Finite Group Databasebeta

← #29665235

Subgroups — #29665235

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456948C211yesmaximal, characteristic, complemented
2#476×D61C211yesmaximal, characteristic, complemented
3#21455284C211yesmaximal, characteristic
4#21454840C211yesmaximal, characteristic, complemented
5#21445231C211yesmaximal, characteristic, complemented
6#21445230C211yesmaximal, characteristic, complemented
7#21445229C211yesmaximal, characteristic, complemented
8#21445228C211yesmaximal, characteristic, complemented
9#21445193C211yesmaximal, characteristic, complemented
10#21445192C211yesmaximal, characteristic, complemented
11#21445191C211yesmaximal, characteristic, complemented
12#21444588C211yesmaximal, characteristic, complemented
13#21444498C211yesmaximal, characteristic, complemented
14#21444467C211yesmaximal, characteristic, complemented
15C61×#1671C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17#16840313C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840158C2211yescharacteristic, complemented
21#16840110C2211yescharacteristic, complemented
22#16840109C2211yescharacteristic, complemented
23#16840082—21no—
24#16840082—21no—
25#16840053C2211yescharacteristic
26#16840052C2211yescharacteristic
27#16840051—21no—
28#16840051—21no—
29#16840051—21no—
30#16840050—21no—
31#16840049C2211yescharacteristic
32#16839980—21no—
33#16839979—21no—
34#16839964C2211yescharacteristic
35#16839958C2211yescharacteristic
36#16839957C2211yescharacteristic
37C8×D244C2211yescharacteristic
38D4×#92418C2211yescharacteristic
39#16839827C2211yescharacteristic
40C8×C22⋊D61—21no—
41#16839825—21no—
42#16839814C2211yescharacteristic
43#16839813C2211yescharacteristic
44#106×D61—21no—
45#16839784C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839762—21no—
48#102×D61—21no—
49C2×#14519689—21no—
50C22⋊C8×D61—21no—
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839510C2211yescharacteristic, complemented
57#16839508C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic, complemented
59#16839505C2211yescharacteristic, complemented
60C61×#498C2211yescharacteristic, complemented
61C61×#476C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#332C2211yescharacteristic, complemented
67C2×#11758226—21no—
68C2×Q8×D61—21no—
69C2×D4×D61D411yescharacteristic
70C22×C4×D61—21no—
71#14519804—44no—
72C2×#11758201—21no—
73C2×D488—21no—
74C2×#11758199—21no—
75C2×#11758189—21no—
76C22×#92418—21no—
77C2×C8×D61—21no—
78C2×C8×D61—21no—
79C2×C8×D61—21no—
80#14519777C2311yescharacteristic
81#14519776—21no—
82#14519776—21no—
83#14519775—21no—
84#14519774C2311yescharacteristic
85#14519766—21no—
86#14519752—21no—
87#14519752—21no—
88C4×D244C2311yescharacteristic
89D4×Dic61D411yescharacteristic
90#14519739—21no—
91#14519739—21no—
92#14519739D411yescharacteristic
93C4×C22⋊D61—21no—
94#14519737—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61D411yescharacteristic
98C22⋊C4×D61—21no—
99C42×D61C2311yescharacteristic
100#14519709—21no—
101#14519708—21no—
102#14519708—21no—
103#14519708—21no—
104#14519708—21no—
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108#14519706—21no—
109#14519706—21no—
110#14519706D411yescharacteristic
111#14519706D411yescharacteristic
112#14519701—21no—
113#14519699—21no—
114#14519699—21no—
115#14519699—21no—
116#14519699—21no—
117#14519698—21no—
118#14519697—21no—
119#14519697—21no—
120#14519697—21no—
121#14519697—21no—
122C22⋊#92418—21no—
123#14519689—21no—
124#14519689—21no—
125#14519689—21no—
126#14519689D411yescharacteristic
127#14519689D411yescharacteristic
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C2×SD16×C61—21no—
131C22×C8×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133C4⋊Q8×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#106×C61—21no—
136#106×C61—21no—
137#106×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#103×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C2×C22⋊D61—21no—
151C22×Dic61—21no—
152#11758226—42no—
153#11758226—42no—
154Q8×D61—82no—
155D4×D61—22no—
156D4×D61SD1614yes—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—41no—
161C2×D244C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—41no—
166C2×C4×D61—41no—
167C2×C4×D61—81no—
168C2×C4×D61—81no—
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61#4111yescharacteristic
172C2×Dic122—21no—
173C2×Dic122—41no—
174SD16×C61—42no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—81no—
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×D411yescharacteristic
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—21no—
189#11758201—42no—
190#11758199—42no—
191#11758198—21no—
192#11758198—21no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—21no—
199#11758196—41no—
200#11758196—41no—
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61#4111yescharacteristic
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—81no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210D488—42no—
211#11758189—42no—
212C8×D61—42no—
213C8×D61—81no—
214C8×D61—81no—
215C23×C61—21no—
216C22×D61—21no—
217C22×D61—24no—
218C22×D61—41no—
219C22×D61—81no—
220C22×D61—81no—
221C22×D61#12111yescharacteristic
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—81no—
225D4×C61—21no—
226D4×C61C2×SD1612yes—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—24no—
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—81no—
239C2×Dic61#12111yescharacteristic
240C2×Dic61#12111yescharacteristic
241C2×Dic61#12111yescharacteristic
242D244—21no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—81no—
248D244—82no—
249D244C2×SD1612yes—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—81no—
256C4×D61—81no—
257C4×D61—82no—
258C4×D61—82no—
259C4×D61C2×SD1612yes—
260C4×D61#13512yes—
261Dic122—82no—
262Dic122—81no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—41no—
266C8×C61—41no—
267C8×C61—41no—
268#92418—21no—
269#92418—21no—
270#92418—41no—
271#92418—41no—
272#92418—41no—
273#19391—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61C2×#12111yescharacteristic
