Finite Group Databasebeta

← #29669534

Subgroups — #29669534

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458796C211yesmaximal, characteristic, complemented
2#21457684C211yesmaximal, characteristic, complemented
3#21457550C211yesmaximal, characteristic
4#21445442C211yesmaximal, characteristic, complemented
5#21445291C211yesmaximal, characteristic, complemented
6#21445247C211yesmaximal, characteristic, complemented
7#21445205C211yesmaximal, characteristic, complemented
8#21445162C211yesmaximal, characteristic, complemented
9#21445102C211yesmaximal, characteristic, complemented
10#21445096C211yesmaximal, characteristic, complemented
11#21445018C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic, complemented
13#21444653C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3386C211yesmaximal, characteristic, complemented
16C2×#14519798—21no—
17#16840434C2211yescharacteristic
18#16840416C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840136—21no—
21#16840136C2211yescharacteristic
22#16840135C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840110C2211yescharacteristic
25#16840090—21no—
26#16840047C2211yescharacteristic
27#16839993—21no—
28#16839992—21no—
29#16839972C2211yescharacteristic
30#16839964—21no—
31#16839964C2211yescharacteristic
32#16839964C2211yescharacteristic
33#16839962—21no—
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839927—21no—
37#16839908—21no—
38#16839901—21no—
39Q8×#92418C2211yescharacteristic
40C4×#11758201—21no—
41C4×#11758201C2211yescharacteristic
42C8×D244C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519701—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839509C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#557C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#501—21no—
59C61×#500C2211yescharacteristic
60C61×#478—21no—
61C61×#477—21no—
62C61×#475C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#440—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758226—21no—
68C2×#11758224—21no—
69#14519798—44no—
70C2×#11758202—21no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78C2×#11758201D411yescharacteristic
79C2×#11758199—21no—
80C22×#92418—21no—
81C2×C8×D61—21no—
82#14519777C2311yescharacteristic
83#14519774—21no—
84#14519774C2311yescharacteristic
85#14519761—41no—
86#14519757—21no—
87#14519752—21no—
88#14519752—41no—
89#14519751—21no—
90C4×D244—21no—
91C4×D244C2311yescharacteristic
92C2×C4⋊Dic61—21no—
93#14519708—21no—
94#14519708—41no—
95#14519708D411yescharacteristic
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#125×C61—21no—
123#125×C61—41no—
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#107×C61—21no—
133#107×C61—21no—
134#107×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
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C2×C22⋊D61—41no—
156C22×Dic61—41no—
157#11758226—42no—
158#11758226—81no—
159#11758224—42no—
160#11758224—81no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—41no—
165C2×D244—41no—
166C2×D244C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170SD16×C61SD1614yescomplemented
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×C61—41no—
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—41no—
192C42×C61C2411yescharacteristic
193#11758202—42no—
194#11758201—41no—
195#11758201—42no—
196#11758201—81no—
197#11758201—81no—
198#11758201—81no—
199#11758201—81no—
200#11758199—42no—
201#11758198—21no—
202#11758198—41no—
203#11758198—41no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—41no—
207C4⋊Dic61C2×D411yescharacteristic
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—81no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215#11758190—42no—
