Finite Group Databasebeta

← #29670841

Subgroups — #29670841

Order 15616.

612 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458835C211yesmaximal, characteristic, complemented
2#21458237C211yesmaximal, characteristic, complemented
3#21458234C211yesmaximal, characteristic, complemented
4#21445440C211yesmaximal, characteristic, complemented
5#21445345C211yesmaximal, characteristic, complemented
6#21445225C211yesmaximal, characteristic, complemented
7#21445194C211yesmaximal, characteristic, complemented
8#21445158C211yesmaximal, characteristic, complemented
9#21445047C211yesmaximal, characteristic, complemented
10#21444990C211yesmaximal, characteristic, complemented
11#21444946C211yesmaximal, characteristic, complemented
12#21444901C211yesmaximal, characteristic, complemented
13#21444656C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15C61×#3305C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17#16840477C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840160C2211yescharacteristic
20#16840149—21no—
21#16840134C2211yescharacteristic
22#16840134C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840101—21no—
25#16840086—21no—
26#16840085—21no—
27#16840083—21no—
28#16840057C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840049C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16839996—21no—
33#16839988—21no—
34#16839977—21no—
35#16839967C2211yescharacteristic
36#16839959—21no—
37#16839954C2211yescharacteristic
38#16839893C2211yescharacteristic
39#16839862—21no—
40C4×#11758199C2211yescharacteristic
41C8×D244C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839798—21no—
44#16839797—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#555C2211yescharacteristic
57C61×#550C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#482—21no—
60C61×#477C2211yescharacteristic
61C61×#466—21no—
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C61×#416—21no—
67C2×#11758200—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—41no—
71C2×#11758199—21no—
72C2×#11758199D411yescharacteristic
73C22×#92418—21no—
74C2×C8×D61—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519772—41no—
78#14519765—21no—
79#14519763—41no—
80#14519753—21no—
81#14519751—21no—
82#14519748—21no—
83C4×Dic122—21no—
84C4×D244C2311yescharacteristic
85#14519736—21no—
86#14519735—21no—
87#14519735—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710D411yescharacteristic
93#14519709—21no—
94#14519709—21no—
95#14519708—21no—
96#14519708—21no—
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519701—21no—
102#14519701—41no—
103#14519700—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C2×SD16×C61D411yescharacteristic
117C2×D8×C61—21no—
118C22×C8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120#127×C61—21no—
121#127×C61C2311yescharacteristic
122#124×C61—41no—
123#121×C61—21no—
124#121×C61—21no—
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#117×C61—21no—
133#107×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144#41×C61—42no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C2×C22⋊D61—41no—
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164SD16×C61—41no—
165D8×C61—81no—
166C2×C8×C61—21no—
167C2×C8×C61—81no—
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—41no—
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C42×C61—21no—
182C42×C61—41no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—41no—
185#11758200—42no—
186#11758200—81no—
187#11758200—81no—
188#11758200—82no—
189#11758199—22no—
190#11758199—81no—
191#11758198—21no—
192#11758198—41no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—41no—
201C4×Dic61—41no—
202C4×Dic61—41no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—42no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C8×D61—42no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—41no—
214C22×D61—21no—
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—81no—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—81no—
227D4×C61—82no—
228D4×C61—161no—
229D4×C61—161no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—82no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C22⋊D61—161no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247D244—21no—
248D244—21no—
249C4×D61—41no—
250C4×D61—41no—
251Dic122—41no—
252Dic122—41no—
253Dic122—81no—
254C8×C61—21no—
255C8×C61—21no—
256C8×C61—41no—
257#92418—21no—
258#92418—41no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#92418#13512yes—
263#19854—611nomaximal, Sylow
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61D4211yescharacteristic
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—41no—
277C4×C61—21no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61C2×#13511yescharacteristic
284C4×C61#57511yescharacteristic
285C4×C61#57711yescharacteristic
286C4×C61#57711yescharacteristic
287Dic61—41no—
288Dic61—81no—
289Dic61—81no—
290#3397—611no—
291#3341—611no—
292#3340—611no—
293#3305D6111yescharacteristic, complemented
294#1714—611no—
295#1696—611no—
296#1677—611no—
297#1671—611no—
298#1665—611no—
299#1646—611no—
300#1638—611no—
301#1631—611no—
302#1624—611no—
303#1582—611no—
304#1580—611no—
305C2×C61—81no—
306C2×C61—81no—
307C2×C61—161no—
308C2×C61#326711yescharacteristic
309C2×C61#328411yescharacteristic
310C2×C61#328811yescharacteristic
311D61—81no—
312#556—611no—
313#555—611no—
314#555C2×D6111yescharacteristic
315#550C2×D6111yescharacteristic
