Finite Group Databasebeta

← #26220196

Subgroups — #26220196

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×#13094052C211yesmaximal, characteristic
2C2×#15689122C211yesmaximal, characteristic, complemented
3C2×#15689127C211yesmaximal, characteristic, complemented
4C2×M16×Dic47C211yesmaximal, characteristic, complemented
5#20245876C211yesmaximal, characteristic, complemented
6C2×#15689024C211yesmaximal, characteristic, complemented
7#20244267C212yesmaximal, complemented
8#20244262C212yesmaximal, complemented
9#20244252C212yesmaximal, complemented
10#20244240C212yesmaximal, complemented
11C2×C47×#428C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C23×D188C2211yescharacteristic
14C2×#13094052—24no—
15C2×#13094052—24no—
16C2×#13094052C418yescomplemented
17C22×#10625325C2212yes—
18C22×D376C2212yes—
19C22×#10625329C2211yescharacteristic
20C22×(C2×D47)⋊C4—22no—
21C22×C4×Dic47C2211yescharacteristic
22#15689127—24no—
23#15689122—24no—
24C2×#13093947C2212yescomplemented
25C2×#13093944C2212yes—
26C2×#13093944C2212yes—
27C2×#13093944C2212yescomplemented
28M16×Dic47—24no—
29C2×#13093934C2212yescomplemented
30C2×C8×Dic47C2212yescomplemented
31#15689028C2211yescharacteristic, complemented
32#15689028C2211yescharacteristic, complemented
33#15689024—22no—
34#15689024C2212yes—
35#15689024C2212yes—
36#15689024C2211yescharacteristic, complemented
37#15689024C2211yescharacteristic, complemented
38#15689000—22no—
39C4×(C2×D47)⋊C4—22no—
40C22×M16×C47C2211yescharacteristic
41C2×#117×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C47×#428—24no—
44C2×#107×C47C2212yescomplemented
45C2×#106×C47C2212yescomplemented
46C24×D47—21no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350—22no—
51C2×#10625350—22no—
52C2×#10625350C2×C414yes—
53C22×Dic94C2311yescharacteristic
54C22×D188—22no—
55C22×D188—22no—
56C22×D188—22no—
57C22×D188—22no—
58C22×D188—21no—
59C22×D188—22no—
60C22×D188C2×C414yes—
61C22×D188C2312yes—
62C22×D188C2311yescharacteristic
63C22×C22⋊D47—21no—
64C23×Dic47D412yes—
65C22×C4×D47—21no—
66#13094052—232no—
67C2×#10625325—24no—
68C2×#10625325—24no—
69C2×#10625325C2×C418yescomplemented
70C2×D376—24no—
71C2×D376—24no—
72C2×D376C2×C418yescomplemented
73C2×#10625329—22no—
74C2×#10625329—22no—
75C2×#10625329D414yes—
76C22×C47⋊C8C2312yes—
77C2×(C2×D47)⋊C4—22no—
78C2×(C2×D47)⋊C4—22no—
79C2×(C2×D47)⋊C4—24no—
80C2×(C2×D47)⋊C4—24no—
81C2×(C2×D47)⋊C4—48no—
82C2×C4×Dic47—22no—
83C2×C4×Dic47—22no—
84C2×C4×Dic47D412yes—
85C2×C4×Dic47C2311yescharacteristic
86C2×C4×Dic47C2311yescharacteristic
87#13093947—24no—
88#13093944—24no—
89#13093944—24no—
90#13093944—24no—
91#13093934—24no—
92C8×Dic47—24no—
93C47⋊#95—22no—
94C47⋊#95—22no—
95C23×C4×C47—21no—
96C23×C4×C47C2311yescharacteristic
97C2×M16×C47C2×C414yes—
98C2×M16×C47D414yes—
99C2×M16×C47D414yes—
100C22×C8×C47C2312yes—
101#117×C47—22no—
102#117×C47—22no—
103C2×C4⋊C4×C47—22no—
104C2×C4⋊C4×C47—22no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47C2312yes—
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C22⋊C4×C47—21no—
110C2×C42×C47—21no—
111#107×C47—24no—
112#106×C47—24no—
113C24×C47C2×D411yescharacteristic
114C23×D47—24no—
115C23×D47—21no—
116C23×D47—22no—
117C23×D47—44no—
118C23×D47—44no—
119C23×D47—42no—
120C23×D47—42no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47—44no—
125C22×C4×C47—42no—
126C22×C4×C47C22×C412yes—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D412yes—
