Finite Group Databasebeta

← #26138809

Subgroups — #26138809

Order 12032.

1862 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
1C22×#13093980C211yesmaximal, characteristic
2C2×#15688981C211yesmaximal, characteristic
3C2×C22⋊C4×Dic47C211yesmaximal, characteristic
4#20241637C211yesmaximal, characteristic
5#20241635C211yesmaximal, characteristic
6#20241635C211yesmaximal, characteristic
7Dic47⋊#95C212yesmaximal, complemented
8Dic47⋊#95C212yesmaximal, complemented
9#20241614C212yesmaximal, complemented
10#20241613C212yesmaximal, complemented
11C47×#1440C211yesmaximal, characteristic
12C24×Dic47C2211yescharacteristic
13C2×#13093980—24no—
14C2×#13093980—24no—
15C2×#13093980C418yescomplemented
16C22×Dic47⋊C4C2212yes—
17C22×Dic47⋊C4C2211yescharacteristic
18C22×Dic47⋊C4C2211yescharacteristic
19C22×C22⋊Dic47C2211yescharacteristic
20C22×C22⋊Dic47C2211yescharacteristic
21C22×C22⋊Dic47C2211yescharacteristic
22C22×C4×Dic47C2212yes—
23C22×C4×Dic47C2211yescharacteristic
24C22×C4×Dic47C2211yescharacteristic
25#15688981—24no—
26C22⋊C4×Dic47—24no—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C22×C22⋊C4×C47C2211yescharacteristic
36C22×C22⋊C4×C47C2211yescharacteristic
37C22×C22⋊C4×C47C2211yescharacteristic
38C2×#95×C47C2212yes—
39C2×#95×C47C2212yes—
40C23×Dic47—24no—
41C23×Dic47—24no—
42C23×Dic47C2×C414yes—
43C23×Dic47D414yes—
44C23×Dic47D414yes—
45C23×Dic47C2312yes—
46C23×Dic47C2311yescharacteristic
47C23×Dic47C2311yescharacteristic
48C23×Dic47C2312yes—
49C23×Dic47C2311yescharacteristic
50C23×Dic47C2311yescharacteristic
51#13093980—232no—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—22no—
54C2×Dic47⋊C4—22no—
55C2×Dic47⋊C4—22no—
56C2×Dic47⋊C4—24no—
57C2×Dic47⋊C4—24no—
58C2×Dic47⋊C4—24no—
59C2×Dic47⋊C4C2×C414yes—
60C2×Dic47⋊C4C2×C414yes—
61C2×C22⋊Dic47—22no—
62C2×C22⋊Dic47—22no—
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47—24no—
65C2×C22⋊Dic47—22no—
66C2×C22⋊Dic47—42no—
67C2×C22⋊Dic47C2×C414yes—
68C2×C4×Dic47—22no—
69C2×C4×Dic47—22no—
70C2×C4×Dic47—22no—
71C2×C4×Dic47—22no—
72C2×C4×Dic47—24no—
73C2×C4×Dic47—24no—
74C2×C4×Dic47—24no—
75C2×C4×Dic47C2×C414yes—
76C2×C4×Dic47C2×C414yes—
77C47⋊#95—24no—
78C47⋊#95—24no—
79C47⋊#95—24no—
80C47⋊#95—24no—
81C47⋊#95—24no—
82C47⋊#95—42no—
83C47⋊#95—42no—
84C47⋊#95—42no—
85C25×C47C2311yescharacteristic
86C23×C4×C47C2312yes—
87C23×C4×C47C2311yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C23×C4×C47C2312yes—
90C2×C22⋊C4×C47—22no—
91C2×C22⋊C4×C47—22no—
92C2×C22⋊C4×C47—22no—
93C2×C22⋊C4×C47—24no—
94C2×C22⋊C4×C47—22no—
95C2×C22⋊C4×C47—42no—
96C2×C22⋊C4×C47C2×C414yes—
97#95×C47—24no—
98#95×C47—42no—
99C24×C47—22no—
100C24×C47—22no—
101C24×C47—22no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47—22no—
105C24×C47C22×C412yes—
106C24×C47C2×D412yes—
107C24×C47C2×D412yes—
108C24×C47C2411yescharacteristic
109C22×C4×C47—22no—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47C22×C412yes—
131C22×C4×C47C22×C412yes—
132C22×C4×C47#4114yes—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—28no—
136C22×Dic47—28no—
137C22×Dic47—216no—
138C22×Dic47—216no—
139C22×Dic47—22no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47C22⋊C4116yes—
161C22×Dic47C22×C412yes—
162C22×Dic47C22×C412yes—
163C22×Dic47C22×C412yes—
164C22×Dic47C22×C412yes—
165C22×Dic47C2×D412yes—
166C22×Dic47C2×D412yes—
167C22×Dic47C2×D412yes—
168C22×Dic47C2×D412yes—
169C22×Dic47#4114yes—
170C22×Dic47#4114yes—
171C22×Dic47#4114yes—
172C22⋊C4×C47—28no—
173C22⋊C4×C47—28no—
174C22⋊C4×C47—44no—
175C22⋊Dic47—28no—
176C22⋊Dic47—28no—
177C22⋊Dic47—44no—
178Dic47⋊C4—28no—
179Dic47⋊C4—28no—
180Dic47⋊C4—48no—
181C4×Dic47—28no—
182C4×Dic47—28no—
183C4×Dic47—48no—
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—24no—
