Finite Group Databasebeta

← #29670119

Subgroups — #29670119

Order 15616.

676 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458684C211yesmaximal, characteristic, complemented
2#21458176C211yesmaximal, characteristic, complemented
3#21457643C211yesmaximal, characteristic, complemented
4#21445438C211yesmaximal, characteristic, complemented
5#21445342C211yesmaximal, characteristic, complemented
6#21445226C211yesmaximal, characteristic, complemented
7#21445192C211yesmaximal, characteristic, complemented
8#21445164C211yesmaximal, characteristic, complemented
9#21445024C211yesmaximal, characteristic, complemented
10#21444970C211yesmaximal, characteristic, complemented
11#21444935C211yesmaximal, characteristic, complemented
12#21444889C211yesmaximal, characteristic, complemented
13#21444660C211yesmaximal, characteristic, complemented
14#21444643C211yesmaximal, characteristic, complemented
15C61×#3272C211yesmaximal, characteristic
16#16840469C2211yescharacteristic
17#16840469C2211yescharacteristic, complemented
18#16840167C2211yescharacteristic, complemented
19#16840163—21no—
20#16840158C2211yescharacteristic, complemented
21#16840134C2211yescharacteristic
22#16840114C2211yescharacteristic, complemented
23#16840108C2211yescharacteristic
24#16840092—21no—
25#16840078—21no—
26#16840078—21no—
27#16840060—21no—
28#16840060—21no—
29#16840052C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16840052C2211yescharacteristic, complemented
33#16840044C2211yescharacteristic, complemented
34#16839989—21no—
35#16839989—21no—
36#16839959C2211yescharacteristic
37#16839913—21no—
38C8×Dic122C2211yescharacteristic, complemented
39C4×#11758200C2211yescharacteristic
40D8×Dic61—21no—
41D4×#92418C2211yescharacteristic
42D4×#92418C2211yescharacteristic, complemented
43C2×#14519708—21no—
44C2×#14519698—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51#16839492C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic, complemented
55#16839483C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#547C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#468—21no—
61C61×#461C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#450—21no—
64C61×#449—21no—
65C4×D8×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758200—41no—
69C2×#11758200D411yescharacteristic
70C22×#92418—21no—
71C22×#92418—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519768—21no—
75#14519768—21no—
76#14519756—21no—
77#14519756—21no—
78#14519754—41no—
79#14519754—41no—
80C4×Dic122C2311yescharacteristic
81D4×Dic61—21no—
82D4×Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85#14519711—21no—
86#14519711—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708D411yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519700—21no—
100#14519700—41no—
101#14519700D411yescharacteristic
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×#41×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118AGL(1,ℤ/8)×C61—44no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×D8×C61—21no—
123C2×D8×C61D411yescharacteristic
124C2×M16×C61—21no—
125C4⋊Q8×C61C2311yescharacteristic
126#127×C61C2311yescharacteristic
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—21no—
132#120×C61—41no—
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61—41no—
154C2×D4×C61—42no—
155C2×D4×C61—42no—
156C2×D4×C61—41no—
157C2×D4×C61—81no—
158C2×D4×C61—81no—
159C2×D4×C61C2×D411yescharacteristic
160C2×D4×C61C2×D411yescharacteristic
161C2×D4×C61C2×D411yescharacteristic
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—41no—
166C22×Dic61—41no—
167C22×Dic61—41no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C2×D411yescharacteristic
171SD16×C61—42no—
172SD16×C61—42no—
173SD16×C61—81no—
174D8×C61—42no—
175D8×C61SD1614yescomplemented
176M16×C61—42no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—41no—
