Finite Group Databasebeta

← #27088535

Subgroups — #27088535

Order 12032.

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#20256791C211yesmaximal, characteristic, complemented
2#20256780C211yesmaximal, characteristic
3#20254689C214yesmaximal, complemented
4#20254674C214yesmaximal, complemented
5#20254659C214yesmaximal, complemented
6#20254644C214yesmaximal, complemented
7#20253910C211yesmaximal, characteristic, complemented
8#20253736C211yesmaximal, characteristic, complemented
9#20253711C211yesmaximal, characteristic, complemented
10#20253305C214yesmaximal, complemented
11#20253174C214yesmaximal, complemented
12C2×#15689182C211yesmaximal, characteristic, complemented
13C47×#3160C211yesmaximal, characteristic, complemented
14#15689753C2214yes—
15#15689752—22no—
16#15689750C2214yes—
17#15689748—22no—
18#15689747C2214yes—
19#15689736C2212yes—
20#15689729C2212yes—
21#15689722—22no—
22#15689713—22no—
23#15689680C2211yescharacteristic
24#15689677—24no—
25#15689676C2212yes—
26#15689673C2211yescharacteristic
27#15689660—24no—
28#15689635C2212yes—
29#15689591C2212yes—
30C2×#13093981C2211yescharacteristic
31#117×D47C2212yes—
32#15689533C2214yescomplemented
33#15689529C2214yescomplemented
34#15689493C2212yescomplemented
35#15689489C2212yescomplemented
36#15689485C2214yescomplemented
37#15689478C2212yescomplemented
38#15689478C2212yescomplemented
39#15689474C2212yescomplemented
40#15689470C2212yescomplemented
41#15689467C2214yes—
42#15689466C2214yes—
43#15689460C2214yes—
44#15689455C2214yes—
45#15689450C2214yes—
46#15689449C2214yes—
47#15689443C2214yes—
48#15689438C2214yes—
49#15689433C2214yescomplemented
50#15689429C2214yescomplemented
51#15689426C2214yescomplemented
52#15689422C2214yescomplemented
53#15689417C2214yescomplemented
54#15689412C2214yescomplemented
55#15689407C2214yescomplemented
56#15689402C2214yescomplemented
57#15689182C2214yes—
58#15689182C2212yes—
59#15689182C2212yes—
60#15689159C2214yes—
61C2×Dic47⋊C8C2212yes—
62#15689120C2214yes—
63#15689117C2214yes—
64C2×(C2×D47)⋊C8C2212yes—
65C2×#13093935C2211yescharacteristic
66C47×#533C2211yescharacteristic
67C47×#533C2211yescharacteristic
68C47×#500C2212yescomplemented
69C47×#499C2214yescomplemented
70C47×#498C2212yescomplemented
71C47×#427C2214yes—
72C2×C4×C8×C47C2211yescharacteristic
73C2×#10625350C2312yes—
74C22×C4×D47C2311yescharacteristic
75C22×C47⋊C8C2312yes—
76#13094024C2311yescharacteristic
77#13094024C2312yes—
78#13094023—24no—
79#13094023C2314yes—
80#13094022—24no—
81#13094021C2311yescharacteristic
82#13094019—24no—
83#13094019—24no—
84#13094016C2312yes—
85#13094015—24no—
86#13094015C2314yes—
87#13094014—24no—
88#13094014—24no—
89#13094013—24no—
90#13094012C2312yes—
91#13094009—24no—
92#13094009—24no—
93#13094008—22no—
94#13094004—24no—
95#13094004C2314yes—
96#13094003—22no—
97#13094002—24no—
98#13094002—24no—
99#13094001—24no—
100#13094000—24no—
101#13094000—24no—
102#13093999—24no—
103#13093999C2314yes—
104#13093998—22no—
105#13093994—22no—
106C4×Dic94—22no—
107C4×Dic94—24no—
108#13093989C2314yes—
109C4×D188—24no—
110C4×D188—22no—
111#13093986C2314yes—
112(C22⋊D47)⋊C4—24no—
113#13093983C2312yes—
114#13093982C2314yes—
115#13093981—22no—
116#13093981—22no—
117#13093981D414yes—
118#13093981C2312yes—
119#13093981C2312yes—
120#13093980—22no—
121C2×Dic47⋊C4D412yes—
122C4⋊C4×D47C2314yes—
