Finite Group Databasebeta

← #29665141

Subgroups — #29665141

Order 15616.

594 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic, complemented
3#21454824C211yesmaximal, characteristic, complemented
4#21445399C211yesmaximal, characteristic, complemented
5#21445392C211yesmaximal, characteristic, complemented
6#21445357C211yesmaximal, characteristic, complemented
7#21445350C211yesmaximal, characteristic, complemented
8#21445275C211yesmaximal, characteristic, complemented
9#21445268C211yesmaximal, characteristic, complemented
10#21445233C211yesmaximal, characteristic, complemented
11#21445226C211yesmaximal, characteristic, complemented
12#21444584C211yesmaximal, characteristic, complemented
13#21444524C211yesmaximal, characteristic, complemented
14#21444501C211yesmaximal, characteristic, complemented
15C61×#3251C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17#16840172—21no—
18#16840121—21no—
19#16840121—21no—
20#16840121—21no—
21#16840112C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840090—21no—
26#16840073—21no—
27#16840057C2211yescharacteristic
28#16840057C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840052C2211yescharacteristic
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#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
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×#11758194—21no—
69C2×#11758194D411yescharacteristic
70C2×#11758194D411yescharacteristic
71C22×#92418—21no—
72#14519777—21no—
73#14519777—21no—
74#14519777—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79#14519710—21no—
80#14519710—21no—
81#14519708—21no—
82#14519708—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#14519701—21no—
94#14519701—21no—
95#14519698—21no—
96#14519698—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—82no—
164SD16×C61—81no—
165D8×C61—81no—
166M16×C61—81no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C42×C61—21no—
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61#4111yescharacteristic
188C42×C61C2411yescharacteristic
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—41no—
194C4⋊Dic61—41no—
195#11758194—41no—
196#11758194—41no—
197#11758194—42no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—42no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C23×C61—21no—
208C23×C61—41no—
209C23×C61#12111yescharacteristic
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—41no—
213Q8×C61—41no—
214D4×C61—21no—
215D4×C61—21no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—81no—
221D4×C61—81no—
222D4×C61—82no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—82no—
235C2×C4×C61#12111yescharacteristic
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—82no—
244Dic122—82no—
245Dic122—81no—
246Dic122—81no—
247C8×C61—41no—
248C8×C61—41no—
249#92418—21no—
250#92418—21no—
251#92418—21no—
252#92418—21no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#19382—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61C2×#12111yescharacteristic
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—81no—
276C4×C61#57511yescharacteristic
277C4×C61#57511yescharacteristic
278C4×C61#57711yescharacteristic
279C4×C61#57811yescharacteristic
280Dic61—81no—
281Dic61—81no—
282#3251D6111yescharacteristic
283#3238—611no—
284#3141—611no—
285#2969—611no—
286#1704—611no—
287#1703—611no—
288#1698—611no—
289#1697—611no—
290#1684—611no—
291#1683—611no—
292#1678—611no—
293#1677—611no—
294#1569—611no—
295#1553—611no—
296#1548—611no—
297C2×C61—41no—
298C2×C61—81no—
299C2×C61—161no—
300C2×C61#304011yescharacteristic
301C2×C61#306311yescharacteristic
302C2×C61#306811yescharacteristic
303#532C2×D6111yescharacteristic
304C2×C4⋊Q8—611no—
305#529C2×D6111yescharacteristic
306#501—1221no—
307#489—1221no—
308#489—1221no—
309#489—1221no—
310#487—611no—
311#487—611no—
312#486—611no—
313#486—611no—
314#483—1221no—
315#480—1221no—
316#478—21no—
317#477—611no—
318#477—611no—
319#476—611no—
320#476—611no—
321#475C2×D6111yescharacteristic
322#474C2×D6111yescharacteristic
323#469—21no—
324#468—21no—
325#463—21no—
326#460C2×D6111yescharacteristic
327#459C2×D6111yescharacteristic
328#446—611no—
329C8×Q8—611no—
330#436—611no—
331#436—1221no—
332#435—1221no—
333C8×D4—611no—
334#433—1221no—
335#432—611no—
