Finite Group Databasebeta

← #29662476

Subgroups — #29662476

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456379C211yesmaximal, characteristic, complemented
2#21455879C211yesmaximal, characteristic
3#41×#92418C211yesmaximal, characteristic, complemented
4#21445275C211yesmaximal, characteristic, complemented
5#21445268C211yesmaximal, characteristic, complemented
6#21445265C211yesmaximal, characteristic, complemented
7#21445258C211yesmaximal, characteristic, complemented
8#21445199C211yesmaximal, characteristic, complemented
9#21445192C211yesmaximal, characteristic, complemented
10#21445169C211yesmaximal, characteristic, complemented
11#21445162C211yesmaximal, characteristic, complemented
12#21444565C211yesmaximal, characteristic, complemented
13#21444484C211yesmaximal, characteristic, complemented
14#21444479C211yesmaximal, characteristic, complemented
15C61×#3107C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840158C2211yescharacteristic
18#16840136C2211yescharacteristic
19#16840112C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840086—21no—
22#16840057C2211yescharacteristic
23#16840052C2211yescharacteristic
24#16839993—21no—
25#16839993—21no—
26#16839962C2211yescharacteristic
27#16839957C2211yescharacteristic
28#16839907—21no—
29Q8×#92418C2211yescharacteristic
30Q8×#92418C2211yescharacteristic
31C4×#11758201—21no—
32C4×#11758201—21no—
33C4×#11758201—21no—
34C4×#11758199—21no—
35D4×#92418—21no—
36D4×#92418—21no—
37D4×#92418C2211yescharacteristic
38D4×#92418C2211yescharacteristic
39#16839815—21no—
40#16839815C2211yescharacteristic
41#16839815C2211yescharacteristic
42C2×C4×#92418—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#485—21no—
59C61×#475C2211yescharacteristic
60C61×#474C2211yescharacteristic
61C61×#462C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#452—21no—
64C4×Q16×C61—21no—
65C4×SD16×C61—21no—
66C61×#431C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758201—21no—
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72C22×#92418—21no—
73C22×#92418—21no—
74C22×#92418D411yescharacteristic
75C22×#92418D411yescharacteristic
76#14519777C2311yescharacteristic
77#14519774C2311yescharacteristic
78#14519765—21no—
79#14519751—21no—
80#14519751—21no—
81C4×D244—21no—
82C4×D244—21no—
83#14519710—21no—
84#14519708—21no—
85#14519708—21no—
86#14519708—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93#14519701—21no—
94#14519701—21no—
95#14519701—21no—
96#14519698—21no—
97#14519697—21no—
98#14519697—21no—
99#14519697—21no—
100#14519697—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C4×#92418—21no—
106C4×#92418—21no—
107C4×#92418—21no—
108C4×#92418—21no—
109C4×#92418D411yescharacteristic
110C4×#92418D411yescharacteristic
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61D411yescharacteristic
114C2×Q16×C61—21no—
115C2×SD16×C61—21no—
116C4⋊Q8×C61C2311yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118#125×C61—21no—
119C22⋊Q8×C61—21no—
120C22⋊Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61D411yescharacteristic
124C4×Q8×C61C2311yescharacteristic
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61D411yescharacteristic
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61—21no—
131#117×C61D411yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#107×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61—21no—
144#41×C61—22no—
145#41×C61—22no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61#4111yescharacteristic
157C2×C22⋊D61—41no—
158#11758222—82no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164Q16×C61—42no—
165SD16×C61—42no—
166C2×C8×C61—21no—
167C2×C8×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—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61#4111yescharacteristic
189C42×C61C2411yescharacteristic
190#11758201—42no—
191#11758201—42no—
192#11758201—42no—
193#11758199—42no—
194#11758198—41no—
195#11758198—41no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C2×#92418—22no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—41no—
204C2×#92418—42no—
205C2×#92418—42no—
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—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61#13512yes—
219D4×C61—24no—
220D4×C61—21no—
221D4×C61#13512yes—
222C2×C4×C61—24no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C22⋊D61—82no—
243C2×Dic61—41no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—41no—
248D244—81no—
249C4×D61—82no—
250Dic122—81no—
251C8×C61—41no—
252C8×C61—41no—
253#92418—21no—
254#92418—21no—
255#92418—21no—
256#92418—41no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#92418#13512yes—
261#19127—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—22no—
265C22×C61—21no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61C2×#12111yescharacteristic
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—22no—
275C4×C61—21no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61#47112yes—
281C4×C61C2×#13511yescharacteristic
282C4×C61C2×#13511yescharacteristic
283C4×C61C2×#13511yescharacteristic
284C4×C61#57511yescharacteristic
285Dic61—81no—
286#3167—611no—
287#3107D6111yescharacteristic, complemented
288#3106—611no—
289C8×#41—611no—
290#1684—611no—
291#1683—611no—
292#1682—611no—
293#1681—611no—
294#1672—611no—
295#1671—611no—
296#1666—611no—
297#1665—611no—
298#1564—611no—
299#1544—611no—
300#1543—611no—
301C2×C61—41no—
302C2×C61—42no—
303C2×C61#303511yescharacteristic
304C2×C61#305011yescharacteristic
305C2×C61C2×#47111yescharacteristic
306D61—161no—