123C2×(C2×D47)⋊C4D412yes—
124C22⋊C4×D47—22no—
125C2×C4×Dic47C2311yescharacteristic
126C42×D47—22no—
127C42×D47—22no—
128#13093957C2314yes—
129#13093957C2314yes—
130#13093955C2314yes—
131#13093955C2314yes—
132Dic47⋊C8C2314yes—
133Dic47⋊C8C2314yes—
134#13093949C2314yes—
135#13093949C2314yes—
136#13093947C2314yes—
137#13093947C2314yes—
138#13093946C2314yes—
139#13093946C2314yes—
140#13093944C2314yes—
141#13093944C2314yes—
142(C2×D47)⋊C8C2314yes—
143(C2×D47)⋊C8C2314yes—
144#13093935C2312yes—
145#13093935C2312yes—
146C22×C8×C47C2312yes—
147C4⋊Q8×C47C2312yes—
148C4⋊Q8×C47C2312yes—
149#125×C47C2312yes—
150#125×C47C2312yes—
151C22⋊Q8×C47—22no—
152C22⋊Q8×C47—22no—
153C4×Q8×C47—22no—
154C4×Q8×C47—22no—
155#117×C47C2312yes—
156#117×C47C2312yes—
157C2×C42×C47C2311yescharacteristic
158#107×C47C2314yes—
159#107×C47C2314yes—
160#106×C47C2314yes—
161#106×C47C2314yes—
162C4×C8×C47C2312yes—
163C4×C8×C47C2311yescharacteristic
164C4×C8×C47C2311yescharacteristic
165C23×D47C2×D411yescharacteristic
166C2×Q8×C47—22no—
167C2×Q8×C47—22no—
168C22×C4×C47C2×D412yes—
169C22×C4×C47C2411yescharacteristic
170C2×C22⋊D47—22no—
171C22×Dic47C2×D411yescharacteristic
172C22×Dic47C2×D412yes—
173#10625350—28no—
174C2×D188—24no—
175C2×D188—22no—
176C2×D188C2412yes—
177C2×C4×D47—24no—
178C2×C4×D47—22no—
179C2×C4×D47—24no—
180C2×C4×D47—24no—
181C2×C4×D47C2×D412yes—
182C2×C4×D47C2412yes—
183C2×Dic94—22no—
184C2×Dic94—24no—
185C2×Dic94C2412yes—
186C2×C8×C47—24no—
187C2×C8×C47C2412yes—
188C2×C8×C47C2412yes—
189C4⋊C4×C47—24no—
190C4⋊C4×C47—24no—
191C4⋊C4×C47—22no—
192C4⋊C4×C47—22no—
193C4⋊C4×C47C2414yes—
194C4⋊C4×C47C2414yes—
195C22⋊C4×C47—22no—
196C22⋊C4×C47—22no—
197C42×C47—22no—
198C42×C47—22no—
199C42×C47C2×D412yes—
200C42×C47C2411yescharacteristic
201C42×C47C2411yescharacteristic
202C22⋊Dic47—22no—
203(C2×D47)⋊C4—24no—
204(C2×D47)⋊C4—24no—
205(C2×D47)⋊C4—24no—
206(C2×D47)⋊C4—24no—
207(C2×D47)⋊C4—24no—
208(C2×D47)⋊C4C2×D414yes—
209(C2×D47)⋊C4C2×D414yes—
210C4⋊Dic47—22no—
211C4⋊Dic47—22no—
212C4⋊Dic47—24no—
213C4⋊Dic47—24no—
214C4⋊Dic47C2414yes—
215Dic47⋊C4—24no—
216Dic47⋊C4—24no—
217Dic47⋊C4—24no—
218Dic47⋊C4—24no—
219Dic47⋊C4—24no—
220Dic47⋊C4C2×D414yes—
221Dic47⋊C4C2×D414yes—
222C4×Dic47—22no—
223C4×Dic47—24no—
224C4×Dic47—24no—
225C4×Dic47C2×D412yes—
226C4×Dic47C2412yes—
227C2×C47⋊C8—24no—
228C2×C47⋊C8C2414yes—
229C23×C47C22×D411yescharacteristic
230C22×D47—22no—
231C22×D47—22no—
232C22×D47—42no—
233C22×D47C22×D412yes—
234Q8×C47—82no—
235Q8×C47—82no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—42no—
239C2×C4×C47C22×D412yes—
240C2×C4×C47C22×D412yes—
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C472−1+414yes—
243C2×C4×C47C2511yescharacteristic
244C22⋊D47—44no—
245C2×Dic47—22no—
246C2×Dic47—24no—
247C2×Dic47—22no—
248C2×Dic47—42no—
249C2×Dic47C22×D412yes—
250C2×Dic47C22×D414yes—
251D188—24no—
252D188—82no—
253D188—84no—
254C4×D47—24no—
255C4×D47—24no—
256C4×D47—44no—
257C4×D47—84no—
258C4×D47—84no—
259Dic94—24no—
260Dic94—84no—
261Dic94—82no—
262C8×C47—22no—
263C8×C47—22no—
264C47⋊C8—24no—
265#60859—471nomaximal, Sylow
266C22×C47—22no—
267C22×C47#53612yes—
268C22×C47C23×D411yescharacteristic
269C2×D47—24no—
270C2×D47—22no—
271C2×D47—44no—
272C2×D47—42no—
273C2×D47—82no—
274C4×C47—22no—
275C4×C47—22no—
276C4×C47—44no—
277C4×C47—44no—
278C4×C47#53612yes—
279C4×C47C2×2−1+412yes—
