Finite Group Databasebeta

← #26138806

Subgroups — #26138806

Order 12032.

1862 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2C2×#15688990C211yesmaximal, characteristic, complemented
3C2×C4×Dic47⋊C4C211yesmaximal, characteristic, complemented
4#20241640C212yesmaximal, complemented
5#20241632C212yesmaximal
6#20241626C211yesmaximal, characteristic
7#20241624C211yesmaximal, characteristic
8#20241620C211yesmaximal, characteristic, complemented
9#20241614C212yesmaximal, complemented
10(C2×D47)⋊#95C212yesmaximal
11C47×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C2×#13093982—24no—
14C2×#13093982—24no—
15C2×#13093982C418yescomplemented
16C22×C4⋊Dic47C2211yescharacteristic
17C22×Dic47⋊C4C2212yes—
18C22×(C2×D47)⋊C4C2212yes—
19C22×(C2×D47)⋊C4C2212yes—
20C22×(C2×D47)⋊C4C2211yescharacteristic
21C22×(C2×D47)⋊C4C2211yescharacteristic
22C22×C4×Dic47C2211yescharacteristic
23C22×C4×Dic47C2211yescharacteristic
24C22×C4×Dic47C2211yescharacteristic
25#15688990—24no—
26C4×Dic47⋊C4—24no—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2211yescharacteristic
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C22×C4⋊C4×C47C2211yescharacteristic
36C22×C42×C47C2211yescharacteristic
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2212yes—
39C2×#95×C47C2212yes—
40C24×D47C2311yescharacteristic
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2312yes—
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2311yescharacteristic
46C22×C4×D47—24no—
47C22×C4×D47—24no—
48C22×C4×D47—22no—
49C22×C4×D47—22no—
50C22×C4×D47C2×C414yes—
51#13093982—232no—
52C2×C4⋊Dic47—22no—
53C2×C4⋊Dic47—22no—
54C2×C4⋊Dic47C2×C414yes—
55C2×Dic47⋊C4—24no—
56C2×Dic47⋊C4—24no—
57C2×Dic47⋊C4—24no—
58C2×(C2×D47)⋊C4—22no—
59C2×(C2×D47)⋊C4—22no—
60C2×(C2×D47)⋊C4—22no—
61C2×(C2×D47)⋊C4—22no—
62C2×(C2×D47)⋊C4—24no—
63C2×(C2×D47)⋊C4—24no—
64C2×(C2×D47)⋊C4—24no—
65C2×(C2×D47)⋊C4—24no—
66C2×(C2×D47)⋊C4—42no—
67C2×(C2×D47)⋊C4—42no—
68C2×(C2×D47)⋊C4C2×C414yes—
69C2×(C2×D47)⋊C4C2×C414yes—
70C2×C4×Dic47—22no—
71C2×C4×Dic47—22no—
72C2×C4×Dic47—22no—
73C2×C4×Dic47—42no—
74C2×C4×Dic47C2×C414yes—
75C2×C4×Dic47C2×C414yes—
76C2×C4×Dic47D414yes—
77C2×C4×Dic47D414yes—
78C47⋊#95—22no—
79C47⋊#95—24no—
80C47⋊#95—24no—
81C47⋊#95—24no—
82C47⋊#95—24no—
83C47⋊#95—24no—
84C47⋊#95—24no—
85C47⋊#95—42no—
86C23×C4×C47C2312yes—
87C23×C4×C47C2311yescharacteristic
88C23×C4×C47C2312yes—
89C23×C4×C47C2311yescharacteristic
90C23×C4×C47C2311yescharacteristic
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47—22no—
93C2×C4⋊C4×C47C2×C414yes—
94C2×C42×C47—22no—
95C2×C42×C47—42no—
96#95×C47—22no—
97#95×C47—24no—
98#95×C47—42no—
99C24×C47C2411yescharacteristic
100C23×D47—21no—
101C23×D47—21no—
102C23×D47—22no—
103C23×D47—22no—
104C23×D47—22no—
105C23×D47—22no—
106C23×D47—22no—
107C23×D47C22×C412yes—
108C23×D47#4114yes—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
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×C47—22no—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C22×C412yes—
