Finite Group Databasebeta

← #26220194

Subgroups — #26220194

Order 12032.

1782 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094049C211yesmaximal, characteristic
2C2×#15689163C211yesmaximal, characteristic, complemented
3C2×#15689129C211yesmaximal, characteristic, complemented
4C2×M16×Dic47C211yesmaximal, characteristic, complemented
5#20245879C211yesmaximal, characteristic, complemented
6C2×#15689024C211yesmaximal, characteristic, complemented
7#20244266C212yesmaximal, complemented
8#20244261C212yesmaximal, complemented
9#20244251C212yesmaximal, complemented
10#20244239C212yesmaximal, complemented
11C2×C47×#417C211yesmaximal, characteristic, complemented
12C23×D188C2211yescharacteristic
13C2×#13094049—24no—
14C2×#13094049—24no—
15C2×#13094049C418yescomplemented
16C22×#10625336C2212yes—
17C22×#10625334C2212yes—
18C22×#10625329C2211yescharacteristic
19C2×#13093983C2211yescharacteristic
20C22×(C2×D47)⋊C4—22no—
21C22×C4×Dic47C2211yescharacteristic
22#15689163—24no—
23C2×#13093957C2212yescomplemented
24C2×#13093955C2212yescomplemented
25#15689129—24no—
26C2×#13093946C2212yes—
27C2×#13093946C2212yes—
28M16×Dic47—24no—
29C2×#13093934C2212yescomplemented
30C2×C8×Dic47C2212yescomplemented
31#15689030C2211yescharacteristic, complemented
32#15689030C2211yescharacteristic, complemented
33#15689026C2211yescharacteristic, complemented
34#15689026C2211yescharacteristic, complemented
35#15689024—22no—
36#15689024C2212yes—
37#15689024C2212yes—
38#15689003—22no—
39C4×C22⋊Dic47—22no—
40C22×#41×C47C2211yescharacteristic
41C22×M16×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C47×#417—24no—
44C2×#103×C47C2212yescomplemented
45C2×#102×C47C2212yescomplemented
46C24×D47—21no—
47C22×D188—22no—
48C22×D188—22no—
49C22×D188—22no—
50C22×D188—22no—
51C22×D188—21no—
52C22×D188—22no—
53C22×D188C2×C414yes—
54C22×D188C2312yes—
55C23×Dic47D412yes—
56#13094049—232no—
57C2×#10625336—24no—
58C2×#10625336—24no—
59C2×#10625336C2×C418yescomplemented
60C2×#10625334—24no—
61C2×#10625334—24no—
62C2×#10625334C2×C418yescomplemented
63C2×#10625329C2×C414yes—
64C2×#10625329D414yes—
65C2×#10625329D414yes—
66C22×C47⋊C8C2312yes—
67#13093983—22no—
68#13093983—22no—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C22⋊Dic47—21no—
72C2×C22⋊Dic47—22no—
73C2×C22⋊Dic47—22no—
74C2×(C2×D47)⋊C4—24no—
75C2×(C2×D47)⋊C4—24no—
76C2×(C2×D47)⋊C4—48no—
77C2×C4×Dic47—22no—
78C2×C4×Dic47—22no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47D412yes—
81C2×C4×Dic47C2311yescharacteristic
82C2×C4×Dic47C2311yescharacteristic
83#13093957—24no—
84#13093955—24no—
85#13093946—24no—
86#13093946—24no—
87#13093934—24no—
88C8×Dic47—24no—
89C47⋊#95—22no—
90C47⋊#95—22no—
91C2×#41×C47—22no—
92C2×#41×C47—22no—
93C2×#41×C47—22no—
94C2×#41×C47—22no—
95C2×#41×C47—22no—
96C2×#41×C47C2×C414yes—
97C22×Q8×C47C2311yescharacteristic
98C22×D4×C47—21no—
99C22×D4×C47C2311yescharacteristic
100C23×C4×C47—21no—
101C23×C4×C47—21no—
102C23×C4×C47C2311yescharacteristic
103C2×M16×C47—22no—
104C2×M16×C47—22no—
105C2×M16×C47D414yes—
106C22×C8×C47C2312yes—
107C2×C4⋊C4×C47—22no—
108C2×C4⋊C4×C47—22no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47C2312yes—
111#103×C47—24no—
112#102×C47—24no—
113C24×C47—21no—
114C24×C47C2×D411yescharacteristic
115C23×D47—24no—
116C23×D47—22no—
117C23×D47—44no—
118C23×D47—44no—
119C23×D47—42no—
120C23×D47—42no—
121#41×C47—28no—
122#41×C47—28no—
123#41×C47—44no—
124#41×C47—44no—
