Finite Group Databasebeta

← #29648182

Subgroups — #29648182

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459782C211yesmaximal, characteristic
2#21454837C211yesmaximal, characteristic, complemented
3#21454468C211yesmaximal, characteristic, complemented
4#21444449C212yesmaximal, complemented
5#21444441C212yesmaximal, complemented
6#21444429C211yesmaximal, characteristic, complemented
7#21444426C211yesmaximal, characteristic, complemented
8#21444425C212yesmaximal, complemented
9#21444415C212yesmaximal, complemented
10#21444410C211yesmaximal, characteristic, complemented
11C61×#1524C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic
13#16840482C411yescharacteristic
14#16840477—21no—
15#16840437—22no—
16Q8×D244C411yescharacteristic
17#16840422C412yes—
18#16840409C412yes—
19#16840408—21no—
20D4×D244C411yescharacteristic
21#16840383—21no—
22#16840348—22no—
23#16840312C2211yescharacteristic
24#16840297C2212yes—
25#124×D61C2212yes—
26#16840236—21no—
27C4×#11758222C2211yescharacteristic
28C8×Dic122—21no—
29#16839841—21no—
30#16839838—22no—
31#16839824—22no—
32#16839819C2211yescharacteristic, complemented
33#16839817C2212yescomplemented
34#16839817C2212yescomplemented
35#16839817C2212yescomplemented
36#16839788—21no—
37#16839728—21no—
38C4×#11758194C2211yescharacteristic, complemented
39#16839714C2212yescomplemented
40#16839504C2212yescomplemented
41#16839502C2212yes—
42#16839501C2212yescomplemented
43#16839501C2212yescomplemented
44#16839501C2212yescomplemented
45#16839501C2212yescomplemented
46C61×#529C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic, complemented
48C2×#97×C61C2211yescharacteristic, complemented
49C61×#330C2211yescharacteristic, complemented
50C61×#330C2211yescharacteristic, complemented
51C61×#329C2211yescharacteristic
52C61×#329C2211yescharacteristic
53C2×#11758222C2×C412yes—
54C2×#11758222D411yescharacteristic
55C2×Q8×D61C2×C412yes—
56C22×D244—21no—
57C2×D4×D61C2×C412yes—
58C2×#11758194—21no—
59C2×#11758189—22no—
60#14519777—21no—
61#14519777C2×C411yescharacteristic
62#14519774C2×C411yescharacteristic
63#14519772—22no—
64#14519770—22no—
65#14519766C2×C412yes—
66#14519765—21no—
67#14519765C2312yes—
68#14519764C2×C412yes—
69#14519763—22no—
70#14519763—22no—
71#14519763—22no—
72#14519762—22no—
73#14519762—22no—
74#14519759—22no—
75#14519757—22no—
76#14519757C2×C412yes—
77#14519756—21no—
78#14519755—22no—
79#14519752C2×C412yes—
80#14519751—21no—
81#14519751—21no—
82#14519751—21no—
83#14519750—22no—
84#14519748—22no—
85#14519747—22no—
86#14519745—22no—
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122D411yescharacteristic
89#14519742C2×C412yes—
90C4×D244—21no—
91C4×D244—21no—
92C4×D244—21no—
93C4×D244—21no—
94C4×D244C2×C411yescharacteristic
95C4×D244D411yescharacteristic
96#14519739—22no—
97#14519739—22no—
98#14519739C2×C412yes—
99C4×C22⋊D61C2×C412yes—
100C4×C22⋊D61D412yes—
101#14519736—21no—
102#14519736D411yescharacteristic
103#14519735—22no—
104#14519734D412yes—
105C22⋊C4×D61—22no—
106C22⋊C4×D61—22no—
107C22⋊C4×D61—22no—
108C42×D61C2312yes—
109C4⋊#92418—21no—
110#14519706—22no—
111#14519706—22no—
112#14519706—22no—
113C22⋊#92418—21no—
114#14519689—22no—
115#14519689—22no—
116#14519689—22no—
117#14519688C2312yes—
118#14519688C2312yes—
119C4×#92418—21no—
120C8×Dic61—22no—
121C2×M16×C61—21no—
122C22×C8×C61—21no—
123#126×C61—21no—
124#125×C61C2311yescharacteristic
125#124×C61C2311yescharacteristic
126#123×C61—21no—
127C22⋊Q8×C61C2×C411yescharacteristic
128#121×C61C2×C411yescharacteristic
129C4×Q8×C61C2×C411yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2×C411yescharacteristic
132#117×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#97×C61—21no—
135#97×C61C2311yescharacteristic
136#97×C61C2311yescharacteristic
137#97×C61C2311yescharacteristic
138#97×C61C2311yescharacteristic
139C4×C8×C61—21no—
140C23×D61—22no—
141C2×Q8×C61C22×C411yescharacteristic
142C2×D4×C61—21no—
143C2×D4×C61C22×C411yescharacteristic
144C22×C4×C61—21no—
