Finite Group Databasebeta

← #26215809

Subgroups — #26215809

Order 12032.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689606C211yesmaximal, characteristic
2C2×#15689151C211yesmaximal, characteristic, complemented
3C2×#15689119C211yesmaximal, characteristic
4#20246012C211yesmaximal, characteristic, complemented
5#20243401C212yesmaximal, complemented
6#20243396C212yesmaximal, complemented
7#20243391C212yesmaximal, complemented
8#20243386C212yesmaximal, complemented
9#20243108C211yesmaximal, characteristic, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C2×C47×#417C211yesmaximal, characteristic
12C23×C47⋊C8C2211yescharacteristic
13#15689606—24no—
14#15689606C418yescomplemented
15C2×Q8×Dic47C2212yes—
16C2×D4×Dic47—22no—
17C2×D4×Dic47C2212yes—
18C2×#13093983—21no—
19C2×#13093983C2211yescharacteristic
20C2×#13093983C2211yescharacteristic
21C22×C4⋊Dic47—21no—
22C22×C4⋊Dic47C2211yescharacteristic
23#15689151—24no—
24C2×Dic47⋊C8C2212yes—
25C2×Dic47⋊C8C2212yes—
26#15689119—24no—
27C2×#13093948C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093945C2211yescharacteristic
30C2×#13093945C2211yescharacteristic
31#15689019—21no—
32#15689019—21no—
33#15688996—22no—
34#15688992C2212yes—
35#15688988C2212yes—
36#15688886C2212yes—
37#15688886C2212yes—
38#15688885C2212yes—
39#15688885C2212yes—
40C22×#41×C47C2211yescharacteristic
41C22×M16×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C47×#417—24no—
44C2×#103×C47C2212yes—
45C2×#102×C47C2212yes—
46C23×Dic47—21no—
47C23×Dic47—22no—
48C22×C47⋊C8—22no—
49C22×C47⋊C8—22no—
50C22×C47⋊C8—22no—
51C22×C47⋊C8C2311yescharacteristic
52C22×C47⋊C8C2311yescharacteristic
53Q8×Dic47—24no—
54Q8×Dic47C2×C418yescomplemented
55D4×Dic47—216no—
56D4×Dic47—24no—
57D4×Dic47C2×C418yescomplemented
58#13093983—28no—
59#13093983—22no—
60#13093983—22no—
61#13093983C2×C414yes—
62#13093983D414yes—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—22no—
66C2×C4⋊Dic47—28no—
67C2×C4⋊Dic47C2×C414yes—
68C2×C4⋊Dic47D411yescharacteristic
69C2×C4⋊Dic47D411yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2312yes—
73C2×Dic47⋊C4—22no—
74C2×Dic47⋊C4—22no—
75C2×C22⋊Dic47—22no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47D412yes—
78C2×C4×Dic47—22no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47C2312yes—
81Dic47⋊C8—24no—
82Dic47⋊C8—24no—
83#13093948—22no—
84#13093948—22no—
85#13093945—22no—
86#13093945—22no—
87C47⋊#95—22no—
88C47⋊#95—22no—
89C2×#41×C47—22no—
90C2×#41×C47—22no—
91C2×#41×C47—22no—
92C2×#41×C47C2×C414yes—
93C2×#41×C47D414yes—
94C2×#41×C47Q814yes—
95C22×Q8×C47C2311yescharacteristic
96C22×D4×C47—21no—
97C22×D4×C47C2311yescharacteristic
98C23×C4×C47—21no—
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101C2×M16×C47—22no—
102C2×M16×C47—22no—
103C2×M16×C47—22no—
104C22×C8×C47C2312yes—
105C2×C4⋊C4×C47—22no—
106C2×C4⋊C4×C47—22no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47C2312yes—
109#103×C47—24no—
110#102×C47—24no—
111C24×C47—21no—
112C24×C47C2×D411yescharacteristic
113#41×C47—28no—
114#41×C47—28no—
115#41×C47—28no—
116#41×C47C4⋊C4116yescomplemented
