Finite Group Databasebeta

← #29669834

Subgroups — #29669834

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#21457730C211yesmaximal, characteristic, complemented
2#21457634C211yesmaximal, characteristic
3D4×#11758201C211yesmaximal, characteristic, complemented
4#21445473C211yesmaximal, characteristic, complemented
5#21445387C211yesmaximal, characteristic, complemented
6#21445244C211yesmaximal, characteristic, complemented
7#21445239C211yesmaximal, characteristic, complemented
8#21445162C211yesmaximal, characteristic, complemented
9#21445068C211yesmaximal, characteristic, complemented
10#21445053C211yesmaximal, characteristic, complemented
11#21444996C211yesmaximal, characteristic, complemented
12#21444929C211yesmaximal, characteristic, complemented
13#21444653C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3367C211yesmaximal, characteristic, complemented
16C22×#11758201—21no—
17#16840417C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840165C2211yescharacteristic, complemented
20#16840136C2211yescharacteristic
21#16840114C2211yescharacteristic, complemented
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840103—21no—
25#16840077—21no—
26#16840042C2211yescharacteristic
27#16840039C2211yescharacteristic
28#16840001—21no—
29#16839997—21no—
30#16839993—21no—
31#16839972C2211yescharacteristic, complemented
32#16839964C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839922—21no—
36#16839912—21no—
37#16839901—21no—
38#16839894—21no—
39C4×#11758201C2211yescharacteristic
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42D4×#92418C2211yescharacteristic, complemented
43#16839756—21no—
44#102×D61—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic, complemented
57D4×Q8×C61C2211yescharacteristic
58C61×#489—21no—
59C61×#487C2211yescharacteristic, complemented
60C61×#477—21no—
61C61×#474C2211yescharacteristic, complemented
62C61×#463—21no—
63C61×#461C2211yescharacteristic
64C61×#446C2211yescharacteristic, complemented
65C61×#440—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70C2×#11758202—21no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—44no—
76C2×#11758201D411yescharacteristic
77C2×#11758199—21no—
78C22×#92418—21no—
79C22×#92418—21no—
80C2×C8×D61—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519759—41no—
84#14519756—21no—
85#14519752—21no—
86#14519751—21no—
87#14519748—21no—
88#14519747—41no—
89C4×D244C2311yescharacteristic
90C4⋊C4×D61—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710—41no—
94#14519708D411yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519701—21no—
99#14519701D411yescharacteristic
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×SD16×C61—21no—
118C2×D8×C61D411yescharacteristic
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—41no—
157C2×C22⋊D61—41no—
158C2×C22⋊D61—41no—
159D4×D61—42no—
160D4×D61—81no—
161#11758222—42no—
162#11758222—81no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—42no—
167C2×D244—41no—
168C2×D244—81no—
169C2×D244C2×D411yescharacteristic
170C2×C4×D61—21no—
171C2×C4×D61—41no—
172C2×Dic122—21no—
173SD16×C61—42no—
174D8×C61—22no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—41no—
191C22⋊C4×C61—41no—
192C22⋊C4×C61—41no—
193C42×C61—21no—
194C42×C61—41no—
195C42×C61C2411yescharacteristic
196C22⋊Dic61—41no—
197#11758202—81no—
198#11758201—22no—
199#11758201—42no—
200#11758201—42no—
201#11758201—42no—
202#11758201—81no—
203#11758199—81no—
204#11758198—21no—
205#11758198—41no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61C2×D411yescharacteristic
209#11758196—41no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—42no—
215C2×#92418—81no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218C8×D61—81no—
219C23×C61—21no—
220C23×C61—21no—
221C22×D61—21no—
222C22×D61—41no—
223C22×D61—82no—
224C22×D61—82no—
225C22×D61—161no—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61—41no—
232Q8×C61—82no—
233D4×C61—21no—
234D4×C61—21no—
235D4×C61—21no—
236D4×C61#13512yes—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—82no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C22⋊D61—82no—
252C22⋊D61—82no—
253C2×Dic61—21no—
254C2×Dic61—41no—
255D244—21no—
256D244—41no—
257D244—41no—
258D244—41no—
259D244—41no—
260D244—81no—
261D244—82no—
262D244AGL(1,ℤ/8)12yes—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—82no—
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—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61D4211yescharacteristic
285C2×D61—22no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—164no—
292C4×C61—21no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—81no—
298C4×C61—81no—
299C4×C61C2×#13511yescharacteristic
300C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C4×C61#57611yescharacteristic
302C4×C61#57711yescharacteristic
303Dic61—41no—
304Dic61—81no—
305#3367D6111yescharacteristic, complemented
306#3302—611no—
307#3294—611no—
308D4×SD16—611no—
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—81no—
322C2×C61#326411yescharacteristic
323C2×C61#328211yescharacteristic
324C2×C61#328911yescharacteristic
325D61—42no—
326D61—161no—
327C22×SD16—1221no—
328#556C2×D6111yescharacteristic, complemented
329D4×Q8C2×D6111yescharacteristic
330#547—611no—
331D42—611no—
332#500—611no—
333#492—611no—
334#489—21no—
335#487—611no—
336#487C2×D6111yescharacteristic, complemented
337#486—611no—
338#486—611no—
339#485—1221no—
340#480—1221no—
341#477—21no—
342#474—611no—
343#474—611no—