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C2×C22⋊D61—22no—
149C2×C22⋊D61—22no—
150C2×C22⋊D61C22⋊C412yes—
151Q8×D61—82no—
152D4×D61—82no—
153#11758222—42no—
154#11758222—82no—
155C2×D244—22no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—44no—
159C2×D244C22⋊C411yescharacteristic
160C2×D244C22×C412yes—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61C22×C412yes—
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2×D412yes—
167C2×Dic122—22no—
168C2×Dic122C22⋊C411yescharacteristic
169C2×Dic122C22×C412yes—
170M16×C61—42no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—42no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C22×C411yescharacteristic
181C4⋊C4×C61C22×C411yescharacteristic
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C42×C61—21no—
188C42×C61C22×C411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2411yescharacteristic
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61C22⋊C412yes—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198C22⋊C412yes—
203#11758198C22⋊C412yes—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—22no—
207C4⋊Dic61C22⋊C411yescharacteristic
208C4⋊Dic61C22⋊C411yescharacteristic
209#11758196—22no—
210#11758196C22⋊C412yes—
211#11758196C22⋊C412yes—
212C4×Dic61—22no—
213C4×Dic61C22×C412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216#11758194—42no—
217C2×#92418—22no—
218C2×#92418—22no—
219#11758189—44no—
220C23×C61—21no—
221C23×C61C2×C22⋊C411yescharacteristic
222C22×D61—22no—
223C22×D61—44no—
224C22×D61—82no—
225C22×D61—82no—
226C22×D61C2×C22⋊C412yes—
227Q8×C61—41no—
228D4×C61—41no—
229D4×C61—81no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—42no—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C22⋊D61—42no—
244C22⋊D61—82no—
245C22⋊D61—82no—
246C2×Dic61—22no—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C412yes—
249C2×Dic61C2×C22⋊C412yes—
250D244—22no—
251D244—42no—
252D244—44no—
253D244—42no—
254D244—81no—
255C4×D61—42no—
256C4×D61—42no—
257C4×D61—42no—
258C4×D61—82no—
259C4×D61—82no—
260C4×D61—82no—
261Dic122—22no—
262Dic122—42no—
263Dic122—82no—
264C8×C61—22no—
265C8×C61—41no—
266C8×C61—41no—
267C8×C61—41no—
268#92418—42no—
269#92418—42no—
270#17681—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61C22×C22⋊C411yescharacteristic
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—164no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61#42112yes—
294Dic61—22no—
295Dic61—22no—
296Dic61—22no—
297Dic61—42no—
298#3514—611no—
299#2971—611no—
300#2923—611no—
301#1535—612no—
302#1533—612no—
303#1530—611no—
304#1529—612no—
305#1529—611no—
306#1527—612no—
307#1525—611no—
308#1524D6111yescharacteristic, complemented
309C2×C61—41no—
310C2×C61—81no—
311C2×C61#288811yescharacteristic
312C2×C61C2×#42111yescharacteristic
313C2×C61#290111yescharacteristic
314D61—42no—
315D61—82no—
316#559—611no—
317#556—611no—
318#555—1221no—
319#551—1222no—
320D4×Q8—611no—
321#548—612no—
322#546—612no—
323#546—1221no—
324D42—611no—
325#542—1221no—
326#538—1222no—
327#532—611no—
328#529—612no—
329#529C2×D6111yescharacteristic
330C2×#124—612no—
331#518—1221no—
332C4×#41—611no—
333C8×Q8—1221no—
334#436—1221no—
335#436—1222no—
336#433—1222no—
337#432—611no—
338#432—612no—
339#432—612no—
340#432—612no—
341#424—1221no—
342#408—1221no—
343C4×M16—611no—
344C4×M16—612no—
345C4×M16C2×D6111yescharacteristic, complemented
346C2×#97C2×D6111yescharacteristic, complemented
347#330—612no—
348#330C2×D6111yescharacteristic, complemented
349#330C2×D6111yescharacteristic, complemented
350#329—612no—
351#329—612no—
352#329—612no—
353#329—612no—
354#329—612no—
355#329C2×D6111yescharacteristic
356#329C2×D6111yescharacteristic
357C61#1768111yescharacteristic, complemented, Sylow
358C2×#41—611no—
359C2×#41—612no—
360C22×Q8—612no—
361C22×D4—612no—
362C22×D4—1221no—
363C2×M16—21no—
364C2×M16—1222no—
365C2×M16—1221no—
366C22×C8—21no—
367C4⋊Q8—611no—
368C4⋊Q8—1221no—
369#127—611no—
370#126—21no—
371#126—1222no—
