Finite Group Databasebeta

← #30548535

Subgroups — #30548535

Order 15616.

2138 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459020C211yesmaximal, characteristic, complemented
2#21456105C214yesmaximal, complemented
3#21456057C214yesmaximal, complemented
4#21456012C218yesmaximal, complemented
5#21455755C212yesmaximal, complemented
6#21455637C212yesmaximal, complemented
7#21455295C212yesmaximal, complemented
8#21455069C212yesmaximal, complemented
9C4×#14519803C214yesmaximal, complemented
10#21454457C211yesmaximal, characteristic, complemented
11C2×C61×#540C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519803C2214yes—
14#16840243—22no—
15#16840242C2214yescomplemented
16#16840239C2212yescomplemented
17#16840236C2212yescomplemented
18#16840235—22no—
19C4×#11758222C2212yes—
20C4×#11758222C2212yes—
21#16840230—22no—
22C2×#14519736C2211yescharacteristic
23#117×D61—22no—
24#16840083C2218yescomplemented
25#16840078C2214yescomplemented
26#16840073C2214yescomplemented
27#16840066C2218yescomplemented
28#16840059C2218yescomplemented
29#16839959C2218yescomplemented
30#16839954C2218yescomplemented
31#16839905C2218yescomplemented
32#16839900C2214yescomplemented
33#16839895C2214yescomplemented
34#16839872C2218yescomplemented
35#16839862C2218yescomplemented
36C4×#11758200C2218yescomplemented
37C4×#11758199C2218yescomplemented
38#16839823C2214yescomplemented
39#16839823C2214yescomplemented
40#16839802C2214yescomplemented
41#16839786C2212yescomplemented
42#16839767C2214yescomplemented
43#16839761C2214yes—
44#16839726C2212yes—
45C4×#11758194C2212yescomplemented
46C61×#540—22no—
47C61×#540—22no—
48C61×#540C2214yes—
49C61×#540C2214yescomplemented
50C2×#127×C61C2211yescharacteristic, complemented
51C2×#125×C61C2211yescharacteristic, complemented
52C2×#123×C61—22no—
53C2×#121×C61—22no—
54C2×#121×C61C2212yes—
55C2×C4×D4×C61C2212yescomplemented
56C2×#117×C61C2211yescharacteristic
57C2×#11758222—21no—
58C2×#11758222—21no—
59C2×#11758222—28no—
60C2×#11758222C2312yes—
61C2×#11758222C2312yes—
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C22⋊D61—21no—
65C22×C4×D61—21no—
66#14519803—216no—
67C2×#11758200C2318yes—
68C2×#11758199C2318yes—
69C2×#11758194C2314yes—
70C4×Dic122C2314yes—
71C4×Dic122C2314yes—
72#14519742—24no—
73C4×D244C2314yes—
74C4×D244C2314yes—
75#14519739—24no—
76C4×C22⋊D61—24no—
77C4×C22⋊D61—24no—
78#14519737—24no—
79#14519737—24no—
80#14519736—21no—
81#14519736—21no—
82#14519736C2312yes—
83#14519736C2312yes—
84#14519735—24no—
85#14519734—22no—
86#14519734—22no—
87#14519733—24no—
88C2×C4⋊Dic61C2312yes—
89C4⋊C4×D61—24no—
90C2×C22⋊Dic61—21no—
91C22⋊C4×D61—24no—
92C2×C4×Dic61—21no—
93C42×D61—22no—
94C42×D61—22no—
95#14519710C2318yes—
96#14519708C2318yes—
97C4⋊#92418C2318yes—
98#14519700C2318yes—
99#14519698C2318yes—
100#14519698C2318yes—
101#14519697C2318yes—
102C22⋊#92418C2314yes—
103C22⋊#92418C2314yes—
104#14519688C2314yes—
105C4×#92418C2314yes—
106C22×D4×C61—22no—
107C22×D4×C61—22no—
108C22×D4×C61C2312yes—
109C23×C4×C61—22no—
110C23×C4×C61C2311yescharacteristic
111#127×C61—22no—
112#127×C61C2314yes—
113#125×C61—22no—
114#125×C61C2314yes—
115#123×C61—28no—
116#123×C61—28no—
117#121×C61—22no—
118#121×C61—22no—
119#121×C61—28no—
120#121×C61—28no—
121#121×C61C2314yes—
122#121×C61C2314yes—
123C4×D4×C61—24no—
124C4×D4×C61C2318yes—
