Finite Group Databasebeta

← #29676300

Subgroups — #29676300

Order 15616.

666 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458805C211yesmaximal, characteristic, complemented
2#21458403C211yesmaximal, characteristic, complemented
3#21458050C211yesmaximal, characteristic, complemented
4#21445488C211yesmaximal, characteristic, complemented
5#21445378C211yesmaximal, characteristic, complemented
6#21445330C211yesmaximal, characteristic, complemented
7#21445308C211yesmaximal, characteristic, complemented
8#21445164C211yesmaximal, characteristic, complemented
9#21445073C211yesmaximal, characteristic, complemented
10#21445054C211yesmaximal, characteristic, complemented
11#21444998C211yesmaximal, characteristic, complemented
12#21444927C211yesmaximal, characteristic, complemented
13#21444733C211yesmaximal, characteristic, complemented
14#21444660C211yesmaximal, characteristic, complemented
15C61×#3272C211yesmaximal, characteristic
16#16840482C2211yescharacteristic
17#16840482C2211yescharacteristic, complemented
18#16840165C2211yescharacteristic, complemented
19#16840165C2211yescharacteristic, complemented
20#16840145—21no—
21#16840142C2211yescharacteristic, complemented
22#16840108C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840097—21no—
25#16840083—21no—
26#16840083—21no—
27#16840044C2211yescharacteristic, complemented
28#16840042C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840006—21no—
31#16840006—21no—
32#16839983—21no—
33#16839983—21no—
34#16839967C2211yescharacteristic
35#16839967C2211yescharacteristic, complemented
36#16839959C2211yescharacteristic
37#16839913—21no—
38C8×Dic122C2211yescharacteristic, complemented
39#16839856—21no—
40C4×#11758200C2211yescharacteristic
41#16839840C2211yescharacteristic
42#16839840C2211yescharacteristic, complemented
43#16839802—21no—
44#16839767—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, 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×Dic244—21no—
68C2×Dic244—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200D411yescharacteristic
73C2×#11758194—21no—
74C2×#11758194—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519770—41no—
78#14519770—41no—
79#14519764—21no—
80#14519764—21no—
81#14519756—21no—
82#14519756—21no—
83C4×Dic122C2311yescharacteristic
84#14519742—21no—
85#14519742—21no—
86#14519736—21no—
87#14519736—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—41no—
93#14519708—21no—
94#14519708D411yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700D411yescharacteristic
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
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—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122C2×D411yescharacteristic
169SD16×C61—42no—
170SD16×C61—42no—
171SD16×C61—81no—
172D8×C61—22no—
173D8×C61—42no—
174M16×C61—42no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C42×C61C2411yescharacteristic
190C22⋊Dic61—41no—
191C22⋊Dic61—41no—
192C22⋊Dic61—41no—
193C22⋊Dic61—41no—
194#11758200—41no—
195#11758200—81no—
196#11758200—81no—
197#11758200—81no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205#11758196—41no—
206C4×Dic61—21no—
207C4×Dic61—41no—
208C4×Dic61—41no—
209#11758194—42no—
210#11758194—81no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217Dic244—42no—
218Dic244—81no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—41no—
223C23×C61—82no—
224C23×C61—82no—
225C23×C61—161no—
226Q8×C61—41no—
227Q8×C61—41no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—81no—
239D4×C61—82no—
240D4×C61—82no—