372#126—1222no—
373#125C22×D6111yescharacteristic
374#124—612no—
375#124—612no—
376#124—612no—
377#124—1221no—
378#124—1222no—
379#124—1222no—
380#124—1222no—
381#124C22×D6111yescharacteristic
382#123—21no—
383#123—1222no—
384#123—1222no—
385#123—1222no—
386C22⋊Q8—612no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1221no—
390C22⋊Q8C4×D6111yescharacteristic
391#121—612no—
392#121—1222no—
393#121—1222no—
394#121—1221no—
395#121—1221no—
396#121—1221no—
397#121C4×D6111yescharacteristic
398#120—1222no—
399#120—1222no—
400C4×Q8—611no—
401C4×Q8—611no—
402C4×Q8—612no—
403C4×Q8C4×D6111yescharacteristic
404C4×D4—21no—
405C4×D4—612no—
406C4×D4—612no—
407C4×D4—612no—
408C4×D4—611no—
409C4×D4—611no—
410C4×D4—1221no—
411C4×D4—1221no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1221no—
415C4×D4—1221no—
416C4×D4C4×D6111yescharacteristic
417#117—21no—
418#117—611no—
419#117—612no—
420#117—1222no—
421#117—1221no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1222no—
425C2×C42—612no—
426C2×C42C22×D6111yescharacteristic
427C4⋊C8—1222no—
428C4⋊C8—1222no—
429C4⋊C8—1222no—
430C4⋊C8—1221no—
431C22⋊C8—1222no—
432C22⋊C8—1222no—
433C22⋊C8—1222no—
434C22⋊C8—1221no—
435#97—21no—
436#97—612no—
437#97—612no—
438#97C22×D6111yescharacteristic
439#97C22×D6111yescharacteristic
440#97C22×D6111yescharacteristic
441#97C22×D6111yescharacteristic
442C4×C8—21no—
443C4×C8—1222no—
444C4×C8—1221no—
445C24—1222no—
446#41—4882no—
447#41—2442no—
448C2×Q8—611no—
449C2×Q8—612no—
450C2×Q8—1222no—
451C2×Q8—4882no—
452C2×Q8C2×C4×D6111yescharacteristic
453C2×D4—21no—
454C2×D4—612no—
455C2×D4—612no—
456C2×D4—611no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—4882no—
463C2×D4—2444no—
464C2×D4C2×C4×D6111yescharacteristic
465C22×C4—21no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4C2×C4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475M16—42no—
476M16—2444no—
477M16—2442no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—42no—
483C2×C8—1222no—
484C2×C8—1222no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—612no—
490C4⋊C4—612no—
491C4⋊C4—611no—
492C4⋊C4—611no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4C2×C4×D6111yescharacteristic
498C4⋊C4C2×C4×D6111yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—612no—
505C22⋊C4—612no—
506C22⋊C4—612no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C42—21no—
517C42—612no—
518C42—612no—
519C42—612no—
520C42—1222no—
521C42C2×C4×D6111yescharacteristic
522C42D4×D6111yescharacteristic
523C42D4×D6111yescharacteristic
524C42C23×D6111yescharacteristic
525C23—21no—
526C23—612no—
527C23—1222no—
528C23—4882no—
529C23—4882no—
530C23—2444no—
531C23C22⋊C4×D6111yescharacteristic
532Q8—41no—
533Q8—1222no—
534Q8—4882no—
535Q8—2442no—
536D4—41no—
537D4—81no—
538D4—1222no—
539D4—4881no—
540D4—4882no—
541D4—4882no—
542D4—2442no—
543D4—2444no—
544D4—2442no—
545D4—2442no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—1222no—
557C2×C4—4882no—
558C2×C4—4882no—
559C2×C4—4882no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4C22⋊C4×D6111yescharacteristic
564C2×C4C22⋊C4×D6111yescharacteristic
565C2×C4C22⋊C4×D6111yescharacteristic
566C2×C4C22×C4×D6111yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C2×D4×D6111yescharacteristic
569C8—22no—
570C8—41no—
571C8—41no—
572C8—41no—
573C8—2442no—
574C8—2442no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—1222no—
582C22—1222no—
583C22—1222no—
584C22—9764no—
585C22—2442no—
586C22—2442no—
587C22—2442no—
588C22C2×C22⋊C4×D6111yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—1222no—
598C4—1222no—
599C4—1222no—
600C4—2442no—
601C4#1683977712yes—
602C2—41no—
603C2—81no—
604C2—4882no—
605C2—2442no—
606C2#2145429611yescharacteristic
607C2C2×#1683977711yescharacteristic
608C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1524C2
socleC22×C61C22×C22⋊C4
prime radical#21459782C2
cosocleC42×C61C24
hypercenter#1524D61
hypocenterC61#17681
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1524

