Finite Group Databasebeta

← #29663478

Subgroups — #29663478

Order 15616.

606 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457156C211yesmaximal, characteristic, complemented
2#21456374C211yesmaximal, characteristic, complemented
3#21454795C211yesmaximal, characteristic, complemented
4#21445327C211yesmaximal, characteristic, complemented
5#21445320C211yesmaximal, characteristic, complemented
6#21445289C211yesmaximal, characteristic, complemented
7#21445282C211yesmaximal, characteristic, complemented
8#21445279C211yesmaximal, characteristic, complemented
9#21445272C211yesmaximal, characteristic, complemented
10#21445261C211yesmaximal, characteristic, complemented
11#21445254C211yesmaximal, characteristic, complemented
12C2×#16839514C211yesmaximal, characteristic, complemented
13#21444517C211yesmaximal, characteristic, complemented
14#21444514C211yesmaximal, characteristic, complemented
15C61×#3247C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840158C2211yescharacteristic
18#16840134C2211yescharacteristic
19#16840114C2211yescharacteristic
20#16840108C2211yescharacteristic
21#16840066—21no—
22#16840047C2211yescharacteristic
23#16840042C2211yescharacteristic
24#16840001—21no—
25#16839989—21no—
26#16839989—21no—
27#16839989—21no—
28#16839981—21no—
29#16839972C2211yescharacteristic
30#16839967C2211yescharacteristic
31#16839893C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33C4×#11758200—21no—
34C4×#11758200—21no—
35#16839840C2211yescharacteristic
36#16839834—21no—
37D4×#92418—21no—
38D4×#92418C2211yescharacteristic
39#16839823—21no—
40#16839819C2211yescharacteristic
41C2×C4⋊#92418C2211yescharacteristic
42#16839718—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839514C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839509C2211yescharacteristic
51#16839509C2211yescharacteristic
52#16839507C2211yescharacteristic
53#16839507C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#478—21no—
59C61×#477C2211yescharacteristic
60C61×#476C2211yescharacteristic
61C61×#466—21no—
62C61×#465—21no—
63C61×#463—21no—
64C61×#460C2211yescharacteristic
65C61×#459C2211yescharacteristic
66C2×C4⋊C8×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758194—21no—
72C22×#92418—21no—
73C22×#92418—21no—
74C22×#92418D411yescharacteristic
75C22×#92418D411yescharacteristic
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519769—21no—
79#14519756—21no—
80#14519756—21no—
81C4×Dic122—21no—
82C4×Dic122—21no—
83#14519736—21no—
84#14519710—21no—
85#14519708—21no—
86#14519708—21no—
87#14519708—21no—
88C4⋊#92418—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519698—21no—
101#14519698—21no—
102#14519698—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106#14519688—21no—
107#14519688—21no—
108C4×#92418—21no—
109C4×#92418D411yescharacteristic
110C4×#92418D411yescharacteristic
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×SD16×C61—21no—
115C2×D8×C61—21no—
116C22×C8×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—82no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61#4111yescharacteristic
158C2×Dic122—21no—
159C2×Dic122—21no—
160SD16×C61—81no—
161D8×C61—81no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—81no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—41no—
179C22⋊C4×C61—41no—
180C42×C61—21no—
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61#4111yescharacteristic
184C42×C61C2411yescharacteristic
185C22⋊Dic61—41no—
186#11758202—81no—
187#11758200—42no—
188#11758200—42no—
189#11758200—81no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194#11758196—41no—
195#11758196—41no—
196C4×Dic61—41no—
197#11758194—42no—
198C2×#92418—22no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—41no—
204C2×#92418—42no—
205C2×#92418—81no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C23×C61—21no—
211C23×C61—41no—
212C23×C61#12111yescharacteristic
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—81no—
216D4×C61—21no—
217D4×C61—21no—
218D4×C61—42no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—81no—
224D4×C61—82no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—82no—
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61#12111yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×Dic61—41no—
244C2×Dic61—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—41no—
248Dic122—41no—
249C8×C61—41no—
250C8×C61—41no—
251#92418—21no—
252#92418—21no—
253#92418—21no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#92418#13512yes—
259#19229—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61#47112yes—
269C22×C61C2×#12111yescharacteristic
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61C2×#13511yescharacteristic
279C4×C61#57511yescharacteristic
280C4×C61#57611yescharacteristic
281C4×C61#57811yescharacteristic
282Dic61—81no—
283Dic61—81no—
284#3248—611no—
285#3247D6111yescharacteristic
286#3167—611no—
287#2966—611no—
288#1692—611no—
289#1691—611no—
290#1686—611no—
291#1685—611no—
292#1684—611no—
293#1683—611no—
294#1682—611no—
295#1681—611no—
296C2×#335—611no—
297#1552—611no—
298#1551—611no—
299C2×C61—22no—
300C2×C61—81no—
301C2×C61—161no—
302C2×C61#303511yescharacteristic
303C2×C61#306111yescharacteristic
304C2×C61C2×#47111yescharacteristic
305#533—611no—
306#532C2×D6111yescharacteristic
307#529C2×D6111yescharacteristic
308#498—611no—
