Finite Group Databasebeta

← #26223646

Subgroups — #26223646

Order 12032.

844 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×#15689467C211yesmaximal, characteristic
2#20246071C211yesmaximal, characteristic
3#20245742C211yesmaximal, characteristic
4#20244021C211yesmaximal, characteristic, complemented
5#20244020C211yesmaximal, characteristic, complemented
6#20243983C211yesmaximal, characteristic, complemented
7#20243968C211yesmaximal, characteristic, complemented
8#20243694C211yesmaximal, characteristic
9#20243579C211yesmaximal, characteristic
10#20243481C211yesmaximal, characteristic
11#20243286C211yesmaximal, characteristic, complemented
12#20243221C211yesmaximal, characteristic, complemented
13C4×#13093949C211yesmaximal, characteristic, complemented
14#20243155C211yesmaximal, characteristic
15C4×#106×C47C211yesmaximal, characteristic, complemented
16C2×#13094004C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18C2×#13093982C2211yescharacteristic
19C2×#13093981C2211yescharacteristic
20C2×C4⋊C4×D47C2211yescharacteristic
21#15689467C418yescomplemented
22C2×#13093957C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093949C2211yescharacteristic
25C2×#13093949C2211yescharacteristic
26C2×#13093947C2211yescharacteristic
27C2×#13093947C2211yescharacteristic
28C2×(C2×D47)⋊C8C2211yescharacteristic
29C2×(C2×D47)⋊C8C2211yescharacteristic
30C2×#13093935C2211yescharacteristic
31#15689028C2211yescharacteristic
32#15689022C2211yescharacteristic
33#15689021—21no—
34#15689015—21no—
35#15689014—21no—
36#15689008—21no—
37#15688980C2211yescharacteristic
38C4⋊C4×Dic47C2211yescharacteristic
39C4⋊C4×Dic47C2211yescharacteristic
40C4×Dic47⋊C4—21no—
41C4×C4⋊Dic47C2211yescharacteristic
42#15688974—21no—
43C4×(C2×D47)⋊C4—21no—
44#15688970—21no—
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#106×C47C2211yescharacteristic
54C2×#106×C47C2211yescharacteristic
55C2×C4×C8×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×Dic94C2311yescharacteristic
61C22×C4×D47—21no—
62C22×C4×D47C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65#13094004—22no—
66#13094004C2×C414yes—
67C4×Dic94—22no—
68C4×Dic94—22no—
69#13093982—22no—
70#13093982C2×C414yes—
71#13093981—22no—
72#13093981—22no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4D411yescharacteristic
81C2×Dic47⋊C4D411yescharacteristic
82C4⋊C4×D47—22no—
83C4⋊C4×D47—22no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×(C2×D47)⋊C4D411yescharacteristic
88C2×(C2×D47)⋊C4D411yescharacteristic
89C2×C4×Dic47—21no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47—21no—
92C2×C4×Dic47—21no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47C2311yescharacteristic
95#13093957C2×C414yescomplemented
96#13093955—22no—
97#13093949—22no—
98#13093949C2×C414yescomplemented
99#13093947—22no—
100#13093947C2×C414yescomplemented
101(C2×D47)⋊C8—22no—
102(C2×D47)⋊C8C2×C414yes—
103#13093935—22no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C22×C8×C47C2311yescharacteristic
112C22×C8×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C42×C47—21no—
118C2×C42×C47—21no—
119C2×C42×C47C2311yescharacteristic
120#106×C47—22no—
