Finite Group Databasebeta

← #29670134

Subgroups — #29670134

Order 15616.

684 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#21458800C211yesmaximal, characteristic, complemented
2#21458692C211yesmaximal, characteristic, complemented
3#21457643C211yesmaximal, characteristic, complemented
4#21445445C211yesmaximal, characteristic, complemented
5#21445318C211yesmaximal, characteristic, complemented
6#21445233C211yesmaximal, characteristic, complemented
7#21445176C211yesmaximal, characteristic, complemented
8#21445158C211yesmaximal, characteristic, complemented
9#21444998C211yesmaximal, characteristic, complemented
10#21444946C211yesmaximal, characteristic, complemented
11#21444914C211yesmaximal, characteristic, complemented
12#21444889C211yesmaximal, characteristic, complemented
13#21444660C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3274C211yesmaximal, characteristic
16#16840491C2211yescharacteristic, complemented
17#16840469C2211yescharacteristic
18#16840162—21no—
19#16840160C2211yescharacteristic, complemented
20#16840134—21no—
21#16840134C2211yescharacteristic, complemented
22#16840134C2211yescharacteristic, complemented
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840078—21no—
26#16840067—21no—
27#16840064—21no—
28#16840057C2211yescharacteristic, complemented
29#16840052C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16839989—21no—
32#16839967—21no—
33#16839967C2211yescharacteristic
34#16839959—21no—
35#16839959C2211yescharacteristic
36#16839959C2211yescharacteristic, complemented
37#16839913—21no—
38Q8×#92418C2211yescharacteristic, complemented
39C8×Dic122C2211yescharacteristic, complemented
40SD16×Dic61—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519698—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839506C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#550C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#493C2211yescharacteristic
60C61×#469—21no—
61C61×#460C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C61×#447—21no—
65C4×SD16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—41no—
73C2×#11758200D411yescharacteristic
74C22×#92418—21no—
75#14519777—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519768—21no—
79#14519768—21no—
80#14519768—41no—
81#14519756—21no—
82#14519754—41no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122C2311yescharacteristic
86D4×Dic61—21no—
87C2×C4⋊Dic61—21no—
88#14519710—21no—
89#14519710—21no—
90#14519709—21no—
91#14519709—21no—
92#14519708—21no—
93#14519708D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519700—21no—
101#14519700D411yescharacteristic
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×#41×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118AGL(1,ℤ/8)×C61—44no—
119C2×SD16×C61—21no—
120C2×SD16×C61D411yescharacteristic
121C2×D8×C61—21no—
122C2×D8×C61—21no—
123C2×D8×C61—21no—
124C2×M16×C61—21no—
125#127×C61—21no—
126#127×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128#121×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—41no—
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×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61—41no—
158C2×D4×C61—41no—
159C2×D4×C61—42no—
160C2×D4×C61—42no—
161C2×D4×C61—81no—
162C2×D4×C61—81no—
163C2×D4×C61C2×D411yescharacteristic
164C2×D4×C61C2×D411yescharacteristic
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—41no—
168C22×Dic61—41no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122C2×D411yescharacteristic
172SD16×C61—42no—
173SD16×C61SD1614yescomplemented
174D8×C61—42no—
175D8×C61—42no—
176D8×C61—81no—
177M16×C61—42no—
178C2×C8×C61—21no—
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×D411yescharacteristic
181C4⋊C4×C61—21no—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61C2411yescharacteristic
191C22⋊Dic61—41no—
192#11758200—41no—
