Finite Group Databasebeta

← #36525768

Subgroups — #36525768

Order 22784.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030256C211yesmaximal, characteristic
2#26029354C211yesmaximal, characteristic, complemented
3#26029289C211yesmaximal, characteristic
4#26019157C211yesmaximal, characteristic, complemented
5#26019139C211yesmaximal, characteristic, complemented
6#26019127C211yesmaximal, characteristic, complemented
7#26019112C211yesmaximal, characteristic, complemented
8#26019092C211yesmaximal, characteristic, complemented
9#26019091C211yesmaximal, characteristic, complemented
10#26019067C211yesmaximal, characteristic, complemented
11#26019049C211yesmaximal, characteristic, complemented
12#26019033C211yesmaximal, characteristic, complemented
13#26018644C211yesmaximal, characteristic
14#26018552C211yesmaximal, characteristic, complemented
15C89×#1497C211yesmaximal, characteristic, complemented
16C2×C4×D356C2211yescharacteristic
17#19151926C411yescharacteristic
18#19151926C411yescharacteristic
19#19151885C411yescharacteristic
20#19151885C411yescharacteristic
21#19151800C411yescharacteristic
22#19151800C411yescharacteristic
23#19151748C411yescharacteristic
24#19151748C411yescharacteristic
25#19151729—21no—
26#19151729C2211yescharacteristic, complemented
27#19151729C2211yescharacteristic, complemented
28Q8×#10623494—21no—
29Q8×#10623494C2211yescharacteristic, complemented
30Q8×#10623494C2211yescharacteristic, complemented
31C4×#13083561C2211yescharacteristic
32C4×#13083561C2211yescharacteristic
33C4×#13083561C2211yescharacteristic
34C4×#13083561C2211yescharacteristic
35#19151675C2211yescharacteristic
36#19151675C2211yescharacteristic
37#19151671—21no—
38#19151671—21no—
39#19151663C2211yescharacteristic
40#19151663C2211yescharacteristic
41#19151659—21no—
42#19151659—21no—
43#19151651C2211yescharacteristic
44#19151651C2211yescharacteristic
45#19151651C2211yescharacteristic
46#19151560—21no—
47C22⋊C8×D89—21no—
48#19151350C2211yescharacteristic, complemented
49#19151350C2211yescharacteristic, complemented
50#19151348C2211yescharacteristic, complemented
51#19151348C2211yescharacteristic, complemented
52#19151327C2211yescharacteristic
53#19151327C2211yescharacteristic
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151320C2211yescharacteristic, complemented
57#19151320C2211yescharacteristic, complemented
58#19151319C2211yescharacteristic, complemented
59C8×#10623494C2211yescharacteristic, complemented
60#533×C89C2211yescharacteristic
61#432×C89C2211yescharacteristic
62#431×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic, complemented
64Q8⋊C8×C89C2211yescharacteristic, complemented
65Q8⋊C8×C89C2211yescharacteristic, complemented
66Q8⋊C8×C89C2211yescharacteristic, complemented
67C22×D356C2×C411yescharacteristic
68C22×C4×D89—21no—
69C2×#13083561C2×C411yescharacteristic
70C2×#13083561C2×C411yescharacteristic
71C2×#13083561C2×C411yescharacteristic
