Finite Group Databasebeta

← #29669836

Subgroups — #29669836

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458537C211yesmaximal, characteristic, complemented
2#21457738C211yesmaximal, characteristic, complemented
3#21457632C211yesmaximal, characteristic
4#21445470C211yesmaximal, characteristic, complemented
5#21445385C211yesmaximal, characteristic, complemented
6#21445244C211yesmaximal, characteristic, complemented
7#21445242C211yesmaximal, characteristic, complemented
8#21445157C211yesmaximal, characteristic, complemented
9#21445068C211yesmaximal, characteristic, complemented
10#21445052C211yesmaximal, characteristic, complemented
11#21444998C211yesmaximal, characteristic, complemented
12#21444928C211yesmaximal, characteristic, complemented
13#21444656C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15D4×SD16×C61C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic, complemented
17#16840416C2211yescharacteristic
18#16840165C2211yescharacteristic, complemented
19#16840124—21no—
20#16840114C2211yescharacteristic, complemented
21#16840113C2211yescharacteristic, complemented
22#16840108C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840106—21no—
25#16840073—21no—
26#16840054—21no—
27#16840042C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840039C2211yescharacteristic, complemented
30#16840002—21no—
31#16839996—21no—
32#16839977—21no—
33#16839972C2211yescharacteristic, complemented
34#16839967C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839921—21no—
37#16839896—21no—
38#16839893C2211yescharacteristic, complemented
39#16839862—21no—
40C4×#11758199C2211yescharacteristic
41C8×D244C2211yescharacteristic, complemented
42D4×#92418C2211yescharacteristic
43#16839755—21no—
44C2×#14519698—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic, complemented
55#16839484C2211yescharacteristic, complemented
56C22×SD16×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#465—21no—
61C61×#461C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C61×#448—21no—
65C4×SD16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758224—21no—
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758199—21no—
71C2×#11758199D411yescharacteristic
72C22×#92418—21no—
73C2×C8×D61—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519772—41no—
77#14519765—21no—
78#14519761—41no—
79#14519757—21no—
80#14519753—21no—
81#14519751—21no—
82C4×Dic122—21no—
83C4×D244C2311yescharacteristic
84#14519735—21no—
85C2×C4⋊Dic61—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—41no—
90#14519710D411yescharacteristic
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519702—21no—
96#14519702—21no—
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519700—21no—
101#14519698—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—41no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—44no—
123C2×SD16×C61D411yescharacteristic
124C22×C8×C61—21no—
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—41no—
130#121×C61—21no—
131#121×C61—21no—
132#120×C61—41no—
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—42no—
157C2×D4×C61—42no—
158C2×D4×C61—81no—
159C2×D4×C61—81no—
160C2×D4×C61C2×D411yescharacteristic
161C2×D4×C61C2×D411yescharacteristic
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—41no—
165C2×C22⋊D61—41no—
166C22×Dic61—41no—
167#11758224—42no—
168#11758224—81no—
169C2×D244C2×D411yescharacteristic
170C2×C4×D61—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173SD16×C61—22no—
174SD16×C61—42no—
175SD16×C61—42no—
176SD16×C61—42no—
177SD16×C61—81no—
178C2×C8×C61—21no—
179C2×C8×C61—42no—
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—41no—
192C22⋊C4×C61—41no—
193C42×C61—21no—
194C42×C61C2411yescharacteristic
195#11758202—81no—
196#11758200—42no—
197#11758199—22no—
198#11758199—81no—
199#11758198—21no—
200#11758198—41no—
201#11758198—41no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—41no—