125C2×Q8×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47C22×C412yes—
131C2×D4×C47—24no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—42no—
138C2×D4×C47—42no—
139C2×D4×C47C22×C412yes—
140C22×C4×C47—21no—
141C22×C4×C47—24no—
142C22×C4×C47—22no—
143C22×C4×C47—42no—
144C22×C4×C47—42no—
145C22×C4×C47—44no—
146C22×C4×C47—42no—
147C22×C4×C47C22×C412yes—
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D412yes—
152C22×C4×C47#4112yes—
153C22×C4×C47#4112yes—
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—24no—
156C22×Dic47—24no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—42no—
160C22×Dic47C2×D412yes—
161C22×Dic47C2×D412yes—
162C2×D188—28no—
163C2×D188—28no—
164C2×D188—22no—
165C2×D188—22no—
166C2×D188—22no—
167C2×D188—22no—
168C2×D188—24no—
169C2×D188—22no—
170C2×D188—44no—
171C2×D188—42no—
172C2×D188—44no—
173C2×D188—42no—
174C2×D188C22×C414yes—
175M16×C47—28no—
176C2×C8×C47—22no—
177C2×C8×C47—22no—
178C2×C8×C47C2×D414yes—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47—42no—
182C22⋊Dic47—44no—
183C22⋊Dic47—42no—
184C22⋊Dic47—44no—
185#10625336—216no—
186#10625334—216no—
187(C2×D47)⋊C4—48no—
188(C2×D47)⋊C4—48no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4×Dic47—24no—
194C4×Dic47—21no—
195C4×Dic47—21no—
196C4×Dic47—21no—
197C4×Dic47—21no—
198C4×Dic47—42no—
199#10625329C22⋊C4116yescomplemented
200C2×C47⋊C8C22×C414yes—
201C2×C47⋊C8C2×D414yes—
202C2×C47⋊C8C2×D414yes—
203C23×C47—22no—
204C23×C47—22no—
205C23×C47—22no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47—42no—
209C23×C47—42no—
210C23×C47C4×D412yes—
211C23×C47#12112yes—
212C23×C47#12712yes—
213C23×C47C22×D411yescharacteristic
214C22×D47—24no—
215C22×D47—416no—
216C22×D47—42no—
217C22×D47—42no—
218C22×D47—44no—
219C22×D47—44no—
220C22×D47—48no—
221C22×D47—84no—
222C22×D47—84no—
223C22×D47—88no—
224Q8×C47—22no—
225Q8×C47—22no—
226Q8×C47—41no—
227Q8×C47—41no—
228D4×C47—22no—
229D4×C47—22no—
230D4×C47—48no—
231D4×C47—41no—
232D4×C47—41no—
233C2×C4×C47—22no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—48no—
237C2×C4×C47—42no—
238C2×C4×C47—44no—
239C2×C4×C47—84no—
240C2×C4×C47C2×C22⋊C414yes—
241C2×C4×C47C4×D412yes—
242C2×C4×C47C4×D414yes—
243C2×C4×C47#12114yes—
244C2×C4×C47#12112yes—
245C2×C4×C47#12414yes—
246C2×C4×C47#12712yes—
247C2×C4×C47C23×C411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C22×D411yescharacteristic
250C2×C4×C47C22×D411yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—48no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47AGL(1,ℤ/8)14yescomplemented
264D188—24no—
265D188—24no—
266D188—42no—
267D188—42no—
268D188—48no—
269C8×C47—24no—
270C47⋊C8C2×C22⋊C418yescomplemented
271#21236—471nomaximal, Sylow
272C22×C47—22no—
273C22×C47—22no—
274C22×C47—24no—
275C22×C47—24no—
276C22×C47—44no—
277C22×C47—42no—
278C22×C47—81no—
279C22×C47—81no—
280C22×C47—82no—
281C22×C47#39014yes—
282C22×C47C2×C4×D411yescharacteristic
283C22×C47C2×#12111yescharacteristic
284C22×C47C2×#12711yescharacteristic
285C22×C47C2×AGL(1,ℤ/8)12yes—
286C22×C47#57512yes—
287C2×D47—48no—
288C2×D47—44no—
289C2×D47—816no—
290C2×D47—82no—
291C2×D47—82no—
292C2×D47—84no—
293C4×C47—42no—
294C4×C47—82no—
295C4×C47#39014yes—
296C4×C47C22×C22⋊C411yescharacteristic
297C4×C47C2×C4×D411yescharacteristic
298C4×C47C2×#12111yescharacteristic
299C4×C47C2×#12411yescharacteristic
300Dic47—24no—
301Dic47—42no—
302Dic47—81no—
303Dic47—81no—
304C22×AGL(1,ℤ/8)—471no—
305C2×#428—471no—
306C2×#417D4711yescharacteristic, complemented