216#11758190—81no—
217C8×D61—42no—
218C23×C61—21no—
219C22×D61—21no—
220C22×D61—41no—
221Q8×C61—21no—
222Q8×C61—21no—
223Q8×C61—21no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—82no—
229Q8×C61C2×SD1612yes—
230D4×C61—21no—
231D4×C61C2×SD1612yes—
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—
248C2×Dic61—82no—
249D244—21no—
250D244—21no—
251D244—41no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—82no—
256D244—81no—
257D244—161no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—82no—
261Dic122—41no—
262Dic122—41no—
263C8×C61—21no—
264C8×C61—41no—
265C8×C61C2×SD1612yes—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#19767—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61D4211yescharacteristic
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C4×C61—22no—
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61C22×SD1611yescharacteristic
292C4×C61#57511yescharacteristic
293C4×C61#57611yescharacteristic
294C4×C61#57711yescharacteristic
295Dic61—41no—
296Dic61—81no—
297#3392—611no—
298#3386D6111yescharacteristic, complemented
299#3298—611no—
300#3284—611no—
301#1714—611no—
302#1686—611no—
303#1680—611no—
304#1673—611no—
305#1665—611no—
306#1654—611no—
307#1653—611no—
308#1642—611no—
309#1627—611no—
310#1582—611no—
311SD16⋊C8—611no—
312C2×C61—42no—
313C2×C61#326811yescharacteristic
314C2×C61D4×SD1611yescharacteristic
315C2×C61#328911yescharacteristic
316D61—81no—
317D61—161no—
318C2×#135—1221no—
319#557C2×D6111yescharacteristic
320#550—611no—
321D4×Q8C2×D6111yescharacteristic
322#547—611no—
323#501—21no—
324#500C2×D6111yescharacteristic
325#498—611no—
326#492—611no—
327#492—611no—
328#492—1221no—
329#487—611no—
330#486—611no—
331#483—1221no—
332#478—21no—
333#477—21no—
334#475—611no—
335#475C2×D6111yescharacteristic
336#474C2×D6111yescharacteristic
337#465—1221no—
338#465—1221no—
339#462—611no—
340#461—611no—
341#461—611no—
342#461—1221no—
343#460—611no—
344#460—1221no—
345#459—611no—
346#452—1221no—
347#449—1221no—
348#448—1221no—
349#446C2×D6111yescharacteristic
350C8×Q8—611no—
351#440—21no—
352C4×SD16—611no—
353C4×SD16—1221no—
354C4×SD16C2×D6111yescharacteristic
355C8×D4—611no—
356C8×D4—611no—
357C2×#103—1221no—
358#335—611no—
359#334—611no—
360#332—611no—
361#332—611no—
362#332—611no—
363#331—611no—
364Q8⋊C8—611no—
365Q8⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
368#322—611no—
369C61#1976711yescharacteristic, complemented, Sylow
370C2×#41—1221no—
371C2×#41—1221no—
372C22×Q8—21no—
373#135—2444no—
374C2×Q16—1221no—
375C2×SD16—611no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×SD16—1221no—
382C2×SD16—1221no—
383C2×SD16C22⋊D6111yescharacteristic
384C2×D8—1221no—
385C22×C8—1221no—
386C22×C8—1221no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—611no—
390C4⋊Q8C22×D6111yescharacteristic
391C4⋊Q8C22×D6111yescharacteristic
392#127—611no—
393#127—1221no—
394#125—21no—
395#125—41no—
396#123—2441no—
397C22⋊Q8—21no—
398C22⋊Q8—41no—
399C22⋊Q8—1221no—
400#121—1221no—
401#121—1221no—
402#121—2441no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8C22×D6111yescharacteristic
406C4×Q8C22×D6111yescharacteristic
407C4×D4—21no—
408C4×D4—611no—
409C4×D4—1221no—
410C4×D4C22×D6111yescharacteristic
411C2×C4⋊C4—1221no—
412#107—21no—
413#107—21no—
414#107—21no—
415#106—21no—
416#106—611no—
417#106—1221no—
418#106—2441no—
419#106C22⋊D6111yescharacteristic
420C4⋊C8—21no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—1221no—
425C4⋊C8C22×D6111yescharacteristic
426#103—21no—
427#103—611no—
428#103—1221no—
429#103—1221no—
430#103—1221no—
431#103—1221no—
432#103—1221no—
433#103—1221no—
434#103C22⋊D6111yescharacteristic
435#102—21no—
436#102—611no—
