Finite Group Databasebeta

← #29665156

Subgroups — #29665156

Order 15616.

606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic, complemented
3#21454824C211yesmaximal, characteristic, complemented
4#21445385C211yesmaximal, characteristic, complemented
5#21445378C211yesmaximal, characteristic, complemented
6#21445371C211yesmaximal, characteristic, complemented
7#21445364C211yesmaximal, characteristic, complemented
8#21445261C211yesmaximal, characteristic, complemented
9#21445254C211yesmaximal, characteristic, complemented
10#21445247C211yesmaximal, characteristic, complemented
11#21445240C211yesmaximal, characteristic, complemented
12#21444584C211yesmaximal, characteristic, complemented
13#21444524C211yesmaximal, characteristic, complemented
14#21444492C211yesmaximal, characteristic, complemented
15C61×#3251C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17#16840157—21no—
18#16840132—21no—
19#16840121—21no—
20#16840121—21no—
21#16840112C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16839972C2211yescharacteristic
26#16839972C2211yescharacteristic
27#16839967C2211yescharacteristic
28#16839967C2211yescharacteristic
29#16839917—21no—
30#16839900—21no—
31#16839893C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33#16839840—21no—
34#16839840C2211yescharacteristic
35#16839834—21no—
36D4×#92418C2211yescharacteristic
37#16839823—21no—
38#16839819C2211yescharacteristic
39#16839815C2211yescharacteristic
40#16839763—21no—
41#16839763—21no—
42#16839718—21no—
43C4×#11758194C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839509C2211yescharacteristic
51#16839509C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#478—21no—
59C61×#475C2211yescharacteristic
60C61×#474C2211yescharacteristic
61C61×#469—21no—
62C61×#468—21no—
63C61×#463—21no—
64C61×#460C2211yescharacteristic
65C61×#459C2211yescharacteristic
66C61×#423C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758194—21no—
73C2×#11758194D411yescharacteristic
74C2×#11758194D411yescharacteristic
75C22×#92418—21no—
76#14519777—21no—
77#14519777—21no—
78#14519777—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C4⋊#92418—21no—
84C4⋊#92418—21no—
85C4⋊#92418C2311yescharacteristic
86C4⋊#92418C2311yescharacteristic
87C4⋊#92418C2311yescharacteristic
88C4⋊#92418C2311yescharacteristic
89#14519701—21no—
90#14519701—21no—
91#14519701—21no—
92#14519701—21no—
93#14519700—21no—
94#14519700—21no—
95#14519700—21no—
96#14519700—21no—
97#14519698—21no—
98#14519698—21no—
99#14519698—21no—
100#14519698—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519688—21no—
106#14519688—21no—
107#14519688D411yescharacteristic
108#14519688D411yescharacteristic
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×SD16×C61—21no—
115C2×D8×C61—21no—
116C2×M16×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118#127×C61C2311yescharacteristic
119#126×C61—21no—
120#125×C61D411yescharacteristic
121#124×C61—21no—
122#124×C61D411yescharacteristic
123#123×C61—21no—
124C22⋊Q8×C61D411yescharacteristic
125#121×C61—21no—
126#121×C61—21no—
127#121×C61D411yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#107×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143#102×C61—21no—
144#41×C61—42no—
145#41×C61—42no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61#4111yescharacteristic
159C22×Dic61—41no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—41no—
163C2×Dic122—81no—
164C2×Dic122—81no—
165SD16×C61—81no—
166D8×C61—81no—
167M16×C61—81no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61#4111yescharacteristic
189C42×C61C2411yescharacteristic
190#11758202—81no—
191#11758202—81no—
192#11758200—81no—
193#11758200—81no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61—41no—
200#11758194—41no—
201#11758194—41no—
202#11758194—42no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—42no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C23×C61—21no—
213C23×C61—41no—
214C23×C61#12111yescharacteristic
215Q8×C61—21no—
216Q8×C61—21no—
217Q8×C61—41no—
218Q8×C61—41no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—81no—
226D4×C61—81no—
227D4×C61—82no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
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—82no—
240C2×C4×C61#12111yescharacteristic
241C2×C4×C61#12111yescharacteristic
242C2×C4×C61#12111yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—82no—
249Dic122—41no—
250Dic122—41no—
251Dic122—41no—
252Dic122—41no—
253Dic122—82no—
254C8×C61—41no—
255C8×C61—41no—
256#92418—21no—
257#92418—21no—
258#92418—21no—
259#92418—21no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#19383—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61C2×#12111yescharacteristic
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61#57511yescharacteristic
284C4×C61#57511yescharacteristic
285C4×C61#57611yescharacteristic
286C4×C61#57711yescharacteristic
287Dic61—81no—
288Dic61—81no—
289#3251D6111yescharacteristic
290#3218—611no—
291#3141—611no—
292#2969—611no—
293#1702—611no—
294#1701—611no—
295#1700—611no—
296#1699—611no—
297#1682—611no—
298#1681—611no—
299#1680—611no—
300#1679—611no—
301#1569—611no—
302#1553—611no—
303#1546—611no—
304C2×C61—41no—
305C2×C61—81no—
306C2×C61—161no—
307C2×C61#304011yescharacteristic
308C2×C61#306211yescharacteristic
309C2×C61#306911yescharacteristic
310#532C2×D6111yescharacteristic