193C22⋊Dic61—41no—
194#11758200—41no—
195#11758200—81no—
196#11758200—82no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—21no—
205#11758196—41no—
206#11758196—41no—
207C4×Dic61—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—42no—
213C2×#92418—81no—
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—41no—
220C23×C61—82no—
221C23×C61—82no—
222C23×C61—161no—
223Q8×C61—41no—
224Q8×C61—41no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—81no—
234D4×C61—82no—
235D4×C61—82no—
236D4×C61—82no—
237D4×C61C2×SD1612yes—
238D4×C61C2×SD1612yes—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—82no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—161no—
258C2×Dic61—161no—
259Dic122—21no—
260Dic122—21no—
261Dic122—81no—
262Dic122—81no—
263C8×C61—21no—
264C8×C61—41no—
265C8×C61C2×SD1612yes—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#19806—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—164no—
286C22×C61D4211yescharacteristic
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—81no—
292C4×C61C22×SD1611yescharacteristic
293C4×C61#57511yescharacteristic
294C4×C61#57711yescharacteristic
295C4×C61#57711yescharacteristic
296Dic61—41no—
297Dic61—41no—
298Dic61—81no—
299Dic61—81no—
300#3380—611no—
301#3335—611no—
302#3295—611no—
303#3272D6111yescharacteristic
304#1714—611no—
305#1695—611no—
306#1677—611no—
307#1671—611no—
308#1666—611no—
309#1643—611no—
310#1635—611no—
311#1629—611no—
312#1622—611no—
313#1583—611no—
314#1580—611no—
315C2×C61—42no—
316C2×C61—42no—
317C2×C61—81no—
318C2×C61—161no—
319C2×C61#326711yescharacteristic
320C2×C61D4×SD1611yescharacteristic
321C2×C61#328811yescharacteristic
322C2×AGL(1,ℤ/8)—21no—
323#554—611no—
324#554—611no—
325#547C2×D6111yescharacteristic
326D42C2×D6111yescharacteristic
327#500—611no—
328#499—1221no—
329#498—611no—
330#492—611no—
331#492C2×D6111yescharacteristic
332#487—611no—
333#486—611no—
334#483—1221no—
335#481—1221no—
336#481—1221no—
337#478—1221no—
338#478—1221no—
339#476—611no—
340#476—611no—
341#476—611no—
342#476—611no—
343#475—611no—
344#468—21no—
345#465—1221no—
346#465—1221no—
347#461C2×D6111yescharacteristic
348#460—611no—
349#459C2×D6111yescharacteristic
350#450—21no—
351#450—1221no—
352#449—21no—
353C8×Q8—611no—
354C4×SD16—611no—
355C4×D8—1221no—
356C4×D8C2×D6111yescharacteristic
357#436C2×D6111yescharacteristic
358C8×D4—611no—
359C8×D4—611no—
360C2×#106—1221no—
361C2×#102—1221no—
362#334—611no—
363#334—611no—
364#333—611no—
365#333—611no—
366#332—611no—
367#331—611no—
368Q8⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8—611no—
372#322—611no—
373C61#1980611yescharacteristic, complemented, Sylow
374C2×#41—21no—
375C22×D4—21no—
376C22×D4—21no—
377AGL(1,ℤ/8)—44no—
378C2×SD16—21no—
379C2×SD16—21no—
380C2×SD16—21no—
381C2×SD16—611no—
382C2×SD16—1221no—
383C2×SD16—2441no—
384C2×D8—21no—
385C2×D8C22⋊D6111yescharacteristic
386C2×M16—21no—
387C22×C8—1221no—
388C22×C8—1221no—
389C4⋊Q8—611no—
390C4⋊Q8—611no—
391C4⋊Q8C22×D6111yescharacteristic
392#127C22×D6111yescharacteristic
393#125—1221no—
394#125—1221no—
395#123—41no—
396C22⋊Q8—21no—
397C22⋊Q8—1221no—
398C22⋊Q8—1221no—
399C22⋊Q8—2441no—
400C22⋊Q8—2441no—
401#121—21no—
402#121—21no—
403#121—21no—
404#120—41no—
405C4×Q8—611no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4C22×D6111yescharacteristic
409C4×D4C22×D6111yescharacteristic
410C4×D4C22×D6111yescharacteristic
411C2×C4⋊C4—21no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414#107—1221no—
415#107—1221no—
416#107C22⋊D6111yescharacteristic
417#106—611no—
418#106—1221no—
419#106—1221no—
420#106—1221no—
421#106—1221no—
422#106—1221no—
423#106—1221no—
424#106—1221no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—1221no—
429C4⋊C8C22×D6111yescharacteristic
430#103—21no—
431#103—611no—
432#103—1221no—
433#103—2441no—