187C23×C47—24no—
188C23×C47—24no—
189C23×C47—24no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—24no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—24no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47—42no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47C2×C22⋊C414yes—
205C23×C47C23×C411yescharacteristic
206C23×C47C22×D411yescharacteristic
207C23×C47C22×D411yescharacteristic
208C23×C47C2×#4112yes—
209C23×C47C2×#4112yes—
210C23×C47C2×#4112yes—
211C23×C47C2×#4112yes—
212C23×C47H(5,𝔽2)12yes—
213C23×C472−1+412yes—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—24no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—42no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—42no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—24no—
241C2×Dic47—24no—
242C2×Dic47—232no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—42no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—42no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47C2×C22⋊C414yes—
268C2×Dic47C2×C22⋊C414yes—
269C2×Dic47#11714yes—
270C2×Dic47#11714yes—
271C2×Dic47#12318yes—
272C2×Dic47#12318yes—
273#13939—471nomaximal, Sylow
274C22×C47—24no—
275C22×C47—24no—
276C22×C47—24no—
277C22×C47—44no—
278C22×C47—44no—
279C22×C47—44no—
280C22×C47—44no—
281C22×C47—44no—
282C22×C47—44no—
283C22×C47C22×C22⋊C411yescharacteristic
284C22×C47C2×#11711yescharacteristic
285C22×C47C2×#11711yescharacteristic
286C22×C47C4×#4112yes—
287C22×C47#52012yes—
288C22×C47C2×#12312yes—
289C22×C47C2×#12312yes—
290C22×C47#52511yescharacteristic
291C22×C47#52511yescharacteristic
292C22×C47#53212yes—
293C22×C47#53511yescharacteristic
294C22×C47#53611yescharacteristic
295C22×C47#53712yes—
296C22×C47#55212yes—
297C22×C47#55212yes—
298C22×C47#55312yes—
299C22×C47#55312yes—
300C22×C47#55512yes—
301C22×C47#55512yes—
302C22×C47#55612yes—
303C22×C47#55612yes—
304C4×C47—42no—
305C4×C47—44no—
306C4×C47—42no—
307C4×C47—82no—
308Dic47—24no—
309Dic47—42no—
310Dic47—44no—
311Dic47—42no—
312Dic47—82no—
313Dic47#38118yes—
314C22×#117—471no—
315C2×#380—471no—
316C2×C4×C22⋊C4—471no—
317#1440—471no—
318#1440—471no—
319#1440—471no—
320#1440D4711yescharacteristic
321C4⋊#95—472no—
322C4⋊#95—472no—
323#1434—472no—
324#1434—472no—
325C2×C47—44no—
326C2×C47C2×#38111yescharacteristic
327C2×C47#229711yescharacteristic
328C2×C47#231111yescharacteristic
329C2×C47#234612yes—
330C2×C47#234712yes—
331C2×C47#248512yes—
332C2×C47#248512yes—
333C2×C47#249912yes—
334C2×C47#250012yes—
335C24×C4—471no—
336C2×#117—478no—
337C2×#117—944no—
338C2×#117—944no—
339C22×C4⋊C4—471no—
340C22×C4⋊C4—471no—
341C22×C4⋊C4—472no—
342C22×C22⋊C4—471no—
343C22×C22⋊C4—471no—
344C22×C22⋊C4—471no—
345C22×C22⋊C4C2×D4711yescharacteristic
346C22×C22⋊C4C2×D4711yescharacteristic
347C22×C22⋊C4C2×D4711yescharacteristic
348C22×C42—471no—
349C22×C42—471no—
350C22×C42—472no—
351#380—944no—
352C4×C22⋊C4—944no—
353C2×#95—472no—
354C2×#95—472no—
355C2×#95—472no—
356C2×#95—472no—
357C2×#95—472no—
358C2×#95—472no—
359C2×#95—472no—
360C2×#95—472no—
361C2×#95C2×D4712yes—
362C2×#95C2×D4712yes—
363C47#1393911yescharacteristic, complemented, Sylow
364C25C22×D4711yescharacteristic
365C23×C4—471no—
366C23×C4—474no—
367C23×C4—474no—
368C23×C4—474no—
369C23×C4—471no—
370C23×C4—471no—
371C23×C4—471no—
372C23×C4—472no—
373C23×C4—472no—
374C23×C4—944no—
375C23×C4—944no—
376C23×C4C22×D4712yes—
377C23×C4C22×D4711yescharacteristic
378C23×C4C22×D4711yescharacteristic
379C23×C4C22×D4712yes—
380#117—9432no—
381C2×C4⋊C4—474no—
382C2×C4⋊C4—474no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4—944no—
389C2×C4⋊C4—944no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—474no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—942no—
