Finite Group Databasebeta

← #26224400

Subgroups — #26224400

Order 12032.

826 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689482C211yesmaximal, characteristic
2#20246134C211yesmaximal, characteristic, complemented
3#20246101C211yesmaximal, characteristic
4#20244349C211yesmaximal, characteristic, complemented
5#20244082C211yesmaximal, characteristic
6#20244075C211yesmaximal, characteristic, complemented
7#20244068C211yesmaximal, characteristic
8#20244061C211yesmaximal, characteristic, complemented
9#20243521C211yesmaximal, characteristic, complemented
10#20243220C211yesmaximal, characteristic, complemented
11#20243215C211yesmaximal, characteristic
12#103×Dic47C211yesmaximal, characteristic, complemented
13#102×Dic47C211yesmaximal, characteristic
14#20243074C211yesmaximal, characteristic, complemented
15C47×#1958C211yesmaximal, characteristic
16C2×#13094024C2211yescharacteristic
17C2×Q8×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19#15689482C418yescomplemented
20C2×#13093948C2211yescharacteristic
21C2×#13093948C2211yescharacteristic
22C2×#13093945C2211yescharacteristic
23C2×#13093945C2211yescharacteristic
24C2×#13093935C2211yescharacteristic
25C2×#13093934C2211yescharacteristic
26C2×#13093934C2211yescharacteristic
27C2×C8×Dic47C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29#15689030C2211yescharacteristic
30#15689027—21no—
31#15689027—21no—
32#15689026C2211yescharacteristic
33#15689019C2211yescharacteristic
34#15689019C2211yescharacteristic
35#15689016—21no—
36#15689016—21no—
37#15689008—21no—
38#15689008—21no—
39#15688998—21no—
40#15688998—21no—
41#15688993C2211yescharacteristic
42C4⋊C4×Dic47C2211yescharacteristic
43C4⋊C4×Dic47C2211yescharacteristic
44#15688969C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×#124×C47C2211yescharacteristic
54C2×#103×C47C2211yescharacteristic
55C2×#103×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C2×#102×C47C2211yescharacteristic
58C2×C4×C8×C47C2211yescharacteristic
59C47×#382C2211yescharacteristic
60C22×Dic94—21no—
61C23×Dic47—21no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094024—22no—
65#13094024C2×C414yescomplemented
66Q8×Dic47—22no—
67Q8×Dic47—22no—
68D4×Dic47—22no—
69D4×Dic47—22no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88#13093948C2×C414yescomplemented
89#13093948C2×C414yescomplemented
90#13093945C2×C414yes—
91#13093945C2×C414yes—
92#13093935C2×C414yescomplemented
93#13093934—22no—
94#13093934—22no—
95C8×Dic47—22no—
96C8×Dic47—22no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95D411yescharacteristic
104C47⋊#95D411yescharacteristic
105C47⋊#95D411yescharacteristic
106C47⋊#95D411yescharacteristic
107C22×Q8×C47C2311yescharacteristic
108C22×D4×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110C22×C8×C47C2311yescharacteristic
111#124×C47—22no—
112#124×C47C2×C414yes—
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C22⋊C4×C47—21no—
116C2×C22⋊C4×C47—21no—
117C2×C42×C47C2311yescharacteristic
118#103×C47—22no—
119#103×C47—22no—
120#102×C47—22no—
121#102×C47—22no—
122C4×C8×C47—22no—
123#95×C47—21no—
124#95×C47—21no—
125C24×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47C22×C412yes—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47C22×C412yes—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—42no—
149C22×Dic47—81no—
150C22×Dic47—81no—
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154C22×Dic47C2×D411yescharacteristic
155C2×Dic94—24no—
156C2×Dic94—81no—
157C2×Dic94—81no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47#4112yes—
163C2×C8×C47#4112yes—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C22⋊C4×C47—24no—
169C22⋊C4×C47—24no—
170C42×C47C22×C412yes—
171C42×C47#4112yes—
172C22⋊Dic47—42no—
173C22⋊Dic47—81no—
174C22⋊Dic47—81no—
175C4⋊Dic47—24no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—42no—
