Finite Group Databasebeta

← #29669641

Subgroups — #29669641

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458684C211yesmaximal, characteristic, complemented
2#21458135C211yesmaximal, characteristic, complemented
3#21457643C211yesmaximal, characteristic, complemented
4#21445471C211yesmaximal, characteristic, complemented
5#21445350C211yesmaximal, characteristic, complemented
6#21445268C211yesmaximal, characteristic, complemented
7#21445260C211yesmaximal, characteristic, complemented
8#21445218C211yesmaximal, characteristic, complemented
9#21445032C211yesmaximal, characteristic, complemented
10#21444970C211yesmaximal, characteristic, complemented
11#21444935C211yesmaximal, characteristic, complemented
12#21444889C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15C61×#3294C211yesmaximal, characteristic
16#16840469C2211yescharacteristic
17#16840469C2211yescharacteristic, complemented
18#16840167C2211yescharacteristic, complemented
19#16840163—21no—
20#16840134C2211yescharacteristic
21#16840114C2211yescharacteristic, complemented
22#16840112C2211yescharacteristic, complemented
23#16840108C2211yescharacteristic
24#16840090—21no—
25#16840078—21no—
26#16840060—21no—
27#16840060—21no—
28#16840052C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic, complemented
32#16840018—21no—
33#16839989—21no—
34#16839969C2211yescharacteristic, complemented
35#16839959C2211yescharacteristic
36#16839924—21no—
37#16839913—21no—
38C8×Dic122C2211yescharacteristic, complemented
39C4×#11758200C2211yescharacteristic
40D8×Dic61—21no—
41#16839840C2211yescharacteristic, complemented
42D4×#92418C2211yescharacteristic
43#16839763—21no—
44C2×#14519698—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51#16839492C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic, complemented
55#16839484C2211yescharacteristic, complemented
56C61×#547C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#480—21no—
60C61×#474C2211yescharacteristic
61C61×#466—21no—
62C61×#459C2211yescharacteristic
63C61×#447—21no—
64C4×D8×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#102×C61—21no—
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758200—41no—
70C2×#11758200D411yescharacteristic
71C2×#11758194—21no—
72C22×#92418—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519768—21no—
76#14519768—21no—
77#14519756—21no—
78#14519756—21no—
79#14519754—41no—
80#14519754—41no—
81C4×Dic122C2311yescharacteristic
82D4×Dic61—21no—
83D4×Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86#14519711—21no—
87#14519711—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708D411yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519700—41no—
101#14519700D411yescharacteristic
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×#41×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×D8×C61—21no—
119C2×D8×C61D411yescharacteristic
120C22×C8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#123×C61—41no—
124C22⋊Q8×C61—21no—
125#121×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128#120×C61—41no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#102×C61—21no—
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C24×C61—41no—
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61—42no—
154C2×D4×C61—42no—
155C2×D4×C61—41no—
156C2×D4×C61—81no—
157C2×D4×C61—81no—
158C2×D4×C61C2×D411yescharacteristic
159C2×D4×C61C2×D411yescharacteristic
160C2×D4×C61C2×D411yescharacteristic
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—41no—
165C22×Dic61—41no—
166C22×Dic61—41no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C2×D411yescharacteristic
170D8×C61—22no—
171D8×C61—81no—
172C2×C8×C61—21no—
173C2×C8×C61—81no—
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189C22⋊Dic61—41no—
190#11758202—81no—
