Finite Group Databasebeta

← #26224402

Subgroups — #26224402

Order 12032.

886 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689481C211yesmaximal, characteristic
2C4×#13094023C211yesmaximal, characteristic, complemented
3C4×#13094011C211yesmaximal, characteristic
4#20244359C211yesmaximal, characteristic, complemented
5#20244203C211yesmaximal, characteristic
6#20244073C211yesmaximal, characteristic
7#20244068C211yesmaximal, characteristic
8#20244065C211yesmaximal, characteristic, complemented
9#20244062C211yesmaximal, characteristic, complemented
10#20243520C211yesmaximal, characteristic, complemented
11#20243221C211yesmaximal, characteristic, complemented
12#20243219C211yesmaximal, characteristic, complemented
13#20243213C211yesmaximal, characteristic
14#97×Dic47C211yesmaximal, characteristic, complemented
15C47×#1762C211yesmaximal, characteristic
16C2×#13094023C2211yescharacteristic
17C2×#13094011C2211yescharacteristic
18C2×C4×Dic94—21no—
19C2×C4×Dic94C2211yescharacteristic
20#15689481C418yescomplemented
21C2×#13093949C2211yescharacteristic
22C2×#13093949C2211yescharacteristic
23C2×#13093947C2211yescharacteristic
24C2×#13093947C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093935C2211yescharacteristic
28C2×#13093934C2211yescharacteristic
29C2×#13093934C2211yescharacteristic
30C2×C8×Dic47C2211yescharacteristic
31C2×C8×Dic47C2211yescharacteristic
32#15689028C2211yescharacteristic
33#15689027—21no—
34#15689026C2211yescharacteristic
35#15689022C2211yescharacteristic
36#15689020—21no—
37#15689019C2211yescharacteristic
38#15689017—21no—
39#15689017—21no—
40#15688992C2211yescharacteristic
41#15688988C2211yescharacteristic
42C4×Dic47⋊C4—21no—
43C4×C4⋊Dic47C2211yescharacteristic
44C4×C22⋊Dic47—21no—
45C4×C22⋊Dic47—21no—
46C42×Dic47C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688885C2211yescharacteristic
52#15688885C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C2×#102×C47C2211yescharacteristic
55C2×#102×C47C2211yescharacteristic
56C2×#97×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94—21no—
61C22×Dic94C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64#13094023—22no—
65#13094023C2×C414yescomplemented
66#13094011—22no—
67#13094011C2×C414yes—
68C4×Dic94—22no—
69C4×Dic94—22no—
70C4×Dic94—44no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×C22⋊Dic47—21no—
83C2×C22⋊Dic47—21no—
84C2×C4×Dic47—21no—
85C2×C4×Dic47—21no—
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47D411yescharacteristic
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91C2×C4×Dic47C2311yescharacteristic
92#13093949—22no—
93#13093949C2×C414yescomplemented
94#13093947—22no—
95#13093947C2×C414yescomplemented
96#13093945—22no—
97#13093945C2×C414yes—
98#13093935—22no—
99#13093934—22no—
100#13093934C2×C414yescomplemented
101C8×Dic47—22no—
102C8×Dic47—22no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C22×D4×C47C2311yescharacteristic
111C23×C4×C47—21no—
112C22×C8×C47C2311yescharacteristic
113C22×C8×C47C2311yescharacteristic
114C4×D4×C47—22no—
115C4×D4×C47—22no—
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C4⋊C4×C47C2311yescharacteristic
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C22⋊C4×C47—21no—
120C2×C42×C47—21no—
121C2×C42×C47C2311yescharacteristic
122#102×C47—22no—
123#102×C47C2×C414yes—
124#97×C47—22no—
125#95×C47—21no—
126C24×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47C22×C412yes—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—42no—
137C22×C4×C47—42no—
138C22×C4×C47—42no—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2411yescharacteristic
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149C22×Dic47C2×D411yescharacteristic
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—24no—
156C2×Dic94—42no—
157C2×Dic94—42no—
158C2×Dic94C22×C412yes—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47C22×C412yes—
164C2×C8×C47#4112yes—
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47C22×C412yes—
171C22⋊C4×C47—42no—
172C42×C47—42no—
173C42×C47#4112yes—
174C42×C47#4112yes—
175C22⋊Dic47—24no—
176C22⋊Dic47—24no—
177C22⋊Dic47—42no—
178C22⋊Dic47—42no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—42no—
