Finite Group Databasebeta

← #26243755

Subgroups — #26243755

Order 12032.

1610 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689159C211yesmaximal, characteristic, complemented
2C2×#15689119C211yesmaximal, characteristic, complemented
3C2×#15689119C211yesmaximal, characteristic, complemented
4C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
5#20245771C211yesmaximal, characteristic, complemented
6#20244588C212yesmaximal, complemented
7#20244581C212yesmaximal, complemented
8#20244574C212yesmaximal, complemented
9#20244567C212yesmaximal, complemented
10#20243104C211yesmaximal, characteristic, complemented
11C2×C47×#525C211yesmaximal, characteristic
12C23×C47⋊C8C2211yescharacteristic
13C23×C47⋊C8C2211yescharacteristic
14C2×#13093983C2211yescharacteristic
15C2×#13093983C2211yescharacteristic
16#15689159—24no—
17C2×#13093957C2212yescomplemented
18C2×#13093955C2212yescomplemented
19#15689119—24no—
20#15689119—24no—
21C2×#13093948C2211yescharacteristic, complemented
22C2×#13093948C2212yescomplemented
23C2×#13093948C2211yescharacteristic, complemented
24C2×#13093945C2211yescharacteristic, complemented
25C2×#13093945C2212yescomplemented
26C2×#13093945C2211yescharacteristic, complemented
27C2×C22⋊(C47⋊C8)—24no—
28C2×C22⋊(C47⋊C8)C2212yes—
29C2×C22⋊(C47⋊C8)C2212yes—
30#15688986—21no—
31#15688983C2212yes—
32#15688983C2212yes—
33#15688979—21no—
34C4×C22⋊Dic47—22no—
35#15688885C2212yescomplemented
36#15688885C2212yescomplemented
37C22×#41×C47C2211yescharacteristic
38C24×C4×C47C2211yescharacteristic
39C47×#525—24no—
40C47×#525—24no—
41C2×#123×C47—21no—
42C2×C22⋊Q8×C47C2212yes—
43C2×#121×C47C2212yes—
44C2×#120×C47—21no—
45C2×C4×D4×C47—22no—
46C2×#117×C47C2211yescharacteristic
47C22×C47⋊C8—22no—
48C22×C47⋊C8—22no—
49C22×C47⋊C8—22no—
50C22×C47⋊C8—22no—
51C22×C47⋊C8—22no—
52C22×C47⋊C8D414yescomplemented
53C22×C47⋊C8C2312yes—
54C22×C47⋊C8C2311yescharacteristic
55C22×C47⋊C8C2311yescharacteristic
56#13093983—22no—
57#13093983—22no—
58#13093983—22no—
59#13093983—22no—
60C2×C4⋊Dic47C2311yescharacteristic
61C2×C4⋊Dic47C2311yescharacteristic
62C2×C4⋊Dic47C2312yes—
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47—22no—
65C2×C22⋊Dic47—21no—
66C2×C22⋊Dic47—21no—
67C2×C4×Dic47—21no—
68C2×C4×Dic47—21no—
69#13093957—24no—
70#13093955—24no—
71#13093948—24no—
72#13093948—22no—
73#13093948—22no—
74#13093945—24no—
75#13093945—22no—
76#13093945—22no—
77C22⋊(C47⋊C8)—24no—
78C22⋊(C47⋊C8)—24no—
79C47⋊#95—22no—
80C47⋊#95—22no—
81C25×C47D411yescharacteristic
82C2×#41×C47—22no—
83C2×#41×C47—22no—
84C2×#41×C47—22no—
85C2×#41×C47—22no—
86C2×#41×C47—22no—
87C2×#41×C47D414yes—
88C22×Q8×C47C2311yescharacteristic
89C22×D4×C47—22no—
90C22×D4×C47—21no—
91C22×D4×C47C2311yescharacteristic
92C23×C4×C47—21no—
93C23×C4×C47—22no—
94C23×C4×C47—22no—
95C23×C4×C47—22no—
96C23×C4×C47—22no—
97C23×C4×C47—22no—
98C23×C4×C47—22no—
99C23×C4×C47D412yes—
100C23×C4×C47D411yescharacteristic
101C23×C4×C47C2312yes—
102C23×C4×C47C2311yescharacteristic
103#123×C47—28no—
104C22⋊Q8×C47—24no—
105C22⋊Q8×C47—24no—
106#121×C47—24no—
107#121×C47—24no—
108#120×C47—28no—
109C4×D4×C47—216no—
110#117×C47—22no—
111#117×C47D414yes—
112C2×C4⋊C4×C47—22no—
113C2×C4⋊C4×C47C2312yes—
114C2×C22⋊C4×C47—21no—
115C2×C22⋊C4×C47—22no—
116C2×C22⋊C4×C47—22no—
117C2×C42×C47—21no—
118C24×C47—22no—
119C24×C47—24no—
120C24×C47—22no—
121C24×C47—22no—
122C24×C47—21no—
123C24×C47—42no—
124C24×C47C2×D412yes—
125C24×C47C2×D411yescharacteristic
