Finite Group Databasebeta

← #30547696

Subgroups — #30547696

Order 15616.

2054 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459641C211yesmaximal, characteristic, complemented
2#21459025C211yesmaximal, characteristic
3#21455680C213yesmaximal, complemented
4#21455675C213yesmaximal, complemented
5#21455653C213yesmaximal, complemented
6#21455648C213yesmaximal, complemented
7#21455640C213yesmaximal, complemented
8#21455615C213yesmaximal, complemented
9#21455605C213yesmaximal, complemented
10C2×#16840065C213yesmaximal, complemented
11#21454412C211yesmaximal, characteristic, complemented
12#21454385C211yesmaximal, characteristic, complemented
13#21454346C211yesmaximal, characteristic
14#21454342C211yesmaximal, characteristic
15C61×#2913C211yesmaximal, characteristic, complemented
16C2×#14519823C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16840491—21no—
19#16840488—21no—
20#16840471—23no—
21#16840469—23no—
22#16840393C2213yescomplemented
23#16840390C2213yescomplemented
24#16840315C2213yescomplemented
25C2×#14519768C2213yescomplemented
26#16840245C2211yescharacteristic
27#16840244—21no—
28#16840243C2213yes—
29#16840236C2213yes—
30#16840234—23no—
31#41×Dic61C2211yescharacteristic
32C4×#11758226—21no—
33#16840230C2213yes—
34C2×#14519739C2213yes—
35C4×D4×D61—23no—
36#16840072C2213yescomplemented
37#16840068C2216yescomplemented
38#16840068C2211yescharacteristic, complemented
39#16840065C2216yescomplemented
40#16840065C2216yescomplemented
41#16840065C2216yescomplemented
42#16840065C2213yescomplemented
43#16840061C2216yescomplemented
44#16840061C2213yescomplemented
45#16840056C2213yescomplemented
46#16840051C2213yescomplemented
47#16840051C2216yescomplemented
48#16840046C2213yescomplemented
49#16840041C2216yescomplemented
50#16840041C2213yescomplemented
51#16839805C2213yescomplemented
52#16839802C2213yescomplemented
53#16839798C2213yescomplemented
54C2×#14519708C2213yescomplemented
55#16839790C2213yescomplemented
56#16839788C2213yescomplemented
57#16839785C2213yescomplemented
58C2×#14519706C2213yescomplemented
59#16839768C2213yes—
60#16839766C2213yes—
61#16839764C2213yes—
62#16839761C2213yes—
63#16839759C2213yes—
64#16839756C2213yes—
65C2×#14519699C2213yes—
66C2×#14519697C2213yes—
67C2×#131×C61C2211yescharacteristic
68C61×#520C2211yescharacteristic
69C61×#520C2211yescharacteristic
70C61×#428C2213yescomplemented
71C61×#428C2213yescomplemented
72C61×#427C2213yescomplemented
73C2×#107×C61C2213yescomplemented
74#14519823—24no—
75#14519823Q818yescomplemented
76C2×#11758226—21no—
77C2×#11758222C2313yes—
78C22×D244C2313yes—
79C2×D4×D61—23no—
80C22×C4×D61—23no—
81#14519784—24no—
82C2×#11758194C2313yes—
83C22×#92418C2313yes—
84#14519777—23no—
85#14519776—23no—
86#14519776C2313yes—
87#14519776C2313yes—
88#14519769—23no—
89#14519768—23no—
90#14519768—23no—
91#14519768—23no—
92#14519768C2313yes—
93#14519768C2316yes—
94#14519768C2311yescharacteristic
95#14519756—26no—
96#14519754—23no—
97#14519754—23no—
98#14519754—26no—
99Q8×Dic61—21no—
100Q8×Dic61D411yescharacteristic
101Q8×Dic61D411yescharacteristic
102C4×Dic122—23no—
103C4×Dic122—23no—
104#14519742C2311yescharacteristic
105#14519742C2313yes—
106C4×D244—23no—
107C4×D244—23no—
108D4×Dic61—23no—
109D4×Dic61D413yes—
110D4×Dic61D413yes—
111#14519739—23no—
112#14519739—23no—
113#14519739C2313yes—
114#14519739C2316yes—
115#14519739C2313yes—
