Finite Group Databasebeta

← #29676880

Subgroups — #29676880

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458557C211yesmaximal, characteristic, complemented
2#21458555C211yesmaximal, characteristic, complemented
3#21458050C211yesmaximal, characteristic
4#21445419C211yesmaximal, characteristic, complemented
5#21445394C211yesmaximal, characteristic, complemented
6#21445392C211yesmaximal, characteristic, complemented
7#21445385C211yesmaximal, characteristic, complemented
8#21445384C211yesmaximal, characteristic, complemented
9#21445089C211yesmaximal, characteristic, complemented
10#21445088C211yesmaximal, characteristic, complemented
11#21444935C211yesmaximal, characteristic
12#21444896C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444707C211yesmaximal, characteristic, complemented
15C61×#3324C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic, complemented
17#16840482C2211yescharacteristic
18#16840158C2211yescharacteristic, complemented
19#16840134C2211yescharacteristic
20#16840121—21no—
21#16840120—21no—
22#16840113C2211yescharacteristic, complemented
23#16840112C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic
25#16840083—21no—
26#16840052C2211yescharacteristic
27#16840052C2211yescharacteristic
28#16840047—21no—
29#16840046—21no—
30#16840042C2211yescharacteristic
31#16840006—21no—
32#16839982—21no—
33#16839981—21no—
34#16839972C2211yescharacteristic, complemented
35#16839969C2211yescharacteristic, complemented
36#16839959C2211yescharacteristic
37#16839913—21no—
38#16839893C2211yescharacteristic, complemented
39#16839892C2211yescharacteristic, complemented
40#16839867—21no—
41#16839866—21no—
42C4×#11758200C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839767—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#482—21no—
60C61×#474C2211yescharacteristic
61C61×#468—21no—
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C4×SD16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#418—21no—
67C2×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758200—41no—
71C2×#11758200D411yescharacteristic
72C2×#11758194—21no—
73#14519777—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519776—21no—
77#14519776—41no—
78#14519770—41no—
79#14519764—21no—
80#14519756—21no—
81Q8×Dic61—21no—
82C4×Dic122—21no—
83C4×Dic122C2311yescharacteristic
84#14519742—21no—
85#14519736—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—41no—
90#14519708D411yescharacteristic
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519706—21no—
96#14519702—21no—
97#14519702—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519700—21no—
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#14519688—21no—
112#14519687—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×D4×C61—21no—
116C22×D4×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C2×M16×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125#121×C61—21no—
126#121×C61—21no—
127#120×C61—41no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#107×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61D411yescharacteristic
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C24×C61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
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—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—41no—
163C2×Dic122C2×D411yescharacteristic
164SD16×C61—41no—
165SD16×C61—81no—
166M16×C61—81no—
167C2×C8×C61—21no—
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C42×C61—21no—
183C42×C61—41no—
184C42×C61C2411yescharacteristic
185C22⋊Dic61—41no—
186C22⋊Dic61—41no—
187#11758202—42no—
188#11758202—42no—
189#11758200—41no—
190#11758200—82no—
191#11758200—81no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—41no—
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—21no—
199#11758196—41no—
200#11758196—41no—
201C4×Dic61—21no—
202C4×Dic61—41no—
203C4×Dic61—41no—
204#11758194—81no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—41no—
214C23×C61—82no—
215C23×C61—82no—
216C23×C61—161no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—41no—
220Q8×C61—41no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—81no—
230D4×C61—82no—
231D4×C61—82no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
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
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248Dic122—21no—
249Dic122—21no—
250Dic122—41no—
251Dic122—41no—
252Dic122—81no—
253Dic122—161no—
254C8×C61—21no—
255C8×C61—21no—
256C8×C61—41no—
257#92418—22no—
258#92418—21no—
259#92418—21no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#20289—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×C61—164no—
274C22×C61D4211yescharacteristic
275C4×C61—21no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—81no—
281C4×C61#57511yescharacteristic
282C4×C61#57511yescharacteristic
283C4×C61#57711yescharacteristic
284C4×C61#57711yescharacteristic
285Dic61—41no—
286Dic61—41no—
287Dic61—81no—
288Dic61—81no—
289#3369—611no—
290#3369—611no—
291#3324—611no—
292#3324D6111yescharacteristic
293#1708—611no—
294#1703—611no—
295#1703—611no—
296#1702—611no—
297#1702—611no—
298#1652—611no—
299#1652—611no—
300#1629—611no—
301#1623—611no—
302#1590—611no—
303#1590—611no—
304C2×C61—81no—
305C2×C61—81no—
306C2×C61—161no—
307C2×C61#327511yescharacteristic
308C2×C61#327511yescharacteristic
309C2×C61#329111yescharacteristic
310Q82—611no—
311#556—611no—
312#556C2×D6111yescharacteristic
313D42C2×D6111yescharacteristic
314#498—611no—
315#492—611no—
316#489—1221no—