307C2×#418—471no—
308C2×C4×M16—471no—
309#2297—471no—
310C2×#390—471no—
311#1973—472no—
312#1972—472no—
313#1970—472no—
314#1968—472no—
315C2×C47—24no—
316C2×C47—82no—
317C2×C47C2×#39011yescharacteristic
318C2×C47C4×AGL(1,ℤ/8)12yes—
319C2×C47#305512yes—
320C2×C47#315111yescharacteristic
321C2×C47#315311yescharacteristic
322D47—84no—
323C22×#41C2×D4711yescharacteristic
324C23×D4—471no—
325C2×AGL(1,ℤ/8)—478no—
326C2×AGL(1,ℤ/8)—944no—
327C2×AGL(1,ℤ/8)—944no—
328C22×SD16—472no—
329C22×D8—472no—
330C22×M16—471no—
331C22×M16C2×D4711yescharacteristic
332C2×#117—471no—
333C22×C4⋊C4C2×D4711yescharacteristic
334C22×C22⋊C4—942no—
335C22×C42—471no—
336#428—944no—
337C2×#107—472no—
338C2×#106—472no—
339#418—944no—
340#417—24no—
341C2×#103C2×D4712yescomplemented
342C2×#102—472no—
343C2×#102—472no—
344C2×#102C2×D4712yescomplemented
345C4×M16—944no—
346C2×#97—472no—
347C2×C4×C8—472no—
348#391—471no—
349#391—471no—
350#390—471no—
351#390—471no—
352#390—472no—
353#390—472no—
354#390—942no—
355#386—942no—
356C4×C22⋊C4—942no—
357C47#2123611yescharacteristic, complemented, Sylow
358C25—941no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—22no—
364C2×#41C4×D4714yes—
365C22×Q8C22×D4711yescharacteristic
366C22×D4—21no—
367C22×D4—474no—
368C22×D4—472no—
369C22×D4—942no—
370C22×D4—942no—
371C22×D4—942no—
372C22×D4—941no—
373C22×D4—942no—
374C22×D4—942no—
375C22×D4C22×D4711yescharacteristic
376C23×C4—21no—
377C23×C4—21no—
378C23×C4—472no—
379C23×C4C22×D4711yescharacteristic
380AGL(1,ℤ/8)—9432no—
381C2×SD16—478no—
382C2×SD16—944no—
383C2×SD16—944no—
384C2×D8—478no—
385C2×D8—944no—
386C2×D8—944no—
387C2×M16—22no—
388C2×M16—22no—
389C2×M16—474no—
390C2×M16—474no—
391C2×M16—474no—
392C2×M16C22⋊D4714yes—
393C22×C8—472no—
394C22×C8C22×D4712yes—
395#117—942no—
396#117—942no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—471no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4C22×D4712yes—
403C2×C22⋊C4—944no—
404C2×C22⋊C4—944no—
405C2×C22⋊C4—942no—
406C2×C22⋊C4—942no—
407C2×C22⋊C4—941no—
408C2×C22⋊C4—1888no—
409C2×C42—472no—
410C2×C42—471no—
411C2×C42—471no—
412C2×C42—942no—
413C2×C42—942no—
414C2×C42—941no—
415#107—944no—
416#106—944no—
417#103—24no—
418#102—24no—
419#102—944no—
420#102—944no—
421#97—944no—
422C4×C8—944no—
423#95—942no—
424#95—942no—
425C24—21no—
426C24—944no—
427C24—942no—
428C24—1884no—
429C24—1884no—
430C24—1882no—
431C24—1882no—
432C24D4×D4711yescharacteristic
433#41—28no—
434#41—28no—
435#41—44no—
436#41—44no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D4712yes—
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—474no—
452C2×D4—948no—
453C2×D4—944no—
454C2×D4—948no—
455C2×D4—942no—
456C2×D4—942no—
457C2×D4—942no—
458C2×D4—942no—
459C2×D4—942no—
460C2×D4—1884no—
461C2×D4—1882no—
462C2×D4—1884no—
463C2×D4—1882no—
464C2×D4C2×C4×D4712yes—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—472no—
473C22×C4—472no—
474C22×C4—944no—
475C22×C4—944no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—1882no—
479C22×C4C2×C4×D4712yes—
480C22×C4#1062535012yes—
481C22×C4D4×D4712yes—
482C22×C4D4×D4712yes—
483C22×C4D4×D4711yescharacteristic
484C22×C4#1062535212yes—
485C22×C4C2×C22⋊D4712yes—
486C22×C4C23×D4711yescharacteristic
487SD16—9416no—
488D8—9416no—
489M16—28no—
490M16—4716no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—474no—
494C2×C8—474no—
495C2×C8—474no—
496C2×C8C2×C22⋊D4714yes—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—941no—