185C4⋊Dic47C22×C412yes—
186Dic47⋊C4—24no—
187Dic47⋊C4—42no—
188Dic47⋊C4—42no—
189Dic47⋊C4—42no—
190Dic47⋊C4—42no—
191C4×Dic47—21no—
192C4×Dic47—21no—
193C4×Dic47—22no—
194C4×Dic47—22no—
195C4×Dic47—41no—
196C4×Dic47—41no—
197C4×Dic47—42no—
198C4×Dic47—42no—
199C4×Dic47C22×C412yes—
200C4×Dic47#4112yes—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8C22×C412yes—
206C2×C47⋊C8#4112yes—
207C23×C47—22no—
208C23×C47—41no—
209C23×C47—41no—
210C23×C47—41no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47C22×D411yescharacteristic
215D4×C47—22no—
216D4×C47—22no—
217D4×C47—41no—
218D4×C47—41no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—22no—
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×C47—41no—
232C2×C4×C47—84no—
233C2×C4×C47#12412yes—
234C2×C4×C47#12412yes—
235C2×C4×C47C23×C411yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—22no—
250C2×Dic47—22no—
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×Dic47C4×D412yes—
262C2×Dic47C4×D412yes—
263C2×Dic47AGL(1,ℤ/8)12yes—
264C2×Dic47#13712yes—
265Dic94—22no—
266Dic94—22no—
267Dic94—48no—
268Dic94—41no—
269Dic94—41no—
270C8×C47—22no—
271C8×C47—22no—
272C47⋊C8—22no—
273C47⋊C8—22no—
274#21655—471nomaximal, Sylow
275C22×C47—42no—
276C22×C47—42no—
277C22×C47—42no—
278C22×C47—81no—
279C22×C47—81no—
280C22×C47—81no—
281C22×C47—81no—
282C22×C47C2×C4×D411yescharacteristic
283C22×C47C2×#12411yescharacteristic
284C22×C47#53211yescharacteristic
285C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C47C2×#13711yescharacteristic
287C22×C47#57511yescharacteristic
288C22×C47#57511yescharacteristic
289C4×C47—22no—
290C4×C47—22no—
291C4×C47—42no—
292C4×C47—81no—
293C4×C47—81no—
294C4×C47C4×#4111yescharacteristic
295C4×C47C4×#4111yescharacteristic
296C4×C47#52911yescharacteristic
297C4×C47#52911yescharacteristic
298Dic47—22no—
299Dic47—22no—
300Dic47—22no—
301Dic47—22no—
302Dic47—42no—
303Dic47—42no—
304Dic47—81no—
305Dic47—81no—
306C2×#489—471no—
307C4×C4⋊Q8—471no—
308C4×#124—471no—
309#1993—471no—
310#1959—471no—
311#1934—471no—
312#1934—471no—
313#1933—471no—
314#1933—471no—
315#1836—471no—
316#1763—471no—
317#1763—471no—
318#1762—471no—
319#1762D4711yescharacteristic
320C4×#97—471no—
321C2×C47—82no—
322C2×C47C4×#12411yescharacteristic
323C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
324C2×C47C4×#13711yescharacteristic
325C2×C47#314011yescharacteristic
326C2×C47#314111yescharacteristic
327C2×C47#318011yescharacteristic
328C2×C47#318111yescharacteristic
329C2×C4⋊Q8—471no—
330C2×#124—471no—
331C2×C4×Q8—471no—
332C2×C4×Q8—941no—
333C2×C4×D4C2×D4711yescharacteristic
334#489—478no—
335C2×#103—471no—
336C2×#103—471no—
337C2×#103—471no—
338C2×#103—471no—
339C2×#102—471no—
340C2×#102—471no—
341C2×#102C2×D4711yescharacteristic
342C2×#102C2×D4711yescharacteristic
343C2×#97—471no—
344C2×#97—471no—
345C2×#97—471no—
346C2×#97C2×D4711yescharacteristic
347C2×C4×C8—471no—
348C2×C4×C8—471no—
349#391—471no—
350#391—941no—
351#390—471no—
352#389—471no—
353#389—471no—
354#389—941no—
355#388—941no—
356#388—941no—
357#384—471no—
358#382—471no—
359C4×C4⋊C4—471no—
360C4×C4⋊C4—941no—
361C4×C4⋊C4C2×D4711yescharacteristic
362C4×C22⋊C4—941no—
363C4×C22⋊C4—941no—
364C43—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#340—471no—
370#340—471no—
371#340C2×D4711yescharacteristic
372#340C2×D4711yescharacteristic
373C47#2165511yescharacteristic, complemented, Sylow
374C22×Q8—471no—
375C22×Q8—941no—
376C22×D4C22×D4711yescharacteristic
377C23×C4—21no—
378C22×C8—471no—
379C22×C8—471no—
380C22×C8C22×D4711yescharacteristic
381C22×C8C22×D4711yescharacteristic
382C4⋊Q8—474no—
383C4⋊Q8—942no—
384#124—474no—
385#124—942no—
386C4×Q8—942no—
387C4×Q8—942no—
388C4×Q8—1884no—
389C4×D4—22no—
390C4×D4—22no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4—941no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4—941no—
400C2×C4⋊C4C22×D4711yescharacteristic
401C2×C4⋊C4C22×D4711yescharacteristic
402C2×C4⋊C4C22×D4711yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—941no—
405C2×C22⋊C4—941no—
406C2×C22⋊C4—941no—
407C2×C22⋊C4—941no—
408C2×C42—21no—
409C2×C42—471no—
410C2×C42—471no—
