Finite Group Databasebeta

← #26217157

Subgroups — #26217157

Order 12032.

1422 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251977C211yesmaximal, characteristic, complemented
2C2×#15689140C211yesmaximal, characteristic
3#20251907C211yesmaximal, characteristic
4#20246012C211yesmaximal, characteristic, complemented
5#20243621C214yesmaximal, complemented
6#20243614C214yesmaximal, complemented
7C2×#15688892C211yesmaximal, characteristic, complemented
8#20243097C211yesmaximal, characteristic, complemented
9C2×C47×#520C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C2×#13093983C2211yescharacteristic
13C2×#13093983C2211yescharacteristic
14#15689166C2212yescomplemented
15#15689166C2212yescomplemented
16#15689163—21no—
17#15689163—21no—
18#15689159—22no—
19#15689140C418yescomplemented
20C2×#13093951C2214yes—
21#15689132—21no—
22#15689128—21no—
23#15689124C2212yes—
24#15689124C2212yes—
25#15689119—22no—
26#15689019—22no—
27#15688996—22no—
28#15688992C2212yescomplemented
29#15688988C2212yescomplemented
30#15688892C2214yescomplemented
31#15688882C2214yescomplemented
32C22×#41×C47C2211yescharacteristic
33C47×#520—22no—
34C47×#520—22no—
35C47×#520—22no—
36C47×#520C2214yes—
37C2×C4×Q8×C47—21no—
38C2×C4×D4×C47—21no—
39C2×C4×D4×C47—22no—
40C2×#117×C47—21no—
41C2×#117×C47C2211yescharacteristic
42C22×C4⋊C4×C47—21no—
43C22×C4⋊C4×C47C2211yescharacteristic
44C2×#10625329—21no—
45C2×#10625329—22no—
46C2×#10625329—21no—
47C2×#10625329—22no—
48C2×#10625329—22no—
49C2×#10625329C2×C414yescomplemented
50C2×#10625329C2312yes—
51C2×#10625329C2312yes—
52C22×C47⋊C8—21no—
53C22×C47⋊C8—21no—
54#13093983—22no—
55#13093983—21no—
56#13093983—21no—
57#13093983—21no—
58#13093983—21no—
59#13093983C2×C414yescomplemented
60C2×C4⋊Dic47—21no—
61C2×C4⋊Dic47C2311yescharacteristic
62C2×C4⋊Dic47C2311yescharacteristic
63C2×Dic47⋊C4—24no—
64C2×C22⋊Dic47—21no—
65C2×C22⋊Dic47—21no—
66C2×C4×Dic47—21no—
67C2×C4×Dic47C2312yes—
68#13093957—22no—
69#13093957—22no—
70#13093955—22no—
71#13093955—22no—
72#13093951C2×C4116yescomplemented
73#13093948—22no—
74#13093948—22no—
75#13093945—22no—
76#13093945—22no—
77C47⋊#95—24no—
78C2×#41×C47—21no—
79C2×#41×C47—21no—
80C2×#41×C47—22no—
81C2×#41×C47—22no—
82C2×#41×C47C2×C414yes—
83C2×#41×C47D412yes—
84C2×#41×C47Q814yes—
85C2×#41×C47C2312yes—
86C22×Q8×C47D411yescharacteristic
87C22×D4×C47—21no—
88C22×D4×C47D411yescharacteristic
89C23×C4×C47—21no—
90C23×C4×C47—22no—
91C23×C4×C47D412yes—
92C23×C4×C47C2311yescharacteristic
93C4×Q8×C47—24no—
94C4×Q8×C47—24no—
95C4×D4×C47—28no—
96C4×D4×C47—24no—
97C4×D4×C47—24no—
98C4×D4×C47—28no—
99#117×C47—24no—
100#117×C47—21no—
101#117×C47—21no—
102#117×C47—24no—
103#117×C47D412yes—
104#117×C47C2312yes—
105C2×C4⋊C4×C47—22no—
106C2×C4⋊C4×C47—24no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47—24no—
112C2×C4⋊C4×C47D412yes—
113C2×C4⋊C4×C47D411yescharacteristic
114C2×C4⋊C4×C47D411yescharacteristic
115C2×C4⋊C4×C47C2312yes—
116C2×C22⋊C4×C47—22no—
117C2×C22⋊C4×C47—21no—
118C2×C42×C47—22no—
119C2×C42×C47—21no—
120C24×C47—21no—
121C24×C47C2×D411yescharacteristic
122#41×C47—24no—
123#41×C47—24no—
124#41×C47—24no—
125#41×C47—24no—
126#41×C47—24no—
127#41×C47—22no—
128#41×C47—22no—
129#41×C47C4⋊C418yescomplemented
130#41×C47C22×C414yes—
131#41×C47C2×Q814yes—
132C2×Q8×C47—21no—
133C2×Q8×C47—22no—
134C2×Q8×C47C4⋊C414yescomplemented
135C2×Q8×C47C2×D412yes—
136C2×D4×C47—24no—
137C2×D4×C47—22no—