434#102—21no—
435#102—611no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102C22⋊D6111yescharacteristic
444#102C22⋊D6111yescharacteristic
445C22⋊C8—21no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C4×C8—611no—
449C4×C8—611no—
450C4×C8—1221no—
451C4×C8C22×D6111yescharacteristic
452C24—41no—
453#41—42no—
454#41—81no—
455C2×Q8—21no—
456C2×Q8—611no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—41no—
467C2×D4—81no—
468C2×D4—81no—
469C2×D4D4×D6111yescharacteristic
470C2×D4C2×C22⋊D6111yescharacteristic
471C2×D4C2×C22⋊D6111yescharacteristic
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—41no—
476C22×C4—2441no—
477C22×C4—2441no—
478SD16—42no—
479SD16—42no—
480SD16—81no—
481SD16—4881no—
482SD16—4882no—
483SD16—2441no—
484D8—42no—
485D8#1175820014yescomplemented
486M16—42no—
487C2×C8—21no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—4881no—
495C2×C8—2442no—
496C2×C8C2×C22⋊D6111yescharacteristic
497C2×C8C2×C22⋊D6111yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—41no—
501C4⋊C4—611no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4D4×D6111yescharacteristic
512C4⋊C4C2×C22⋊D6111yescharacteristic
513C4⋊C4C2×C22⋊D6111yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—2441no—
520C22⋊C4—2441no—
521C42—1221no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—21no—
525C23—21no—
526C23—41no—
527C23—82no—
528C23—82no—
529C23—161no—
530Q8—41no—
531Q8—41no—
532Q8—1221no—
533Q8—1221no—
534Q8—4881no—
535Q8—4881no—
536D4—21no—
537D4—21no—
538D4—21no—
539D4—21no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—81no—
545D4—82no—
546D4—82no—
547D4—82no—
548D4C2×#1175820012yes—
549D4C2×#1175820012yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—82no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—9761no—
564C2×C4—9761no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C22×C22⋊D6111yescharacteristic
569C2×C4#1451982211yescharacteristic
570C8—21no—
571C8—41no—
572C8—1221no—
573C8—1221no—
574C8—2441no—
575C8—2441no—
576C8—2441no—
577C8—2441no—
578C8C2×#1175820012yes—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—164no—
592C22D4×C22⋊D6111yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—4881no—
599C4—4881no—
600C4—2441no—
601C4—2441no—
602C4C22×#1175820011yescharacteristic
603C4#1684055811yescharacteristic
604C4#1684057211yescharacteristic
605C4#1684057311yescharacteristic
606C2—42no—
607C2—42no—
608C2—81no—
609C2—161no—
610C2#2145733811yescharacteristic
611C2D4×#1175820011yescharacteristic
612C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3272C2
socleC22×C61D42
hypercenter#3272D61
hypocenterC61#19806
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3272

Complements

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

NH ≅ G/Ntype
1. #21458684C2semidirect
2. #21458176C2semidirect
3. #21457643C2semidirect
4. #21445438C2semidirect
5. #21445342C2semidirect
6. #21445226C2semidirect
7. #21445192C2semidirect
8. #21445164C2semidirect
9. #21445024C2semidirect
10. #21444970C2semidirect
11. #21444935C2semidirect
12. #21444889C2semidirect
13. #21444660C2semidirect
14. #21444643C2semidirect
17. #16840469C22semidirect
18. #16840167C22semidirect
20. #16840158C22semidirect
22. #16840114C22semidirect
32. #16840052C22semidirect
33. #16840044C22semidirect
38. C8×Dic122C22semidirect
42. D4×#92418C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839510C22semidirect
48. #16839510C22semidirect
49. #16839508C22semidirect
51. #16839492C22semidirect
54. D4⋊#92418C22semidirect
55. #16839483C22semidirect
175. D8×C61SD16semidirect
373. C61#19806semidirect
485. D8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214586842
#214581762
#214576432
#214454382
#214453422
#214452262
#214451922
#214451642
#214450242
#214449702
#214449352
#214448892
#214446602
#214446432
C61×#32722
#1980661