Finite Group Databasebeta

← #26264440

Subgroups — #26264440

Order 12032.

806 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689461C211yesmaximal, characteristic
2C2×#15689362C211yesmaximal, characteristic
3#20247841C211yesmaximal, characteristic
4#20244859C211yesmaximal, characteristic, complemented
5#20244806C211yesmaximal, characteristic, complemented
6#20244766C211yesmaximal, characteristic, complemented
7#20244710C211yesmaximal, characteristic, complemented
8#20244156C211yesmaximal, characteristic, complemented
9#20244029C211yesmaximal, characteristic, complemented
10#20243729C211yesmaximal, characteristic, complemented
11#20243700C211yesmaximal, characteristic
12#20243568C211yesmaximal, characteristic
13#20243393C211yesmaximal, characteristic, complemented
14#20243104C211yesmaximal, characteristic
15C47×#2362C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18C2×#13094003C2211yescharacteristic
19C2×#13093983C2211yescharacteristic
20#15689461—24no—
21#15689362—24no—
22C2×#13093957C2211yescharacteristic
23C2×#13093957C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×C4⋊(C47⋊C8)C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093948C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093948C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32C2×#13093947C2211yescharacteristic
33C2×C22⋊(C47⋊C8)C2211yescharacteristic
34#15689048—21no—
35#15689048—21no—
36#15689029C2211yescharacteristic
37#15689029C2211yescharacteristic
38#15689008—21no—
39#15689008—21no—
40#15688993C2211yescharacteristic
41#15688989C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688878C2211yescharacteristic
49#15688878C2211yescharacteristic
50C2×C22⋊Q8×C47C2211yescharacteristic
51C2×C4×Q8×C47C2211yescharacteristic
52C2×#117×C47C2211yescharacteristic
53C47×#391C2211yescharacteristic
54C47×#389C2211yescharacteristic
55C47×#388—21no—
56C47×#388—21no—
57C47×#387—21no—
58C4×C4⋊C4×C47—21no—
59C4×C4⋊C4×C47C2211yescharacteristic
60C4×C22⋊C4×C47C2211yescharacteristic
61C22×Dic94C2311yescharacteristic
62C2×#10625337—22no—
63C2×#10625337—22no—
64C2×#10625337—41no—
65C22×C47⋊C8—22no—
66C22×C47⋊C8—22no—
67C22×C47⋊C8—41no—
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72#13094003—22no—
73#13094003—22no—
74#13093983—22no—
75#13093983—22no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47—21no—
78C2×C4⋊Dic47C2311yescharacteristic
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×Dic47⋊C4—21no—
82C2×C22⋊Dic47—21no—
83C2×C22⋊Dic47—21no—
84C2×C4×Dic47—21no—
85#13093957—22no—
86#13093957—22no—
87#13093955—22no—
88C4⋊(C47⋊C8)—22no—
89C4⋊(C47⋊C8)—41no—
90#13093948—22no—
91#13093948—22no—
92#13093948—22no—
93#13093947—22no—
94#13093947—22no—
95#13093947—41no—
96C22⋊(C47⋊C8)—22no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C22×Q8×C47C2311yescharacteristic
104C23×C4×C47C2311yescharacteristic
105C22⋊Q8×C47—22no—
106C22⋊Q8×C47D414yes—
107C4×Q8×C47—22no—
108C4×Q8×C47—41no—
109#117×C47—22no—
110#117×C47—41no—
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—21no—
113C2×C4⋊C4×C47—21no—
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C22⋊C4×C47—21no—
118C2×C22⋊C4×C47—21no—
119C2×C22⋊C4×C47D411yescharacteristic
120C2×C22⋊C4×C47D411yescharacteristic
121C2×C42×C47—21no—
122C2×C42×C47—21no—
123C2×C42×C47—21no—
124C2×C42×C47C2311yescharacteristic
125C2×C42×C47C2311yescharacteristic
126#95×C47—21no—
127#95×C47—21no—
128#95×C47—21no—
129#95×C47—21no—
130#95×C47D411yescharacteristic
131#95×C47D411yescharacteristic
132C24×C47C2×D411yescharacteristic
133C2×Q8×C47—21no—
134C2×Q8×C47—21no—
135C2×Q8×C47—21no—
136C2×Q8×C47—21no—
137C2×Q8×C47—21no—
138C2×Q8×C47C2×D412yes—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D412yes—
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47#4112yes—
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94—21no—