183C4⋊Dic47—81no—
184C4⋊Dic47—81no—
185C4⋊Dic47C22×C412yes—
186C4⋊Dic47C22×C412yes—
187Dic47⋊C4—81no—
188Dic47⋊C4—81no—
189C4×Dic47—21no—
190C4×Dic47—21no—
191C4×Dic47—42no—
192C4×Dic47—81no—
193C4×Dic47—81no—
194C4×Dic47#4112yes—
195C4×Dic47#4112yes—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8C22×C412yes—
201C2×C47⋊C8C22×C412yes—
202C23×C47—22no—
203C23×C47—41no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47—41no—
209C23×C47C22×D411yescharacteristic
210Q8×C47—22no—
211Q8×C47—22no—
212Q8×C47—41no—
213Q8×C47—41no—
214D4×C47—22no—
215D4×C47—22no—
216D4×C47—41no—
217D4×C47—41no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—22no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47C4×D412yes—
232C2×C4×C47C4×D412yes—
233C2×C4×C47C23×C411yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—41no—
252C2×Dic47—41no—
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—84no—
262C2×Dic47#12412yes—
263C2×Dic47#12412yes—
264Dic94—84no—
265C8×C47—22no—
266C8×C47—22no—
267C47⋊C8—22no—
268C47⋊C8—22no—
269#21654—471nomaximal, Sylow
270C22×C47—42no—
271C22×C47—42no—
272C22×C47—42no—
273C22×C47—81no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47—81no—
277C22×C47C2×C4×D411yescharacteristic
278C22×C47C2×#12411yescharacteristic
279C22×C47#53211yescharacteristic
280C22×C47#57611yescharacteristic
281C22×C47#57711yescharacteristic
282C22×C47#57711yescharacteristic
283C22×C47#57811yescharacteristic
284C4×C47—22no—
285C4×C47—22no—
286C4×C47—42no—
287C4×C47—81no—
288C4×C47—81no—
289C4×C47C4×#4111yescharacteristic
290C4×C47C4×#4111yescharacteristic
291C4×C47#52911yescharacteristic
292C4×C47#52911yescharacteristic
293Dic47—22no—
294Dic47—22no—
295Dic47—41no—
296Dic47—41no—
297Dic47—42no—
298Dic47—42no—
299Dic47—81no—
300Dic47—81no—
301C2×#489—471no—
302#2331—471no—
303#2326—471no—
304#1990—471no—
305#1958D4711yescharacteristic
306#1936—471no—
307#1935—471no—
308#1934—471no—
309#1933—471no—
310#1836—471no—
311#1763—471no—
312#1762—471no—
313C4×#103—471no—
314C4×#102—471no—
315#1726—471no—
316C2×C47—82no—
317C2×C47C4×#12411yescharacteristic
318C2×C47#294611yescharacteristic
319C2×C47#294711yescharacteristic
320C2×C47#313611yescharacteristic
321C2×C47#313711yescharacteristic
322C2×C47#317311yescharacteristic
323C2×C47#317511yescharacteristic
324C2×C4⋊Q8—471no—
325C2×#124C2×D4711yescharacteristic
326C2×C4×Q8—471no—
327C2×C4×D4—471no—
328#489—478no—
329C2×#103—471no—
330C2×#103—471no—
331C2×#103C2×D4711yescharacteristic
332C2×#103C2×D4711yescharacteristic
333C2×#102—471no—
334C2×#102—471no—
335C2×#102C2×D4711yescharacteristic
336C2×#102C2×D4711yescharacteristic
337C2×#97—471no—
338C2×#97—471no—
339C2×#97—471no—
340C2×C4×C8—471no—
341C2×C4×C8—471no—
342C2×C4×C8C2×D4711yescharacteristic
343#391—471no—
344#391—941no—
345#391—941no—
346#390—471no—
347#389—471no—
348#389—471no—
349#388—941no—
350#388—941no—
351#387—941no—
352#387—941no—
353#385—941no—
354#385—941no—
355#384—471no—
356#382C2×D4711yescharacteristic
357C4×C4⋊C4—471no—
358C4×C4⋊C4—471no—
359#376—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368C47#2165411yescharacteristic, complemented, Sylow
369C22×Q8—941no—
370C22×Q8C22×D4711yescharacteristic
371C22×D4C22×D4711yescharacteristic
372C23×C4—941no—
373C22×C8—471no—
374C22×C8—471no—
375C22×C8C22×D4711yescharacteristic
376C22×C8C22×D4711yescharacteristic
377C4⋊Q8—474no—
378C4⋊Q8—942no—
379#124—22no—
380#124C4×D4714yes—
381C4×Q8—942no—
382C4×Q8—942no—
383C4×D4—942no—
384C4×D4—942no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—941no—
400C2×C22⋊C4—941no—
401C2×C22⋊C4—941no—
402C2×C42—471no—
403C2×C42—471no—
404C2×C42—941no—
405C2×C42—941no—
406C2×C42—941no—
407C2×C42C22×D4711yescharacteristic
408#103—22no—
409#103—22no—
410#103—474no—
411#103—474no—
412#102—22no—
