Finite Group Databasebeta

← #36525689

Subgroups — #36525689

Order 22784.

592 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030332C211yesmaximal, characteristic
2#26029408C211yesmaximal, characteristic
3#26029341C211yesmaximal, characteristic, complemented
4#26019158C211yesmaximal, characteristic, complemented
5#26019157C211yesmaximal, characteristic, complemented
6#26019105C211yesmaximal, characteristic, complemented
7#26019099C211yesmaximal, characteristic, complemented
8#26019092C211yesmaximal, characteristic, complemented
9#26019091C211yesmaximal, characteristic, complemented
10#26019068C211yesmaximal, characteristic, complemented
11#26019067C211yesmaximal, characteristic, complemented
12#26019035C211yesmaximal, characteristic, complemented
13#26018713C211yesmaximal, characteristic
14#26018544C211yesmaximal, characteristic, complemented
15C2×Q8⋊C8×C89C211yesmaximal, characteristic, complemented
16#19152132C2211yescharacteristic
17#19151928C411yescharacteristic
18#19151911C411yescharacteristic
19#19151905C411yescharacteristic
20#19151905C411yescharacteristic
21#19151820C411yescharacteristic
22#19151820C411yescharacteristic
23#19151754C411yescharacteristic
24#19151737C411yescharacteristic
25#19151729—21no—
26#19151729—21no—
27#19151729C2211yescharacteristic
28#19151729C2211yescharacteristic
29#19151725C2211yescharacteristic
30Q8×#10623494C2211yescharacteristic
31Q8×#10623494C2211yescharacteristic
32#19151722C2211yescharacteristic
33C4×#13083564C2211yescharacteristic
34C4×#13083564C2211yescharacteristic
35C4×#13083561C2211yescharacteristic
36C4×#13083561C2211yescharacteristic
37#19151675C2211yescharacteristic
38#19151669—21no—
39#19151663C2211yescharacteristic
40C8×C22⋊D89—21no—
41#19151658—21no—
42#19151655—21no—
43#19151655C2211yescharacteristic
44#19151651—21no—
45#19151651C2211yescharacteristic
46#19151651C2211yescharacteristic
47#19151623—21no—
48#19151350C2211yescharacteristic
49#19151350C2211yescharacteristic
50#19151348C2211yescharacteristic
51#19151348C2211yescharacteristic
52#19151330C2211yescharacteristic
53#19151330C2211yescharacteristic
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151320C2211yescharacteristic
57#19151320C2211yescharacteristic
58#19151320C2211yescharacteristic
59#19151320C2211yescharacteristic
60C2×C4×Q8×C89C2211yescharacteristic
61C2×C4⋊C8×C89C2211yescharacteristic
62C2×C4×C8×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic
64Q8⋊C8×C89C2211yescharacteristic
65Q8⋊C8×C89C2211yescharacteristic
66Q8⋊C8×C89C2211yescharacteristic
67C2×#13083564C2×C411yescharacteristic
68C2×#13083564C2×C411yescharacteristic
69C2×#13083561C2×C411yescharacteristic
70C2×#13083561C2×C411yescharacteristic
71C2×#13083557—21no—
72C2×C8×D89—21no—
73#15668006—21no—
74#15668004C2×C411yescharacteristic
75#15668000C2×C411yescharacteristic
76#15667995—21no—
77#15667991C2×C411yescharacteristic
78#15667986C2×C411yescharacteristic
79C4×Dic178C2311yescharacteristic
80C4×D356C2311yescharacteristic
81C4×C22⋊D89—21no—
82#15667969—21no—
83#15667945C2×C411yescharacteristic
84#15667945C2×C411yescharacteristic
85#15667943C2×C411yescharacteristic
86#15667943C2×C411yescharacteristic
87C4⋊#10623494—21no—
88C4⋊#10623494—21no—
89C4⋊#10623494—21no—
90C4⋊#10623494—21no—
91C4⋊#10623494C2×C411yescharacteristic
92C4⋊#10623494C2×C411yescharacteristic
93C4⋊#10623494D411yescharacteristic
94C4⋊#10623494D411yescharacteristic
95C4⋊#10623494C2311yescharacteristic
