411yescharacteristic
25#19151728C2211yescharacteristic
26#19151728C2211yescharacteristic
27#19151727—21no—
28#19151727—21no—
29C8×Dic178C2211yescharacteristic
30C8×Dic178C2211yescharacteristic
31C4×#13083564C2211yescharacteristic
32C4×#13083564C2211yescharacteristic
33C4×#13083562C2211yescharacteristic
34C4×#13083562C2211yescharacteristic
35#19151656—21no—
36#19151656—21no—
37#19151655C2211yescharacteristic
38C4×#13083558C2211yescharacteristic
39C2×C4×#10623494C2211yescharacteristic
40#19151350C2211yescharacteristic
41#19151350C2211yescharacteristic
42#19151348C2211yescharacteristic
43#19151348C2211yescharacteristic
44Q8⋊#10623494C2211yescharacteristic
45Q8⋊#10623494C2211yescharacteristic
46#19151328C2211yescharacteristic
47#19151328C2211yescharacteristic
48#19151328C2211yescharacteristic
49#19151328C2211yescharacteristic
50D4⋊#10623494C2211yescharacteristic
51D4⋊#10623494C2211yescharacteristic
52#19151320C2211yescharacteristic
53#19151320C2211yescharacteristic
54#19151319C2211yescharacteristic
55#19151319C2211yescharacteristic
56#529×C89C2211yescharacteristic
57#446×C89C2211yescharacteristic
58C8×Q8×C89C2211yescharacteristic
59#436×C89—21no—
60#436×C89C2211yescharacteristic
61#435×C89—21no—
62C8×D4×C89C2211yescharacteristic
63#433×C89—21no—
64#431×C89C2211yescharacteristic
65#424×C89—21no—
66#423×C89C2211yescharacteristic
67C2×#13083564C2×C411yescharacteristic
68C2×#13083564C2×C411yescharacteristic
69C2×#13083562C2×C411yescharacteristic
70C2×#13083562C2×C411yescharacteristic
71C2×#13083558C2×C411yescharacteristic
72C2×#13083558C2×C411yescharacteristic
73C22×#10623494—21no—
74#15668012C2×C411yescharacteristic
75#15668012C2×C411yescharacteristic
76#15668004—21no—
77#15667991—21no—
78#15667991C2×C411yescharacteristic
79#15667991C2×C411yescharacteristic
80C4×Dic178—21no—
81C4×Dic178C2311yescharacteristic
82C4×Dic178C2311yescharacteristic
83#15667971—21no—
84#15667945C2×C411yescharacteristic
85#15667945C2×C411yescharacteristic
86#15667943C2×C411yescharacteristic
87#15667943C2×C411yescharacteristic
88C4⋊#10623494C2311yescharacteristic
89C4⋊#10623494C2311yescharacteristic
90#15667941—21no—
91#15667941—21no—
92#15667941—21no—
93#15667941—21no—
94#15667936C2×C411yescharacteristic
95#15667936C2×C411yescharacteristic
96#15667935C2×C411yescharacteristic
97#15667935C2×C411yescharacteristic
98#15667935C2×C411yescharacteristic
99#15667935C2×C411yescharacteristic
100#15667933C2×C411yescharacteristic
101#15667933C2×C411yescharacteristic
102C22⋊#10623494—21no—
103#15667923—21no—
104#15667923C2×C411yescharacteristic
105#15667923C2×C411yescharacteristic
106#15667922—21no—
107#15667922—21no—
108C4×#10623494—21no—
109C4×#10623494C2311yescharacteristic
110C4×#10623494C2311yescharacteristic
111C4×#10623494C2311yescharacteristic
112C4×#10623494C2311yescharacteristic
113C8×Dic89—21no—
114C8×Dic89—21no—
115C2×M16×C89—21no—
116C2×M16×C89—21no—
117C22×C8×C89—21no—
118#126×C89—21no—
119#125×C89C2×C411yescharacteristic
120#124×C89C2×C411yescharacteristic
121#123×C89—21no—
122C22⋊Q8×C89C2×C411yescharacteristic
123#121×C89C2×C411yescharacteristic
124C4×Q8×C89C2311yescharacteristic
