Finite Group Databasebeta

← #29652515

Subgroups — #29652515

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455763C211yesmaximal, characteristic, complemented
2#21455755C211yesmaximal, characteristic, complemented
3#21450093C211yesmaximal, characteristic, complemented
4#21444867C214yesmaximal, complemented
5#21444833C214yesmaximal, complemented
6#21444352C211yesmaximal, characteristic, complemented
7#21444348C211yesmaximal, characteristic, complemented
8#21444146C211yesmaximal, characteristic, complemented
9C2×C61×#538C211yesmaximal, characteristic
10C2×#14519736C2211yescharacteristic
11#16840100—22no—
12#16840095C2212yescomplemented
13#16840090C2212yescomplemented
14#16840083—22no—
15#16840078C2212yescomplemented
16#16840073C2212yescomplemented
17#16839802—21no—
18#16839802—21no—
19#16839771—21no—
20#16839767—21no—
21#16839726C2211yescharacteristic
22#16839726C2211yescharacteristic
23#16839698C2212yescomplemented
24#16839687C2212yescomplemented
25#16839650—22no—
26#16839635—21no—
27#16839625C2211yescharacteristic
28#16839618—21no—
29#16839499C2214yescomplemented
30#16839496C2214yescomplemented
31#16839486C2214yescomplemented
32C61×#538—22no—
33C61×#538—22no—
34C61×#538—22no—
35C61×#538C2214yes—
36C2×#126×C61—21no—
37C2×#124×C61—21no—
38C2×#123×C61—21no—
39C2×C22⋊Q8×C61C2211yescharacteristic
40C2×#121×C61—21no—
41C2×#121×C61C2211yescharacteristic
42C2×#120×C61—21no—
43C2×C4×D4×C61—21no—
44C22×C22⋊C4×C61C2211yescharacteristic
45C2×#11758194—21no—
46C2×#11758194—21no—
47#14519736—21no—
48#14519736—21no—
49#14519736—21no—
50#14519736—21no—
51C2×C4⋊Dic61C2311yescharacteristic
52C2×C4⋊Dic61C2311yescharacteristic
53C2×#11758196—22no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61—21no—
57C2×C22⋊Dic61—22no—
58C2×C4×Dic61—21no—
59#14519710—22no—
60#14519710—22no—
61#14519708—22no—
62#14519708—22no—
63#14519701—22no—
64#14519701—22no—
65#14519698—22no—
66#14519698—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C22⋊#92418C2312yes—
70C22⋊#92418C2312yes—
71#14519684—22no—
72#14519684—21no—
73#14519684C2312yes—
74C25×C61D411yescharacteristic
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61—21no—
77C22×D4×C61—21no—
78C22×D4×C61—21no—
79C22×D4×C61D411yescharacteristic
80C23×C4×C61—21no—
81C23×C4×C61D411yescharacteristic
82C23×C4×C61C2311yescharacteristic
83#126×C61—24no—
84#126×C61—24no—
85#124×C61—24no—
86#124×C61—24no—
87#123×C61—24no—
88#123×C61—24no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61D412yes—
92C22⋊Q8×C61C2312yes—
93#121×C61—24no—
94#121×C61—24no—
95#121×C61—21no—
96#121×C61—21no—
97#121×C61D412yes—
98#121×C61C2312yes—
99#120×C61—24no—
100#120×C61—24no—
101C4×D4×C61—24no—
102C4×D4×C61—24no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61D412yes—
117C2×C22⋊C4×C61C2312yes—
118C2×C42×C61—21no—
119C24×C61—21no—
120C24×C61—21no—
121C24×C61—21no—
122C24×C61—42no—
123C24×C61—42no—
124C24×C61—81no—
125C24×C61C2×D411yescharacteristic
126C24×C61C2×D412yes—
127C2×Q8×C61—21no—
128C2×Q8×C61—42no—
129C2×Q8×C61C2×D412yes—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—24no—
133C2×D4×C61—24no—
134C2×D4×C61—21no—
135C2×D4×C61—42no—
136C2×D4×C61—44no—
137C2×D4×C61—82no—
138C2×D4×C61—82no—
139C2×D4×C61C2×D412yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—24no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—81no—
148C22×C4×C61—82no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—22no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61C2×D412yes—
165C4⋊C4×C61C2×D412yes—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C42×C61—24no—
178C22⋊Dic61—42no—
179C22⋊Dic61—44no—
180C22⋊Dic61—41no—
181C22⋊Dic61—41no—
182C22⋊Dic61—82no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187#11758196—82no—
188C4×Dic61—41no—
189C4×Dic61—41no—
190#11758194—42no—
191#11758194—42no—
192C2×#92418—22no—
193C2×#92418—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61—41no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—81no—
211C23×C61—81no—
212C23×C61—88no—
213C23×C61#12011yescharacteristic
214C23×C61#12011yescharacteristic
215C23×C61C22×D411yescharacteristic
216Q8×C61—42no—
217Q8×C61—81no—
218D4×C61—42no—
219D4×C61—44no—
220D4×C61—44no—
221D4×C61—81no—
222D4×C61—84no—
223D4×C61—84no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—81no—
240C2×C4×C61—84no—
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252#92418—42no—
253#92418—42no—
254#18149—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—44no—
263C22×C61—44no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—82no—
271C22×C61—84no—
272C22×C61—84no—
273C22×C61—84no—
274C22×C61#45812yes—
275C22×C61C2×#12011yescharacteristic
276C22×C61#57512yes—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—42no—
285Dic61—82no—
286Dic61—81no—
287Dic61—81no—
288C2×#538D6111yescharacteristic
289#3095—611no—
290#3094—611no—
291#2482—611no—
292#1618—614no—
293#1612—614no—
294#1511—611no—