126#41×C47—28no—
127#41×C47—28no—
128#41×C47—44no—
129#41×C47—44no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47—21no—
133C2×Q8×C47—21no—
134C2×Q8×C47—21no—
135C2×Q8×C47C2×D412yes—
136C2×D4×C47—28no—
137C2×D4×C47—24no—
138C2×D4×C47—21no—
139C2×D4×C47—21no—
140C2×D4×C47—21no—
141C2×D4×C47—21no—
142C2×D4×C47—21no—
143C2×D4×C47—44no—
144C2×D4×C47—42no—
145C2×D4×C47—42no—
146C2×D4×C47—44no—
147C2×D4×C47C2×D412yes—
148C22×C4×C47—22no—
149C22×C4×C47—22no—
150C22×C4×C47—22no—
151C22×C4×C47—22no—
152C22×C4×C47—21no—
153C22×C4×C47—24no—
154C22×C4×C47—24no—
155C22×C4×C47—24no—
156C22×C4×C47—24no—
157C22×C4×C47—22no—
158C22×C4×C47—22no—
159C22×C4×C47—22no—
160C22×C4×C47—22no—
161C22×C4×C47—22no—
162C22×C4×C47—22no—
163C22×C4×C47—44no—
164C22×C4×C47—44no—
165C22×C4×C47—42no—
166C22×C4×C47—44no—
167C22×C4×C47—42no—
168C22×C4×C47—44no—
169C22×C4×C47C2×D412yes—
170C22×C4×C47C2×D411yescharacteristic
171C22×C4×C47C2×D412yes—
172C22×C4×C47C2×D412yes—
173C22×C4×C47C2×D412yes—
174C22×C4×C47#4112yes—
175C22×C4×C47#4112yes—
176C22×C4×C47#4112yes—
177C22×C4×C47C2411yescharacteristic
178C22×Dic47—22no—
179C22×Dic47—21no—
180C22×Dic47—21no—
181C4⋊C4×C47—28no—
182C4⋊C4×C47—22no—
183C4⋊C4×C47C2×D414yes—
184C22⋊C4×C47—28no—
185C22⋊C4×C47—28no—
186C22⋊C4×C47—24no—
187C42×C47—24no—
188C22⋊Dic47—44no—
189C22⋊Dic47—42no—
190C22⋊Dic47—42no—
191C22⋊Dic47—44no—
192C4⋊Dic47—22no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47—22no—
196C4⋊Dic47—21no—
197C4⋊Dic47—21no—
198C4×Dic47—42no—
199C4×Dic47—42no—
200C2×C47⋊C8—28no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—22no—
206C2×C47⋊C8—22no—
207C2×C47⋊C8—22no—
208C2×C47⋊C8—44no—
209C2×C47⋊C8C2×D412yes—
210C2×C47⋊C8C2×D412yes—
211C23×C47—22no—
212C23×C47—22no—
213C23×C47—28no—
214C23×C47—22no—
215C23×C47—22no—
216C23×C47—22no—
217C23×C47—22no—
218C23×C47—22no—
219C23×C47—22no—
220C23×C47—22no—
221C23×C47—48no—
222C23×C47—48no—
223C23×C47—44no—
224C23×C47—44no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—42no—
228C23×C47—42no—
229C23×C47#12012yes—
230C23×C47#12112yes—
231C23×C47#12312yes—
232C23×C47#13514yes—
233C23×C47C22×D411yescharacteristic
234Q8×C47—22no—
235Q8×C47—22no—
236Q8×C47—41no—
237Q8×C47—41no—
238D4×C47—22no—
239D4×C47—22no—
240D4×C47—416no—
241D4×C47—48no—
242D4×C47—41no—
243D4×C47—41no—
244C2×C4×C47—24no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—22no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47—44no—
254C2×C4×C47—42no—
255C2×C4×C47—42no—
256C2×C4×C47—42no—
257C2×C4×C47—42no—
258C2×C4×C47—48no—
259C2×C4×C47—48no—
260C2×C4×C47#12012yes—
261C2×C4×C47#12114yes—
262C2×C4×C47#12114yes—
263C2×C4×C47#12112yes—
264C2×C4×C47#12314yes—
265C2×C4×C47#12312yes—
266C2×C4×C47#12414yes—
267C2×C4×C47C22×D411yescharacteristic
268C2×C4×C47C22×D411yescharacteristic
269C2×C4×C47C22×D411yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×C4×C47C2×#4111yescharacteristic
272C2×C4×C47C2×#4111yescharacteristic
273C2×Dic47—42no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—42no—
277C2×Dic47—42no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C47⋊C8—22no—
281C47⋊C8—22no—
282C47⋊C8—42no—
283#23479—471nomaximal, Sylow
284C22×C47—24no—
285C22×C47—24no—
286C22×C47—22no—
287C22×C47—22no—
288C22×C47—24no—
289C22×C47—24no—
290C22×C47—24no—
291C22×C47—416no—
292C22×C47—42no—
293C22×C47—42no—
294C22×C47—42no—
295C22×C47—42no—
296C22×C47—42no—
297C22×C47—44no—
298C22×C47—82no—