131C22×C4×C47#4112yes—
132C22×C4×C47#4112yes—
133C22×C4×C47C2411yescharacteristic
134C2×C22⋊D47—24no—
135C2×C22⋊D47—42no—
136C2×C22⋊D47—42no—
137C22×Dic47—24no—
138C22×Dic47—21no—
139C22×Dic47—24no—
140C22×Dic47—42no—
141C22×Dic47C2×D412yes—
142C22×Dic47C2×D412yes—
143#10625350—28no—
144#10625350—28no—
145#10625350—44no—
146#10625350—44no—
147C2×D188—28no—
148C2×D188—28no—
149C2×D188—22no—
150C2×D188—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188—22no—
154C2×D188—22no—
155C2×D188—22no—
156C2×D188—21no—
157C2×D188—24no—
158C2×D188—22no—
159C2×D188—21no—
160C2×D188—44no—
161C2×D188—42no—
162C2×D188—44no—
163C2×D188—42no—
164C2×D188C22×C412yes—
165C2×D188C22×C414yes—
166C2×C4×D47—24no—
167C2×C4×D47—42no—
168C2×C4×D47—42no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94C22×C412yes—
175M16×C47C22⋊C4116yescomplemented
176C2×C8×C47C22×C414yes—
177C2×C8×C47C2×D414yes—
178C2×C8×C47C2×D414yes—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—22no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—22no—
183C4⋊C4×C47—21no—
184C4⋊C4×C47—21no—
185C4⋊C4×C47—42no—
186C22⋊C4×C47—42no—
187C42×C47—42no—
188(C2×D47)⋊C4—44no—
189(C2×D47)⋊C4—48no—
190(C2×D47)⋊C4—44no—
191(C2×D47)⋊C4—48no—
192C4×Dic47—24no—
193C4×Dic47—21no—
194C4×Dic47—21no—
195C4×Dic47—21no—
196C4×Dic47—21no—
197#10625329—28no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—22no—
200C2×C47⋊C8C2×D414yes—
201D376—216no—
202#10625325—216no—
203C23×C47—22no—
204C23×C47—22no—
205C23×C47C4×D412yes—
206C23×C47#12112yes—
207C23×C47#12712yes—
208C23×C47C22×D411yescharacteristic
209C22×D47—22no—
210C22×D47—24no—
211C22×D47—416no—
212C22×D47—41no—
213C22×D47—42no—
214C22×D47—41no—
215C22×D47—42no—
216C22×D47—44no—
217C22×D47—42no—
218C22×D47—42no—
219C22×D47—44no—
220C22×D47—48no—
221C22×D47—84no—
222C22×D47—84no—
223C22×D47—88no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—42no—
227C2×C4×C47—44no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—84no—
231C2×C4×C47C2×C22⋊C414yes—
232C2×C4×C47C4×D414yes—
233C2×C4×C47C4×D412yes—
234C2×C4×C47#12114yes—
235C2×C4×C47#12112yes—
236C2×C4×C47#12414yes—
237C2×C4×C47#12712yes—
238C2×C4×C47C23×C411yescharacteristic
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C22⋊D47—48no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—48no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47AGL(1,ℤ/8)14yescomplemented
255D188—24no—
256D188—22no—
257D188—24no—
258D188—22no—
259D188—42no—
260D188—41no—
261D188—42no—
262D188—41no—
263D188—48no—
264C4×D47—48no—
265Dic94—22no—
266Dic94—22no—
267Dic94—41no—
268Dic94—41no—
269C8×C47C2×C22⋊C418yescomplemented
270C47⋊C8—24no—
271#21236—471nomaximal, Sylow
272C22×C47—22no—
273C22×C47—22no—
274C22×C47—24no—
275C22×C47—24no—
276C22×C47#39014yes—
277C22×C47C2×C4×D411yescharacteristic
278C22×C47C2×#12111yescharacteristic
279C22×C47C2×#12711yescharacteristic
280C22×C47C2×AGL(1,ℤ/8)12yes—
281C22×C47#57512yes—
282C2×D47—44no—
283C2×D47—48no—
284C2×D47—42no—
285C2×D47—44no—
286C2×D47—816no—
287C2×D47—81no—
288C2×D47—82no—
289C2×D47—81no—
290C2×D47—82no—
291C2×D47—82no—
292C2×D47—84no—
293C4×C47—81no—
294C4×C47—82no—
295C4×C47—81no—
296C4×C47#39014yes—
297C4×C47C22×C22⋊C411yescharacteristic
298C4×C47C2×C4×D411yescharacteristic
299C4×C47C2×#12111yescharacteristic
300C4×C47C2×#12411yescharacteristic
301Dic47—24no—
302Dic47—42no—
303Dic47—42no—
