Finite Group Databasebeta

← #26249298

Subgroups — #26249298

Order 12032.

1484 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689456C211yesmaximal, characteristic, complemented
2C22×#13093945C211yesmaximal, characteristic, complemented
3#20247705C211yesmaximal, characteristic, complemented
4#20244624C212yesmaximal, complemented
5#20244567C212yesmaximal, complemented
6#20243812C211yesmaximal, characteristic, complemented
7#20243681C211yesmaximal, characteristic, complemented
8#20243635C212yesmaximal, complemented
9#20243328C212yesmaximal, complemented
10#20243319C211yesmaximal, characteristic, complemented
11C2×C47×#546C211yesmaximal, characteristic
12C23×C47⋊C8C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14#15689456—24no—
15C2×#13093957C2212yescomplemented
16C2×#13093948C2212yescomplemented
17C2×#13093948C2212yescomplemented
18C2×#13093945—24no—
19C2×#13093945C2212yes—
20C2×#13093945C2212yes—
21C2×#13093945C2212yes—
22C2×#13093945C2212yes—
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×C22⋊(C47⋊C8)C2212yescomplemented
25#15689062—21no—
26#15689059—21no—
27#15689021C2212yescomplemented
28#15689003—21no—
29#15688995—21no—
30#15688983C2211yescharacteristic
31#15688983C2212yescomplemented
32#15688885C2212yescomplemented
33#15688878C2212yescomplemented
34C22×#41×C47C2211yescharacteristic
35C23×D4×C47C2211yescharacteristic
36C47×#546—24no—
37C47×#546—24no—
38C2×#124×C47C2211yescharacteristic
39C2×#123×C47—21no—
40C2×C22⋊Q8×C47C2211yescharacteristic
41C2×#121×C47—21no—
42C2×#121×C47C2211yescharacteristic
43C2×#120×C47—21no—
44C2×C4×D4×C47C2211yescharacteristic
45C2×C4×D4×C47C2211yescharacteristic
46C22×C22⋊C4×C47—21no—
47C23×Dic47—21no—
48C22×C47⋊C8—22no—
49C22×C47⋊C8—42no—
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2312yes—
52C2×C4⋊Dic47—22no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47—21no—
55C2×C4⋊Dic47C2312yes—
56C2×C4⋊Dic47C2312yes—
57C2×Dic47⋊C4—22no—
58C2×C22⋊Dic47—21no—
59C2×C22⋊Dic47—22no—
60#13093957—24no—
61#13093948—24no—
62#13093948—42no—
63#13093945—24no—
64#13093945—24no—
65#13093945—24no—
66#13093945—42no—
67C22⋊(C47⋊C8)—22no—
68C22⋊(C47⋊C8)—42no—
69C47⋊#95—21no—
70C47⋊#95—22no—
71C47⋊#95—22no—
72C25×C47—21no—
73C2×#41×C47—22no—
74C2×#41×C47—22no—
75C2×#41×C47—24no—
76C2×#41×C47—42no—
77C22×Q8×C47C2311yescharacteristic
78C22×D4×C47—24no—
79C22×D4×C47—21no—
80C22×D4×C47—21no—
81C22×D4×C47—21no—
82C22×D4×C47—22no—
83C22×D4×C47—42no—
84C22×D4×C47D411yescharacteristic
85C22×D4×C47D414yes—
86C22×D4×C47C2311yescharacteristic
87C22×D4×C47C2311yescharacteristic
88C23×C4×C47—21no—
89C23×C4×C47—21no—
90C23×C4×C47C2311yescharacteristic
91C23×C4×C47C2311yescharacteristic
92#124×C47—22no—
93#124×C47—22no—
94#123×C47—28no—
95C22⋊Q8×C47—22no—
96C22⋊Q8×C47D414yes—
97#121×C47—28no—
98#121×C47—22no—
99#121×C47—22no—
100#120×C47—28no—
101C4×D4×C47—22no—
102C4×D4×C47—22no—
103C4×D4×C47—22no—
104C4×D4×C47—22no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47D411yescharacteristic
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C22⋊C4×C47—28no—
109C2×C22⋊C4×C47—21no—
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47D411yescharacteristic
115C2×C22⋊C4×C47D411yescharacteristic
116C2×C42×C47C2311yescharacteristic
117C24×C47—21no—
118C24×C47—21no—
119C24×C47—24no—
120C24×C47—44no—
121C24×C47—42no—
122C24×C47—42no—
123C24×C47—82no—
