Finite Group Databasebeta

← #29664009

Subgroups — #29664009

Order 15616.

630 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456144C211yesmaximal, characteristic, complemented
2#21455970C211yesmaximal, characteristic
3#41×#92418C211yesmaximal, characteristic, complemented
4#21445279C211yesmaximal, characteristic, complemented
5#21445272C211yesmaximal, characteristic, complemented
6#21445265C211yesmaximal, characteristic, complemented
7#21445258C211yesmaximal, characteristic, complemented
8#21445251C211yesmaximal, characteristic, complemented
9#21445244C211yesmaximal, characteristic, complemented
10#21445237C211yesmaximal, characteristic, complemented
11#21445230C211yesmaximal, characteristic, complemented
12#21444580C211yesmaximal, characteristic, complemented
13#21444529C211yesmaximal, characteristic, complemented
14#21444525C211yesmaximal, characteristic, complemented
15C61×#3116C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840119—21no—
18#16840119—21no—
19#16840114C2211yescharacteristic
20#16840114C2211yescharacteristic
21#16840110C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840086—21no—
24#16840047C2211yescharacteristic
25#16840042C2211yescharacteristic
26#16839962C2211yescharacteristic
27#16839957C2211yescharacteristic
28#16839907—21no—
29Q8×#92418C2211yescharacteristic
30Q8×#92418C2211yescharacteristic
31#16839872—21no—
32#16839860—21no—
33D4×#92418—21no—
34D4×#92418—21no—
35D4×#92418C2211yescharacteristic
36D4×#92418C2211yescharacteristic
37#16839819C2211yescharacteristic
38#16839815—21no—
39#16839815C2211yescharacteristic
40#16839756—21no—
41#16839756—21no—
42C2×C4×#92418—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839507C2211yescharacteristic
53#16839507C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#485—21no—
59C61×#477C2211yescharacteristic
60C61×#476C2211yescharacteristic
61C61×#462C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#452—21no—
64C61×#441—21no—
65C61×#440—21no—
66C61×#432C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758201—21no—
69C2×#11758199—21no—
70C22×#92418—21no—
71C22×#92418—21no—
72C22×#92418D411yescharacteristic
73C22×#92418D411yescharacteristic
74#14519777C2311yescharacteristic
75#14519774C2311yescharacteristic
76#14519765—21no—
77#14519751—21no—
78#14519751—21no—
79C4×D244—21no—
80C4×D244—21no—
81#14519710—21no—
82#14519708—21no—
83C4⋊#92418—21no—
84C4⋊#92418—21no—
85C4⋊#92418C2311yescharacteristic
86C4⋊#92418C2311yescharacteristic
87C4⋊#92418C2311yescharacteristic
88C4⋊#92418C2311yescharacteristic
89#14519701—21no—
90#14519700—21no—
91#14519700—21no—
92#14519700—21no—
93#14519700—21no—
94#14519698—21no—
95#14519697—21no—
96#14519697—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—
105#14519688—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—
143#97×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—42no—
159#11758222—42no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×C4×D61—41no—
164C2×Dic122—21no—
165Q16×C61—42no—
166SD16×C61—42no—
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×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—41no—
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61#4111yescharacteristic
190C42×C61C2411yescharacteristic
191#11758201—42no—
192#11758199—42no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—41no—
202C2×#92418—41no—
203C2×#92418—42no—
204C2×#92418—42no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—21no—
210C23×C61#12111yescharacteristic
211C22×D61—41no—
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—81no—
216Q8×C61#13512yes—
217D4×C61—24no—
218D4×C61—21no—
219D4×C61#13512yes—
220C2×C4×C61—24no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
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—41no—
249Dic122—41no—
250C8×C61—41no—
251C8×C61—41no—
252#92418—21no—
253#92418—21no—
254#92418—21no—
255#92418—21no—
256#92418—41no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#19278—611nomaximal, Sylow
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—22no—
264C22×C61—21no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61C2×#12111yescharacteristic
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61C2×#13511yescharacteristic
281C4×C61C2×#13511yescharacteristic
282C4×C61#57511yescharacteristic
283C4×C61#57511yescharacteristic
284Dic61—81no—
285#3131—611no—
286#3116D6111yescharacteristic, complemented
287#3115—611no—
288C8×#41—611no—
289#1684—611no—
290#1683—611no—
291#1682—611no—
292#1681—611no—
293#1680—611no—
294#1679—611no—
295#1678—611no—
296#1677—611no—
297#1568—611no—
298#1554—611no—
299#1553—611no—
300C2×C61—41no—
301C2×C61—42no—
302C2×C61#304011yescharacteristic
303C2×C61#305011yescharacteristic
304C2×C61#306711yescharacteristic
305D61—161no—
306#533C2×D6111yescharacteristic
307#532—611no—
308C4×#41C2×D6111yescharacteristic
309#488—1221no—