138C2×D4×C47—24no—
139C2×D4×C47—22no—
140C2×D4×C47—21no—
141C2×D4×C47—22no—
142C2×D4×C47C4⋊C414yescomplemented
143C2×D4×C47C2×D412yes—
144C22×C4×C47—21no—
145C22×C4×C47—24no—
146C22×C4×C47—22no—
147C22×C4×C47—24no—
148C22×C4×C47—22no—
149C22×C4×C47—21no—
150C22×C4×C47—22no—
151C22×C4×C47—21no—
152C22×C4×C47—28no—
153C22×C4×C47—22no—
154C22×C4×C47—82no—
155C22×C4×C47—84no—
156C22×C4×C47C22×C412yes—
157C22×C4×C47C2×D414yes—
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D411yescharacteristic
162C22×C4×C47C2×D411yescharacteristic
163C22×C4×C47C2×Q812yes—
164C22×C4×C47#4112yes—
165C22×C4×C47#4112yes—
166C22×C4×C47C2411yescharacteristic
167C22×Dic47—22no—
168C22×Dic47—21no—
169C22×Dic47—21no—
170C4⋊C4×C47—28no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—21no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47—22no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47C2×D412yes—
178C4⋊C4×C47C2×D412yes—
179C22⋊C4×C47—28no—
180C22⋊C4×C47—22no—
181C22⋊C4×C47—22no—
182C42×C47—28no—
183C42×C47—22no—
184C42×C47—22no—
185C22⋊Dic47—24no—
186C22⋊Dic47—41no—
187C22⋊Dic47—41no—
188C4⋊Dic47—24no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193Dic47⋊C4—84no—
194C4×Dic47—22no—
195C4×Dic47—41no—
196C4×Dic47—41no—
197C4×Dic47C22×C414yescomplemented
198#10625329—24no—
199#10625329—22no—
200#10625329—22no—
201#10625329—22no—
202#10625329—42no—
203#10625329C22×C414yescomplemented
204C2×C47⋊C8—24no—
205C2×C47⋊C8—22no—
206C2×C47⋊C8—22no—
207C2×C47⋊C8—42no—
208C23×C47—22no—
209C23×C47—24no—
210C23×C47—41no—
211C23×C47—41no—
212C23×C47—42no—
213C23×C47—42no—
214C23×C47C4×D412yes—
215C23×C47#12011yescharacteristic
216C23×C47#12011yescharacteristic
217C23×C47C22⋊Q812yes—
218C23×C47C22×D411yescharacteristic
219Q8×C47—22no—
220Q8×C47—24no—
221Q8×C47C2×C4⋊C414yes—
222D4×C47—24no—
223D4×C47—24no—
224D4×C47—28no—
225D4×C47—22no—
226D4×C47—24no—
227D4×C47C2×C4⋊C414yes—
228C2×C4×C47—22no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—28no—
233C2×C4×C47—22no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—42no—
237C2×C4×C47—44no—
238C2×C4×C47—42no—
239C2×C4×C47—44no—
240C2×C4×C47—82no—
241C2×C4×C47—88no—
242C2×C4×C47C2×C4⋊C412yes—
243C2×C4×C47C2×C4⋊C412yes—
244C2×C4×C47C4×D412yes—
245C2×C4×C47C4×D412yes—
246C2×C4×C47C4×D412yes—
247C2×C4×C47#12011yescharacteristic
248C2×C4×C47#12011yescharacteristic
249C2×C4×C47C22⋊Q812yes—
250C2×C4×C47C22⋊Q812yes—
251C2×C4×C47C22⋊Q812yes—
252C2×C4×C47#12312yes—
253C2×C4×C47#12312yes—
254C2×C4×C47C23×C411yescharacteristic
255C2×C4×C47C22×D411yescharacteristic
256C2×C4×C47C22×D411yescharacteristic
257C2×C4×C47C22×Q811yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×C4×C47C2×#4111yescharacteristic
260C2×Dic47—24no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C47⋊C8—24no—
268C47⋊C8—42no—
269#20864—471nomaximal, Sylow
270C22×C47—21no—
271C22×C47—21no—
272C22×C47—21no—
273C22×C47—21no—
274C22×C47—21no—
275C22×C47—21no—
276C22×C47—24no—
277C22×C47—42no—
278C22×C47—42no—
279C22×C47—48no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47#38512yes—
283C22×C47#38512yes—
284C22×C47C2×C4×D411yescharacteristic
285C22×C47C2×#12011yescharacteristic
286C22×C47C2×C22⋊Q811yescharacteristic
287C4×C47—24no—
288C4×C47—44no—
289C4×C47—42no—
290C4×C47—42no—
291C4×C47#38512yes—
292C4×C47#38512yes—
293C4×C47C22×C4⋊C411yescharacteristic
294C4×C47C2×C4×D411yescharacteristic
