Finite Group Databasebeta

← #29670343

Subgroups — #29670343

Order 15616.

660 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458504C211yesmaximal, characteristic, complemented
2#21458457C211yesmaximal, characteristic, complemented
3#21458053C211yesmaximal, characteristic
4#21457510C211yesmaximal, characteristic
5#21445471C211yesmaximal, characteristic, complemented
6#21445365C211yesmaximal, characteristic, complemented
7#21445261C211yesmaximal, characteristic, complemented
8#21445245C211yesmaximal, characteristic, complemented
9#21445165C211yesmaximal, characteristic, complemented
10#21445078C211yesmaximal, characteristic, complemented
11#21445070C211yesmaximal, characteristic, complemented
12#21444995C211yesmaximal, characteristic
13#21444659C211yesmaximal, characteristic, complemented
14#21444634C211yesmaximal, characteristic, complemented
15C61×#1626C211yesmaximal, characteristic, complemented
16C2×SD16×D61—21no—
17#16840485C2211yescharacteristic
18Q8×D244C2211yescharacteristic
19#16840397C2211yescharacteristic
20#16840167C2211yescharacteristic, complemented
21#16840134C2211yescharacteristic
22#16840119—21no—
23#16840114C2211yescharacteristic, complemented
24#16840112—21no—
25#16840112C2211yescharacteristic, complemented
26#16840108C2211yescharacteristic
27#16840084—21no—
28#16840066—21no—
29#16840047—21no—
30#16840047C2211yescharacteristic, complemented
31#16840043C2211yescharacteristic
32#16840039C2211yescharacteristic
33#16840005—21no—
34#16839990—21no—
35#16839984—21no—
36#16839972C2211yescharacteristic, complemented
37#16839968C2211yescharacteristic
38#16839964—21no—
39#16839964C2211yescharacteristic
40#16839914—21no—
41#16839911—21no—
42#16839899—21no—
43Q8×#92418C2211yescharacteristic, complemented
44Q8×#92418C2211yescharacteristic, complemented
45C4×#11758202—21no—
46C4×#11758201—21no—
47C4×#11758201C2211yescharacteristic
48C4×#11758201C2211yescharacteristic
49#16839839C2211yescharacteristic
50#16839827C2211yescharacteristic
51#16839765—21no—
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839507C2211yescharacteristic, complemented
56#16839506C2211yescharacteristic
57#16839491C2211yescharacteristic
58#16839491C2211yescharacteristic
59#16839484C2211yescharacteristic, complemented
60C61×#557C2211yescharacteristic, complemented
61C61×#486C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#460C2211yescharacteristic
64C61×#332C2211yescharacteristic, complemented
65C61×Q8⋊C8C2211yescharacteristic, complemented
66C61×Q8⋊C8C2211yescharacteristic, complemented
67C2×Q8×D61—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×#11758202—21no—
71C2×#11758190—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201D411yescharacteristic
76C2×#11758201D411yescharacteristic
77SD16×D61—44no—
78C2×#11758189—21no—
79C2×C8×D61—21no—
80#14519777—21no—
81#14519777C2311yescharacteristic
82#14519777C2311yescharacteristic
83#14519772—41no—
84#14519765—21no—
85#14519757—21no—
86#14519757—41no—
87#14519753—21no—
88#14519748—21no—
89#14519748—21no—
90#14519745—41no—
91C4×Dic122—21no—
92C4×D244—21no—
93C4×D244C2311yescharacteristic
94C4×D244C2311yescharacteristic
95#14519735—21no—
96#14519711—21no—
97#14519711—41no—
98#14519710—21no—
99#14519708—21no—
100#14519708D411yescharacteristic
101#14519708D411yescharacteristic
102C4⋊#92418—21no—
103C4⋊#92418—21no—
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106#14519702—21no—
107#14519701—21no—
108#14519701—21no—
109#14519701—21no—
110#14519701—21no—
111#14519701D411yescharacteristic
112#14519701D411yescharacteristic
113#14519700—21no—
114#14519700—21no—
115#14519700—21no—
116#14519699—21no—
117#14519698—21no—
118#14519698—21no—
119#14519697—21no—
120#14519697D411yescharacteristic
121#14519697D411yescharacteristic
122#14519689—21no—
123#14519689—21no—
124C4×#92418—21no—
125C4×#92418—21no—
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C2×SD16×C61—21no—
129C2×SD16×C61—21no—
130C4⋊Q8×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133#125×C61—21no—
134#125×C61—41no—
135C4×Q8×C61—21no—
136C4×Q8×C61—21no—
137C4×Q8×C61C2311yescharacteristic
138C4×Q8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#102×C61—21no—
142#102×C61—21no—
143#102×C61—41no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—41no—
154Q8×D61—42no—
155Q8×D61—81no—
156D4×D61—42no—
157D4×D61—42no—
158D4×D61—81no—
159D4×D61—81no—
160C2×D244C2×D411yescharacteristic
161C2×D244C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168SD16×C61—42no—
169SD16×C61—81no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—41no—
185C42×C61C2411yescharacteristic
186C22⋊Dic61—41no—
187#11758202—42no—
188#11758201—22no—
189#11758201—22no—
190#11758201—42no—
191#11758200—42no—
192#11758200—81no—
193#11758198—21no—
194#11758198—21no—
195#11758198—41no—
196#11758198—41no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61C2×D411yescharacteristic
204C4⋊Dic61C2×D411yescharacteristic
205#11758196—41no—
206C4×Dic61—41no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213#11758190—42no—
214#11758189—81no—
215C8×D61—42no—
216C23×C61—41no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—82no—
220C22×D61—82no—
221C22×D61—161no—
222Q8×C61—21no—
223Q8×C61—21no—
224Q8×C61—21no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61AGL(1,ℤ/8)12yes—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—81no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C22⋊D61—82no—
242C22⋊D61—82no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247D244—21no—
248D244—21no—
249D244—21no—
250D244#13512yes—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—82no—
256Dic122—41no—
257Dic122—41no—
258Dic122—41no—
259Dic122—41no—
260Dic122—82no—
261C8×C61—41no—
262C8×C61—41no—
263C8×C61—41no—
264#92418—21no—
