Finite Group Databasebeta

← #29665271

Subgroups — #29665271

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456958C211yesmaximal, characteristic, complemented
2#21455912C211yesmaximal, characteristic
3#21455829C211yesmaximal, characteristic, complemented
4#21454840C211yesmaximal, characteristic, complemented
5#21445257C211yesmaximal, characteristic, complemented
6#21445256C211yesmaximal, characteristic, complemented
7#21445231C211yesmaximal, characteristic, complemented
8#21445230C211yesmaximal, characteristic, complemented
9#21445215C211yesmaximal, characteristic, complemented
10#21445197C211yesmaximal, characteristic, complemented
11#21445196C211yesmaximal, characteristic, complemented
12#21444598C211yesmaximal, characteristic, complemented
13#21444528C211yesmaximal, characteristic, complemented
14#21444480C211yesmaximal, characteristic, complemented
15C61×#1675C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840313C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16840158C2211yescharacteristic, complemented
20#16840110C2211yescharacteristic, complemented
21#16840109C2211yescharacteristic, complemented
22#16840094—21no—
23#16840089—21no—
24#16840089—21no—
25#16840082—21no—
26#16840065—21no—
27#16840065—21no—
28#16840049C2211yescharacteristic
29#16840049C2211yescharacteristic
30#16839968C2211yescharacteristic
31#16839967C2211yescharacteristic
32#16839958C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839914—21no—
35#16839897—21no—
36#16839873—21no—
37#16839871—21no—
38#16839853—21no—
39#16839849—21no—
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42#16839827C2211yescharacteristic
43C8×C22⋊D61—21no—
44#16839825—21no—
45#16839818C2211yescharacteristic
46#16839814C2211yescharacteristic
47#16839814C2211yescharacteristic
48#16839813C2211yescharacteristic
49C2×#14519689—21no—
50C22⋊C8×D61—21no—
51#16839515C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839511C2211yescharacteristic, complemented
56#16839509C2211yescharacteristic, complemented
57#16839507C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic, complemented
59#16839505C2211yescharacteristic, complemented
60C61×#500C2211yescharacteristic, complemented
61C61×#476C2211yescharacteristic
62C61×#476C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#333C2211yescharacteristic, complemented
67C2×#11758226—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×D488—21no—
73C2×#11758199—21no—
74C22×#92418—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77#14519777C2311yescharacteristic
78#14519774C2311yescharacteristic
79#14519768—21no—
80#14519766—21no—
81#14519757—21no—
82#14519753—21no—
83#14519752—21no—
84#14519752—21no—
85Q8×Dic61—21no—
86#14519742—21no—
87C4×D244C2311yescharacteristic
88D4×Dic61D411yescharacteristic
89#14519739—21no—
90#14519739—21no—
91#14519739D411yescharacteristic
92C4×C22⋊D61—21no—
93#14519737—21no—
94#14519735—21no—
95C4⋊C4×D61D411yescharacteristic
96C22⋊C4×D61—21no—
97C42×D61C2311yescharacteristic
98#14519708—21no—
99#14519708—21no—
100#14519708—21no—
101#14519708—21no—
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105#14519706D411yescharacteristic
106#14519706D411yescharacteristic
107#14519702—21no—
108#14519700—21no—
109#14519699—21no—
110#14519699—21no—
111#14519699—21no—
112#14519697—21no—
113#14519697—21no—
114#14519697—21no—
115#14519697—21no—
116#14519697—21no—
117C22⋊#92418—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689D411yescharacteristic
124#14519689D411yescharacteristic
125#14519687—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C22×C8×C61—21no—
130C4⋊Q8×C61C2311yescharacteristic
131C4⋊Q8×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#107×C61—21no—
134#107×C61—21no—
135#106×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×D4×C61C2×D411yescharacteristic
148C22×C4×C61—21no—
149C2×C22⋊D61—21no—
150C22×Dic61—21no—
151#11758226—82no—
152D4×D61—22no—
153D4×D61—22no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—41no—
158C2×D244C2×D411yescharacteristic
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—41no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61#4111yescharacteristic
168C2×Dic122—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—81no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C42×C61C2411yescharacteristic
182C22⋊Dic61—21no—
183#11758200—42no—
184#11758199—42no—
185#11758198—21no—
186#11758198—21no—
187#11758198—41no—
188#11758198—41no—
189#11758198—41no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—41no—
193C4⋊Dic61C2×D411yescharacteristic
194#11758196—21no—
195#11758196—41no—
196C4×Dic61—41no—
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61#4111yescharacteristic
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—81no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206D488—42no—
207#11758190—42no—
208C8×D61—81no—
209C8×D61—81no—
210C23×C61—21no—
211C22×D61—21no—
212C22×D61—24no—
213C22×D61—41no—
214C22×D61—81no—
215C22×D61—81no—
216C22×D61#12111yescharacteristic
217Q8×C61—81no—
218Q8×C61—81no—
219D4×C61—21no—
220D4×C61#13512yes—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C22⋊D61—24no—
230C2×Dic61—21no—
231C2×Dic61—41no—
232C2×Dic61—81no—
233C2×Dic61#12111yescharacteristic
234C2×Dic61#12111yescharacteristic
235C2×Dic61#12111yescharacteristic
236D244—21no—
237D244—41no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—81no—
242D244—82no—
243D244#13512yes—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—21no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—81no—
252C4×D61—82no—
253C4×D61—81no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61—41no—
261#92418—21no—
262#92418—21no—