96C4⋊#10623494C2311yescharacteristic
97#15667941—21no—
98#15667941—21no—
99#15667936C2×C411yescharacteristic
100#15667936C2×C411yescharacteristic
101#15667936C2×C411yescharacteristic
102#15667936C2×C411yescharacteristic
103#15667935C2×C411yescharacteristic
104#15667935C2×C411yescharacteristic
105#15667932C2×C411yescharacteristic
106#15667932C2×C411yescharacteristic
107C22⋊#10623494—21no—
108C22⋊#10623494C2×C411yescharacteristic
109C22⋊#10623494C2×C411yescharacteristic
110C22⋊#10623494D411yescharacteristic
111C22⋊#10623494D411yescharacteristic
112#15667924—21no—
113#15667924—21no—
114#15667923—21no—
115#15667923—21no—
116#15667923—21no—
117#15667922—21no—
118C4×#10623494—21no—
119C4×#10623494—21no—
120C4×#10623494C2311yescharacteristic
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C8×Dic89—21no—
125C22×Q8×C89C2×C411yescharacteristic
126C22×C8×C89—21no—
127C22×C8×C89—21no—
128C4×Q8×C89—21no—
129C4×Q8×C89—21no—
130C4×Q8×C89C2×C411yescharacteristic
131C4×Q8×C89C2×C411yescharacteristic
132C4×Q8×C89C2311yescharacteristic
133C4×Q8×C89C2311yescharacteristic
134C2×C4⋊C4×C89—21no—
135C2×C4⋊C4×C89C2×C411yescharacteristic
136C2×C42×C89—21no—
137C2×C42×C89C2311yescharacteristic
138C4⋊C8×C89—21no—
139C4⋊C8×C89C2311yescharacteristic
140C4⋊C8×C89C2311yescharacteristic
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#13083564—41no—
150#13083561—22no—
151#13083561—41no—
152#13083557—81no—
153C2×#10623494—21no—
154C2×#10623494—21no—
155C2×#10623494C22×C411yescharacteristic
156C2×#10623494C22×C411yescharacteristic
157C2×#10623494C22×C411yescharacteristic
158C2×#10623494C22×C411yescharacteristic
159C2×#10623494C2×D411yescharacteristic
160C2×#10623494C2×D411yescharacteristic
161C2×#10623494C2×D411yescharacteristic
162C2×#10623494C2×D411yescharacteristic
163C8×D89—81no—
164C4⋊Dic89C22×C411yescharacteristic
165C4⋊Dic89C22×C411yescharacteristic
166#13083553—21no—
167#13083553—21no—
168#13083553—21no—
169#13083553—21no—
170C22⋊Dic89—21no—
171#13083551—21no—
172#13083551—21no—
173#13083551—21no—
174#13083551—21no—
175C4×Dic89—21no—
176C2×Q8×C89—21no—
177C2×Q8×C89—22no—
178C2×Q8×C89—41no—
179C2×Q8×C89C22×C411yescharacteristic
180C2×Q8×C89C22×C411yescharacteristic
181C22×C4×C89—21no—
182C22×C4×C89—21no—
183C22×C4×C89C22×C411yescharacteristic
184C22×C4×C89C2×D411yescharacteristic
185C22×C4×C89#4111yescharacteristic
186C2×C8×C89—21no—
187C2×C8×C89—21no—
188C2×C8×C89—21no—
189C2×C8×C89—21no—
190C2×C8×C89—42no—
191C2×C8×C89—81no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89C22×C411yescharacteristic
198C4⋊C4×C89C22×C411yescharacteristic
199C42×C89—21no—
200C42×C89—21no—
201C42×C89—21no—
202C42×C89—21no—
203C42×C89C22×C411yescharacteristic
204C42×C89C2×D411yescharacteristic
205C42×C89#4111yescharacteristic
206C42×C89C2411yescharacteristic
207C23×C89C4×D411yescharacteristic
208C22×D89—21no—
209Q8×C89—21no—
210Q8×C89—21no—
211Q8×C89—21no—
212Q8×C89—42no—
213Q8×C89#13112yes—
214C2×C4×C89—21no—
215C2×C4×C89—22no—
216C2×C4×C89—21no—
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—42no—
224C2×C4×C89—42no—
225C2×C4×C89C4×D411yescharacteristic
226C2×C4×C89C4×D411yescharacteristic
227C2×C4×C89C4×D411yescharacteristic
228C2×C4×C89C23×C411yescharacteristic
229C2×C4×C89C22×D411yescharacteristic
230C2×C4×C89C2×#4111yescharacteristic
231C22⋊D89—42no—
232C2×Dic89—21no—
233C2×Dic89—21no—
234C2×Dic89—21no—
235D356—21no—
236D356—21no—
237C4×D89—41no—
238C4×D89—41no—
239Dic178—21no—
