Finite Group Databasebeta

← #26198722

Subgroups — #26198722

Order 12032.

618 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253970C211yesmaximal, characteristic, complemented
2#20253176C211yesmaximal, characteristic
3#41×C47⋊C8C211yesmaximal, characteristic
4#20242848C211yesmaximal, characteristic, complemented
5#20242844C211yesmaximal, characteristic, complemented
6#20242841C211yesmaximal, characteristic, complemented
7#20242837C211yesmaximal, characteristic, complemented
8#20242790C211yesmaximal, characteristic, complemented
9#20242785C211yesmaximal, characteristic, complemented
10#20242756C211yesmaximal, characteristic, complemented
11#20242751C211yesmaximal, characteristic, complemented
12C2×#15688875C211yesmaximal, characteristic, complemented
13#20242046C211yesmaximal, characteristic, complemented
14#20242045C211yesmaximal, characteristic, complemented
15C47×#3076C211yesmaximal, characteristic
16#15689680C2211yescharacteristic
17#15689526C2211yescharacteristic
18#15689502C2211yescharacteristic
19#15689475C2211yescharacteristic
20#15689473C2211yescharacteristic
21#15689430—21no—
22#15689423—21no—
23#15689418C2211yescharacteristic
24#15689413C2211yescharacteristic
25#15689408C2211yescharacteristic
26#15689403C2211yescharacteristic
27#15689362—21no—
28#15689362—21no—
29Q8×C47⋊C8C2211yescharacteristic
30Q8×C47⋊C8C2211yescharacteristic
31C4×#10625337—21no—
32C4×#10625337—21no—
33D4×C47⋊C8—21no—
34D4×C47⋊C8—21no—
35D4×C47⋊C8C2211yescharacteristic
36D4×C47⋊C8C2211yescharacteristic
37#15689176—21no—
38#15689176C2211yescharacteristic
39#15689159—21no—
40C2×C4⋊(C47⋊C8)C2211yescharacteristic
41#15689119—21no—
42C2×C4×C47⋊C8—21no—
43C2×C4×C47⋊C8C2211yescharacteristic
44#15688875C2211yescharacteristic
45#15688875C2211yescharacteristic
46#15688875C2211yescharacteristic
47#15688875C2211yescharacteristic
48#15688872C2211yescharacteristic
49#15688872C2211yescharacteristic
50#15688871C2211yescharacteristic
51#15688871C2211yescharacteristic
52#15688869C2211yescharacteristic
53#15688869C2211yescharacteristic
54#15688868C2211yescharacteristic
55#15688868C2211yescharacteristic
56C47×#533C2211yescharacteristic
57C4×#41×C47C2211yescharacteristic
58C47×#479—21no—
59C47×#478—21no—
60C47×#477C2211yescharacteristic
61C47×#476C2211yescharacteristic
62C47×#475C2211yescharacteristic
63C47×#474C2211yescharacteristic
64C47×#427—21no—
65C2×C4⋊C8×C47C2211yescharacteristic
66C47×#416—21no—
67C2×#10625337—21no—
68C2×#10625337—21no—
69C22×C47⋊C8—21no—
70C22×C47⋊C8—21no—
71C22×C47⋊C8—21no—
72C22×C47⋊C8D411yescharacteristic
73C22×C47⋊C8D411yescharacteristic
74#13094024C2311yescharacteristic
75#13094024C2311yescharacteristic
76#13094016—21no—
77#13094003—21no—
78#13094003—21no—
79C4×Dic94—21no—
80C4×Dic94—21no—
81#13093983—21no—
82#13093957—21no—
83#13093957—21no—
84#13093957—21no—
85#13093957—21no—
86#13093955—21no—
87#13093955—21no—
88C4⋊(C47⋊C8)—21no—
89C4⋊(C47⋊C8)—21no—
90C4⋊(C47⋊C8)—21no—
91C4⋊(C47⋊C8)C2311yescharacteristic
92C4⋊(C47⋊C8)C2311yescharacteristic
93C4⋊(C47⋊C8)C2311yescharacteristic
94C4⋊(C47⋊C8)C2311yescharacteristic
95#13093948—21no—
96#13093948—21no—
97#13093948—21no—
98#13093948—21no—
99#13093947—21no—
100#13093947—21no—