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—84no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61#46512yes—
288C4×C61—21no—
289C4×C61—41no—
290C4×C61—81no—
291C4×C61—81no—
292C4×C61C22×SD1611yescharacteristic
293C4×C61C2×#13511yescharacteristic
294C4×C61#57711yescharacteristic
295C4×C61#57711yescharacteristic
296Dic61—21no—
297Dic61—21no—
298Dic61—41no—
299Dic61—81no—
300Dic61#46512yes—
301#3227—611no—
302C2×#476—611no—
303#3036—611no—
304#2972—611no—
305#1677—611no—
306#1677—611no—
307#1677—611no—
308#1677—611no—
309#1671—611no—
310#1671—611no—
311#1671—611no—
312#1671D6111yescharacteristic, complemented
313#1570—611no—
314#1548—611no—
315#1540—611no—
316C2×C61—42no—
317C2×C61#304211yescharacteristic
318C2×C61C2×#46511yescharacteristic
319C2×C61#306811yescharacteristic
320D61—22no—
321D61—42no—
322D61—161no—
323C2×AGL(1,ℤ/8)—1221no—
324#532—611no—
325C2×C4⋊Q8—611no—
326C2×C4×D4—611no—
327#498—611no—
328#498C2×D6111yescharacteristic, complemented
329#486—611no—
330#486—611no—
331#481—1221no—
332#481—1221no—
333#476—611no—
334#476—611no—
335#476—611no—
336#476—1221no—
337#476—1221no—
338#476—1221no—
339#476—1221no—
340#476C2×D6111yescharacteristic
341#463—1221no—
342#463—1221no—
343#461—611no—
344#461C2×D6111yescharacteristic
345#459—611no—
346#459—611no—
347C8×D4—611no—
348C8×D4—611no—
349C8×D4—611no—
350C8×D4—1221no—
351C8×D4—1221no—
352C8×D4C2×D6111yescharacteristic
353#431—611no—
354#431—611no—
355C2×#106—1221no—
356#423—611no—
357C2×C4⋊C8—611no—
358#417—1221no—
359C2×#102—1221no—
360C2×C22⋊C8—1221no—
361C2×C22⋊C8—1221no—
362#334—611no—
363#334—611no—
364#334—611no—
365#334C2×D6111yescharacteristic, complemented
366#333—611no—
367#333—611no—
368#333—611no—
369#333C2×D6111yescharacteristic, complemented
370#332—611no—
371#332C2×D6111yescharacteristic, complemented
372#331—611no—
373#331—611no—
374C61#1939111yescharacteristic, complemented, Sylow
375C2×#41—1221no—
376C22×Q8—1221no—
377C22×D4—611no—
378C23×C4—1221no—
379AGL(1,ℤ/8)—2444no—
380C2×SD16—21no—
381C2×SD16—1221no—
382C2×D8—1221no—
383C2×D8—1221no—
384C2×M16—1221no—
385C22×C8—21no—
386C22×C8—1221no—
387C22×C8—1221no—
388C22×C8—1221no—
389C22×C8—1221no—
390C4⋊Q8—611no—
391C4⋊Q8—1221no—
392C4⋊Q8—1221no—
393C4⋊Q8—1221no—
394C4⋊Q8C22×D6111yescharacteristic
395C4⋊Q8C22×D6111yescharacteristic
396#127—611no—
397#124—1221no—
398#121—1221no—
399#121—1221no—
400C4×D4—611no—
401C4×D4—611no—
402C4×D4—611no—
403C4×D4—1221no—
404C4×D4—1221no—
405C4×D4—1221no—
406C4×D4—1221no—
407C4×D4C22×D6111yescharacteristic
408C2×C4⋊C4—611no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C42—611no—
413#106—21no—
414#106—21no—
415#106—21no—
416#106—1221no—
417#106—1221no—
418#106—1221no—
419#106—1221no—
420#106—1221no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—611no—
425C4⋊C8—611no—
426C4⋊C8—1221no—
427C4⋊C8—1221no—
428C4⋊C8C22×D6111yescharacteristic
429C4⋊C8C22×D6111yescharacteristic
430C4⋊C8C22×D6111yescharacteristic
431#103—21no—
432#103—1221no—
433#102—21no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443C22⋊C8—21no—
444C22⋊C8—611no—
445C22⋊C8—611no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C22⋊C8—1221no—
449C22⋊C8—1221no—
450C4×C8—611no—
451C4×C8—1221no—
452C4×C8C22×D6111yescharacteristic
453C24—1221no—
454#41—2442no—
455#41—2442no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—1221no—
459C2×Q8—4882no—
460C2×Q8—2441no—
461C2×D4—614no—
462C2×D4—611no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—2441no—
469C2×D4D4×D6111yescharacteristic
470C22×C4—21no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—4881no—
479C22×C4—4881no—
480C22×C4—2441no—
481C22×C4—2441no—
482SD16—42no—
483SD16—2442no—
484D8—2442no—
485D8—2442no—
486M16—2442no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—81no—
491C2×C8—611no—
492C2×C8—611no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—4881no—
497C2×C8—4881no—
498C2×C8—4881no—
499C2×C8—2442no—
500C2×C8D4×D6111yescharacteristic
501C2×C8D4×D6111yescharacteristic
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—611no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—2441no—
511C4⋊C4—2441no—
512C4⋊C4D4×D6111yescharacteristic
513C22⋊C4—21no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C22⋊C4—1221no—
517C22⋊C4—2441no—
518C22⋊C4—2441no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—611no—
525C23—1224no—
526C23—1221no—
527C23—4881no—
528C23—4881no—
529C23—2441no—
530Q8—41no—
531Q8—41no—
532Q8—81no—
533Q8—4881no—
534Q8—4882no—
535D4—21no—
536D4—612no—
537D4—1224no—
538D4—1221no—
539D4—4881no—
540D4—4882no—
541D4—2441no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4SD16×D6112yes—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—611no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—611no—
555C2×C4—611no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—4881no—
560C2×C4—4881no—
561C2×C4—4882no—
562C2×C4—4882no—
563C2×C4—4881no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4C2×D4×D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C8—21no—
572C8—21no—
573C8—41no—
574C8—41no—
575C8—41no—
576C8—1221no—
577C8—1221no—
578C8—2441no—
579C8—2441no—
580C8—2441no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—612no—
585C22—1221no—
586C22—1221no—
587C22—1222no—
588C22—4884no—
589C22—4881no—
590C22—4881no—
591C22—4881no—
592C22—2441no—
593C22—2441no—
594C22#121×D6111yescharacteristic
595C4—21no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—612no—
600C4—1221no—
601C4—1221no—
602C4—4881no—
603C4—2441no—
604C4C2×SD16×D6111yescharacteristic
605C4#135×D6111yescharacteristic
606C4#1684056811yescharacteristic
607C4#1684057111yescharacteristic
608C2—42no—
609C2—1222no—
610C2—9761no—
611C2—2442no—
612C2#2145532611yescharacteristic
613C2#465×D6111yescharacteristic
614C2#2145554211yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1671C2
socleC22×C61C2×#121
prime radical#21455284C2
hypercenter#1671D61
hypocenterC61#19391
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1671

