Finite Group Databasebeta

← #26142952

Subgroups — #26142952

Order 12032.

1610 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094017C211yesmaximal, characteristic
2C2×#15688987C211yesmaximal, characteristic, complemented
3C2×C4×Dic47⋊C4C211yesmaximal, characteristic, complemented
4#20241640C212yesmaximal, complemented
5#20241639C212yesmaximal, complemented
6#20241631C212yesmaximal
7#20241631C212yesmaximal
8#20241625C211yesmaximal, characteristic
9#20241623C211yesmaximal, characteristic, complemented
10#20241621C211yesmaximal, characteristic, complemented
11C47×#1437C211yesmaximal, characteristic
12C2×#13094017—24no—
13C2×#13094017—24no—
14C2×#13094017C418yescomplemented
15C22×C4⋊Dic47C2211yescharacteristic
16C22×Dic47⋊C4C2212yes—
17C22×Dic47⋊C4C2211yescharacteristic
18C22×Dic47⋊C4C2211yescharacteristic
19C22×Dic47⋊C4C2212yes—
20C22×C22⋊Dic47C2211yescharacteristic
21C22×C22⋊Dic47C2211yescharacteristic
22C22×C22⋊Dic47C2212yes—
23C22×C4×Dic47C2211yescharacteristic
24C22×C4×Dic47C2211yescharacteristic
25#15688987—24no—
26C4×Dic47⋊C4—24no—
27C2×C47⋊#95C2211yescharacteristic
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×C47C2212yes—
37C22×C42×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C2×#95×C47C2212yes—
40C23×Dic47C2312yes—
41C23×Dic47C2311yescharacteristic
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2312yes—
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2311yescharacteristic
46#13094017—232no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47C2×C414yes—
50C2×Dic47⋊C4—22no—
51C2×Dic47⋊C4—22no—
52C2×Dic47⋊C4—24no—
53C2×Dic47⋊C4—24no—
54C2×Dic47⋊C4—22no—
55C2×Dic47⋊C4—22no—
56C2×Dic47⋊C4—24no—
57C2×Dic47⋊C4—24no—
58C2×Dic47⋊C4—42no—
59C2×Dic47⋊C4—42no—
60C2×Dic47⋊C4C2×C414yes—
61C2×Dic47⋊C4C2×C414yes—
62C2×C22⋊Dic47—22no—
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47—22no—
65C2×C22⋊Dic47—22no—
66C2×C22⋊Dic47—24no—
67C2×C22⋊Dic47—42no—
68C2×C22⋊Dic47—42no—
69C2×C22⋊Dic47C2×C414yes—
70C2×C22⋊Dic47C2×C414yes—
71C2×C4×Dic47—22no—
72C2×C4×Dic47—42no—
73C2×C4×Dic47C2×C414yes—
74C2×C4×Dic47D414yes—
75C2×C4×Dic47D414yes—
76C47⋊#95—22no—
77C47⋊#95—24no—
78C47⋊#95—24no—
79C47⋊#95—24no—
80C47⋊#95—24no—
81C47⋊#95—42no—
82C47⋊#95—42no—
83C47⋊#95—42no—
84C25×C47C2311yescharacteristic
85C23×C4×C47C2312yes—
86C23×C4×C47C2311yescharacteristic
87C23×C4×C47C2312yes—
88C23×C4×C47C2311yescharacteristic
89C2×C22⋊C4×C47—22no—
90C2×C22⋊C4×C47—22no—
91C2×C22⋊C4×C47—24no—
92C2×C22⋊C4×C47—42no—
93C2×C22⋊C4×C47—42no—
94C2×C22⋊C4×C47C2×C414yes—
95C2×C42×C47—22no—
96C2×C42×C47—42no—
97#95×C47—22no—
98#95×C47—42no—
99C24×C47—21no—
100C24×C47—21no—
101C24×C47—22no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47—22no—
105C24×C47—22no—
106C24×C47—22no—
107C24×C47C22×C412yes—
108C24×C47C2411yescharacteristic
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
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×C47C22×C412yes—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47#4114yes—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—22no—
143C22×Dic47—22no—
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×Dic47C22×C412yes—
159C22×Dic47C22×C412yes—
160C22×Dic47C22×C412yes—
161C22×Dic47C22×C412yes—
162C22×Dic47C2×D412yes—
163C22×Dic47C2×D412yes—
164C22×Dic47C2×D412yes—
165C22×Dic47C2×D412yes—
166C22×Dic47#4114yes—
167C22⋊C4×C47—28no—
168C22⋊C4×C47—48no—
169C42×C47—44no—
170C22⋊Dic47—28no—
171C22⋊Dic47—28no—
172C22⋊Dic47—48no—
173C4⋊Dic47—28no—
174Dic47⋊C4—216no—
175Dic47⋊C4—28no—
176Dic47⋊C4—28no—
177Dic47⋊C4—48no—
178C4×Dic47—44no—
179C4×Dic47C22⋊C4116yescomplemented
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—24no—
184C23×C47—24no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—41no—
