Finite Group Databasebeta

← #29676909

Subgroups — #29676909

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458537C211yesmaximal, characteristic, complemented
2#21458405C211yesmaximal, characteristic, complemented
3#21458118C211yesmaximal, characteristic
4#21445413C211yesmaximal, characteristic, complemented
5#21445375C211yesmaximal, characteristic, complemented
6#21445368C211yesmaximal, characteristic, complemented
7#21445332C211yesmaximal, characteristic, complemented
8#21445305C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic, complemented
10#21445101C211yesmaximal, characteristic, complemented
11#21445087C211yesmaximal, characteristic, complemented
12#21445082C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444702C211yesmaximal, characteristic, complemented
15C61×#3276C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17#16840417C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840124—21no—
22#16840114C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840097—21no—
25#16840096—21no—
26#16840084—21no—
27#16840054—21no—
28#16840047C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840039C2211yescharacteristic
31#16840039C2211yescharacteristic
32#16840006—21no—
33#16840005—21no—
34#16839977—21no—
35#16839967C2211yescharacteristic
36#16839967C2211yescharacteristic
37#16839905—21no—
38#16839893C2211yescharacteristic
39#16839862—21no—
40C4×#11758199C2211yescharacteristic
41#16839841C2211yescharacteristic
42#16839840C2211yescharacteristic
43#16839801—21no—
44#16839761—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C2×#137×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58C61×#547C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#470—21no—
61C61×#462C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#452—21no—
64C61×#451—21no—
65C4×SD16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758199D411yescharacteristic
71C2×#11758194—21no—
72C2×#11758189—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519772—41no—
76#14519765—21no—
77#14519759—41no—
78#14519756—21no—
79#14519753—21no—
80#14519748—21no—
81C4×Dic122—21no—
82C4×D244C2311yescharacteristic
83#14519735—21no—
84C4⋊C4×D61—21no—
85#14519711—21no—
86#14519711—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—41no—
93#14519708—41no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519701—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519698—21no—
106#14519698D411yescharacteristic
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C22×Q8×C61—21no—
117#137×C61—44no—
118C2×Q16×C61—21no—
119C2×Q16×C61—21no—
120C2×Q16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61D411yescharacteristic
123C2×M16×C61—21no—
124C4⋊Q8×C61—21no—
125C4⋊Q8×C61C2311yescharacteristic
126C4⋊Q8×C61C2311yescharacteristic
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131#121×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×C4×C61—41no—
160C2×C22⋊D61—41no—
161#11758222—42no—
162#11758222—81no—
163C2×D244C2×D411yescharacteristic
164C2×C4×D61—21no—
165C2×C4×D61—41no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168Q16×C61—42no—
169Q16×C61—42no—
170Q16×C61—81no—
171SD16×C61—22no—
172SD16×C61—42no—
173M16×C61—42no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
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—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61—21no—
192C42×C61C2411yescharacteristic
193C22⋊Dic61—41no—
194#11758200—42no—
195#11758200—81no—
196#11758199—41no—
197#11758198—21no—
198#11758198—41no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C2×D411yescharacteristic
205#11758196—41no—
206#11758196—41no—
207C4×Dic61—41no—
208#11758194—81no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215#11758189—42no—
216C23×C61—21no—
217C23×C61—21no—
218C22×D61—21no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—82no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—82no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—82no—
242C2×C4×C61—82no—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C22⋊D61—82no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250D244—21no—
251D244—21no—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—82no—
255Dic122—41no—
256Dic122—41no—
257Dic122—41no—
258Dic122—41no—
259C8×C61—21no—
260C8×C61—41no—
261C8×C61AGL(1,ℤ/8)12yes—
262#92418—22no—
263#92418—21no—
264#92418—21no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#20291—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61D4211yescharacteristic
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C4×C61—21no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C4×C61#57511yescharacteristic
288C4×C61#57711yescharacteristic
289C4×C61#57711yescharacteristic
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293#3367—611no—
294#3353—611no—
295#3330—611no—
296#3276D6111yescharacteristic, complemented
297#1707—611no—
298#1700—611no—
299#1699—611no—
300#1693—611no—
301#1689—611no—
302#1654—611no—
303#1654—611no—
304#1652—611no—
305#1651—611no—
306#1590—611no—
307#1589—611no—
308C2×C61—81no—
309C2×C61—81no—
310C2×C61#327211yescharacteristic
311C2×C61#327511yescharacteristic
312C2×C61#329311yescharacteristic
313D61—81no—
314C2×#137—21no—
315#556—611no—
316D4×Q8C2×D6111yescharacteristic
317#547—611no—