134C22×C4×C47C2×D412yes—
135C22×C4×C47C2×D412yes—
136C22×C4×C47#4114yes—
137C22×C4×C47#4114yes—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—21no—
144C22×Dic47—21no—
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×Dic47C22×C412yes—
160C22×Dic47C22×C412yes—
161C22×Dic47C22×C412yes—
162C22×Dic47C2×D412yes—
163C22×Dic47C2×D412yes—
164C22×Dic47C2×D412yes—
165C22×Dic47C2×D412yes—
166C22×Dic47#4114yes—
167C2×C4×D47—216no—
168C2×C4×D47—216no—
169C2×C4×D47—28no—
170C2×C4×D47—28no—
171C2×C4×D47—28no—
172C4⋊C4×C47—28no—
173C42×C47—44no—
174(C2×D47)⋊C4—28no—
175(C2×D47)⋊C4—28no—
176(C2×D47)⋊C4—48no—
177(C2×D47)⋊C4—48no—
178C4⋊Dic47—28no—
179Dic47⋊C4—216no—
180C4×Dic47—28no—
181C4×Dic47—44no—
182C4×Dic47C22⋊C4116yescomplemented
183C23×C47C23×C411yescharacteristic
184C23×C47C22×D411yescharacteristic
185C23×C47C22×D411yescharacteristic
186C23×C47C2×#4112yes—
187C23×C47C2×#4112yes—
188C23×C47C2×#4112yes—
189C23×C47C2×#4112yes—
190C23×C47H(5,𝔽2)12yes—
191C23×C472−1+412yes—
192C22×D47—22no—
193C22×D47—22no—
194C22×D47—24no—
195C22×D47—24no—
196C22×D47—24no—
197C22×D47—24no—
198C22×D47—24no—
199C22×D47—24no—
200C22×D47—24no—
201C22×D47—24no—
202C22×D47—42no—
203C22×D47—41no—
204C22×D47—41no—
205C22×D47—41no—
206C22×D47—41no—
207C22×D47—42no—
208C22×D47—42no—
209C22×D47—42no—
210C22×D47—42no—
211C22×D47—42no—
212C22×D47#11714yes—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—24no—
216C2×C4×C47—24no—
217C2×C4×C47—24no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—24no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—42no—
227C2×C4×C47—42no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
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×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47C2×C22⋊C414yes—
241C2×C4×C47#11714yes—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47—41no—
258C2×Dic47—41no—
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×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47C2×C22⋊C414yes—
270C2×Dic47C2×C22⋊C414yes—
271C4×D47—232no—
272#13938—471nomaximal, Sylow
273C22×C47C22×C22⋊C411yescharacteristic
274C22×C47C2×#11711yescharacteristic
275C22×C47C2×#11711yescharacteristic
276C22×C47C4×#4112yes—
277C22×C47#52012yes—
278C22×C47#52512yes—
279C22×C47#52512yes—
280C22×C47#52512yes—
281C22×C47#53211yescharacteristic
282C22×C47#53211yescharacteristic
283C22×C47#53511yescharacteristic
284C22×C47#53611yescharacteristic
285C22×C47#53712yes—
286C22×C47#55112yes—
287C22×C47#55112yes—
288C22×C47#55212yes—
289C22×C47#55212yes—
290C22×C47#55312yes—
291C22×C47#55312yes—
292C22×C47#55612yes—
293C22×C47#55612yes—
294C2×D47—28no—
295C2×D47—24no—
296C2×D47—42no—
297C2×D47—44no—
298C2×D47—44no—
299C2×D47—44no—
300C2×D47—44no—
301C2×D47—44no—
302C2×D47—42no—
303C4×C47—24no—
304C4×C47—24no—
305C4×C47—42no—
306C4×C47—44no—
307C4×C47—82no—
308Dic47—24no—
309Dic47—24no—
310Dic47—44no—
311Dic47—42no—
