Finite Group Databasebeta

← #29671372

Subgroups — #29671372

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458716C211yesmaximal, characteristic, complemented
2#21457678C211yesmaximal, characteristic, complemented
3#21457643C211yesmaximal, characteristic, complemented
4#21445418C211yesmaximal, characteristic, complemented
5#21445318C211yesmaximal, characteristic, complemented
6#21445288C211yesmaximal, characteristic, complemented
7#21445244C211yesmaximal, characteristic, complemented
8#21445226C211yesmaximal, characteristic, complemented
9#21445024C211yesmaximal, characteristic, complemented
10#21444998C211yesmaximal, characteristic, complemented
11#21444970C211yesmaximal, characteristic, complemented
12#21444921C211yesmaximal, characteristic, complemented
13#21444707C211yesmaximal, characteristic, complemented
14D8×#92418C211yesmaximal, characteristic, complemented
15C61×#3272C211yesmaximal, characteristic
16#16840469C2211yescharacteristic
17D4×Dic122C2211yescharacteristic, complemented
18#16840170—21no—
19#16840160C2211yescharacteristic, complemented
20#16840114C2211yescharacteristic, complemented
21#16840114C2211yescharacteristic, complemented
22#16840108C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840092—21no—
25#16840078—21no—
26#16840064—21no—
27#16840052C2211yescharacteristic
28#16840052C2211yescharacteristic, complemented
29#16840050—21no—
30#16840044C2211yescharacteristic, complemented
31#16840042C2211yescharacteristic
32#16839989—21no—
33#16839989—21no—
34#16839967C2211yescharacteristic
35#16839959C2211yescharacteristic
36#16839913—21no—
37#16839900—21no—
38#16839892C2211yescharacteristic, complemented
39#16839874—21no—
40C4×#11758200C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42D4×#92418C2211yescharacteristic, complemented
43C2×#14519700—21no—
44C2×#14519698—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51#16839492C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#547C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#468—21no—
61C61×#461C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#450—21no—
64C61×#449—21no—
65C4×D8×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200D411yescharacteristic
71C22×#92418—21no—
72C22×#92418—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519776—21no—
76#14519768—21no—
77#14519756—21no—
78#14519756—21no—
79#14519754—41no—
80#14519754—41no—
81C4×Dic122C2311yescharacteristic
82D4×Dic61—21no—
83D4×Dic61—21no—
84C2×C4⋊Dic61—21no—
85#14519710—21no—
86#14519709—21no—
87#14519709—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708D411yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519706—21no—
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519700—41no—
101#14519700D411yescharacteristic
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519687—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118AGL(1,ℤ/8)×C61—44no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×D8×C61—21no—
123C2×D8×C61D411yescharacteristic
124C2×M16×C61—21no—
125C4⋊Q8×C61C2311yescharacteristic
126#127×C61C2311yescharacteristic
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—21no—
132#120×C61—41no—
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61—41no—
154C2×D4×C61—42no—
155C2×D4×C61—42no—
156C2×D4×C61—41no—
157C2×D4×C61—81no—
158C2×D4×C61—81no—
159C2×D4×C61C2×D411yescharacteristic
160C2×D4×C61C2×D411yescharacteristic
161C2×D4×C61C2×D411yescharacteristic
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—41no—
166C22×Dic61—41no—
167C22×Dic61—41no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—42no—
171C2×Dic122—41no—
172C2×Dic122—81no—
173C2×Dic122C2×D411yescharacteristic
174SD16×C61—42no—
175SD16×C61—42no—
176SD16×C61—81no—
177D8×C61—42no—
178D8×C61SD1614yescomplemented
179M16×C61—42no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—41no—
193C22⋊C4×C61—41no—
