Finite Group Databasebeta

← #29669841

Subgroups — #29669841

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#21458533C211yesmaximal, characteristic, complemented
2#21457626C211yesmaximal, characteristic
3SD16×D244C211yesmaximal, characteristic, complemented
4#21445474C211yesmaximal, characteristic, complemented
5#21445389C211yesmaximal, characteristic, complemented
6#21445240C211yesmaximal, characteristic, complemented
7#21445239C211yesmaximal, characteristic, complemented
8#21445160C211yesmaximal, characteristic, complemented
9#21445066C211yesmaximal, characteristic, complemented
10#21445054C211yesmaximal, characteristic, complemented
11#21444997C211yesmaximal, characteristic, complemented
12#21444929C211yesmaximal, characteristic, complemented
13#21444656C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15C61×#3302C211yesmaximal, characteristic, complemented
16C2×SD16×D61—21no—
17Q8×D244C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840165C2211yescharacteristic, complemented
20#16840135C2211yescharacteristic
21#16840121—21no—
22#16840112C2211yescharacteristic, complemented
23#16840110C2211yescharacteristic
24#16840110C2211yescharacteristic
25#16840075—21no—
26#16840057—21no—
27#16840042C2211yescharacteristic, complemented
28#16840039C2211yescharacteristic
29#16839996—21no—
30#16839992—21no—
31#16839974—21no—
32#16839967C2211yescharacteristic
33#16839964C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839914—21no—
37#16839902—21no—
38#16839895—21no—
39#16839893C2211yescharacteristic, complemented
40#16839860—21no—
41C4×#11758199C2211yescharacteristic
42C8×D244C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519697—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic, complemented
57C61×#547C2211yescharacteristic
58C61×#487C2211yescharacteristic, complemented
59C61×#485—21no—
60C61×#474C2211yescharacteristic
61C61×#467—21no—
62C61×#462C2211yescharacteristic, complemented
63C61×#452—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic, complemented
66C61×#416—21no—
67C2×Q8×D61—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70C2×#11758190—21no—
71C2×#11758190—21no—
72C2×#11758200—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75SD16×D61—44no—
76C2×#11758199—21no—
77C2×#11758199D411yescharacteristic
78C22×#92418—21no—
79C2×C8×D61—21no—
80#14519777—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519757—21no—
84#14519757—41no—
85#14519752—21no—
86#14519751—21no—
87#14519747—41no—
88C4×D244—21no—
89C4×D244C2311yescharacteristic
90#14519710—21no—
91#14519710—21no—
92#14519710—41no—
93#14519710D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519700—21no—
102#14519698—21no—
103#14519698D411yescharacteristic
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×Q16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C22×C8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124#123×C61—41no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—21no—
127#121×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133C2×C4⋊C4×C61—21no—
134#107×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—41no—
152C2×D4×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—41no—
157C2×C22⋊D61—41no—
158Q8×D61—42no—
159Q8×D61—81no—
160D4×D61—42no—
161D4×D61—81no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—42no—
166C2×D244—41no—
167C2×D244—81no—
168C2×D244C2×D411yescharacteristic
169C2×C4×D61—21no—
170C2×C4×D61—41no—
171C2×Dic122—21no—
172C2×Dic122—41no—
173Q16×C61—81no—
174SD16×C61—22no—
175C2×C8×C61—21no—
176C2×C8×C61—81no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—41no—
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—
193C22⋊C4×C61—41no—
194C42×C61—21no—
195C42×C61—41no—
196C42×C61C2411yescharacteristic
197#11758201—42no—
198#11758201—42no—
199#11758200—42no—
200#11758199—22no—
201#11758199—81no—
202#11758198—21no—
203#11758198—41no—
204#11758198—41no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—41no—
208C4⋊Dic61C2×D411yescharacteristic
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—81no—
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216#11758190—42no—
217#11758190—81no—
218C8×D61—42no—
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—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—21no—
235D4×C61—82no—
236D4×C61AGL(1,ℤ/8)12yes—
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—
252C2×Dic61—21no—
253C2×Dic61—41no—
254D244—21no—
255D244—41no—
256D244—41no—
257D244—41no—
258D244—41no—
259D244—81no—
260D244—82no—
261D244#13512yes—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—82no—
