Finite Group Databasebeta

← #29656871

Subgroups — #29656871

Order 15616.

1224 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#21456810C211yesmaximal, characteristic, complemented
2#21456553C211yesmaximal, characteristic, complemented
3#21455819C211yesmaximal, characteristic, complemented
4#21445103C212yesmaximal, complemented
5#21444998C212yesmaximal, complemented
6#21444908C212yesmaximal, complemented
7#21444873C212yesmaximal, complemented
8#21444524C211yesmaximal, characteristic, complemented
9#21444262C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic, complemented
11C61×#3496C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic, complemented
13#16840158C2211yescharacteristic, complemented
14#16840149—21no—
15#16840138C2211yescharacteristic, complemented
16#16840124—21no—
17#16840110C2211yescharacteristic, complemented
18#16840108C2211yescharacteristic, complemented
19#16840075—21no—
20#16840042C2212yescomplemented
21#16840013—22no—
22#16839967C2212yescomplemented
23#16839957C2212yescomplemented
24#16839957C2212yescomplemented
25#16839905—22no—
26#16839896—21no—
27#16839872—22no—
28C4×#11758199—22no—
29#16839819C2211yescharacteristic, complemented
30#16839815C2211yescharacteristic, complemented
31C4×#11758194C2211yescharacteristic, complemented
32#16839508C2212yescomplemented
33#16839506C2212yescomplemented
34D4⋊#92418C2212yescomplemented
35D4⋊#92418C2212yescomplemented
36D4⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38C2×H(5,𝔽2)×C61—21no—
39C61×#547—21no—
40C61×#547C2212yes—
41C61×#546—22no—
42C61×#546—22no—
43D42×C61—21no—
44D42×C61C2212yes—
45C61×#537—21no—
46C61×#534—22no—
47C61×#532C2211yescharacteristic
48C61×#532C2211yescharacteristic
49C2×#121×C61—22no—
50C2×C4×D4×C61C2211yescharacteristic
51C2×#11758222—21no—
52C2×#11758200—22no—
53C2×#11758199—22no—
54C2×#11758199—22no—
55C2×#11758199—22no—
56C2×#11758194—21no—
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519765—21no—
60#14519751—21no—
61#14519751—21no—
62C4×D244—21no—
63C4×D244—21no—
64#14519710—22no—
65#14519708—22no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519700—22no—
69#14519698—22no—
70#14519698—22no—
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74C22⋊#92418—21no—
75C22⋊#92418—21no—
76#14519688—21no—
77#14519688—21no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81H(5,𝔽2)×C61—48no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61—21no—
85C2×#41×C61—21no—
86C22×D4×C61—24no—
87C22×D4×C61—22no—
88C22×D4×C61—22no—
89C22×D4×C61—22no—
90C22×D4×C61D411yescharacteristic
91C23×C4×C61—21no—
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61C2311yescharacteristic
94#127×C61D411yescharacteristic
95#127×C61C2311yescharacteristic
96#124×C61—21no—
97#124×C61—21no—
98#123×C61—24no—
99#123×C61—24no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102#121×C61—24no—
103#121×C61—22no—
104#121×C61—24no—
105#121×C61—22no—
106#121×C61—21no—
107#121×C61—21no—
108#121×C61—22no—
109#121×C61—22no—
110#121×C61D412yes—
111#121×C61D412yes—
112#120×C61—24no—
113#120×C61—24no—
114C4×D4×C61—21no—
115C4×D4×C61—21no—
116C4×D4×C61—21no—
117C4×D4×C61—21no—
118C4×D4×C61D412yes—
119C4×D4×C61D412yes—
120C4×D4×C61C2312yes—
121C4×D4×C61C2312yes—
122C2×C4⋊C4×C61—22no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C22⋊C4×C61—24no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61C2311yescharacteristic
127C24×C61—21no—
128C24×C61—24no—
129#41×C61—44no—
130#41×C61—44no—
131#41×C61—48no—
132#41×C61—44no—
133#41×C61—44no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×D4×C61—24no—
137C2×D4×C61—24no—
138C2×D4×C61—24no—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61—24no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—48no—
148C2×D4×C61—44no—
149C2×D4×C61—416no—
150C2×D4×C61—44no—
151C2×D4×C61—82no—
152C2×D4×C61—82no—
153C2×D4×C61C2×D412yes—
154C2×D4×C61C2×D412yes—
155C2×D4×C61C2×D412yes—
156C22×C4×C61—24no—