241D4×C61—82no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—82no—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260Dic122—21no—
261Dic122—21no—
262Dic122—41no—
263Dic122—41no—
264Dic122—41no—
265Dic122—41no—
266C8×C61—21no—
267C8×C61—41no—
268C8×C61AGL(1,ℤ/8)12yes—
269#92418—21no—
270#92418—21no—
271#92418—41no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#20250—611nomaximal, Sylow
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—164no—
289C22×C61D4211yescharacteristic
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
296C4×C61#57511yescharacteristic
297C4×C61#57611yescharacteristic
298C4×C61#57711yescharacteristic
299Dic61—41no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303#3393—611no—
304#3353—611no—
305#3324—611no—
306#3272D6111yescharacteristic
307#1722—611no—
308#1701—611no—
309#1693—611no—
310#1689—611no—
311#1666—611no—
312#1650—611no—
313#1647—611no—
314#1639—611no—
315#1628—611no—
316#1594—611no—
317#1583—611no—
318C2×C61—81no—
319C2×C61—81no—
320C2×C61—81no—
321C2×C61—161no—
322C2×C61#326711yescharacteristic
323C2×C61#327411yescharacteristic
324C2×C61#328911yescharacteristic
325C2×AGL(1,ℤ/8)—21no—
326#556—611no—
327#556—611no—
328#547C2×D6111yescharacteristic
329D42C2×D6111yescharacteristic
330#500—611no—
331#500—611no—
332#495—1221no—
333#494—611no—
334#492C2×D6111yescharacteristic
335#486—611no—
336#486—611no—
337#484—1221no—
338#482—1221no—
339#482—1221no—
340#475—611no—
341#474—611no—
342#474—611no—
343#468—21no—
344#468—1221no—
345#468—1221no—
346#464—1221no—
347#464—1221no—
348#461—611no—
349#461—611no—
350#461C2×D6111yescharacteristic
351#460—611no—
352#459C2×D6111yescharacteristic
353#450—21no—
354#450—1221no—
355#449—21no—
356C8×Q8—611no—
357C4×Q16—1221no—
358C4×SD16—611no—
359C4×D8C2×D6111yescharacteristic
360#436—611no—
361#436—611no—
362#436C2×D6111yescharacteristic
363#428—1221no—
364#418—1221no—
365#334—611no—
366#334—611no—
367#332—611no—
368#332—611no—
369#332—611no—
370#331—611no—
371Q8⋊C8—611no—
372D4⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375#322—611no—
376C61#2025011yescharacteristic, complemented, Sylow
377C2×#41—21no—
378C22×D4—21no—
379C22×D4—21no—
380AGL(1,ℤ/8)—44no—
381C2×Q16—1221no—
382C2×Q16—1221no—
383C2×SD16—21no—
384C2×SD16—21no—
385C2×SD16—21no—
386C2×SD16—611no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×D8—21no—
391C2×D8C22⋊D6111yescharacteristic
392C2×M16—21no—
393C2×M16—1221no—
394C2×M16—1221no—
395C4⋊Q8—611no—
396C4⋊Q8—611no—
397C4⋊Q8C22×D6111yescharacteristic
398#127C22×D6111yescharacteristic
399#126—2441no—
400#126—2441no—
401#124—1221no—
402#124—1221no—
403#123—41no—
404C22⋊Q8—21no—
405C22⋊Q8—1221no—
406C22⋊Q8—1221no—
407#121—21no—
408#121—21no—
409#121—21no—
410#120—41no—
411C4×Q8—611no—
412C4×Q8—1221no—
413C4×Q8—1221no—
414C4×D4C22×D6111yescharacteristic
415C4×D4C22×D6111yescharacteristic
416C4×D4C22×D6111yescharacteristic
417#117—1221no—
418#117—1221no—
419C2×C4⋊C4—21no—
420#107—1221no—
421#107—1221no—
422#107—1221no—
423#107—1221no—
424#107—2441no—
425#107C22⋊D6111yescharacteristic
426#106—611no—
427#106—1221no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—1221no—
432C4⋊C8C22×D6111yescharacteristic
433#103—21no—
434#103—611no—
435#103—1221no—
436#103—1221no—
437#103—1221no—
438#102—21no—
439#102—611no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—2441no—
446#102C22⋊D6111yescharacteristic
447#102C22⋊D6111yescharacteristic
448C22⋊C8—21no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451C4×C8—611no—