280Dic47—24no—
281Dic47—22no—
282Dic47—44no—
283Dic47—42no—
284#3458—471no—
285#3455—471no—
286#3244—474no—
287#3243—474no—
288#3242—474no—
289#3241—474no—
290#3162—471no—
291#3160D4711yescharacteristic, complemented
292#3135—471no—
293#3131—471no—
294#3091—474no—
295#3076—474no—
296C2×#433—471no—
297C2×C47—22no—
298C2×C47C2×#53611yescharacteristic
299C2×C47#358412yes—
300D47—42no—
301D47—82no—
302#544—474no—
303#544—474no—
304#544—942no—
305#543—942no—
306#542—474no—
307#541—472no—
308#540—472no—
309#539—942no—
310#538—942no—
311#533—472no—
312#533—471no—
313#533—944no—
314#533C2×D4711yescharacteristic
315#533C2×D4711yescharacteristic
316#532—471no—
317#529—944no—
318#525—472no—
319C4×#41—472no—
320C2×#117—471no—
321C2×#117—472no—
322#501—474no—
323#501—474no—
324#500C2×D4712yescomplemented
325#499C2×D4714yescomplemented
326#498C2×D4712yescomplemented
327#491—472no—
328#490—472no—
329#489—474no—
330#488—472no—
331#488—472no—
332#487—472no—
333#486—472no—
334#485—474no—
335#485—474no—
336#484—474no—
337#483—474no—
338#482—474no—
339#482—474no—
340#481—474no—
341#480—474no—
342#479—474no—
343#479—474no—
344#478—474no—
345#478—474no—
346#477—474no—
347#476—474no—
348#475—474no—
349#474—474no—
350#433—474no—
351#433—472no—
352#433—472no—
353#427—474no—
354#427C2×D4714yes—
355C2×C4⋊C8—472no—
356#416—474no—
357#416—474no—
358C2×C22⋊C8—472no—
359C2×#97—471no—
360C2×C4×C8C2×D4711yescharacteristic
361C47#6085911yescharacteristic, complemented, Sylow
362C2×#41—472no—
363C23×C4—471no—
364C22×C8—472no—
365C22×C8C22×D4712yes—
366C4⋊Q8—474no—
367C4⋊Q8—472no—
368C4⋊Q8—471no—
369C4⋊Q8—944no—
370C4⋊Q8—944no—
371C4⋊Q8C22×D4712yes—
372C4⋊Q8C22×D4712yes—
373#127—471no—
374#126—944no—
375#126—944no—
376#125—474no—
377#125—472no—
378#125—944no—
379#125—944no—
380#125—944no—
381#125C22×D4712yes—
382#125C22×D4712yes—
383#124—472no—
384#124—944no—
385#123—942no—
386#123—944no—
387#123—944no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—474no—
391C22⋊Q8—944no—
392C22⋊Q8—944no—
393C22⋊Q8—944no—
394C22⋊Q8—944no—
395C22⋊Q8—944no—
396C22⋊Q8—944no—
397C22⋊Q8—942no—
398#121—474no—
399#121—944no—
400#121—942no—
401#120—942no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—474no—
405C4×Q8—944no—
406C4×Q8—942no—
407C4×D4—474no—
408C4×D4—944no—
409C4×D4—944no—
410C4×D4—942no—
411#117—474no—
412#117—472no—
413#117—472no—
414#117—472no—
415#117—474no—
416#117—942no—
417#117—942no—
418#117—942no—
419#117C22×D4712yes—
420#117C22×D4712yes—
421C2×C4⋊C4—472no—
422C2×C4⋊C4—474no—
423C2×C22⋊C4—472no—
424C2×C22⋊C4—942no—
425C2×C42—471no—
426C2×C42—942no—
427C2×C42—942no—
428C2×C42C22×D4711yescharacteristic
429#107—474no—
430#107—474no—
431#107C22×D4714yes—
432#107C22×D4714yes—
433#106—474no—
434#106—474no—
435#106C22×D4714yes—
436#106C22×D4714yes—
437C4⋊C8—474no—
438C4⋊C8—474no—
439#103—474no—
440#103—474no—
441#103—474no—
442#103—474no—
443#102—474no—
444#102—474no—
445#102—474no—
446#102—474no—
447C22⋊C8—474no—
448C22⋊C8—474no—
449#97—472no—
450#97—472no—
451C4×C8C22×D4712yes—
452C4×C8C22×D4711yescharacteristic
453C4×C8C22×D4711yescharacteristic
454C24—471no—
455#41—948no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—472no—
459C2×Q8—944no—
460C2×Q8—942no—
461C2×D4—472no—
462C2×D4—944no—
463C2×D4—942no—
464C2×D4—942no—
465C22×C4—471no—
466C22×C4—472no—
467C22×C4—472no—
