Finite Group Databasebeta

← #29670351

Subgroups — #29670351

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#21458457C211yesmaximal, characteristic, complemented
2#21458044C211yesmaximal, characteristic
3SD16×D244C211yesmaximal, characteristic
4#21445474C211yesmaximal, characteristic, complemented
5#21445364C211yesmaximal, characteristic, complemented
6#21445265C211yesmaximal, characteristic, complemented
7#21445239C211yesmaximal, characteristic, complemented
8#21445167C211yesmaximal, characteristic, complemented
9#21445075C211yesmaximal, characteristic, complemented
10#21445066C211yesmaximal, characteristic, complemented
11#21444997C211yesmaximal, characteristic
12#21444915C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15C61×#3364C211yesmaximal, characteristic, complemented
16C2×SD16×D61—21no—
17Q8×D244C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840165C2211yescharacteristic, complemented
20#16840135C2211yescharacteristic
21#16840112—21no—
22#16840112C2211yescharacteristic, complemented
23#16840110C2211yescharacteristic
24#16840110C2211yescharacteristic
25#16840086—21no—
26#16840047—21no—
27#16840047C2211yescharacteristic, complemented
28#16840039C2211yescharacteristic
29#16840010—21no—
30#16839992—21no—
31#16839967C2211yescharacteristic
32#16839964—21no—
33#16839964C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839962C2211yescharacteristic
36#16839914—21no—
37#16839912—21no—
38#16839902—21no—
39Q8×#92418C2211yescharacteristic, complemented
40C4×#11758201—21no—
41C4×#11758201C2211yescharacteristic
42#16839840C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#16839766—21no—
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C61×#557C2211yescharacteristic, complemented
57C61×#556C2211yescharacteristic
58C61×#488—21no—
59C61×#487C2211yescharacteristic, complemented
60C61×#479—21no—
61C61×#474C2211yescharacteristic
62C61×#464—21no—
63C61×#462C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×Q16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×Q8×D61—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70C2×#11758190—21no—
71C2×#11758190—21no—
72C2×#11758200—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201D411yescharacteristic
77SD16×D61—44no—
78C2×#11758194—21no—
79C2×C8×D61—21no—
80#14519777—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519757—21no—
84#14519757—41no—
85#14519752—21no—
86#14519751—21no—
87#14519747—41no—
88C4×D244—21no—
89C4×D244C2311yescharacteristic
90#14519710—21no—
91#14519710—41no—
92#14519708—21no—
93#14519708D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×Q16×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121#126×C61—41no—
122#125×C61—21no—
123#125×C61—41no—
124#124×C61—21no—
125C22⋊Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#107×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C2×C22⋊D61—41no—
153Q8×D61—42no—
154Q8×D61—81no—
155D4×D61—42no—
156D4×D61—81no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—42no—
161C2×D244—41no—
162C2×D244—81no—
163C2×D244C2×D411yescharacteristic
164C2×C4×D61—21no—
165C2×C4×D61—41no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168Q16×C61—42no—
169SD16×C61—22no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—41no—
192C42×C61—41no—
193C42×C61C2411yescharacteristic
194#11758201—22no—
195#11758201—42no—
196#11758201—42no—
197#11758201—81no—
198#11758200—42no—
199#11758198—21no—
200#11758198—41no—
201#11758198—41no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—41no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758194—81no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213#11758190—42no—
214#11758190—81no—
215C8×D61—42no—
216C23×C61—21no—