336#431—611no—
337#423C2×D6111yescharacteristic
338#417—1221no—
339#417—1221no—
340#405—1221no—
341C4×M16—611no—
342#335—611no—
343#335—611no—
344#334—611no—
345#334—611no—
346#333—611no—
347#333—611no—
348#333—611no—
349#333—611no—
350#332—611no—
351#332—611no—
352#331—611no—
353#331—611no—
354C61#1938211yescharacteristic, complemented, Sylow
355C2×#41—21no—
356C22×Q8—1221no—
357C2×SD16—21no—
358C2×D8—21no—
359C2×M16—21no—
360C2×M16—611no—
361C2×M16—611no—
362C2×M16—1221no—
363C22×C8—1221no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8—1221no—
369C4⋊Q8C22×D6111yescharacteristic
370#127C22×D6111yescharacteristic
371#126—21no—
372#125C22⋊D6111yescharacteristic
373#124—21no—
374#124C22⋊D6111yescharacteristic
375#123—21no—
376C22⋊Q8C22⋊D6111yescharacteristic
377#121—21no—
378#121—21no—
379#121C22⋊D6111yescharacteristic
380C4×Q8C22×D6111yescharacteristic
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4C22×D6111yescharacteristic
385#117—21no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C42C22×D6111yescharacteristic
389#107—21no—
390#107—1221no—
391#107—1221no—
392#106—21no—
393#106—1221no—
394#106—1221no—
395C4⋊C8—21no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8C22×D6111yescharacteristic
403C4⋊C8C22×D6111yescharacteristic
404#103—21no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#102—21no—
412#102—21no—
413#102—21no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424#97—611no—
425#97—611no—
426#97—1221no—
427#97—1221no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—1221no—
431C4×C8—1221no—
432#41—42no—
433#41—42no—
434C2×Q8—21no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—4882no—
438C2×Q8—2441no—
439C2×Q8C2×C22⋊D6111yescharacteristic
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—41no—
445C2×D4C2×C22⋊D6111yescharacteristic
446C22×C4—21no—
447C22×C4—41no—
448C22×C4—2441no—
449C22×C4D4×D6111yescharacteristic
450C22×C4#1175822411yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452SD16—81no—
453D8—81no—
454M16—81no—
455M16—2441no—
456M16—2441no—
457M16—2442no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—2442no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4C2×C22⋊D6111yescharacteristic
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C42—21no—
491C42D4×D6111yescharacteristic
492C42#1175822411yescharacteristic
493C42C2×C22⋊D6111yescharacteristic
494C42C23×D6111yescharacteristic
495C23—21no—
496C23—41no—
497C23#1451974911yescharacteristic
498Q8—21no—
499Q8—21no—
500Q8—41no—
501Q8—41no—
502Q8—4881no—
503Q8—4881no—
504Q8—4882no—
505D4—21no—
506D4—21no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—81no—
512D4—81no—
513D4—82no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—82no—
526C2×C4—4882no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4#1451974911yescharacteristic
530C2×C4#1451974911yescharacteristic
531C2×C4#1451974911yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C2×C4C2×#1175822411yescharacteristic
535C8—41no—
536C8—41no—
537C8—1221no—
538C8—1221no—
539C8—1221no—
540C8—1221no—
541C8—2441no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22C2×#1451974911yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—4881no—
564C4—4881no—
565C4#1684055811yescharacteristic
566C4#1684055811yescharacteristic
567C4#1684057311yescharacteristic
568C4#1684057711yescharacteristic
569C2—41no—
570C2—81no—
571C2—161no—
572C2#2145531111yescharacteristic
573C2#2145551211yescharacteristic
574C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3251C2
socleC22×C61C2×#121
hypercenter#3251D61
hypocenterC61#19382
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. #21457037C2semidirect
2. #21456212C2semidirect
3. #21454824C2semidirect
4. #21445399C2semidirect
5. #21445392C2semidirect
6. #21445357C2semidirect
7. #21445350C2semidirect
8. #21445275C2semidirect
9. #21445268C2semidirect
10. #21445233C2semidirect
11. #21445226C2semidirect
12. #21444584C2semidirect
13. #21444524C2semidirect
14. #21444501C2semidirect
354. C61#19382semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214562122
#214548242
#214453992
#214453922
#214453572
#214453502
#214452752
#214452682
#214452332
#214452262
#214445842
#214445242
#214445012
C61×#32512
#1938261