72C2×#13083561C2×C411yescharacteristic
73C2×#13083557—21no—
74C2×#13083557—21no—
75C2×C8×D89—21no—
76C2×C8×D89—21no—
77C4×D356—21no—
78C4×D356—21no—
79C4×D356C2×C411yescharacteristic
80C4×D356C2×C411yescharacteristic
81C4×D356C2311yescharacteristic
82C4×D356C2311yescharacteristic
83C2×C4⋊Dic89C2×C411yescharacteristic
84C2×#13083551—21no—
85#15667943C2×C411yescharacteristic
86#15667943C2×C411yescharacteristic
87#15667943C2×C411yescharacteristic
88#15667943C2×C411yescharacteristic
89C4⋊#10623494—21no—
90C4⋊#10623494—21no—
91C4⋊#10623494—21no—
92C4⋊#10623494—21no—
93C4⋊#10623494C2×C411yescharacteristic
94C4⋊#10623494C2×C411yescharacteristic
95C4⋊#10623494D411yescharacteristic
96C4⋊#10623494D411yescharacteristic
97C4⋊#10623494C2311yescharacteristic
98C4⋊#10623494C2311yescharacteristic
99#15667936C2×C411yescharacteristic
100#15667936C2×C411yescharacteristic
101#15667936C2×C411yescharacteristic
102#15667936C2×C411yescharacteristic
103#15667932C2×C411yescharacteristic
104#15667932C2×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#15667924—21no—
115#15667924—21no—
116#15667923—21no—
117#15667923—21no—
118C4×#10623494—21no—
119C4×#10623494—21no—
120C4×#10623494—21no—
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C4×#10623494C2311yescharacteristic
125C4⋊Q8×C89C2×C411yescharacteristic
126C4⋊Q8×C89C2×C411yescharacteristic
127#125×C89—21no—
128C22⋊Q8×C89—21no—
129C22⋊Q8×C89C2×C411yescharacteristic
130C22⋊Q8×C89C2×C411yescharacteristic
131C4×Q8×C89—21no—
132C4×Q8×C89C2311yescharacteristic
133C4×Q8×C89C2311yescharacteristic
134#117×C89—21no—
135C2×C42×C89C2311yescharacteristic
136C4⋊C8×C89—21no—
137C4⋊C8×C89C2311yescharacteristic
138C4⋊C8×C89C2311yescharacteristic
139C22⋊C8×C89—21no—
140C22⋊C8×C89—21no—
141#97×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C23×D89—21no—
145C2×D356—22no—
146C2×D356—21no—
147C2×D356—41no—
148C2×D356C22×C411yescharacteristic
149C2×D356C22×C411yescharacteristic
150C22×Dic89—21no—
151C2×C4×D89—21no—
152C2×C4×D89—21no—
153C2×C4×D89—21no—
154C2×C4×D89—21no—
155C2×C4×D89—81no—
156C2×C4×D89—81no—
157#13083561—22no—
158#13083561—22no—
159#13083561—41no—
160#13083561—41no—
161#13083557—81no—
162#13083557—81no—
163C2×#10623494—21no—
164C2×#10623494—21no—
165C2×#10623494C22×C411yescharacteristic
166C2×#10623494C22×C411yescharacteristic
167C2×#10623494C22×C411yescharacteristic
168C2×#10623494C22×C411yescharacteristic
169C2×#10623494C2×D411yescharacteristic
170C2×#10623494C2×D411yescharacteristic
171C2×#10623494C2×D411yescharacteristic
172C2×#10623494C2×D411yescharacteristic
173C8×D89—81no—
174C8×D89—81no—
175C4⋊Dic89—21no—
176C4⋊Dic89C22×C411yescharacteristic
177C4⋊Dic89C22×C411yescharacteristic
178#13083551—21no—
179#13083551—21no—
180#13083551—21no—
181#13083551—21no—
182C2×Q8×C89—21no—
183C2×Q8×C89C22×C411yescharacteristic
184C2×Q8×C89C22×C411yescharacteristic
185C22×C4×C89C22×C411yescharacteristic
186C22×C4×C89C2×D411yescharacteristic
