Finite Group Databasebeta

← #30552270

Subgroups — #30552270

Order 15616.

1654 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459222C211yesmaximal, characteristic, complemented
2#21459211C211yesmaximal, characteristic
3#21457120C214yesmaximal, complemented
4#21457105C214yesmaximal, complemented
5#21457090C214yesmaximal, complemented
6#21457075C214yesmaximal, complemented
7#21456341C211yesmaximal, characteristic, complemented
8#21456167C211yesmaximal, characteristic, complemented
9#21456142C211yesmaximal, characteristic, complemented
10#21455736C214yesmaximal, complemented
11#21455605C214yesmaximal, complemented
12C2×#16839821C211yesmaximal, characteristic, complemented
13C61×#3160C211yesmaximal, characteristic, complemented
14#16840392C2214yes—
15#16840391—22no—
16#16840389C2214yes—
17#16840387—22no—
18#16840386C2214yes—
19#16840375C2212yes—
20#16840368C2212yes—
21#16840361—22no—
22#16840352—22no—
23#16840319C2211yescharacteristic
24#16840316—24no—
25#16840315C2212yes—
26#16840312C2211yescharacteristic
27#16840299—24no—
28#16840274C2212yes—
29#16840230C2212yes—
30C2×#14519734C2211yescharacteristic
31#117×D61C2212yes—
32#16840172C2214yescomplemented
33#16840168C2214yescomplemented
34#16840132C2212yescomplemented
35#16840128C2212yescomplemented
36#16840124C2214yescomplemented
37#16840117C2212yescomplemented
38#16840117C2212yescomplemented
39#16840113C2212yescomplemented
40#16840109C2212yescomplemented
41#16840106C2214yes—
42#16840105C2214yes—
43#16840099C2214yes—
44#16840094C2214yes—
45#16840089C2214yes—
46#16840088C2214yes—
47#16840082C2214yes—
48#16840077C2214yes—
49#16840072C2214yescomplemented
50#16840068C2214yescomplemented
51#16840065C2214yescomplemented
52#16840061C2214yescomplemented
53#16840056C2214yescomplemented
54#16840051C2214yescomplemented
55#16840046C2214yescomplemented
56#16840041C2214yescomplemented
57#16839821C2214yes—
58#16839821C2212yes—
59#16839821C2212yes—
60#16839798C2214yes—
61C2×#14519706C2212yes—
62#16839759C2214yes—
63#16839756C2214yes—
64C2×#14519689C2212yes—
65C2×#14519688C2211yescharacteristic
66C61×#533C2211yescharacteristic
67C61×#533C2211yescharacteristic
68C61×#500C2212yescomplemented
69C61×#499C2214yescomplemented
70C61×#498C2212yescomplemented
71C61×#427C2214yes—
72C2×C4×C8×C61C2211yescharacteristic
73C2×#11758222C2312yes—
74C22×C4×D61C2311yescharacteristic
75C22×#92418C2312yes—
76#14519777C2311yescharacteristic
77#14519777C2312yes—
78#14519776—24no—
79#14519776C2314yes—
80#14519775—24no—
81#14519774C2311yescharacteristic
82#14519772—24no—
83#14519772—24no—
84#14519769C2312yes—
85#14519768—24no—
86#14519768C2314yes—
87#14519767—24no—
88#14519767—24no—
89#14519766—24no—
90#14519765C2312yes—
91#14519762—24no—
92#14519762—24no—
93#14519761—22no—
94#14519757—24no—
95#14519757C2314yes—
96#14519756—22no—
97#14519755—24no—
98#14519755—24no—
99#14519754—24no—
100#14519753—24no—
101#14519753—24no—
102#14519752—24no—
103#14519752C2314yes—
104#14519751—22no—
105#14519747—22no—
106C4×Dic122—22no—
107C4×Dic122—24no—
108#14519742C2314yes—
109C4×D244—24no—
110C4×D244—22no—
111#14519739C2314yes—
112#14519737—24no—
113#14519736C2312yes—
114#14519735C2314yes—
115#14519734—22no—
116#14519734—22no—
117#14519734D414yes—
118#14519734C2312yes—
119#14519734C2312yes—
120#14519733—22no—
121C2×#11758196D412yes—
122C4⋊C4×D61C2314yes—
123C2×#11758198D412yes—