468C22×C4—472no—
469C22×C4—944no—
470C22×C4—944no—
471C22×C4—944no—
472C22×C4—942no—
473C22×C4D4×D4712yes—
474C22×C4C23×D4711yescharacteristic
475C2×C8—24no—
476C2×C8—474no—
477C2×C8—944no—
478C2×C8C23×D4712yes—
479C2×C8C23×D4712yes—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—474no—
485C4⋊C4—474no—
486C4⋊C4—474no—
487C4⋊C4—944no—
488C4⋊C4—944no—
489C4⋊C4—944no—
490C4⋊C4—944no—
491C4⋊C4—944no—
492C4⋊C4—944no—
493C4⋊C4—944no—
494C4⋊C4—942no—
495C4⋊C4—942no—
496C4⋊C4C23×D4714yes—
497C4⋊C4C23×D4714yes—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—474no—
501C22⋊C4—474no—
502C22⋊C4—944no—
503C22⋊C4—944no—
504C22⋊C4—944no—
505C22⋊C4—944no—
506C22⋊C4—942no—
507C22⋊C4—944no—
508C42—22no—
509C42—22no—
510C42—472no—
511C42—472no—
512C42—944no—
513C42—944no—
514C42—942no—
515C42D4×D4712yes—
516C42C23×D4711yescharacteristic
517C42C23×D4711yescharacteristic
518C23—472no—
519C23—942no—
520C23—942no—
521C23—1882no—
522C23C2×D4×D4711yescharacteristic
523Q8—82no—
524Q8—82no—
525Q8—944no—
526Q8—3762no—
527Q8—3764no—
528D4—944no—
529D4—3762no—
530D4—3764no—
531D4—1884no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—474no—
536C2×C4—472no—
537C2×C4—942no—
538C2×C4—944no—
539C2×C4—944no—
540C2×C4—942no—
541C2×C4—944no—
542C2×C4—3764no—
543C2×C4—3764no—
544C2×C4—1882no—
545C2×C4—1884no—
546C2×C4C2×D4×D4712yes—
547C2×C4C2×D4×D4712yes—
548C2×C4C2×D4×D4711yescharacteristic
549C2×C4#1309407214yes—
550C2×C4C24×D4711yescharacteristic
551C8—22no—
552C8—22no—
553C8—944no—
554C22—22no—
555C22—944no—
556C22—942no—
557C22—3762no—
558C22—1884no—
559C22—1882no—
560C22#1568969812yes—
561C22C22×D4×D4711yescharacteristic
562C4—22no—
563C4—22no—
564C4—44no—
565C4—44no—
566C4—944no—
567C4—942no—
568C4—1884no—
569C4—1882no—
570C4#1568969812yes—
571C4C2×#1309407212yes—
572C2—22no—
573C2—3762no—
574C2—1882no—
575C2C2×#1568969811yescharacteristic
576C2#2025815712yes—
subgroupG/H
centerC2×C4C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#3160C2
socleC22×C47C23×D4
prime radical#20256780C2
hypercenter#3160D47
hypocenterC47#60859
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
1. #20256791C2semidirect
3. #20254689C2semidirect
4. #20254674C2semidirect
5. #20254659C2semidirect
6. #20254644C2semidirect
7. #20253910C2semidirect
8. #20253736C2semidirect
9. #20253711C2semidirect
10. #20253305C2semidirect
11. #20253174C2semidirect
12. C2×#15689182C2semidirect
13. C47×#3160C2semidirect
32. #15689533C22semidirect
33. #15689529C22semidirect
34. #15689493C22semidirect
35. #15689489C22semidirect
36. #15689485C22semidirect
37. #15689478C22semidirect
38. #15689478C22semidirect
39. #15689474C22semidirect
40. #15689470C22semidirect
49. #15689433C22semidirect
50. #15689429C22semidirect
51. #15689426C22semidirect
52. #15689422C22semidirect
53. #15689417C22semidirect
54. #15689412C22semidirect
55. #15689407C22semidirect
56. #15689402C22semidirect
68. C47×#500C22semidirect
69. C47×#499C22semidirect
70. C47×#498C22semidirect
291. #3160D47semidirect
324. #500C2×D47semidirect
325. #499C2×D47semidirect
326. #498C2×D47semidirect
361. C47#60859semidirect

Maximal subgroups

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

subgroupindex
#202567912
#202567802
#202546892
#202546742
#202546592
#202546442
#202539102
#202537362
#202537112
#202533052
#202531742
C2×#156891822
C47×#31602
#6085947