Complements

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

NH ≅ G/Ntype
1. #21456948C2semidirect
2. #476×D61C2semidirect
4. #21454840C2semidirect
5. #21445231C2semidirect
6. #21445230C2semidirect
7. #21445229C2semidirect
8. #21445228C2semidirect
9. #21445193C2semidirect
10. #21445192C2semidirect
11. #21445191C2semidirect
12. #21444588C2semidirect
13. #21444498C2semidirect
14. #21444467C2semidirect
15. C61×#1671C2semidirect
20. #16840158C22semidirect
21. #16840110C22semidirect
22. #16840109C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839510C22semidirect
56. #16839510C22semidirect
57. #16839508C22semidirect
58. #16839505C22semidirect
59. #16839505C22semidirect
60. C61×#498C22semidirect
64. C61×#334C22semidirect
65. C61×#333C22semidirect
66. C61×#332C22semidirect
312. #1671D61semidirect
328. #498C2×D61semidirect
365. #334C2×D61semidirect
369. #333C2×D61semidirect
371. #332C2×D61semidirect
374. C61#19391semidirect

Maximal subgroups

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

subgroupindex
#214569482
#476×D612
#214552842
#214548402
#214452312
#214452302
#214452292
#214452282
#214451932
#214451922
#214451912
#214445882
#214444982
#214444672
C61×#16712
#1939161