124C22⋊C4×D61—22no—
125C2×C4×Dic61C2311yescharacteristic
126C42×D61—22no—
127C42×D61—22no—
128#14519710C2314yes—
129#14519710C2314yes—
130#14519708C2314yes—
131#14519708C2314yes—
132#14519706C2314yes—
133#14519706C2314yes—
134#14519702C2314yes—
135#14519702C2314yes—
136#14519700C2314yes—
137#14519700C2314yes—
138#14519699C2314yes—
139#14519699C2314yes—
140#14519697C2314yes—
141#14519697C2314yes—
142#14519689C2314yes—
143#14519689C2314yes—
144#14519688C2312yes—
145#14519688C2312yes—
146C22×C8×C61C2312yes—
147C4⋊Q8×C61C2312yes—
148C4⋊Q8×C61C2312yes—
149#125×C61C2312yes—
150#125×C61C2312yes—
151C22⋊Q8×C61—22no—
152C22⋊Q8×C61—22no—
153C4×Q8×C61—22no—
154C4×Q8×C61—22no—
155#117×C61C2312yes—
156#117×C61C2312yes—
157C2×C42×C61C2311yescharacteristic
158#107×C61C2314yes—
159#107×C61C2314yes—
160#106×C61C2314yes—
161#106×C61C2314yes—
162C4×C8×C61C2312yes—
163C4×C8×C61C2311yescharacteristic
164C4×C8×C61C2311yescharacteristic
165C23×D61C2×D411yescharacteristic
166C2×Q8×C61—22no—
167C2×Q8×C61—22no—
168C22×C4×C61C2×D412yes—
169C22×C4×C61C2411yescharacteristic
170C2×C22⋊D61—22no—
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D412yes—
173#11758222—28no—
174C2×D244—24no—
175C2×D244—22no—
176C2×D244C2412yes—
177C2×C4×D61—24no—
178C2×C4×D61—22no—
179C2×C4×D61—24no—
180C2×C4×D61—24no—
181C2×C4×D61C2×D412yes—
182C2×C4×D61C2412yes—
183C2×Dic122—22no—
184C2×Dic122—24no—
185C2×Dic122C2412yes—
186C2×C8×C61—24no—
187C2×C8×C61C2412yes—
188C2×C8×C61C2412yes—
189C4⋊C4×C61—24no—
190C4⋊C4×C61—24no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61C2414yes—
194C4⋊C4×C61C2414yes—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—22no—
197C42×C61—22no—
198C42×C61—22no—
199C42×C61C2×D412yes—
200C42×C61C2411yescharacteristic
201C42×C61C2411yescharacteristic
202C22⋊Dic61—22no—
203#11758198—24no—
204#11758198—24no—
205#11758198—24no—
206#11758198—24no—
207#11758198—24no—
208#11758198C2×D414yes—
209#11758198C2×D414yes—
210C4⋊Dic61—22no—
211C4⋊Dic61—22no—
212C4⋊Dic61—24no—
213C4⋊Dic61—24no—
214C4⋊Dic61C2414yes—
215#11758196—24no—
216#11758196—24no—
217#11758196—24no—
218#11758196—24no—
219#11758196—24no—
220#11758196C2×D414yes—
221#11758196C2×D414yes—
222C4×Dic61—22no—
223C4×Dic61—24no—
224C4×Dic61—24no—
225C4×Dic61C2×D412yes—
226C4×Dic61C2412yes—
227C2×#92418—24no—
228C2×#92418C2414yes—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—42no—
233C22×D61C22×D412yes—
234Q8×C61—82no—
235Q8×C61—82no—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—42no—
239C2×C4×C61C22×D412yes—
240C2×C4×C61C22×D412yes—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C612−1+414yes—
243C2×C4×C61C2511yescharacteristic
244C22⋊D61—44no—
245C2×Dic61—22no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—42no—
249C2×Dic61C22×D412yes—
250C2×Dic61C22×D414yes—
251D244—24no—
252D244—82no—
253D244—84no—
254C4×D61—24no—
255C4×D61—24no—
256C4×D61—44no—
257C4×D61—84no—
258C4×D61—84no—
259Dic122—24no—
260Dic122—84no—
261Dic122—82no—
262C8×C61—22no—
263C8×C61—22no—
264#92418—24no—
265#60859—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61#53612yes—
268C22×C61C23×D411yescharacteristic
269C2×D61—24no—
270C2×D61—22no—
271C2×D61—44no—
272C2×D61—42no—
273C2×D61—82no—
274C4×C61—22no—
275C4×C61—22no—
276C4×C61—44no—
277C4×C61—44no—
278C4×C61#53612yes—
279C4×C61C2×2−1+412yes—
280Dic61—24no—
281Dic61—22no—