265Dic122—41no—
266Dic122—41no—
267Dic122—82no—
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#3367—611no—
306#3302D6111yescharacteristic, complemented
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×Q8—611no—
330#547C2×D6111yescharacteristic
331D42—611no—
332#500—611no—
333#492—611no—
334#489—1221no—
335#487—611no—
336#487C2×D6111yescharacteristic, complemented
337#486—611no—
338#486—611no—
339#485—21no—
340#480—1221no—
341#477—1221no—
342#474—611no—
343#474—611no—
344#474C2×D6111yescharacteristic
345#467—21no—
346#466—1221no—
347#465—1221no—
348#463—1221no—
349#462C2×D6111yescharacteristic, complemented
350#461—611no—
351#461—611no—
352#460—611no—
353#459—611no—
354#452—21no—
355#450—1221no—
356#448—1221no—
357#447—1221no—
358#446—611no—
359#440—1221no—
360C4×SD16C2×D6111yescharacteristic
361C4×D8—611no—
362C8×D4—611no—
363C8×D4—611no—
364C8×D4C2×D6111yescharacteristic, complemented
365#416—21no—
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—21no—
380C22×Q8—1221no—
381C22×D4—1221no—
382C22×D4—1221no—
383C2×Q16—21no—
384C2×SD16—1221no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—2444no—
390C2×SD16C22⋊D6111yescharacteristic
391C2×D8—611no—
392C2×D8—1221no—
393C22×C8—21no—
394C22×C8—1221no—
395C22×C8—1221no—
396C4⋊Q8—611no—
397C4⋊Q8—1221no—
398C4⋊Q8C22×D6111yescharacteristic
399C4⋊Q8C22×D6111yescharacteristic
400#127—611no—
401#126—41no—
402#124—21no—
403#123—41no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8—1221no—
407C22⋊Q8—2441no—
408#121—21no—
409#121—1221no—
410#121—1221no—
411#120—2441no—
412C4×Q8—21no—
413C4×Q8C22×D6111yescharacteristic
414C4×D4—611no—
415C4×D4—1221no—
416C4×D4C22×D6111yescharacteristic
417C4×D4C22×D6111yescharacteristic
418#117—21no—
419C2×C4⋊C4—21no—
420#107—21no—
421#107—611no—
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—21no—
433#103—21no—
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—2441no—
458#41—42no—
459#41—81no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—1221no—
463C2×Q8—4881no—
464C2×Q8—2441no—
465C2×Q8—2442no—
466C2×Q8C2×C22⋊D6111yescharacteristic
467C2×D4—41no—
468C2×D4—611no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—4881no—
473C2×D4—4881no—
474C2×D4—2441no—
475C2×D4—2442no—
476C2×D4—2441no—
477C2×D4—2442no—
478C2×D4D4×D6111yescharacteristic
479C2×D4C2×C22⋊D6111yescharacteristic
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—1221no—
484C22×C4—2441no—
485Q16—81no—
486SD16—22no—
487SD16—4881no—
488SD16—2442no—
489SD16—2442no—
490SD16—2442no—
491SD16—2442no—
492D8—1222no—
493D8—4881no—
494C2×C8—21no—
495C2×C8—81no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
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—4882no—
545Q8—2441no—
546Q8—2441no—
547D4—21no—
548D4—21no—
549D4—21no—
550D4—82no—
551D4—612no—
552D4—1221no—
553D4—4881no—
554D4—4882no—
555D4—4882no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559D4—2441no—
560D4#1451980212yes—
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—
612C4#135×D6111yescharacteristic
613C4C2×#1451980211yescharacteristic
614C4#1684056511yescharacteristic
615C4#1684056911yescharacteristic
616C2—42no—
617C2—81no—
618C2—9761no—
619C2—2442no—
620C2#2145730011yescharacteristic
621C2#2145752011yescharacteristic
622C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3302C2
socleC22×C61D42
prime radical#21457626C2
hypercenter#3302D61
hypocenterC61#19787
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3302

Complements

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

NH ≅ G/Ntype
1. #21458533C2semidirect
3. SD16×D244C2semidirect
4. #21445474C2semidirect
5. #21445389C2semidirect
6. #21445240C2semidirect
7. #21445239C2semidirect
8. #21445160C2semidirect
9. #21445066C2semidirect
10. #21445054C2semidirect
11. #21444997C2semidirect
12. #21444929C2semidirect
13. #21444656C2semidirect
14. #21444641C2semidirect
15. C61×#3302C2semidirect
19. #16840165C22semidirect
22. #16840112C22semidirect
27. #16840042C22semidirect
39. #16839893C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839484C22semidirect
56. C61×#556C22semidirect
58. C61×#487C22semidirect
62. C61×#462C22semidirect
65. C8×D4×C61C22semidirect
306. #3302D61semidirect
328. #556C2×D61semidirect
336. #487C2×D61semidirect
349. #462C2×D61semidirect
364. C8×D4C2×D61semidirect
378. C61#19787semidirect

Maximal subgroups

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

subgroupindex
#214585332
#214576262
SD16×D2442
#214454742
#214453892
#214452402
#214452392
#214451602
#214450662
#214450542
#214449972
#214449292
#214446562
#214446412
C61×#33022
#1978761