240Dic178—21no—
241C8×C89—41no—
242C8×C89—41no—
243C8×C89—41no—
244C8×C89—41no—
245#10623494—21no—
246#10623494—21no—
247#10623494—21no—
248#10623494—21no—
249#10623494—21no—
250#10623494—21no—
251#10623494—21no—
252#10623494—21no—
253#10623494—41no—
254#10623494—41no—
255C22×C89—21no—
256C22×C89—21no—
257C22×C89#44412yes—
258C22×C89C2×C4×D411yescharacteristic
259C2×D89—41no—
260C2×D89—41no—
261C2×D89—41no—
262C4×C89—21no—
263C4×C89—22no—
264C4×C89—21no—
265C4×C89—21no—
266C4×C89—21no—
267C4×C89—41no—
268C4×C89—41no—
269C4×C89—41no—
270C4×C89C2×#13111yescharacteristic
271C4×C89#56811yescharacteristic
272C4×C89#57711yescharacteristic
273C4×C89#57811yescharacteristic
274Dic89—41no—
275Dic89—41no—
276Dic89—41no—
277#18023—891nomaximal, Sylow
278C2×C89—22no—
279C2×C89#293311yescharacteristic
280C2×C89#294711yescharacteristic
281C2×C89C2×#44411yescharacteristic
282D89—81no—
283#3108—891no—
284#2994—891no—
285#2984—891no—
286#1593—891no—
287#1592—891no—
288#1584—891no—
289#1584—891no—
290#1583—891no—
291#1582—891no—
292#1579—891no—
293SD16⋊C8—891no—
294#1573—891no—
295#1494—891no—
296C2×Q8⋊C8D8911yescharacteristic, complemented
297#1452—891no—
298C89#1802311yescharacteristic, complemented, Sylow
299#529—891no—
300C2×C4×Q8C2×D8911yescharacteristic
301#483—891no—
302#480—891no—
303#479—891no—
304#479—891no—
305#464—891no—
306#464—891no—
307#451—891no—
308#448—891no—
309#446—891no—
310#446—891no—
311#446—891no—
312#446—1781no—
313#446—1781no—
314C8×Q8—891no—
315C8×Q8—891no—
316C8×Q8—891no—
317C4×Q16—891no—
318C4×Q16—891no—
319C4×SD16—891no—
320C4×SD16—891no—
321#436—891no—
322#435—1781no—
323C8×D4—891no—
324C8×D4—1781no—
325#433—1781no—
326#432—891no—
327#432—1781no—
328#431—891no—
329#431—891no—
330#431—1781no—
331#424—1781no—
332C2×C4⋊C8C2×D8911yescharacteristic
333C2×C4×C8C2×D8911yescharacteristic
334#335—891no—
335#335—891no—
336#334—891no—
337#334—891no—
338Q8⋊C8—891no—
339Q8⋊C8—891no—
340Q8⋊C8C2×D8911yescharacteristic
341Q8⋊C8C2×D8911yescharacteristic
342Q8⋊C8C2×D8911yescharacteristic
343Q8⋊C8C2×D8911yescharacteristic
344D4⋊C8—891no—
345D4⋊C8—891no—
346#322—891no—
347#322—891no—
348#322—891no—
349#322—891no—
350C22×Q8C4×D8911yescharacteristic
351C2×Q16—891no—
352C2×Q16—891no—
353C2×SD16—891no—
354C2×SD16—891no—
355C2×M16—1781no—
356C22×C8—21no—
357C22×C8—21no—
358C22×C8—1781no—
359#126—1781no—
360#125—891no—
361#124—891no—
362#123—1781no—
363C22⋊Q8—891no—
364#121—891no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—891no—
368C4×Q8C4×D8911yescharacteristic
369C4×Q8C4×D8911yescharacteristic
370C4×Q8C22×D8911yescharacteristic
371C4×Q8C22×D8911yescharacteristic
372C4×D4—891no—
373C4×D4—1781no—
374#117—1781no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C4×D8911yescharacteristic
377C2×C42—21no—
378C2×C42C22×D8911yescharacteristic
379#107—891no—
380#107—891no—
381#106—891no—
382#106—891no—
383C4⋊C8—21no—
384C4⋊C8—891no—
385C4⋊C8—891no—
386C4⋊C8—891no—
387C4⋊C8—891no—
388C4⋊C8—891no—
389C4⋊C8—891no—
390C4⋊C8—1781no—
391C4⋊C8—1781no—
392C4⋊C8—1781no—
393C4⋊C8—1781no—
394C4⋊C8—1781no—
395C4⋊C8—1781no—
396C4⋊C8C22×D8911yescharacteristic
397C4⋊C8C22×D8911yescharacteristic
398#103—891no—
399#103—891no—
400#103—891no—
401#103—891no—
402#103—891no—
403#103—891no—
404#102—891no—
405#102—891no—
406C22⋊C8—891no—
407C22⋊C8—891no—
408C22⋊C8—891no—
409C22⋊C8—891no—
410C22⋊C8—1781no—
411C22⋊C8—1781no—
412C22⋊C8—1781no—
413#97—1781no—
414#97—1781no—
415#97—1781no—