194C42×C61C2411yescharacteristic
195C22⋊Dic61—41no—
196C22⋊Dic61—41no—
197#11758200—41no—
198#11758200—81no—
199#11758200—81no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—41no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—21no—
207#11758196—41no—
208#11758196—41no—
209C4×Dic61—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—81no—
215C2×#92418—81no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—41no—
222C23×C61—82no—
223C23×C61—82no—
224C23×C61—161no—
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—81no—
236D4×C61—82no—
237D4×C61—82no—
238D4×C61—82no—
239D4×C61C2×SD1612yes—
240D4×C61C2×SD1612yes—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—82no—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—82no—
260C2×Dic61—161no—
261Dic122—21no—
262Dic122—21no—
263Dic122—41no—
264Dic122—41no—
265Dic122—82no—
266Dic122—81no—
267C8×C61—21no—
268C8×C61—41no—
269C8×C61C2×SD1612yes—
270#92418—21no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#19891—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—164no—
290C22×C61D4211yescharacteristic
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—81no—
296C4×C61C22×SD1611yescharacteristic
297C4×C61#57511yescharacteristic
298C4×C61#57611yescharacteristic
299C4×C61#57711yescharacteristic
300Dic61—41no—
301Dic61—41no—
302Dic61—81no—
303Dic61—81no—
304#3384—611no—
305#3298—611no—
306#3295—611no—
307#3272D6111yescharacteristic
308#1708—611no—
309#1691—611no—
310#1686—611no—
311#1679—611no—
312#1677—611no—
313#1643—611no—
314#1639—611no—
315#1635—611no—
316#1627—611no—
317#1590—611no—
318C8×D8—611no—
319C2×C61—42no—
320C2×C61—42no—
321C2×C61—81no—
322C2×C61—161no—
323C2×C61#327311yescharacteristic
324C2×C61D4×SD1611yescharacteristic
325C2×C61#329011yescharacteristic
326C2×AGL(1,ℤ/8)—21no—
327#554—611no—
328D4×Q8—611no—
329#547C2×D6111yescharacteristic
330D42C2×D6111yescharacteristic
331#501—1221no—
332#498—611no—
333#492C2×D6111yescharacteristic
334#487—611no—
335#487—611no—
336#486—611no—
337#486—611no—
338#483—1221no—
339#481—1221no—
340#478—1221no—
341#476—611no—
342#476—611no—
343#476—1221no—
344#475—611no—
345#474—611no—
346#468—21no—
347#465—1221no—
348#465—1221no—
349#461—611no—
350#461C2×D6111yescharacteristic
351#460—611no—
352#459C2×D6111yescharacteristic
353#450—21no—
354#450—1221no—
355#449—21no—
356#448—1221no—
357#446—611no—
358#442—1221no—
359C4×SD16—611no—
360C4×D8C2×D6111yescharacteristic
361#436C2×D6111yescharacteristic
362C8×D4—611no—
363C8×D4—611no—
364C2×#103—1221no—
365C2×#102—1221no—
366#335—611no—
367#334—611no—
368#333—611no—
369#332—611no—
370#332—611no—
371#331—611no—
372Q8⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375D4⋊C8—611no—
376C82—611no—
377C61#1989111yescharacteristic, complemented, Sylow
378C2×#41—21no—
379C22×Q8—1221no—
380C22×D4—21no—
381C22×D4—21no—
382AGL(1,ℤ/8)—44no—
383C2×SD16—21no—
384C2×SD16—21no—
385C2×SD16—21no—
386C2×SD16—611no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×D8—21no—
390C2×D8C22⋊D6111yescharacteristic
391C2×M16—21no—
392C22×C8—1221no—
393C22×C8—1221no—
394C4⋊Q8—611no—
395C4⋊Q8—611no—
396C4⋊Q8—1221no—
397C4⋊Q8C22×D6111yescharacteristic
398#127C22×D6111yescharacteristic
399#125—1221no—
400#123—41no—
401C22⋊Q8—21no—
402C22⋊Q8—1221no—
403C22⋊Q8—1221no—
404C22⋊Q8—2441no—
405C22⋊Q8—2441no—
406#121—21no—
407#121—21no—
408#121—21no—
409#120—41no—
410C4×Q8—611no—
411C4×D4—1221no—
412C4×D4—1221no—
413C4×D4C22×D6111yescharacteristic
414C4×D4C22×D6111yescharacteristic
415C4×D4C22×D6111yescharacteristic
416C2×C4⋊C4—21no—
417C2×C4⋊C4—1221no—
418#107—1221no—
419#107C22⋊D6111yescharacteristic
420#106—611no—
421#106—1221no—
422#106—1221no—
423#106—1221no—
424#106—1221no—
425#106—1221no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8C22×D6111yescharacteristic
431#103—21no—
432#103—611no—
433#103—1221no—
434#103—1221no—