187C22×C4×C89#4111yescharacteristic
188C2×C8×C89—21no—
189C2×C8×C89—21no—
190C2×C8×C89—21no—
191C2×C8×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89—21no—
198C4⋊C4×C89—21no—
199C4⋊C4×C89C22×C411yescharacteristic
200C4⋊C4×C89C22×C411yescharacteristic
201C22⋊C4×C89—21no—
202C22⋊C4×C89—21no—
203C42×C89—21no—
204C42×C89—21no—
205C42×C89C22×C411yescharacteristic
206C42×C89C2×D411yescharacteristic
207C42×C89#4111yescharacteristic
208C42×C89C2411yescharacteristic
209C23×C89C4×D411yescharacteristic
210C22×D89—21no—
211C22×D89—21no—
212C22×D89—42no—
213C22×D89—42no—
214C22×D89—81no—
215Q8×C89—21no—
216Q8×C89—21no—
217Q8×C89—21no—
218Q8×C89—21no—
219Q8×C89—42no—
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—21no—
227C2×C4×C89—21no—
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
235C2×Dic89—21no—
236C2×Dic89—21no—
237C2×Dic89—81no—
238D356—21no—
239D356—21no—
240D356—21no—
241D356—42no—
242D356#13112yes—
243C4×D89—41no—
244C4×D89—41no—
245C4×D89—41no—
246C4×D89—41no—
247C4×D89—81no—
248C4×D89—81no—
249C8×C89—41no—
250C8×C89—41no—
251C8×C89—41no—
252C8×C89—41no—
253#10623494—21no—
254#10623494—21no—
255#10623494—21no—
256#10623494—21no—
257#10623494—21no—
258#10623494—21no—
259#10623494—21no—
260#10623494—21no—
261#10623494—41no—
262#10623494—41no—
263C22×C89—21no—
264C22×C89—21no—
265C22×C89—21no—
266C22×C89C2×C4×D411yescharacteristic
267C2×D89—22no—
268C2×D89—41no—
269C2×D89—41no—
270C2×D89—41no—
271C2×D89—41no—
272C2×D89—41no—
273C2×D89—84no—
274C4×C89—21no—
275C4×C89—21no—
276C4×C89—21no—
277C4×C89—21no—
278C4×C89—41no—
279C4×C89—41no—
280C4×C89—41no—
281C4×C89—41no—
282C4×C89C2×#13111yescharacteristic
283C4×C89#56811yescharacteristic
284C4×C89#57711yescharacteristic
285C4×C89#57711yescharacteristic
286Dic89—41no—
287Dic89—41no—
288#18028—891nomaximal, Sylow
289C2×C89—41no—
290C2×C89#293311yescharacteristic
291C2×C89#294911yescharacteristic
292C2×C89#295511yescharacteristic
293D89—42no—
294D89—81no—
295#3101—891no—
296#2986—891no—
297#2974—891no—
298#1592—891no—
299#1590—891no—
300#1588—891no—
301#1585—891no—
302#1583—891no—
303#1582—891no—
304SD16⋊C8—891no—
305C8×SD16—891no—
306#1572—891no—
307#1497D8911yescharacteristic, complemented
308#1478—891no—
309#1454—891no—
310C89#1802811yescharacteristic, complemented, Sylow
311#533C2×D8911yescharacteristic
312C2×C4×D4—891no—
313#483—891no—
314#483—891no—
315#476—891no—
316#476—891no—
317#461—891no—
318#461—891no—
319#450—891no—
320#450—891no—
321#446—891no—
322#446—891no—
323#446—1781no—
324C8×Q8—891no—
325C8×Q8—891no—
326C8×Q8—1781no—
327C4×SD16—891no—
328C4×SD16—891no—
329C4×SD16—891no—
330C4×SD16—891no—
331#436—891no—
332#436—891no—
333#435—1781no—
334#435—1781no—
335C8×D4—891no—
336C8×D4—891no—
337#433—1781no—
338#433—1781no—
339#432C2×D8911yescharacteristic
340#431—891no—
341#431—891no—
342#431—891no—
343#431C2×D8911yescharacteristic
344#407—1781no—
345C2×C22⋊C8—1781no—
346#335—891no—