Complements

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

NH ≅ G/Ntype
2. #21454837C2semidirect
3. #21454468C2semidirect
4. #21444449C2semidirect
5. #21444441C2semidirect
6. #21444429C2semidirect
7. #21444426C2semidirect
8. #21444425C2semidirect
9. #21444415C2semidirect
10. #21444410C2semidirect
11. C61×#1524C2semidirect
32. #16839819C22semidirect
33. #16839817C22semidirect
34. #16839817C22semidirect
35. #16839817C22semidirect
38. C4×#11758194C22semidirect
39. #16839714C22semidirect
40. #16839504C22semidirect
42. #16839501C22semidirect
43. #16839501C22semidirect
44. #16839501C22semidirect
45. #16839501C22semidirect
47. C4×M16×C61C22semidirect
48. C2×#97×C61C22semidirect
49. C61×#330C22semidirect
50. C61×#330C22semidirect
308. #1524D61semidirect
345. C4×M16C2×D61semidirect
346. C2×#97C2×D61semidirect
348. #330C2×D61semidirect
349. #330C2×D61semidirect
357. C61#17681semidirect

Maximal subgroups

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

subgroupindex
#214597822
#214548372
#214544682
#214444492
#214444412
#214444292
#214444262
#214444252
#214444152
#214444102
C61×#15242
#1768161