282Dic61—44no—
283Dic61—42no—
284#3458—611no—
285#3455—611no—
286#3244—614no—
287#3243—614no—
288#3242—614no—
289#3241—614no—
290#3162—611no—
291#3160D6111yescharacteristic, complemented
292#3135—611no—
293#3131—611no—
294#3091—614no—
295#3076—614no—
296C2×#433—611no—
297C2×C61—22no—
298C2×C61C2×#53611yescharacteristic
299C2×C61#358412yes—
300D61—42no—
301D61—82no—
302#544—614no—
303#544—614no—
304#544—1222no—
305#543—1222no—
306#542—614no—
307#541—612no—
308#540—612no—
309#539—1222no—
310#538—1222no—
311#533—612no—
312#533—611no—
313#533—1224no—
314#533C2×D6111yescharacteristic
315#533C2×D6111yescharacteristic
316#532—611no—
317#529—1224no—
318#525—612no—
319C4×#41—612no—
320C2×#117—611no—
321C2×#117—612no—
322#501—614no—
323#501—614no—
324#500C2×D6112yescomplemented
325#499C2×D6114yescomplemented
326#498C2×D6112yescomplemented
327#491—612no—
328#490—612no—
329#489—614no—
330#488—612no—
331#488—612no—
332#487—612no—
333#486—612no—
334#485—614no—
335#485—614no—
336#484—614no—
337#483—614no—
338#482—614no—
339#482—614no—
340#481—614no—
341#480—614no—
342#479—614no—
343#479—614no—
344#478—614no—
345#478—614no—
346#477—614no—
347#476—614no—
348#475—614no—
349#474—614no—
350#433—614no—
351#433—612no—
352#433—612no—
353#427—614no—
354#427C2×D6114yes—
355C2×C4⋊C8—612no—
356#416—614no—
357#416—614no—
358C2×C22⋊C8—612no—
359C2×#97—611no—
360C2×C4×C8C2×D6111yescharacteristic
361C61#6085911yescharacteristic, complemented, Sylow
362C2×#41—612no—
363C23×C4—611no—
364C22×C8—612no—
365C22×C8C22×D6112yes—
366C4⋊Q8—614no—
367C4⋊Q8—612no—
368C4⋊Q8—611no—
369C4⋊Q8—1224no—
370C4⋊Q8—1224no—
371C4⋊Q8C22×D6112yes—
372C4⋊Q8C22×D6112yes—
373#127—611no—
374#126—1224no—
375#126—1224no—
376#125—614no—
377#125—612no—
378#125—1224no—
379#125—1224no—
380#125—1224no—
381#125C22×D6112yes—
382#125C22×D6112yes—
383#124—612no—
384#124—1224no—
385#123—1222no—
386#123—1224no—
387#123—1224no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—614no—
391C22⋊Q8—1224no—
392C22⋊Q8—1224no—
393C22⋊Q8—1224no—
394C22⋊Q8—1224no—
395C22⋊Q8—1224no—
396C22⋊Q8—1224no—
397C22⋊Q8—1222no—
398#121—614no—
399#121—1224no—
400#121—1222no—
401#120—1222no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—614no—
405C4×Q8—1224no—
406C4×Q8—1222no—
407C4×D4—614no—
408C4×D4—1224no—
409C4×D4—1224no—
410C4×D4—1222no—
411#117—614no—
412#117—612no—
413#117—612no—
414#117—612no—
415#117—614no—
416#117—1222no—
417#117—1222no—
418#117—1222no—
419#117C22×D6112yes—
420#117C22×D6112yes—
421C2×C4⋊C4—612no—
422C2×C4⋊C4—614no—
423C2×C22⋊C4—612no—
424C2×C22⋊C4—1222no—
425C2×C42—611no—
426C2×C42—1222no—
427C2×C42—1222no—
428C2×C42C22×D6111yescharacteristic
429#107—614no—
430#107—614no—
431#107C22×D6114yes—
432#107C22×D6114yes—
433#106—614no—
434#106—614no—
435#106C22×D6114yes—
436#106C22×D6114yes—
437C4⋊C8—614no—
438C4⋊C8—614no—
439#103—614no—
440#103—614no—
441#103—614no—
442#103—614no—
443#102—614no—
444#102—614no—
445#102—614no—
446#102—614no—
447C22⋊C8—614no—
448C22⋊C8—614no—
449#97—612no—
450#97—612no—
451C4×C8C22×D6112yes—
452C4×C8C22×D6111yescharacteristic
453C4×C8C22×D6111yescharacteristic
454C24—611no—
455#41—1228no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—612no—
459C2×Q8—1224no—
460C2×Q8—1222no—
461C2×D4—612no—
462C2×D4—1224no—
463C2×D4—1222no—
464C2×D4—1222no—
465C22×C4—611no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—1224no—