347#335—891no—
348#334—891no—
349#334—891no—
350Q8⋊C8C2×D8911yescharacteristic, complemented
351Q8⋊C8C2×D8911yescharacteristic, complemented
352Q8⋊C8C2×D8911yescharacteristic, complemented
353Q8⋊C8C2×D8911yescharacteristic, complemented
354D4⋊C8—891no—
355D4⋊C8—891no—
356D4⋊C8—891no—
357D4⋊C8—891no—
358#322—891no—
359#322—891no—
360#322—891no—
361C82—891no—
362C22×D4—891no—
363C23×C4—1781no—
364C2×SD16—891no—
365C2×SD16—891no—
366C2×SD16—891no—
367C2×SD16—891no—
368C2×M16—1781no—
369C2×M16—1781no—
370C22×C8—1781no—
371C22×C8—1781no—
372C4⋊Q8C4×D8911yescharacteristic
373C4⋊Q8C4×D8911yescharacteristic
374#125—21no—
375C22⋊Q8—21no—
376C22⋊Q8C4×D8911yescharacteristic
377C22⋊Q8C4×D8911yescharacteristic
378C4×Q8—21no—
379C4×Q8C22×D8911yescharacteristic
380C4×Q8C22×D8911yescharacteristic
381C4×D4—891no—
382C4×D4—891no—
383C4×D4—891no—
384C4×D4—891no—
385C4×D4—1781no—
386C4×D4—1781no—
387#117—21no—
388C2×C4⋊C4—891no—
389C2×C22⋊C4—1781no—
390C2×C42C22×D8911yescharacteristic
391#106—891no—
392#106—891no—
393#106—891no—
394#106—891no—
395C4⋊C8—21no—
396C4⋊C8—891no—
397C4⋊C8—891no—
398C4⋊C8—891no—
399C4⋊C8—891no—
400C4⋊C8—891no—
401C4⋊C8—891no—
402C4⋊C8—1781no—
403C4⋊C8—1781no—
404C4⋊C8—1781no—
405C4⋊C8—1781no—
406C4⋊C8C22×D8911yescharacteristic
407C4⋊C8C22×D8911yescharacteristic
408#103—891no—
409#103—891no—
410#103—891no—
411#103—891no—
412#102—891no—
413#102—891no—
414#102—891no—
415#102—891no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8—891no—
419C22⋊C8—891no—
420C22⋊C8—891no—
421C22⋊C8—891no—
422C22⋊C8—1781no—
423C22⋊C8—1781no—
424C22⋊C8—1781no—
425C22⋊C8—1781no—
426C22⋊C8—1781no—
427#97—21no—
428#97—1781no—
429#97—1781no—
430C4×C8—891no—
431C4×C8—891no—
432C4×C8—891no—
433C4×C8—891no—
434C4×C8—1781no—
435C4×C8—1781no—
436C4×C8—1781no—
437C4×C8C22×D8911yescharacteristic
438C4×C8C22×D8911yescharacteristic
439C24—1781no—
440C2×Q8—21no—
441C2×Q8C2×C4×D8911yescharacteristic
442C2×Q8C2×C4×D8911yescharacteristic
443C2×D4—891no—
444C2×D4—891no—
445C2×D4—3561no—
446C2×D4—1782no—
447C2×D4—1781no—
448C22×C4—1781no—
449C22×C4—1781no—
450C22×C4—1781no—
451C22×C4—1781no—
452C22×C4—1781no—
453C22×C4—7121no—
454C22×C4—7121no—
455C22×C4C2×C4×D8911yescharacteristic
456C22×C4D4×D8911yescharacteristic
457C22×C4#1308357311yescharacteristic
458SD16—3561no—
459SD16—3561no—
460SD16—1782no—
461SD16—1782no—
462M16—7121no—
463M16—7121no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—891no—
469C2×C8—891no—
470C2×C8—891no—
471C2×C8—891no—
472C2×C8—891no—
473C2×C8—891no—
474C2×C8—891no—
475C2×C8—891no—
476C2×C8—1781no—
477C2×C8—1781no—
478C2×C8—7121no—
479C2×C8—7121no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—891no—
488C4⋊C4—891no—
489C4⋊C4—1781no—
490C4⋊C4C2×C4×D8911yescharacteristic
491C4⋊C4C2×C4×D8911yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—1781no—
495C22⋊C4—1781no—
496C22⋊C4—1781no—
497C22⋊C4—1781no—
498C42—21no—
499C42—21no—