309#492—611no—
310#487—611no—
311#486—611no—
312#479—1221no—
313#478—21no—
314#477C2×D6111yescharacteristic
315#476C2×D6111yescharacteristic
316#475—611no—
317#474—611no—
318#467—1221no—
319#466—21no—
320#465—21no—
321#465—1221no—
322#465—1221no—
323#465—1221no—
324#464—1221no—
325#463—21no—
326#462—611no—
327#461—611no—
328#460C2×D6111yescharacteristic
329#459C2×D6111yescharacteristic
330#446—611no—
331C8×Q8—611no—
332C4×SD16—1221no—
333C4×SD16—1221no—
334#436—611no—
335#435—1221no—
336C8×D4—611no—
337C8×D4—1221no—
338#433—1221no—
339#432—611no—
340C2×C4⋊C8—611no—
341C2×C4⋊C8C2×D6111yescharacteristic
342#405—1221no—
343C2×C4×C8—611no—
344#335—611no—
345#335—611no—
346#335—611no—
347#335—611no—
348#333—611no—
349#333—611no—
350#332—611no—
351#332—611no—
352#332—611no—
353#332—611no—
354#331—611no—
355#331—611no—
356C61#1922911yescharacteristic, complemented, Sylow
357C2×#41—21no—
358C2×Q16—1221no—
359C2×SD16—21no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×SD16—1221no—
363C2×D8—21no—
364C2×M16—1221no—
365C22×C8—21no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—1221no—
369C22×C8—1221no—
370C4⋊Q8—611no—
371C4⋊Q8—611no—
372C4⋊Q8C22×D6111yescharacteristic
373#127C22×D6111yescharacteristic
374#126—21no—
375#125—1221no—
376#125C22⋊D6111yescharacteristic
377#124—21no—
378#124C22⋊D6111yescharacteristic
379#123—21no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8C22⋊D6111yescharacteristic
383#121—21no—
384#121—21no—
385#121C22⋊D6111yescharacteristic
386C4×Q8—1221no—
387C4×Q8—1221no—
388C4×Q8C22×D6111yescharacteristic
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4C22×D6111yescharacteristic
393#117—21no—
394#117—1221no—
395C2×C42C22×D6111yescharacteristic
396#107—21no—
397#107—1221no—
398#106—21no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402C4⋊C8—21no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—1221no—
408C4⋊C8—1221no—
409C4⋊C8—1221no—
410C4⋊C8C22×D6111yescharacteristic
411C4⋊C8C22×D6111yescharacteristic
412#103—21no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#102—21no—
419#102—21no—
420#102—21no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427#97—1221no—
428#97—1221no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—611no—
433C4×C8—1221no—
434#41—82no—
435C2×Q8—21no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8C2×C22⋊D6111yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—41no—
444C2×D4C2×C22⋊D6111yescharacteristic
445C22×C4—21no—
446C22×C4—41no—
447C22×C4D4×D6111yescharacteristic
448C22×C4#1175822411yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450Q16—4881no—
451SD16—81no—
452SD16—4881no—
453SD16—2442no—
454SD16—2442no—
455D8—81no—
456M16—2442no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—81no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—1222no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—4881no—
470C2×C8—2441no—
471C2×C8—2442no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C4⋊C4C2×C22⋊D6111yescharacteristic
485C4⋊C4C2×C22⋊D6111yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—2441no—
494C42—21no—
495C42—2441no—
496C42D4×D6111yescharacteristic
497C42#1175822411yescharacteristic
498C42C2×C22⋊D6111yescharacteristic
499C42C23×D6111yescharacteristic
500C23—21no—
501C23—41no—
502C23#1451974911yescharacteristic
503Q8—21no—
504Q8—21no—
505Q8—81no—
506Q8—2441no—
507Q8—2441no—
508Q8—2441no—
509Q8—2441no—
510D4—21no—
511D4—21no—
512D4—42no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—81no—
518D4—82no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—82no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4#1451974911yescharacteristic
534C2×C4#1451974911yescharacteristic
535C2×C4#1451974911yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C22×C22⋊D6111yescharacteristic
538C2×C4C2×#1175822411yescharacteristic
539C8—41no—
540C8—41no—
541C8—612no—
542C8—1221no—
543C8—1221no—
544C8—1221no—
545C8—2441no—
546C8—2441no—
547C8—2441no—
548C8—2441no—
549C22—21no—
550C22—21no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#1684002312yes—
558C22C2×#1451974911yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—4881no—
568C4—4881no—
569C4#135×D6111yescharacteristic
570C4#1684055911yescharacteristic
571C4#1684056511yescharacteristic
572C4#1684057711yescharacteristic
573C2—22no—
574C2—81no—
575C2—161no—
576C2#2145527011yescharacteristic
577C2#2145548611yescharacteristic
578C2C2×#1684002311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3247C2
socleC22×C61C2×#121
hypercenter#3247D61
hypocenterC61#19229
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3247

Complements

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

NH ≅ G/Ntype
1. #21457156C2semidirect
2. #21456374C2semidirect
3. #21454795C2semidirect
4. #21445327C2semidirect
5. #21445320C2semidirect
6. #21445289C2semidirect
7. #21445282C2semidirect
8. #21445279C2semidirect
9. #21445272C2semidirect
10. #21445261C2semidirect
11. #21445254C2semidirect
12. C2×#16839514C2semidirect
13. #21444517C2semidirect
14. #21444514C2semidirect
356. C61#19229semidirect

Maximal subgroups

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

subgroupindex
#214571562
#214563742
#214547952
#214453272
#214453202
#214452892
#214452822
#214452792
#214452722
#214452612
#214452542
C2×#168395142
#214445172
#214445142
C61×#32472
#1922961