304C22×AGL(1,ℤ/8)—471no—
305C2×#428D4711yescharacteristic, complemented
306C2×#417—471no—
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×C47C2×#39011yescharacteristic
317C2×C47C4×AGL(1,ℤ/8)12yes—
318C2×C47#305512yes—
319C2×C47#315111yescharacteristic
320C2×C47#315311yescharacteristic
321D47—82no—
322D47—84no—
323C22×#41—471no—
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×#117C2×D4711yescharacteristic
333C22×C4⋊C4C2×D4711yescharacteristic
334C22×C22⋊C4—942no—
335C22×C42—471no—
336#428—24no—
337C2×#107C2×D4712yescomplemented
338C2×#106C2×D4712yescomplemented
339#418—944no—
340#417—944no—
341C2×#103—472no—
342C2×#102—472no—
343C2×#102—472no—
344C2×#102—472no—
345C4×M16—944no—
346C2×#97—472no—
347C2×C4×C8—472no—
348#391—471no—
349#391—471no—
350#390—472no—
351#390—472no—
352#390—471no—
353#390—471no—
354#390—942no—
355#386—942no—
356C4×C22⋊C4—942no—
357C47#2123611yescharacteristic, complemented, Sylow
358C25—941no—
359C2×#41—474no—
360C2×#41—942no—
361C2×#41—942no—
362C2×#41—942no—
363C2×#41—942no—
364C2×#41—942no—
365C22×Q8—471no—
366C22×D4—474no—
367C22×D4—471no—
368C22×D4—472no—
369C22×D4—941no—
370C22×D4—942no—
371C22×D4—942no—
372C22×D4—941no—
373C22×D4—942no—
374C22×D4—942no—
375C22×D4—942no—
376C23×C4—21no—
377C23×C4—472no—
378C23×C4—941no—
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—474no—
388C2×M16—942no—
389C2×M16—942no—
390C2×M16C4×D4714yes—
391C2×M16D18814yes—
392C2×M16C22⋊D4714yes—
393C22×C8—472no—
394C22×C8C22×D4712yes—
395#117—22no—
396#117—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4C22×D4712yes—
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C4⋊C4C22×D4711yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—944no—
405C2×C22⋊C4—944no—
406C2×C22⋊C4—942no—
407C2×C22⋊C4—942no—
408C2×C22⋊C4—1888no—
409C2×C42—21no—
410C2×C42—472no—
411C2×C42—471no—
412C2×C42—471no—
413C2×C42—942no—
414C2×C42—942no—
415#107—24no—
416#106—24no—
417#103—944no—
418#102—944no—
419#102—944no—
420#102—944no—
421#97—944no—
422C4×C8—944no—
423#95—942no—
424#95—942no—
425C24—944no—
426C24—941no—
427C24—942no—
428C24—1884no—
429C24—1884no—
430C24—1882no—
431C24—1882no—
432C24D4×D4711yescharacteristic
433#41—948no—
434#41—948no—
435#41—1884no—
436#41—1884no—
437C2×Q8—472no—
438C2×Q8—941no—
439C2×Q8—941no—
440C2×Q8—941no—
441C2×Q8—941no—
442C2×Q8—941no—
443C2×D4—472no—
444C2×D4—474no—
445C2×D4—944no—
446C2×D4—948no—
447C2×D4—948no—
448C2×D4—944no—
449C2×D4—942no—
450C2×D4—941no—
451C2×D4—941no—
452C2×D4—942no—
453C2×D4—941no—
454C2×D4—942no—
455C2×D4—942no—
456C2×D4—941no—
457C2×D4—942no—
458C2×D4—941no—
459C2×D4—1882no—
460C2×D4—1882no—
461C2×D4—1882no—
462C2×D4—1884no—
463C2×D4—1884no—
464C2×D4—1882no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—472no—
471C22×C4—472no—
472C22×C4—944no—
473C22×C4—944no—
474C22×C4—941no—
475C22×C4—944no—
476C22×C4—1882no—
477C22×C4—1882no—
478C22×C4—1882no—
479C22×C4C2×C4×D4712yes—
480C22×C4C2×D18812yes—
481C22×C4D4×D4711yescharacteristic
482C22×C4D4×D4712yes—
483C22×C4#1062535412yes—
484C22×C4#1062535412yes—
485C22×C4C2×C22⋊D4712yes—
486C22×C4C23×D4711yescharacteristic
487SD16—9416no—
488D8—9416no—
489M16—948no—
490M16(C2×D47)⋊C4116yescomplemented
491C2×C8—474no—
492C2×C8—942no—
493C2×C8—942no—
494C2×C8C2×C4×D4714yes—
495C2×C8C2×D18814yes—
496C2×C8C2×C22⋊D4714yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—1884no—
506C22⋊C4—1888no—
507C22⋊C4—1884no—
