Finite Group Databasebeta

← #36524997

Subgroups — #36524997

Order 22784.

614 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030816C211yesmaximal, characteristic
2#26029228C211yesmaximal, characteristic, complemented
3#26029221C211yesmaximal, characteristic, complemented
4#26019163C211yesmaximal, characteristic, complemented
5#26019158C211yesmaximal, characteristic, complemented
6#26019157C211yesmaximal, characteristic, complemented
7#26019151C211yesmaximal, characteristic, complemented
8#26019073C211yesmaximal, characteristic, complemented
9#26019068C211yesmaximal, characteristic, complemented
10#26019067C211yesmaximal, characteristic, complemented
11#26019061C211yesmaximal, characteristic, complemented
12#26019013C211yesmaximal, characteristic, complemented
13#26018713C211yesmaximal, characteristic, complemented
14C2×#19151320C211yesmaximal, characteristic, complemented
15C89×#1494C211yesmaximal, characteristic, complemented
16#19152132C2211yescharacteristic
17#19151997C411yescharacteristic
18#19151980C411yescharacteristic
19#19151955C411yescharacteristic
20#19151952C411yescharacteristic
21#19151837C411yescharacteristic
22#19151832C411yescharacteristic
23#19151829C411yescharacteristic
24#19151825C411yescharacteristic
25#19151729C2211yescharacteristic
26#19151725C2211yescharacteristic
27Q8×#10623494C2211yescharacteristic
28#19151722C2211yescharacteristic
29C4×#13083564C2211yescharacteristic
30C4×#13083561C2211yescharacteristic
31C4×#13083562C2211yescharacteristic
32C4×#13083559C2211yescharacteristic
33#19151676C2211yescharacteristic
34#19151675C2211yescharacteristic
35#19151670—21no—
36#19151666—21no—
37D4×#10623494—21no—
38D4×#10623494C2211yescharacteristic
39#19151663C2211yescharacteristic
40#19151661—21no—
41#19151660—21no—
42#19151659—21no—
43#19151655C2211yescharacteristic
44#19151555—21no—
45#19151554—21no—
46C2×C4×#10623494C2211yescharacteristic
47C2×C4×#10623494C2211yescharacteristic
48#19151350C2211yescharacteristic
49#19151350C2211yescharacteristic
50#19151348C2211yescharacteristic
51#19151348C2211yescharacteristic
52#19151330C2211yescharacteristic
53#19151330C2211yescharacteristic
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151320C2211yescharacteristic
57#19151320C2211yescharacteristic
58#19151320C2211yescharacteristic
59#19151320C2211yescharacteristic
60#529×C89C2211yescharacteristic
61#432×C89C2211yescharacteristic
62C2×C4×C8×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic
64Q8⋊C8×C89C2211yescharacteristic
65D4⋊C8×C89C2211yescharacteristic
66D4⋊C8×C89C2211yescharacteristic
67C2×#13083564C2×C411yescharacteristic
68C2×#13083562C2×C411yescharacteristic
69C2×#13083561C2×C411yescharacteristic
70C2×#13083559C2×C411yescharacteristic
71C2×#13083558—21no—
72C2×#13083557—21no—
73C22×#10623494—21no—
74C22×#10623494C2×C411yescharacteristic
75C22×#10623494C2×C411yescharacteristic
76C22×#10623494D411yescharacteristic
77C22×#10623494D411yescharacteristic
78C2×C8×D89—21no—
79#15668006—21no—
80#15668004C2×C411yescharacteristic
81#15668000C2×C411yescharacteristic
82#15667995—21no—
83#15667991C2×C411yescharacteristic
84#15667986C2×C411yescharacteristic
85C4×Dic178C2311yescharacteristic
86C4×D356C2311yescharacteristic
87C4×C22⋊D89—21no—
88#15667969—21no—
89#15667945C2×C411yescharacteristic
90#15667945C2×C411yescharacteristic
91#15667943C2×C411yescharacteristic
92#15667943C2×C411yescharacteristic
93C4⋊#10623494—21no—
94C4⋊#10623494—21no—
95C4⋊#10623494C2311yescharacteristic