437#102—1221no—
438#102—1221no—
439#102—2441no—
440#102C22⋊D6111yescharacteristic
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443#97—21no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—1221no—
447C4×C8C22×D6111yescharacteristic
448#41—4881no—
449#41—4881no—
450#41—2442no—
451#41—2442no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—41no—
455C2×Q8—42no—
456C2×Q8—81no—
457C2×Q8—1221no—
458C2×Q8D4×D6111yescharacteristic
459C2×Q8C2×C22⋊D6111yescharacteristic
460C2×D4—611no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—2441no—
465C2×D4—2441no—
466C2×D4—2441no—
467C2×D4C2×C22⋊D6111yescharacteristic
468C22×C4—21no—
469C22×C4—41no—
470C22×C4—1221no—
471C22×C4—2441no—
472C22×C4—2441no—
473Q16—2442no—
474SD16—4881no—
475SD16—4881no—
476SD16—4881no—
477SD16—4881no—
478SD16—4881no—
479SD16—2441no—
480SD16—2442no—
481SD16—2442no—
482SD16#1175820114yescomplemented
483D8—2442no—
484C2×C8—21no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—4881no—
492C2×C8—2442no—
493C2×C8C2×C22⋊D6111yescharacteristic
494C2×C8C2×C22⋊D6111yescharacteristic
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—611no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—2441no—
509C4⋊C4D4×D6111yescharacteristic
510C4⋊C4C2×C22⋊D6111yescharacteristic
511C4⋊C4C2×C22⋊D6111yescharacteristic
512C22⋊C4—21no—
513C22⋊C4—41no—
514C22⋊C4—1221no—
515C22⋊C4—2441no—
516C22⋊C4—2441no—
517C42—21no—
518C42—21no—
519C42—41no—
520C42C23×D6111yescharacteristic
521C23—21no—
522C23—1221no—
523C23—2441no—
524Q8—21no—
525Q8—21no—
526Q8—21no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—82no—
532Q8—2441no—
533Q8—2441no—
534Q8C2×#1175820112yes—
535D4—21no—
536D4—1221no—
537D4—1221no—
538D4—9761no—
539D4—4881no—
540D4—4882no—
541D4—4882no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546D4C2×#1175820112yes—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—82no—
557C2×C4—1221no—
558C2×C4—4882no—
559C2×C4—4882no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4#1451982211yescharacteristic
566C8—21no—
567C8—41no—
568C8—1221no—
569C8—1221no—
570C8—2441no—
571C8—2441no—
572C8—2441no—
573C8—2441no—
574C8C2×#1175820112yes—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22—2441no—
582C22—2441no—
583C22—2441no—
584C22D4×C22⋊D6111yescharacteristic
585C4—22no—
586C4—21no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—81no—
592C4—81no—
593C4—4881no—
594C4—2441no—
595C4C22×#1175820111yescharacteristic
596C4#1684055811yescharacteristic
597C4#1684056611yescharacteristic
598C4#1684057011yescharacteristic
599C2—42no—
600C2—9761no—
601C2—4881no—
602C2#2145735311yescharacteristic
603C2D4×#1175820111yescharacteristic
604C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3386C2
socleC22×C61D42
prime radical#21457550C2
hypercenter#3386D61
hypocenterC61#19767
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3386

Complements

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

NH ≅ G/Ntype
1. #21458796C2semidirect
2. #21457684C2semidirect
4. #21445442C2semidirect
5. #21445291C2semidirect
6. #21445247C2semidirect
7. #21445205C2semidirect
8. #21445162C2semidirect
9. #21445102C2semidirect
10. #21445096C2semidirect
11. #21445018C2semidirect
12. #21444922C2semidirect
13. #21444653C2semidirect
14. SD16⋊#92418C2semidirect
15. C61×#3386C2semidirect
170. SD16×C61SD16semidirect
298. #3386D61semidirect
369. C61#19767semidirect
482. SD16#11758201semidirect

Maximal subgroups

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

subgroupindex
#214587962
#214576842
#214575502
#214454422
#214452912
#214452472
#214452052
#214451622
#214451022
#214450962
#214450182
#214449222
#214446532
SD16⋊#924182
C61×#33862
#1976761