310#488—1221no—
311#487—611no—
312#487—611no—
313#486—611no—
314#486—611no—
315#485—21no—
316#482—1221no—
317#477C2×D6111yescharacteristic
318#476C2×D6111yescharacteristic
319#475—611no—
320#474—611no—
321#462C2×D6111yescharacteristic
322#461C2×D6111yescharacteristic
323#460—611no—
324#459—611no—
325#452—21no—
326#449—1221no—
327C8×Q8—611no—
328C8×Q8—611no—
329#442—1221no—
330#441—21no—
331#440—21no—
332#440—1221no—
333C8×D4—611no—
334C8×D4—611no—
335C8×D4—1221no—
336C8×D4—1221no—
337#432—611no—
338#432C2×D6111yescharacteristic
339#431—611no—
340#431—1221no—
341#416—1221no—
342#416—1221no—
343C2×C4×C8—611no—
344C2×C4×C8—1221no—
345#335—611no—
346#335—611no—
347#334—611no—
348#334—611no—
349#333—611no—
350#333—611no—
351#332—611no—
352#332—611no—
353#332—611no—
354#332—611no—
355#331—611no—
356#331—611no—
357C61#1927811yescharacteristic, complemented, Sylow
358C2×#41—1221no—
359C2×#41C22⋊D6111yescharacteristic
360C2×Q16—21no—
361C2×SD16—21no—
362C2×SD16—1221no—
363C2×D8—1221no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—1221no—
367C22×C8—1221no—
368C4⋊Q8—611no—
369C4⋊Q8C22×D6111yescharacteristic
370C4⋊Q8C22×D6111yescharacteristic
371#127—611no—
372#125—21no—
373#124—1221no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376#121—1221no—
377#121—1221no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8C22⋊D6111yescharacteristic
381C4×Q8C22×D6111yescharacteristic
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—1221no—
385C4×D4—1221no—
386C4×D4C22⋊D6111yescharacteristic
387C4×D4C22×D6111yescharacteristic
388#117—21no—
389#117—21no—
390#117C22⋊D6111yescharacteristic
391C2×C42—21no—
392C2×C42C22×D6111yescharacteristic
393#107—21no—
394#107—1221no—
395#106—21no—
396#106—1221no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8C22×D6111yescharacteristic
404C4⋊C8C22×D6111yescharacteristic
405#103—21no—
406#103—21no—
407#103—21no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#102—21no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421C22⋊C8—21no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426#97—21no—
427#97—1221no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—1221no—
433C4×C8—1221no—
434C4×C8—1221no—
435#41—22no—
436#41—22no—
437#41—2442no—
438#41—2442no—
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—
459D8—2442no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—2441no—
471C2×C8—2441no—
472C2×C8—2442no—
473C2×C8—2442no—
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—41no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—41no—
496C42D4×D6111yescharacteristic
497C42#1175822411yescharacteristic
498C42C2×C22⋊D6111yescharacteristic
499C42C23×D6111yescharacteristic
500C23—21no—
501C23—2441no—
502C23#1451974911yescharacteristic
503Q8—21no—
504Q8—41no—
505Q8—41no—
506Q8—81no—
507Q8—2441no—
508Q8—2441no—
509Q8#1451979712yes—
510D4—24no—
511D4—21no—
512D4—4881no—
513D4—4882no—
514D4—2441no—
515D4—2441no—
516D4—2441no—
517D4—2441no—
518D4#1451979712yes—
519C2×C4—24no—
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—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—4882no—
534C2×C4—2441no—
535C2×C4#1451974911yescharacteristic
536C2×C4#1451974911yescharacteristic
537C2×C4#1451974911yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4C22×C22⋊D6111yescharacteristic
540C2×C4C2×#1175822411yescharacteristic
541C8—41no—
542C8—41no—
543C8—1221no—
544C8—1221no—
545C8—1221no—
546C8—1221no—
547C8—2441no—
548C8—2441no—
549C8—2441no—
550C8—2441no—
551C22—21no—
552C22—22no—
553C22—21no—
554C22—21no—
555C22—41no—
556C22—41no—
557C22—4881no—
558C22—4881no—
559C22—4881no—
560C22C2×#1451974911yescharacteristic
561C4—21no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—81no—
569C4—81no—
570C4—4881no—
571C4C2×#1451979711yescharacteristic
572C4C2×#1451979711yescharacteristic
573C4#1684056111yescharacteristic
574C4#1684056111yescharacteristic
575C2—41no—
576C2—42no—
577C2—9761no—
578C2#2145531511yescharacteristic
579C2#2145539711yescharacteristic
580C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3116C2
socleC22×C61C2×#121
prime radical#21455970C2
hypercenter#3116D61
hypocenterC61#19278
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
1. #21456144C2semidirect
3. #41×#92418C2semidirect
4. #21445279C2semidirect
5. #21445272C2semidirect
6. #21445265C2semidirect
7. #21445258C2semidirect
8. #21445251C2semidirect
9. #21445244C2semidirect
10. #21445237C2semidirect
11. #21445230C2semidirect
12. #21444580C2semidirect
13. #21444529C2semidirect
14. #21444525C2semidirect
15. C61×#3116C2semidirect
286. #3116D61semidirect
357. C61#19278semidirect

Maximal subgroups

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

subgroupindex
#214561442
#214559702
#41×#924182
#214452792
#214452722
#214452652
#214452582
#214452512
#214452442
#214452372
#214452302
#214445802
#214445292
#214445252
C61×#31162
#1927861