208C4⋊Dic61C2×D411yescharacteristic
209#11758196—41no—
210C4×Dic61—41no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—81no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218C8×D61—81no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—41no—
222C23×C61—82no—
223C23×C61—82no—
224C23×C61—161no—
225C22×D61—21no—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—82no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—81no—
238D4×C61—82no—
239D4×C61—82no—
240D4×C61#13512yes—
241D4×C61AGL(1,ℤ/8)12yes—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—82no—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C22⋊D61—82no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—82no—
260D244—21no—
261D244—21no—
262C4×D61—41no—
263C4×D61—41no—
264Dic122—41no—
265Dic122—41no—
266Dic122—41no—
267Dic122—41no—
268C8×C61—21no—
269C8×C61—21no—
270C8×C61—41no—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#19787—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—164no—
288C22×C61D4211yescharacteristic
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—41no—
292C4×C61—21no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—81no—
298C4×C61C2×#13511yescharacteristic
299C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C4×C61#57611yescharacteristic
301C4×C61#57711yescharacteristic
302Dic61—41no—
303Dic61—81no—
304Dic61—81no—
305#3367—611no—
306#3302—611no—
307#3294—611no—
308D4×SD16D6111yescharacteristic, complemented
309#1719—611no—
310#1702—611no—
311#1679—611no—
312#1679—611no—
313#1665—611no—
314#1649—611no—
315#1647—611no—
316#1639—611no—
317#1628—611no—
318#1582—611no—
319#1580—611no—
320C2×C61—42no—
321C2×C61—42no—
322C2×C61—161no—
323C2×C61#326411yescharacteristic
324C2×C61#328211yescharacteristic
325C2×C61#328911yescharacteristic
326D61—81no—
327C22×SD16—21no—
328#556—611no—
329D4×Q8C2×D6111yescharacteristic
330#547—611no—
331D42C2×D6111yescharacteristic
332#500—611no—
333#492C2×D6111yescharacteristic
334#489—1221no—
335#487—611no—
336#487—611no—
337#486—611no—
338#486—611no—
339#485—1221no—
340#480—1221no—
341#477—1221no—
342#474—611no—
343#474—611no—
344#474—611no—
345#467—1221no—
346#466—1221no—
347#465—21no—
348#463—1221no—
349#462—611no—
350#461—611no—
351#461C2×D6111yescharacteristic
352#460C2×D6111yescharacteristic
353#459—611no—
354#452—1221no—
355#450—21no—
356#448—21no—
357#447—1221no—
358#446—611no—
359#440—1221no—
360C4×SD16C2×D6111yescharacteristic
361C4×D8—611no—
362C8×D4—611no—
363C8×D4—611no—
364C8×D4C2×D6111yescharacteristic
365#416—1221no—
366C2×#102—1221no—
367#334—611no—
368#334—611no—
369#332—611no—
370#332—611no—
371#332—611no—
372#331—611no—
373Q8⋊C8—611no—
374D4⋊C8—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377#322—611no—
378C61#1978711yescharacteristic, complemented, Sylow
379C2×#41—1221no—
380C22×Q8—21no—
381C22×D4—21no—
382C22×D4—21no—
383C2×Q16—1221no—
384C2×SD16—21no—
385C2×SD16—21no—
386C2×SD16—21no—
387C2×SD16—21no—
388C2×SD16—44no—
389C2×SD16—1221no—
390C2×SD16C22⋊D6111yescharacteristic
391C2×D8—611no—
392C2×D8—1221no—
393C22×C8—21no—
394C22×C8—1221no—
395C22×C8—1221no—
396C4⋊Q8—21no—
397C4⋊Q8—611no—
398C4⋊Q8—611no—
399C4⋊Q8C22×D6111yescharacteristic
400#127C22×D6111yescharacteristic
401#126—2441no—
402#124—1221no—
403#123—2441no—
404C22⋊Q8—21no—
405C22⋊Q8—41no—
406C22⋊Q8—1221no—
407C22⋊Q8—1221no—
408#121—21no—
409#121—21no—
410#121—1221no—
411#120—41no—
412C4×Q8—1221no—
413C4×Q8C22×D6111yescharacteristic
414C4×D4—21no—
415C4×D4—611no—
416C4×D4C22×D6111yescharacteristic
417C4×D4C22×D6111yescharacteristic
418#117—1221no—
419C2×C4⋊C4—1221no—
420#107—611no—
421#107—1221no—
422#107—1221no—
423#107—1221no—
424#107—2441no—
425#106C22⋊D6111yescharacteristic
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8C22×D6111yescharacteristic
431#103—21no—
432#103—1221no—
433#103—1221no—
434#103—1221no—