116C4×C22⋊D61—26no—
117#14519737—23no—
118#14519737—23no—
119#14519736C2313yes—
120#14519736C2313yes—
121#14519735—23no—
122#14519734—23no—
123C2×C4⋊Dic61—23no—
124C2×C4⋊Dic61C2313yes—
125C2×#11758196—23no—
126C4⋊C4×D61—23no—
127C2×#11758198—23no—
128C22⋊C4×D61—26no—
129C2×C4×Dic61C2313yes—
130C42×D61—23no—
131#14519710C2313yes—
132#14519710C2313yes—
133#14519708C2313yes—
134#14519708C2316yes—
135#14519708C2311yescharacteristic
136#14519706C2311yescharacteristic
137#14519706C2313yes—
138#14519706C2313yes—
139#14519706C2313yes—
140#14519706C2316yes—
141#14519702C2311yescharacteristic
142#14519702C2313yes—
143#14519700C2311yescharacteristic
144#14519700C2313yes—
145#14519699C2313yes—
146#14519699C2316yes—
147#14519699C2313yes—
148#14519697C2313yes—
149#14519697C2316yes—
150#14519697C2313yes—
151C2×#41×C61C2311yescharacteristic
152#131×C61—24no—
153C2×M16×C61C2313yes—
154C22×C8×C61C2313yes—
155C4×Q8×C61—21no—
156C4×Q8×C61—21no—
157C4×D4×C61—23no—
158C4×D4×C61—23no—
159#117×C61C2313yes—
160#117×C61C2313yes—
161C2×C4⋊C4×C61C2313yes—
162C2×C4⋊C4×C61C2313yes—
163#107×C61C2311yescharacteristic
164#107×C61C2316yes—
165#107×C61C2313yes—
166#106×C61C2313yes—
167#106×C61C2313yes—
168C23×D61—23no—
169#41×C61C2×Q814yes—
170#41×C61#4114yes—
171C2×Q8×C61C2×D411yescharacteristic
172C2×D4×C61C2×D413yes—
173C22×C4×C61—23no—
174C22×C4×C61—23no—
175C22×C4×C61C2413yes—
176C2×C22⋊D61—23no—
177C22×Dic61—23no—
178C22×Dic61C2×D413yes—
179C22×Dic61C2×D413yes—
180#11758226—28no—
181D4×D61—224no—
182#11758222—26no—
183#11758222C2×Q8112yes—
184C2×D244—23no—
185C2×D244—26no—
186C2×D244C2×Q8112yes—
187C2×D244C2413yes—
188C2×C4×D61—23no—
189C2×C4×D61—23no—
190C2×C4×D61—26no—
191C2×C4×D61—23no—
192C2×C4×D61—86no—
193C2×Dic122—23no—
194C2×Dic122—23no—
195C2×Dic122C2411yescharacteristic
196M16×C61—26no—
197C2×C8×C61—26no—
198C2×C8×C61C2413yes—
199C2×C8×C61C2411yescharacteristic
200C4⋊C4×C61—23no—
201C4⋊C4×C61—23no—
202C4⋊C4×C61C2411yescharacteristic
203C4⋊C4×C61C2411yescharacteristic
204C4⋊C4×C61C2413yes—
205C4⋊C4×C61C2413yes—
206C22⋊C4×C61—23no—
207C22⋊C4×C61—23no—
208C42×C61—23no—
209C42×C61—23no—
210C22⋊Dic61—23no—
211C22⋊Dic61C2×D416yes—
212#11758198—23no—
213#11758198—26no—
214#11758198—23no—
215C4⋊Dic61—23no—
216C4⋊Dic61—23no—
217C4⋊Dic61—26no—
218C4⋊Dic61—21no—
219C4⋊Dic61C2×D413yes—
220C4⋊Dic61C2×D413yes—
221C4⋊Dic61C2413yes—
222C4⋊Dic61C2411yescharacteristic
223#11758196—23no—
224#11758196—23no—
225#11758196—26no—
226#11758196—21no—
227#11758196—21no—
228#11758196—23no—
229#11758196—23no—
230C4×Dic61—23no—
231C4×Dic61—21no—
232C4×Dic61—23no—
233C4×Dic61C2×D413yes—
234C4×Dic61C2×D413yes—
235C4×Dic61C2411yescharacteristic
236C4×Dic61C2413yes—
237#11758194—26no—
238C2×#92418—26no—
239C2×#92418C2411yescharacteristic
240C2×#92418C2413yes—
241C23×C61C22×D413yes—
242C22×D61—23no—
243C22×D61—212no—
244C22×D61—83no—
245C22×D61—83no—
246Q8×C61C22⋊Q814yes—
247D4×C61C22⋊Q8112yes—
248C2×C4×C61—21no—
249C2×C4×C61—23no—
250C2×C4×C61—23no—
251C2×C4×C61—21no—
252C2×C4×C61—83no—
253C2×C4×C61—83no—
254C2×C4×C61C22×D413yes—
255C2×C4×C61C22×Q816yes—
256C2×C4×C61C2×#4116yes—
257C2×C4×C61C2511yescharacteristic
258C22⋊D61—212no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—23no—
262C2×Dic61—43no—
263C2×Dic61—43no—
264C2×Dic61—83no—
265C2×Dic61C22×D411yescharacteristic
266C2×Dic61C22×D413yes—
267C2×Dic61C22×D411yescharacteristic
