Finite Group Databasebeta

← #29670346

Subgroups — #29670346

Order 15616.

660 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458502C211yesmaximal, characteristic, complemented
2#21458461C211yesmaximal, characteristic, complemented
3#21458050C211yesmaximal, characteristic
4#21445470C211yesmaximal, characteristic, complemented
5#21445368C211yesmaximal, characteristic, complemented
6#21445261C211yesmaximal, characteristic, complemented
7#21445242C211yesmaximal, characteristic, complemented
8#21445164C211yesmaximal, characteristic, complemented
9#21445073C211yesmaximal, characteristic, complemented
10#21445068C211yesmaximal, characteristic, complemented
11#21444998C211yesmaximal, characteristic
12#21444914C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15D4×SD16×C61C211yesmaximal, characteristic
16#16840491C2211yescharacteristic, complemented
17#16840482C2211yescharacteristic
18#16840165C2211yescharacteristic, complemented
19#16840115—21no—
20#16840114—21no—
21#16840114C2211yescharacteristic, complemented
22#16840113C2211yescharacteristic, complemented
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840083—21no—
26#16840072—21no—
27#16840044—21no—
28#16840044C2211yescharacteristic, complemented
29#16840042C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16840006—21no—
32#16839982—21no—
33#16839972C2211yescharacteristic, complemented
34#16839967—21no—
35#16839967C2211yescharacteristic
36#16839959C2211yescharacteristic
37#16839913—21no—
38Q8×#92418C2211yescharacteristic, complemented
39C8×Dic122C2211yescharacteristic, complemented
40#16839854—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839767—21no—
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C22×SD16×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#465—21no—
61C61×#461C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C61×#448—21no—
65C4×SD16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200D411yescharacteristic
71C2×#11758194—21no—
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519770—41no—
76#14519768—21no—
77#14519768—41no—
78#14519764—21no—
79#14519756—21no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519736—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—41no—
88#14519708—21no—
89#14519708D411yescharacteristic
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519706—21no—
95#14519702—21no—
96#14519702—21no—
97#14519700—21no—
98#14519700—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—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C22×Q8×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—44no—
123C2×SD16×C61D411yescharacteristic
124C22×C8×C61—21no—
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—41no—
130#121×C61—21no—
131#121×C61—21no—
132#120×C61—41no—
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—42no—
157C2×D4×C61—42no—
158C2×D4×C61—81no—
159C2×D4×C61—81no—
160C2×D4×C61C2×D411yescharacteristic
161C2×D4×C61C2×D411yescharacteristic
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—41no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122C2×D411yescharacteristic
168SD16×C61—22no—
169SD16×C61—42no—
170SD16×C61—42no—
171SD16×C61—42no—
172SD16×C61—81no—
173C2×C8×C61—21no—
174C2×C8×C61—42no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
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—41no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61C2411yescharacteristic
190C22⋊Dic61—41no—
191C22⋊Dic61—41no—
192#11758202—42no—
193#11758200—22no—
194#11758200—42no—
195#11758200—81no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205#11758196—41no—
206C4×Dic61—21no—
207C4×Dic61—41no—
208C4×Dic61—41no—
209#11758194—81no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—41no—
219C23×C61—82no—
220C23×C61—82no—
221C23×C61—161no—
222Q8×C61—21no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—82no—
226Q8×C61AGL(1,ℤ/8)12yes—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—81no—
235D4×C61—82no—
236D4×C61—82no—
237D4×C61#13512yes—
238C2×C4×C61—21no—
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—82no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255Dic122—21no—
256Dic122—21no—
257Dic122—41no—
258Dic122—41no—
259Dic122—41no—
260Dic122—41no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—41no—
264#92418—21no—
265#92418—21no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#19821—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61—164no—
281C22×C61D4211yescharacteristic
282C4×C61—22no—
283C4×C61—21no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61C2×#13511yescharacteristic
289C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
290C4×C61#57711yescharacteristic
291C4×C61#57711yescharacteristic
292Dic61—41no—
293Dic61—41no—
294Dic61—81no—
295Dic61—81no—
296#3364—611no—
297#3359—611no—
298#3324—611no—
299D4×SD16D6111yescharacteristic
300#1719—611no—
301#1699—611no—
302#1682—611no—
303#1679—611no—
304#1666—611no—
305#1650—611no—
306#1649—611no—
307#1639—611no—
308#1626—611no—
309#1583—611no—
310SD16⋊C8—611no—
311C2×C61—42no—
312C2×C61—81no—
313C2×C61—161no—
314C2×C61#326511yescharacteristic
315C2×C61#328211yescharacteristic
316C2×C61#328911yescharacteristic
317C22×SD16—21no—
318#557—611no—
319#556—611no—
320D4×Q8C2×D6111yescharacteristic
321D42C2×D6111yescharacteristic
322#500—611no—
323#492C2×D6111yescharacteristic
324#488—1221no—
325#487—611no—
326#487—611no—
327#487—1221no—
328#486—611no—
329#486—611no—
330#482—1221no—