157C22×C4×C61—21no—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61—82no—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C2×C22⋊D61—41no—
167#11758222—82no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×C4×D61—41no—
172C2×Dic122—21no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C2×D412yes—
178C4⋊C4×C61C2×D412yes—
179C4⋊C4×C61C2×D412yes—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2411yescharacteristic
192#11758200—82no—
193#11758199—44no—
194#11758199—44no—
195#11758199—82no—
196#11758198—41no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200#11758194—42no—
201C2×#92418—21no—
202C2×#92418—22no—
203C2×#92418—21no—
204C2×#92418—22no—
205C23×C61—24no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—48no—
210C23×C61—81no—
211C23×C61—81no—
212C23×C61—84no—
213C23×C61—84no—
214C23×C61#12011yescharacteristic
215C22×D61—41no—
216Q8×C61—42no—
217Q8×C61—42no—
218D4×C61—24no—
219D4×C61—22no—
220D4×C61—22no—
221D4×C61—22no—
222D4×C61—42no—
223D4×C61—42no—
224D4×C61—44no—
225D4×C61—44no—
226D4×C61—48no—
227D4×C61—48no—
228D4×C61—42no—
229D4×C61—42no—
230D4×C61—81no—
231D4×C61—84no—
232D4×C61AGL(1,ℤ/8)14yes—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—44no—
240C2×C4×C61—48no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—82no—
246C2×C4×C61—81no—
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61#12011yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C22⋊D61—82no—
254C2×Dic61—41no—
255D244—42no—
256D244—42no—
257D244—42no—
258C4×D61—82no—
259Dic122—42no—
260#92418—22no—
261#92418—41no—
262#92418—42no—
263#92418—42no—
264#18598—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—24no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—84no—
278C22×C61—168no—
279C22×C61C2×#12011yescharacteristic
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—42no—
288C4×C61—42no—
289C4×C61—42no—
290C4×C61C2×AGL(1,ℤ/8)12yes—
291C4×C61#57612yes—
292Dic61—81no—
293#3496D6111yescharacteristic, complemented
294#3214—611no—
295#3184—611no—
296#3100—611no—
297#1654—612no—
298#1639—612no—
299#1625—612no—
300#1619—612no—
301#1553—611no—
302#1490—611no—
303#1484—611no—
304C2×C61—41no—
305C2×C61—44no—
306C2×C61—82no—
307C2×C61—82no—
308C2×C61#300311yescharacteristic
309C2×C61#300811yescharacteristic
310C2×C61#301811yescharacteristic
311D61—161no—
312C2×H(5,𝔽2)—21no—
313#547—21no—
314#547C2×D6112yes—
315#546—22no—
316#546—22no—
317D42—21no—
318D42C2×D6112yes—
319#537—21no—
320#534—22no—
321#532—611no—
322#532C2×D6111yescharacteristic
323#532C2×D6111yescharacteristic
324C2×#121—22no—
325C2×C4×D4C2×D6111yescharacteristic
326#498—611no—
327#496—1221no—
328#493—611no—
329#489—1221no—
330#486—611no—
331#486—611no—
332#480—1221no—
333#474—612no—
334#469—1222no—
335#461—612no—
336#459—612no—
337#459—612no—
338#449—1222no—
339#447—1221no—
340#442—1222no—
341C4×D8—1222no—
342#432—611no—
343#431—611no—
344C4×M16—611no—
345#332—612no—
346#331—612no—
347D4⋊C8—612no—
348D4⋊C8—612no—
349D4⋊C8—612no—
350D4⋊C8—612no—
351C61#1859811yescharacteristic, complemented, Sylow
352H(5,𝔽2)—48no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41—21no—
356C2×#41—21no—
357C2×#41—1221no—
358C22×D4—24no—
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4C22⋊D6111yescharacteristic
363C23×C4—21no—
364C2×SD16—1222no—
365C2×D8—1222no—
366C2×D8—1222no—
367C2×D8—1222no—
368C2×M16—1221no—
369C4⋊Q8—611no—
370C4⋊Q8C22⋊D6111yescharacteristic
371C4⋊Q8C22×D6111yescharacteristic
372#127—611no—
373#127C22⋊D6111yescharacteristic
374#127C22×D6111yescharacteristic
375#124—21no—
376#124—21no—
377#124—1221no—
378#123—24no—
379#123—24no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382#121—24no—
383#121—24no—
384#121—22no—
385#121—22no—
386#121—21no—
387#121—21no—
388#121—22no—
389#121—22no—
390#121—1221no—
391#121—1221no—
392#121C22⋊D6112yes—
393#121C22⋊D6112yes—
394#120—24no—
395#120—24no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4—21no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4C22⋊D6112yes—