265#92418—21no—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418—41no—
270#19821—611nomaximal, Sylow
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61D4211yescharacteristic
275C2×D61—22no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—164no—
282C4×C61—22no—
283C4×C61—21no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61C2×#13511yescharacteristic
289C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
290C4×C61#57711yescharacteristic
291C4×C61#57711yescharacteristic
292Dic61—41no—
293Dic61—41no—
294Dic61—81no—
295Dic61—81no—
296#3364—611no—
297#3359—611no—
298#3324—611no—
299D4×SD16—611no—
300#1719—611no—
301#1699—611no—
302#1682—611no—
303#1679—611no—
304#1666—611no—
305#1650—611no—
306#1649—611no—
307#1639—611no—
308#1626D6111yescharacteristic, complemented
309#1583—611no—
310SD16⋊C8—611no—
311C2×C61—161no—
312C2×C61#326511yescharacteristic
313C2×C61#328211yescharacteristic
314C2×C61#328911yescharacteristic
315D61—42no—
316D61—81no—
317C22×SD16—1221no—
318#557C2×D6111yescharacteristic, complemented
319#556—611no—
320D4×Q8—611no—
321D42—611no—
322#500—611no—
323#492—611no—
324#488—1221no—
325#487—611no—
326#487—611no—
327#487—1221no—
328#486—611no—
329#486C2×D6111yescharacteristic
330#482—1221no—
331#479—1221no—
332#475—611no—
333#475—1221no—
334#474—611no—
335#474—611no—
336#468—1221no—
337#465—1221no—
338#464—1221no—
339#462—611no—
340#461—611no—
341#461—611no—
342#461—1221no—
343#460C2×D6111yescharacteristic
344#460C2×D6111yescharacteristic
345#450—1221no—
346#450—1221no—
347#448—1221no—
348C8×Q8—611no—
349C8×Q8—611no—
350C4×Q16—1221no—
351C4×SD16—611no—
352C4×SD16—611no—
353C4×SD16—1221no—
354#436—611no—
355C8×D4—611no—
356#418—1221no—
357#335—611no—
358#334—611no—
359#332—611no—
360#332—611no—
361#332C2×D6111yescharacteristic, complemented
362#331—611no—
363Q8⋊C8C2×D6111yescharacteristic, complemented
364Q8⋊C8C2×D6111yescharacteristic, complemented
365D4⋊C8—611no—
366D4⋊C8—611no—
367#322—611no—
368C61#1982111yescharacteristic, complemented, Sylow
369C22×Q8—1221no—
370C22×D4—1221no—
371C22×D4—1221no—
372C2×Q16—1221no—
373C2×SD16—21no—
374C2×SD16—21no—
375C2×SD16—611no—
376C2×SD16—611no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×SD16—2444no—
382C2×M16—1221no—
383C22×C8—1221no—
384C4⋊Q8—21no—
385C4⋊Q8—611no—
386C4⋊Q8—611no—
387C4⋊Q8—1221no—
388C4⋊Q8C22×D6111yescharacteristic
389#127C22×D6111yescharacteristic
390#126—2441no—
391#125—21no—
392#125—41no—
393#124—1221no—
394C22⋊Q8—1221no—
395C22⋊Q8—1221no—
396C22⋊Q8—2441no—
397#121—1221no—
398#121—1221no—
399#120—2441no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—1221no—
403C4×Q8C22×D6111yescharacteristic
404C4×Q8C22×D6111yescharacteristic
405C4×D4—611no—
406C4×D4—611no—
407C4×D4—1221no—
408#117—1221no—
409#107—1221no—
410#107—1221no—
411#107—2441no—
412#106—611no—
413#106—611no—
414#106—1221no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419C4⋊C8C22×D6111yescharacteristic
420C4⋊C8C22×D6111yescharacteristic
421#103—611no—
422#103—611no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#103—1221no—
431#102—21no—
432#102—21no—
433#102—41no—
434#102—611no—
435#102—611no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440C22⋊C8—1221no—
441C22⋊C8—1221no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—1221no—
445C4×C8—1221no—
446C4×C8C22×D6111yescharacteristic
447C24—2441no—
448C2×Q8—21no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—4881no—
452C2×Q8—2441no—
453C2×Q8—2442no—
454C2×Q8D4×D6111yescharacteristic
455C2×Q8D4×D6111yescharacteristic
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—611no—
460C2×D4—611no—
461C2×D4—4881no—
462C2×D4—4881no—
463C2×D4—2441no—
464C2×D4—2441no—
465C2×D4—2442no—
466C2×D4—2442no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—2441no—
470Q16—2442no—
471SD16—42no—
472SD16—81no—
473SD16—1222no—
474SD16—1222no—
475SD16—4881no—
476SD16—2442no—
477SD16—2442no—
478SD16—2442no—
479M16—4881no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—2442no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—611no—
498C4⋊C4—611no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—2441no—
504C4⋊C4—2441no—
505C4⋊C4D4×D6111yescharacteristic
506C4⋊C4D4×D6111yescharacteristic
507C22⋊C4—1221no—
508C22⋊C4—1221no—
509C22⋊C4—2441no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—2441no—
517C42C23×D6111yescharacteristic
518C23—41no—
519C23—1221no—
520C23—1221no—
521C23—9761no—
522C23—4882no—
523C23—4882no—
524Q8—21no—
525Q8—21no—
526Q8—21no—
527Q8—41no—
528Q8—41no—
529Q8—4882no—
530Q8—2441no—
531Q8—2441no—
532Q8—2441no—
533Q8—2441no—
534Q8#1451980412yes—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—81no—
540D4—612no—
541D4—1221no—
542D4—1221no—
543D4—1221no—
544D4—4882no—
545D4—4882no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—4882no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4#1451982211yescharacteristic
563C8—41no—
564C8—41no—
565C8—41no—
566C8—1221no—
567C8—1221no—
568C8—1221no—
569C8—1221no—
570C8—2441no—
571C8—2441no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—1222no—
576C22—9764no—
577C22—2441no—
578C22—2441no—
579C22—2441no—
580C22—2441no—
581C22—2441no—
582C22#1684039411yescharacteristic
583C4—22no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—4881no—
590C4—4881no—
591C4—2441no—
592C4—2441no—
593C4#135×D6111yescharacteristic
594C4C2×#1451980411yescharacteristic
595C4#1684057111yescharacteristic
596C4#1684057211yescharacteristic
597C2—161no—
598C2—4881no—
599C2—2442no—
600C2#2145730711yescharacteristic
601C2#2145752911yescharacteristic
602C2#2145759611yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1626C2
socleC22×C61D42
prime radical#16840397C22
hypercenter#1626D61
hypocenterC61#19821
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1626