117C2×Q8×C47—21no—
118C2×Q8×C47—21no—
119C2×Q8×C47—21no—
120C2×Q8×C47C22×C412yes—
121C2×Q8×C47C2×D412yes—
122C2×Q8×C47C2×Q812yes—
123C2×D4×C47—24no—
124C2×D4×C47—24no—
125C2×D4×C47—24no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47C22×C412yes—
130C2×D4×C47C2×D412yes—
131C2×D4×C47C2×Q812yes—
132C22×C4×C47—21no—
133C22×C4×C47—24no—
134C22×C4×C47—24no—
135C22×C4×C47—24no—
136C22×C4×C47—22no—
137C22×C4×C47—42no—
138C22×C4×C47—81no—
139C22×C4×C47—81no—
140C22×C4×C47C22×C412yes—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D412yes—
143C22×C4×C47C2×D412yes—
144C22×C4×C47C2×Q812yes—
145C22×C4×C47#4112yes—
146C22×C4×C47#4112yes—
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—22no—
149C22×Dic47—24no—
150C22×Dic47—28no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—82no—
154C22×Dic47—82no—
155C22×Dic47—81no—
156C22×Dic47—81no—
157C22×Dic47C2×D412yes—
158C22×Dic47C2×D412yes—
159M16×C47—44no—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47—22no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—42no—
166C22⋊Dic47—28no—
167C22⋊Dic47—42no—
168C22⋊Dic47C22⋊C418yes—
169C4⋊Dic47—24no—
170C4⋊Dic47—22no—
171C4⋊Dic47—28no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47C22⋊C414yes—
177C4⋊Dic47C22⋊C414yes—
178C4⋊Dic47C22×C414yes—
179Dic47⋊C4—82no—
180Dic47⋊C4—82no—
181C4×Dic47—28no—
182C4×Dic47—42no—
183C4×Dic47C22×C414yes—
184C4×Dic47C2×D414yes—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—44no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47—41no—
203C23×C47C2×C22⋊C412yes—
204C23×C47#12112yes—
205C23×C47C22⋊Q812yes—
206C23×C47C22×D411yescharacteristic
207Q8×C47—22no—
208Q8×C47—22no—
209Q8×C47—22no—
210Q8×C47C2×C4⋊C414yes—
211D4×C47—216no—
212D4×C47—22no—
213D4×C47—22no—
214D4×C47—22no—
215D4×C47C2×C4⋊C414yes—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—216no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47—84no—
224C2×C4×C47C2×C22⋊C412yes—
225C2×C4×C47C2×C4⋊C414yes—
226C2×C4×C47#11714yes—
227C2×C4×C47#12114yes—
228C2×C4×C47#12112yes—
229C2×C4×C47C22⋊Q814yes—
230C2×C4×C47C22⋊Q812yes—
231C2×C4×C47C23×C411yescharacteristic
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C22×Q811yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—24no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—88no—
251C2×Dic47—84no—
252C2×Dic47C2×C22⋊C414yes—
253C2×Dic47C2×C22⋊C414yes—
254C8×C47—42no—
255C47⋊C8—41no—
256C47⋊C8—41no—
257#20652—471nomaximal, Sylow
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—22no—
264C22×C47—24no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—42no—
271C22×C47#38014yes—
272C22×C47C22×C22⋊C411yescharacteristic
273C22×C47C2×#12111yescharacteristic
274C22×C47C2×C22⋊Q811yescharacteristic
275C22×C47#57611yescharacteristic
276C22×C47#57711yescharacteristic
277C22×C47#57711yescharacteristic
278C22×C47#57811yescharacteristic
279C4×C47—24no—
280C4×C47—82no—
281C4×C47#38014yes—
282C4×C47C22×C4⋊C411yescharacteristic
283C4×C47C2×#11711yescharacteristic
284C4×C47C2×#12111yescharacteristic
285C4×C47C2×C22⋊Q811yescharacteristic
286Dic47—24no—
287Dic47—24no—