311C2×C4⋊Q8—611no—
312#529C2×D6111yescharacteristic
313#497—1221no—
314#491—1221no—
315#489—1221no—
316#489—1221no—
317#487—611no—
318#487—611no—
319#486—611no—
320#486—611no—
321#478—21no—
322#475C2×D6111yescharacteristic
323#474C2×D6111yescharacteristic
324#469—21no—
325#468—21no—
326#463—21no—
327#462—611no—
328#462—611no—
329#461—611no—
330#461—611no—
331#460C2×D6111yescharacteristic
332#459C2×D6111yescharacteristic
333#451—1221no—
334#448—1221no—
335#446—611no—
336C8×Q8—611no—
337#436—611no—
338#436—1221no—
339#435—1221no—
340C8×D4—611no—
341#433—1221no—
342#432—611no—
343#431—611no—
344#423C2×D6111yescharacteristic
345#417—1221no—
346#417—1221no—
347#405—1221no—
348C4×M16—611no—
349#335—611no—
350#335—611no—
351#334—611no—
352#334—611no—
353#332—611no—
354#332—611no—
355#332—611no—
356#332—611no—
357#332—611no—
358#332—611no—
359#331—611no—
360#331—611no—
361C61#1938311yescharacteristic, complemented, Sylow
362C2×#41—21no—
363C22×Q8—1221no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×SD16—21no—
367C2×SD16—1221no—
368C2×SD16—1221no—
369C2×D8—21no—
370C2×M16—21no—
371C2×M16—611no—
372C2×M16—611no—
373C2×M16—1221no—
374C22×C8—1221no—
375C4⋊Q8—611no—
376C4⋊Q8—611no—
377C4⋊Q8—1221no—
378C4⋊Q8—1221no—
379C4⋊Q8—1221no—
380C4⋊Q8C22×D6111yescharacteristic
381#127C22×D6111yescharacteristic
382#126—21no—
383#125C22⋊D6111yescharacteristic
384#124—21no—
385#124C22⋊D6111yescharacteristic
386#123—21no—
387C22⋊Q8C22⋊D6111yescharacteristic
388#121—21no—
389#121—21no—
390#121C22⋊D6111yescharacteristic
391C4×Q8C22×D6111yescharacteristic
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4C22×D6111yescharacteristic
396#117—21no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C42C22×D6111yescharacteristic
400#107—21no—
401#106—21no—
402C4⋊C8—21no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—1221no—
408C4⋊C8—1221no—
409C4⋊C8C22×D6111yescharacteristic
410C4⋊C8C22×D6111yescharacteristic
411#103—21no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#102—21no—
421#102—21no—
422#102—21no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427C22⋊C8—1221no—
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431#97—611no—
432#97—611no—
433#97—1221no—
434#97—1221no—
435C4×C8—611no—
436C4×C8—611no—
437C4×C8—1221no—
438C4×C8—1221no—
439#41—42no—
440#41—42no—
441C2×Q8—21no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—4881no—
445C2×Q8—4881no—
446C2×Q8—2441no—
447C2×Q8C2×C22⋊D6111yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4C2×C22⋊D6111yescharacteristic
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—2441no—
457C22×C4D4×D6111yescharacteristic
458C22×C4#1175822411yescharacteristic
459C22×C4C2×C22⋊D6111yescharacteristic
460Q16—4881no—
461Q16—4881no—
462SD16—81no—
463SD16—4881no—
464SD16—4881no—
465D8—81no—
466M16—81no—
467M16—2441no—
468M16—2441no—
469M16—2442no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—611no—
473C2×C8—611no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—2442no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—2441no—
492C4⋊C4—2441no—
493C4⋊C4C2×C22⋊D6111yescharacteristic
494C4⋊C4C2×C22⋊D6111yescharacteristic
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C42—21no—
503C42D4×D6111yescharacteristic
504C42#1175822411yescharacteristic
505C42C2×C22⋊D6111yescharacteristic
506C42C23×D6111yescharacteristic
507C23—21no—
508C23—41no—
509C23#1451974911yescharacteristic
510Q8—21no—
511Q8—21no—
512Q8—41no—
513Q8—41no—
514Q8—4882no—
515Q8—2441no—
516Q8—2441no—
517Q8—2441no—
518Q8—2441no—
519D4—21no—
520D4—21no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—41no—
525D4—81no—
526D4—81no—
527D4—82no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—4882no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4#1451974911yescharacteristic
544C2×C4#1451974911yescharacteristic
545C2×C4#1451974911yescharacteristic
546C2×C4C2×D4×D6111yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4C2×#1175822411yescharacteristic
549C8—41no—
550C8—41no—
551C8—1221no—
552C8—1221no—
553C8—1221no—
554C8—1221no—
555C8—2441no—
556C8—2441no—
557C8—2441no—
558C8—2441no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22C2×#1451974911yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—81no—
577C4—4881no—
578C4—4881no—
579C4#1684055811yescharacteristic
580C4#1684055811yescharacteristic
581C4#1684056511yescharacteristic
582C4#1684057011yescharacteristic
583C2—41no—
584C2—81no—
585C2—161no—
586C2#2145531111yescharacteristic
587C2#2145550111yescharacteristic
588C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3251C2
socleC22×C61C2×#121
hypercenter#3251D61
hypocenterC61#19383
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3251

Complements

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

NH ≅ G/Ntype
1. #21456843C2semidirect
2. #21456212C2semidirect
3. #21454824C2semidirect
4. #21445385C2semidirect
5. #21445378C2semidirect
6. #21445371C2semidirect
7. #21445364C2semidirect
8. #21445261C2semidirect
9. #21445254C2semidirect
10. #21445247C2semidirect
11. #21445240C2semidirect
12. #21444584C2semidirect
13. #21444524C2semidirect
14. #21444492C2semidirect
361. C61#19383semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214562122
#214548242
#214453852
#214453782
#214453712
#214453642
#214452612
#214452542
#214452472
#214452402
#214445842
#214445242
#214444922
C61×#32512
#1938361