Complements

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

NH ≅ G/Ntype
1. #21458504C2semidirect
2. #21458457C2semidirect
5. #21445471C2semidirect
6. #21445365C2semidirect
7. #21445261C2semidirect
8. #21445245C2semidirect
9. #21445165C2semidirect
10. #21445078C2semidirect
11. #21445070C2semidirect
13. #21444659C2semidirect
14. #21444634C2semidirect
15. C61×#1626C2semidirect
20. #16840167C22semidirect
23. #16840114C22semidirect
25. #16840112C22semidirect
30. #16840047C22semidirect
36. #16839972C22semidirect
43. Q8×#92418C22semidirect
44. Q8×#92418C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839509C22semidirect
55. #16839507C22semidirect
59. #16839484C22semidirect
60. C61×#557C22semidirect
64. C61×#332C22semidirect
65. C61×Q8⋊C8C22semidirect
66. C61×Q8⋊C8C22semidirect
308. #1626D61semidirect
318. #557C2×D61semidirect
361. #332C2×D61semidirect
363. Q8⋊C8C2×D61semidirect
364. Q8⋊C8C2×D61semidirect
368. C61#19821semidirect

Maximal subgroups

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

subgroupindex
#214585042
#214584572
#214580532
#214575102
#214454712
#214453652
#214452612
#214452452
#214451652
#214450782
#214450702
#214449952
#214446592
#214446342
C61×#16262
#1982161