125C4×D4×C89—21no—
126C4×D4×C89C2311yescharacteristic
127#117×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
140#97×C89—21no—
141C4×C8×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C2×Dic178—21no—
145C2×Dic178C22×C411yescharacteristic
146C2×Dic178C22×C411yescharacteristic
147#13083564—41no—
148#13083564—41no—
149#13083562—41no—
150#13083562—41no—
151#13083558—41no—
152#13083558—41no—
153C2×#10623494—21no—
154C2×#10623494—21no—
155C2×#10623494—21no—
156C2×#10623494—21no—
157C2×#10623494—42no—
158C2×#10623494C22×C411yescharacteristic
159C2×#10623494C22×C411yescharacteristic
160C2×#10623494C22×C411yescharacteristic
161C2×#10623494C22×C411yescharacteristic
162C4⋊Dic89—21no—
163C4⋊Dic89—21no—
164C4⋊Dic89—21no—
165C4⋊Dic89C22×C411yescharacteristic
166C4⋊Dic89C22×C411yescharacteristic
167#13083553—21no—
168#13083553—21no—
169#13083553—21no—
170#13083553—21no—
171C22⋊Dic89—21no—
172C22⋊Dic89—21no—
173C4×Dic89—21no—
174C4×Dic89—21no—
175C2×Q8×C89C22×C411yescharacteristic
176C2×D4×C89—21no—
177C2×D4×C89C22×C411yescharacteristic
178C22×C4×C89—21no—
179C22×C4×C89C22×C411yescharacteristic
180C22×C4×C89C2×D411yescharacteristic
181C22×C4×C89#4111yescharacteristic
182M16×C89—81no—
183M16×C89—81no—
184C2×C8×C89—21no—
185C2×C8×C89—21no—
186C2×C8×C89—81no—
187C2×C8×C89C2×D411yescharacteristic
188C2×C8×C89C2×D411yescharacteristic
189C2×C8×C89C2×D411yescharacteristic
190C2×C8×C89C2×D411yescharacteristic
191C4⋊C4×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89C22×C411yescharacteristic
196C4⋊C4×C89C22×C411yescharacteristic
197C22⋊C4×C89—21no—
198C22⋊C4×C89—21no—
199C22⋊C4×C89—21no—
200C22⋊C4×C89—21no—
201C22⋊C4×C89—21no—
202C42×C89—21no—
203C42×C89C22×C411yescharacteristic
204C42×C89C2×D411yescharacteristic
205C42×C89#4111yescharacteristic
206C42×C89C2411yescharacteristic
207C23×C89—21no—
208C23×C89C4×D411yescharacteristic
209Q8×C89—21no—
210Q8×C89—21no—
211D4×C89—21no—
212D4×C89—21no—
213D4×C89—81no—
214C2×C4×C89—21no—
215C2×C4×C89—21no—
216C2×C4×C89—21no—
217C2×C4×C89—21no—
218C2×C4×C89—21no—
219C2×C4×C89—21no—
220C2×C4×C89—21no—
221C2×C4×C89—41no—
222C2×C4×C89—41no—
223C2×C4×C89—41no—
224C2×C4×C89C4×D411yescharacteristic
225C2×C4×C89C4×D411yescharacteristic
226C2×C4×C89C4×D411yescharacteristic
227C2×C4×C89C23×C411yescharacteristic
228C2×C4×C89C22×D411yescharacteristic
229C2×C4×C89C2×#4111yescharacteristic
230C2×Dic89—21no—
231C2×Dic89—21no—
232C2×Dic89—21no—
233C2×Dic89—21no—
234Dic178—21no—
235Dic178—21no—
236Dic178—21no—
237Dic178—21no—
238Dic178—81no—
239C8×C89—21no—
240C8×C89—21no—
241C8×C89—21no—
242C8×C89—21no—
243C8×C89—41no—
244C8×C89—41no—
245#10623494—22no—
246#10623494—21no—
247#10623494—21no—
248#10623494—21no—
249#10623494—21no—
250#10623494—41no—
251#10623494—41no—
252#10623494—41no—
253C22×C89—21no—
254C22×C89—21no—
255C22×C89—21no—
256C22×C89—41no—
257C22×C89—41no—
258C22×C89—41no—
259C22×C89C2×C4×D411yescharacteristic
260C4×C89—21no—
261C4×C89—21no—
262C4×C89—21no—
263C4×C89—21no—
264C4×C89—41no—
265C4×C89—41no—
266C4×C89—41no—
267C4×C89#56811yescharacteristic
268C4×C89#56811yescharacteristic
269C4×C89#57711yescharacteristic
