Finite Group Databasebeta

← #29669514

Subgroups — #29669514

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#21458733C211yesmaximal, characteristic, complemented
2#21458674C211yesmaximal, characteristic, complemented
3#21457643C211yesmaximal, characteristic
4D4×#11758200C211yesmaximal, characteristic
5#21445439C211yesmaximal, characteristic, complemented
6#21445290C211yesmaximal, characteristic, complemented
7#21445234C211yesmaximal, characteristic, complemented
8#21445216C211yesmaximal, characteristic, complemented
9#21445196C211yesmaximal, characteristic, complemented
10#21445028C211yesmaximal, characteristic, complemented
11#21444987C211yesmaximal, characteristic, complemented
12#21444895C211yesmaximal, characteristic, complemented
13#21444652C211yesmaximal, characteristic, complemented
14#21444628C211yesmaximal, characteristic, complemented
15C61×#1629C211yesmaximal, characteristic
16C22×#11758200—21no—
17#16840492C2211yescharacteristic, complemented
18#16840469C2211yescharacteristic
19D4×Dic122C2211yescharacteristic
20#16840169—21no—
21#16840165C2211yescharacteristic, complemented
22#16840159—21no—
23#16840158C2211yescharacteristic, complemented
24#16840134C2211yescharacteristic
25#16840113C2211yescharacteristic, complemented
26#16840108C2211yescharacteristic
27#16840078—21no—
28#16840072—21no—
29#16840058C2211yescharacteristic, complemented
30#16840052C2211yescharacteristic
31#16840050—21no—
32#16840048C2211yescharacteristic, complemented
33#16840046—21no—
34#16840040—21no—
35#16839989—21no—
36#16839989—21no—
37#16839967C2211yescharacteristic
38#16839959C2211yescharacteristic
39#16839959C2211yescharacteristic
40#16839913—21no—
41#16839913—21no—
42#16839900—21no—
43#16839889C2211yescharacteristic, complemented
44#16839886C2211yescharacteristic, complemented
45#16839859—21no—
46#16839853—21no—
47C4×#11758200C2211yescharacteristic
48C4×#11758200C2211yescharacteristic
49D4×#92418C2211yescharacteristic
50D4×#92418C2211yescharacteristic
51C2×#14519698—21no—
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839509C2211yescharacteristic, complemented
57#16839494C2211yescharacteristic, complemented
58#16839494C2211yescharacteristic, complemented
59#16839484C2211yescharacteristic, complemented
60D42×C61C2211yescharacteristic
61C61×#492C2211yescharacteristic
62C61×#476C2211yescharacteristic
63C61×#476C2211yescharacteristic
64C61×#331C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic
66C61×D4⋊C8C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—44no—
74C2×#11758200D411yescharacteristic
75C2×#11758200D411yescharacteristic
76C22×#92418—21no—
77C22×#92418—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519776—21no—
81#14519776—21no—
82#14519776—21no—
83#14519775—41no—
84#14519768—21no—
85#14519756—21no—
86#14519756—21no—
87#14519754—41no—
88#14519754—41no—
89C4×Dic122C2311yescharacteristic
90C4×Dic122C2311yescharacteristic
91#14519742—21no—
92#14519742—21no—
93D4×Dic61—21no—
94D4×Dic61—21no—
95C2×C4⋊Dic61—21no—
96#14519711—21no—
97#14519710—21no—
98#14519710—21no—
99#14519709—21no—
100#14519708—21no—
101#14519708—21no—
102#14519708—21no—
103#14519708D411yescharacteristic
104#14519708D411yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107#14519706—21no—
108#14519706—21no—
109#14519702—21no—
110#14519702—21no—
111#14519702—41no—
112#14519700—21no—
113#14519700D411yescharacteristic
114#14519700D411yescharacteristic
115#14519698—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519698—21no—
120#14519698D411yescharacteristic
121#14519698D411yescharacteristic
122C22⋊#92418—21no—
123C22⋊#92418—21no—
124#14519687—21no—
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C22×D4×C61—21no—
129C22×D4×C61—21no—
130C2×SD16×C61—41no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133#121×C61—21no—
134#121×C61—21no—
135#120×C61—41no—
136C4×D4×C61C2311yescharacteristic
137C4×D4×C61C2311yescharacteristic
