Finite Group Databasebeta

← #29663486

Subgroups — #29663486

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#21457145C211yesmaximal, characteristic
2#21456379C211yesmaximal, characteristic, complemented
3#21454795C211yesmaximal, characteristic, complemented
4#21445323C211yesmaximal, characteristic, complemented
5#21445316C211yesmaximal, characteristic, complemented
6#21445293C211yesmaximal, characteristic, complemented
7#21445286C211yesmaximal, characteristic, complemented
8#21445275C211yesmaximal, characteristic, complemented
9#21445268C211yesmaximal, characteristic, complemented
10#21445265C211yesmaximal, characteristic, complemented
11#21445258C211yesmaximal, characteristic, complemented
12C2×#16839514C211yesmaximal, characteristic, complemented
13#21444519C211yesmaximal, characteristic, complemented
14#21444512C211yesmaximal, characteristic, complemented
15C61×#3248C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840158C2211yescharacteristic
18#16840136C2211yescharacteristic
19#16840112C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840059—21no—
22#16840057C2211yescharacteristic
23#16840052C2211yescharacteristic
24#16839996—21no—
25#16839993—21no—
26#16839993—21no—
27#16839993—21no—
28#16839974—21no—
29#16839962C2211yescharacteristic
30#16839957C2211yescharacteristic
31#16839893C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33C4×#11758201—21no—
34C4×#11758201—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#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#479—21no—
59C61×#475C2211yescharacteristic
60C61×#474C2211yescharacteristic
61C61×#467—21no—
62C61×#465—21no—
63C61×#464—21no—
64C61×#462C2211yescharacteristic
65C61×#461C2211yescharacteristic
66C2×C4⋊C8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758201—21no—
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72C2×#11758194—21no—
73C22×#92418—21no—
74C22×#92418—21no—
75C22×#92418D411yescharacteristic
76C22×#92418D411yescharacteristic
77#14519777C2311yescharacteristic
78#14519774C2311yescharacteristic
79#14519765—21no—
80#14519751—21no—
81#14519751—21no—
82C4×D244—21no—
83C4×D244—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#14519701—21no—
97#14519701—21no—
98#14519698—21no—
99#14519697—21no—
100#14519697—21no—
101#14519697—21no—
102#14519697—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×Q16×C61—21no—
114C2×SD16×C61—21no—
115C22×C8×C61—21no—
116C4⋊Q8×C61C2311yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118#126×C61—21no—
119#125×C61—21no—
120#125×C61D411yescharacteristic
121#124×C61D411yescharacteristic
122#123×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61D411yescharacteristic
126#121×C61D411yescharacteristic
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
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#103×C61—21no—
142#103×C61—21no—
143#102×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61#4111yescharacteristic
153C2×C22⋊D61—41no—
154#11758222—82no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×C4×D61—41no—
159C2×Dic122—21no—
160Q16×C61—81no—
161SD16×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×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—41no—
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—
183C42×C61—21no—
184C42×C61—41no—
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61#4111yescharacteristic
188C42×C61C2411yescharacteristic
189#11758201—42no—
190#11758201—42no—
191#11758201—81no—
192#11758199—81no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
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#12111yescharacteristic
212C22×D61—41no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—42no—
222C2×C4×C61—22no—
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#12111yescharacteristic
235C2×C4×C61#12111yescharacteristic
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C22⋊D61—82no—
241C2×Dic61—41no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—81no—
247C4×D61—82no—
248Dic122—81no—
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#47112yes—
266C22×C61C2×#12111yescharacteristic
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
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×C61—81no—
279C4×C61C2×#13511yescharacteristic
280C4×C61#57511yescharacteristic
281C4×C61#57611yescharacteristic
282C4×C61#57811yescharacteristic
283Dic61—81no—
