Finite Group Databasebeta

← #29669510

Subgroups — #29669510

Order 15616.

668 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458727C211yesmaximal, characteristic, complemented
2Q8×#11758200C211yesmaximal, characteristic, complemented
3#21457643C211yesmaximal, characteristic
4#21445438C211yesmaximal, characteristic, complemented
5#21445288C211yesmaximal, characteristic, complemented
6#21445233C211yesmaximal, characteristic, complemented
7#21445218C211yesmaximal, characteristic, complemented
8#21445192C211yesmaximal, characteristic, complemented
9#21445024C211yesmaximal, characteristic, complemented
10#21444984C211yesmaximal, characteristic, complemented
11#21444935C211yesmaximal, characteristic
12#21444896C211yesmaximal, characteristic, complemented
13#21444655C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15D4×SD16×C61C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic, complemented
17#16840469C2211yescharacteristic
18#16840169—21no—
19#16840167C2211yescharacteristic, complemented
20#16840158—21no—
21#16840158C2211yescharacteristic, complemented
22#16840134C2211yescharacteristic
23#16840114C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic
25#16840078—21no—
26#16840071—21no—
27#16840057C2211yescharacteristic, complemented
28#16840052—21no—
29#16840052C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16840044—21no—
33#16840044C2211yescharacteristic, complemented
34#16840040—21no—
35#16839989—21no—
36#16839959C2211yescharacteristic
37#16839913—21no—
38#16839892C2211yescharacteristic, complemented
39Q8×#92418C2211yescharacteristic, complemented
40#16839863—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519698—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56C22×SD16×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#465—21no—
61C61×#461C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C61×#448—21no—
65C4×SD16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758200—41no—
69C2×#11758200D411yescharacteristic
70C22×#92418—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519776—21no—
75#14519776—41no—
76#14519768—21no—
77#14519756—21no—
78#14519754—41no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122C2311yescharacteristic
82D4×Dic61—21no—
83C2×C4⋊Dic61—21no—
84#14519711—21no—
85#14519711—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519700—21no—
101#14519700—41no—
102#14519700D411yescharacteristic
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519687—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—44no—
123C2×SD16×C61D411yescharacteristic
124C22×C8×C61—21no—
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—41no—
130#121×C61—21no—
131#121×C61—21no—
132#120×C61—41no—
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—42no—
157C2×D4×C61—42no—
158C2×D4×C61—81no—
159C2×D4×C61—81no—
160C2×D4×C61C2×D411yescharacteristic
161C2×D4×C61C2×D411yescharacteristic
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—41no—
165C22×Dic61—41no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—41no—
169C2×Dic122C2×D411yescharacteristic
170SD16×C61—42no—
171SD16×C61—42no—
172SD16×C61—42no—
173SD16×C61—81no—
174SD16×C61SD1614yescomplemented
175C2×C8×C61—21no—
176C2×C8×C61—42no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C22⋊C4×C61—41no—
190C42×C61—21no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—41no—
193#11758200—22no—
194#11758200—81no—
195#11758200—82no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205#11758196—41no—
206C4×Dic61—21no—
207C4×Dic61—41no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—81no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—41no—
218C23×C61—82no—
219C23×C61—82no—