500C42C2×C4×D8911yescharacteristic
501C42D4×D8911yescharacteristic
502C42#1308357311yescharacteristic
503C42C23×D8911yescharacteristic
504C23—3562no—
505C23—3562no—
506C23—1781no—
507C23—1781no—
508C23—7121no—
509C23#1566797211yescharacteristic
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—21no—
514Q8—42no—
515D4—892no—
516D4—3562no—
517D4—1781no—
518D4—1781no—
519D4—1781no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—3561no—
530C2×C4—3561no—
531C2×C4—3561no—
532C2×C4—3561no—
533C2×C4—1781no—
534C2×C4—1781no—
535C2×C4—7121no—
536C2×C4—7121no—
537C2×C4—7121no—
538C2×C4#1566797211yescharacteristic
539C2×C4#1566797211yescharacteristic
540C2×C4#1566797211yescharacteristic
541C2×C4C22×C4×D8911yescharacteristic
542C2×C4C2×D4×D8911yescharacteristic
543C2×C4C2×#1308357311yescharacteristic
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—3561no—
549C8—3561no—
550C8—1781no—
551C8—1781no—
552C8—1781no—
553C8—1781no—
554C8—1781no—
555C8—1781no—
556C8—1781no—
557C8—1781no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—3561no—
562C22—3561no—
563C22—3561no—
564C22—3561no—
565C22—3561no—
566C22—1782no—
567C22—7124no—
568C22C2×#1566797211yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—3561no—
578C4—3561no—
579C4#131×D8911yescharacteristic
580C4#1915236311yescharacteristic
581C4#1915240711yescharacteristic
582C4#1915240711yescharacteristic
583C2—41no—
584C2—3562no—
585C2—7121no—
586C2#2602898211yescharacteristic
587C2#2602910711yescharacteristic
588C2#2602915111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1497C2
socleC22×C89C2×C4×D4
prime radicalC22×D356C2×C4
cosocleC42×C89C24
hypercenter#1497D89
hypocenterC89#18028
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1497

Complements

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

NH ≅ G/Ntype
2. #26029354C2semidirect
4. #26019157C2semidirect
5. #26019139C2semidirect
6. #26019127C2semidirect
7. #26019112C2semidirect
8. #26019092C2semidirect
9. #26019091C2semidirect
10. #26019067C2semidirect
11. #26019049C2semidirect
12. #26019033C2semidirect
14. #26018552C2semidirect
15. C89×#1497C2semidirect
26. #19151729C22semidirect
27. #19151729C22semidirect
29. Q8×#10623494C22semidirect
30. Q8×#10623494C22semidirect
48. #19151350C22semidirect
49. #19151350C22semidirect
50. #19151348C22semidirect
51. #19151348C22semidirect
56. #19151320C22semidirect
57. #19151320C22semidirect
58. #19151319C22semidirect
59. C8×#10623494C22semidirect
63. Q8⋊C8×C89C22semidirect
64. Q8⋊C8×C89C22semidirect
65. Q8⋊C8×C89C22semidirect
66. Q8⋊C8×C89C22semidirect
307. #1497D89semidirect
310. C89#18028semidirect
350. Q8⋊C8C2×D89semidirect
351. Q8⋊C8C2×D89semidirect
352. Q8⋊C8C2×D89semidirect
353. Q8⋊C8C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260302562
#260293542
#260292892
#260191572
#260191392
#260191272
#260191122
#260190922
#260190912
#260190672
#260190492
#260190332
#260186442
#260185522
C89×#14972
#1802889