470C22×C4—1224no—
471C22×C4—1224no—
472C22×C4—1222no—
473C22×C4D4×D6112yes—
474C22×C4C23×D6111yescharacteristic
475C2×C8—24no—
476C2×C8—614no—
477C2×C8—1224no—
478C2×C8C23×D6112yes—
479C2×C8C23×D6112yes—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—614no—
485C4⋊C4—614no—
486C4⋊C4—614no—
487C4⋊C4—1224no—
488C4⋊C4—1224no—
489C4⋊C4—1224no—
490C4⋊C4—1224no—
491C4⋊C4—1224no—
492C4⋊C4—1224no—
493C4⋊C4—1224no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4C23×D6114yes—
497C4⋊C4C23×D6114yes—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—614no—
501C22⋊C4—614no—
502C22⋊C4—1224no—
503C22⋊C4—1224no—
504C22⋊C4—1224no—
505C22⋊C4—1224no—
506C22⋊C4—1222no—
507C22⋊C4—1224no—
508C42—22no—
509C42—22no—
510C42—612no—
511C42—612no—
512C42—1224no—
513C42—1224no—
514C42—1222no—
515C42D4×D6112yes—
516C42C23×D6111yescharacteristic
517C42C23×D6111yescharacteristic
518C23—612no—
519C23—1222no—
520C23—1222no—
521C23—2442no—
522C23C2×D4×D6111yescharacteristic
523Q8—82no—
524Q8—82no—
525Q8—1224no—
526Q8—4882no—
527Q8—4884no—
528D4—1224no—
529D4—4882no—
530D4—4884no—
531D4—2444no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—614no—
536C2×C4—612no—
537C2×C4—1222no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1222no—
541C2×C4—1224no—
542C2×C4—4884no—
543C2×C4—4884no—
544C2×C4—2442no—
545C2×C4—2444no—
546C2×C4C2×D4×D6112yes—
547C2×C4C2×D4×D6112yes—
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4#1451982514yes—
550C2×C4C24×D6111yescharacteristic
551C8—22no—
552C8—22no—
553C8—1224no—
554C22—22no—
555C22—1224no—
556C22—1222no—
557C22—4882no—
558C22—2444no—
559C22—2442no—
560C22#1684033712yes—
561C22C22×D4×D6111yescharacteristic
562C4—22no—
563C4—22no—
564C4—44no—
565C4—44no—
566C4—1224no—
567C4—1222no—
568C4—2444no—
569C4—2442no—
570C4#1684033712yes—
571C4C2×#1451982512yes—
572C2—22no—
573C2—4882no—
574C2—2442no—
575C2C2×#1684033711yescharacteristic
576C2#2146058812yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3160C2
socleC22×C61C23×D4
prime radical#21459211C2
hypercenter#3160D61
hypocenterC61#60859
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
1. #21459222C2semidirect
3. #21457120C2semidirect
4. #21457105C2semidirect
5. #21457090C2semidirect
6. #21457075C2semidirect
7. #21456341C2semidirect
8. #21456167C2semidirect
9. #21456142C2semidirect
10. #21455736C2semidirect
11. #21455605C2semidirect
12. C2×#16839821C2semidirect
13. C61×#3160C2semidirect
32. #16840172C22semidirect
33. #16840168C22semidirect
34. #16840132C22semidirect
35. #16840128C22semidirect
36. #16840124C22semidirect
37. #16840117C22semidirect
38. #16840117C22semidirect
39. #16840113C22semidirect
40. #16840109C22semidirect
49. #16840072C22semidirect
50. #16840068C22semidirect
51. #16840065C22semidirect
52. #16840061C22semidirect
53. #16840056C22semidirect
54. #16840051C22semidirect
55. #16840046C22semidirect
56. #16840041C22semidirect
68. C61×#500C22semidirect
69. C61×#499C22semidirect
70. C61×#498C22semidirect
291. #3160D61semidirect
324. #500C2×D61semidirect
325. #499C2×D61semidirect
326. #498C2×D61semidirect
361. C61#60859semidirect

Maximal subgroups

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

subgroupindex
#214592222
#214592112
#214571202
#214571052
#214570902
#214570752
#214563412
#214561672
#214561422
#214557362
#214556052
C2×#168398212
C61×#31602
#6085961