Finite Group Databasebeta

← #29676865

Subgroups — #29676865

Order 15616.

670 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#21458848C211yesmaximal, characteristic, complemented
2#21458544C211yesmaximal, characteristic, complemented
3#21458050C211yesmaximal, characteristic, complemented
4#21445424C211yesmaximal, characteristic, complemented
5#21445385C211yesmaximal, characteristic, complemented
6#21445378C211yesmaximal, characteristic, complemented
7#21445308C211yesmaximal, characteristic, complemented
8#21445300C211yesmaximal, characteristic, complemented
9#21445012C211yesmaximal, characteristic, complemented
10#21444998C211yesmaximal, characteristic, complemented
11#21444927C211yesmaximal, characteristic, complemented
12#21444913C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444707C211yesmaximal, characteristic, complemented
15C61×#3274C211yesmaximal, characteristic
16#16840490C2211yescharacteristic, complemented
17#16840482C2211yescharacteristic
18#16840153—21no—
19#16840142C2211yescharacteristic, complemented
20#16840126—21no—
21#16840108C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840108C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic, complemented
25#16840083—21no—
26#16840062—21no—
27#16840042C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840006—21no—
30#16839983—21no—
31#16839972C2211yescharacteristic, complemented
32#16839971—21no—
33#16839967C2211yescharacteristic
34#16839959—21no—
35#16839959C2211yescharacteristic
36#16839959C2211yescharacteristic, complemented
37#16839913—21no—
38#16839893C2211yescharacteristic, complemented
39#16839892C2211yescharacteristic, complemented
40#16839868—21no—
41#16839862—21no—
42C4×#11758200C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839767—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839506C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic, 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×Dic244—21no—
68C2×Dic244—21no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200D411yescharacteristic
74C2×#11758194—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519775—21no—
78#14519770—41no—
79#14519769—21no—
80#14519767—41no—
81#14519764—21no—
82#14519756—21no—
83C4×Dic122—21no—
84C4×Dic122C2311yescharacteristic
85#14519742—21no—
86#14519742—21no—
87#14519736—21no—
88#14519710—21no—
89#14519710—21no—
90#14519708D411yescharacteristic
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519706—21no—
96#14519700—21no—
97#14519700—21no—
98#14519700D411yescharacteristic
99#14519698—21no—
100#14519698—21no—
101#14519698—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—41no—
108#14519698—41no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519688—21no—
112#14519687—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
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—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C2×D411yescharacteristic
171SD16×C61—22no—
172SD16×C61—42no—
173D8×C61—42no—
174D8×C61—42no—
175D8×C61—81no—
176M16×C61—42no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61C2411yescharacteristic
190C22⋊Dic61—41no—
191C22⋊Dic61—41no—
192#11758202—81no—
193#11758200—41no—
194#11758200—42no—
195#11758200—81no—
196#11758200—81no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—21no—
203#11758196—41no—
204#11758196—41no—
205#11758196—41no—
206#11758196—41no—
207C4×Dic61—21no—
208C4×Dic61—41no—
209C4×Dic61—41no—
210#11758194—81no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217Dic244—42no—
218Dic244—81no—
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×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—81no—
241D4×C61—81no—
242D4×C61—82no—
243D4×C61—82no—
244D4×C61—82no—
245D4×C61—161no—
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—
262Dic122—21no—
263Dic122—21no—
264Dic122—41no—
265Dic122—41no—
266Dic122—41no—
267Dic122—41no—
268C8×C61—21no—
269C8×C61—41no—
270C8×C61AGL(1,ℤ/8)12yes—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#20288—611nomaximal, Sylow
278C22×C61—41no—
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—21no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
298C4×C61#57511yescharacteristic
299C4×C61#57611yescharacteristic
300C4×C61#57711yescharacteristic
301Dic61—41no—
302Dic61—41no—
303Dic61—81no—
304Dic61—81no—
305#3398—611no—
306#3368—611no—
307#3324—611no—
308#3274D6111yescharacteristic
309#1709—611no—
310#1702—611no—
311#1701—611no—
312#1689—611no—
313#1688—611no—
314#1641—611no—
315#1639—611no—
316#1628—611no—
317#1626—611no—
318#1590—611no—
319#1590—611no—
320C2×C61—81no—
321C2×C61—81no—
322C2×C61—161no—
323C2×C61—161no—
324C2×C61#327411yescharacteristic
325C2×C61#327511yescharacteristic
326C2×C61#329011yescharacteristic
327C2×AGL(1,ℤ/8)—21no—
328#557—611no—
329#556—611no—
330#550C2×D6111yescharacteristic
331D42C2×D6111yescharacteristic
332#497—1221no—
333#494—611no—
334#493C2×D6111yescharacteristic
335#490—1221no—