138#106×C61—21no—
139#106×C61—21no—
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#102×C61—21no—
143#102×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C2×Q8×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—41no—
151C2×D4×C61—41no—
152C2×D4×C61—42no—
153C2×D4×C61—42no—
154C2×D4×C61—81no—
155C2×D4×C61—81no—
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×Dic61—41no—
161C22×Dic61—41no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—42no—
165C2×Dic122—41no—
166C2×Dic122—41no—
167C2×Dic122—81no—
168C2×Dic122C2×D411yescharacteristic
169C2×Dic122C2×D411yescharacteristic
170SD16×C61—82no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—41no—
181C42×C61C2411yescharacteristic
182C22⋊Dic61—41no—
183C22⋊Dic61—41no—
184#11758200—22no—
185#11758200—42no—
186#11758200—42no—
187#11758200—42no—
188#11758200—81no—
189#11758200—81no—
190#11758200SD1614yescomplemented
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—21no—
199#11758196—21no—
200#11758196—41no—
201#11758196—41no—
202#11758196—41no—
203#11758196—41no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—41no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—42no—
210C2×#92418—81no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
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—81no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—81no—
229D4×C61—82no—
230D4×C61—82no—
231D4×C61C2×SD1612yes—
232D4×C61#13712yes—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—82no—
250C2×Dic61—161no—
251Dic122—21no—
252Dic122—21no—
253Dic122—21no—
254Dic122—41no—
255Dic122—41no—
256Dic122—82no—
257Dic122—81no—
258Dic122—161no—
259Dic122C2×SD1612yes—
260C8×C61—41no—
261C8×C61—41no—
262C8×C61—41no—
263#92418—21no—
264#92418—21no—
265#92418—21no—
266#92418—41no—
267#92418—41no—
268#92418C2×SD1612yes—
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—21no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61C22×SD1611yescharacteristic
286C4×C61C2×#13711yescharacteristic
287C4×C61#57711yescharacteristic
288C4×C61#57711yescharacteristic
289Dic61—22no—
290Dic61—41no—
291Dic61—41no—
292Dic61—41no—
293Dic61—81no—
294Dic61—81no—
295#3385—611no—
296Q8×SD16—611no—
297#3295—611no—
298D4×SD16—611no—
299#1714—611no—
300#1686—611no—
301#1678—611no—
302#1675—611no—
303#1671—611no—
304#1643—611no—
305#1637—611no—
306#1629D6111yescharacteristic
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—1221no—
317Q82—611no—
318#554—611no—
319D4×Q8—611no—
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—1221no—
334#476C2×D6111yescharacteristic
335#476C2×D6111yescharacteristic
336#475—611no—
337#475—1221no—
338#474—1221no—
339#465—1221no—
340#465—1221no—
341#461—611no—
342#460—611no—
343#460—611no—
344#450—1221no—
345#450—1221no—
346#448—1221no—
347#446—611no—
348C8×Q8—611no—
349#440—1221no—
350C4×SD16—611no—
351C4×SD16—611no—
352C4×SD16—1221no—
353C8×D4—611no—
354C8×D4—611no—
355C2×#102—1221no—
356#335—611no—
357#334—611no—
358#333—611no—
359#333—611no—
360#332—611no—
361#331C2×D6111yescharacteristic
362Q8⋊C8—611no—
363Q8⋊C8—611no—
364D4⋊C8C2×D6111yescharacteristic
365D4⋊C8C2×D6111yescharacteristic
366#322—611no—
367C61#1976511yescharacteristic, complemented, Sylow
368C22×Q8—1221no—
369C22×D4—21no—
370C22×D4—21no—
371C2×SD16—41no—
372C2×SD16—611no—
373C2×SD16—611no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×SD16—2444no—
380C22×C8—1221no—
381C22×C8—1221no—
382C4⋊Q8—611no—
383C4⋊Q8—611no—
384C4⋊Q8—1221no—
385C4⋊Q8—1221no—
386C4⋊Q8—1221no—
387C4⋊Q8—2441no—
388C4⋊Q8C22×D6111yescharacteristic
389#127C22×D6111yescharacteristic
390#125—1221no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—2441no—
394C22⋊Q8—2441no—
395#121—21no—
396#121—21no—
397#120—41no—
398C4×Q8—611no—
399C4×Q8—611no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×D4—1221no—
403C4×D4—1221no—
404C4×D4C22×D6111yescharacteristic
405C4×D4C22×D6111yescharacteristic
406C2×C4⋊C4—1221no—
