Finite Group Databasebeta

← #29669519

Subgroups — #29669519

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458170C211yesmaximal, characteristic, complemented
2#21457684C211yesmaximal, characteristic, complemented
3#21457540C211yesmaximal, characteristic
4#21445453C211yesmaximal, characteristic, complemented
5#21445313C211yesmaximal, characteristic, complemented
6#21445240C211yesmaximal, characteristic, complemented
7#21445196C211yesmaximal, characteristic, complemented
8#21445191C211yesmaximal, characteristic, complemented
9#21445102C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21445026C211yesmaximal, characteristic, complemented
12#21444968C211yesmaximal, characteristic, complemented
13#21444653C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3384C211yesmaximal, characteristic, complemented
16C2×#14519796—21no—
17#16840416C2211yescharacteristic
18#16840416C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840142C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840090—21no—
25#16840090—21no—
26#16840080—21no—
27#16840049C2211yescharacteristic
28#16840047C2211yescharacteristic
29#16840047C2211yescharacteristic
30#16840004—21no—
31#16839993—21no—
32#16839993—21no—
33#16839967C2211yescharacteristic
34#16839967C2211yescharacteristic
35#16839964C2211yescharacteristic
36#16839925—21no—
37#16839925—21no—
38#16839901—21no—
39C4×#11758201C2211yescharacteristic
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43C2×#14519708—21no—
44C2×#14519701—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#554C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#501—21no—
59C61×#498C2211yescharacteristic
60C61×#478—21no—
61C61×#476—21no—
62C61×#476C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#442—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758224—21no—
68C2×#11758224—21no—
69#14519796—44no—
70C2×Dic244—21no—
71C2×Dic244—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201D411yescharacteristic
76C22×#92418—21no—
77C22×#92418—21no—
78C2×C8×D61—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519761—41no—
82#14519761—41no—
83#14519757—21no—
84#14519757—21no—
85#14519751—21no—
86#14519751—21no—
87C4×D244C2311yescharacteristic
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—41no—
96#14519708D411yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701—41no—
106#14519701D411yescharacteristic
107#14519700—21no—
108#14519700—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×D8×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121#125×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132#107×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135#106×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
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×D4×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C22×C4×C61—41no—
157C2×C22⋊D61—41no—
158C2×C22⋊D61—41no—
159C22×Dic61—41no—
160C22×Dic61—41no—
161#11758224—42no—
162#11758224—42no—
163#11758224—81no—
164#11758224—81no—
165C2×D244C2×D411yescharacteristic
166C2×C4×D61—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169D8×C61SD1614yescomplemented
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61C2411yescharacteristic
190#11758201—41no—
191#11758201—81no—
192#11758200—42no—
193#11758200—42no—
194#11758198—21no—
195#11758198—41no—
196#11758198—41no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203C4⋊Dic61C2×D411yescharacteristic
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—42no—
209C2×#92418—81no—
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212Dic244—42no—
213Dic244—81no—
214C8×D61—42no—
215C23×C61—21no—
216C23×C61—21no—
217C22×D61—21no—
218Q8×C61—21no—
219Q8×C61—21no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—81no—
223Q8×C61—82no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61C2×SD1612yes—
227D4×C61C2×SD1612yes—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—82no—
239C2×C4×C61—161no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C22⋊D61—82no—
244C22⋊D61—82no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—82no—
249C2×Dic61—82no—
250D244—21no—
251D244—21no—
252C4×D61—41no—
253C4×D61—41no—
254Dic122—41no—
255Dic122—41no—
256Dic122—41no—
257Dic122—41no—
258C8×C61—21no—
259C8×C61—41no—
260C8×C61C2×SD1612yes—
261#92418—21no—
262#92418—21no—
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#19766—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61D4211yescharacteristic
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C4×C61—21no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61C22×SD1611yescharacteristic
286C4×C61#57511yescharacteristic
287C4×C61#57711yescharacteristic
288C4×C61#57711yescharacteristic
289Dic61—41no—
290Dic61—81no—
291Dic61—81no—
292#3384D6111yescharacteristic, complemented
293#3335—611no—
294#3298—611no—
295#3283—611no—
296#1716—611no—
297#1690—611no—
298#1679—611no—
299#1671—611no—
300#1671—611no—
301#1654—611no—
302#1651—611no—
303#1643—611no—
304#1635—611no—
305#1582—611no—
306#1580—611no—
307C2×C61—42no—
308C2×C61—42no—
309C2×C61#326711yescharacteristic
310C2×C61D4×SD1611yescharacteristic
311C2×C61#328911yescharacteristic
312D61—81no—
313C2×#135—1221no—
314#554C2×D6111yescharacteristic
315D4×Q8C2×D6111yescharacteristic
316#547—611no—
317#547—611no—
318#501—21no—
319#498—611no—
320#498—611no—
321#498C2×D6111yescharacteristic
322#494—611no—
323#487—611no—