96C4⋊#10623494C2311yescharacteristic
97#15667941—21no—
98#15667941—21no—
99#15667936C2×C411yescharacteristic
100#15667936C2×C411yescharacteristic
101#15667935C2×C411yescharacteristic
102#15667935C2×C411yescharacteristic
103#15667933C2×C411yescharacteristic
104#15667933C2×C411yescharacteristic
105#15667932C2×C411yescharacteristic
106#15667932C2×C411yescharacteristic
107C22⋊#10623494—21no—
108C22⋊#10623494—21no—
109C22⋊#10623494—21no—
110#15667924—21no—
111#15667924—21no—
112#15667923—21no—
113#15667923—21no—
114#15667922—21no—
115C4×#10623494—21no—
116C4×#10623494C2×C411yescharacteristic
117C4×#10623494C2×C411yescharacteristic
118C4×#10623494D411yescharacteristic
119C4×#10623494D411yescharacteristic
120C4×#10623494C2311yescharacteristic
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C8×Dic89—21no—
125C22×C8×C89—21no—
126#126×C89—21no—
127#125×C89C2×C411yescharacteristic
128#124×C89C2×C411yescharacteristic
129#123×C89—21no—
130C22⋊Q8×C89C2×C411yescharacteristic
131#121×C89C2×C411yescharacteristic
132C4×Q8×C89C2311yescharacteristic
133C4×D4×C89—21no—
134C4×D4×C89C2311yescharacteristic
135#117×C89—21no—
136C2×C42×C89C2311yescharacteristic
137C4⋊C8×C89C2311yescharacteristic
138C4⋊C8×C89C2311yescharacteristic
139C22⋊C8×C89—21no—
140#97×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—22no—
150#13083561—22no—
151#13083559—22no—
152#13083558—42no—
153#13083557—42no—
154C2×#10623494—22no—
155C2×#10623494—22no—
156C2×#10623494—21no—
157C2×#10623494—21no—
158C2×#10623494—41no—
159C2×#10623494—41no—
160C2×#10623494—42no—
161C2×#10623494C22×C411yescharacteristic
162C2×#10623494C22×C411yescharacteristic
163C2×#10623494C22×C411yescharacteristic
164C2×#10623494C22×C411yescharacteristic
165C2×#10623494C2×D411yescharacteristic
166C2×#10623494C2×D411yescharacteristic
167C2×#10623494C2×D411yescharacteristic
168C2×#10623494C2×D411yescharacteristic
169C8×D89—42no—
170C4⋊Dic89C22×C411yescharacteristic
171C4⋊Dic89C22×C411yescharacteristic
172#13083553—21no—
173#13083553—21no—
174#13083553—21no—
175#13083553—21no—
176C22⋊Dic89—21no—
177#13083551—21no—
178#13083551—21no—
179#13083551—21no—
180#13083551—21no—
181C4×Dic89—21no—
182C2×Q8×C89C22×C411yescharacteristic
183C2×D4×C89—21no—
184C2×D4×C89C22×C411yescharacteristic
185C22×C4×C89—21no—
186C22×C4×C89C22×C411yescharacteristic
187C22×C4×C89C2×D411yescharacteristic
188C22×C4×C89#4111yescharacteristic
189C2×C8×C89—21no—
190C2×C8×C89—21no—
191C2×C8×C89—21no—
192C2×C8×C89—21no—
193C2×C8×C89—42no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89—21no—
198C4⋊C4×C89C22×C411yescharacteristic
199C4⋊C4×C89C22×C411yescharacteristic
200C22⋊C4×C89—21no—
201C22⋊C4×C89—21no—
202C22⋊C4×C89—21no—
203C22⋊C4×C89—21no—
204C22⋊C4×C89—21no—
205C42×C89—21no—
206C42×C89C22×C411yescharacteristic
207C42×C89C2×D411yescharacteristic
208C42×C89#4111yescharacteristic
209C42×C89C2411yescharacteristic
210C23×C89—21no—
211C23×C89C4×D411yescharacteristic
212C22×D89—21no—
213Q8×C89—21no—
214Q8×C89—21no—
215D4×C89—21no—
216D4×C89—21no—
217D4×C89—42no—
218C2×C4×C89—22no—
219C2×C4×C89—21no—
220C2×C4×C89—21no—
221C2×C4×C89—21no—
222C2×C4×C89—21no—
223C2×C4×C89—21no—
224C2×C4×C89—21no—
225C2×C4×C89—21no—
226C2×C4×C89—41no—
227C2×C4×C89—41no—
228C2×C4×C89C4×D411yescharacteristic
229C2×C4×C89C4×D411yescharacteristic
230C2×C4×C89C4×D411yescharacteristic
231C2×C4×C89C23×C411yescharacteristic
232C2×C4×C89C22×D411yescharacteristic
233C2×C4×C89C2×#4111yescharacteristic
234C22⋊D89—42no—
235C2×Dic89—21no—
236C2×Dic89—21no—