411C2×C42—471no—
412C2×C42—471no—
413C2×C42—471no—
414C2×C42—471no—
415C2×C42—941no—
416C2×C42—941no—
417C2×C42C22×D4711yescharacteristic
418#103—474no—
419#103—474no—
420#103—942no—
421#103—942no—
422#102—22no—
423#102—474no—
424#102—942no—
425#102C4×D4714yes—
426#97—22no—
427#97—474no—
428#97—942no—
429#97—942no—
430C4×C8—942no—
431C4×C8—942no—
432#95—21no—
433#95—941no—
434#95—941no—
435#95—941no—
436#95—941no—
437#95—941no—
438#95—941no—
439#95—941no—
440C24—21no—
441C2×Q8—472no—
442C2×Q8—944no—
443C2×Q8—941no—
444C2×Q8—941no—
445C2×Q8—941no—
446C2×Q8—941no—
447C2×Q8—941no—
448C2×Q8—1882no—
449C2×Q8—1882no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4C2×C4×D4712yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—471no—
463C22×C4—471no—
464C22×C4—471no—
465C22×C4—471no—
466C22×C4—941no—
467C22×C4—941no—
468C22×C4—941no—
469C22×C4—941no—
470C22×C4D4×D4711yescharacteristic
471C22×C4D4×D4711yescharacteristic
472C22×C4C23×D4711yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—472no—
478C2×C8—472no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8C2×C4×D4712yes—
484C2×C8#1062535012yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—472no—
491C4⋊C4—944no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4—941no—
496C4⋊C4—941no—
497C4⋊C4—1882no—
498C4⋊C4—1882no—
499C4⋊C4—1882no—
500C4⋊C4—1882no—
501C4⋊C4—1882no—
502C4⋊C4C2×C4×D4712yes—
503C22⋊C4—42no—
504C22⋊C4—944no—
505C22⋊C4—944no—
506C22⋊C4—1882no—
507C22⋊C4—1882no—
508C42—42no—
509C42—472no—
510C42—472no—
511C42—942no—
512C42—942no—
513C42—941no—
514C42—941no—
515C42—1881no—
516C42—1881no—
517C42—1882no—
518C42—1882no—
519C42#1062535012yes—
520C42#1062535212yes—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C2×D4×D4711yescharacteristic
529Q8—942no—
530Q8—942no—
531Q8—1888no—
532Q8—1881no—
533Q8—1881no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
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—84no—
552C2×C4—472no—
553C2×C4—472no—
554C2×C4—472no—
555C2×C4—472no—
556C2×C4—941no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—941no—
561C2×C4—941no—
562C2×C4—941no—
563C2×C4—941no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4#1309401012yes—
577C2×C4#1309401012yes—
578C2×C4C22×C4×D4711yescharacteristic
579C2×C4#41×D4711yescharacteristic
580C2×C4#41×D4711yescharacteristic
581C2×C4C2×#1062535211yescharacteristic
582C2×C4C2×#1062535011yescharacteristic
583C2×C4C2×#1062535011yescharacteristic
584C8—22no—
585C8—22no—
586C8—942no—
587C8—942no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D4711yescharacteristic
596C22C2×#1309401011yescharacteristic
597C22#1568967011yescharacteristic
598C22AGL(1,ℤ/8)×D4711yescharacteristic
599C22#137×D4711yescharacteristic
600C22#1568991911yescharacteristic
601C22#1568991911yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—942no—
608C4—942no—
609C4—942no—
610C4—942no—
611C4—3761no—
612C4—3761no—
613C4—1882no—
614C4—1882no—
615C4C4×#1062535211yescharacteristic
616C4C4×#1062535011yescharacteristic
617C4#1568965111yescharacteristic
618C4#1568965311yescharacteristic
619C2—82no—
620C2C4×#1309401011yescharacteristic
621C2#2025219911yescharacteristic
622C2#2025221011yescharacteristic
623C2#2025377211yescharacteristic
624C2#2025377911yescharacteristic
625C2#2025408411yescharacteristic
626C2#2025409311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1762C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1762D47
hypocenterC47#21655
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C4×#13094023C2semidirect
4. #20244359C2semidirect
8. #20244065C2semidirect
9. #20244062C2semidirect
10. #20243520C2semidirect
11. #20243221C2semidirect
12. #20243219C2semidirect
14. #97×Dic47C2semidirect
20. #15689481C4semidirect
65. #13094023C2×C4semidirect
93. #13093949C2×C4semidirect
95. #13093947C2×C4semidirect
100. #13093934C2×C4semidirect
373. C47#21655semidirect

Maximal subgroups

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

subgroupindex
C2×#156894812
C4×#130940232
C4×#130940112
#202443592
#202442032
#202440732
#202440682
#202440652
#202440622
#202435202
#202432212
#202432192
#202432132
#97×Dic472
C47×#17622
#2165547