101#13093945—21no—
102#13093945—21no—
103C22⋊(C47⋊C8)—21no—
104C22⋊(C47⋊C8)—21no—
105C22⋊(C47⋊C8)—21no—
106C4×C47⋊C8—21no—
107C4×C47⋊C8—21no—
108C4×C47⋊C8—21no—
109C4×C47⋊C8D411yescharacteristic
110C4×C47⋊C8D411yescharacteristic
111C4×C47⋊C8C2311yescharacteristic
112C4×C47⋊C8C2311yescharacteristic
113C2×#41×C47D411yescharacteristic
114C22×C8×C47—21no—
115C4⋊Q8×C47C2311yescharacteristic
116C4⋊Q8×C47C2311yescharacteristic
117#125×C47—21no—
118C22⋊Q8×C47—21no—
119C22⋊Q8×C47—21no—
120C4×Q8×C47—21no—
121C4×Q8×C47—21no—
122C4×Q8×C47D411yescharacteristic
123C4×Q8×C47C2311yescharacteristic
124C4×D4×C47—21no—
125C4×D4×C47—21no—
126C4×D4×C47D411yescharacteristic
127C4×D4×C47C2311yescharacteristic
128#117×C47—21no—
129#117×C47—21no—
130#117×C47D411yescharacteristic
131C2×C42×C47—21no—
132C2×C42×C47C2311yescharacteristic
133#107×C47—21no—
134#107×C47—21no—
135#106×C47—21no—
136#106×C47—21no—
137C4⋊C8×C47—21no—
138C4⋊C8×C47C2311yescharacteristic
139C4⋊C8×C47C2311yescharacteristic
140#103×C47—21no—
141#103×C47—21no—
142#102×C47—21no—
143#102×C47—21no—
144#41×C47—22no—
145#41×C47Q1614yescomplemented
146C2×Q8×C47—21no—
147C2×Q8×C47—21no—
148C2×Q8×C47C2×D411yescharacteristic
149C2×D4×C47—21no—
150C2×D4×C47C2×D411yescharacteristic
151C22×C4×C47—21no—
152C22×C4×C47—21no—
153C22×C4×C47—21no—
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47#4111yescharacteristic
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—42no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—41no—
169C4⋊C4×C47—41no—
170C4⋊C4×C47C2×D411yescharacteristic
171C4⋊C4×C47C2×D411yescharacteristic
172C22⋊C4×C47—21no—
173C22⋊C4×C47—21no—
174C22⋊C4×C47—21no—
175C22⋊C4×C47—41no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47—41no—
180C42×C47C2×D411yescharacteristic
181C42×C47C2×D411yescharacteristic
182C42×C47#4111yescharacteristic
183C42×C47C2411yescharacteristic
184C22⋊Dic47—41no—
185#10625337—42no—
186#10625337—42no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191Dic47⋊C4—41no—
192Dic47⋊C4—41no—
193C4×Dic47—41no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—41no—
200C2×C47⋊C8—42no—
201C2×C47⋊C8—42no—
202C2×C47⋊C8—42no—
203C2×C47⋊C8C2×D411yescharacteristic
204C2×C47⋊C8C2×D411yescharacteristic
205C2×C47⋊C8C2×D411yescharacteristic
206C2×C47⋊C8C2×D411yescharacteristic
207C23×C47—21no—
208C23×C47#12111yescharacteristic
209Q8×C47—21no—
210Q8×C47—41no—
211Q8×C47—41no—
212Q8×C47—81no—
213Q8×C47C2×Q1612yes—
214D4×C47—24no—
215D4×C47—21no—
216D4×C47C2×Q1612yes—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—24no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—41no—
224C2×C4×C47—42no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47#12111yescharacteristic
230C2×C4×C47#12111yescharacteristic
231C2×C4×C47#12111yescharacteristic
232C2×C4×C47C2×Q1612yes—
233C2×C4×C47#13512yes—
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×Dic47—41no—
238C2×Dic47—41no—
239Dic94—41no—
240Dic94—41no—
241Dic94—81no—
242C8×C47—41no—
243C8×C47—41no—
244C47⋊C8—21no—
245C47⋊C8—21no—
246C47⋊C8—21no—
247C47⋊C8—41no—
248C47⋊C8—41no—
249C47⋊C8—41no—
250C47⋊C8—41no—
251C47⋊C8#13512yes—
252#19126—471nomaximal, Sylow
253C22×C47—21no—
254C22×C47—21no—
