Finite Group Databasebeta

← #26257558

Subgroups — #26257558

Order 12032.

784 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689427C211yesmaximal, characteristic, complemented
2C2×#15689180C211yesmaximal, characteristic
3C4×#13094003C211yesmaximal, characteristic
4#20244824C211yesmaximal, characteristic
5#20244819C211yesmaximal, characteristic
6#20244591C211yesmaximal, characteristic
7#20244584C211yesmaximal, characteristic
8#20244346C211yesmaximal, characteristic, complemented
9#20244154C211yesmaximal, characteristic, complemented
10#20244128C211yesmaximal, characteristic, complemented
11#20244123C211yesmaximal, characteristic, complemented
12#20244077C211yesmaximal, characteristic, complemented
13#20244077C211yesmaximal, characteristic, complemented
14#20243400C211yesmaximal, characteristic, complemented
15C47×#2851C211yesmaximal, characteristic
16C2×#13094003C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18#15689427—24no—
19#15689180—24no—
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25C2×C4⋊(C47⋊C8)C2211yescharacteristic
26C2×#13093947C2211yescharacteristic
27C2×#13093947C2211yescharacteristic
28C2×#13093947C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×C22⋊(C47⋊C8)C2211yescharacteristic
31C2×C22⋊(C47⋊C8)C2211yescharacteristic
32C2×#13093935C2211yescharacteristic
33C2×#13093935C2211yescharacteristic
34#15689029C2211yescharacteristic
35#15689021C2211yescharacteristic
36#15689007—21no—
37#15688996—21no—
38#15688983C2211yescharacteristic
39C4×Dic47⋊C4—21no—
40C4×C4⋊Dic47C2211yescharacteristic
41C4×C22⋊Dic47—21no—
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50C2×#125×C47C2211yescharacteristic
51C22×C42×C47C2211yescharacteristic
52C47×#389C2211yescharacteristic
53C47×#389C2211yescharacteristic
54C47×#387—21no—
55C47×#387—21no—
56C47×#385—21no—
57C47×#385—21no—
58C47×#382C2211yescharacteristic
59C47×#380C2211yescharacteristic
60C47×#380C2211yescharacteristic
61C22×Dic94C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66#13094003—22no—
67#13094003—22no—
68C4×Dic94—22no—
69C4×Dic94—22no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80#13093957—22no—
81#13093957—22no—
82#13093955—22no—
83#13093955—22no—
84C4⋊(C47⋊C8)—22no—
85C4⋊(C47⋊C8)D414yes—
86#13093947—22no—
87#13093947—22no—
88#13093947—22no—
89#13093947—22no—
90C22⋊(C47⋊C8)—22no—
91C22⋊(C47⋊C8)—22no—
92#13093935—22no—
93#13093935—22no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C23×C4×C47D411yescharacteristic
100C23×C4×C47D411yescharacteristic
101C23×C4×C47C2311yescharacteristic
102#125×C47—22no—
103#125×C47—22no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C22⋊C4×C47—21no—
113C2×C22⋊C4×C47—21no—
114C2×C42×C47—22no—
115C2×C42×C47—22no—
116C2×C42×C47D411yescharacteristic
117C2×C42×C47D411yescharacteristic
118C2×C42×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124#95×C47—21no—
125#95×C47—21no—
126C24×C47C2×D411yescharacteristic
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—41no—
141C22×C4×C47—41no—
142C22×C4×C47—41no—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94#4112yes—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—42no—
168C4⋊C4×C47—42no—
169C4⋊C4×C47—42no—
170C4⋊C4×C47—42no—
171C22⋊C4×C47—42no—
172C22⋊C4×C47—42no—
173C42×C47—22no—
174C42×C47—22no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47C2×D412yes—
178C42×C47#4112yes—
179C22⋊Dic47—24no—
180C22⋊Dic47—42no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47#4112yes—
187Dic47⋊C4—24no—