295#1510—611no—
296#1462—611no—
297C2×C61—22no—
298C2×C61—22no—
299C2×C61—41no—
300C2×C61—82no—
301C2×C61—82no—
302C2×C61#300511yescharacteristic
303C2×C61#301911yescharacteristic
304C2×C61C2×#45811yescharacteristic
305#538—22no—
306#538—22no—
307#538—22no—
308#538C2×D6114yes—
309C2×#126—21no—
310C2×#124—21no—
311C2×#123—21no—
312C2×C22⋊Q8C2×D6111yescharacteristic
313C2×#121—21no—
314C2×#121C2×D6111yescharacteristic
315C2×#120—21no—
316C2×C4×D4—21no—
317C2×#117—611no—
318C22×C22⋊C4C2×D6111yescharacteristic
319#485—1222no—
320#484—612no—
321#483—612no—
322#482—1222no—
323#481—612no—
324#480—612no—
325#428—1221no—
326#428—1221no—
327#419—1221no—
328#418—1221no—
329#407—611no—
330#407—611no—
331#399—612no—
332#397—612no—
333#388—1222no—
334#385—1221no—
335#381—611no—
336#379—1221no—
337#328—614no—
338#327—614no—
339#323—614no—
340C61#1814911yescharacteristic, complemented, Sylow
341C25C22⋊D6111yescharacteristic
342C22×Q8C22⋊D6111yescharacteristic
343C22×D4—21no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4C22⋊D6111yescharacteristic
347C23×C4—21no—
348C23×C4C22⋊D6111yescharacteristic
349C23×C4C22×D6111yescharacteristic
350C2×M16—1221no—
351C2×M16—1221no—
352#126—24no—
353#126—24no—
354#124—24no—
355#124—24no—
356#123—24no—
357#123—24no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8C22⋊D6112yes—
361C22⋊Q8C22×D6112yes—
362#121—24no—
363#121—24no—
364#121—21no—
365#121—21no—
366#121C22⋊D6112yes—
367#121C22×D6112yes—
368#120—24no—
369#120—24no—
370C4×D4—24no—
371C4×D4—24no—
372#117—1221no—
373#117—1221no—
374#117—1221no—
375#117—1221no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4C22⋊D6111yescharacteristic
382C2×C4⋊C4C22⋊D6111yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4C22⋊D6112yes—
397C2×C22⋊C4C22×D6112yes—
398C2×C42—21no—
399C2×C42—1221no—
400#107—1222no—
401#107—1222no—
402#106—1222no—
403#106—1222no—
404#103—1222no—
405#103—1222no—
406#102—1222no—
407#102—1222no—
408C22⋊C8—612no—
409C22⋊C8—612no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412#95—612no—
413#95—1221no—
414#95—1222no—
415C24—21no—
416C24—21no—
417C24—21no—
418C24—42no—
419C24—42no—
420C24—81no—
421C24C2×C22⋊D6111yescharacteristic
422C24C2×C22⋊D6112yes—
423C2×Q8—21no—
424C2×Q8—42no—
425C2×Q8C2×C22⋊D6112yes—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4C2×C22⋊D6112yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—82no—
445C22×C4—1222no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6112yes—
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C23×D6111yescharacteristic
455M16—2442no—
456M16—2442no—
457C2×C8—1222no—
458C2×C8—1222no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—4882no—
469C4⋊C4C2×C22⋊D6112yes—
470C4⋊C4C2×C22⋊D6112yes—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—4882no—
483C22⋊C4—2442no—
484C22⋊C4—2444no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C42—24no—
488C42—2441no—
489C42—2441no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—81no—
507C23—81no—
508C23—88no—
509C23#1451974611yescharacteristic
510C23#1451974611yescharacteristic
511C23C22×C22⋊D6111yescharacteristic
512Q8—42no—
513Q8—81no—
514D4—42no—
515D4—44no—
516D4—44no—
517D4—81no—
518D4—84no—
519D4—84no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—81no—
536C2×C4—84no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4#1451974611yescharacteristic
545C2×C4#1451974611yescharacteristic
546C2×C4C22×C22⋊D6111yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C8—2442no—
549C8—2442no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—44no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—82no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22#1683995212yes—
570C22C2×#1451974611yescharacteristic
571C22#1684056012yes—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—4882no—
581C4—4881no—
582C4—4881no—
583C2—22no—
584C2—22no—
585C2—41no—
586C2—82no—
587C2—82no—
588C2#2145506111yescharacteristic
589C2#2145517611yescharacteristic
590C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#538C2
socleC22×C61C2×#120
hypercenterC2×#538D61
hypocenterC61#18149
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#538

Complements

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

NH ≅ G/Ntype
1. #21455763C2semidirect
2. #21455755C2semidirect
3. #21450093C2semidirect
4. #21444867C2semidirect
5. #21444833C2semidirect
6. #21444352C2semidirect
7. #21444348C2semidirect
8. #21444146C2semidirect
12. #16840095C22semidirect
13. #16840090C22semidirect
15. #16840078C22semidirect
16. #16840073C22semidirect
23. #16839698C22semidirect
24. #16839687C22semidirect
29. #16839499C22semidirect
30. #16839496C22semidirect
31. #16839486C22semidirect
340. C61#18149semidirect

Maximal subgroups

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

subgroupindex
#214557632
#214557552
#214500932
#214448672
#214448332
#214443522
#214443482
#214441462
C2×C61×#5382
#1814961