268C2×Dic61C22×D413yes—
269D244—212no—
270D244—23no—
271D244C22×Q816yes—
272C4×D61—212no—
273C4×D61—46no—
274C4×D61—83no—
275C4×D61—86no—
276Dic122—21no—
277Dic122—83no—
278Dic122—83no—
279Dic122C22×Q812yes—
280C8×C61—21no—
281C8×C61—23no—
282#92418—21no—
283#92418—23no—
284#60496—611nomaximal, Sylow
285C22×C61—23no—
286C22×C61—23no—
287C22×C61C2×C22⋊Q816yes—
288C22×C61C23×D411yescharacteristic
289C2×D61—26no—
290C2×D61—43no—
291C2×D61—43no—
292C2×D61—812no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—43no—
296C4×C61—43no—
297C4×C61C2×C22⋊Q816yes—
298C4×C61C23×Q811yescharacteristic
299C4×C61C22×#4111yescharacteristic
300Dic61—23no—
301Dic61—21no—
302Dic61—23no—
303Dic61—21no—
304Dic61—22no—
305Dic61—43no—
306Dic61—41no—
307#3504—611no—
308C4×H(5,𝔽2)—611no—
309#3084—613no—
310#3083—613no—
311#3081—613no—
312#3081—613no—
313#3080—613no—
314#3077—613no—
315#3076—613no—
316C2×#478—613no—
317#2913D6111yescharacteristic, complemented
318#2912—611no—
319#2906—611no—
320#2897—611no—
321#2897—611no—
322C2×C61—26no—
323C2×C61C22×C22⋊Q811yescharacteristic
324C2×C61#358411yescharacteristic
325C2×C61#358511yescharacteristic
326D61—46no—
327C2×H(5,𝔽2)—611no—
328C2×#131—611no—
329C2×#131C2×D6111yescharacteristic
330#557—1221no—
331#557—1221no—
332#554—1223no—
333#554—1223no—
334#544—613no—
335#544—613no—
336#533—613no—
337C2×#125—613no—
338#520—611no—
339#520—613no—
340#520—1221no—
341#520C2×D6111yescharacteristic
342#520C2×D6111yescharacteristic
343#518—613no—
344#518—1223no—
345C4×#41—611no—
346C4×#41—613no—
347C4×#41—1221no—
348C2×C4×D4—613no—
349C2×C4×D4—1223no—
350#479—611no—
351#479—616no—
352#479—613no—
353#478—613no—
354#478—616no—
355#478—616no—
356#478—616no—
357#478—613no—
358#478—616no—
359#477—613no—
360#476—613no—
361#476—616no—
362#475—613no—
363#474—613no—
364#474—616no—
365#428—613no—
366#428—613no—
367#428C2×D6113yescomplemented
368#428C2×D6113yescomplemented
369#427—613no—
370#427C2×D6113yescomplemented
371C2×#107C2×D6113yescomplemented
372C2×#106—613no—
373#424—613no—
374#424—613no—
375#423—613no—
376C2×C4⋊C8—613no—
377#418—613no—
378#418—613no—
379#417—613no—
380#417—613no—
381#416—613no—
382#416—613no—
383C2×#102—613no—
384C2×#102—613no—
385C61#6049611yescharacteristic, complemented, Sylow
386H(5,𝔽2)—618no—
387H(5,𝔽2)—1224no—
388C2×#41—613no—
389C2×#41—1221no—
390C2×#41C22×D6111yescharacteristic
391C22×D4—613no—
392C22×D4—1223no—
393C23×C4—1223no—
394#131—24no—
395#131—1224no—
396C2×M16—613no—
397C2×M16C22×D6113yes—
398C22×C8—613no—
399C22×C8C22×D6113yes—
400C4⋊Q8—613no—
401C4⋊Q8—613no—
402C4⋊Q8—1223no—
403C4⋊Q8—1223no—
404#125—613no—
405#125—616no—
406#125—611no—
407#125—1223no—
408#125—1223no—
409#125—1223no—
410#125—1223no—
411C22⋊Q8—1226no—
412C22⋊Q8—1223no—
413C22⋊Q8—1223no—
414C22⋊Q8—1226no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—611no—
418C4×Q8—611no—
419C4×Q8—611no—
420C4×Q8—613no—
421C4×Q8—1223no—
422C4×Q8—1223no—
423C4×Q8—1221no—
424C4×D4—23no—
425C4×D4—23no—
426C4×D4—613no—
427C4×D4—613no—
428C4×D4—613no—
429C4×D4—616no—
430C4×D4—613no—
431C4×D4—1223no—
432C4×D4—1223no—
433C4×D4—1223no—
434C4×D4—1223no—
435C4×D4—1223no—
436C4×D4—1223no—
437C4×D4—1226no—
438C4×D4—1223no—
439#117—613no—
440#117—613no—
441#117—1223no—
442#117—1223no—
443#117C22×D6113yes—
444#117C22×D6113yes—
445C2×C4⋊C4—613no—
446C2×C4⋊C4—1223no—
447C2×C4⋊C4—1223no—
448C2×C4⋊C4—1223no—