452C4×C8—611no—
453C4×C8—1221no—
454C4×C8C22×D6111yescharacteristic
455C24—41no—
456#41—42no—
457#41—81no—
458C2×Q8—21no—
459C2×Q8—611no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—41no—
470C2×D4—81no—
471C2×D4—81no—
472C2×D4D4×D6111yescharacteristic
473C2×D4C2×C22⋊D6111yescharacteristic
474C2×D4C2×C22⋊D6111yescharacteristic
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—41no—
479Q16—4881no—
480Q16—2442no—
481SD16—42no—
482SD16—42no—
483SD16—81no—
484SD16—4881no—
485SD16—4881no—
486SD16—4881no—
487SD16—2441no—
488D8—22no—
489D8—42no—
490M16—42no—
491M16—4881no—
492M16—2442no—
493C2×C8—21no—
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8C2×C22⋊D6111yescharacteristic
501C2×C8C2×C22⋊D6111yescharacteristic
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—611no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—2441no—
512C4⋊C4—2441no—
513C4⋊C4D4×D6111yescharacteristic
514C4⋊C4C2×C22⋊D6111yescharacteristic
515C4⋊C4C2×C22⋊D6111yescharacteristic
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—2441no—
522C22⋊C4—2441no—
523C22⋊C4—2441no—
524C22⋊C4—2441no—
525C42—1221no—
526C42—2441no—
527C42—2441no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—21no—
531C23—21no—
532C23—41no—
533C23—82no—
534C23—82no—
535C23—161no—
536Q8—41no—
537Q8—41no—
538Q8—1221no—
539Q8—1221no—
540Q8—2441no—
541Q8—2441no—
542Q8—2441no—
543Q8—2441no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—21no—
549D4—21no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—81no—
555D4—82no—
556D4—82no—
557D4—82no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—82no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4#1451982211yescharacteristic
576C8—21no—
577C8—41no—
578C8—1221no—
579C8—1221no—
580C8—2441no—
581C8—2441no—
582C8—2441no—
583C8—2441no—
584C8#1451980312yes—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—164no—
598C22D4×C22⋊D6111yescharacteristic
599C4—21no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—81no—
604C4—4881no—
605C4—4881no—
606C4—2441no—
607C4—2441no—
608C4C2×#1451980311yescharacteristic
609C4#1684055811yescharacteristic
610C4#1684056511yescharacteristic
611C4#1684057211yescharacteristic
612C2—81no—
613C2—81no—
614C2—81no—
615C2—161no—
616C2#2145733811yescharacteristic
617C2#2145743511yescharacteristic
618C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3272C2
socleC22×C61D42
hypercenter#3272D61
hypocenterC61#20250
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. #21458805C2semidirect
2. #21458403C2semidirect
3. #21458050C2semidirect
4. #21445488C2semidirect
5. #21445378C2semidirect
6. #21445330C2semidirect
7. #21445308C2semidirect
8. #21445164C2semidirect
9. #21445073C2semidirect
10. #21445054C2semidirect
11. #21444998C2semidirect
12. #21444927C2semidirect
13. #21444733C2semidirect
14. #21444660C2semidirect
17. #16840482C22semidirect
18. #16840165C22semidirect
19. #16840165C22semidirect
21. #16840142C22semidirect
27. #16840044C22semidirect
35. #16839967C22semidirect
38. C8×Dic122C22semidirect
42. #16839840C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839509C22semidirect
48. #16839509C22semidirect
49. #16839508C22semidirect
51. #16839492C22semidirect
54. D4⋊#92418C22semidirect
55. #16839483C22semidirect
376. C61#20250semidirect

Maximal subgroups

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

subgroupindex
#214588052
#214584032
#214580502
#214454882
#214453782
#214453302
#214453082
#214451642
#214450732
#214450542
#214449982
#214449272
#214447332
#214446602
C61×#32722
#2025061