121#106×C47C2×C414yescomplemented
122C4×C8×C47—22no—
123#95×C47—21no—
124#95×C47—21no—
125C23×D47C2×D411yescharacteristic
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2411yescharacteristic
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47C2×D411yescharacteristic
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140C2×C4×D47—21no—
141C2×C4×D47—21no—
142C2×C4×D47—21no—
143C2×C4×D47—21no—
144C2×C4×D47—24no—
145C2×C4×D47—81no—
146C2×C4×D47—81no—
147C2×C4×D47C22×C412yes—
148C2×C4×D47#4112yes—
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94—21no—
154C2×Dic94C22×C412yes—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47C22×C412yes—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—21no—
166C4⋊C4×C47C22×C412yes—
167C4⋊C4×C47C22×C412yes—
168C4⋊C4×C47#4112yes—
169C42×C47—81no—
170C42×C47—81no—
171C42×C47#4112yes—
172C42×C47#4112yes—
173(C2×D47)⋊C4—24no—
174(C2×D47)⋊C4—24no—
175(C2×D47)⋊C4—22no—
176(C2×D47)⋊C4—22no—
177(C2×D47)⋊C4—81no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—24no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—81no—
184C4⋊Dic47C22×C412yes—
185C4⋊Dic47#4112yes—
186Dic47⋊C4—24no—
187Dic47⋊C4—22no—
188Dic47⋊C4—22no—
189Dic47⋊C4—42no—
190C4×Dic47—21no—
191C4×Dic47—21no—
192C4×Dic47—24no—
193C4×Dic47—42no—
194C4×Dic47—81no—
195C4×Dic47—81no—
196C4×Dic47—81no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8C22×C412yes—
203C23×C47C22×D411yescharacteristic
204C22×D47—21no—
205C22×D47—21no—
206C22×D47—21no—
207C22×D47—21no—
208C22×D47—21no—
209C22×D47—21no—
210C22×D47C4×D412yes—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47#12312yes—
228C2×C4×C47#12312yes—
229C2×C4×C47C23×C411yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47C2×#4111yescharacteristic
232C2×C4×C47C2×#4111yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×Dic47—21no—
236C2×Dic47—21no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—22no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47C4×D412yes—
256C4×D47—22no—
257C4×D47—22no—
258C4×D47—22no—
259C4×D47—22no—
260C4×D47—84no—
261Dic94—22no—
262Dic94—22no—
263Dic94—41no—
264Dic94—41no—
265C8×C47—22no—
266C8×C47—41no—
267C47⋊C8—22no—
268C47⋊C8—41no—
269#21591—471nomaximal, Sylow
270C22×C47C2×C4×D411yescharacteristic
271C22×C47C2×#12311yescharacteristic
272C22×C47#52511yescharacteristic
273C22×C47#57711yescharacteristic
274C22×C47#57711yescharacteristic
275C22×C47#57711yescharacteristic
276C22×C47#57811yescharacteristic
277C2×D47—22no—
278C2×D47—22no—
279C2×D47—22no—
280C2×D47—41no—
281C2×D47—41no—
282C2×D47—41no—
283C2×D47—41no—
284C4×C47—22no—
285C4×C47—22no—
286C4×C47—42no—
287C4×C47—42no—
288C4×C47—42no—
289C4×C47—81no—
290C4×C47C4×#4111yescharacteristic
291C4×C47C4×#4111yescharacteristic
292C4×C47#52911yescharacteristic
293C4×C47#52911yescharacteristic
294Dic47—22no—
295Dic47—41no—
296Dic47—41no—
297Dic47—42no—
298Dic47—42no—
299Dic47—42no—
300Dic47—81no—
301C2×#485—471no—
302#2323—471no—
303#2274—471no—
304#1927—471no—
305#1927—471no—
306#1921—471no—
307#1919—471no—
308#1872—471no—
309#1853—471no—
310#1825—471no—
311#1776—471no—