217C22×D61—21no—
218C22×D61—41no—
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×C61—41no—
228Q8×C61—41no—
229Q8×C61AGL(1,ℤ/8)12yes—
230D4×C61—21no—
231D4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C22⋊D61—82no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247D244—21no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—81no—
253D244—82no—
254D244#13512yes—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—82no—
258Dic122—41no—
259Dic122—41no—
260Dic122—82no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—41no—
264#92418—21no—
265#92418—21no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#19821—611nomaximal, Sylow
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61D4211yescharacteristic
275C2×D61—22no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—164no—
282C4×C61—22no—
283C4×C61—21no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61C2×#13511yescharacteristic
291C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C61#57711yescharacteristic
293C4×C61#57711yescharacteristic
294Dic61—41no—
295Dic61—81no—
296#3364D6111yescharacteristic, complemented
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#1626—611no—
309#1583—611no—
310SD16⋊C8—611no—
311C2×C61—81no—
312C2×C61#326511yescharacteristic
313C2×C61#328211yescharacteristic
314C2×C61#328911yescharacteristic
315D61—42no—
316D61—161no—
317C22×SD16—1221no—
318#557C2×D6111yescharacteristic, complemented
319#556C2×D6111yescharacteristic
320D4×Q8—611no—
321D42—611no—
322#500—611no—
323#492—611no—
324#488—21no—
325#487—611no—
326#487—1221no—
327#487C2×D6111yescharacteristic, complemented
328#486—611no—
329#486—611no—
330#482—1221no—
331#479—21no—
332#475—611no—
333#475—1221no—
334#474—611no—
335#474C2×D6111yescharacteristic
336#468—1221no—
337#465—1221no—
338#464—21no—
339#462C2×D6111yescharacteristic, complemented
340#461—611no—
341#461—611no—
342#461—1221no—
343#460—611no—
344#460—611no—
345#450—1221no—
346#450—1221no—
347#448—1221no—
348C8×Q8—611no—
349C8×Q8C2×D6111yescharacteristic, complemented
350C4×Q16—21no—
351C4×SD16—611no—
352C4×SD16—1221no—
353C4×SD16C2×D6111yescharacteristic
354#436—611no—
355C8×D4—611no—
356#418—1221no—
357#335—611no—
358#334—611no—
359#332—611no—
360#332—611no—
361#332—611no—
362#331—611no—
363Q8⋊C8—611no—
364Q8⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8—611no—
367#322—611no—
368C61#1982111yescharacteristic, complemented, Sylow
369C22×Q8—1221no—
370C22×D4—1221no—
371C22×D4—1221no—
372C2×Q16—21no—
373C2×SD16—611no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—2444no—
381C2×SD16C22⋊D6111yescharacteristic
382C2×M16—1221no—
383C22×C8—1221no—
384C4⋊Q8—21no—
385C4⋊Q8—611no—
386C4⋊Q8—1221no—
387C4⋊Q8C22×D6111yescharacteristic
388C4⋊Q8C22×D6111yescharacteristic
389#127—611no—
390#126—41no—
391#125—21no—
392#125—41no—
393#124—21no—
394C22⋊Q8—21no—
395C22⋊Q8—1221no—
396C22⋊Q8—2441no—
397#121—1221no—
398#121—1221no—
399#120—2441no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8C22×D6111yescharacteristic
404C4×Q8C22×D6111yescharacteristic
405C4×D4—611no—
406C4×D4—1221no—
407C4×D4C22×D6111yescharacteristic
408#117—21no—
409#107—21no—
410#107—1221no—
411#107—2441no—
412#106—611no—
413#106—1221no—
414#106C22⋊D6111yescharacteristic
415C4⋊C8—21no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—1221no—
420C4⋊C8C22×D6111yescharacteristic
421#103—21no—
422#103—21no—
423#103—21no—
424#103—611no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#103C22⋊D6111yescharacteristic
431#102—21no—
432#102—611no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—2441no—
439#102C22⋊D6111yescharacteristic
440C22⋊C8—1221no—
441C22⋊C8—1221no—
442C4×C8—21no—
443C4×C8—611no—