288Dic47—44no—
289Dic47—81no—
290Dic47—81no—
291C2×#520—471no—
292C2×#424—471no—
293C2×#417D4711yescharacteristic
294C2×#416—471no—
295#2317—471no—
296#1805—472no—
297#1804—472no—
298#1803—472no—
299#1802—472no—
300#1736—471no—
301C2×#340—471no—
302C2×C47—24no—
303C2×C47—44no—
304C2×C47C2×#38011yescharacteristic
305C2×C47#289711yescharacteristic
306C2×C47#289811yescharacteristic
307C2×C47#304511yescharacteristic
308C2×C47#304611yescharacteristic
309C2×C47#308311yescharacteristic
310C2×C47#308411yescharacteristic
311C22×#41C2×D4711yescharacteristic
312C22×M16C2×D4711yescharacteristic
313C23×C8—471no—
314#520—478no—
315#520—944no—
316C2×C4×Q8—472no—
317C2×C4×D4—472no—
318C2×C4×D4—942no—
319C2×#117—471no—
320C2×#117—471no—
321C2×#117—941no—
322C22×C4⋊C4—471no—
323C22×C4⋊C4—941no—
324C22×C4⋊C4C2×D4711yescharacteristic
325#424—944no—
326C2×C4⋊C8—472no—
327C2×C4⋊C8—472no—
328#417—24no—
329#416—944no—
330C2×#103—471no—
331C2×#103—471no—
332C2×#103C2×D4712yes—
333C2×#102—471no—
334C2×#102—471no—
335C2×#102C2×D4712yes—
336#389—941no—
337#389—941no—
338#385—942no—
339#384—472no—
340#382—472no—
341#340—472no—
342#340—472no—
343#340—472no—
344#340—472no—
345C47#2065211yescharacteristic, complemented, Sylow
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41Dic9414yes—
350C2×#41C4×D4714yes—
351C2×#41C22⋊D4714yes—
352C22×Q8C22×D4711yescharacteristic
353C22×D4—21no—
354C22×D4C22×D4711yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—942no—
358C23×C4—941no—
359C23×C4C22×D4711yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—22no—
363C22×C8—471no—
364C22×C8—471no—
365C22×C8—942no—
366C22×C8—942no—
367C22×C8—942no—
368C22×C8C22×D4712yes—
369C4×Q8—478no—
370C4×Q8—944no—
371C4×D4—478no—
372C4×D4—944no—
373C4×D4—9416no—
374#117—474no—
375#117—474no—
376#117—948no—
377#117—942no—
378#117—942no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—474no—
385C2×C4⋊C4—472no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—948no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4—942no—
393C2×C4⋊C4—942no—
394C2×C4⋊C4C22×D4712yes—
395C2×C22⋊C4—472no—
396C2×C22⋊C4—942no—
397C2×C22⋊C4—941no—
398C2×C42—472no—
399C2×C42—942no—
400C2×C42—941no—
401C4⋊C8—944no—
402C4⋊C8—944no—
403#103—24no—
404#103—942no—
405#103—942no—
406#102—24no—
407#102—942no—
408#102—942no—
409#95—942no—
410#95—942no—
411C24—21no—
412C24D4×D4711yescharacteristic
413#41—28no—
414#41—28no—
415#41—28no—
416#41Dic47⋊C4116yescomplemented
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8C2×Dic9412yes—
421C2×Q8C2×C4×D4712yes—
422C2×Q8C2×C22⋊D4712yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×Dic9412yes—
430C2×D4C2×C4×D4712yes—
431C2×D4C2×C22⋊D4712yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—472no—
441C22×C4—472no—
442C22×C4—948no—
443C22×C4—944no—
444C22×C4—942no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—3762no—
448C22×C4—3762no—
449C22×C4—3761no—
450C22×C4—3761no—
451C22×C4C2×Dic9412yes—
452C22×C4C2×C4×D4712yes—
453C22×C4D4×D4711yescharacteristic
454C22×C4D4×D4712yes—
455C22×C4#1062535212yes—
456C22×C4#1062535212yes—
457C22×C4C2×C22⋊D4712yes—