316#500—611no—
317#498—611no—
318#496—1221no—
319#492—611no—
320#492—611no—
321#487—611no—
322#486C2×D6111yescharacteristic
323#485—1221no—
324#482—21no—
325#482—1221no—
326#482—1221no—
327#482—1221no—
328#477—611no—
329#477C2×D6111yescharacteristic
330#476—611no—
331#476—611no—
332#474—611no—
333#466—21no—
334#466—1221no—
335#465—1221no—
336#463—1221no—
337#461—611no—
338#460—1221no—
339#459—611no—
340#459C2×D6111yescharacteristic
341#449—21no—
342#446—611no—
343#440—1221no—
344C4×SD16C2×D6111yescharacteristic
345C4×D8—611no—
346C8×D4—611no—
347C8×D4—611no—
348C8×D4C2×D6111yescharacteristic
349#427—1221no—
350#427—1221no—
351#416—21no—
352#334—611no—
353#334—611no—
354#333—611no—
355#333—611no—
356#332—611no—
357#331—611no—
358Q8⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362#322—611no—
363C61#1985411yescharacteristic, complemented, Sylow
364C2×#41—21no—
365C2×SD16—1221no—
366C2×SD16—1221no—
367C2×SD16—1221no—
368C2×SD16—2441no—
369C2×SD16C22⋊D6111yescharacteristic
370C2×D8—21no—
371C2×D8—611no—
372C2×D8—1221no—
373C22×C8—21no—
374C22×C8—1221no—
375C22×C8—1221no—
376C4⋊Q8—611no—
377C4⋊Q8—611no—
378C4⋊Q8C22×D6111yescharacteristic
379#127—21no—
380#127C22×D6111yescharacteristic
381#126—2441no—
382#124—41no—
383#124—1221no—
384#124—2441no—
385C22⋊Q8—1221no—
386#121—21no—
387#121—21no—
388#121—21no—
389#121—41no—
390#121—1221no—
391#121—1221no—
392C4×Q8—1221no—
393C4×Q8C22×D6111yescharacteristic
394C4×D4—21no—
395C4×D4—611no—
396C4×D4C22×D6111yescharacteristic
397C4×D4C22×D6111yescharacteristic
398#117—21no—
399#117—21no—
400#117—1221no—
401#117—1221no—
402#117—1221no—
403#107—21no—
404#107—611no—
405#107—1221no—
406#107—1221no—
407#107—1221no—
408#107—1221no—
409#106—1221no—
410#106—1221no—
411#106—1221no—
412#106—1221no—
413#106C22⋊D6111yescharacteristic
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—1221no—
418C4⋊C8C22×D6111yescharacteristic
419#103—21no—
420#103—1221no—
421#103—1221no—
422#103—2441no—
423#103C22⋊D6111yescharacteristic
424#102—21no—
425#102—21no—
426#102—21no—
427#102—611no—
428#102—611no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102C22⋊D6111yescharacteristic
434C22⋊C8—21no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437#97—1221no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8C22×D6111yescharacteristic
441#41—42no—
442#41—81no—
443C2×Q8—21no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8C2×C22⋊D6111yescharacteristic
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—611no—
454C2×D4—2441no—
455C2×D4D4×D6111yescharacteristic
456C2×D4C2×C22⋊D6111yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—41no—
460C22×C4—1221no—
461SD16—41no—
462SD16—4881no—
463SD16—4881no—
464SD16—4882no—
465SD16—2442no—
466D8—81no—
467D8—1222no—
468D8—4881no—
469C2×C8—21no—
470C2×C8—81no—
471C2×C8—611no—
472C2×C8—611no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—2442no—
478C2×C8—2442no—
479C2×C8C2×C22⋊D6111yescharacteristic
480C2×C8C2×C22⋊D6111yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—611no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—2441no—
490C4⋊C4D4×D6111yescharacteristic
491C4⋊C4C2×C22⋊D6111yescharacteristic
492C4⋊C4C2×C22⋊D6111yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—1221no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C42—21no—
504C42—41no—
505C42—2441no—
506C42—2441no—
507C42C23×D6111yescharacteristic
508C23—21no—
509C23—21no—
510C23—41no—
511C23—1221no—
512Q8—21no—
513Q8—21no—
514Q8—81no—
515Q8—4881no—
516Q8—2441no—
517Q8—2441no—
518D4—21no—
519D4—21no—
520D4—21no—
521D4—21no—
522D4—41no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—81no—
527D4—82no—
528D4—161no—
529D4—161no—
530D4—1221no—
531D4—1221no—
532D4—9761no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—82no—
543C2×C4—1221no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C22×C22⋊D6111yescharacteristic
550C2×C4#1451982211yescharacteristic
551C8—21no—
552C8—21no—
553C8—41no—
554C8—612no—
555C8—1221no—
556C8—2441no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—2441no—
570C22—2441no—
571C22—2441no—
572C22D4×C22⋊D6111yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—81no—
579C4—4881no—
580C4—4881no—
581C4—2441no—
582C4#135×D6111yescharacteristic
583C4#1684056011yescharacteristic
584C4#1684056911yescharacteristic
585C4#1684057311yescharacteristic
586C2—81no—
587C2—81no—
588C2—161no—
589C2—4881no—
590C2#2145734011yescharacteristic
591C2#2145754311yescharacteristic
592C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3305C2
socleC22×C61D42
hypercenter#3305D61
hypocenterC61#19854
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3305

Complements

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

NH ≅ G/Ntype
1. #21458835C2semidirect
2. #21458237C2semidirect
3. #21458234C2semidirect
4. #21445440C2semidirect
5. #21445345C2semidirect
6. #21445225C2semidirect
7. #21445194C2semidirect
8. #21445158C2semidirect
9. #21445047C2semidirect
10. #21444990C2semidirect
11. #21444946C2semidirect
12. #21444901C2semidirect
13. #21444656C2semidirect
14. #21444641C2semidirect
15. C61×#3305C2semidirect
293. #3305D61semidirect
363. C61#19854semidirect

Maximal subgroups

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

subgroupindex
#214588352
#214582372
#214582342
#214454402
#214453452
#214452252
#214451942
#214451582
#214450472
#214449902
#214449462
#214449012
#214446562
#214446412
C61×#33052
#1985461