124C24×C47C2×D411yescharacteristic
125C24×C47C2×D411yescharacteristic
126#41×C47—48no—
127#41×C47—44no—
128#41×C47—44no—
129C2×Q8×C47—21no—
130C2×Q8×C47—22no—
131C2×Q8×C47—22no—
132C2×Q8×C47C2×D412yes—
133C2×D4×C47—28no—
134C2×D4×C47—24no—
135C2×D4×C47—22no—
136C2×D4×C47—22no—
137C2×D4×C47—22no—
138C2×D4×C47—24no—
139C2×D4×C47—24no—
140C2×D4×C47—21no—
141C2×D4×C47—21no—
142C2×D4×C47—22no—
143C2×D4×C47—22no—
144C2×D4×C47—22no—
145C2×D4×C47—44no—
146C2×D4×C47—48no—
147C2×D4×C47—44no—
148C2×D4×C47—44no—
149C2×D4×C47—82no—
150C2×D4×C47C2×D412yes—
151C2×D4×C47C2×D412yes—
152C2×D4×C47#4114yes—
153C22×C4×C47—22no—
154C22×C4×C47—22no—
155C22×C4×C47—24no—
156C22×C4×C47—21no—
157C22×C4×C47—21no—
158C22×C4×C47—24no—
159C22×C4×C47—21no—
160C22×C4×C47—21no—
161C22×C4×C47—22no—
162C22×C4×C47—22no—
163C22×C4×C47—44no—
164C22×C4×C47—82no—
165C22×C4×C47C2×D411yescharacteristic
166C22×C4×C47C2×D411yescharacteristic
167C22×C4×C47C2×D411yescharacteristic
168C22×C4×C47C2×D411yescharacteristic
169C22×C4×C47C2×D412yes—
170C22×C4×C47C2×D412yes—
171C22×C4×C47C2411yescharacteristic
172C22×Dic47—22no—
173C22×Dic47—22no—
174C22×Dic47—42no—
175C22×Dic47—82no—
176C4⋊C4×C47—24no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47C2×D412yes—
180C22⋊C4×C47—24no—
181C22⋊C4×C47—24no—
182C22⋊C4×C47—24no—
183C22⋊C4×C47—24no—
184C22⋊C4×C47—22no—
185C22⋊C4×C47—24no—
186C22⋊C4×C47—22no—
187C42×C47—21no—
188C42×C47—21no—
189C22⋊Dic47—42no—
190C22⋊Dic47—82no—
191C4⋊Dic47—22no—
192C4⋊Dic47—22no—
193C4⋊Dic47—22no—
194C4⋊Dic47—22no—
195C4⋊Dic47—42no—
196Dic47⋊C4—82no—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—22no—
200C2×C47⋊C8—22no—
201C2×C47⋊C8—22no—
202C2×C47⋊C8—22no—
203C2×C47⋊C8—44no—
204C23×C47—22no—
205C23×C47—22no—
206C23×C47—22no—
207C23×C47—24no—
208C23×C47—22no—
209C23×C47—21no—
210C23×C47—22no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—22no—
214C23×C47—48no—
215C23×C47—41no—
216C23×C47—41no—
217C23×C47—42no—
218C23×C47—42no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47—42no—
222C23×C47—48no—
223C23×C47—84no—
224C23×C47—88no—
225C23×C47—88no—
226C23×C47#12012yes—
227C23×C47#13512yes—
228C23×C47#13712yes—
229C23×C47C22×D411yescharacteristic
230Q8×C47—24no—
231Q8×C47—41no—
232Q8×C47—41no—
233D4×C47—22no—
234D4×C47—22no—
235D4×C47—22no—
236D4×C47—24no—
237D4×C47—48no—
238D4×C47—48no—
239D4×C47—44no—
240D4×C47—42no—
241D4×C47—84no—
242D4×C47#12314yes—
243C2×C4×C47—22no—
244C2×C4×C47—22no—
245C2×C4×C47—22no—
246C2×C4×C47—24no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—22no—
250C2×C4×C47—21no—
251C2×C4×C47—22no—
252C2×C4×C47—22no—
253C2×C4×C47—21no—
254C2×C4×C47—21no—
255C2×C4×C47—22no—
256C2×C4×C47—48no—
257C2×C4×C47—42no—
258C2×C4×C47—42no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47—42no—
262C2×C4×C47—84no—
263C2×C4×C47#12012yes—
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47C22×D411yescharacteristic
266C2×C4×C47C2×#4112yes—
267C2×C4×C47H(5,𝔽2)12yes—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47—42no—
273C2×Dic47—42no—
274C2×Dic47—84no—
275C47⋊C8—42no—
276C47⋊C8—42no—
277#24039—471nomaximal, Sylow
278C22×C47—22no—
279C22×C47—22no—
280C22×C47—24no—
281C22×C47—24no—
282C22×C47—24no—
283C22×C47—42no—
284C22×C47—42no—
285C22×C47—44no—
286C22×C47—44no—
287C22×C47—41no—
288C22×C47—42no—
289C22×C47—41no—
290C22×C47—42no—
291C22×C47—41no—
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—42no—