125#117×C61D412yes—
126#117×C61D412yes—
127#117×C61C2312yes—
128#117×C61C2312yes—
129C2×C4⋊C4×C61—22no—
130C2×C4⋊C4×C61D412yes—
131C2×C4⋊C4×C61C2312yes—
132C2×C22⋊C4×C61—22no—
133C2×C22⋊C4×C61—22no—
134C2×C22⋊C4×C61D412yes—
135C2×C42×C61C2311yescharacteristic
136C2×C42×C61C2311yescharacteristic
137C24×C61—22no—
138C24×C61C2×D411yescharacteristic
139C23×D61—21no—
140C2×D4×C61—24no—
141C2×D4×C61—24no—
142C2×D4×C61—22no—
143C2×D4×C61—28no—
144C2×D4×C61—28no—
145C2×D4×C61—84no—
146C2×D4×C61—84no—
147C2×D4×C61C2414yes—
148C22×C4×C61—24no—
149C22×C4×C61—22no—
150C22×C4×C61—28no—
151C22×C4×C61—84no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—28no—
161C2×C22⋊D61—22no—
162C2×C22⋊D61—22no—
163C22×Dic61—21no—
164C22×Dic61—22no—
165#11758222—28no—
166#11758222—28no—
167#11758222—24no—
168#11758222—24no—
169#11758222—24no—
170#11758222#4118yes—
171C2×D244—21no—
172C2×D244—24no—
173C2×D244C2412yes—
174C2×C4×D61—28no—
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—24no—
178C2×C4×D61—24no—
179C2×Dic122—21no—
180C2×Dic122—24no—
181C2×Dic122C2412yes—
182C4⋊C4×C61—28no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61C2×D414yes—
185C4⋊C4×C61C2×D414yes—
186C4⋊C4×C61C2414yes—
187C22⋊C4×C61—28no—
188C22⋊C4×C61—24no—
189C22⋊C4×C61—24no—
190C22⋊C4×C61C2×D414yes—
191C22⋊C4×C61C2×D414yes—
192C42×C61C2×D412yes—
193C42×C61C2×D412yes—
194C42×C61C2412yes—
195C42×C61C2412yes—
196C22⋊Dic61—22no—
197C22⋊Dic61—22no—
198#11758200—216no—
199#11758199—216no—
200#11758198—24no—
201#11758198—24no—
202C4⋊Dic61—22no—
203C4⋊Dic61C2414yes—
204#11758196—24no—
205#11758196—24no—
206C4×Dic61—22no—
207C4×Dic61—24no—
208C4×Dic61—24no—
209C4×Dic61—22no—
210#11758194—28no—
211C2×#92418C2418yes—
212C23×C61—22no—
213C23×C61—24no—
214C23×C61—42no—
215C23×C61—44no—
216C23×C61—44no—
217C23×C61—44no—
218C23×C61C22×D411yescharacteristic
219C23×C61C22×D411yescharacteristic
220C23×C61C22×D411yescharacteristic
221C22×D61—22no—
222C22×D61—24no—
223C22×D61—82no—
224D4×C61—28no—
225D4×C61—44no—
226D4×C61—88no—
227D4×C61—84no—
228D4×C61—84no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—44no—
238C2×C4×C61—88no—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D414yes—
241C2×C4×C61C22×D414yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4114yes—
245C2×C4×C61C2511yescharacteristic
246C22⋊D61—28no—
247C22⋊D61—28no—
248C2×Dic61—24no—
249C2×Dic61—22no—
250C2×Dic61—24no—
251C2×Dic61—82no—
252D244—24no—
253D244—22no—
254D244C2×#4114yes—
255C4×D61—28no—
256C4×D61—28no—
257C4×D61—84no—
258C4×D61—84no—
259Dic122—24no—
260Dic122—22no—
261Dic122C2×#4114yes—
262#92418—28no—
263#60568—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—44no—
270C22×C61—44no—
271C22×C61—44no—
272C22×C61—82no—
273C22×C61—88no—
274C22×C61#52512yes—
275C22×C61C23×D411yescharacteristic
276C2×D61—24no—
277C2×D61—44no—
278C2×D61—84no—
279C4×C61—24no—
280C4×C61—24no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—44no—
284C4×C61#52512yes—
285C4×C61C22×#4112yes—
286Dic61—24no—
287Dic61—44no—
288C2×#540D6111yescharacteristic, complemented
289#3428—611no—
290#3126—614no—
291#3122—614no—
292#3119—618no—
293#3094—612no—
294#3080—612no—
295#3038—612no—
296#3006—612no—
297C4×AGL(1,ℤ/8)—614no—
298#2921—611no—
299C2×C61—22no—
300C2×C61—41no—
301C2×C61—41no—
302C2×C61—84no—
303C2×C61C2×#52511yescharacteristic
