Finite Group Databasebeta

← #29670136

Subgroups — #29670136

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#21458798C211yesmaximal, characteristic, complemented
2#21458694C211yesmaximal, characteristic, complemented
3#21457435C211yesmaximal, characteristic
4#21445449C211yesmaximal, characteristic, complemented
5#21445320C211yesmaximal, characteristic, complemented
6#21445232C211yesmaximal, characteristic, complemented
7#21445177C211yesmaximal, characteristic, complemented
8#21445160C211yesmaximal, characteristic, complemented
9#21444998C211yesmaximal, characteristic, complemented
10#21444945C211yesmaximal, characteristic, complemented
11#21444913C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15C61×#3295C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17#16840490C2211yescharacteristic, complemented
18#16840433C2211yescharacteristic
19#16840161—21no—
20#16840158C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic
22#16840136C2211yescharacteristic, complemented
23#16840135C2211yescharacteristic, complemented
24#16840133—21no—
25#16840108C2211yescharacteristic
26#16840066—21no—
27#16840062—21no—
28#16840054C2211yescharacteristic, complemented
29#16840042C2211yescharacteristic
30#16840013—21no—
31#16839967C2211yescharacteristic
32#16839966—21no—
33#16839962C2211yescharacteristic, complemented
34#16839961—21no—
35#16839959C2211yescharacteristic
36#16839957C2211yescharacteristic
37#16839905—21no—
38#16839896—21no—
39Q8×#92418C2211yescharacteristic, complemented
40C8×Dic122C2211yescharacteristic, complemented
41#16839849—21no—
42C4×#11758200—21no—
43C4×#11758200C2211yescharacteristic
44#16839840C2211yescharacteristic
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C61×#554C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#481—21no—
60C61×#476C2211yescharacteristic
61C61×#465—21no—
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#102×C61—21no—
67C2×#11758222—21no—
68#14519803—44no—
69C2×#11758190—21no—
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200D411yescharacteristic
74C2×#11758201—21no—
75C2×#11758201—41no—
76C2×#11758199—21no—
77C2×#11758199—21no—
78C2×#11758199—21no—
79C2×#11758194—21no—
80#14519777C2311yescharacteristic
81#14519775—21no—
82#14519774C2311yescharacteristic
83#14519773—21no—
84#14519769—21no—
85#14519767—41no—
86#14519751—21no—
87#14519750—41no—
88C4×Dic122—21no—
89C4×Dic122C2311yescharacteristic
90#14519742—21no—
91#14519739—21no—
92#14519710—21no—
93#14519710—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708D411yescharacteristic
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519706—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700D411yescharacteristic
105#14519698—21no—
106#14519698—21no—
107#14519698—41no—
108#14519698D411yescharacteristic
109#14519697—21no—
110#14519697—21no—
111C22⋊#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C8×Dic61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61D411yescharacteristic
120C22×C8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#125×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126#121×C61—21no—
127#121×C61—21no—
128#120×C61—41no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C2×C4⋊C4×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61—42no—
154C2×D4×C61—42no—
155C2×D4×C61—81no—
156C2×D4×C61—81no—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C2×C22⋊D61—41no—
163C2×C22⋊D61—41no—
164#11758222—42no—
165#11758222—81no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×C4×D61—41no—
170C2×Dic122—21no—
171C2×Dic122C2×D411yescharacteristic
172SD16×C61—41no—
173SD16×C61—81no—
174C2×C8×C61—21no—
175C2×C8×C61—81no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C22⋊C4×C61—41no—
190C42×C61—21no—
191C42×C61C2411yescharacteristic
192#11758201—81no—
193#11758201—82no—
194#11758200—42no—
195#11758200—81no—
196#11758200SD1614yescomplemented
197#11758199—42no—
198#11758199—42no—
199#11758199—81no—
200#11758198—41no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—21no—
205#11758196—41no—
206#11758196—41no—
207#11758196—41no—
208C4×Dic61—21no—
209C4×Dic61—41no—
210#11758194—42no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217#11758190—42no—
218#11758190—81no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—41no—
222C23×C61—82no—
223C23×C61—82no—
224C23×C61—161no—
225C22×D61—41no—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—81no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—81no—
237D4×C61—82no—
238D4×C61—82no—
239D4×C61C2×SD1612yes—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—161no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C22⋊D61—82no—
254C22⋊D61—161no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—81no—
263C4×D61—82no—
264Dic122—21no—
265Dic122—41no—
266Dic122—41no—
267Dic122C2×SD1612yes—
268C8×C61—21no—
269C8×C61—21no—
270C8×C61—41no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418C2×SD1612yes—
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—164no—
288C22×C61D4211yescharacteristic
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—81no—
292C4×C61—21no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—81no—
298C4×C61C22×SD1611yescharacteristic
299C4×C61#57511yescharacteristic
300C4×C61#57611yescharacteristic