295C22×C47—816no—
296C22×C47—82no—
297C22×C47—82no—
298C22×C47#45214yes—
299C22×C47C2×#12011yescharacteristic
300C22×C47#54612yes—
301C22×C47C2×#13511yescharacteristic
302C22×C47C2×#13711yescharacteristic
303C22×C47#57612yes—
304C4×C47—22no—
305C4×C47—41no—
306C4×C47—42no—
307C4×C47—41no—
308C4×C47—41no—
309C4×C47—42no—
310C4×C47C2×#12311yescharacteristic
311C4×C47#53411yescharacteristic
312C4×C47#54612yes—
313Dic47—82no—
314Dic47—82no—
315C2×#546D4711yescharacteristic
316C2×#484—471no—
317C22×#102—471no—
318#2487—471no—
319#2042—472no—
320#2033—472no—
321#1892—471no—
322#1869—471no—
323#1862—472no—
324#1788—472no—
325#1786—471no—
326C2×C47—24no—
327C2×C47—44no—
328C2×C47—81no—
329C2×C47—82no—
330C2×C47#240511yescharacteristic
331C2×C47C2×#45211yescharacteristic
332C2×C47#300811yescharacteristic
333C2×C47#329712yes—
334C2×C47#331412yes—
335C22×#41C2×D4711yescharacteristic
336C23×D4C2×D4711yescharacteristic
337C23×C8—471no—
338#546—24no—
339#546—24no—
340C2×#124C2×D4711yescharacteristic
341C2×#123—21no—
342C2×C22⋊Q8C2×D4711yescharacteristic
343C2×#121—21no—
344C2×#121C2×D4711yescharacteristic
345C2×#120—21no—
346C2×C4×D4C2×D4711yescharacteristic
347C2×C4×D4C2×D4711yescharacteristic
348C22×C4⋊C4—471no—
349C22×C22⋊C4—21no—
350#484—944no—
351C2×#107—472no—
352C2×#103—472no—
353C2×#103—472no—
354C2×#102—472no—
355C2×#102—472no—
356C2×#102—472no—
357C2×#102—472no—
358C2×#102—944no—
359C2×C22⋊C8—471no—
360C2×C22⋊C8—472no—
361#400—941no—
362#399—941no—
363#389—472no—
364#386—941no—
365#385—941no—
366#380—471no—
367#380—472no—
368#340—472no—
369#336—472no—
370C47#2403911yescharacteristic, complemented, Sylow
371C25—21no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41—24no—
375C2×#41—42no—
376C22×Q8C22×D4711yescharacteristic
377C22×D4—24no—
378C22×D4—21no—
379C22×D4—21no—
380C22×D4—21no—
381C22×D4—22no—
382C22×D4—42no—
383C22×D4C22⋊D4711yescharacteristic
384C22×D4C22⋊D4714yes—
385C22×D4C22×D4711yescharacteristic
386C22×D4C22×D4711yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4—941no—
390C23×C4C22×D4711yescharacteristic
391C23×C4C22×D4711yescharacteristic
392C22×C8—472no—
393C22×C8—472no—
394C22×C8—942no—
395C22×C8—1882no—
396#124—22no—
397#124—22no—
398#123—28no—
399C22⋊Q8—22no—
400C22⋊Q8C22⋊D4714yes—
401#121—28no—
402#121—22no—
403#121—22no—
404#120—28no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—472no—
411C2×C4⋊C4—472no—
412C2×C4⋊C4—942no—
413C2×C4⋊C4—942no—
414C2×C4⋊C4—942no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4C22⋊D4711yescharacteristic
417C2×C4⋊C4C22×D4711yescharacteristic
418C2×C22⋊C4—28no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—942no—
425C2×C22⋊C4—941no—
426C2×C22⋊C4C22⋊D4711yescharacteristic
427C2×C22⋊C4C22⋊D4711yescharacteristic
428C2×C42C22×D4711yescharacteristic
429#107—944no—
430#103—944no—
431#103—1882no—
432#102—944no—
433#102—944no—
434#102—944no—
435#102—1882no—
436C22⋊C8—942no—
437C22⋊C8—1882no—
438#95—942no—
439#95—941no—
440#95—942no—
441C24—21no—
442C24—21no—
443C24—24no—
444C24—44no—
445C24—42no—
446C24—42no—
447C24—82no—
448C24C2×C22⋊D4711yescharacteristic
449C24C2×C22⋊D4711yescharacteristic
450#41—48no—
451#41—44no—
452#41—44no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8C2×C22⋊D4712yes—
457C2×D4—28no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—44no—