304C2×C61#358512yes—
305D61—44no—
306C22×#41—611no—
307C2×AGL(1,ℤ/8)—614no—
308#540—22no—
309#540—22no—
310#540C2×D6114yes—
311#540C2×D6114yescomplemented
312C2×#127C2×D6111yescharacteristic, complemented
313C2×#125C2×D6111yescharacteristic, complemented
314C2×#123—22no—
315C2×#121—22no—
316C2×#121C2×D6112yes—
317#520—614no—
318#520—1222no—
319#519—612no—
320#518—612no—
321#518—1222no—
322C4×#41—612no—
323C4×#41—612no—
324C4×#41—1222no—
325C2×C4×D4C2×D6112yescomplemented
326C2×#117—611no—
327C2×#117—1222no—
328C2×#117C2×D6111yescharacteristic
329#482—618no—
330#481—614no—
331#480—614no—
332#479—618no—
333#478—618no—
334#460—618no—
335#459—618no—
336#449—618no—
337#448—614no—
338#447—614no—
339#442—618no—
340#440—618no—
341C4×SD16—618no—
342C4×D8—618no—
343#433—614no—
344#433—614no—
345#428—614no—
346#423—612no—
347#418—614no—
348#417—614no—
349#407—612no—
350C4×M16—612no—
351C61#6056811yescharacteristic, complemented, Sylow
352C2×#41—612no—
353C2×#41—612no—
354C2×#41—1228no—
355C2×#41—1221no—
356C2×#41—1221no—
357C22×Q8—611no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—611no—
361C22×D4—1221no—
362C22×D4C22×D6112yes—
363C23×C4—22no—
364C23×C4—1221no—
365C23×C4C22×D6111yescharacteristic
366AGL(1,ℤ/8)—12216no—
367C2×SD16—618no—
368C2×D8—618no—
369C2×M16—614no—
370#127—22no—
371#127C22×D6114yes—
372#125—22no—
373#125C22×D6114yes—
374#123—28no—
375#123—28no—
376#121—22no—
377#121—22no—
378#121—28no—
379#121—28no—
380#121C22×D6114yes—
381#121C22×D6114yes—
382C4×Q8—614no—
383C4×Q8—614no—
384C4×Q8—1224no—
385C4×D4—24no—
386C4×D4—614no—
387C4×D4—614no—
388C4×D4—1224no—
389C4×D4—1224no—
390C4×D4—1224no—
391C4×D4—1224no—
392C4×D4—1224no—
393C4×D4C22×D6118yes—
394#117—612no—
395#117—612no—
396#117—1221no—
397#117—1221no—
398#117—1224no—
399#117—1224no—
400#117—1222no—
401#117—1222no—
402#117C22⋊D6112yes—
403#117C22⋊D6112yes—
404#117C22×D6112yes—
405#117C22×D6112yes—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—1224no—
409C2×C4⋊C4C22⋊D6112yes—
410C2×C4⋊C4C22×D6112yes—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1224no—
415C2×C22⋊C4C22⋊D6112yes—
416C2×C42—1221no—
417C2×C42—1222no—
418C2×C42—1222no—
419C2×C42C22×D6111yescharacteristic
420C2×C42C22×D6111yescharacteristic
421#107—618no—
422#106—618no—
423C4⋊C8—618no—
424#103—618no—
425#102—618no—
426#102—618no—
427#102—618no—
428C22⋊C8—614no—
429C22⋊C8—614no—
430#97—614no—
431C4×C8—614no—
432C24—22no—
433C24—1221no—
434C24C2×C22⋊D6111yescharacteristic
435#41—618no—
436#41—1224no—
437#41—1224no—
438#41—1228no—
439#41—1228no—
440#41—1224no—
441C2×Q8—612no—
442C2×Q8—1224no—
443C2×Q8—1221no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—28no—
448C2×D4—28no—
449C2×D4—84no—
450C2×D4—84no—
451C2×D4—612no—
452C2×D4—1224no—
453C2×D4—1228no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1221no—
457C2×D4C23×D6114yes—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—28no—
461C22×C4—84no—
462C22×C4—1228no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1224no—
468C22×C4—1224no—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6112yes—
473C22×C4C2×C22⋊D6112yes—
474C22×C4C23×D6111yescharacteristic
475C22×C4C23×D6111yescharacteristic
476C22×C4C23×D6111yescharacteristic
477SD16—12216no—
478D8—12216no—
479M16—1228no—
480C2×C8—618no—
481C4⋊C4—28no—
482C4⋊C4—22no—
483C4⋊C4—614no—