295C4×C47C2×C22⋊Q811yescharacteristic
296C4×C47C2×#12311yescharacteristic
297Dic47—44no—
298Dic47—81no—
299Dic47—81no—
300C2×#520D4711yescharacteristic
301#2911—471no—
302C2×#421—471no—
303#2896—471no—
304#2317—471no—
305#1860—474no—
306#1859—474no—
307C2×#343—471no—
308#1733—471no—
309C2×C47—21no—
310C2×C47—22no—
311C2×C47—21no—
312C2×C47—22no—
313C2×C47—44no—
314C2×C47C2×#38511yescharacteristic
315C2×C47#301911yescharacteristic
316C2×C47#302011yescharacteristic
317C22×#41C2×D4711yescharacteristic
318C22×M16—471no—
319C22×M16—471no—
320#520—22no—
321#520—22no—
322#520—22no—
323#520C2×D4714yes—
324C2×C4×Q8—21no—
325C2×C4×D4—21no—
326C2×C4×D4—22no—
327C2×#117—21no—
328C2×#117—471no—
329C2×#117—471no—
330C2×#117C2×D4711yescharacteristic
331C22×C4⋊C4—21no—
332C22×C4⋊C4C2×D4711yescharacteristic
333#428—472no—
334#428—472no—
335#428—941no—
336#428—941no—
337#427—942no—
338#421—478no—
339C2×#104—474no—
340#419—941no—
341#418—941no—
342#417—472no—
343#417—472no—
344#416—942no—
345#389—942no—
346#385—942no—
347#384—472no—
348#382—472no—
349#343—474no—
350#339—474no—
351C47#2086411yescharacteristic, complemented, Sylow
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—21no—
355C2×#41—21no—
356C2×#41Dic9414yes—
357C2×#41C4×D4714yes—
358C2×#41C22⋊D4712yes—
359C2×#41C22×D4712yes—
360C22×Q8C22⋊D4711yescharacteristic
361C22×D4—21no—
362C22×D4C22⋊D4711yescharacteristic
363C23×C4—21no—
364C23×C4—22no—
365C23×C4C22⋊D4712yes—
366C23×C4C22×D4711yescharacteristic
367C2×M16—474no—
368C2×M16—472no—
369C2×M16—472no—
370C2×M16—942no—
371C2×M16—941no—
372C2×M16—941no—
373C2×M16—942no—
374C2×M16—942no—
375C22×C8—941no—
376C22×C8—941no—
377C4×Q8—24no—
378C4×Q8—24no—
379C4×D4—28no—
380C4×D4—24no—
381C4×D4—24no—
382C4×D4—28no—
383#117—24no—
384#117—21no—
385#117—21no—
386#117—24no—
387#117—474no—
388#117—942no—
389#117—941no—
390#117—941no—
391#117—941no—
392#117—941no—
393#117C22⋊D4712yes—
394#117C22×D4712yes—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—471no—
403C2×C4⋊C4—471no—
404C2×C4⋊C4—941no—
405C2×C4⋊C4—944no—
406C2×C4⋊C4C22⋊D4712yes—
407C2×C4⋊C4C22⋊D4711yescharacteristic
408C2×C4⋊C4C22⋊D4711yescharacteristic
409C2×C4⋊C4C22×D4712yes—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—941no—
413C2×C22⋊C4—941no—
414C2×C42—22no—
415C2×C42—21no—
416C2×C42—472no—
417C2×C42—941no—
418#107—942no—
419#107—942no—
420#106—942no—
421#106—942no—
422#104—4716no—
423#103—942no—
424#103—942no—
425#102—942no—
426#102—942no—
427#95—944no—
428C24—21no—
429C24C2×C22⋊D4711yescharacteristic
430#41—24no—
431#41—24no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—22no—
436#41—22no—
437#41Dic47⋊C418yescomplemented
438#41C2×Dic9414yes—
439#41C2×C4×D4714yes—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8Dic47⋊C414yescomplemented
443C2×Q8C2×C22⋊D4712yes—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4Dic47⋊C414yescomplemented
451C2×D4C2×C22⋊D4712yes—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4—942no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4C2×Dic9412yes—
468C22×C4C2×C4×D4712yes—
469C22×C4#1062535012yes—
470C22×C4#1062535012yes—
471C22×C4C2×C22⋊D4714yes—
472C22×C4C2×C22⋊D4711yescharacteristic
473C22×C4C2×C22⋊D4711yescharacteristic
474C22×C4C2×C22⋊D4711yescharacteristic
475C22×C4C2×C22⋊D4711yescharacteristic
476C22×C4C2×C22⋊D4711yescharacteristic
477C22×C4C23×D4711yescharacteristic
478M16—474no—
479M16—944no—