220C23×C61—161no—
221Q8×C61—21no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—82no—
225Q8×C61C2×SD1612yes—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—81no—
233D4×C61—82no—
234D4×C61—82no—
235D4×C61C2×SD1612yes—
236D4×C61#13712yes—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—82no—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—161no—
255Dic122—21no—
256Dic122—21no—
257Dic122—81no—
258Dic122—81no—
259Dic122—161no—
260C8×C61—21no—
261C8×C61—41no—
262C8×C61C2×SD1612yes—
263#92418—21no—
264#92418—21no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#19765—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—164no—
280C22×C61D4211yescharacteristic
281C4×C61—22no—
282C4×C61—21no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—81no—
287C4×C61C22×SD1611yescharacteristic
288C4×C61C2×#13711yescharacteristic
289C4×C61#57711yescharacteristic
290C4×C61#57711yescharacteristic
291Dic61—41no—
292Dic61—41no—
293Dic61—81no—
294Dic61—81no—
295#3385—611no—
296Q8×SD16—611no—
297#3295—611no—
298D4×SD16D6111yescharacteristic
299#1714—611no—
300#1686—611no—
301#1678—611no—
302#1675—611no—
303#1671—611no—
304#1643—611no—
305#1637—611no—
306#1629—611no—
307#1623—611no—
308#1582—611no—
309SD16⋊C8—611no—
310C2×C61—42no—
311C2×C61—42no—
312C2×C61—161no—
313C2×C61#326611yescharacteristic
314C2×C61D4×SD1611yescharacteristic
315C2×C61#328811yescharacteristic
316C22×SD16—21no—
317Q82—611no—
318#554—611no—
319D4×Q8C2×D6111yescharacteristic
320D42C2×D6111yescharacteristic
321#501—1221no—
322#500—611no—
323#498—611no—
324#498—1221no—
325#492—611no—
326#492C2×D6111yescharacteristic
327#487—611no—
328#486—611no—
329#481—1221no—
330#479—1221no—
331#477—611no—
332#476—611no—
333#476—611no—
334#476—611no—
335#476—1221no—
336#475—611no—
337#475—1221no—
338#474—1221no—
339#465—21no—
340#465—1221no—
341#461C2×D6111yescharacteristic
342#460—611no—
343#460C2×D6111yescharacteristic
344#450—21no—
345#450—1221no—
346#448—21no—
347#446—611no—
348C8×Q8—611no—
349#440—1221no—
350C4×SD16—611no—
351C4×SD16—1221no—
352C4×SD16C2×D6111yescharacteristic
353C8×D4—611no—
354C8×D4C2×D6111yescharacteristic
355C2×#102—1221no—
356#335—611no—
357#334—611no—
358#333—611no—
359#333—611no—
360#332—611no—
361#331—611no—
362Q8⋊C8—611no—
363Q8⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366#322—611no—
367C61#1976511yescharacteristic, complemented, Sylow
368C22×Q8—21no—
369C22×D4—21no—
370C22×D4—21no—
371C2×SD16—21no—
372C2×SD16—21no—
373C2×SD16—21no—
374C2×SD16—21no—
375C2×SD16—44no—
376C2×SD16—611no—
377C2×SD16—1221no—
378C2×SD16—2441no—
379C2×SD16C22⋊D6111yescharacteristic
380C22×C8—21no—
381C22×C8—1221no—
382C4⋊Q8—21no—
383C4⋊Q8—611no—
384C4⋊Q8—611no—
385C4⋊Q8—1221no—
386C4⋊Q8—1221no—
387C4⋊Q8—2441no—
388C4⋊Q8C22×D6111yescharacteristic
389#127C22×D6111yescharacteristic
390#125—1221no—
391C22⋊Q8—21no—
392C22⋊Q8—41no—
393C22⋊Q8—1221no—
394C22⋊Q8—2441no—
395#121—21no—
396#121—21no—
397#120—41no—
398C4×Q8—611no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8C22×D6111yescharacteristic
402C4×D4—21no—
403C4×D4—1221no—
404C4×D4C22×D6111yescharacteristic
405C4×D4C22×D6111yescharacteristic
406C2×C4⋊C4—1221no—
407#107—1221no—
408#107—1221no—
409#107—1221no—
410#106—611no—
411#106—1221no—
412#106—1221no—
413#106—1221no—
414#106—1221no—
415#106—1221no—
416#106—1221no—
417#106C22⋊D6111yescharacteristic
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—1221no—
422C4⋊C8—1221no—
423C4⋊C8C22×D6111yescharacteristic
424#103—21no—
425#103—611no—
426#103—1221no—
427#103—1221no—
428#103—2441no—
429#103C22⋊D6111yescharacteristic
430#102—21no—
431#102—611no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102C22⋊D6111yescharacteristic