336#486—611no—
337#486—611no—
338#486—611no—
339#486—611no—
340#482—1221no—
341#478—1221no—
342#474—611no—
343#474—611no—
344#469—21no—
345#468—1221no—
346#464—1221no—
347#462—611no—
348#462—1221no—
349#461—611no—
350#460—611no—
351#460—611no—
352#460—1221no—
353#460C2×D6111yescharacteristic
354#459C2×D6111yescharacteristic
355#450—1221no—
356#449—21no—
357#447—21no—
358#446—611no—
359#446—611no—
360#441—1221no—
361#440—1221no—
362C4×SD16—611no—
363C4×SD16C2×D6111yescharacteristic
364#436—611no—
365#436C2×D6111yescharacteristic
366#418—1221no—
367#334—611no—
368#334—611no—
369#332—611no—
370#332—611no—
371#331—611no—
372#331—611no—
373Q8⋊C8—611no—
374Q8⋊C8—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377C82—611no—
378C61#2028811yescharacteristic, complemented, Sylow
379C2×#41—21no—
380C22×D4—21no—
381C22×D4—21no—
382AGL(1,ℤ/8)—44no—
383C2×Q16—1221no—
384C2×Q16—1221no—
385C2×Q16—1221no—
386C2×SD16—21no—
387C2×SD16—611no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—1221no—
391C2×SD16C22⋊D6111yescharacteristic
392C2×D8—21no—
393C2×D8—21no—
394C2×D8—21no—
395C2×M16—21no—
396C2×M16—1221no—
397C4⋊Q8—611no—
398C4⋊Q8—611no—
399C4⋊Q8—1221no—
400#127—21no—
401#127C22×D6111yescharacteristic
402#127C22×D6111yescharacteristic
403#126—2441no—
404#125—1221no—
405#125—2441no—
406#124—1221no—
407C22⋊Q8—1221no—
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×Q8—1221no—
417C4×Q8C22×D6111yescharacteristic
418C4×D4—21no—
419C4×D4C22×D6111yescharacteristic
420C4×D4C22×D6111yescharacteristic
421#117—1221no—
422#107—1221no—
423#107—1221no—
424#106—611no—
425#106C22⋊D6111yescharacteristic
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431C4⋊C8C22×D6111yescharacteristic
432#103—611no—
433#103—1221no—
434#103—1221no—
435#103C22⋊D6111yescharacteristic
436#102—21no—
437#102—21no—
438#102—611no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—2441no—
448#102—2441no—
449#102C22⋊D6111yescharacteristic
450C22⋊C8—21no—
451C22⋊C8—1221no—
452#97—1221no—
453#97—1221no—
454C4×C8—611no—
455C4×C8—611no—
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—
483Q16—4881no—
484Q16—4881no—
485Q16—2442no—
486SD16—22no—
487SD16—42no—
488SD16—4881no—
489SD16—4881no—
490SD16—2441no—
491SD16—2442no—
492D8—42no—
493D8—42no—
494D8—81no—
495M16—42no—
496M16—4881no—
497C2×C8—21no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8C2×C22⋊D6111yescharacteristic
505C2×C8C2×C22⋊D6111yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—611no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—2441no—
514C4⋊C4—2441no—
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—2441no—
525C22⋊C4—2441no—
526C42—21no—
527C42—1221no—
528C42—2441no—
529C42—2441no—
530C42C23×D6111yescharacteristic
531C23—21no—
532C23—21no—
533C23—41no—
534C23—41no—
535C23—82no—
536C23—82no—
537C23—161no—
538Q8—21no—
539Q8—21no—
540Q8—1221no—
541Q8—1221no—
542Q8—2441no—
543Q8—2441no—
544Q8—2441no—
545Q8—2441no—
546D4—21no—
547D4—21no—
548D4—21no—
549D4—21no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—81no—
559D4—81no—
560D4—82no—
561D4—82no—
562D4—82no—
563D4—161no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—82no—
573C2×C4—1221no—
574C2×C4—1221no—
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—
588C8#1451980312yes—
589C22—41no—
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—21no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—4881no—
609C4—4881no—
610C4—2441no—
611C4—2441no—
612C4C2×#1451980311yescharacteristic
613C4#1684055811yescharacteristic
614C4#1684056511yescharacteristic
615C4#1684057211yescharacteristic
616C2—81no—
617C2—81no—
618C2—161no—
619C2—161no—
620C2#2145743511yescharacteristic
621C2#2145745011yescharacteristic
622C2#2145760111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3274C2
socleC22×C61D42
hypercenter#3274D61
hypocenterC61#20288
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. #21458848C2semidirect
2. #21458544C2semidirect
3. #21458050C2semidirect
4. #21445424C2semidirect
5. #21445385C2semidirect
6. #21445378C2semidirect
7. #21445308C2semidirect
8. #21445300C2semidirect
9. #21445012C2semidirect
10. #21444998C2semidirect
11. #21444927C2semidirect
12. #21444913C2semidirect
13. #21444710C2semidirect
14. #21444707C2semidirect
16. #16840490C22semidirect
19. #16840142C22semidirect
23. #16840108C22semidirect
24. #16840108C22semidirect
31. #16839972C22semidirect
36. #16839959C22semidirect
38. #16839893C22semidirect
39. #16839892C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839509C22semidirect
48. #16839509C22semidirect
50. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. C8×#92418C22semidirect
378. C61#20288semidirect

Maximal subgroups

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

subgroupindex
#214588482
#214585442
#214580502
#214454242
#214453852
#214453782
#214453082
#214453002
#214450122
#214449982
#214449272
#214449132
#214447102
#214447072
C61×#32742
#2028861