480M16—942no—
481M16—942no—
482M16—942no—
483M16—1882no—
484C2×C8—944no—
485C2×C8—942no—
486C2×C8—942no—
487C2×C8—1882no—
488C4⋊C4—28no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—944no—
496C4⋊C4—941no—
497C4⋊C4—941no—
498C4⋊C4—941no—
499C4⋊C4—941no—
500C4⋊C4—3764no—
501C4⋊C4C2×C22⋊D4712yes—
502C4⋊C4C2×C22⋊D4712yes—
503C22⋊C4—28no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—944no—
507C22⋊C4—1881no—
508C22⋊C4—1881no—
509C42—28no—
510C42—22no—
511C42—22no—
512C42—474no—
513C42—942no—
514C42—1881no—
515C42—1881no—
516C23—24no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23C4×C22⋊D4712yes—
523C23#1309399311yescharacteristic
524C23#1309399311yescharacteristic
525C23#1309400312yes—
526C23C22×C22⋊D4711yescharacteristic
527Q8—24no—
528Q8—22no—
529Q8C2×Dic47⋊C414yes—
530D4—28no—
531D4—24no—
532D4—24no—
533D4—24no—
534D4—22no—
535D4C2×Dic47⋊C414yes—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—82no—
549C2×C4—88no—
550C2×C4—944no—
551C2×C4—1882no—
552C2×C4—1882no—
553C2×C4—1881no—
554C2×C4—1881no—
555C2×C4—1882no—
556C2×C4—1882no—
557C2×C4C2×Dic47⋊C412yes—
558C2×C4C2×Dic47⋊C412yes—
559C2×C4C4×C22⋊D4712yes—
560C2×C4C4×C22⋊D4712yes—
561C2×C4C4×C22⋊D4712yes—
562C2×C4#1309399311yescharacteristic
563C2×C4#1309399311yescharacteristic
564C2×C4#1309400312yes—
565C2×C4#1309400312yes—
566C2×C4#1309400312yes—
567C2×C4#1309400712yes—
568C2×C4#1309400712yes—
569C2×C4C22×C4×D4711yescharacteristic
570C2×C4C22×C22⋊D4711yescharacteristic
571C2×C4C22×C22⋊D4711yescharacteristic
572C2×C4C22×Dic9411yescharacteristic
573C2×C4C2×#1062535011yescharacteristic
574C2×C4C2×#1062535011yescharacteristic
575C8—944no—
576C8—1882no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—48no—
587C22—42no—
588C22—42no—
589C22#1568899612yes—
590C22#1568899612yes—
591C22C2×C4×C22⋊D4711yescharacteristic
592C22C2×#1309399311yescharacteristic
593C22C2×#1309400311yescharacteristic
594C4—24no—
595C4—44no—
596C4—42no—
597C4—42no—
598C4—3761no—
599C4—3761no—
600C4—1884no—
601C4#1568899612yes—
602C4#1568899612yes—
603C4C22×Dic47⋊C411yescharacteristic
604C4C2×C4×C22⋊D4711yescharacteristic
605C4C2×#1309400311yescharacteristic
606C4C2×#1309400711yescharacteristic
607C2—21no—
608C2—22no—
609C2—22no—
610C2—21no—
611C2—44no—
612C2C2×#1568899611yescharacteristic
613C2#2025274511yescharacteristic
614C2#2025275211yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#520C2
socleC22×C47C2×#120
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#520D47
hypocenterC47#20864
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. #20251977C2semidirect
4. #20246012C2semidirect
5. #20243621C2semidirect
6. #20243614C2semidirect
7. C2×#15688892C2semidirect
8. #20243097C2semidirect
14. #15689166C22semidirect
15. #15689166C22semidirect
19. #15689140C4semidirect
28. #15688992C22semidirect
29. #15688988C22semidirect
30. #15688892C22semidirect
31. #15688882C22semidirect
49. C2×#10625329C2×C4semidirect
59. #13093983C2×C4semidirect
72. #13093951C2×C4semidirect
129. #41×C47C4⋊C4semidirect
134. C2×Q8×C47C4⋊C4semidirect
142. C2×D4×C47C4⋊C4semidirect
197. C4×Dic47C22×C4semidirect
203. #10625329C22×C4semidirect
351. C47#20864semidirect
437. #41Dic47⋊C4semidirect
442. C2×Q8Dic47⋊C4semidirect
450. C2×D4Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
#202519772
C2×#156891402
#202519072
#202460122
#202436212
#202436142
C2×#156888922
#202430972
C2×C47×#5202
#2086447