191#11758200—22no—
192#11758200—81no—
193#11758200—82no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61—41no—
200C4⋊Dic61C2×D411yescharacteristic
201#11758196—21no—
202#11758196—41no—
203#11758196—41no—
204C4×Dic61—21no—
205#11758194—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—81no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—41no—
217C23×C61—82no—
218C23×C61—82no—
219C23×C61—161no—
220Q8×C61—41no—
221Q8×C61—41no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—81no—
231D4×C61—82no—
232D4×C61—82no—
233D4×C61—82no—
234D4×C61#13512yes—
235D4×C61#13712yes—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—82no—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—161no—
255C2×Dic61—161no—
256Dic122—21no—
257Dic122—21no—
258Dic122—81no—
259Dic122—81no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—41no—
263#92418—21no—
264#92418—21no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#19774—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—164no—
283C22×C61D4211yescharacteristic
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61C2×#13511yescharacteristic
290C4×C61C2×#13711yescharacteristic
291C4×C61#57611yescharacteristic
292C4×C61#57711yescharacteristic
293Dic61—41no—
294Dic61—41no—
295Dic61—81no—
296Dic61—81no—
297#3380—611no—
298#3331—611no—
299#3295—611no—
300#3294D6111yescharacteristic
301#1719—611no—
302#1697—611no—
303#1683—611no—
304#1682—611no—
305#1675—611no—
306#1644—611no—
307#1635—611no—
308#1629—611no—
309#1622—611no—
310#1581—611no—
311SD16⋊C8—611no—
312C2×C61—42no—
313C2×C61—42no—
314C2×C61—81no—
315C2×C61—161no—
316C2×C61#326611yescharacteristic
317C2×C61#328011yescharacteristic
318C2×C61#328811yescharacteristic
319#554—611no—
320#554—611no—
321#547C2×D6111yescharacteristic
322D42C2×D6111yescharacteristic
323#500—611no—
324#499—1221no—
325#492—611no—
326#487—611no—
327#487—611no—
328#486—611no—
329#486C2×D6111yescharacteristic
330#483—1221no—
331#481—1221no—
332#480—21no—
333#478—1221no—
334#478—1221no—
335#476—611no—
336#476—611no—
337#476—611no—
338#476—611no—
339#474C2×D6111yescharacteristic
340#470—1221no—
341#466—21no—
342#465—1221no—
343#462—611no—
344#460—611no—
345#459C2×D6111yescharacteristic
346#452—1221no—
347#450—1221no—
348#447—21no—
349C8×Q8—611no—
350C4×SD16—611no—
351C4×D8—1221no—
352C4×D8C2×D6111yescharacteristic
353#436—611no—
354C8×D4—611no—
355C8×D4C2×D6111yescharacteristic
356#417—1221no—
357C2×#102—21no—
358C2×#102—1221no—
359#335—611no—
360#335—611no—
361#333—611no—
362#333—611no—
363#332—611no—
364#331—611no—
365Q8⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369#322—611no—
370C61#1977411yescharacteristic, complemented, Sylow
371C2×#41—21no—
372C22×D4—21no—
373C22×D4—21no—
374C2×Q16—1221no—
375C2×SD16—611no—
376C2×SD16—1221no—
377C2×SD16—2441no—
378C2×D8—21no—
379C2×D8C22⋊D6111yescharacteristic
380C2×M16—1221no—
381C22×C8—21no—
382C22×C8—1221no—
383C4⋊Q8—611no—
384C4⋊Q8—611no—
385C4⋊Q8C22×D6111yescharacteristic
386#127C22×D6111yescharacteristic
387#125—1221no—
388#125—1221no—
389#123—41no—
390C22⋊Q8—21no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—2441no—
394C22⋊Q8—2441no—
395#121—21no—
396#121—21no—
397#121—21no—
398#120—41no—
399C4×Q8—611no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4C22×D6111yescharacteristic
403C4×D4C22×D6111yescharacteristic
404C4×D4C22×D6111yescharacteristic
405C2×C4⋊C4—21no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408#107—21no—
409#107—1221no—
410#107—1221no—
411#107C22⋊D6111yescharacteristic
412#106—611no—
413#106—1221no—
414#106—1221no—
415#106—1221no—
416#106—1221no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—1221no—
421C4⋊C8C22×D6111yescharacteristic
422#103—611no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—2441no—