344#474C2×D6111yescharacteristic, complemented
345#467—1221no—
346#466—1221no—
347#465—1221no—
348#463—21no—
349#462—611no—
350#461—611no—
351#461C2×D6111yescharacteristic
352#460—611no—
353#459—611no—
354#452—1221no—
355#450—1221no—
356#448—1221no—
357#447—1221no—
358#446C2×D6111yescharacteristic, complemented
359#440—21no—
360C4×SD16—611no—
361C4×D8C2×D6111yescharacteristic
362C8×D4—611no—
363C8×D4—611no—
364C8×D4—611no—
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—1221no—
382C22×D4—1221no—
383C2×Q16—1221no—
384C2×SD16—21no—
385C2×SD16—611no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—2444no—
391C2×D8—1221no—
392C2×D8C22⋊D6111yescharacteristic
393C22×C8—1221no—
394C22×C8—1221no—
395C22×C8—1221no—
396C4⋊Q8—21no—
397C4⋊Q8—611no—
398C4⋊Q8C22×D6111yescharacteristic
399C4⋊Q8C22×D6111yescharacteristic
400#127—611no—
401#126—41no—
402#124—21no—
403#123—2441no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8—41no—
407C22⋊Q8—1221no—
408#121—1221no—
409#121—1221no—
410#121—1221no—
411#120—2441no—
412C4×Q8—21no—
413C4×Q8C22×D6111yescharacteristic
414C4×D4—21no—
415C4×D4—611no—
416C4×D4C22×D6111yescharacteristic
417C4×D4C22×D6111yescharacteristic
418#117—21no—
419C2×C4⋊C4—1221no—
420#107—21no—
421#107—1221no—
422#107—1221no—
423#107—2441no—
424#107C22⋊D6111yescharacteristic
425#106—611no—
426C4⋊C8—21no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8C22×D6111yescharacteristic
431#103—21no—
432#103—21no—
433#103—21no—
434#103—611no—
435#103—1221no—
436#103—1221no—
437#103—1221no—
438#103—1221no—
439#102—21no—
440#102—611no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—2441no—
448#102C22⋊D6111yescharacteristic
449#102C22⋊D6111yescharacteristic
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452C22⋊C8—1221no—
453#97—21no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8C22×D6111yescharacteristic
457C24—2441no—
458#41—4881no—
459#41—2442no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—41no—
463C2×Q8—42no—
464C2×Q8—81no—
465C2×Q8—1221no—
466C2×Q8D4×D6111yescharacteristic
467C2×D4—611no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—4881no—
472C2×D4—4881no—
473C2×D4—2441no—
474C2×D4—2442no—
475C2×D4—2441no—
476C2×D4—2441no—
477C2×D4—2442no—
478C2×D4C2×C22⋊D6111yescharacteristic
479C2×D4C2×C22⋊D6111yescharacteristic
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—1221no—
484C22×C4—2441no—
485Q16—4881no—
486SD16—42no—
487SD16—1222no—
488SD16—4881no—
489SD16—2442no—
490SD16—2442no—
491SD16—2442no—
492D8—22no—
493D8—4881no—
494C2×C8—21no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—4881no—
502C2×C8—4881no—
503C2×C8—2442no—
504C2×C8C2×C22⋊D6111yescharacteristic
505C2×C8C2×C22⋊D6111yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—611no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
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—41no—
525C22⋊C4—1221no—
526C22⋊C4—2441no—
527C22⋊C4—2441no—
528C42—21no—
529C42—41no—
530C42C23×D6111yescharacteristic
531C23—21no—
532C23—21no—
533C23—1221no—
534C23—9761no—
535C23—4882no—
536C23—4882no—
537C23—2441no—
538Q8—21no—
539Q8—21no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—82no—
545Q8—2441no—
546Q8—2441no—
547D4—21no—
548D4—21no—
549D4—21no—
550D4—612no—
551D4—1221no—
552D4—4881no—
553D4—4882no—
554D4—4882no—
555D4—4882no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559D4—2441no—
560D4#1451979712yes—
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—41no—
571C2×C4—82no—
572C2×C4—1221no—
573C2×C4—4882no—
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—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—1222no—
596C22—9764no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—2441no—
601C22—2441no—
602C22D4×C22⋊D6111yescharacteristic
603C4—21no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—4881no—
611C4—2441no—
612C4C2×#1451979711yescharacteristic
613C4AGL(1,ℤ/8)×D6111yescharacteristic
614C4#1684056611yescharacteristic
615C4#1684057011yescharacteristic
616C2—42no—
617C2—81no—
618C2—9761no—
619C2—2442no—
620C2#2145729311yescharacteristic
621C2#2145752711yescharacteristic
622C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3367C2
socleC22×C61D42
prime radical#21457634C2
hypercenter#3367D61
hypocenterC61#19787
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3367

Complements

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

NH ≅ G/Ntype
1. #21457730C2semidirect
3. D4×#11758201C2semidirect
4. #21445473C2semidirect
5. #21445387C2semidirect
6. #21445244C2semidirect
7. #21445239C2semidirect
8. #21445162C2semidirect
9. #21445068C2semidirect
10. #21445053C2semidirect
11. #21444996C2semidirect
12. #21444929C2semidirect
13. #21444653C2semidirect
14. #21444643C2semidirect
15. C61×#3367C2semidirect
19. #16840165C22semidirect
21. #16840114C22semidirect
31. #16839972C22semidirect
42. D4×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839508C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839483C22semidirect
56. C61×#556C22semidirect
59. C61×#487C22semidirect
61. C61×#474C22semidirect
64. C61×#446C22semidirect
305. #3367D61semidirect
328. #556C2×D61semidirect
336. #487C2×D61semidirect
344. #474C2×D61semidirect
358. #446C2×D61semidirect
378. C61#19787semidirect

Maximal subgroups

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

subgroupindex
#214577302
#214576342
D4×#117582012
#214454732
#214453872
#214452442
#214452392
#214451622
#214450682
#214450532
#214449962
#214449292
#214446532
#214446432
C61×#33672
#1978761