484C4⋊C4—1222no—
485C4⋊C4—1224no—
486C4⋊C4—1224no—
487C4⋊C4C2×C22⋊D6114yes—
488C4⋊C4C2×C22⋊D6114yes—
489C4⋊C4C23×D6114yes—
490C22⋊C4—28no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4—1224no—
496C22⋊C4—1224no—
497C22⋊C4C2×C22⋊D6114yes—
498C22⋊C4C2×C22⋊D6114yes—
499C42—1222no—
500C42—1222no—
501C42—1224no—
502C42—1224no—
503C42C2×C22⋊D6112yes—
504C42C2×C22⋊D6112yes—
505C42C23×D6112yes—
506C42C23×D6112yes—
507C23—22no—
508C23—24no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—1224no—
514C23—1222no—
515C23—4882no—
516C23C22×C22⋊D6111yescharacteristic
517C23C22×C22⋊D6111yescharacteristic
518C23C22×C22⋊D6111yescharacteristic
519Q8—614no—
520Q8—1224no—
521Q8—1222no—
522D4—28no—
523D4—44no—
524D4—88no—
525D4—84no—
526D4—84no—
527D4—614no—
528D4—1224no—
529D4—1228no—
530D4—1228no—
531D4—1222no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—88no—
542C2×C4—1228no—
543C2×C4—1224no—
544C2×C4—1222no—
545C2×C4—1228no—
546C2×C4—1224no—
547C2×C4—4884no—
548C2×C4—4884no—
549C2×C4—4882no—
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4C22×C22⋊D6114yes—
552C2×C4C22×C22⋊D6114yes—
553C2×C4C22×C22⋊D6111yescharacteristic
554C2×C4C22×C22⋊D6111yescharacteristic
555C2×C4#41×D6114yes—
556C2×C4C24×D6111yescharacteristic
557C8—1228no—
558C22—22no—
559C22—21no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—82no—
567C22—88no—
568C22—1224no—
569C22—4884no—
570C22—2444no—
571C22#1684026912yes—
572C22C23×C22⋊D6111yescharacteristic
573C4—21no—
574C4—21no—
575C4—24no—
576C4—24no—
577C4—44no—
578C4—1224no—
579C4—2444no—
580C4#1684026912yes—
581C4C2×#41×D6112yes—
582C2—22no—
583C2—41no—
584C2—41no—
585C2—84no—
586C2—2444no—
587C2C2×#1684026911yescharacteristic
588C2#2146059912yes—
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#540C2
socleC22×C61C23×D4
hypercenterC2×#540D61
hypocenterC61#60568
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. #21459020C2semidirect
2. #21456105C2semidirect
3. #21456057C2semidirect
4. #21456012C2semidirect
5. #21455755C2semidirect
6. #21455637C2semidirect
7. #21455295C2semidirect
8. #21455069C2semidirect
9. C4×#14519803C2semidirect
10. #21454457C2semidirect
11. C2×C61×#540C2semidirect
15. #16840242C22semidirect
16. #16840239C22semidirect
17. #16840236C22semidirect
24. #16840083C22semidirect
25. #16840078C22semidirect
26. #16840073C22semidirect
27. #16840066C22semidirect
28. #16840059C22semidirect
29. #16839959C22semidirect
30. #16839954C22semidirect
31. #16839905C22semidirect
32. #16839900C22semidirect
33. #16839895C22semidirect
34. #16839872C22semidirect
35. #16839862C22semidirect
36. C4×#11758200C22semidirect
37. C4×#11758199C22semidirect
38. #16839823C22semidirect
39. #16839823C22semidirect
40. #16839802C22semidirect
41. #16839786C22semidirect
42. #16839767C22semidirect
45. C4×#11758194C22semidirect
49. C61×#540C22semidirect
50. C2×#127×C61C22semidirect
51. C2×#125×C61C22semidirect
55. C2×C4×D4×C61C22semidirect
288. C2×#540D61semidirect
311. #540C2×D61semidirect
312. C2×#127C2×D61semidirect
313. C2×#125C2×D61semidirect
325. C2×C4×D4C2×D61semidirect
351. C61#60568semidirect

Maximal subgroups

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

subgroupindex
#214590202
#214561052
#214560572
#214560122
#214557552
#214556372
#214552952
#214550692
C4×#145198032
#214544572
C2×C61×#5402
#6056861