Finite Group Databasebeta

← #29669511

Subgroups — #29669511

Order 15616.

668 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458725C211yesmaximal, characteristic, complemented
2#21457645C211yesmaximal, characteristic
3D4×#11758201C211yesmaximal, characteristic
4#21445444C211yesmaximal, characteristic, complemented
5#21445293C211yesmaximal, characteristic, complemented
6#21445235C211yesmaximal, characteristic, complemented
7#21445216C211yesmaximal, characteristic, complemented
8#21445191C211yesmaximal, characteristic, complemented
9#21445025C211yesmaximal, characteristic, complemented
10#21444984C211yesmaximal, characteristic, complemented
11#21444934C211yesmaximal, characteristic
12#21444897C211yesmaximal, characteristic, complemented
13#21444656C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15Q8×SD16×C61C211yesmaximal, characteristic, complemented
16C22×#11758201—21no—
17#16840472C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840168—21no—
20#16840165C2211yescharacteristic, complemented
21#16840136C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840114C2211yescharacteristic, complemented
24#16840109C2211yescharacteristic
25#16840082—21no—
26#16840069—21no—
27#16840057C2211yescharacteristic, complemented
28#16840052C2211yescharacteristic
29#16840049C2211yescharacteristic
30#16840039—21no—
31#16839994—21no—
32#16839993—21no—
33#16839964C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839962C2211yescharacteristic
36#16839912—21no—
37#16839911—21no—
38#16839901—21no—
39#16839893C2211yescharacteristic, complemented
40#16839860—21no—
41C4×#11758201C2211yescharacteristic
42C8×D244C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44#102×D61—21no—
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56Q82×C61C2211yescharacteristic, complemented
57D4×Q8×C61C2211yescharacteristic
58C61×#498—21no—
59C61×#498C2211yescharacteristic, complemented
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#475—21no—
63C61×#475C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C22×D244—21no—
68C2×D4×D61—21no—
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×#11758201—41no—
75C2×#11758201—44no—
76C2×#11758201D411yescharacteristic
77C22×#92418—21no—
78C2×C8×D61—21no—
79#14519777C2311yescharacteristic
80#14519774C2311yescharacteristic
81#14519769—21no—
82#14519755—41no—
83#14519753—21no—
84#14519752—21no—
85#14519751—21no—
86#14519747—41no—
87C4×D244—21no—
88C4×D244C2311yescharacteristic
89C4⋊C4×D61—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708D411yescharacteristic
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519701—21no—
101#14519701—21no—
102#14519701D411yescharacteristic
103#14519700—41no—
104#14519699—21no—
105#14519699—21no—
106#14519698—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#106×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×Q8×C61C2×D411yescharacteristic
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—41no—
157C2×C22⋊D61—41no—
158D4×D61—42no—
159D4×D61—81no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—42no—
164C2×D244—41no—
165C2×D244—81no—
166C2×D244C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170SD16×C61—22no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
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—41no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—41no—
191C42×C61C2411yescharacteristic
192#11758201—42no—
193#11758201—42no—
194#11758201—42no—
195#11758201—81no—
196#11758201—81no—
197#11758201—82no—
198#11758201SD1614yescomplemented
199#11758198—21no—
200#11758198—41no—
201#11758198—41no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C2×D411yescharacteristic
205#11758196—41no—
206#11758196—41no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—42no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C8×D61—81no—
215C23×C61—21no—
216C22×D61—21no—