439C22⋊C8—21no—
440C22⋊C8—1221no—
441#97—1221no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—1221no—
445C4×C8C22×D6111yescharacteristic
446C24—41no—
447C2×Q8—21no—
448C2×Q8—41no—
449C2×Q8—42no—
450C2×Q8—81no—
451C2×Q8—611no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—2441no—
455C2×Q8C2×C22⋊D6111yescharacteristic
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4D4×D6111yescharacteristic
466C2×D4C2×C22⋊D6111yescharacteristic
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—41no—
470C22×C4—2441no—
471SD16—42no—
472SD16—42no—
473SD16—42no—
474SD16—81no—
475SD16—1222no—
476SD16—4881no—
477SD16—4882no—
478SD16#1175820014yescomplemented
479C2×C8—21no—
480C2×C8—42no—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—4881no—
488C2×C8C2×C22⋊D6111yescharacteristic
489C2×C8C2×C22⋊D6111yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—611no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—2441no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4D4×D6111yescharacteristic
505C4⋊C4C2×C22⋊D6111yescharacteristic
506C4⋊C4C2×C22⋊D6111yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—2441no—
512C42—21no—
513C42—1221no—
514C42—2441no—
515C42C23×D6111yescharacteristic
516C23—21no—
517C23—21no—
518C23—41no—
519C23—82no—
520C23—82no—
521C23—161no—
522Q8—21no—
523Q8—41no—
524Q8—41no—
525Q8—82no—
526Q8—1221no—
527Q8—1221no—
528Q8—9761no—
529Q8—4881no—
530Q8—4881no—
531Q8C2×#1175820012yes—
532D4—21no—
533D4—21no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—81no—
539D4—82no—
540D4—82no—
541D4C2×#1175820012yes—
542D4#1451980612yes—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—9761no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C22×C22⋊D6111yescharacteristic
560C2×C4#1451982211yescharacteristic
561C8—21no—
562C8—41no—
563C8—1221no—
564C8—1221no—
565C8—2441no—
566C8—2441no—
567C8—2441no—
568C8—2441no—
569C8C2×#1175820012yes—
570C22—22no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—164no—
580C22D4×C22⋊D6111yescharacteristic
581C4—22no—
582C4—21no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—4881no—
588C4—4881no—
589C4—2441no—
590C4—2441no—
591C4C22×#1175820011yescharacteristic
592C4C2×#1451980611yescharacteristic
593C4#1684056811yescharacteristic
594C4#1684057311yescharacteristic
595C2—42no—
596C2—42no—
597C2—161no—
598C2#2145732911yescharacteristic
599C2D4×#1175820011yescharacteristic
600C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×SD16×C61C2
socleC22×C61D42
prime radicalD42×C61C22
hypercenterD4×SD16D61
hypocenterC61#19765
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×SD16

Complements

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

NH ≅ G/Ntype
1. #21458727C2semidirect
2. Q8×#11758200C2semidirect
4. #21445438C2semidirect
5. #21445288C2semidirect
6. #21445233C2semidirect
7. #21445218C2semidirect
8. #21445192C2semidirect
9. #21445024C2semidirect
10. #21444984C2semidirect
12. #21444896C2semidirect
13. #21444655C2semidirect
14. SD16⋊#92418C2semidirect
16. Q8×Dic122C22semidirect
19. #16840167C22semidirect
21. #16840158C22semidirect
23. #16840114C22semidirect
27. #16840057C22semidirect
33. #16840044C22semidirect
38. #16839892C22semidirect
39. Q8×#92418C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839510C22semidirect
48. #16839510C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. #16839483C22semidirect
174. SD16×C61SD16semidirect
367. C61#19765semidirect
478. SD16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214587272
Q8×#117582002
#214576432
#214454382
#214452882
#214452332
#214452182
#214451922
#214450242
#214449842
#214449352
#214448962
#214446552
SD16⋊#924182
D4×SD16×C612
#1976561