188C23×C47—41no—
189C23×C47—41no—
190C23×C47—41no—
191C23×C47—42no—
192C23×C47—42no—
193C23×C47—42no—
194C23×C47—42no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47C23×C411yescharacteristic
202C23×C47C22×D411yescharacteristic
203C23×C47C22×D411yescharacteristic
204C23×C47C2×#4112yes—
205C23×C47C2×#4112yes—
206C23×C47H(5,𝔽2)12yes—
207C23×C47H(5,𝔽2)12yes—
208C23×C472−1+412yes—
209C23×C472−1+412yes—
210C2×C4×C47—24no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—24no—
214C2×C4×C47—22no—
215C2×C4×C47—24no—
216C2×C4×C47—24no—
217C2×C4×C47—24no—
218C2×C4×C47—24no—
219C2×C4×C47—42no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—42no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
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×C47C2×C22⋊C414yes—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—24no—
237C2×Dic47—24no—
238C2×Dic47—24no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—24no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—24no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—42no—
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×Dic47C2×C22⋊C414yes—
267C2×Dic47C2×C22⋊C414yes—
268#14291—471nomaximal, Sylow
269C22×C47—42no—
270C22×C47—42no—
271C22×C47—44no—
272C22×C47—44no—
273C22×C47—42no—
274C22×C47—82no—
275C22×C47—82no—
276C22×C47—82no—
277C22×C47—82no—
278C22×C47C22×C22⋊C411yescharacteristic
279C22×C47C4×#4111yescharacteristic
280C22×C47C4×#4111yescharacteristic
281C22×C47#52012yes—
282C22×C47#52012yes—
283C22×C47#52512yes—
284C22×C47#53212yes—
285C22×C47#53511yescharacteristic
286C22×C47#53511yescharacteristic
287C22×C47#53611yescharacteristic
288C22×C47#53611yescharacteristic
289C22×C47#53712yes—
290C22×C47#53712yes—
291C22×C47#53812yes—
292C22×C47#53812yes—
293C22×C47#53912yes—
294C22×C47#53912yes—
295C22×C47#54112yes—
296C22×C47#54112yes—
297C22×C47#54212yes—
298C22×C47#54212yes—
299C4×C47—24no—
300C4×C47—42no—
301C4×C47—44no—
302C4×C47—82no—
303Dic47—24no—
304Dic47—24no—
305Dic47—42no—
306Dic47—44no—
307Dic47—42no—
308Dic47—82no—
309C22×#126—471no—
310C2×#382—471no—
311C2×C4×C4⋊C4—471no—
312#1441—472no—
313#1441—472no—
314#1439—472no—
315#1439—472no—
316#1437—471no—
317#1437D4711yescharacteristic
318#1436—471no—
319#1436—471no—
320C2×C47—82no—
321C2×C47#229711yescharacteristic
322C2×C47#231111yescharacteristic
323C2×C47#231111yescharacteristic
324C2×C47#233412yes—
325C2×C47#233412yes—
326C2×C47#263612yes—
327C2×C47#263712yes—
328C2×C47#266212yes—
329C2×C47#266412yes—
330C2×#126—478no—
331C2×#126—944no—
332C2×#126—944no—
333C22×C4⋊C4—471no—
334C22×C4⋊C4—471no—
335C22×C4⋊C4—471no—
336C22×C4⋊C4—472no—
337C22×C4⋊C4—472no—
338C22×C22⋊C4—471no—
339C22×C22⋊C4—471no—
340C22×C22⋊C4—472no—
341C22×C22⋊C4C2×D4711yescharacteristic
342C22×C22⋊C4C2×D4712yes—
343C22×C42—471no—
344C22×C42—471no—
345C22×C42C2×D4711yescharacteristic
346#382—944no—
347C4×C4⋊C4—944no—
348C2×#95—471no—
349C2×#95—472no—
350C2×#95—472no—
351C2×#95—472no—
352C2×#95—472no—
353C2×#95—472no—
354C2×#95—472no—
355C2×#95—472no—
356C2×#95C2×D4711yescharacteristic
357C2×#95C2×D4712yes—
358C47#1429111yescharacteristic, complemented, Sylow
359C25C22×D4711yescharacteristic
360C23×C4—471no—
361C23×C4—471no—
362C23×C4—471no—
363C23×C4—471no—
364C23×C4—472no—
365C23×C4—472no—
366C23×C4C22×D4712yes—
367C23×C4C22×D4711yescharacteristic
368C23×C4C22×D4712yes—
369C23×C4C22×D4711yescharacteristic
370#126—9432no—
371C2×C4⋊C4—474no—
372C2×C4⋊C4—474no—
373C2×C4⋊C4—474no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4—944no—
379C2×C4⋊C4—944no—
380C2×C4⋊C4—944no—
381C2×C4⋊C4—942no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—944no—
384C2×C4⋊C4—1882no—
385C2×C4⋊C4—1882no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—474no—