458C22×C4C23×D4711yescharacteristic
459M16—44no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—941no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—1884no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4—474no—
474C4⋊C4—474no—
475C4⋊C4—474no—
476C4⋊C4—948no—
477C4⋊C4—944no—
478C4⋊C4—942no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—3762no—
484C4⋊C4—3762no—
485C22⋊C4—478no—
486C22⋊C4—948no—
487C22⋊C4—1882no—
488C42—474no—
489C42—474no—
490C42—948no—
491C42—1882no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C22⋊C4×D4712yes—
504C23#1309399612yes—
505C23#1309400112yes—
506C23C2×D4×D4711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8C2×Dic47⋊C414yes—
511D4—216no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C2×Dic47⋊C414yes—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—84no—
524C2×C4—474no—
525C2×C4—474no—
526C2×C4—942no—
527C2×C4—942no—
528C2×C4—942no—
529C2×C4—942no—
530C2×C4—944no—
531C2×C4—3768no—
532C2×C4—3764no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1882no—
540C2×C4—1882no—
541C2×C4C22⋊C4×D4712yes—
542C2×C4C2×Dic47⋊C414yes—
543C2×C4#1309398014yes—
544C2×C4#1309399612yes—
545C2×C4#1309399614yes—
546C2×C4#1309400112yes—
547C2×C4#1309400114yes—
548C2×C4C22×C4×D4711yescharacteristic
549C2×C4C2×D4×D4711yescharacteristic
550C2×C4C22×C22⋊D4711yescharacteristic
551C2×C4C22×Dic9411yescharacteristic
552C2×C4C2×#1062535211yescharacteristic
553C2×C4C2×#1062535211yescharacteristic
554C8—42no—
555C8—1881no—
556C8—1881no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#1568898114yes—
571C22C2×C22⋊C4×D4711yescharacteristic
572C22C2×#1309399611yescharacteristic
573C22C2×#1309400111yescharacteristic
574C22#1568992711yescharacteristic
575C22#1568993211yescharacteristic
576C22#1568993311yescharacteristic
577C22#1568993911yescharacteristic
578C4—24no—
579C4—82no—
580C4—944no—
581C4—944no—
582C4—3761no—
583C4—3761no—
584C4—1884no—
585C4#1568898114yes—
586C4C22×Dic47⋊C411yescharacteristic
587C4C2×#1309398011yescharacteristic
588C4C2×#1309399611yescharacteristic
589C4C2×#1309400111yescharacteristic
590C2—24no—
591C2—44no—
592C2C2×#1568898111yescharacteristic
593C2#2025191211yescharacteristic
594C2#2025191911yescharacteristic
595C2#2025293111yescharacteristic
596C2#2025294211yescharacteristic
597C2#2025324311yescharacteristic
598C2#2025325411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#417C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#417D47
hypocenterC47#20652
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×#15689151C2semidirect
4. #20246012C2semidirect
5. #20243401C2semidirect
6. #20243396C2semidirect
7. #20243391C2semidirect
8. #20243386C2semidirect
9. #20243108C2semidirect
10. C2×#15688885C2semidirect
14. #15689606C4semidirect
54. Q8×Dic47C2×C4semidirect
57. D4×Dic47C2×C4semidirect
116. #41×C47C4⋊C4semidirect
345. C47#20652semidirect
416. #41Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156896062
C2×#156891512
C2×#156891192
#202460122
#202434012
#202433962
#202433912
#202433862
#202431082
C2×#156888852
C2×C47×#4172
#2065247