312C4×#106D4711yescharacteristic, complemented
313#1763—471no—
314C4×#103—471no—
315#1750—471no—
316C2×C47C4×#12311yescharacteristic
317C2×C47#294711yescharacteristic
318C2×C47#295211yescharacteristic
319C2×C47#309711yescharacteristic
320C2×C47#309911yescharacteristic
321C2×C47#312111yescharacteristic
322C2×C47#312311yescharacteristic
323D47—42no—
324C2×C22⋊Q8—471no—
325C2×C4×Q8—471no—
326C2×#117—471no—
327C2×#117—471no—
328C22×C4⋊C4—471no—
329#485—478no—
330C2×#107—471no—
331C2×#106—471no—
332C2×#106C2×D4711yescharacteristic
333C2×#106C2×D4711yescharacteristic
334C2×#103—471no—
335C2×#103—471no—
336C2×#103—471no—
337C2×#103—471no—
338C2×C22⋊C8—471no—
339C2×C22⋊C8—471no—
340C2×#97—471no—
341C2×C4×C8C2×D4711yescharacteristic
342#391—471no—
343#389—471no—
344#389—941no—
345#388—941no—
346#388—941no—
347#387—941no—
348#380—471no—
349C4×C4⋊C4—471no—
350C4×C4⋊C4—471no—
351C4×C4⋊C4—471no—
352C4×C4⋊C4—941no—
353C4×C4⋊C4—941no—
354C4×C4⋊C4C2×D4711yescharacteristic
355C4×C4⋊C4C2×D4711yescharacteristic
356C4×C22⋊C4—941no—
357C4×C22⋊C4—941no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363#340—471no—
364#340C2×D4711yescharacteristic
365#340C2×D4711yescharacteristic
366#336—471no—
367#336—471no—
368C47#2159111yescharacteristic, complemented, Sylow
369C22×Q8—471no—
370C23×C4—471no—
371C23×C4—941no—
372C22×C8—471no—
373C22×C8—471no—
374C22×C8C22×D4711yescharacteristic
375C22×C8C22×D4711yescharacteristic
376C22⋊Q8—474no—
377C22⋊Q8—942no—
378C4×Q8—942no—
379C4×Q8—942no—
380#117—474no—
381#117—942no—
382#117—942no—
383#117—942no—
384C2×C4⋊C4—471no—
385C2×C4⋊C4—471no—
386C2×C4⋊C4—471no—
387C2×C4⋊C4—471no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—942no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4—941no—
395C2×C4⋊C4C22×D4711yescharacteristic
396C2×C4⋊C4C22×D4711yescharacteristic
397C2×C4⋊C4C22×D4711yescharacteristic
398C2×C4⋊C4C22×D4711yescharacteristic
399C2×C22⋊C4—471no—
400C2×C22⋊C4—471no—
401C2×C22⋊C4—941no—
402C2×C22⋊C4—941no—
403C2×C22⋊C4—941no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—471no—
407C2×C42—941no—
408C2×C42—941no—
409C2×C42—941no—
410C2×C42—941no—
411C2×C42—941no—
412C2×C42C22×D4711yescharacteristic
413#107—474no—
414#106—22no—
415#106—942no—
416#106C4×D4714yescomplemented
417#103—474no—
418#103—474no—
419#103—942no—
420#103—942no—
421C22⋊C8—474no—
422C22⋊C8—942no—
423#97—942no—
424C4×C8—22no—
425#95—21no—
426#95—21no—
427#95—941no—
428#95—941no—
429#95—941no—
430#95—941no—
431#95—941no—
432#95—941no—
433#95—941no—
434C24—471no—
435C2×Q8—472no—
436C2×Q8—941no—
437C2×Q8—941no—
438C2×Q8—941no—
439C2×Q8—941no—
440C2×Q8—941no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—471no—
446C22×C4—472no—
447C22×C4—472no—
448C22×C4—471no—
449C22×C4—471no—
450C22×C4—944no—
451C22×C4—941no—
452C22×C4—941no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—3761no—
460C22×C4—3761no—
461C22×C4D4×D4711yescharacteristic
462C22×C4D4×D4711yescharacteristic
463C22×C4C23×D4711yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—472no—
470C2×C8—941no—
471C2×C8—941no—
472C2×C8—941no—
473C2×C8—941no—
474C2×C8—941no—
475C2×C8C2×C4×D4712yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—472no—