427#102—21no—
428#102—21no—
429#102—21no—
430#102—21no—
431#102—611no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102C22⋊D6111yescharacteristic
440#102C22⋊D6111yescharacteristic
441C22⋊C8—21no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—1221no—
447C4×C8C22×D6111yescharacteristic
448C24—41no—
449#41—42no—
450#41—81no—
451C2×Q8—21no—
452C2×Q8—611no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—41no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4D4×D6111yescharacteristic
466C2×D4C2×C22⋊D6111yescharacteristic
467C2×D4C2×C22⋊D6111yescharacteristic
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—41no—
472C22×C4—2441no—
473C22×C4—2441no—
474Q16—4881no—
475SD16—1222no—
476SD16—4881no—
477SD16—4882no—
478D8—22no—
479D8—81no—
480M16—4881no—
481C2×C8—21no—
482C2×C8—81no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—4881no—
490C2×C8C2×C22⋊D6111yescharacteristic
491C2×C8C2×C22⋊D6111yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—611no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4—2441no—
505C4⋊C4D4×D6111yescharacteristic
506C4⋊C4C2×C22⋊D6111yescharacteristic
507C4⋊C4C2×C22⋊D6111yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—2441no—
514C22⋊C4—2441no—
515C42—1221no—
516C42C23×D6111yescharacteristic
517C23—21no—
518C23—21no—
519C23—21no—
520C23—41no—
521C23—82no—
522C23—82no—
523C23—161no—
524Q8—41no—
525Q8—41no—
526Q8—1221no—
527Q8—1221no—
528Q8—4881no—
529Q8—4881no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—21no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—81no—
539D4—82no—
540D4—82no—
541D4—82no—
542D4#1451979612yes—
543D4#1451980712yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—82no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—9761no—
558C2×C4—9761no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4#1451982211yescharacteristic
564C8—21no—
565C8—21no—
566C8—41no—
567C8—1221no—
568C8—1221no—
569C8—2441no—
570C8—2441no—
571C8—2441no—
572C8—2441no—
573C22—22no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—164no—
586C22D4×C22⋊D6111yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—81no—
592C4—4881no—
593C4—4881no—
594C4—2441no—
595C4—2441no—
596C4C2×#1451979611yescharacteristic
597C4C2×#1451980711yescharacteristic
598C4#1684056311yescharacteristic
599C4#1684057311yescharacteristic
600C2—42no—
601C2—42no—
602C2—81no—
603C2—161no—
604C2#2145732511yescharacteristic
605C2#2145750411yescharacteristic
606C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3294C2
socleC22×C61D42
hypercenter#3294D61
hypocenterC61#19774
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3294

Complements

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

NH ≅ G/Ntype
1. #21458684C2semidirect
2. #21458135C2semidirect
3. #21457643C2semidirect
4. #21445471C2semidirect
5. #21445350C2semidirect
6. #21445268C2semidirect
7. #21445260C2semidirect
8. #21445218C2semidirect
9. #21445032C2semidirect
10. #21444970C2semidirect
11. #21444935C2semidirect
12. #21444889C2semidirect
13. #21444649C2semidirect
14. #21444631C2semidirect
17. #16840469C22semidirect
18. #16840167C22semidirect
21. #16840114C22semidirect
22. #16840112C22semidirect
31. #16840052C22semidirect
34. #16839969C22semidirect
38. C8×Dic122C22semidirect
41. #16839840C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839510C22semidirect
48. #16839510C22semidirect
49. #16839508C22semidirect
51. #16839492C22semidirect
54. D4⋊#92418C22semidirect
55. #16839484C22semidirect
370. C61#19774semidirect

Maximal subgroups

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

subgroupindex
#214586842
#214581352
#214576432
#214454712
#214453502
#214452682
#214452602
#214452182
#214450322
#214449702
#214449352
#214448892
#214446492
#214446312
C61×#32942
#1977461