444C4×C8—611no—
445C4×C8—1221no—
446C4×C8C22×D6111yescharacteristic
447C24—2441no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—1221no—
451C2×Q8—4881no—
452C2×Q8—2441no—
453C2×Q8—2442no—
454C2×Q8D4×D6111yescharacteristic
455C2×Q8C2×C22⋊D6111yescharacteristic
456C2×D4—611no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—4881no—
461C2×D4—4881no—
462C2×D4—2441no—
463C2×D4—2442no—
464C2×D4—2441no—
465C2×D4—2442no—
466C2×D4C2×C22⋊D6111yescharacteristic
467C22×C4—21no—
468C22×C4—1221no—
469C22×C4—2441no—
470Q16—42no—
471SD16—22no—
472SD16—1222no—
473SD16—4881no—
474SD16—4881no—
475SD16—2442no—
476SD16—2442no—
477SD16—2442no—
478SD16—2442no—
479M16—4881no—
480C2×C8—21no—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—2442no—
488C2×C8C2×C22⋊D6111yescharacteristic
489C2×C8C2×C22⋊D6111yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—611no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—2441no—
504C4⋊C4D4×D6111yescharacteristic
505C4⋊C4C2×C22⋊D6111yescharacteristic
506C4⋊C4C2×C22⋊D6111yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—1221no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—41no—
517C42C23×D6111yescharacteristic
518C23—21no—
519C23—1221no—
520C23—9761no—
521C23—4882no—
522C23—4882no—
523C23—2441no—
524Q8—21no—
525Q8—21no—
526Q8—21no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—4882no—
532Q8—2441no—
533Q8—2441no—
534Q8#1451980212yes—
535D4—21no—
536D4—21no—
537D4—612no—
538D4—1221no—
539D4—4881no—
540D4—4882no—
541D4—4882no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—1221no—
556C2×C4—4882no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C22×C22⋊D6111yescharacteristic
562C2×C4#1451982211yescharacteristic
563C8—21no—
564C8—21no—
565C8—41no—
566C8—1221no—
567C8—1221no—
568C8—2441no—
569C8—2441no—
570C8—2441no—
571C8—2441no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—1222no—
576C22—9764no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—2441no—
581C22—2441no—
582C22D4×C22⋊D6111yescharacteristic
583C4—22no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—81no—
590C4—81no—
591C4—4881no—
592C4—2441no—
593C4#135×D6111yescharacteristic
594C4C2×#1451980211yescharacteristic
595C4#1684056911yescharacteristic
596C4#1684057011yescharacteristic
597C2—81no—
598C2—9761no—
599C2—2442no—
600C2#2145731511yescharacteristic
601C2#2145752011yescharacteristic
602C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3364C2
socleC22×C61D42
prime radicalD4×D244C22
hypercenter#3364D61
hypocenterC61#19821
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3364

Complements

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

NH ≅ G/Ntype
1. #21458457C2semidirect
4. #21445474C2semidirect
5. #21445364C2semidirect
6. #21445265C2semidirect
7. #21445239C2semidirect
8. #21445167C2semidirect
9. #21445075C2semidirect
10. #21445066C2semidirect
12. #21444915C2semidirect
13. #21444663C2semidirect
14. #21444629C2semidirect
15. C61×#3364C2semidirect
19. #16840165C22semidirect
22. #16840112C22semidirect
27. #16840047C22semidirect
39. Q8×#92418C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
52. Q8⋊#92418C22semidirect
55. #16839484C22semidirect
56. C61×#557C22semidirect
59. C61×#487C22semidirect
63. C61×#462C22semidirect
64. C8×Q8×C61C22semidirect
296. #3364D61semidirect
318. #557C2×D61semidirect
327. #487C2×D61semidirect
339. #462C2×D61semidirect
349. C8×Q8C2×D61semidirect
368. C61#19821semidirect

Maximal subgroups

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

subgroupindex
#214584572
#214580442
SD16×D2442
#214454742
#214453642
#214452652
#214452392
#214451672
#214450752
#214450662
#214449972
#214449152
#214446632
#214446292
C61×#33642
#1982161