501C4⋊C4—941no—
502C4⋊C4—941no—
503C4⋊C4—941no—
504C22⋊C4—1888no—
505C22⋊C4—1888no—
506C22⋊C4—1884no—
507C22⋊C4—1884no—
508C22⋊C4—1882no—
509C42—944no—
510C42—941no—
511C42—941no—
512C42—941no—
513C42—941no—
514C42—1882no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—944no—
523C23—3768no—
524C23—3764no—
525C23—3764no—
526C23—18816no—
527C23—1888no—
528C23—1882no—
529C23—1884no—
530C23—1882no—
531C23—1884no—
532C23(C22⋊D47)⋊C412yes—
533C23#1309399712yes—
534C23#1309402012yes—
535C23C2×D4×D4711yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545D4—944no—
546D4—944no—
547D4—1888no—
548D4—1882no—
549D4—1882no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—474no—
558C2×C4—942no—
559C2×C4—942no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—1888no—
564C2×C4—1882no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1882no—
568C2×C4C22⋊C4×D4714yes—
569C2×C4(C22⋊D47)⋊C412yes—
570C2×C4C4×C22⋊D4714yes—
571C2×C4#1309399614yes—
572C2×C4#1309399712yes—
573C2×C4#1309401014yes—
574C2×C4#1309402012yes—
575C2×C4C22×C4×D4711yescharacteristic
576C2×C4C2×D4×D4711yescharacteristic
577C2×C4C2×D4×D4711yescharacteristic
578C2×C4C22×C22⋊D4711yescharacteristic
579C2×C4C2×#1062535211yescharacteristic
580C2×C4C2×#1062535011yescharacteristic
581C8—24no—
582C8—478no—
583C22—24no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—44no—
588C22—42no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22—37616no—
593C22—3764no—
594C22—3762no—
595C22—3762no—
596C22—1888no—
597C22—1884no—
598C22#1568902314yes—
599C22C2×(C22⋊D47)⋊C411yescharacteristic
600C22C2×#1309399711yescharacteristic
601C22C2×#1309402011yescharacteristic
602C22AGL(1,ℤ/8)×D4712yes—
603C22#1568992212yes—
604C4—42no—
605C4—82no—
606C4—944no—
607C4—3761no—
608C4—3761no—
609C4—1882no—
610C4#1568902314yes—
611C4C2×C22⋊C4×D4711yescharacteristic
612C4C2×C4×C22⋊D4711yescharacteristic
613C4C2×#1309399611yescharacteristic
614C4C2×#1309401011yescharacteristic
615C2—24no—
616C2—82no—
617C2—3764no—
618C2C2×#1568902311yescharacteristic
619C2#2025220112yes—
620C2#2025301012yes—
621C2#2025383611yescharacteristic
622C2#2025385511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#417C2
socleC23×C47C22×D4
prime radicalC22×#13094049C2
cosocleC22×C4×C47C24
hypercenterC2×#417D47
hypocenterC47#21236
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#15689163C2semidirect
3. C2×#15689129C2semidirect
4. C2×M16×Dic47C2semidirect
5. #20245879C2semidirect
6. C2×#15689024C2semidirect
7. #20244266C2semidirect
8. #20244261C2semidirect
9. #20244251C2semidirect
10. #20244239C2semidirect
11. C2×C47×#417C2semidirect
15. C2×#13094049C4semidirect
23. C2×#13093957C22semidirect
24. C2×#13093955C22semidirect
29. C2×#13093934C22semidirect
30. C2×C8×Dic47C22semidirect
31. #15689030C22semidirect
32. #15689030C22semidirect
33. #15689026C22semidirect
34. #15689026C22semidirect
44. C2×#103×C47C22semidirect
45. C2×#102×C47C22semidirect
59. C2×#10625336C2×C4semidirect
62. C2×#10625334C2×C4semidirect
199. #10625329C22⋊C4semidirect
263. C2×Dic47AGL(1,ℤ/8)semidirect
270. C47⋊C8C2×C22⋊C4semidirect
306. C2×#417D47semidirect
341. C2×#103C2×D47semidirect
344. C2×#102C2×D47semidirect
357. C47#21236semidirect

Maximal subgroups

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

subgroupindex
C22×#130940492
C2×#156891632
C2×#156891292
C2×M16×Dic472
#202458792
C2×#156890242
#202442662
#202442612
#202442512
#202442392
C2×C47×#4172
#2123647