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#19394—611nomaximal, Sylow
267C22×C61—22no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61C2×#12111yescharacteristic
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—22no—
274C2×D61—21no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—84no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61C2×#13511yescharacteristic
286C4×C61C2×#13511yescharacteristic
287C4×C61#57711yescharacteristic
288C4×C61#57711yescharacteristic
289Dic61—21no—
290Dic61—21no—
291Dic61—21no—
292Dic61—41no—
293Dic61—81no—
294#3228—611no—
295#3109—611no—
296#3101—611no—
297#2972—611no—
298#1681—611no—
299#1681—611no—
300#1677—611no—
301#1677—611no—
302#1675—611no—
303#1675D6111yescharacteristic, complemented
304#1671—611no—
305#1671—611no—
306#1572—611no—
307#1554—611no—
308#1543—611no—
309C2×C61—42no—
310C2×C61#304211yescharacteristic
311C2×C61#305011yescharacteristic
312C2×C61#306811yescharacteristic
313D61—41no—
314D61—42no—
315D61—161no—
316#533—611no—
317#532—611no—
318C2×C4×D4—611no—
319#500C2×D6111yescharacteristic, complemented
320#498—611no—
321#486—611no—
322#486—611no—
323#483—1221no—
324#482—1221no—
325#482—1221no—
326#481—1221no—
327#478—1221no—
328#478—1221no—
329#476—611no—
330#476—611no—
331#476C2×D6111yescharacteristic
332#476C2×D6111yescharacteristic
333#461—611no—
334#461—611no—
335#459—611no—
336#459—611no—
337#450—1221no—
338#447—1221no—
339#442—1221no—
340#442—1221no—
341C4×SD16—1221no—
342C4×SD16—1221no—
343C8×D4—611no—
344C8×D4—611no—
345C8×D4—611no—
346C8×D4—1221no—
347C8×D4—1221no—
348C8×D4C2×D6111yescharacteristic
349#432—611no—
350#431—611no—
351#431—611no—
352#431—611no—
353C2×C22⋊C8—1221no—
354C2×C22⋊C8—1221no—
355#335—611no—
356#335C2×D6111yescharacteristic, complemented
357#334—611no—
358#334—611no—
359#333—611no—
360#333—611no—
361#333C2×D6111yescharacteristic, complemented
362#333C2×D6111yescharacteristic, complemented
363#332—611no—
364#332—611no—
365#331—611no—
366#331—611no—
367C61#1939411yescharacteristic, complemented, Sylow
368C2×#41—1221no—
369C22×D4—611no—
370C23×C4—1221no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×D8—1221no—
374C2×D8—1221no—
375C22×C8—21no—
376C22×C8—1221no—
377C22×C8—1221no—
378C22×C8—1221no—
379C4⋊Q8—611no—
380C4⋊Q8C22×D6111yescharacteristic
381C4⋊Q8C22×D6111yescharacteristic
382#127—611no—
383#125—1221no—
384#124—1221no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387#121—1221no—
388#121—1221no—
389C4×Q8—1221no—
390C4×Q8—1221no—
391C4×D4—611no—
392C4×D4—611no—
393C4×D4—611no—
394C4×D4—1221no—
395C4×D4—1221no—
396C4×D4—1221no—
397C4×D4—1221no—
398C4×D4C22×D6111yescharacteristic
399#117—1221no—
400C2×C4⋊C4—611no—
401C2×C22⋊C4—1221no—
402C2×C42—611no—
403#107—21no—
404#107—21no—
405#106—21no—
406#106—21no—
407#106—1221no—
408#106—1221no—
409#106—1221no—
410#106—1221no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8C22×D6111yescharacteristic
417C4⋊C8C22×D6111yescharacteristic
418C4⋊C8C22×D6111yescharacteristic
419#103—1221no—
420#103—1221no—
421#102—21no—
422#102—21no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431C22⋊C8—21no—
432C22⋊C8—611no—
433C22⋊C8—611no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440#97—1221no—
441C4×C8—611no—
442C4×C8—1221no—
443C4×C8—1221no—
444C4×C8C22×D6111yescharacteristic
445C24—1221no—
446#41—4882no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—1221no—
450C2×D4—611no—
451C2×D4—1221no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—2441no—
458C2×D4D4×D6111yescharacteristic
459C22×C4—21no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—4881no—
468C22×C4—4881no—
469C22×C4—2441no—
470SD16—2442no—
471SD16—2442no—
472D8—2442no—
473D8—2442no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—81no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—4881no—
484C2×C8—4881no—
485C2×C8—4881no—
486C2×C8D4×D6111yescharacteristic
487C2×C8D4×D6111yescharacteristic
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—611no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—2441no—
497C4⋊C4—2441no—
498C4⋊C4D4×D6111yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C22⋊C4—2441no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—2441no—
510C42C23×D6111yescharacteristic
511C23—21no—
512C23—611no—
513C23—1224no—
514C23—1221no—
515C23—4881no—
516C23—4881no—
517C23—2441no—
518Q8—81no—
519Q8—81no—
520Q8—2441no—
521Q8—2441no—
522D4—21no—
523D4—612no—
524D4—1224no—
525D4—1221no—
526D4—4881no—
527D4—4882no—
528D4—2441no—
529D4—2441no—
530D4—2441no—
531D4—2441no—
532D4#1451980012yes—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—4881no—
547C2×C4—4881no—
548C2×C4—4882no—
549C2×C4—4881no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4#41×D6111yescharacteristic
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—1221no—
563C8—1221no—
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—1221no—
571C22—1222no—
572C22—1221no—
573C22—1221no—
574C22—4884no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—2441no—
579C22—2441no—
580C22#121×D6111yescharacteristic
581C4—21no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—1221no—
586C4—1221no—
587C4—1221no—
588C4—4881no—
589C4—2441no—
590C4#135×D6111yescharacteristic
591C4C2×#1451980011yescharacteristic
592C4#1684056811yescharacteristic
593C4#1684057111yescharacteristic
594C2—42no—
595C2—9761no—
596C2—2441no—
597C2—2442no—
598C2#2145532611yescharacteristic
599C2#2145540211yescharacteristic
600C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1675C2
socleC22×C61C2×#121
prime radical#21455912C2
hypercenter#1675D61
hypocenterC61#19394
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1675