416#97—1781no—
417C4×C8—21no—
418C4×C8—891no—
419C4×C8—891no—
420C4×C8—891no—
421C4×C8—891no—
422C4×C8—1781no—
423C4×C8—1781no—
424C4×C8—1781no—
425C4×C8C22×D8911yescharacteristic
426C4×C8C22×D8911yescharacteristic
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8—891no—
431C2×Q8C2×C4×D8911yescharacteristic
432C2×Q8C2×C4×D8911yescharacteristic
433C2×D4—891no—
434C2×D4—1781no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—1781no—
438C22×C4C2×C4×D8911yescharacteristic
439C22×C4D4×D8911yescharacteristic
440C22×C4#1308357311yescharacteristic
441Q16—3561no—
442Q16—1782no—
443SD16—3561no—
444SD16—1782no—
445M16—7121no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—42no—
451C2×C8—81no—
452C2×C8—891no—
453C2×C8—891no—
454C2×C8—891no—
455C2×C8—891no—
456C2×C8—891no—
457C2×C8—891no—
458C2×C8—891no—
459C2×C8—891no—
460C2×C8—1781no—
461C2×C8—1781no—
462C2×C8—7121no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—891no—
469C4⋊C4—891no—
470C4⋊C4—1781no—
471C4⋊C4—1781no—
472C4⋊C4—1781no—
473C4⋊C4—1781no—
474C4⋊C4C2×C4×D8911yescharacteristic
475C4⋊C4C2×C4×D8911yescharacteristic
476C22⋊C4—1781no—
477C22⋊C4—1781no—
478C22⋊C4—1781no—
479C22⋊C4—1781no—
480C22⋊C4—1781no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—1781no—
486C42C2×C4×D8911yescharacteristic
487C42D4×D8911yescharacteristic
488C42#1308357311yescharacteristic
489C42C23×D8911yescharacteristic
490C23—1781no—
491C23#1566797211yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—42no—
496Q8—1781no—
497Q8—1781no—
498Q8#1566801812yes—
499D4—3562no—
500D4—1781no—
501D4—1781no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—3561no—
514C2×C4—3561no—
515C2×C4—1781no—
516C2×C4—1781no—
517C2×C4—1781no—
518C2×C4#1566797211yescharacteristic
519C2×C4#1566797211yescharacteristic
520C2×C4#1566797211yescharacteristic
521C2×C4C22×C4×D8911yescharacteristic
522C2×C4C2×D4×D8911yescharacteristic
523C2×C4C2×#1308357311yescharacteristic
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—3561no—
529C8—3561no—
530C8—1781no—
531C8—1781no—
532C8—1781no—
533C8—1781no—
534C8—1781no—
535C8—1781no—
536C8—1781no—
537C8—1781no—
538C22—21no—
539C22—21no—
540C22—3561no—
541C22—3561no—
542C22—3561no—
543C22#1915171712yes—
544C22C2×#1566797211yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—3561no—
554C4—3561no—
555C4—3561no—
556C4C2×#1566801811yescharacteristic
557C4#1915236411yescharacteristic
558C4#1915240411yescharacteristic
559C4#1915241111yescharacteristic
560C2—22no—
561C2—7121no—
562C2#2602897311yescharacteristic
563C2#2602908911yescharacteristic
564C2C2×#1915171711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC2×Q8⋊C8×C89C2
socleC22×C89C2×C4×D4
prime radical#15667986C2×C4
cosocleC42×C89C24
hypercenterC2×Q8⋊C8D89
hypocenterC89#18023
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #26029341C2semidirect
4. #26019158C2semidirect
5. #26019157C2semidirect
6. #26019105C2semidirect
7. #26019099C2semidirect
8. #26019092C2semidirect
9. #26019091C2semidirect
10. #26019068C2semidirect
11. #26019067C2semidirect
12. #26019035C2semidirect
14. #26018544C2semidirect
15. C2×Q8⋊C8×C89C2semidirect
296. C2×Q8⋊C8D89semidirect
298. C89#18023semidirect

Maximal subgroups

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

subgroupindex
#260303322
#260294082
#260293412
#260191582
#260191572
#260191052
#260190992
#260190922
#260190912
#260190682
#260190672
#260190352
#260187132
#260185442
C2×Q8⋊C8×C892
#1802389