403C4×D4C22⋊D6112yes—
404C4×D4C22×D6112yes—
405C4×D4C22×D6112yes—
406C2×C4⋊C4—22no—
407C2×C4⋊C4C22⋊D6111yescharacteristic
408C2×C22⋊C4—24no—
409C2×C22⋊C4—21no—
410C2×C42C22×D6111yescharacteristic
411#107—1222no—
412#106—1222no—
413C4⋊C8—612no—
414C4⋊C8—612no—
415#103—1222no—
416#102—1222no—
417#102—1222no—
418#102—1222no—
419#102—1222no—
420#102—1222no—
421C22⋊C8—1221no—
422C22⋊C8—1221no—
423#97—1221no—
424#97—1221no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—1221no—
428C24—21no—
429C24—24no—
430#41—44no—
431#41—44no—
432#41—48no—
433#41—44no—
434#41—44no—
435#41—4882no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—1221no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—48no—
451C2×D4—44no—
452C2×D4—416no—
453C2×D4—44no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—2441no—
460C2×D4C2×C22⋊D6112yes—
461C2×D4C2×C22⋊D6112yes—
462C2×D4C2×C22⋊D6112yes—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—82no—
470C22×C4—2441no—
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474SD16—4882no—
475D8—4882no—
476D8—2444no—
477D8—2444no—
478M16—2442no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1222no—
482C2×C8—1222no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4C2×C22⋊D6112yes—
490C4⋊C4C2×C22⋊D6112yes—
491C4⋊C4C2×C22⋊D6112yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C42C2×C22⋊D6111yescharacteristic
503C42C2×C22⋊D6111yescharacteristic
504C42C2×C22⋊D6111yescharacteristic
505C42C23×D6111yescharacteristic
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—48no—
511C23—81no—
512C23—81no—
513C23—84no—
514C23—84no—
515C23—2441no—
516C23#1451974611yescharacteristic
517Q8—42no—
518Q8—42no—
519Q8—2442no—
520D4—24no—
521D4—22no—
522D4—22no—
523D4—22no—
524D4—42no—
525D4—42no—
526D4—44no—
527D4—44no—
528D4—48no—
529D4—48no—
530D4—42no—
531D4—42no—
532D4—81no—
533D4—84no—
534D4—4882no—
535D4—2442no—
536D4—2442no—
537D4—2442no—
538D4#1451980314yes—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—4882no—
554C2×C4—2441no—
555C2×C4#1451974611yescharacteristic
556C2×C4#1451974611yescharacteristic
557C2×C4#1451974611yescharacteristic
558C2×C4C22×C22⋊D6111yescharacteristic
559C2×C4C22×C22⋊D6111yescharacteristic
560C2×C4C22×C22⋊D6111yescharacteristic
561C8—1222no—
562C8—2441no—
563C8—2442no—
564C8—2442no—
565C22—21no—
566C22—21no—
567C22—24no—
568C22—21no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—84no—
578C22—168no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22C2×#1451974611yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—4881no—
591C4C2×#1451980312yes—
592C4#1684056512yes—
593C2—41no—
594C2—44no—
595C2—82no—
596C2—82no—
597C2—9761no—
598C2#2145504611yescharacteristic
599C2#2145508711yescharacteristic
600C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18598
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #21456810C2semidirect
2. #21456553C2semidirect
3. #21455819C2semidirect
4. #21445103C2semidirect
5. #21444998C2semidirect
6. #21444908C2semidirect
7. #21444873C2semidirect
8. #21444524C2semidirect
9. #21444262C2semidirect
10. #21444236C2semidirect
11. C61×#3496C2semidirect
12. #16840312C22semidirect
13. #16840158C22semidirect
15. #16840138C22semidirect
17. #16840110C22semidirect
18. #16840108C22semidirect
20. #16840042C22semidirect
22. #16839967C22semidirect
23. #16839957C22semidirect
24. #16839957C22semidirect
29. #16839819C22semidirect
30. #16839815C22semidirect
31. C4×#11758194C22semidirect
32. #16839508C22semidirect
33. #16839506C22semidirect
34. D4⋊#92418C22semidirect
35. D4⋊#92418C22semidirect
36. D4⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
293. #3496D61semidirect
351. C61#18598semidirect

Maximal subgroups

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

subgroupindex
#214568102
#214565532
#214558192
#214451032
#214449982
#214449082
#214448732
#214445242
#214442622
#214442362
C61×#34962
#1859861