312Dic47—82no—
313C22×#117—471no—
314C2×#383—471no—
315C2×C4×C4⋊C4—471no—
316#1441—472no—
317#1439—472no—
318#1437—471no—
319#1437—471no—
320#1436—471no—
321#1436D4711yescharacteristic, complemented
322#1434—472no—
323C22⋊#95—472no—
324C2×C47#228011yescharacteristic
325C2×C47#229711yescharacteristic
326C2×C47#231111yescharacteristic
327C2×C47#234312yes—
328C2×C47#234712yes—
329C2×C47#259812yes—
330C2×C47#259912yes—
331C2×C47#260112yes—
332C2×C47#260112yes—
333D47—44no—
334C24×C4—471no—
335C2×#117—478no—
336C2×#117—944no—
337C2×#117—944no—
338C22×C4⋊C4—471no—
339C22×C4⋊C4—472no—
340C22×C4⋊C4C2×D4711yescharacteristic
341C22×C22⋊C4—471no—
342C22×C22⋊C4—471no—
343C22×C22⋊C4—472no—
344C22×C22⋊C4—472no—
345C22×C42—471no—
346C22×C42—471no—
347C22×C42—471no—
348C22×C42C2×D4711yescharacteristic
349#383—944no—
350C4×C4⋊C4—944no—
351C2×#95—472no—
352C2×#95—472no—
353C2×#95—472no—
354C2×#95—472no—
355C2×#95—472no—
356C2×#95—471no—
357C2×#95—472no—
358C2×#95—472no—
359C2×#95C2×D4712yes—
360C2×#95C2×D4711yescharacteristic
361C2×#95C2×D4712yes—
362C47#1393811yescharacteristic, complemented, Sylow
363C25—471no—
364C23×C4—471no—
365C23×C4—471no—
366C23×C4—471no—
367C23×C4—472no—
368C23×C4—474no—
369C23×C4—472no—
370C23×C4—942no—
371C23×C4—942no—
372C23×C4—944no—
373C23×C4—944no—
374C23×C4C22×D4712yes—
375C23×C4C22×D4711yescharacteristic
376C23×C4C22×D4712yes—
377C23×C4C22×D4711yescharacteristic
378C23×C4C22×D4711yescharacteristic
379#117—9432no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—474no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—944no—
386C2×C4⋊C4—944no—
387C2×C4⋊C4—944no—
388C2×C4⋊C4C4×D4714yes—
389C2×C22⋊C4—474no—
390C2×C22⋊C4—474no—
391C2×C22⋊C4—942no—
392C2×C22⋊C4—942no—
393C2×C22⋊C4—942no—
394C2×C22⋊C4—942no—
395C2×C22⋊C4—944no—
396C2×C22⋊C4—944no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—944no—
399C2×C22⋊C4—1882no—
400C2×C22⋊C4—1882no—
401C2×C42—22no—
402C2×C42—42no—
403C2×C42—474no—
404C2×C42—474no—
405C2×C42—474no—
406C2×C42—474no—
407C2×C42—942no—
408C2×C42—942no—
409C2×C42—942no—
410C2×C42—1882no—
411#95—24no—
412#95—22no—
413#95—42no—
414#95—944no—
415#95—944no—
416#95—944no—
417#95—944no—
418#95—944no—
419#95—942no—
420#95—944no—
421#95—1882no—
422C24—472no—
423C24—474no—
424C24—941no—
425C24—941no—
426C24—942no—
427C24—942no—
428C24—942no—
429C24—942no—
430C24—942no—
431C24C23×D4711yescharacteristic
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
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—472no—
458C22×C4—472no—
459C22×C4—472no—
460C22×C4—472no—
461C22×C4—474no—
462C22×C4—948no—
463C22×C4—948no—
464C22×C4—9416no—
465C22×C4—9416no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—948no—
471C22×C4—942no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
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×C4C2×C4×D4712yes—
489C22×C4C2×C4×D4712yes—
490C22×C4C2×C4×D4712yes—
491C22×C4#1062535014yes—
492C22×C4D4×D4712yes—