307#533C2×D6111yescharacteristic
308#532—611no—
309C4×#41C2×D6111yescharacteristic
310#498—611no—
311#492—611no—
312#487—611no—
313#486—611no—
314#485—21no—
315#482—1221no—
316#477—611no—
317#476—611no—
318#475C2×D6111yescharacteristic
319#474C2×D6111yescharacteristic
320#465—1221no—
321#465—1221no—
322#462C2×D6111yescharacteristic
323#461C2×D6111yescharacteristic
324#460—611no—
325#459—611no—
326#452—21no—
327#449—1221no—
328C8×Q8—611no—
329C8×Q8—611no—
330C4×Q16—21no—
331C4×SD16—21no—
332C4×SD16—1221no—
333C4×SD16—1221no—
334C4×SD16—1221no—
335C4×D8—1221no—
336C8×D4—611no—
337C8×D4—611no—
338C8×D4—1221no—
339C8×D4—1221no—
340#431—611no—
341#431—611no—
342#431—1221no—
343#431C2×D6111yescharacteristic
344C2×C4×C8—611no—
345C2×C4×C8—1221no—
346#335—611no—
347#335—611no—
348#334—611no—
349#334—611no—
350#333—611no—
351#333—611no—
352#332—611no—
353#332—611no—
354#332—611no—
355#332—611no—
356#331—611no—
357#331—611no—
358C61#1912711yescharacteristic, complemented, Sylow
359C2×#41—1221no—
360C2×#41C22⋊D6111yescharacteristic
361C2×Q16—21no—
362C2×SD16—21no—
363C2×SD16—1221no—
364C2×SD16—1221no—
365C2×SD16—1221no—
366C2×D8—1221no—
367C22×C8—611no—
368C22×C8—611no—
369C22×C8—1221no—
370C22×C8—1221no—
371C4⋊Q8—611no—
372C4⋊Q8C22×D6111yescharacteristic
373C4⋊Q8C22×D6111yescharacteristic
374#127—611no—
375#125—21no—
376#124—1221no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379#121—1221no—
380#121—1221no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8C22⋊D6111yescharacteristic
384C4×Q8C22×D6111yescharacteristic
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—1221no—
388C4×D4—1221no—
389C4×D4C22⋊D6111yescharacteristic
390C4×D4C22×D6111yescharacteristic
391#117—21no—
392#117—21no—
393#117C22⋊D6111yescharacteristic
394C2×C42—21no—
395C2×C42C22×D6111yescharacteristic
396#107—21no—
397#107—1221no—
398#106—21no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—1221no—
407C4⋊C8—1221no—
408C4⋊C8C22×D6111yescharacteristic
409C4⋊C8C22×D6111yescharacteristic
410#103—21no—
411#103—21no—
412#103—21no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#102—21no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—21no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427C4×C8—21no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—1221no—
433C4×C8—1221no—
434C4×C8—1221no—
435C4×C8—1221no—
436#41—22no—
437#41—22no—
438#41—4882no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—1221no—
442C2×Q8C2×C22⋊D6111yescharacteristic
443C2×D4—21no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—2441no—
448C2×D4C2×C22⋊D6111yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—2441no—
453C22×C4D4×D6111yescharacteristic
454C22×C4#1175822411yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456Q16—42no—
457SD16—42no—
458SD16—2442no—
459SD16—2442no—
460SD16—2442no—
461D8—2442no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—1222no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—2442no—
474C2×C8—2441no—
475C2×C8—2442no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4C2×C22⋊D6111yescharacteristic
487C4⋊C4C2×C22⋊D6111yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—41no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—41no—
498C42D4×D6111yescharacteristic
499C42#1175822411yescharacteristic
500C42C2×C22⋊D6111yescharacteristic
501C42C23×D6111yescharacteristic
502C23—21no—
503C23—2441no—
504C23#1451974911yescharacteristic
505Q8—21no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—4881no—
511Q8#1451979712yes—
512D4—24no—
513D4—21no—
514D4—4881no—
515D4—4882no—
516D4—2441no—
517D4—2441no—
518D4—2441no—
519D4—2441no—
520D4#1451979712yes—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—4882no—
536C2×C4—2441no—
537C2×C4#1451974911yescharacteristic
538C2×C4#1451974911yescharacteristic
539C2×C4#1451974911yescharacteristic
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4C22×C22⋊D6111yescharacteristic
542C2×C4C2×#1175822411yescharacteristic
543C8—41no—
544C8—41no—
545C8—612no—
546C8—1221no—
547C8—1221no—
548C8—1221no—
549C8—2441no—
550C8—2441no—
551C8—2441no—
552C8—2441no—
553C22—21no—
554C22—22no—
555C22—21no—
556C22—21no—
557C22—41no—
558C22—41no—
559C22—4881no—
560C22—4881no—
561C22—4881no—
562C22C2×#1451974911yescharacteristic
563C4—21no—
564C4—22no—
565C4—21no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—4881no—
572C4#1684002712yes—
573C4#135×D6111yescharacteristic
574C4C2×#1451979711yescharacteristic
575C4C2×#1451979711yescharacteristic
576C4#1684055911yescharacteristic
577C2—41no—
578C2—42no—
579C2—9761no—
580C2#2145527011yescharacteristic
581C2#2145539711yescharacteristic
582C2C2×#1684002711yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3107C2
socleC22×C61C2×#121
prime radical#21455879C2
hypercenter#3107D61
hypocenterC61#19127
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
1. #21456379C2semidirect
3. #41×#92418C2semidirect
4. #21445275C2semidirect
5. #21445268C2semidirect
6. #21445265C2semidirect
7. #21445258C2semidirect
8. #21445199C2semidirect
9. #21445192C2semidirect
10. #21445169C2semidirect
11. #21445162C2semidirect
12. #21444565C2semidirect
13. #21444484C2semidirect
14. #21444479C2semidirect
15. C61×#3107C2semidirect
287. #3107D61semidirect
358. C61#19127semidirect

Maximal subgroups

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

subgroupindex
#214563792
#214558792
#41×#924182
#214452752
#214452682
#214452652
#214452582
#214451992
#214451922
#214451692
#214451622
#214445652
#214444842
#214444792
C61×#31072
#1912761