449C2×C4⋊C4C22×D6113yes—
450C2×C4⋊C4C22×D6113yes—
451C2×C22⋊C4—1226no—
452C2×C22⋊C4—1223no—
453C2×C42—613no—
454C2×C42—1223no—
455#107—613no—
456#107—613no—
457#107C22×D6113yes—
458#107C22×D6116yes—
459#107C22×D6111yescharacteristic
460#106—613no—
461#106—616no—
462#106—611no—
463#106C22×D6113yes—
464#106C22×D6113yes—
465C4⋊C8—611no—
466C4⋊C8—613no—
467C4⋊C8—613no—
468C4⋊C8—613no—
469C4⋊C8—616no—
470#103—611no—
471#103—613no—
472#103—611no—
473#103—613no—
474#102—613no—
475#102—616no—
476#102—613no—
477#102—613no—
478#102—616no—
479#102—613no—
480C24—1223no—
481#41—6112no—
482#41—1228no—
483#41—1226no—
484#41#1175822214yes—
485#41Q8×D6114yes—
486C2×Q8—611no—
487C2×Q8—1223no—
488C2×Q8—1223no—
489C2×Q8D4×D6111yescharacteristic
490C2×D4—6112no—
491C2×D4—613no—
492C2×D4—1223no—
493C2×D4—12224no—
494C2×D4—1223no—
495C2×D4—1226no—
496C2×D4D4×D6113yes—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—613no—
500C22×C4—613no—
501C22×C4—1223no—
502C22×C4—1226no—
503C22×C4—1223no—
504C22×C4—1223no—
505C22×C4—1223no—
506C22×C4—4886no—
507C22×C4C23×D6113yes—
508M16—26no—
509M16—1226no—
510C2×C8—26no—
511C2×C8—611no—
512C2×C8—613no—
513C2×C8—1226no—
514C2×C8C23×D6113yes—
515C2×C8C23×D6111yescharacteristic
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—613no—
519C4⋊C4—613no—
520C4⋊C4—613no—
521C4⋊C4—611no—
522C4⋊C4—1223no—
523C4⋊C4—1223no—
524C4⋊C4—1223no—
525C4⋊C4—1226no—
526C4⋊C4—1226no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1223no—
530C4⋊C4—1223no—
531C4⋊C4—1221no—
532C4⋊C4—1223no—
533C4⋊C4C23×D6111yescharacteristic
534C4⋊C4C23×D6113yes—
535C4⋊C4C23×D6113yes—
536C4⋊C4C23×D6111yescharacteristic
537C22⋊C4—23no—
538C22⋊C4—23no—
539C22⋊C4—616no—
540C22⋊C4—1223no—
541C22⋊C4—1223no—
542C22⋊C4—1226no—
543C22⋊C4—1223no—
544C42—23no—
545C42—23no—
546C42—613no—
547C42—613no—
548C42—611no—
549C42—613no—
550C42—1221no—
551C42—1223no—
552C42—1223no—
553C23—12212no—
554C23—1223no—
555C23—4883no—
556C23—4883no—
557C23C2×D4×D6113yes—
558Q8—612no—
559Q8—1221no—
560Q8—4883no—
561Q8—4883no—
562Q8#1451975514yes—
563D4—616no—
564D4—12212no—
565D4—12212no—
566D4—1223no—
567D4#14519755112yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—23no—
571C2×C4—23no—
572C2×C4—83no—
573C2×C4—83no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—613no—
577C2×C4—613no—
578C2×C4—1221no—
579C2×C4—12212no—
580C2×C4—1221no—
581C2×C4—1223no—
582C2×C4—4883no—
583C2×C4—4886no—
584C2×C4—4883no—
585C2×C4—2446no—
586C2×C4—2443no—
587C2×C4—2443no—
588C2×C4C2×D4×D6113yes—
589C2×C4C2×Q8×D6116yes—
590C2×C4C2×#1175822216yes—
591C2×C4C24×D6111yescharacteristic
592C8—21no—
593C8—23no—
594C8—1221no—
595C8—1223no—
596C22—23no—
597C22—23no—
598C22—1226no—
599C22—48812no—
600C22—2443no—
601C22—2443no—
602C22C2×#1451975516yes—
603C22C22×D4×D6111yescharacteristic
604C4—41no—
605C4—41no—
606C4—43no—
607C4—43no—
608C4—1222no—
609C4—1223no—
610C4—1221no—
611C4—1223no—
612C4—1221no—
613C4—2443no—
614C4—2441no—
615C4C2×#1451975516yes—
616C4C22×Q8×D6111yescharacteristic
617C4C22×#1175822211yescharacteristic
618C2—26no—
619C2—2446no—
620C2C22×#1451975511yescharacteristic
621C2#2146058911yescharacteristic
622C2#2146059411yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2913C2
socleC22×C61C23×D4
prime radicalC2×#14519823C22
hypercenter#2913D61
hypocenterC61#60496
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2913