237C2×Dic89—21no—
238D356—21no—
239D356—21no—
240C4×D89—41no—
241C4×D89—41no—
242Dic178—21no—
243Dic178—21no—
244C8×C89—41no—
245C8×C89—41no—
246C8×C89—41no—
247C8×C89—41no—
248#10623494—21no—
249#10623494—21no—
250#10623494—21no—
251#10623494—21no—
252#10623494—21no—
253#10623494—21no—
254#10623494—41no—
255#10623494—41no—
256#10623494#13112yes—
257#10623494#13112yes—
258C22×C89—21no—
259C22×C89—21no—
260C22×C89—41no—
261C22×C89—41no—
262C22×C89—41no—
263C22×C89#44412yes—
264C22×C89C2×C4×D411yescharacteristic
265C2×D89—41no—
266C2×D89—41no—
267C2×D89—41no—
268C4×C89—21no—
269C4×C89—21no—
270C4×C89—21no—
271C4×C89—21no—
272C4×C89—41no—
273C4×C89—41no—
274C4×C89—41no—
275C4×C89C2×#13111yescharacteristic
276C4×C89C2×#13111yescharacteristic
277C4×C89#57511yescharacteristic
278C4×C89#57511yescharacteristic
279Dic89—41no—
280Dic89—41no—
281Dic89—41no—
282#17963—891nomaximal, Sylow
283C2×C89—22no—
284C2×C89—81no—
285C2×C89#293111yescharacteristic
286C2×C89#294311yescharacteristic
287C2×C89C2×#44411yescharacteristic
288D89—81no—
289#3168—891no—
290#2966—891no—
291#2966—891no—
292#1594—891no—
293#1593—891no—
294#1592—891no—
295#1592—891no—
296#1580—891no—
297#1579—891no—
298SD16⋊C8—891no—
299SD16⋊C8—891no—
300#1568—891no—
301#1494—891no—
302#1494D8911yescharacteristic, complemented
303C2×#322—891no—
304C89#1796311yescharacteristic, complemented, Sylow
305#529—891no—
306#529C2×D8911yescharacteristic
307#499—891no—
308#495—891no—
309#488—891no—
310#488—891no—
311#467—891no—
312#466—891no—
313#465—891no—
314#465—891no—
315#446—891no—
316#446—891no—
317C8×Q8—891no—
318C8×Q8—891no—
319C4×Q16—891no—
320C4×SD16—891no—
321C4×SD16—891no—
322C4×D8—891no—
323#436—891no—
324#436—891no—
325#435—1781no—
326#435—1781no—
327C8×D4—891no—
328C8×D4—891no—
329C8×D4—1781no—
330C8×D4—1781no—
331#433—1781no—
332#433—1781no—
333#432—891no—
334#432C2×D8911yescharacteristic
335#405—1781no—
336#405—1781no—
337C2×C4×C8—891no—
338C2×C4×C8—891no—
339C2×C4×C8C2×D8911yescharacteristic
340#335—891no—
341#335—891no—
342#334—891no—
343#334—891no—
344Q8⋊C8—891no—
345Q8⋊C8—891no—
346Q8⋊C8C2×D8911yescharacteristic
347Q8⋊C8C2×D8911yescharacteristic
348D4⋊C8—891no—
349D4⋊C8—891no—
350D4⋊C8C2×D8911yescharacteristic
351D4⋊C8C2×D8911yescharacteristic
352#322—891no—
353#322—891no—
354#322—891no—
355#322—891no—
356C2×Q16—891no—
357C2×SD16—891no—
358C2×SD16—891no—
359C2×D8—891no—
360C2×M16—1781no—
361C2×M16—1781no—
362C22×C8—21no—
363C22×C8—891no—
364C22×C8—891no—
365C22×C8—891no—
366C22×C8—891no—
367C22×C8—1781no—
368C22×C8—1781no—
369#126—21no—
370#126—1781no—
371#125—891no—
372#125C4×D8911yescharacteristic
373#124—891no—
374#124C4×D8911yescharacteristic
375#123—21no—
376#123—1781no—
377C22⋊Q8—891no—
378C22⋊Q8C4×D8911yescharacteristic
379#121—891no—
380#121C4×D8911yescharacteristic
381C4×Q8—891no—
382C4×Q8C22×D8911yescharacteristic
383C4×D4—21no—
384C4×D4—891no—
385C4×D4—1781no—
386C4×D4C22×D8911yescharacteristic
387#117—21no—
388#117—1781no—
389C2×C42C22×D8911yescharacteristic
390#107—891no—
391#107—891no—
392#106—891no—
393#106—891no—
394C4⋊C8—891no—
395C4⋊C8—891no—
396C4⋊C8—1781no—
397C4⋊C8—1781no—
398C4⋊C8—1781no—
399C4⋊C8—1781no—
400C4⋊C8C22×D8911yescharacteristic
401C4⋊C8C22×D8911yescharacteristic
402#103—891no—
403#103—891no—
404#103—891no—
405#103—891no—
406#102—891no—
407#102—891no—
408#102—891no—
409#102—891no—
410C22⋊C8—21no—
411C22⋊C8—1781no—
412C22⋊C8—1781no—
413C22⋊C8—1781no—
414C22⋊C8—1781no—
415C22⋊C8—1781no—
416#97—21no—
417#97—1781no—
418#97—1781no—