482C4⋊C4—472no—
483C4⋊C4—944no—
484C4⋊C4—944no—
485C4⋊C4—942no—
486C4⋊C4—942no—
487C4⋊C4—941no—
488C4⋊C4—941no—
489C4⋊C4—941no—
490C4⋊C4—941no—
491C4⋊C4—3761no—
492C4⋊C4—1882no—
493C4⋊C4C2×C4×D4712yes—
494C4⋊C4C2×C4×D4712yes—
495C4⋊C4#1062535012yes—
496C22⋊C4—944no—
497C22⋊C4—944no—
498C22⋊C4—942no—
499C22⋊C4—942no—
500C22⋊C4—3761no—
501C42—81no—
502C42—81no—
503C42—944no—
504C42—941no—
505C42—941no—
506C42—3761no—
507C42—3761no—
508C42—3761no—
509C42—1882no—
510C42#1062535012yes—
511C42#1062535412yes—
512C23—472no—
513C23—941no—
514C23—941no—
515C23—941no—
516C23—941no—
517C23—941no—
518C23—941no—
519C23C2×D4×D4711yescharacteristic
520Q8—942no—
521Q8—942no—
522Q8—1881no—
523Q8—1881no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—472no—
541C2×C4—941no—
542C2×C4—941no—
543C2×C4—942no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—942no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—942no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—941no—
556C2×C4—3764no—
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#1309400912yes—
567C2×C4#1309400912yes—
568C2×C4C22×C4×D4711yescharacteristic
569C2×C4#41×D4711yescharacteristic
570C2×C4#41×D4711yescharacteristic
571C2×C4C2×#1062535011yescharacteristic
572C2×C4C2×#1062535011yescharacteristic
573C2×C4C2×#1062535411yescharacteristic
574C8—22no—
575C8—41no—
576C8—942no—
577C8—1881no—
578C22—942no—
579C22—942no—
580C22—942no—
581C22—1881no—
582C22—1881no—
583C22—1881no—
584C22—1881no—
585C22C4×D4×D4711yescharacteristic
586C22C2×#1309400911yescharacteristic
587C22#1568963211yescharacteristic
588C22#1568992911yescharacteristic
589C22#1568992911yescharacteristic
590C22#1568993011yescharacteristic
591C22#1568993711yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—942no—
599C4—3761no—
600C4—1882no—
601C4—1882no—
602C4—1881no—
603C4—1882no—
604C4—1881no—
605C4C4×#1062535011yescharacteristic
606C4C4×#1062535411yescharacteristic
607C4#1568965111yescharacteristic
608C4#1568966011yescharacteristic
609C2—1882no—
610C2C4×#1309400911yescharacteristic
611C2#2025222411yescharacteristic
612C2#2025226611yescharacteristic
613C2#2025336011yescharacteristic
614C2#2025338211yescharacteristic
615C2#2025361311yescharacteristic
616C2#2025363711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#106×C47C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenterC4×#106D47
hypocenterC47#21591
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
4. #20244021C2semidirect
5. #20244020C2semidirect
6. #20243983C2semidirect
7. #20243968C2semidirect
11. #20243286C2semidirect
12. #20243221C2semidirect
13. C4×#13093949C2semidirect
15. C4×#106×C47C2semidirect
21. #15689467C4semidirect
95. #13093957C2×C4semidirect
98. #13093949C2×C4semidirect
100. #13093947C2×C4semidirect
121. #106×C47C2×C4semidirect
312. C4×#106D47semidirect
368. C47#21591semidirect
416. #106C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156894672
#202460712
#202457422
#202440212
#202440202
#202439832
#202439682
#202436942
#202435792
#202434812
#202432862
#202432212
C4×#130939492
#202431552
C4×#106×C472
#2159147