270C4×C89#57811yescharacteristic
271Dic89—41no—
272Dic89—41no—
273Dic89—41no—
274Dic89—41no—
275#17988—891nomaximal, Sylow
276C2×C89—41no—
277C2×C89—81no—
278C2×C89#293211yescharacteristic
279C2×C89#294711yescharacteristic
280C2×C89#295611yescharacteristic
281#3172—891no—
282#2993D8911yescharacteristic
283#2985—891no—
284#1588—891no—
285#1587—891no—
286#1587—891no—
287#1585—891no—
288#1584—891no—
289#1584—891no—
290#1583—891no—
291#1582—891no—
292#1569—891no—
293#1494—891no—
294#1493—891no—
295#1451—891no—
296C89#1798811yescharacteristic, complemented, Sylow
297#533—891no—
298#529C2×D8911yescharacteristic
299#501—891no—
300#497—891no—
301#491—891no—
302#489—891no—
303#470—891no—
304#470—891no—
305#468—891no—
306#468—891no—
307#446—891no—
308#446—891no—
309#446—1781no—
310#446—1781no—
311#446C2×D8911yescharacteristic
312C8×Q8—891no—
313C8×Q8—891no—
314C8×Q8C2×D8911yescharacteristic
315C4×Q16—891no—
316C4×Q16—891no—
317C4×SD16—891no—
318C4×SD16—891no—
319#436—21no—
320#436C2×D8911yescharacteristic
321#435—21no—
322C8×D4C2×D8911yescharacteristic
323#433—21no—
324#433—1781no—
325#433—1781no—
326#432—891no—
327#431C2×D8911yescharacteristic
328#424—21no—
329#423C2×D8911yescharacteristic
330C4×M16—891no—
331C2×C4×C8—891no—
332#335—891no—
333#335—891no—
334#334—891no—
335#334—891no—
336Q8⋊C8—891no—
337Q8⋊C8—891no—
338Q8⋊C8—891no—
339Q8⋊C8—891no—
340Q8⋊C8—891no—
341Q8⋊C8—891no—
342D4⋊C8—891no—
343D4⋊C8—891no—
344#322—891no—
345#322—891no—
346#322—891no—
347#322—891no—
348C2×Q16—891no—
349C2×Q16—891no—
350C2×SD16—891no—
351C2×SD16—891no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—891no—
355C2×M16—891no—
356C22×C8—21no—
357C22×C8—1781no—
358C4⋊Q8—891no—
359C4⋊Q8—891no—
360#126—21no—
361#125—1781no—
362#125C4×D8911yescharacteristic
363#124C4×D8911yescharacteristic
364#123—21no—
365C22⋊Q8—891no—
366C22⋊Q8—891no—
367C22⋊Q8—1781no—
368C22⋊Q8C4×D8911yescharacteristic
369#121C4×D8911yescharacteristic
370C4×Q8—891no—
371C4×Q8—891no—
372C4×Q8—1781no—
373C4×Q8C22×D8911yescharacteristic
374C4×D4—21no—
375C4×D4C22×D8911yescharacteristic
376#117—21no—
377#117—1781no—
378C2×C42C22×D8911yescharacteristic
379#107—891no—
380#107—891no—
381#106—891no—
382#106—891no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—891no—
387C4⋊C8—891no—
388C4⋊C8—1781no—
389C4⋊C8—1781no—
390C4⋊C8—1781no—
391C4⋊C8—1781no—
392C4⋊C8C22⋊D8911yescharacteristic
393C4⋊C8C22⋊D8911yescharacteristic
394C4⋊C8C22×D8911yescharacteristic
395C4⋊C8C22×D8911yescharacteristic
396#103—891no—
397#103—891no—
398#103—891no—
399#103—891no—
400#103—891no—
401#103—891no—
402#102—891no—
403#102—891no—
404C22⋊C8—21no—
405C22⋊C8—21no—
406C22⋊C8—1781no—
407C22⋊C8C22⋊D8911yescharacteristic
408C22⋊C8C22⋊D8911yescharacteristic
409#97—21no—
410#97—891no—
411#97—891no—
412#97—1781no—
413#97—1781no—
414#97—1781no—
415C4×C8—21no—
416C4×C8—891no—
417C4×C8—891no—
418C4×C8—891no—
419C4×C8—891no—
420C4×C8—1781no—
421C4×C8—1781no—
422C4×C8—1781no—
423C4×C8C22×D8911yescharacteristic
424C4×C8C22×D8911yescharacteristic
425C2×Q8—891no—
426C2×Q8—891no—
427C2×Q8—1781no—
428C2×Q8C2×C4×D8911yescharacteristic
429C2×D4—21no—