392C2×C22⋊C4—474no—
393C2×C22⋊C4—942no—
394C2×C22⋊C4—942no—
395C2×C22⋊C4—942no—
396C2×C22⋊C4—942no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—1882no—
399C2×C22⋊C4—1882no—
400C2×C22⋊C4C4×D4714yes—
401C2×C42—22no—
402C2×C42—42no—
403C2×C42—474no—
404C2×C42—474no—
405C2×C42—474no—
406C2×C42—942no—
407C2×C42—1882no—
408#95—22no—
409#95—42no—
410#95—944no—
411#95—944no—
412#95—944no—
413#95—944no—
414#95—942no—
415#95—1882no—
416#95—1882no—
417#95—1882no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24C2×C4×D4712yes—
427C24C23×D4711yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
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—472no—
448C22×C4—472no—
449C22×C4—472no—
450C22×C4—472no—
451C22×C4—472no—
452C22×C4—472no—
453C22×C4—472no—
454C22×C4—472no—
455C22×C4—474no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—941no—
464C22×C4—941no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4C2×C4×D4712yes—
482C22×C4C2×C4×D4712yes—
483C22×C4#1062535014yes—
484C22×C4D4×D4712yes—
485C22×C4D4×D4712yes—
486C4⋊C4—948no—
487C4⋊C4—948no—
488C4⋊C4—9416no—
489C4⋊C4—948no—
490C4⋊C4—1888no—
491C22⋊C4—28no—
492C22⋊C4—48no—
493C22⋊C4—948no—
494C22⋊C4—948no—
495C22⋊C4—1888no—
496C42—44no—
497C42—4716no—
498C42—1884no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23C22×C4×D4711yescharacteristic
521C23C2×D4×D4711yescharacteristic
522C23C2×D4×D4711yescharacteristic
523C23#41×D4712yes—
524C23C2×#1062535012yes—
525C23#1309406912yes—
526C23#1309406912yes—
527C23#1309407112yes—
528C23#1309407112yes—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—474no—
553C2×C4—474no—
554C2×C4—942no—
555C2×C4—942no—
556C2×C4—944no—
557C2×C4—944no—
558C2×C4—944no—
559C2×C4—942no—
560C2×C4—942no—
561C2×C4—944no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—944no—
567C2×C4—944no—
568C2×C4—944no—
569C2×C4—944no—
570C2×C4—1882no—
571C2×C4—1882no—
572C2×C4—1882no—
573C2×C4—1882no—
574C2×C4—1882no—
575C2×C4—1882no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1882no—
586C2×C4C22⋊C4×D4714yes—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C2×C22⋊C4×D4711yescharacteristic
597C22#1568959111yescharacteristic
598C22C4×#1062535011yescharacteristic
599C22#1568960212yes—
600C22#1568960212yes—
601C22#1568963212yes—
602C22#1568967012yes—
603C22#1568968611yescharacteristic
604C22#1568968811yescharacteristic
605C22#1568969311yescharacteristic
606C22#1568969511yescharacteristic
607C22#1568970012yes—
608C22#1568970212yes—
609C22#1568970512yes—
610C22#1568971112yes—
611C22#1568971512yes—
612C22#1568971612yes—
613C22#1568973012yes—
614C22#1568973512yes—
615C22#1568973712yes—
616C22#1568973812yes—
617C4—24no—
618C4—44no—
619C4—42no—
620C4—82no—
621C4—944no—
622C4—944no—
623C4—3762no—
624C4—1882no—
625C4—1884no—
626C4—1882no—
627C2—82no—
628C2#2024587411yescharacteristic
629C2#2024597511yescharacteristic
630C2#2024597811yescharacteristic
631C2#2024616112yes—
632C2#2024616512yes—
633C2#2024928912yes—
634C2#2024930012yes—
635C2#2024952912yes—
636C2#2024955312yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1437C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenter#1437D47
hypocenterC47#14291
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#15688987C2semidirect
3. C2×C4×Dic47⋊C4C2semidirect
4. #20241640C2semidirect
5. #20241639C2semidirect
9. #20241623C2semidirect
10. #20241621C2semidirect
14. C2×#13094017C4semidirect
179. C4×Dic47C22⋊C4semidirect
358. C47#14291semidirect

Maximal subgroups

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

subgroupindex
C22×#130940172
C2×#156889872
C2×C4×Dic47⋊C42
#202416402
#202416392
#202416312
#202416252
#202416232
#202416212
C47×#14372
#1429147