217C22×D61—41no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61—161no—
221Q8×C61—21no—
222Q8×C61—21no—
223Q8×C61—21no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—82no—
227Q8×C61—81no—
228Q8×C61—161no—
229Q8×C61C2×SD1612yes—
230D4×C61—21no—
231D4×C61#13712yes—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—82no—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C22⋊D61—82no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248D244—21no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—41no—
253D244—81no—
254D244—82no—
255D244C2×SD1612yes—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—161no—
259Dic122—81no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—41no—
263#92418—21no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#92418C2×SD1612yes—
269#19765—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61D4211yescharacteristic
274C2×D61—22no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—164no—
281C4×C61—22no—
282C4×C61—21no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61C22×SD1611yescharacteristic
290C4×C61C2×#13711yescharacteristic
291C4×C61#57711yescharacteristic
292C4×C61#57711yescharacteristic
293Dic61—41no—
294Dic61—81no—
295#3385—611no—
296Q8×SD16D6111yescharacteristic, complemented
297#3295—611no—
298D4×SD16—611no—
299#1714—611no—
300#1686—611no—
301#1678—611no—
302#1675—611no—
303#1671—611no—
304#1643—611no—
305#1637—611no—
306#1629—611no—
307#1623—611no—
308#1582—611no—
309SD16⋊C8—611no—
310C2×C61—42no—
311C2×C61#326611yescharacteristic
312C2×C61D4×SD1611yescharacteristic
313C2×C61#328811yescharacteristic
314D61—42no—
315D61—161no—
316C22×SD16—1221no—
317Q82C2×D6111yescharacteristic, complemented
318#554—611no—
319D4×Q8C2×D6111yescharacteristic
320D42—611no—
321#501—1221no—
322#500—611no—
323#498—21no—
324#498C2×D6111yescharacteristic, complemented
325#492—611no—
326#492—611no—
327#487—611no—
328#486—611no—
329#481—1221no—
330#479—1221no—
331#477—611no—
332#476—21no—
333#476—611no—
334#476—611no—
335#476C2×D6111yescharacteristic
336#475—21no—
337#475C2×D6111yescharacteristic, complemented
338#474—1221no—
339#465—1221no—
340#465—1221no—
341#461—611no—
342#460—611no—
343#460—611no—
344#450—1221no—
345#450—1221no—
346#448—1221no—
347#446—611no—
348C8×Q8C2×D6111yescharacteristic, complemented
349#440—1221no—
350C4×SD16—21no—
351C4×SD16—611no—
352C4×SD16C2×D6111yescharacteristic
353C8×D4—611no—
354C8×D4—611no—
355C2×#102—1221no—
356#335—611no—
357#334—611no—
358#333—611no—
359#333—611no—
360#332—611no—
361#331—611no—
362Q8⋊C8—611no—
363Q8⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366#322—611no—
367C61#1976511yescharacteristic, complemented, Sylow
368C22×Q8—21no—
369C22×D4—1221no—
370C22×D4—1221no—
371C2×SD16—611no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—2441no—
378C2×SD16—2444no—
379C2×SD16C22⋊D6111yescharacteristic
380C22×C8—1221no—
381C22×C8—1221no—
382C4⋊Q8—21no—
383C4⋊Q8—21no—
384C4⋊Q8—21no—
385C4⋊Q8—41no—
386C4⋊Q8—611no—
387C4⋊Q8C22×D6111yescharacteristic
388C4⋊Q8C22×D6111yescharacteristic
389#127—611no—
390#125—1221no—
391C22⋊Q8—21no—
392C22⋊Q8—41no—
393C22⋊Q8—1221no—
394C22⋊Q8—2441no—
395#121—1221no—
396#121—1221no—
397#120—2441no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8C22×D6111yescharacteristic
401C4×Q8C22×D6111yescharacteristic
402C4×D4—21no—
403C4×D4—611no—
404C4×D4—1221no—
405C4×D4C22×D6111yescharacteristic
406C2×C4⋊C4—1221no—
407#107—1221no—
408#107—1221no—
409#107—1221no—
410#106—21no—
411#106—21no—
412#106—21no—
413#106—21no—
414#106—611no—
415#106—1221no—
416#106—1221no—
417#106C22⋊D6111yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—1221no—
423C4⋊C8C22×D6111yescharacteristic
424#103—21no—
425#103—611no—
426#103—1221no—
427#103—1221no—
428#103—2441no—
429#103C22⋊D6111yescharacteristic
430#102—21no—
431#102—611no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102C22⋊D6111yescharacteristic