165C4⋊C4×C47—24no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—42no—
171C4⋊C4×C47—42no—
172C4⋊C4×C47C2×D412yes—
173C4⋊C4×C47#4112yes—
174C22⋊C4×C47—24no—
175C22⋊C4×C47—22no—
176C22⋊C4×C47—41no—
177C22⋊C4×C47—42no—
178C42×C47—21no—
179C42×C47—21no—
180C42×C47—21no—
181C42×C47—21no—
182C42×C47—42no—
183C42×C47—42no—
184C42×C47—42no—
185C22⋊Dic47—42no—
186C22⋊Dic47—42no—
187#10625337—44no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—42no—
195C4⋊Dic47—42no—
196Dic47⋊C4—42no—
197C4×Dic47—42no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—44no—
210C2×C47⋊C8#4112yes—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—21no—
216C23×C47—21no—
217C23×C47#12112yes—
218C23×C47C22×D411yescharacteristic
219Q8×C47—22no—
220Q8×C47—22no—
221Q8×C47—41no—
222Q8×C47—41no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—21no—
234C2×C4×C47—22no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47#12112yes—
253C2×C4×C47#13512yes—
254C2×C4×C47C22×D411yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47H(5,𝔽2)11yescharacteristic
259C2×C4×C472−1+411yescharacteristic
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272Dic94—41no—
273Dic94—41no—
274Dic94—41no—
275Dic94—41no—
276C47⋊C8—22no—
277C47⋊C8—41no—
278C47⋊C8—41no—
279C47⋊C8—41no—
280#25201—471nomaximal, Sylow
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47C2×#12111yescharacteristic
289C22×C47#54611yescharacteristic
290C22×C47#54711yescharacteristic
291C22×C47C2×#13511yescharacteristic
292C22×C47#57511yescharacteristic
293C22×C47#57811yescharacteristic
294C22×C47#57811yescharacteristic
295C4×C47—22no—
296C4×C47—22no—
297C4×C47—41no—
298C4×C47—41no—
299C4×C47—42no—
300C4×C47—42no—
301C4×C47—41no—
302C4×C47—42no—
303C4×C47—81no—
304C4×C47#52911yescharacteristic
305C4×C47#53911yescharacteristic
306C4×C47#54611yescharacteristic
307C4×C47#54711yescharacteristic
308Dic47—81no—
309Dic47—81no—
310Dic47—81no—
311Dic47—81no—
312C2×#485—471no—
313C2×#467—471no—
314#2499—471no—
315#2362D4711yescharacteristic
316#2083—471no—
317#2074—471no—
318#2066—471no—
319#2056—471no—
320#1949—471no—
321#1928—471no—
322#1878—471no—
323#1873—471no—
324#1850—471no—
325#1803—471no—
326#1735—471no—
327C2×C47—42no—
328C2×C47#242511yescharacteristic
329C2×C47#303511yescharacteristic
330C2×C47#304411yescharacteristic
331C2×C47#330011yescharacteristic
332C2×C47#331511yescharacteristic
333C2×C47#334211yescharacteristic
334C2×C47#334911yescharacteristic
335C22×Q16—471no—
336C23×C8—471no—
337C2×C22⋊Q8—471no—
338C2×C22⋊Q8C2×D4711yescharacteristic
339C2×C4×Q8C2×D4711yescharacteristic
340C2×#117—471no—
341C2×#117C2×D4711yescharacteristic
342#485—944no—
343#467—944no—
344C2×#107—471no—
345C2×#107—471no—
346C2×#106—471no—
347C2×C4⋊C8—471no—
348C2×C4⋊C8—471no—
349C2×#103—471no—
350C2×#103—471no—
351C2×#103—471no—
352C2×#103—471no—
353C2×#103—471no—
354C2×#103—471no—
355C2×C22⋊C8—471no—
356#397—941no—
357#397—941no—
358#391—471no—
359#391—471no—
360#391C2×D4711yescharacteristic
361#389C2×D4711yescharacteristic
362#388—21no—
363#388—21no—
364#387—21no—
365#387—941no—
366#387—941no—
367#384—471no—
368#382—471no—
369C4×C4⋊C4—21no—
370C4×C4⋊C4C2×D4711yescharacteristic
371C4×C22⋊C4C2×D4711yescharacteristic
372#340—471no—
373#340—471no—
374#340—471no—
375#340—471no—
376#340—471no—
377#340—471no—
378#336—471no—
379#336—471no—
380C47#2520111yescharacteristic, complemented, Sylow
381C22×Q8—471no—
382C22×Q8C22×D4711yescharacteristic
383C23×C4C22×D4711yescharacteristic
384C2×Q16—942no—
385C2×Q16—942no—
386C2×Q16—1881no—
387C22×C8—471no—
388C22×C8—471no—
389C22×C8—471no—
390C22×C8—471no—
391C22×C8—942no—
392C22×C8—942no—
393C22×C8—1881no—
394C22⋊Q8—22no—
395C22⋊Q8—942no—
396C22⋊Q8—942no—
397C22⋊Q8D18814yes—
398C4×Q8—22no—
399C4×Q8—41no—
400#117—22no—
401#117—41no—
402#117—942no—