Complements

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

NH ≅ G/Ntype
1. #21456958C2semidirect
3. #21455829C2semidirect
4. #21454840C2semidirect
5. #21445257C2semidirect
6. #21445256C2semidirect
7. #21445231C2semidirect
8. #21445230C2semidirect
9. #21445215C2semidirect
10. #21445197C2semidirect
11. #21445196C2semidirect
12. #21444598C2semidirect
13. #21444528C2semidirect
14. #21444480C2semidirect
15. C61×#1675C2semidirect
19. #16840158C22semidirect
20. #16840110C22semidirect
21. #16840109C22semidirect
51. #16839515C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839511C22semidirect
56. #16839509C22semidirect
57. #16839507C22semidirect
58. #16839505C22semidirect
59. #16839505C22semidirect
60. C61×#500C22semidirect
64. C61×#335C22semidirect
65. C61×#333C22semidirect
66. C61×#333C22semidirect
303. #1675D61semidirect
319. #500C2×D61semidirect
356. #335C2×D61semidirect
361. #333C2×D61semidirect
362. #333C2×D61semidirect
367. C61#19394semidirect

Maximal subgroups

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

subgroupindex
#214569582
#214559122
#214558292
#214548402
#214452572
#214452562
#214452312
#214452302
#214452152
#214451972
#214451962
#214445982
#214445282
#214444802
C61×#16752
#1939461