419#97—1781no—
420C4×C8—21no—
421C4×C8—891no—
422C4×C8—891no—
423C4×C8—891no—
424C4×C8—891no—
425C4×C8—891no—
426C4×C8—891no—
427C4×C8—891no—
428C4×C8—891no—
429C4×C8—1781no—
430C4×C8—1781no—
431C4×C8C22×D8911yescharacteristic
432C4×C8C22×D8911yescharacteristic
433C2×Q8—891no—
434C2×Q8C2×C4×D8911yescharacteristic
435C2×D4—21no—
436C2×D4—891no—
437C2×D4—1781no—
438C2×D4C2×C4×D8911yescharacteristic
439C22×C4—21no—
440C22×C4—1781no—
441C22×C4C2×C4×D8911yescharacteristic
442C22×C4D4×D8911yescharacteristic
443C22×C4#1308357311yescharacteristic
444Q16—1782no—
445SD16—1782no—
446SD16—1782no—
447D8—1782no—
448M16—3562no—
449M16—3562no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—42no—
455C2×C8—891no—
456C2×C8—891no—
457C2×C8—891no—
458C2×C8—891no—
459C2×C8—891no—
460C2×C8—891no—
461C2×C8—891no—
462C2×C8—891no—
463C2×C8—3562no—
464C2×C8—3562no—
465C2×C8—3561no—
466C2×C8—3561no—
467C2×C8—1782no—
468C2×C8—1782no—
469C2×C8—1781no—
470C2×C8—1781no—
471C4⋊C4—21no—
472C4⋊C4—21no—
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—21no—
487C22⋊C4—21no—
488C22⋊C4—1781no—
489C22⋊C4—1781no—
490C22⋊C4—1781no—
491C22⋊C4—1781no—
492C22⋊C4—1781no—
493C42—21no—
494C42—1781no—
495C42C2×C4×D8911yescharacteristic
496C42D4×D8911yescharacteristic
497C42#1308357311yescharacteristic
498C42C23×D8911yescharacteristic
499C23—21no—
500C23—1781no—
501C23#1566797211yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—1781no—
505Q8—1781no—
506D4—21no—
507D4—21no—
508D4—42no—
509D4—3562no—
510D4—1781no—
511D4—1781no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—3561no—
523C2×C4—3561no—
524C2×C4—1781no—
525C2×C4—1781no—
526C2×C4—1781no—
527C2×C4#1566797211yescharacteristic
528C2×C4#1566797211yescharacteristic
529C2×C4#1566797211yescharacteristic
530C2×C4C22×C4×D8911yescharacteristic
531C2×C4C2×D4×D8911yescharacteristic
532C2×C4C2×#1308357311yescharacteristic
533C8—41no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—892no—
538C8—892no—
539C8—3561no—
540C8—3561no—
541C8—1781no—
542C8—1781no—
543C8—1781no—
544C8—1781no—
545C8—1781no—
546C8—1781no—
547C22—21no—
548C22—21no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—3561no—
553C22—3561no—
554C22—3561no—
555C22#1915171712yes—
556C22C2×#1566797211yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—3561no—
565C4—3561no—
566C4—3561no—
567C4#131×D8911yescharacteristic
568C4#131×D8911yescharacteristic
569C4#1915239511yescharacteristic
570C4#1915239511yescharacteristic
571C2—22no—
572C2—81no—
573C2—7121no—
574C2#2602896011yescharacteristic
575C2#2602904811yescharacteristic
576C2C2×#1915171711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1494C2
socleC22×C89C2×C4×D4
prime radical#19151832C4
cosocleC42×C89C24
hypercenter#1494D89
hypocenterC89#17963
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
2. #26029228C2semidirect
3. #26029221C2semidirect
4. #26019163C2semidirect
5. #26019158C2semidirect
6. #26019157C2semidirect
7. #26019151C2semidirect
8. #26019073C2semidirect
9. #26019068C2semidirect
10. #26019067C2semidirect
11. #26019061C2semidirect
12. #26019013C2semidirect
13. #26018713C2semidirect
14. C2×#19151320C2semidirect
15. C89×#1494C2semidirect
302. #1494D89semidirect
304. C89#17963semidirect

Maximal subgroups

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

subgroupindex
#260308162
#260292282
#260292212
#260191632
#260191582
#260191572
#260191512
#260190732
#260190682
#260190672
#260190612
#260190132
#260187132
C2×#191513202
C89×#14942
#1796389