413#102—22no—
414#102—474no—
415#102—474no—
416#97—474no—
417#97—942no—
418#97—942no—
419C4×C8—22no—
420C4×C8—942no—
421C4×C8—942no—
422#95—21no—
423#95—21no—
424#95—471no—
425#95—471no—
426#95—471no—
427#95—471no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95—941no—
433#95—941no—
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—944no—
441C2×Q8—3761no—
442C2×Q8—3761no—
443C2×Q8C2×C4×D4712yes—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C2×C4×D4712yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—471no—
456C22×C4—471no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461C22×C4—3761no—
462C22×C4—3761no—
463C22×C4—1882no—
464C22×C4D4×D4711yescharacteristic
465C22×C4D4×D4711yescharacteristic
466C22×C4C23×D4711yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—472no—
472C2×C8—472no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8#1062535212yes—
478C2×C8#1062535212yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—472no—
484C4⋊C4—472no—
485C4⋊C4—944no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—3761no—
493C4⋊C4—3761no—
494C4⋊C4—3761no—
495C4⋊C4—3761no—
496C4⋊C4—1882no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—3761no—
500C22⋊C4—3761no—
501C22⋊C4—1882no—
502C42—472no—
503C42—472no—
504C42—941no—
505C42—941no—
506C42—3761no—
507C42—3761no—
508C42—1882no—
509C42C2×C4×D4712yes—
510C42#1062535212yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×D4×D4711yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523Q8—3764no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—472no—
542C2×C4—472no—
543C2×C4—941no—
544C2×C4—941no—
545C2×C4—941no—
546C2×C4—941no—
547C2×C4—942no—
548C2×C4—942no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—3764no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1881no—
565C2×C4—1881no—
566C2×C4(C22⋊D47)⋊C412yes—
567C2×C4(C22⋊D47)⋊C412yes—
568C2×C4C22×C4×D4711yescharacteristic
569C2×C4#41×D4711yescharacteristic
570C2×C4#41×D4711yescharacteristic
571C2×C4C2×#1062535211yescharacteristic
572C2×C4C2×#1062535211yescharacteristic
573C2×C4C2×#1062535211yescharacteristic
574C8—22no—
575C8—22no—
576C8—942no—
577C8—942no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×(C22⋊D47)⋊C411yescharacteristic
586C22#124×D4711yescharacteristic
587C22#1568967111yescharacteristic
588C22#1568992511yescharacteristic
589C22#1568993011yescharacteristic
590C22#1568993011yescharacteristic
591C22#1568993711yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—942no—
598C4—942no—
599C4—3761no—
600C4—3761no—
601C4—1881no—
602C4—1882no—
603C4—1882no—
604C4—1881no—
605C4C4×#1062535211yescharacteristic
606C4C4×#1062535211yescharacteristic
607C4#1568965411yescharacteristic
608C4#1568965411yescharacteristic
609C2—82no—
610C2#2024593511yescharacteristic
611C2#2025222311yescharacteristic
612C2#2025223411yescharacteristic
613C2#2025374111yescharacteristic
614C2#2025375211yescharacteristic
615C2#2025400211yescharacteristic
616C2#2025402411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1958C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1958D47
hypocenterC47#21654
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. #20246134C2semidirect
4. #20244349C2semidirect
6. #20244075C2semidirect
8. #20244061C2semidirect
9. #20243521C2semidirect
10. #20243220C2semidirect
12. #103×Dic47C2semidirect
14. #20243074C2semidirect
19. #15689482C4semidirect
65. #13094024C2×C4semidirect
88. #13093948C2×C4semidirect
89. #13093948C2×C4semidirect
92. #13093935C2×C4semidirect
368. C47#21654semidirect

Maximal subgroups

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

subgroupindex
C2×#156894822
#202461342
#202461012
#202443492
#202440822
#202440752
#202440682
#202440612
#202435212
#202432202
#202432152
#103×Dic472
#102×Dic472
#202430742
C47×#19582
#2165447