331#479—1221no—
332#475—611no—
333#475—1221no—
334#474—611no—
335#474—611no—
336#468—1221no—
337#465—21no—
338#464—1221no—
339#462—611no—
340#461—611no—
341#461—1221no—
342#461C2×D6111yescharacteristic
343#460—611no—
344#460C2×D6111yescharacteristic
345#450—21no—
346#450—1221no—
347#448—21no—
348C8×Q8—611no—
349C8×Q8—611no—
350C4×Q16—1221no—
351C4×SD16—611no—
352C4×SD16—1221no—
353C4×SD16C2×D6111yescharacteristic
354#436—611no—
355C8×D4C2×D6111yescharacteristic
356#418—1221no—
357#335—611no—
358#334—611no—
359#332—611no—
360#332—611no—
361#332—611no—
362#331—611no—
363Q8⋊C8—611no—
364Q8⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8—611no—
367#322—611no—
368C61#1982111yescharacteristic, complemented, Sylow
369C22×Q8—21no—
370C22×D4—21no—
371C22×D4—21no—
372C2×Q16—1221no—
373C2×SD16—21no—
374C2×SD16—21no—
375C2×SD16—21no—
376C2×SD16—21no—
377C2×SD16—44no—
378C2×SD16—611no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×SD16C22⋊D6111yescharacteristic
382C2×M16—1221no—
383C22×C8—21no—
384C4⋊Q8—21no—
385C4⋊Q8—611no—
386C4⋊Q8—611no—
387C4⋊Q8—1221no—
388C4⋊Q8C22×D6111yescharacteristic
389#127C22×D6111yescharacteristic
390#126—2441no—
391#125—1221no—
392#125—2441no—
393#124—1221no—
394C22⋊Q8—21no—
395C22⋊Q8—41no—
396C22⋊Q8—1221no—
397#121—21no—
398#121—21no—
399#120—41no—
400C4×Q8—611no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8C22×D6111yescharacteristic
405C4×D4—21no—
406C4×D4C22×D6111yescharacteristic
407C4×D4C22×D6111yescharacteristic
408#117—1221no—
409#107—1221no—
410#107—1221no—
411#107—2441no—
412#106—611no—
413#106—1221no—
414#106C22⋊D6111yescharacteristic
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—1221no—
419C4⋊C8—1221no—
420C4⋊C8C22×D6111yescharacteristic
421#103—21no—
422#103—611no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#103C22⋊D6111yescharacteristic
431#102—21no—
432#102—611no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—2441no—
439#102C22⋊D6111yescharacteristic
440C22⋊C8—21no—
441C22⋊C8—1221no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—1221no—
445C4×C8—1221no—
446C4×C8C22×D6111yescharacteristic
447C24—41no—
448C2×Q8—21no—
449C2×Q8—41no—
450C2×Q8—42no—
451C2×Q8—81no—
452C2×Q8—611no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8C2×C22⋊D6111yescharacteristic
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4D4×D6111yescharacteristic
466C2×D4C2×C22⋊D6111yescharacteristic
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—41no—
470Q16—2442no—
471SD16—22no—
472SD16—42no—
473SD16—42no—
474SD16—42no—
475SD16—81no—
476SD16—1222no—
477SD16—4881no—
478SD16—2442no—
479M16—4881no—
480C2×C8—21no—
481C2×C8—42no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8C2×C22⋊D6111yescharacteristic
489C2×C8C2×C22⋊D6111yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—611no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—2441no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4D4×D6111yescharacteristic
505C4⋊C4C2×C22⋊D6111yescharacteristic
506C4⋊C4C2×C22⋊D6111yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C42—21no—
514C42—1221no—
515C42—2441no—
516C42—2441no—
517C42C23×D6111yescharacteristic
518C23—21no—
519C23—21no—
520C23—41no—
521C23—82no—
522C23—82no—
523C23—161no—
524Q8—21no—
525Q8—41no—
526Q8—41no—
527Q8—82no—
528Q8—1221no—
529Q8—1221no—
530Q8—2441no—
531Q8—2441no—
532Q8—2441no—
533Q8—2441no—
534Q8#1451980312yes—
535D4—21no—
536D4—21no—
537D4—21no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—81no—
543D4—82no—
544D4—82no—
545D4#1451979812yes—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—82no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C22×C22⋊D6111yescharacteristic
562C2×C4#1451982211yescharacteristic
563C8—21no—
564C8—21no—
565C8—41no—
566C8—1221no—
567C8—1221no—
568C8—2441no—
569C8—2441no—
570C8—2441no—
571C8—2441no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—164no—
582C22D4×C22⋊D6111yescharacteristic
583C4—22no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—4881no—
590C4—4881no—
591C4—2441no—
592C4—2441no—
593C4C2×#1451979811yescharacteristic
594C4C2×#1451980311yescharacteristic
595C4#1684056811yescharacteristic
596C4#1684057311yescharacteristic
597C2—42no—
598C2—81no—
599C2—161no—
600C2#2145731011yescharacteristic
601C2#2145752611yescharacteristic
602C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×SD16×C61C2
socleC22×C61D42
prime radicalD42×C61C22
hypercenterD4×SD16D61
hypocenterC61#19821
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×SD16

Complements

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

NH ≅ G/Ntype
1. #21458502C2semidirect
2. #21458461C2semidirect
4. #21445470C2semidirect
5. #21445368C2semidirect
6. #21445261C2semidirect
7. #21445242C2semidirect
8. #21445164C2semidirect
9. #21445073C2semidirect
10. #21445068C2semidirect
12. #21444914C2semidirect
13. #21444663C2semidirect
14. #21444631C2semidirect
16. #16840491C22semidirect
18. #16840165C22semidirect
21. #16840114C22semidirect
22. #16840113C22semidirect
28. #16840044C22semidirect
33. #16839972C22semidirect
38. Q8×#92418C22semidirect
39. C8×Dic122C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. #16839484C22semidirect
368. C61#19821semidirect

Maximal subgroups

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

subgroupindex
#214585022
#214584612
#214580502
#214454702
#214453682
#214452612
#214452422
#214451642
#214450732
#214450682
#214449982
#214449142
#214446632
#214446312
D4×SD16×C612
#1982161