318#547C2×D6111yescharacteristic
319#500—611no—
320#498—611no—
321#498—611no—
322#494C2×D6111yescharacteristic
323#489—1221no—
324#487—611no—
325#487—611no—
326#484—1221no—
327#484—1221no—
328#482—1221no—
329#477—1221no—
330#475—611no—
331#474—611no—
332#474—611no—
333#474—611no—
334#470—21no—
335#468—1221no—
336#468—1221no—
337#463—1221no—
338#462C2×D6111yescharacteristic
339#461—611no—
340#461—611no—
341#461C2×D6111yescharacteristic
342#452—21no—
343#451—21no—
344#449—1221no—
345#446—611no—
346#440—1221no—
347C4×SD16C2×D6111yescharacteristic
348C4×D8—611no—
349#436—611no—
350#436—611no—
351#436C2×D6111yescharacteristic
352#428—1221no—
353#417—1221no—
354#335—611no—
355#334—611no—
356#332—611no—
357#332—611no—
358#332—611no—
359#332—611no—
360Q8⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364C82—611no—
365C61#2029111yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C2×#41—1221no—
368C22×Q8—21no—
369#137—44no—
370C2×Q16—21no—
371C2×Q16—21no—
372C2×Q16—21no—
373C2×SD16—21no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16C22⋊D6111yescharacteristic
377C2×D8—611no—
378C2×M16—21no—
379C2×M16—1221no—
380C2×M16—1221no—
381C4⋊Q8—21no—
382C4⋊Q8—611no—
383C4⋊Q8—611no—
384C4⋊Q8C22×D6111yescharacteristic
385C4⋊Q8C22×D6111yescharacteristic
386#126—2441no—
387#124—1221no—
388#123—41no—
389#123—2441no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—41no—
393C22⋊Q8—1221no—
394C22⋊Q8—1221no—
395#121—21no—
396#121—1221no—
397C4×Q8—1221no—
398C4×Q8C22×D6111yescharacteristic
399C4×D4—21no—
400C4×D4—611no—
401C4×D4C22×D6111yescharacteristic
402C4×D4C22×D6111yescharacteristic
403#117—1221no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—1221no—
406#107—611no—
407#107—1221no—
408#107—1221no—
409#107—1221no—
410#107—1221no—
411#107—1221no—
412#106—1221no—
413#106—2441no—
414#106—2441no—
415#106C22⋊D6111yescharacteristic
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—1221no—
420C4⋊C8C22×D6111yescharacteristic
421#103—21no—
422#103—21no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#103C22⋊D6111yescharacteristic
431#102—611no—
432#102—611no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102C22⋊D6111yescharacteristic
437C22⋊C8—21no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440#97—1221no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8C22×D6111yescharacteristic
444#41—42no—
445#41—81no—
446#41—4881no—
447#41—2442no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—41no—
451C2×Q8—42no—
452C2×Q8—81no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8C2×C22⋊D6111yescharacteristic
456C2×D4—41no—
457C2×D4—611no—
458C2×D4—2441no—
459C2×D4D4×D6111yescharacteristic
460C2×D4C2×C22⋊D6111yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—1221no—
466C22×C4—2441no—
467Q16—42no—
468Q16—42no—
469Q16—81no—
470SD16—22no—
471SD16—42no—
472SD16—4881no—
473SD16—2442no—
474D8—2441no—
475M16—42no—
476M16—4881no—
477M16—2442no—
478C2×C8—21no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8C2×C22⋊D6111yescharacteristic
486C2×C8C2×C22⋊D6111yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
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—
511C22⋊C4—2441no—
512C42—21no—
513C42—2441no—
514C42C23×D6111yescharacteristic
515C23—21no—
516C23—21no—
517C23—1221no—
518Q8—21no—
519Q8—21no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—41no—
524Q8—82no—
525Q8—2441no—
526Q8—2441no—
527Q8—2441no—
528Q8—2441no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—82no—
534D4—1221no—
535D4—1221no—
536D4—4882no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—1221no—
550C2×C4—4882no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C2×C4#1451982211yescharacteristic
558C8—21no—
559C8—41no—
560C8—1222no—
561C8—1221no—
562C8—1221no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C8#1451980312yes—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—2441no—
574C22—2441no—
575C22—2441no—
576C22D4×C22⋊D6111yescharacteristic
577C4—21no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—4881no—
585C4—4881no—
586C4—2441no—
587C4C2×#1451980311yescharacteristic
588C4#1684056111yescharacteristic
589C4#1684057111yescharacteristic
590C4#1684057311yescharacteristic
591C2—81no—
592C2—81no—
593C2—4881no—
594C2#2145741511yescharacteristic
595C2#2145744411yescharacteristic
596C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3276C2
socleC22×C61D42
prime radical#21458118C2
hypercenter#3276D61
hypocenterC61#20291
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3276

Complements

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

NH ≅ G/Ntype
1. #21458537C2semidirect
2. #21458405C2semidirect
4. #21445413C2semidirect
5. #21445375C2semidirect
6. #21445368C2semidirect
7. #21445332C2semidirect
8. #21445305C2semidirect
9. #21445103C2semidirect
10. #21445101C2semidirect
11. #21445087C2semidirect
12. #21445082C2semidirect
13. #21444710C2semidirect
14. #21444702C2semidirect
15. C61×#3276C2semidirect
296. #3276D61semidirect
365. C61#20291semidirect

Maximal subgroups

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

subgroupindex
#214585372
#214584052
#214581182
#214454132
#214453752
#214453682
#214453322
#214453052
#214451032
#214451012
#214450872
#214450822
#214447102
#214447022
C61×#32762
#2029161