Finite Group Databasebeta

← #27083949

Subgroups — #27083949

Order 12032.

2054 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094070C211yesmaximal, characteristic
2#20253246C213yesmaximal, complemented
3#20253235C213yesmaximal, complemented
4#20253213C213yesmaximal, complemented
5#20253213C213yesmaximal, complemented
6#20253206C213yesmaximal, complemented
7#20253181C213yesmaximal, complemented
8#20253170C213yesmaximal, complemented
9C2×#15689420C213yesmaximal, complemented
10#20251986C211yesmaximal, characteristic, complemented
11#20251981C211yesmaximal, characteristic, complemented
12#20251956C211yesmaximal, characteristic, complemented
13#20251911C211yesmaximal, characteristic
14#20251911C211yesmaximal, characteristic
15C47×#3504C211yesmaximal, characteristic, complemented
16C2×#13094070C2211yescharacteristic
17C2×#13094031C2211yescharacteristic
18C2×#13094031C2211yescharacteristic
19#15689606C2211yescharacteristic
20#15689605—21no—
21#15689603C2213yes—
22#15689597C2213yes—
23#15689595—23no—
24C4×#10625354—21no—
25C4×#10625350C2213yes—
26C2×C4×D188C2213yes—
27C4×D4×D47—23no—
28#15689427C2213yescomplemented
29#15689427C2216yescomplemented
30#15689427C2211yescharacteristic, complemented
31#15689420C2216yescomplemented
32#15689420C2216yescomplemented
33#15689420C2216yescomplemented
34#15689420C2216yescomplemented
35#15689420C2213yescomplemented
36#15689420C2213yescomplemented
37#15689415C2213yescomplemented
38#15689410C2213yescomplemented
39#15689410C2216yescomplemented
40#15689405C2213yescomplemented
41#15689400C2213yescomplemented
42#15689400C2216yescomplemented
43#15689166C2213yescomplemented
44#15689166C2213yescomplemented
45#15689163C2213yescomplemented
46#15689163C2213yescomplemented
47#15689159C2213yescomplemented
48#15689159C2213yescomplemented
49C2×#13093957C2213yescomplemented
50C2×#13093955C2213yescomplemented
51#15689150C2213yescomplemented
52#15689150C2213yescomplemented
53#15689147C2213yescomplemented
54C2×C4⋊(C47⋊C8)C2213yescomplemented
55#15689127C2213yes—
56#15689127C2213yes—
57#15689122C2213yes—
58#15689122C2213yes—
59#15689117C2213yes—
60#15689117C2213yes—
61C2×#13093944C2213yes—
62C2×#13093944C2213yes—
63C47×#557—21no—
64C47×#557—21no—
65C47×#554—23no—
66C47×#554—23no—
67C47×#544C2213yescomplemented
68C47×#544C2213yescomplemented
69C47×#533C2213yescomplemented
70C2×#125×C47C2213yescomplemented
71C47×#520C2211yescharacteristic
72C47×#520C2211yescharacteristic
73C4×#41×C47C2211yescharacteristic
74#13094070—24no—
75#13094070Q818yescomplemented
76C2×#10625354—21no—
77C2×#10625350C2313yes—
78C22×D188C2313yes—
79C2×D4×D47—23no—
80C22×C4×D47—23no—
81#13094031—24no—
82#13094031—24no—
83C2×#10625329C2313yes—
84C2×#10625329C2313yes—
85C22×C47⋊C8C2313yes—
86C22×C47⋊C8C2313yes—
87Q8×Dic47—21no—
88C4×Dic94C2311yescharacteristic
89C4×Dic94C2313yes—
90C4×D188—23no—
91C4×D188—23no—
92C4×D188C2313yes—
93C4×D188C2316yes—
94C4×D188C2313yes—