299C22×C47—81no—
300C22×C47—81no—
301C22×C47#39414yes—
302C22×C47C2×#12011yescharacteristic
303C22×C47C2×#12111yescharacteristic
304C22×C47C2×#12311yescharacteristic
305C22×C47C2×#13512yes—
306C22×C47#57512yes—
307C4×C47—42no—
308C4×C47—42no—
309C4×C47—44no—
310C4×C47#39414yes—
311C4×C47C2×#12111yescharacteristic
312C4×C47C2×#12111yescharacteristic
313C4×C47C2×#12311yescharacteristic
314C4×C47C2×#12411yescharacteristic
315Dic47—82no—
316Dic47—81no—
317Dic47—81no—
318C2×#525D4711yescharacteristic
319C2×#427—471no—
320C2×#416—471no—
321C2×#416—471no—
322C22×C22⋊C8—471no—
323#2280—471no—
324#2036—472no—
325#2035—472no—
326#2034—472no—
327#2033—472no—
328#1735—471no—
329C2×C47—24no—
330C2×C47—44no—
331C2×C47—82no—
332C2×C47C2×#39411yescharacteristic
333C2×C47#300212yes—
334C2×C47#305011yescharacteristic
335C2×C47#305411yescharacteristic
336C2×C47#309112yes—
337C22×#41C2×D4711yescharacteristic
338C24×C4C2×D4711yescharacteristic
339C23×C8—471no—
340C23×C8—471no—
341#525—24no—
342#525—24no—
343C2×#123—21no—
344C2×C22⋊Q8C2×D4712yes—
345C2×#121C2×D4712yes—
346C2×#120—21no—
347C2×C4×D4—22no—
348C2×#117—471no—
349C2×#117—471no—
350C2×#117C2×D4711yescharacteristic
351#427—944no—
352C2×#107—472no—
353C2×#106—472no—
354#416—944no—
355#416—944no—
356C2×#103—471no—
357C2×#103—472no—
358C2×#103—471no—
359C2×#102—471no—
360C2×#102—472no—
361C2×#102—471no—
362C2×C22⋊C8—472no—
363C2×C22⋊C8—472no—
364C2×C22⋊C8—944no—
365#381—941no—
366#380—472no—
367#380—472no—
368#379—941no—
369C4×C22⋊C4—942no—
370#340—472no—
371#340—472no—
372C47#2347911yescharacteristic, complemented, Sylow
373C25C22⋊D4711yescharacteristic
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41C22⋊D4714yes—
380C22×Q8C22×D4711yescharacteristic
381C22×D4—22no—
382C22×D4—21no—
383C22×D4C22×D4711yescharacteristic
384C23×C4—22no—
385C23×C4—22no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—21no—
391C23×C4C22⋊D4711yescharacteristic
392C23×C4C22⋊D4712yes—
393C23×C4C22×D4712yes—
394C23×C4C22×D4711yescharacteristic
395C22×C8—474no—
396C22×C8—471no—
397C22×C8—471no—
398C22×C8—472no—
399C22×C8—942no—
400C22×C8—942no—
401C22×C8—942no—
402C22×C8—942no—
403C22×C8—942no—
404#123—28no—
405C22⋊Q8—24no—
406C22⋊Q8—24no—
407#121—24no—
408#121—24no—
409#120—28no—
410C4×D4—216no—
411#117—22no—
412#117—942no—
413#117—942no—
414#117—942no—
415#117—942no—
416#117C22⋊D4714yes—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—471no—
419C2×C4⋊C4—471no—
420C2×C4⋊C4—472no—
421C2×C4⋊C4C22×D4712yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—941no—
426C2×C22⋊C4—941no—
427C2×C22⋊C4—942no—
428C2×C22⋊C4—942no—
429C2×C42—21no—
430C2×C42—941no—
431C2×C42—941no—
432#107—944no—
433#106—944no—
434#103—942no—
435#103—942no—
436#103—944no—
437#102—942no—
438#102—942no—
439#102—944no—
440C22⋊C8—944no—
441C22⋊C8—944no—
442#95—942no—
443#95—942no—
444C24—22no—
445C24—24no—
446C24—22no—
447C24—22no—
448C24—21no—
449C24—42no—
450C24C2×C22⋊D4712yes—
451C24C2×C22⋊D4711yescharacteristic
452#41—28no—
453#41—28no—
454#41—44no—
455#41—44no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×C22⋊D4712yes—
462C2×D4—28no—
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—44no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—44no—
473C2×D4C2×C22⋊D4712yes—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—942no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4D4×D4712yes—
499C22×C4#1062535212yes—