193#11758200—42no—
194#11758200—81no—
195#11758200—81no—
196#11758200—81no—
197#11758200—81no—
198#11758200—82no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205C4⋊Dic61—41no—
206C4⋊Dic61C2×D411yescharacteristic
207#11758196—21no—
208#11758196—41no—
209#11758196—41no—
210C4×Dic61—21no—
211C4×Dic61—41no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—81no—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—82no—
224C23×C61—82no—
225C23×C61—161no—
226Q8×C61—21no—
227Q8×C61C2×SD1612yes—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—82no—
240D4×C61—81no—
241D4×C61—81no—
242D4×C61—82no—
243D4×C61—82no—
244D4×C61—161no—
245D4×C61C2×SD1612yes—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—82no—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—161no—
263Dic122—21no—
264Dic122—21no—
265Dic122—41no—
266Dic122—41no—
267Dic122—81no—
268C8×C61—21no—
269C8×C61—41no—
270C8×C61C2×SD1612yes—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#19807—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—164no—
291C22×C61D4211yescharacteristic
292C4×C61—22no—
293C4×C61—21no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61C22×SD1611yescharacteristic
298C4×C61#57511yescharacteristic
299C4×C61#57611yescharacteristic
300C4×C61#57711yescharacteristic
301Dic61—41no—
302Dic61—41no—
303Dic61—81no—
304Dic61—81no—
305#3392—611no—
306#3381—611no—
307#3295—611no—
308#3274D6111yescharacteristic
309#1715—611no—
310#1691—611no—
311#1678—611no—
312#1668—611no—
313#1665—611no—
314#1639—611no—
315#1631—611no—
316#1626—611no—
317#1622—611no—
318#1583—611no—
319SD16⋊C8—611no—
320C2×C61—42no—
321C2×C61—81no—
322C2×C61—161no—
323C2×C61—161no—
324C2×C61#326811yescharacteristic
325C2×C61D4×SD1611yescharacteristic
326C2×C61#328811yescharacteristic
327C2×AGL(1,ℤ/8)—21no—
328#557—611no—
329#554—611no—
330#550C2×D6111yescharacteristic
331D42C2×D6111yescharacteristic
332#499—1221no—
333#498—611no—
334#493C2×D6111yescharacteristic
335#492—611no—
336#492—611no—
337#492—1221no—
338#486—611no—
339#486—611no—
340#481—1221no—
341#479—1221no—
342#478—1221no—
343#477—611no—
344#476—611no—
345#474—611no—
346#469—21no—
347#465—1221no—
348#461—611no—
349#461—1221no—
350#460—611no—
351#460—611no—
352#460—1221no—
353#460C2×D6111yescharacteristic
354#459C2×D6111yescharacteristic
355#450—1221no—
356#449—21no—
357#447—21no—
358C8×Q8—611no—
359C8×Q8—611no—
360C4×SD16—611no—
361C4×SD16—1221no—
362C4×SD16—1221no—
363C4×SD16C2×D6111yescharacteristic
364#436C2×D6111yescharacteristic
365C8×D4—611no—
366C2×#102—1221no—
367#335—611no—
368#334—611no—
369#333—611no—
370#332—611no—
371#331—611no—
372#331—611no—
373Q8⋊C8—611no—
374Q8⋊C8—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377#322—611no—
378C61#1980711yescharacteristic, complemented, Sylow
379C2×#41—21no—
380C22×D4—21no—
381C22×D4—21no—
382AGL(1,ℤ/8)—44no—
383C2×SD16—21no—
384C2×SD16—611no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—2441no—
391C2×SD16C22⋊D6111yescharacteristic
392C2×D8—21no—
393C2×D8—21no—
394C2×D8—21no—
395C2×M16—21no—
396C22×C8—1221no—
397C4⋊Q8—611no—
398C4⋊Q8—611no—
399C4⋊Q8—1221no—
400#127—21no—
401#127C22×D6111yescharacteristic
402#127C22×D6111yescharacteristic
403#125—1221no—
404#125—1221no—
405#125—2441no—
406C22⋊Q8—1221no—
407C22⋊Q8—2441no—
408#121—21no—
409#121—21no—
410#121—21no—
411#121—41no—
412#120—41no—
413C4×Q8—611no—
414C4×Q8—1221no—
415C4×Q8—1221no—
416C4×Q8C22×D6111yescharacteristic
417C4×D4—21no—
418C4×D4—1221no—
419C4×D4C22×D6111yescharacteristic
420C4×D4C22×D6111yescharacteristic
421C2×C4⋊C4—1221no—
422#107—1221no—
423#107—1221no—
424#106—611no—
425#106—1221no—
426#106—1221no—
427#106—1221no—
428#106C22⋊D6111yescharacteristic
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—611no—
432C4⋊C8—1221no—
433C4⋊C8—1221no—
434C4⋊C8C22×D6111yescharacteristic