403#117—942no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4—941no—
412C2×C4⋊C4—941no—
413C2×C4⋊C4C22×D4711yescharacteristic
414C2×C4⋊C4C22×D4711yescharacteristic
415C2×C4⋊C4C22×D4711yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—941no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4C22⋊D4711yescharacteristic
421C2×C22⋊C4C22⋊D4711yescharacteristic
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—941no—
426C2×C42C22×D4711yescharacteristic
427C2×C42C22×D4711yescharacteristic
428#107—942no—
429#107—942no—
430#106—942no—
431C4⋊C8—942no—
432C4⋊C8—1881no—
433#103—942no—
434#103—942no—
435#103—942no—
436#103—942no—
437#103—942no—
438#103—1881no—
439C22⋊C8—942no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—941no—
445#95—941no—
446#95—941no—
447#95—941no—
448#95—941no—
449#95—941no—
450#95C22⋊D4711yescharacteristic
451#95C22⋊D4711yescharacteristic
452C24C2×C22⋊D4711yescharacteristic
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—941no—
459C2×Q8—941no—
460C2×Q8—941no—
461C2×Q8—941no—
462C2×Q8—941no—
463C2×Q8—941no—
464C2×Q8C2×D18812yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4C2×D18812yes—
478C22×C4#1062535012yes—
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C23×D4711yescharacteristic
485Q16—1884no—
486C2×C8—472no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—941no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—1884no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4—1882no—
513C4⋊C4—1882no—
514C4⋊C4—1882no—
515C4⋊C4C2×D18812yes—
516C4⋊C4#1062535212yes—
517C22⋊C4—24no—
518C22⋊C4—22no—
519C22⋊C4—41no—
520C22⋊C4—42no—
521C22⋊C4—1882no—
522C22⋊C4—1882no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—1882no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23#1309399812yes—
538C23C22×C22⋊D4711yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—1881no—
544Q8—1881no—
545Q8—1881no—
546Q8—1881no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4#1309399812yes—
589C2×C4#1309404412yes—
590C2×C4C22×D18811yescharacteristic
591C2×C4#41×D4711yescharacteristic
592C2×C4C2×#1062535211yescharacteristic
593C2×C4C2×#1062535011yescharacteristic
594C2×C4#1309406911yescharacteristic
595C2×C4#1309407111yescharacteristic
596C8—942no—
597C8—1881no—
598C8—1881no—
599C8—1881no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22C2×#1309399811yescharacteristic
608C22#1568976711yescharacteristic
609C22#1568977211yescharacteristic
610C22C2×#1309404411yescharacteristic
611C22#1568992111yescharacteristic
612C22#1568993811yescharacteristic
613C22#1568993811yescharacteristic
614C4—22no—
615C4—22no—
616C4—41no—
617C4—42no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—81no—
623C4—3761no—
624C4—3761no—
625C4—3761no—
626C4—3761no—
627C4#1568965311yescharacteristic
628C4#1568971511yescharacteristic
629C4#1568976811yescharacteristic
630C4#1568977711yescharacteristic
631C2—42no—
632C2#2024693311yescharacteristic
633C2#2025284311yescharacteristic
634C2#2025291811yescharacteristic
635C2#2025527511yescharacteristic
636C2#2025548411yescharacteristic
637C2#2025581411yescharacteristic
638C2#2025591811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2362C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2362D47
hypocenterC47#25201
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
4. #20244859C2semidirect
5. #20244806C2semidirect
6. #20244766C2semidirect
7. #20244710C2semidirect
8. #20244156C2semidirect
9. #20244029C2semidirect
10. #20243729C2semidirect
13. #20243393C2semidirect
380. C47#25201semidirect

Maximal subgroups

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

subgroupindex
C2×#156894612
C2×#156893622
#202478412
#202448592
#202448062
#202447662
#202447102
#202441562
#202440292
#202437292
#202437002
#202435682
#202433932
#202431042
C47×#23622
#2520147