Complements

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

NH ≅ G/Ntype
1. #21459641C2semidirect
3. #21455680C2semidirect
4. #21455675C2semidirect
5. #21455653C2semidirect
6. #21455648C2semidirect
7. #21455640C2semidirect
8. #21455615C2semidirect
9. #21455605C2semidirect
10. C2×#16840065C2semidirect
11. #21454412C2semidirect
12. #21454385C2semidirect
15. C61×#2913C2semidirect
22. #16840393C22semidirect
23. #16840390C22semidirect
24. #16840315C22semidirect
25. C2×#14519768C22semidirect
36. #16840072C22semidirect
37. #16840068C22semidirect
38. #16840068C22semidirect
39. #16840065C22semidirect
40. #16840065C22semidirect
41. #16840065C22semidirect
42. #16840065C22semidirect
43. #16840061C22semidirect
44. #16840061C22semidirect
45. #16840056C22semidirect
46. #16840051C22semidirect
47. #16840051C22semidirect
48. #16840046C22semidirect
49. #16840041C22semidirect
50. #16840041C22semidirect
51. #16839805C22semidirect
52. #16839802C22semidirect
53. #16839798C22semidirect
54. C2×#14519708C22semidirect
55. #16839790C22semidirect
56. #16839788C22semidirect
57. #16839785C22semidirect
58. C2×#14519706C22semidirect
70. C61×#428C22semidirect
71. C61×#428C22semidirect
72. C61×#427C22semidirect
73. C2×#107×C61C22semidirect
75. #14519823Q8semidirect
317. #2913D61semidirect
367. #428C2×D61semidirect
368. #428C2×D61semidirect
370. #427C2×D61semidirect
371. C2×#107C2×D61semidirect
385. C61#60496semidirect

Maximal subgroups

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

subgroupindex
#214596412
#214590252
#214556802
#214556752
#214556532
#214556482
#214556402
#214556152
#214556052
C2×#168400652
#214544122
#214543852
#214543462
#214543422
C61×#29132
#6049661