255C22×C47—22no—
256C22×C47—41no—
257C22×C47—41no—
258C22×C47#46712yes—
259C22×C47C2×#12111yescharacteristic
260C4×C47—21no—
261C4×C47—22no—
262C4×C47—21no—
263C4×C47—21no—
264C4×C47—41no—
265C4×C47—41no—
266C4×C47—81no—
267C4×C47—81no—
268C4×C47#46712yes—
269C4×C47C22×Q1611yescharacteristic
270C4×C47C2×#13511yescharacteristic
271C4×C47C2×#13511yescharacteristic
272C4×C47#57511yescharacteristic
273Dic47—81no—
274Dic47—81no—
275#3170—471no—
276#3076—471no—
277#3076D4711yescharacteristic
278C8×#41—471no—
279#1684—471no—
280#1684—471no—
281#1683—471no—
282#1683—471no—
283#1676—471no—
284#1675—471no—
285#1670—471no—
286#1669—471no—
287C2×#335—471no—
288#1542—471no—
289#1542—471no—
290C2×C47—22no—
291C2×C47—42no—
292C2×C47#303511yescharacteristic
293C2×C47C2×#46711yescharacteristic
294C2×C47#306711yescharacteristic
295#533—471no—
296#533C2×D4711yescharacteristic
297C4×#41C2×D4711yescharacteristic
298#500—471no—
299#494—471no—
300#487—471no—
301#487—471no—
302#479—21no—
303#479—941no—
304#478—21no—
305#478—941no—
306#477—471no—
307#477C2×D4711yescharacteristic
308#476—471no—
309#476C2×D4711yescharacteristic
310#475—471no—
311#475C2×D4711yescharacteristic
312#474—471no—
313#474C2×D4711yescharacteristic
314#467—941no—
315#467—941no—
316C8×Q8—471no—
317C8×Q8—471no—
318C4×Q16—941no—
319C4×Q16—941no—
320C8×D4—471no—
321C8×D4—471no—
322C8×D4—941no—
323C8×D4—941no—
324#431—471no—
325#431—941no—
326#427—21no—
327#427—941no—
328C2×C4⋊C8—471no—
329C2×C4⋊C8C2×D4711yescharacteristic
330#416—21no—
331#416—941no—
332C2×C4×C8—471no—
333C2×C4×C8—941no—
334#335—471no—
335#335—471no—
336#335—471no—
337#335—471no—
338#333—471no—
339#333—471no—
340#333—471no—
341#333—471no—
342#332—471no—
343#332—471no—
344#332—471no—
345#332—471no—
346C47#1912611yescharacteristic, complemented, Sylow
347C2×#41C22⋊D4711yescharacteristic
348C2×Q16—941no—
349C2×Q16—941no—
350C22×C8—21no—
351C22×C8—471no—
352C22×C8—471no—
353C22×C8—941no—
354C22×C8—941no—
355C22×C8—941no—
356C4⋊Q8—471no—
357C4⋊Q8—471no—
358C4⋊Q8C22×D4711yescharacteristic
359C4⋊Q8C22×D4711yescharacteristic
360#125—21no—
361#125—941no—
362C22⋊Q8—21no—
363C22⋊Q8—21no—
364C22⋊Q8—941no—
365C22⋊Q8—941no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—941no—
369C4×Q8—941no—
370C4×Q8C22⋊D4711yescharacteristic
371C4×Q8C22×D4711yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C22⋊D4711yescharacteristic
375C4×D4C22×D4711yescharacteristic
376#117—21no—
377#117—21no—
378#117—941no—
379#117C22⋊D4711yescharacteristic
380C2×C42—21no—
381C2×C42C22×D4711yescharacteristic
382#107—21no—
383#107—21no—
384#107—941no—
385#107—941no—
386#107—941no—
387#107—941no—
388#106—21no—
389#106—21no—
390#106—941no—
391#106—941no—
392C4⋊C8—21no—
393C4⋊C8—471no—
394C4⋊C8—471no—
395C4⋊C8—471no—
396C4⋊C8—471no—
397C4⋊C8—941no—
398C4⋊C8—941no—
399C4⋊C8—941no—
400C4⋊C8C22×D4711yescharacteristic
401C4⋊C8C22×D4711yescharacteristic
402#103—21no—
403#103—21no—
404#103—941no—
405#103—941no—
406#103—941no—
407#103—941no—
408#103—941no—
409#103—941no—
410#102—21no—
411#102—21no—
412#102—941no—
413#102—941no—
414C22⋊C8—941no—
415C22⋊C8—941no—
416C22⋊C8—941no—
417C4×C8—471no—
418C4×C8—471no—
419C4×C8—471no—
420C4×C8—471no—
421C4×C8—941no—