470C2×D4—48no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—82no—
474C2×D4#1062535214yes—
475C2×D4C2×C22⋊D4712yes—
476C2×D4C2×C22⋊D4712yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—82no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—3762no—
492C22×C4—1882no—
493C22×C4C2×C22⋊D4711yescharacteristic
494C22×C4C2×C22⋊D4711yescharacteristic
495C22×C4C2×C22⋊D4711yescharacteristic
496C22×C4C2×C22⋊D4711yescharacteristic
497C22×C4C2×C22⋊D4712yes—
498C22×C4C2×C22⋊D4712yes—
499C22×C4C23×D4711yescharacteristic
500C2×C8—942no—
501C2×C8—942no—
502C2×C8—942no—
503C2×C8—942no—
504C2×C8—942no—
505C2×C8—942no—
506C2×C8—1884no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—942no—
511C4⋊C4—942no—
512C4⋊C4—942no—
513C4⋊C4—942no—
514C4⋊C4—3762no—
515C4⋊C4—1882no—
516C4⋊C4C2×C22⋊D4712yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—3762no—
525C22⋊C4—1882no—
526C42—21no—
527C42—21no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—48no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—48no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—84no—
548C23—88no—
549C23—88no—
550C23#1309399312yes—
551C23#1309404412yes—
552C23#1309405412yes—
553C23C22×C22⋊D4711yescharacteristic
554Q8—24no—
555Q8—41no—
556Q8—41no—
557D4—24no—
558D4—22no—
559D4—22no—
560D4—22no—
561D4—48no—
562D4—48no—
563D4—44no—
564D4—42no—
565D4—84no—
566D4#1309400514yes—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—3764no—
588C2×C4—1882no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1882no—
593C2×C4—1882no—
594C2×C4#1309399312yes—
595C2×C4C22×C22⋊D4711yescharacteristic
596C2×C4C22×C22⋊D4711yescharacteristic
597C2×C4C2×#1062535212yes—
598C2×C4#1309406912yes—
599C8—1882no—
600C8—1882no—
601C22—24no—
602C22—22no—
603C22—24no—
604C22—22no—
605C22—24no—
606C22—44no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—42no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—816no—
619C22—82no—
620C22—82no—
621C22#1568928514yes—
622C22C2×#1309399311yescharacteristic
623C22#1568976712yes—
624C22C2×#1309404411yescharacteristic
625C22C2×#1309405411yescharacteristic
626C22#1568992612yes—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—3762no—
634C4—3762no—
635C4C2×#1309400511yescharacteristic
636C4#1568968211yescharacteristic
637C4#1568976712yes—
638C2—24no—
639C2—44no—
640C2—81no—
641C2—82no—
642C2#2024671111yescharacteristic
643C2C2×#1568928511yescharacteristic
644C2#2025265611yescharacteristic
645C2#2025523812yes—
646C2#2025546912yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#546C2
socleC23×C47C22×D4
hypercenterC2×#546D47
hypocenterC47#24039
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
1. C2×#15689456C2semidirect
2. C22×#13093945C2semidirect
3. #20247705C2semidirect
4. #20244624C2semidirect
5. #20244567C2semidirect
6. #20243812C2semidirect
7. #20243681C2semidirect
8. #20243635C2semidirect
9. #20243328C2semidirect
10. #20243319C2semidirect
15. C2×#13093957C22semidirect
16. C2×#13093948C22semidirect
17. C2×#13093948C22semidirect
24. C2×C22⋊(C47⋊C8)C22semidirect
27. #15689021C22semidirect
31. #15688983C22semidirect
32. #15688885C22semidirect
33. #15688878C22semidirect
370. C47#24039semidirect

Maximal subgroups

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

subgroupindex
C2×#156894562
C22×#130939452
#202477052
#202446242
#202445672
#202438122
#202436812
#202436352
#202433282
#202433192
C2×C47×#5462
#2403947