284#3248D6111yescharacteristic, complemented
285#3247—611no—
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#303511yescharacteristic
302C2×C61#306111yescharacteristic
303C2×C61C2×#47111yescharacteristic
304D61—161no—
305#533C2×D6111yescharacteristic
306#532—611no—
307#529C2×D6111yescharacteristic
308#498—611no—
309#492—611no—
310#487—611no—
311#486—611no—
312#479—21no—
313#478—1221no—
314#477—611no—
315#476—611no—
316#475C2×D6111yescharacteristic
317#474C2×D6111yescharacteristic
318#467—21no—
319#466—1221no—
320#465—21no—
321#465—1221no—
322#465—1221no—
323#465—1221no—
324#464—21no—
325#463—1221no—
326#462C2×D6111yescharacteristic
327#461C2×D6111yescharacteristic
328#460—611no—
329#459—611no—
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—1221no—
358C2×Q16—21no—
359C2×SD16—21no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×SD16—1221no—
363C2×D8—1221no—
364C2×M16—1221no—
365C22×C8—21no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—1221no—
369C22×C8—1221no—
370C4⋊Q8—611no—
371C4⋊Q8C22×D6111yescharacteristic
372C4⋊Q8C22×D6111yescharacteristic
373#127—611no—
374#126—21no—
375#125—21no—
376#125C22⋊D6111yescharacteristic
377#124—1221no—
378#124C22⋊D6111yescharacteristic
379#123—21no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8C22⋊D6111yescharacteristic
383#121—1221no—
384#121—1221no—
385#121C22⋊D6111yescharacteristic
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8C22×D6111yescharacteristic
389C4×D4—21no—
390C4×D4—1221no—
391C4×D4—1221no—
392C4×D4C22×D6111yescharacteristic
393#117—21no—
394#117—21no—
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—21no—
414#103—21no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#102—21no—
419#102—1221no—
420#102—1221no—
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—4882no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—1221no—
438C2×Q8C2×C22⋊D6111yescharacteristic
439C2×D4—21no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—2441no—
444C2×D4C2×C22⋊D6111yescharacteristic
445C22×C4—21no—
446C22×C4—2441no—
447C22×C4D4×D6111yescharacteristic
448C22×C4#1175822411yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450Q16—81no—
451SD16—81no—
452SD16—4881no—
453SD16—2442no—
454SD16—2442no—
455D8—4881no—
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—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
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—2441no—
493C22⋊C4—2441no—
494C42—21no—
495C42—41no—
496C42D4×D6111yescharacteristic
497C42#1175822411yescharacteristic
498C42C2×C22⋊D6111yescharacteristic
499C42C23×D6111yescharacteristic
500C23—21no—
501C23—2441no—
502C23#1451974911yescharacteristic
503Q8—21no—
504Q8—21no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—4881no—
510D4—21no—
511D4—21no—
512D4—42no—
513D4—4881no—
514D4—4882no—
515D4—2441no—
516D4—2441no—
517D4—2441no—
518D4—2441no—
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—41no—
531C2×C4—4882no—
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—4881no—
555C22—4881no—
556C22—4881no—
557C22#1684002712yes—
558C22C2×#1451974911yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—4881no—
569C4#135×D6111yescharacteristic
570C4#1684055911yescharacteristic
571C4#1684056511yescharacteristic
572C4#1684057711yescharacteristic
573C2—22no—
574C2—81no—
575C2—9761no—
576C2#2145527011yescharacteristic
577C2#2145548611yescharacteristic
578C2C2×#1684002711yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3248C2
socleC22×C61C2×#121
prime radical#21457145C2
hypercenter#3248D61
hypocenterC61#19229
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3248

Complements

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

NH ≅ G/Ntype
2. #21456379C2semidirect
3. #21454795C2semidirect
4. #21445323C2semidirect
5. #21445316C2semidirect
6. #21445293C2semidirect
7. #21445286C2semidirect
8. #21445275C2semidirect
9. #21445268C2semidirect
10. #21445265C2semidirect
11. #21445258C2semidirect
12. C2×#16839514C2semidirect
13. #21444519C2semidirect
14. #21444512C2semidirect
15. C61×#3248C2semidirect
284. #3248D61semidirect
356. C61#19229semidirect

Maximal subgroups

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

subgroupindex
#214571452
#214563792
#214547952
#214453232
#214453162
#214452932
#214452862
#214452752
#214452682
#214452652
#214452582
C2×#168395142
#214445192
#214445122
C61×#32482
#1922961