324#487—611no—
325#483—1221no—
326#483—1221no—
327#481—1221no—
328#478—21no—
329#476—21no—
330#476—611no—
331#476C2×D6111yescharacteristic
332#475—611no—
333#475—611no—
334#474C2×D6111yescharacteristic
335#467—1221no—
336#465—1221no—
337#465—1221no—
338#461—611no—
339#461—611no—
340#461—611no—
341#452—1221no—
342#452—1221no—
343#448—1221no—
344#446C2×D6111yescharacteristic
345#442—21no—
346C4×SD16—611no—
347C4×D8C2×D6111yescharacteristic
348C8×D4—611no—
349C8×D4—611no—
350C8×D4—611no—
351C2×#106—1221no—
352C2×#103—1221no—
353#334—611no—
354#334—611no—
355#333—611no—
356#332—611no—
357#332—611no—
358#332—611no—
359Q8⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363#322—611no—
364C61#1976611yescharacteristic, complemented, Sylow
365C2×#41—1221no—
366C2×#41—1221no—
367C22×Q8—21no—
368#135—2444no—
369C2×Q16—1221no—
370C2×Q16—1221no—
371C2×SD16—611no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×D8C22⋊D6111yescharacteristic
376C22×C8—1221no—
377C22×C8—1221no—
378C22×C8—1221no—
379C4⋊Q8—21no—
380C4⋊Q8—611no—
381C4⋊Q8—611no—
382C4⋊Q8C22×D6111yescharacteristic
383C4⋊Q8C22×D6111yescharacteristic
384#125—21no—
385#123—2441no—
386#123—2441no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—41no—
390C22⋊Q8—41no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393#121—1221no—
394#121—1221no—
395C4×Q8C22×D6111yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—611no—
399C4×D4C22×D6111yescharacteristic
400C4×D4C22×D6111yescharacteristic
401C2×C4⋊C4—21no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404#107—21no—
405#107C22⋊D6111yescharacteristic
406#106—21no—
407#106—21no—
408#106—21no—
409#106—611no—
410#106—1221no—
411#106—1221no—
412#106—1221no—
413#106—1221no—
414#106—1221no—
415#106—2441no—
416C4⋊C8—21no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8C22×D6111yescharacteristic
421#103—611no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—2441no—
430#102—21no—
431#102—21no—
432#102—611no—
433#102—1221no—
434#102C22⋊D6111yescharacteristic
435#102C22⋊D6111yescharacteristic
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439#97—21no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8C22×D6111yescharacteristic
443#41—4881no—
444#41—4881no—
445#41—2442no—
446#41—2442no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—41no—
450C2×Q8—42no—
451C2×Q8—81no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8D4×D6111yescharacteristic
455C2×D4—611no—
456C2×D4—2441no—
457C2×D4—2441no—
458C2×D4C2×C22⋊D6111yescharacteristic
459C2×D4C2×C22⋊D6111yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—1221no—
465C22×C4—2441no—
466C22×C4—2441no—
467Q16—4881no—
468Q16—2442no—
469SD16—4881no—
470SD16—2441no—
471SD16—2442no—
472SD16—2442no—
473D8#1175820114yescomplemented
474C2×C8—21no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—4881no—
482C2×C8—2442no—
483C2×C8—2442no—
484C2×C8C2×C22⋊D6111yescharacteristic
485C2×C8C2×C22⋊D6111yescharacteristic
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—611no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C4⋊C4D4×D6111yescharacteristic
502C4⋊C4C2×C22⋊D6111yescharacteristic
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—1221no—
509C22⋊C4—2441no—
510C22⋊C4—2441no—
511C42—21no—
512C42C23×D6111yescharacteristic
513C23—21no—
514C23—21no—
515C23—1221no—
516Q8—21no—
517Q8—21no—
518Q8—41no—
519Q8—41no—
520Q8—81no—
521Q8—82no—
522Q8—2441no—
523Q8—2441no—
524Q8—2441no—
525Q8—2441no—
526D4—21no—
527D4—21no—
528D4—1221no—
529D4—1221no—
530D4—4882no—
531D4—4882no—
532D4C2×#1175820112yes—
533D4C2×#1175820112yes—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—82no—
545C2×C4—161no—
546C2×C4—1221no—
547C2×C4—4882no—
548C2×C4—4882no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C22×C22⋊D6111yescharacteristic
555C2×C4#1451982211yescharacteristic
556C8—21no—
557C8—41no—
558C8—1221no—
559C8—1221no—
560C8—2441no—
561C8—2441no—
562C8—2441no—
563C8—2441no—
564C8C2×#1175820112yes—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—2441no—
572C22—2441no—
573C22—2441no—
574C22D4×C22⋊D6111yescharacteristic
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—81no—
581C4—81no—
582C4—4881no—
583C4—4881no—
584C4—2441no—
585C4C22×#1175820111yescharacteristic
586C4#1684055811yescharacteristic
587C4#1684057011yescharacteristic
588C4#1684057211yescharacteristic
589C2—42no—
590C2—42no—
591C2—4881no—
592C2#2145733811yescharacteristic
593C2D4×#1175820111yescharacteristic
594C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3384C2
socleC22×C61D42
prime radical#21457540C2
hypercenter#3384D61
hypocenterC61#19766
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3384

Complements

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

NH ≅ G/Ntype
1. #21458170C2semidirect
2. #21457684C2semidirect
4. #21445453C2semidirect
5. #21445313C2semidirect
6. #21445240C2semidirect
7. #21445196C2semidirect
8. #21445191C2semidirect
9. #21445102C2semidirect
10. #21445082C2semidirect
11. #21445026C2semidirect
12. #21444968C2semidirect
13. #21444653C2semidirect
14. #21444643C2semidirect
15. C61×#3384C2semidirect
169. D8×C61SD16semidirect
292. #3384D61semidirect
364. C61#19766semidirect
473. D8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214581702
#214576842
#214575402
#214454532
#214453132
#214452402
#214451962
#214451912
#214451022
#214450822
#214450262
#214449682
#214446532
#214446432
C61×#33842
#1976661