407#107—1221no—
408#107—1221no—
409#107—1221no—
410#106—21no—
411#106—21no—
412#106—611no—
413#106—611no—
414#106—1221no—
415#106—1221no—
416#106—1221no—
417#106—1221no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—1221no—
421C4⋊C8—1221no—
422C4⋊C8C22×D6111yescharacteristic
423C4⋊C8C22×D6111yescharacteristic
424#103—611no—
425#103—611no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—2441no—
430#102—21no—
431#102—21no—
432#102—611no—
433#102—611no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439C22⋊C8—1221no—
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—611no—
449C2×Q8—611no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—4881no—
453C2×Q8—2441no—
454C2×Q8—2441no—
455C2×Q8—2442no—
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×D4D4×D6111yescharacteristic
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—2441no—
470C22×C4—2441no—
471SD16—82no—
472SD16—614no—
473SD16—1222no—
474SD16—4881no—
475SD16—4881no—
476SD16—2442no—
477SD16—2442no—
478SD16—2442no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—4881no—
489C2×C8—2442no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—611no—
493C4⋊C4—611no—
494C4⋊C4—1221no—
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⋊C4—2441no—
505C4⋊C4D4×D6111yescharacteristic
506C4⋊C4D4×D6111yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—41no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C42—1221no—
513C42—1221no—
514C42—2441no—
515C42C23×D6111yescharacteristic
516C23—21no—
517C23—21no—
518C23—41no—
519C23—82no—
520C23—82no—
521C23—161no—
522Q8—81no—
523Q8—612no—
524Q8—1221no—
525Q8—1221no—
526Q8—1221no—
527Q8—9761no—
528Q8—4881no—
529Q8—4882no—
530Q8—2441no—
531Q8—2441no—
532D4—21no—
533D4—21no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—81no—
539D4—82no—
540D4—82no—
541D4SD16×D6112yes—
542D4#1451980612yes—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—9761no—
555C2×C4—4882no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4#1451982211yescharacteristic
561C8—41no—
562C8—41no—
563C8—41no—
564C8—612no—
565C8—1221no—
566C8—1221no—
567C8—1221no—
568C8—2441no—
569C8—2441no—
570C22—22no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—164no—
580C22#1684039411yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—1222no—
586C4—4881no—
587C4—4881no—
588C4—2441no—
589C4—2441no—
590C4—2441no—
591C4C2×SD16×D6111yescharacteristic
592C4C2×#1451980611yescharacteristic
593C4#1684057211yescharacteristic
594C4#1684057211yescharacteristic
595C2—42no—
596C2—42no—
597C2—161no—
598C2#2145733311yescharacteristic
599C2#2145751311yescharacteristic
600C2#2145758011yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1629C2
socleC22×C61D42
prime radicalD42×C61C22
hypercenter#1629D61
hypocenterC61#19765
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1629

Complements

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

NH ≅ G/Ntype
1. #21458733C2semidirect
2. #21458674C2semidirect
5. #21445439C2semidirect
6. #21445290C2semidirect
7. #21445234C2semidirect
8. #21445216C2semidirect
9. #21445196C2semidirect
10. #21445028C2semidirect
11. #21444987C2semidirect
12. #21444895C2semidirect
13. #21444652C2semidirect
14. #21444628C2semidirect
17. #16840492C22semidirect
21. #16840165C22semidirect
23. #16840158C22semidirect
25. #16840113C22semidirect
29. #16840058C22semidirect
32. #16840048C22semidirect
43. #16839889C22semidirect
44. #16839886C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839510C22semidirect
56. #16839509C22semidirect
57. #16839494C22semidirect
58. #16839494C22semidirect
59. #16839484C22semidirect
190. #11758200SD16semidirect
367. C61#19765semidirect

Maximal subgroups

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

subgroupindex
#214587332
#214586742
#214576432
D4×#117582002
#214454392
#214452902
#214452342
#214452162
#214451962
#214450282
#214449872
#214448952
#214446522
#214446282
C61×#16292
#1976561