500C22×C4#1062535212yes—
501C22×C4#1062535212yes—
502C22×C4C2×C22⋊D4712yes—
503C22×C4C2×C22⋊D4712yes—
504C22×C4C2×C22⋊D4712yes—
505C22×C4C2×C22⋊D4711yescharacteristic
506C22×C4C23×D4711yescharacteristic
507C2×C8—472no—
508C2×C8—472no—
509C2×C8—948no—
510C2×C8—941no—
511C2×C8—941no—
512C2×C8—941no—
513C2×C8—941no—
514C2×C8—942no—
515C2×C8—942no—
516C2×C8—942no—
517C2×C8—1884no—
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—941no—
521C4⋊C4—941no—
522C4⋊C4—942no—
523C4⋊C4—942no—
524C4⋊C4—941no—
525C4⋊C4—941no—
526C4⋊C4C2×C22⋊D4714yes—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—1884no—
531C22⋊C4—1882no—
532C22⋊C4—1882no—
533C22⋊C4—1884no—
534C42—24no—
535C42—1882no—
536C42—1882no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—48no—
548C23—48no—
549C23—44no—
550C23—44no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23#1309399312yes—
556C23#1309399612yes—
557C23#1309400512yes—
558C23#1309404414yes—
559C23C22×C22⋊D4711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—416no—
567D4—48no—
568D4—41no—
569D4—41no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—1882no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1882no—
593C2×C4#1309399312yes—
594C2×C4#1309399514yes—
595C2×C4#1309399612yes—
596C2×C4#1309399614yes—
597C2×C4#1309400512yes—
598C2×C4#1309400514yes—
599C2×C4#1309401114yes—
600C2×C4C2×D4×D4711yescharacteristic
601C2×C4C22×C22⋊D4711yescharacteristic
602C2×C4C22×C22⋊D4711yescharacteristic
603C2×C4C2×#1062535211yescharacteristic
604C2×C4C2×#1062535211yescharacteristic
605C2×C4C2×#1062535211yescharacteristic
606C8—942no—
607C8—942no—
608C8—1882no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—416no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—81no—
624C22—81no—
625C22—82no—
626C22#1568903714yes—
627C22C2×#1309399311yescharacteristic
628C22C2×#1309399611yescharacteristic
629C22C2×#1309400511yescharacteristic
630C22C2×#1309404412yes—
631C22#1568992112yes—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—3761no—
636C4—3761no—
637C4—3762no—
638C4#1568903714yes—
639C4C2×#1309399511yescharacteristic
640C4C2×#1309399611yescharacteristic
641C4C2×#1309400511yescharacteristic
642C4C2×#1309401111yescharacteristic
643C2—24no—
644C2—44no—
645C2—82no—
646C2C2×#1568903711yescharacteristic
647C2#2025260612yes—
648C2#2025296611yescharacteristic
649C2#2025299411yescharacteristic
650C2#2025329712yes—
subgroupG/H
centerC22×C4C2×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#525C2
socleC23×C47C22×D4
hypercenterC2×#525D47
hypocenterC47#23479
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C2×#15689159C2semidirect
2. C2×#15689119C2semidirect
3. C2×#15689119C2semidirect
4. C22×C22⋊(C47⋊C8)C2semidirect
5. #20245771C2semidirect
6. #20244588C2semidirect
7. #20244581C2semidirect
8. #20244574C2semidirect
9. #20244567C2semidirect
10. #20243104C2semidirect
17. C2×#13093957C22semidirect
18. C2×#13093955C22semidirect
21. C2×#13093948C22semidirect
22. C2×#13093948C22semidirect
23. C2×#13093948C22semidirect
24. C2×#13093945C22semidirect
25. C2×#13093945C22semidirect
26. C2×#13093945C22semidirect
35. #15688885C22semidirect
36. #15688885C22semidirect
52. C22×C47⋊C8D4semidirect
372. C47#23479semidirect

Maximal subgroups

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

subgroupindex
C2×#156891592
C2×#156891192
C22×C22⋊(C47⋊C8)2
#202457712
#202445882
#202445812
#202445742
#202445672
#202431042
C2×C47×#5252
#2347947