508C22⋊C4—1888no—
509C42—42no—
510C42—944no—
511C42—941no—
512C42—941no—
513C42—941no—
514C42—941no—
515C23—22no—
516C23—22no—
517C23—942no—
518C23—944no—
519C23—3768no—
520C23—3764no—
521C23—3764no—
522C23—18816no—
523C23—1888no—
524C23—1881no—
525C23—1882no—
526C23—1884no—
527C23—1882no—
528C23—1881no—
529C23—1882no—
530C23—1882no—
531C23—1884no—
532C23#1309398612yes—
533C23#1309399912yes—
534C23#1309402012yes—
535C23C2×D4×D4711yescharacteristic
536Q8—942no—
537Q8—942no—
538Q8—1881no—
539Q8—1881no—
540D4—942no—
541D4—944no—
542D4—944no—
543D4—942no—
544D4—1888no—
545D4—1888no—
546D4—1882no—
547D4—1881no—
548D4—1882no—
549D4—1881no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
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—942no—
564C2×C4—1888no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1888no—
568C2×C4C2×(C2×D47)⋊C414yes—
569C2×C4#1309398612yes—
570C2×C4#1309398614yes—
571C2×C4#1309399912yes—
572C2×C4#1309399914yes—
573C2×C4#1309401314yes—
574C2×C4#1309402012yes—
575C2×C4C22×C4×D4711yescharacteristic
576C2×C4C2×D4×D4711yescharacteristic
577C2×C4C22×C22⋊D4711yescharacteristic
578C2×C4C22×D18811yescharacteristic
579C2×C4C2×#1062535411yescharacteristic
580C2×C4C2×#1062535411yescharacteristic
581C8—944no—
582C8C2×(C2×D47)⋊C418yescomplemented
583C22—24no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—37616no—
588C22—3764no—
589C22—3762no—
590C22—3762no—
591C22—3761no—
592C22—3762no—
593C22—3761no—
594C22—1888no—
595C22—1884no—
596C22—1882no—
597C22—1884no—
598C22#1568902414yes—
599C22C2×#1309398611yescharacteristic
600C22C2×#1309399911yescharacteristic
601C22C2×#1309402011yescharacteristic
602C22AGL(1,ℤ/8)×D4712yes—
603C22#1568992212yes—
604C4—82no—
605C4—81no—
606C4—81no—
607C4—944no—
608C4—1882no—
609C4—1882no—
610C4#1568902414yes—
611C4C22×(C2×D47)⋊C411yescharacteristic
612C4C2×#1309398611yescharacteristic
613C4C2×#1309399911yescharacteristic
614C4C2×#1309401311yescharacteristic
615C2—24no—
616C2—3764no—
617C2—3762no—
618C2C2×#1568902411yescharacteristic
619C2#2025219512yes—
620C2#2025300812yes—
621C2#2025383611yescharacteristic
622C2#2025385511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC22×#13094052C2
cosocleC22×C4×C47C24
hypercenterC2×#428D47
hypocenterC47#21236
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#15689122C2semidirect
3. C2×#15689127C2semidirect
4. C2×M16×Dic47C2semidirect
5. #20245876C2semidirect
6. C2×#15689024C2semidirect
7. #20244267C2semidirect
8. #20244262C2semidirect
9. #20244252C2semidirect
10. #20244240C2semidirect
11. C2×C47×#428C2semidirect
16. C2×#13094052C4semidirect
24. C2×#13093947C22semidirect
27. C2×#13093944C22semidirect
29. C2×#13093934C22semidirect
30. C2×C8×Dic47C22semidirect
31. #15689028C22semidirect
32. #15689028C22semidirect
36. #15689024C22semidirect
37. #15689024C22semidirect
44. C2×#107×C47C22semidirect
45. C2×#106×C47C22semidirect
69. C2×#10625325C2×C4semidirect
72. C2×D376C2×C4semidirect
175. M16×C47C22⋊C4semidirect
254. C2×Dic47AGL(1,ℤ/8)semidirect
269. C8×C47C2×C22⋊C4semidirect
305. C2×#428D47semidirect
337. C2×#107C2×D47semidirect
338. C2×#106C2×D47semidirect
357. C47#21236semidirect
490. M16(C2×D47)⋊C4semidirect
582. C8C2×(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130940522
C2×#156891222
C2×#156891272
C2×M16×Dic472
#202458762
C2×#156890242
#202442672
#202442622
#202442522
#202442402
C2×C47×#4282
#2123647