435#103—1221no—
436#103—1221no—
437#103—2441no—
438#102—21no—
439#102—611no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—2441no—
447#102C22⋊D6111yescharacteristic
448#102C22⋊D6111yescharacteristic
449C22⋊C8—21no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452#97—1221no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8C22×D6111yescharacteristic
456C24—41no—
457#41—42no—
458#41—81no—
459C2×Q8—21no—
460C2×Q8—611no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—4881no—
464C2×Q8—2441no—
465C2×Q8—2442no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—41no—
474C2×D4—81no—
475C2×D4—81no—
476C2×D4D4×D6111yescharacteristic
477C2×D4C2×C22⋊D6111yescharacteristic
478C2×D4C2×C22⋊D6111yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—2441no—
484C22×C4—2441no—
485SD16—42no—
486SD16—42no—
487SD16—81no—
488SD16—4881no—
489SD16—4881no—
490SD16—2441no—
491D8—42no—
492D8#1175820014yescomplemented
493M16—42no—
494C2×C8—21no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—4881no—
502C2×C8—4881no—
503C2×C8C2×C22⋊D6111yescharacteristic
504C2×C8C2×C22⋊D6111yescharacteristic
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—41no—
508C4⋊C4—611no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C4⋊C4—2441no—
517C4⋊C4D4×D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C42—1221no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—21no—
531C23—21no—
532C23—41no—
533C23—82no—
534C23—82no—
535C23—161no—
536Q8—41no—
537Q8—41no—
538Q8—1221no—
539Q8—1221no—
540Q8—4881no—
541Q8—4882no—
542Q8—2441no—
543Q8—2441no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—81no—
553D4—82no—
554D4—82no—
555D4—82no—
556D4C2×#1175820012yes—
557D4C2×#1175820012yes—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—82no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—9761no—
572C2×C4—4882no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C22×C22⋊D6111yescharacteristic
577C2×C4#1451982211yescharacteristic
578C8—21no—
579C8—41no—
580C8—1221no—
581C8—1221no—
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C8—2441no—
586C8C2×#1175820012yes—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—164no—
600C22D4×C22⋊D6111yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—4881no—
607C4—4881no—
608C4—2441no—
609C4—2441no—
610C4C22×#1175820011yescharacteristic
611C4#1684055811yescharacteristic
612C4#1684056611yescharacteristic
613C4#1684057311yescharacteristic
614C2—42no—
615C2—42no—
616C2—81no—
617C2—161no—
618C2#2145742411yescharacteristic
619C2D4×#1175820011yescharacteristic
620C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3272C2
socleC22×C61D42
hypercenter#3272D61
hypocenterC61#19891
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3272

Complements

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

NH ≅ G/Ntype
1. #21458716C2semidirect
2. #21457678C2semidirect
3. #21457643C2semidirect
4. #21445418C2semidirect
5. #21445318C2semidirect
6. #21445288C2semidirect
7. #21445244C2semidirect
8. #21445226C2semidirect
9. #21445024C2semidirect
10. #21444998C2semidirect
11. #21444970C2semidirect
12. #21444921C2semidirect
13. #21444707C2semidirect
14. D8×#92418C2semidirect
17. D4×Dic122C22semidirect
19. #16840160C22semidirect
20. #16840114C22semidirect
21. #16840114C22semidirect
28. #16840052C22semidirect
30. #16840044C22semidirect
38. #16839892C22semidirect
42. D4×#92418C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839510C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
51. #16839492C22semidirect
54. D4⋊#92418C22semidirect
55. C8×#92418C22semidirect
178. D8×C61SD16semidirect
377. C61#19891semidirect
492. D8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214587162
#214576782
#214576432
#214454182
#214453182
#214452882
#214452442
#214452262
#214450242
#214449982
#214449702
#214449212
#214447072
D8×#924182
C61×#32722
#1989161