399C2×C22⋊C4—942no—
400C2×C22⋊C4—942no—
401C2×C22⋊C4—942no—
402C2×C22⋊C4—1882no—
403C2×C22⋊C4C4×D4714yes—
404C2×C42—474no—
405C2×C42—474no—
406C2×C42—942no—
407C2×C42—942no—
408C2×C42—942no—
409C2×C42—942no—
410C2×C42—944no—
411C2×C42—944no—
412C2×C42—944no—
413#95—24no—
414#95—42no—
415#95—944no—
416#95—944no—
417#95—944no—
418#95—944no—
419#95—944no—
420#95—1882no—
421#95—1882no—
422#95—1882no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C2×C4×D4712yes—
430C24D4×D4712yes—
431C24D4×D4712yes—
432C24C23×D4711yescharacteristic
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—472no—
455C22×C4—472no—
456C22×C4—472no—
457C22×C4—474no—
458C22×C4—474no—
459C22×C4—4716no—
460C22×C4—472no—
461C22×C4—472no—
462C22×C4—472no—
463C22×C4—472no—
464C22×C4—472no—
465C22×C4—474no—
466C22×C4—948no—
467C22×C4—948no—
468C22×C4—9416no—
469C22×C4—9416no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4C2×C4×D4712yes—
494C22×C4C2×C4×D4712yes—
495C22×C4#1062535214yes—
496C4⋊C4—948no—
497C4⋊C4—948no—
498C4⋊C4—1888no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—44no—
502C22⋊C4—948no—
503C22⋊C4—948no—
504C22⋊C4—1884no—
505C42—948no—
506C42—948no—
507C42—1888no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C22⋊C4×D4714yes—
529C23C22×C4×D4711yescharacteristic
530C23C2×D4×D4711yescharacteristic
531C23C2×D4×D4711yescharacteristic
532C23#41×D4712yes—
533C23#41×D4712yes—
534C23#41×D4712yes—
535C23C2×#1062535212yes—
536C23#1309406912yes—
537C23#1309407112yes—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—474no—
563C2×C4—474no—
564C2×C4—474no—
565C2×C4—478no—
566C2×C4—478no—
567C2×C4—474no—
568C2×C4—9432no—
569C2×C4—942no—
570C2×C4—942no—
571C2×C4—944no—
572C2×C4—944no—
573C2×C4—944no—
574C2×C4—944no—
575C2×C4—942no—
576C2×C4—942no—
577C2×C4—944no—
578C2×C4—944no—
579C2×C4—944no—
580C2×C4—1882no—
581C2×C4—1882no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1882no—
586C2×C4—1882no—
587C2×C4—1882no—
588C2×C4—1882no—
589C2×C4—1882no—
590C2×C4—1881no—
591C2×C4—1881no—
592C2×C4—1882no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1882no—
596C2×C4—1882no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22C2×C22⋊C4×D4711yescharacteristic
607C22#117×D4711yescharacteristic
608C22#117×D4711yescharacteristic
609C22C4×#1062535212yes—
610C22#1568960212yes—
611C22#123×D4712yes—
612C22#123×D4712yes—
613C22#1568962911yescharacteristic
614C22#1568963111yescharacteristic
615C22#1568966812yes—
616C22#1568968811yescharacteristic
617C22#1568969511yescharacteristic
618C22#1568970012yes—
619C22#1568980912yes—
620C22#1568981012yes—
621C22#1568981612yes—
622C22#1568981812yes—
623C22#1568983512yes—
624C22#1568983712yes—
625C22#1568984012yes—
626C22#1568984212yes—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—82no—
631C4—478no—
632C4—944no—
633C4—3762no—
634C4—1882no—
635C4—1884no—
636C4—1882no—
637C2—44no—
638C2#381×D4711yescharacteristic
639C2#2024587311yescharacteristic
640C2#2024597511yescharacteristic
641C2#2024627012yes—
642C2#2024627812yes—
643C2#2024767812yes—
644C2#2024768012yes—
645C2#2024783612yes—
646C2#2024784512yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1440C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenter#1440D47
hypocenterC47#13939
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
7. Dic47⋊#95C2semidirect
8. Dic47⋊#95C2semidirect
9. #20241614C2semidirect
10. #20241613C2semidirect
15. C2×#13093980C4semidirect
363. C47#13939semidirect

Maximal subgroups

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

subgroupindex
C22×#130939802
C2×#156889812
C2×C22⋊C4×Dic472
#202416372
#202416352
Dic47⋊#952
#202416142
#202416132
C47×#14402
#1393947