301C4×C61#57711yescharacteristic
302Dic61—22no—
303Dic61—41no—
304Dic61—81no—
305#3392—611no—
306#3381—611no—
307#3295D6111yescharacteristic, complemented
308#3274—611no—
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#326811yescharacteristic
324C2×C61D4×SD1611yescharacteristic
325C2×C61#328811yescharacteristic
326D61—161no—
327C2×AGL(1,ℤ/8)—1221no—
328#557—611no—
329#554C2×D6111yescharacteristic
330#550—611no—
331D42C2×D6111yescharacteristic
332#499—1221no—
333#498—611no—
334#493—611no—
335#492—611no—
336#492—611no—
337#492—1221no—
338#486—611no—
339#486C2×D6111yescharacteristic
340#481—21no—
341#479—1221no—
342#478—1221no—
343#477—611no—
344#476C2×D6111yescharacteristic
345#474—611no—
346#469—1221no—
347#465—21no—
348#461—611no—
349#461—1221no—
350#460—611no—
351#460—611no—
352#460—1221no—
353#460C2×D6111yescharacteristic
354#459—611no—
355#450—21no—
356#449—1221no—
357#447—1221no—
358C8×Q8—611no—
359C8×Q8—611no—
360C4×SD16—611no—
361C4×SD16—1221no—
362C4×SD16—1221no—
363C4×SD16C2×D6111yescharacteristic
364#436—611no—
365C8×D4C2×D6111yescharacteristic
366C2×#102—21no—
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—1221no—
380C22×D4—21no—
381C22×D4—21no—
382AGL(1,ℤ/8)—2444no—
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—1221no—
393C2×D8—1221no—
394C2×D8—1221no—
395C2×M16—1221no—
396C22×C8—21no—
397C4⋊Q8—611no—
398C4⋊Q8—1221no—
399C4⋊Q8C22×D6111yescharacteristic
400#127—611no—
401#127—1221no—
402#127C22×D6111yescharacteristic
403#125—21no—
404#125—1221no—
405#125—2441no—
406C22⋊Q8—21no—
407C22⋊Q8—41no—
408#121—21no—
409#121—21no—
410#121—1221no—
411#121—2441no—
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—21no—
422#107—1221no—
423#107—1221no—
424#106—21no—
425#106—611no—
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—21no—
442#102—21no—
443#102—611no—
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—4881no—
459#41—2442no—
460C2×Q8—21no—
461C2×Q8—611no—
462C2×Q8—1221no—
463C2×Q8C2×C22⋊D6111yescharacteristic
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—81no—
472C2×D4—81no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—2441no—
477C2×D4—2441no—
478C2×D4D4×D6111yescharacteristic
479C2×D4C2×C22⋊D6111yescharacteristic
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—2441no—
484SD16—41no—
485SD16—81no—
486SD16—614no—
487SD16—4881no—
488SD16—4881no—
489SD16—4882no—
490SD16—4881no—
491SD16—2442no—
492SD16—2442no—
493D8—4881no—
494D8—2442no—
495D8—2442no—
496M16—2442no—
497C2×C8—21no—
498C2×C8—81no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8C2×C22⋊D6111yescharacteristic
506C2×C8C2×C22⋊D6111yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—611no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
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—82no—
535C23—82no—
536C23—161no—
537C23—2441no—
538Q8—21no—
539Q8—21no—
540Q8—81no—
541Q8—612no—
542Q8—1221no—
543Q8—2441no—
544Q8—2441no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—81no—
553D4—82no—
554D4—82no—
555D4—9761no—
556D4—4881no—
557D4—4882no—
558D4—2441no—
559D4—2441no—
560D4—2441no—
561D4—2441no—
562D4SD16×D6112yes—
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—41no—
572C2×C4—161no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—4882no—
576C2×C4—2441no—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4#1451982211yescharacteristic
580C8—21no—
581C8—21no—
582C8—41no—
583C8—612no—
584C8—1221no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—164no—
599C22—4881no—
600C22—4881no—
601C22—4881no—
602C22D4×C22⋊D6111yescharacteristic
603C4—21no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—1222no—
610C4—4881no—
611C4—2441no—
612C4C2×SD16×D6111yescharacteristic
613C4#1684056011yescharacteristic
614C4#1684056511yescharacteristic
615C4#1684056811yescharacteristic
616C2—42no—
617C2—81no—
618C2—161no—
619C2—9761no—
620C2#2145735511yescharacteristic
621C2SD16×C22⋊D6111yescharacteristic
622C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3295C2
socleC22×C61D42
prime radical#21457435C2
hypercenter#3295D61
hypocenterC61#19807
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3295

Complements

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

NH ≅ G/Ntype
1. #21458798C2semidirect
2. #21458694C2semidirect
4. #21445449C2semidirect
5. #21445320C2semidirect
6. #21445232C2semidirect
7. #21445177C2semidirect
8. #21445160C2semidirect
9. #21444998C2semidirect
10. #21444945C2semidirect
11. #21444913C2semidirect
12. #21444891C2semidirect
13. #21444663C2semidirect
14. #21444631C2semidirect
15. C61×#3295C2semidirect
17. #16840490C22semidirect
20. #16840158C22semidirect
22. #16840136C22semidirect
23. #16840135C22semidirect
28. #16840054C22semidirect
33. #16839962C22semidirect
39. Q8×#92418C22semidirect
40. C8×Dic122C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839511C22semidirect
48. #16839509C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. #16839484C22semidirect
196. #11758200SD16semidirect
307. #3295D61semidirect
378. C61#19807semidirect

Maximal subgroups

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

subgroupindex
#214587982
#214586942
#214574352
#214454492
#214453202
#214452322
#214451772
#214451602
#214449982
#214449452
#214449132
#214448912
#214446632
#214446312
C61×#32952
#1980761