439C22⋊C8—1221no—
440C22⋊C8—1221no—
441#97—1221no—
442C4×C8—21no—
443C4×C8—611no—
444C4×C8—611no—
445C4×C8C22×D6111yescharacteristic
446C24—2441no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—41no—
450C2×Q8—41no—
451C2×Q8—42no—
452C2×Q8—81no—
453C2×Q8—1221no—
454C2×Q8D4×D6111yescharacteristic
455C2×Q8C2×C22⋊D6111yescharacteristic
456C2×D4—611no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—4881no—
461C2×D4—4881no—
462C2×D4—2441no—
463C2×D4—2442no—
464C2×D4—2441no—
465C2×D4—2442no—
466C2×D4C2×C22⋊D6111yescharacteristic
467C22×C4—21no—
468C22×C4—41no—
469C22×C4—1221no—
470C22×C4—2441no—
471SD16—22no—
472SD16—614no—
473SD16—4881no—
474SD16—4882no—
475SD16—4881no—
476SD16—2442no—
477SD16—2442no—
478SD16—2442no—
479C2×C8—21no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—4881no—
487C2×C8—2442no—
488C2×C8C2×C22⋊D6111yescharacteristic
489C2×C8C2×C22⋊D6111yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—611no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4D4×D6111yescharacteristic
505C4⋊C4C2×C22⋊D6111yescharacteristic
506C4⋊C4C2×C22⋊D6111yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—41no—
509C22⋊C4—1221no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C42—21no—
513C42—21no—
514C42—41no—
515C42C23×D6111yescharacteristic
516C23—21no—
517C23—1221no—
518C23—9761no—
519C23—4882no—
520C23—4882no—
521C23—2441no—
522Q8—21no—
523Q8—21no—
524Q8—21no—
525Q8—41no—
526Q8—41no—
527Q8—82no—
528Q8—81no—
529Q8—161no—
530Q8—4881no—
531Q8SD16×D6112yes—
532D4—21no—
533D4—612no—
534D4—1221no—
535D4—4881no—
536D4—4882no—
537D4—4882no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541D4—2441no—
542D4#1451980812yes—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—1221no—
554C2×C4—9761no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C22×C22⋊D6111yescharacteristic
560C2×C4#1451982211yescharacteristic
561C8—21no—
562C8—21no—
563C8—41no—
564C8—612no—
565C8—1221no—
566C8—2441no—
567C8—2441no—
568C8—2441no—
569C8—2441no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—1222no—
574C22—9764no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—2441no—
579C22—2441no—
580C22D4×C22⋊D6111yescharacteristic
581C4—22no—
582C4—21no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—4881no—
590C4—2441no—
591C4C2×SD16×D6111yescharacteristic
592C4C2×#1451980811yescharacteristic
593C4#1684056811yescharacteristic
594C4#1684057011yescharacteristic
595C2—42no—
596C2—9761no—
597C2—2442no—
598C2#2145733011yescharacteristic
599C2SD16×C22⋊D6111yescharacteristic
600C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×SD16×C61C2
socleC22×C61D42
prime radicalD4×D244C22
hypercenterQ8×SD16D61
hypocenterC61#19765
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. #21458725C2semidirect
4. #21445444C2semidirect
5. #21445293C2semidirect
6. #21445235C2semidirect
7. #21445216C2semidirect
8. #21445191C2semidirect
9. #21445025C2semidirect
10. #21444984C2semidirect
12. #21444897C2semidirect
13. #21444656C2semidirect
14. #21444629C2semidirect
15. Q8×SD16×C61C2semidirect
20. #16840165C22semidirect
23. #16840114C22semidirect
27. #16840057C22semidirect
39. #16839893C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839511C22semidirect
48. #16839510C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
52. Q8⋊#92418C22semidirect
55. #16839484C22semidirect
56. Q82×C61C22semidirect
59. C61×#498C22semidirect
63. C61×#475C22semidirect
64. C8×Q8×C61C22semidirect
198. #11758201SD16semidirect
296. Q8×SD16D61semidirect
317. Q82C2×D61semidirect
324. #498C2×D61semidirect
337. #475C2×D61semidirect
348. C8×Q8C2×D61semidirect
367. C61#19765semidirect

Maximal subgroups

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

subgroupindex
#214587252
#214576452
D4×#117582012
#214454442
#214452932
#214452352
#214452162
#214451912
#214450252
#214449842
#214449342
#214448972
#214446562
#214446292
Q8×SD16×C612
#1976561