188Dic47⋊C4—42no—
189Dic47⋊C4—42no—
190C4×Dic47—42no—
191C4×Dic47—42no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8C2×D412yes—
203C2×C47⋊C8C2×D412yes—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47#13712yes—
211C23×C47C22×D411yescharacteristic
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—44no—
226C2×C4×C47—44no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47#12112yes—
244C2×C4×C47#12112yes—
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C2×#4111yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×C4×C472−1+411yescharacteristic
250C2×C4×C472−1+411yescharacteristic
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263Dic94—22no—
264Dic94—22no—
265Dic94—41no—
266Dic94—41no—
267C47⋊C8—22no—
268C47⋊C8—22no—
269C47⋊C8—41no—
270C47⋊C8—41no—
271#24707—471nomaximal, Sylow
272C22×C47—22no—
273C22×C47—22no—
274C22×C47—22no—
275C22×C47—41no—
276C22×C47—41no—
277C22×C47—41no—
278C22×C47—41no—
279C22×C47C2×#12111yescharacteristic
280C22×C47#52511yescharacteristic
281C22×C47#53611yescharacteristic
282C22×C47C2×#13711yescharacteristic
283C22×C47#57511yescharacteristic
284C22×C47#57511yescharacteristic
285C22×C47#57511yescharacteristic
286C4×C47—22no—
287C4×C47—22no—
288C4×C47—41no—
289C4×C47—41no—
290C4×C47—41no—
291C4×C47—81no—
292C4×C47—81no—
293C4×C47—81no—
294C4×C47—81no—
295C4×C47#54711yescharacteristic
296C4×C47#54811yescharacteristic
297C4×C47#55211yescharacteristic
298C4×C47#55611yescharacteristic
299Dic47—42no—
300Dic47—42no—
301Dic47—81no—
302Dic47—81no—
303C2×#479—471no—
304C2×#432—471no—
305#2851D4711yescharacteristic
306C4×C22⋊Q8—471no—
307#2077—471no—
308#2076—471no—
309#2036—471no—
310#2035—471no—
311#1989—471no—
312#1949—471no—
313#1944—471no—
314#1943—471no—
315#1935—471no—
316#1935—471no—
317#1805—471no—
318C2×C47—42no—
319C2×C47#241511yescharacteristic
320C2×C47#305411yescharacteristic
321C2×C47#305611yescharacteristic
322C2×C47#311111yescharacteristic
323C2×C47#311711yescharacteristic
324C2×C47#323611yescharacteristic
325C2×C47#324011yescharacteristic
326C2×#125C2×D4711yescharacteristic
327C2×C22⋊Q8—471no—
328C2×C4×Q8—471no—
329C22×C42C2×D4711yescharacteristic
330#479—944no—
331#432—944no—
332C2×#107—471no—
333C2×#107—471no—
334C2×#106—471no—
335C2×#106—471no—
336C2×C4⋊C8—471no—
337C2×C4⋊C8—471no—
338C2×#103—471no—
339C2×#103—471no—
340C2×#103—471no—
341C2×#103—471no—
342C2×C22⋊C8—471no—
343C2×C22⋊C8—471no—
344C2×#97—471no—
345C2×#97—471no—
346#391—471no—
347#389—471no—
348#389C2×D4711yescharacteristic
349#389C2×D4711yescharacteristic
350#387—21no—
351#387—21no—
352#387—941no—
353#385—21no—
354#385—21no—
355#385—941no—
356#382C2×D4711yescharacteristic
357#380—471no—
358#380C2×D4711yescharacteristic
359#380C2×D4711yescharacteristic
360C4×C4⋊C4—471no—
361C4×C4⋊C4—941no—
362C4×C22⋊C4—941no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#340—471no—
370#340—471no—
371C47#2470711yescharacteristic, complemented, Sylow
372C22×Q8—471no—
373C23×C4C22⋊D4711yescharacteristic
374C23×C4C22⋊D4711yescharacteristic
375C23×C4C22×D4711yescharacteristic
376C22×C8—471no—
377C22×C8—471no—
378C22×C8—471no—
379C22×C8—471no—
380#125—22no—
381#125—22no—
382C22⋊Q8—942no—
383C22⋊Q8—942no—
384C4×Q8—942no—
385C4×Q8—942no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—471no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C4⋊C4C22×D4711yescharacteristic
399C2×C4⋊C4C22×D4711yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—941no—
403C2×C22⋊C4—941no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—941no—