493C22×C4D4×D4712yes—
494C22×C4#1062535414yes—
495C4⋊C4—28no—
496C4⋊C4—948no—
497C4⋊C4—9416no—
498C22⋊C4—948no—
499C22⋊C4—948no—
500C22⋊C4—1888no—
501C22⋊C4—1888no—
502C42—44no—
503C42—4716no—
504C42—948no—
505C42—1884no—
506C23—474no—
507C23—944no—
508C23—944no—
509C23—944no—
510C23—944no—
511C23—942no—
512C23—942no—
513C23—944no—
514C23—944no—
515C23—944no—
516C23—944no—
517C23—1882no—
518C23—1882no—
519C23—1882no—
520C23—1882no—
521C23—1882no—
522C23—1881no—
523C23—1881no—
524C23—1881no—
525C23—1881no—
526C23—1882no—
527C23C22×C4×D4711yescharacteristic
528C23C2×D4×D4711yescharacteristic
529C23C2×D4×D4711yescharacteristic
530C23#41×D4712yes—
531C23#41×D4712yes—
532C23C2×#1062535012yes—
533C23C2×#1062535412yes—
534C23#1309406912yes—
535C23#1309407112yes—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—474no—
564C2×C4—474no—
565C2×C4—9432no—
566C2×C4—942no—
567C2×C4—942no—
568C2×C4—944no—
569C2×C4—944no—
570C2×C4—944no—
571C2×C4—944no—
572C2×C4—942no—
573C2×C4—942no—
574C2×C4—944no—
575C2×C4—944no—
576C2×C4—942no—
577C2×C4—944no—
578C2×C4—944no—
579C2×C4—1882no—
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×C4C22⋊C4×D4714yes—
594C2×C4#1309398114yes—
595C22—944no—
596C22—948no—
597C22—1884no—
598C22—1882no—
599C22—1884no—
600C22—1884no—
601C22—1884no—
602C22—1884no—
603C22—1882no—
604C22C2×C22⋊C4×D4711yescharacteristic
605C22#117×D4711yescharacteristic
606C22C2×#1309398111yescharacteristic
607C22C4×#1062535412yes—
608C22#1568960212yes—
609C22#1568963212yes—
610C22#1568963212yes—
611C22#1568963612yes—
612C22#1568967211yescharacteristic
613C22#1568967411yescharacteristic
614C22#1568968811yescharacteristic
615C22#1568969511yescharacteristic
616C22#1568970012yes—
617C22#1568980012yes—
618C22#1568980112yes—
619C22#1568980912yes—
620C22#1568981112yes—
621C22#1568982112yes—
622C22#1568982312yes—
623C22#1568984112yes—
624C22#1568984612yes—
625C4—24no—
626C4—24no—
627C4—44no—
628C4—42no—
629C4—82no—
630C4—944no—
631C4—944no—
632C4—3762no—
633C4—1882no—
634C4—1884no—
635C2—1884no—
636C2#2024576911yescharacteristic
637C2#2024587711yescharacteristic
638C2#2024597511yescharacteristic
639C2#2024624712yes—
640C2#2024627712yes—
641C2#2024895312yes—
642C2#2024896412yes—
643C2#2024898412yes—
644C2#2024899212yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1436C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenter#1436D47
hypocenterC47#13938
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
2. C2×#15688990C2semidirect
3. C2×C4×Dic47⋊C4C2semidirect
4. #20241640C2semidirect
8. #20241620C2semidirect
9. #20241614C2semidirect
11. C47×#1436C2semidirect
15. C2×#13093982C4semidirect
182. C4×Dic47C22⋊C4semidirect
321. #1436D47semidirect
362. C47#13938semidirect

Maximal subgroups

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

subgroupindex
C22×#130939822
C2×#156889902
C2×C4×Dic47⋊C42
#202416402
#202416322
#202416262
#202416242
#202416202
#202416142
(C2×D47)⋊#952
C47×#14362
#1393847