435#103—1221no—
436#103—1221no—
437#103—1221no—
438#103C22⋊D6111yescharacteristic
439#102—21no—
440#102—611no—
441#102—611no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—2441no—
449#102C22⋊D6111yescharacteristic
450C22⋊C8—21no—
451C22⋊C8—1221no—
452C22⋊C8—1221no—
453#97—1221no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8C22×D6111yescharacteristic
457C24—41no—
458#41—4881no—
459#41—2442no—
460C2×Q8—21no—
461C2×Q8—41no—
462C2×Q8—42no—
463C2×Q8—81no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8C2×C22⋊D6111yescharacteristic
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—81no—
475C2×D4—81no—
476C2×D4—611no—
477C2×D4—2441no—
478C2×D4D4×D6111yescharacteristic
479C2×D4C2×C22⋊D6111yescharacteristic
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—1221no—
484C22×C4—2441no—
485Q16—4881no—
486SD16—22no—
487SD16—42no—
488SD16—42no—
489SD16—42no—
490SD16—81no—
491SD16—2442no—
492D8—1222no—
493D8—4881no—
494C2×C8—21no—
495C2×C8—42no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—4881no—
503C2×C8—4881no—
504C2×C8C2×C22⋊D6111yescharacteristic
505C2×C8C2×C22⋊D6111yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—611no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—2441no—
516C4⋊C4—2441no—
517C4⋊C4D4×D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—1221no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C22⋊C4—2441no—
528C42—21no—
529C42—2441no—
530C42C23×D6111yescharacteristic
531C23—21no—
532C23—21no—
533C23—41no—
534C23—82no—
535C23—82no—
536C23—161no—
537C23—1221no—
538Q8—21no—
539Q8—21no—
540Q8—41no—
541Q8—41no—
542Q8—82no—
543Q8—2441no—
544Q8—2441no—
545Q8—2441no—
546Q8—2441no—
547D4—21no—
548D4—21no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—81no—
554D4—82no—
555D4—82no—
556D4—1221no—
557D4—1221no—
558D4—4882no—
559D4#1451979812yes—
560D4#1451980312yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—82no—
571C2×C4—1221no—
572C2×C4—4882no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4#1451982211yescharacteristic
580C8—21no—
581C8—21no—
582C8—41no—
583C8—1221no—
584C8—1221no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—164no—
599C22—2441no—
600C22—2441no—
601C22—2441no—
602C22D4×C22⋊D6111yescharacteristic
603C4—21no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—4881no—
610C4—4881no—
611C4—2441no—
612C4C2×#1451979811yescharacteristic
613C4C2×#1451980311yescharacteristic
614C4#1684056511yescharacteristic
615C4#1684056811yescharacteristic
616C2—42no—
617C2—42no—
618C2—161no—
619C2—4881no—
620C2#2145729511yescharacteristic
621C2#2145752611yescharacteristic
622C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×SD16×C61C2
socleC22×C61D42
prime radical#21457632C2
hypercenterD4×SD16D61
hypocenterC61#19787
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×SD16

Complements

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

NH ≅ G/Ntype
1. #21458537C2semidirect
2. #21457738C2semidirect
4. #21445470C2semidirect
5. #21445385C2semidirect
6. #21445244C2semidirect
7. #21445242C2semidirect
8. #21445157C2semidirect
9. #21445068C2semidirect
10. #21445052C2semidirect
11. #21444998C2semidirect
12. #21444928C2semidirect
13. #21444656C2semidirect
14. #21444641C2semidirect
15. D4×SD16×C61C2semidirect
16. #16840485C22semidirect
18. #16840165C22semidirect
20. #16840114C22semidirect
21. #16840113C22semidirect
29. #16840039C22semidirect
33. #16839972C22semidirect
38. #16839893C22semidirect
41. C8×D244C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
54. #16839489C22semidirect
55. #16839484C22semidirect
308. D4×SD16D61semidirect
378. C61#19787semidirect

Maximal subgroups

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

subgroupindex
#214585372
#214577382
#214576322
#214454702
#214453852
#214452442
#214452422
#214451572
#214450682
#214450522
#214449982
#214449282
#214446562
#214446412
D4×SD16×C612
#1978761