407C2×C42—941no—
408C2×C42C22⋊D4711yescharacteristic
409C2×C42C22⋊D4711yescharacteristic
410C2×C42C22×D4711yescharacteristic
411C2×C42C22×D4711yescharacteristic
412#107—942no—
413#107—942no—
414#106—942no—
415#106—942no—
416C4⋊C8—474no—
417C4⋊C8—942no—
418#103—942no—
419#103—942no—
420#103—942no—
421#103—942no—
422C22⋊C8—942no—
423C22⋊C8—942no—
424#97—942no—
425#97—942no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—941no—
433#95—941no—
434#95—941no—
435#95—941no—
436#95—941no—
437C24C2×C22⋊D4711yescharacteristic
438C2×Q8—472no—
439C2×Q8—941no—
440C2×Q8—941no—
441C2×Q8—941no—
442C2×Q8—941no—
443C2×Q8—941no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—941no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—941no—
464C22×C4C2×C22⋊D4711yescharacteristic
465C22×C4C2×C22⋊D4711yescharacteristic
466C22×C4C2×C22⋊D4711yescharacteristic
467C22×C4C2×C22⋊D4711yescharacteristic
468C22×C4C2×C22⋊D4711yescharacteristic
469C22×C4C23×D4711yescharacteristic
470C2×C8—472no—
471C2×C8—472no—
472C2×C8—941no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—472no—
495C4⋊C4—944no—
496C4⋊C4—941no—
497C4⋊C4—941no—
498C4⋊C4—941no—
499C4⋊C4—941no—
500C4⋊C4—941no—
501C4⋊C4—1882no—
502C4⋊C4—1882no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—944no—
506C22⋊C4—1882no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—1882no—
512C42—1882no—
513C42D4×D4712yes—
514C42#1062535212yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#1309405412yes—
522C23C22×C22⋊D4711yescharacteristic
523Q8—942no—
524Q8—942no—
525Q8—1881no—
526Q8—1881no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—942no—
559C2×C4—942no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4#1309399612yes—
571C2×C4#1309399612yes—
572C2×C4C2×D4×D4711yescharacteristic
573C2×C4#41×D4711yescharacteristic
574C2×C4#41×D4711yescharacteristic
575C2×C4C2×#1062535211yescharacteristic
576C2×C4#1309407211yescharacteristic
577C2×C4#1309407211yescharacteristic
578C8—942no—
579C8—942no—
580C8—1881no—
581C8—1881no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1309399611yescharacteristic
590C22#1568963011yescharacteristic
591C22#1568969711yescharacteristic
592C22C2×#1309405411yescharacteristic
593C22#1568992111yescharacteristic
594C22#1568992111yescharacteristic
595C22#1568992111yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—3761no—
606C4—3761no—
607C4—1882no—
608C4—1882no—
609C4#1568977511yescharacteristic
610C4#1568978311yescharacteristic
611C4#1568980711yescharacteristic
612C4#1568984411yescharacteristic
613C2—42no—
614C2#2024681811yescharacteristic
615C2#2025299411yescharacteristic
616C2#2025302011yescharacteristic
617C2#2025349111yescharacteristic
618C2#2025355511yescharacteristic
619C2#2025459411yescharacteristic
620C2#2025462811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2851C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2851D47
hypocenterC47#24707
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. C2×#15689427C2semidirect
8. #20244346C2semidirect
9. #20244154C2semidirect
10. #20244128C2semidirect
11. #20244123C2semidirect
12. #20244077C2semidirect
13. #20244077C2semidirect
14. #20243400C2semidirect
371. C47#24707semidirect

Maximal subgroups

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

subgroupindex
C2×#156894272
C2×#156891802
C4×#130940032
#202448242
#202448192
#202445912
#202445842
#202443462
#202441542
#202441282
#202441232
#202440772
#202434002
C47×#28512
#2470747