430C2×D4C2×C4×D8911yescharacteristic
431C22×C4—21no—
432C22×C4C2×C4×D8911yescharacteristic
433C22×C4C2×C22⋊D8911yescharacteristic
434C22×C4#1308357411yescharacteristic
435Q16—3561no—
436Q16—3561no—
437SD16—3561no—
438SD16—3561no—
439M16—81no—
440M16—81no—
441M16—3561no—
442M16—3561no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—81no—
446C2×C8—891no—
447C2×C8—891no—
448C2×C8—891no—
449C2×C8—891no—
450C2×C8—3562no—
451C2×C8—1781no—
452C2×C8—1781no—
453C2×C8—1781no—
454C2×C8—1781no—
455C2×C8C2×C22⋊D8911yescharacteristic
456C2×C8C2×C22⋊D8911yescharacteristic
457C2×C8C2×C22⋊D8911yescharacteristic
458C2×C8C2×C22⋊D8911yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—891no—
464C4⋊C4—891no—
465C4⋊C4—1781no—
466C4⋊C4—1781no—
467C4⋊C4—1781no—
468C4⋊C4—1781no—
469C4⋊C4—1781no—
470C4⋊C4—1781no—
471C4⋊C4—1781no—
472C4⋊C4C2×C4×D8911yescharacteristic
473C4⋊C4C2×C4×D8911yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—1781no—
480C22⋊C4—1781no—
481C42—21no—
482C42—1781no—
483C42—1781no—
484C42C2×C4×D8911yescharacteristic
485C42C2×C22⋊D8911yescharacteristic
486C42#1308357411yescharacteristic
487C42C23×D8911yescharacteristic
488C23—21no—
489C23C4×C22⋊D8911yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—1781no—
493Q8—1781no—
494Q8—1781no—
495Q8—1781no—
496Q8—7121no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—1781no—
511C2×C4—1781no—
512C2×C4—1781no—
513C2×C4—1781no—
514C2×C4C4×C22⋊D8911yescharacteristic
515C2×C4C4×C22⋊D8911yescharacteristic
516C2×C4C4×C22⋊D8911yescharacteristic
517C2×C4C22×C4×D8911yescharacteristic
518C2×C4C22×C22⋊D8911yescharacteristic
519C2×C4C2×#1308357411yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—3561no—
527C8—3561no—
528C8—3561no—
529C8—1782no—
530C8—1781no—
531C8—1781no—
532C8—1781no—
533C8—1781no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22C2×C4×C22⋊D8911yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—3561no—
549C4—3561no—
550C4—3561no—
551C4—3561no—
552C4#1915236311yescharacteristic
553C4#1915236311yescharacteristic
554C4#1915240911yescharacteristic
555C4#1915241311yescharacteristic
556C2—41no—
557C2—81no—
558C2#2602896811yescharacteristic
559C2#2602909111yescharacteristic
560C2#2602915911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2993C2
socleC22×C89C2×C4×D4
prime radical#121×C89C2×C4
cosocleC42×C89C24
hypercenter#2993D89
hypocenterC89#17988
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2993

Complements

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

NH ≅ G/Ntype
2. #26029346C2semidirect
3. #26019128C2semidirect
4. #26019123C2semidirect
5. #26019121C2semidirect
6. #26019109C2semidirect
7. #26019102C2semidirect
8. #26019100C2semidirect
9. #26019093C2semidirect
10. #26019088C2semidirect
11. #26019017C2semidirect
13. #26018706C2semidirect
14. #26018542C2semidirect
296. C89#17988semidirect

Maximal subgroups

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

subgroupindex
#260308592
#260293462
#260191282
#260191232
#260191212
#260191092
#260191022
#260191002
#260190932
#260190882
#260190172
#260187122
#260187062
#260185422
C89×#29932
#1798889