317#489—1221no—
318#487—611no—
319#487—611no—
320#486—611no—
321#486C2×D6111yescharacteristic
322#482—21no—
323#482—1221no—
324#476—611no—
325#476—611no—
326#475—1221no—
327#475—1221no—
328#474—611no—
329#474C2×D6111yescharacteristic
330#468—21no—
331#468—1221no—
332#464—1221no—
333#464—1221no—
334#462—611no—
335#462—611no—
336#460—611no—
337#460C2×D6111yescharacteristic
338#450—21no—
339#450—1221no—
340#446—611no—
341#446—611no—
342#441—1221no—
343#441—1221no—
344C4×SD16—611no—
345C4×SD16C2×D6111yescharacteristic
346#436—611no—
347#436C2×D6111yescharacteristic
348#418—21no—
349#418—1221no—
350#334—611no—
351#334—611no—
352#333—611no—
353#332—611no—
354#332—611no—
355#331—611no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360C82—611no—
361C61#2028911yescharacteristic, complemented, Sylow
362C22×D4—21no—
363C22×D4—21no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×SD16—21no—
367C2×SD16—611no—
368C2×SD16—1221no—
369C2×SD16—2441no—
370C2×SD16C22⋊D6111yescharacteristic
371C2×M16—21no—
372C2×M16—1221no—
373C4⋊Q8—611no—
374C4⋊Q8—611no—
375C4⋊Q8—1221no—
376C4⋊Q8—1221no—
377C4⋊Q8—2441no—
378C4⋊Q8C22×D6111yescharacteristic
379#127C22×D6111yescharacteristic
380#126—41no—
381#126—2441no—
382#124—21no—
383#124—1221no—
384C22⋊Q8—21no—
385C22⋊Q8—1221no—
386#121—21no—
387#121—21no—
388#120—41no—
389C4×Q8—21no—
390C4×Q8—611no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8—1221no—
394C4×Q8C22×D6111yescharacteristic
395C4×D4C22×D6111yescharacteristic
396C4×D4C22×D6111yescharacteristic
397#117—21no—
398#117—1221no—
399#107—21no—
400#107—1221no—
401#106—611no—
402#106—1221no—
403#106—1221no—
404#106—2441no—
405#106C22⋊D6111yescharacteristic
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—1221no—
410C4⋊C8—1221no—
411C4⋊C8C22×D6111yescharacteristic
412#103—611no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103C22⋊D6111yescharacteristic
420#102—21no—
421#102—21no—
422#102—21no—
423#102—21no—
424#102—611no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102C22⋊D6111yescharacteristic
432C22⋊C8—21no—
433C22⋊C8—1221no—
434#97—1221no—
435#97—1221no—
436C4×C8—611no—
437C4×C8—611no—
438C4×C8C22×D6111yescharacteristic
439C24—41no—
440C2×Q8—21no—
441C2×Q8—611no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—2441no—
445C2×Q8C2×C22⋊D6111yescharacteristic
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—41no—
450C2×D4—41no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—81no—
454C2×D4—81no—
455C2×D4D4×D6111yescharacteristic
456C2×D4C2×C22⋊D6111yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459Q16—2442no—
460Q16—2442no—
461SD16—41no—
462SD16—81no—
463SD16—4882no—
464SD16—4881no—
465SD16—2441no—
466M16—81no—
467M16—4881no—
468C2×C8—21no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8C2×C22⋊D6111yescharacteristic
476C2×C8C2×C22⋊D6111yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—611no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—2441no—
488C4⋊C4—2441no—
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—2441no—
499C22⋊C4—2441no—
500C42—21no—
501C42—41no—
502C42—1221no—
503C42—2441no—
504C42—2441no—
505C42C23×D6111yescharacteristic
506C23—21no—
507C23—21no—
508C23—41no—
509C23—82no—
510C23—82no—
511C23—161no—
512Q8—21no—
513Q8—21no—
514Q8—41no—
515Q8—41no—
516Q8—1221no—
517Q8—1221no—
518Q8—9761no—
519Q8—4881no—
520Q8—2441no—
521Q8—2441no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—81no—
531D4—82no—
532D4—82no—
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—1221no—
543C2×C4—1221no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4C2×D4×D6111yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4#1451982211yescharacteristic
549C8—21no—
550C8—21no—
551C8—41no—
552C8—1222no—
553C8—1221no—
554C8—1221no—
555C8—2441no—
556C8—2441no—
557C8—2441no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—164no—
568C22D4×C22⋊D6111yescharacteristic
569C4—21no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—4881no—
576C4—4881no—
577C4—2441no—
578C4—2441no—
579C4#1684055811yescharacteristic
580C4#1684056011yescharacteristic
581C4#1684057211yescharacteristic
582C4#1684057311yescharacteristic
583C2—81no—
584C2—81no—
585C2—161no—
586C2#2145744611yescharacteristic
587C2#2145745011yescharacteristic
588C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3324C2
socleC22×C61D42
prime radicalD42×C61C22
hypercenter#3324D61
hypocenterC61#20289
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3324

Complements

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

NH ≅ G/Ntype
1. #21458557C2semidirect
2. #21458555C2semidirect
4. #21445419C2semidirect
5. #21445394C2semidirect
6. #21445392C2semidirect
7. #21445385C2semidirect
8. #21445384C2semidirect
9. #21445089C2semidirect
10. #21445088C2semidirect
12. #21444896C2semidirect
13. #21444710C2semidirect
14. #21444707C2semidirect
16. Q8×Dic122C22semidirect
18. #16840158C22semidirect
22. #16840113C22semidirect
23. #16840112C22semidirect
34. #16839972C22semidirect
35. #16839969C22semidirect
38. #16839893C22semidirect
39. #16839892C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839510C22semidirect
48. #16839509C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. C8×#92418C22semidirect
361. C61#20289semidirect

Maximal subgroups

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

subgroupindex
#214585572
#214585552
#214580502
#214454192
#214453942
#214453922
#214453852
#214453842
#214450892
#214450882
#214449352
#214448962
#214447102
#214447072
C61×#33242
#2028961