422C4×C8—941no—
423C4×C8—941no—
424#41—22no—
425#41#1062533714yescomplemented
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—941no—
429C2×Q8—941no—
430C2×Q8C2×C22⋊D4711yescharacteristic
431C2×D4—21no—
432C2×D4C2×C22⋊D4711yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4D4×D4711yescharacteristic
437C22×C4#1062535211yescharacteristic
438C22×C4C2×C22⋊D4711yescharacteristic
439Q16—1882no—
440Q16—1882no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—42no—
444C2×C8—471no—
445C2×C8—471no—
446C2×C8—471no—
447C2×C8—471no—
448C2×C8—942no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—1882no—
454C2×C8—1881no—
455C2×C8—1882no—
456C2×C8—1882no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C4⋊C4—1881no—
470C4⋊C4—1881no—
471C4⋊C4C2×C22⋊D4711yescharacteristic
472C4⋊C4C2×C22⋊D4711yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—41no—
477C22⋊C4—1881no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—1881no—
483C42D4×D4711yescharacteristic
484C42#1062535211yescharacteristic
485C42C2×C22⋊D4711yescharacteristic
486C42C23×D4711yescharacteristic
487C23—21no—
488C23#1309399611yescharacteristic
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493Q8—3761no—
494Q8—1881no—
495Q8—1881no—
496Q8C2×#1062533712yes—
497D4—24no—
498D4—21no—
499D4C2×#1062533712yes—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—1881no—
513C2×C4—1881no—
514C2×C4#1309399611yescharacteristic
515C2×C4#1309399611yescharacteristic
516C2×C4#1309399611yescharacteristic
517C2×C4C2×#1062533712yes—
518C2×C4#1309404412yes—
519C2×C4C2×D4×D4711yescharacteristic
520C2×C4C22×C22⋊D4711yescharacteristic
521C2×C4C2×#1062535211yescharacteristic
522C8—41no—
523C8—41no—
524C8—472no—
525C8—941no—
526C8—941no—
527C8—941no—
528C8—1881no—
529C8—1881no—
530C8—1881no—
531C8—1881no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—41no—
536C22—41no—
537C22#1568936212yes—
538C22C2×#1309399611yescharacteristic
539C4—22no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—3761no—
548C4—3761no—
549C4#1568936212yes—
550C4C22×#1062533711yescharacteristic
551C4#135×D4711yescharacteristic
552C4C2×#1309404411yescharacteristic
553C4#1568992011yescharacteristic
554C2—22no—
555C2—42no—
556C2#2025283911yescharacteristic
557C2C2×#1568936211yescharacteristic
558C2#2025310511yescharacteristic
subgroupG/H
centerC2×C4#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3076C2
socleC22×C47C2×#121
prime radicalC2×#41×C47D4
hypercenter#3076D47
hypocenterC47#19126
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #20253970C2semidirect
4. #20242848C2semidirect
5. #20242844C2semidirect
6. #20242841C2semidirect
7. #20242837C2semidirect
8. #20242790C2semidirect
9. #20242785C2semidirect
10. #20242756C2semidirect
11. #20242751C2semidirect
12. C2×#15688875C2semidirect
13. #20242046C2semidirect
14. #20242045C2semidirect
145. #41×C47Q16semidirect
346. C47#19126semidirect
425. #41#10625337semidirect

Maximal subgroups

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

subgroupindex
#202539702
#202531762
#41×C47⋊C82
#202428482
#202428442
#202428412
#202428372
#202427902
#202427852
#202427562
#202427512
C2×#156888752
#202420462
#202420452
C47×#30762
#1912647