435#103—611no—
436#103—1221no—
437#103—1221no—
438#103C22⋊D6111yescharacteristic
439#102—21no—
440#102—21no—
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—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8—1221no—
455C4×C8—1221no—
456C4×C8C22×D6111yescharacteristic
457C24—41no—
458#41—42no—
459#41—81no—
460C2×Q8—611no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8C2×C22⋊D6111yescharacteristic
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—81no—
477C2×D4—81no—
478C2×D4D4×D6111yescharacteristic
479C2×D4C2×C22⋊D6111yescharacteristic
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—2441no—
484SD16—42no—
485SD16—4881no—
486SD16—4881no—
487SD16—4881no—
488SD16—4881no—
489SD16—4882no—
490SD16—2441no—
491SD16—2442no—
492SD16#1175820014yescomplemented
493D8—42no—
494D8—42no—
495D8—81no—
496M16—42no—
497C2×C8—21no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—4881no—
505C2×C8C2×C22⋊D6111yescharacteristic
506C2×C8C2×C22⋊D6111yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—611no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C4⋊C4—2441no—
517C4⋊C4—2441no—
518C4⋊C4—2441no—
519C4⋊C4D4×D6111yescharacteristic
520C4⋊C4C2×C22⋊D6111yescharacteristic
521C4⋊C4C2×C22⋊D6111yescharacteristic
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—2441no—
527C42—21no—
528C42—1221no—
529C42—2441no—
530C42C23×D6111yescharacteristic
531C23—21no—
532C23—21no—
533C23—41no—
534C23—41no—
535C23—82no—
536C23—82no—
537C23—161no—
538Q8—21no—
539Q8—1221no—
540Q8—1221no—
541Q8—4881no—
542Q8—2441no—
543Q8—2441no—
544Q8C2×#1175820012yes—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—82no—
557D4—81no—
558D4—81no—
559D4—82no—
560D4—82no—
561D4—161no—
562D4C2×#1175820012yes—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—82no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—9761no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4#1451982211yescharacteristic
580C8—21no—
581C8—41no—
582C8—1221no—
583C8—1221no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C8C2×#1175820012yes—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—164no—
602C22D4×C22⋊D6111yescharacteristic
603C4—22no—
604C4—21no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—4881no—
609C4—4881no—
610C4—2441no—
611C4—2441no—
612C4C22×#1175820011yescharacteristic
613C4#1684055811yescharacteristic
614C4#1684056611yescharacteristic
615C4#1684057311yescharacteristic
616C2—42no—
617C2—81no—
618C2—161no—
619C2—161no—
620C2#2145735311yescharacteristic
621C2D4×#1175820011yescharacteristic
622C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3274C2
socleC22×C61D42
hypercenter#3274D61
hypocenterC61#19807
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3274

Complements

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

NH ≅ G/Ntype
1. #21458800C2semidirect
2. #21458692C2semidirect
3. #21457643C2semidirect
4. #21445445C2semidirect
5. #21445318C2semidirect
6. #21445233C2semidirect
7. #21445176C2semidirect
8. #21445158C2semidirect
9. #21444998C2semidirect
10. #21444946C2semidirect
11. #21444914C2semidirect
12. #21444889C2semidirect
13. #21444660C2semidirect
14. SD16⋊#92418C2semidirect
16. #16840491C22semidirect
19. #16840160C22semidirect
21. #16840134C22semidirect
22. #16840134C22semidirect
28. #16840057C22semidirect
36. #16839959C22semidirect
38. Q8×#92418C22semidirect
39. C8×Dic122C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839510C22semidirect
48. #16839507C22semidirect
50. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. #16839483C22semidirect
173. SD16×C61SD16semidirect
378. C61#19807semidirect
492. SD16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214588002
#214586922
#214576432
#214454452
#214453182
#214452332
#214451762
#214451582
#214449982
#214449462
#214449142
#214448892
#214446602
SD16⋊#924182
C61×#32742
#1980761