95D4×Dic47—23no—
96#13093986—23no—
97#13093986—23no—
98C4×C22⋊D47—23no—
99C4×C22⋊D47—23no—
100(C22⋊D47)⋊C4—26no—
101#13093983C2313yes—
102#13093982—23no—
103#13093981—23no—
104C2×C4⋊Dic47C2313yes—
105C4⋊C4×D47—23no—
106C2×(C2×D47)⋊C4—23no—
107C22⋊C4×D47—26no—
108C42×D47—23no—
109#13093957C2311yescharacteristic
110#13093957C2313yes—
111#13093957C2316yes—
112#13093957C2313yes—
113#13093957C2313yes—
114#13093955C2313yes—
115#13093955C2313yes—
116#13093955C2316yes—
117#13093955C2311yescharacteristic
118#13093955C2313yes—
119C4⋊(C47⋊C8)C2311yescharacteristic
120C4⋊(C47⋊C8)C2313yes—
121C4⋊(C47⋊C8)C2313yes—
122C4⋊(C47⋊C8)C2313yes—
123C4⋊(C47⋊C8)C2316yes—
124#13093947C2311yescharacteristic
125#13093947C2311yescharacteristic
126#13093947C2313yes—
127#13093947C2313yes—
128#13093944C2313yes—
129#13093944C2313yes—
130#13093944C2316yes—
131#13093944C2316yes—
132#13093944C2313yes—
133#13093944C2313yes—
134C2×#41×C47C2311yescharacteristic
135C4⋊Q8×C47—23no—
136C4⋊Q8×C47—23no—
137C4⋊Q8×C47C2313yes—
138C4⋊Q8×C47C2313yes—
139#125×C47—23no—
140#125×C47—23no—
141#125×C47—23no—
142#125×C47—23no—
143#125×C47C2313yes—
144#125×C47C2316yes—
145#125×C47C2311yescharacteristic
146C22⋊Q8×C47—23no—
147C22⋊Q8×C47—23no—
148C22⋊Q8×C47—26no—
149C22⋊Q8×C47—26no—
150C4×Q8×C47—21no—
151C4×Q8×C47—21no—
152C4×Q8×C47—23no—
153C4×Q8×C47—23no—
154C4×Q8×C47D411yescharacteristic
155C4×Q8×C47D411yescharacteristic
156C4×D4×C47—23no—
157C4×D4×C47—23no—
158C4×D4×C47D413yes—
159C4×D4×C47D413yes—
160#117×C47C2313yes—
161#117×C47C2313yes—
162#117×C47C2313yes—
163C2×C4⋊C4×C47—23no—
164C2×C4⋊C4×C47—23no—
165C2×C4⋊C4×C47C2313yes—
166C2×C4⋊C4×C47C2313yes—
167C2×C42×C47C2313yes—
168C23×D47—23no—
169#41×C47C2×Q814yes—
170#41×C47#4114yes—
171C2×Q8×C47—23no—
172C2×Q8×C47—23no—
173C2×Q8×C47C2×D411yescharacteristic
174C2×D4×C47C2×D413yes—
175C22×C4×C47—23no—
176C22×C4×C47—23no—
177C22×C4×C47C2×D413yes—
178C22×C4×C47C2×D413yes—
179C22×C4×C47C2413yes—
180C2×C22⋊D47—23no—
181C22×Dic47—23no—
182#10625354—28no—
183D4×D47—224no—
184#10625350—26no—
185#10625350C2×Q8112yes—
186C2×D188—23no—
187C2×D188—26no—
188C2×D188C2×Q8112yes—
189C2×D188C2413yes—
190C2×C4×D47—23no—
191C2×C4×D47—23no—
192C2×C4×D47—23no—
193C2×C4×D47—26no—
194C2×C4×D47—86no—
195C2×Dic94C2411yescharacteristic
196C4⋊C4×C47—23no—
197C4⋊C4×C47—23no—
198C4⋊C4×C47—26no—
199C4⋊C4×C47—26no—
200C4⋊C4×C47—23no—
201C4⋊C4×C47—23no—
202C4⋊C4×C47—21no—
203C4⋊C4×C47—21no—
204C4⋊C4×C47—23no—
205C4⋊C4×C47—23no—
206C4⋊C4×C47C2×D413yes—
207C4⋊C4×C47C2×D413yes—
208C4⋊C4×C47C2413yes—
209C4⋊C4×C47C2413yes—
210C4⋊C4×C47C2411yescharacteristic
211C4⋊C4×C47C2411yescharacteristic
212C22⋊C4×C47—23no—
213C22⋊C4×C47—23no—
214C22⋊C4×C47C2×D416yes—
215C42×C47—23no—
216C42×C47—23no—
217C42×C47C2×D413yes—
218C42×C47C2×D413yes—
219C42×C47C2411yescharacteristic
220C42×C47C2413yes—
221C22⋊Dic47—23no—
222(C2×D47)⋊C4—23no—
223(C2×D47)⋊C4—26no—
224(C2×D47)⋊C4—23no—
225C4⋊Dic47—23no—
226C4⋊Dic47C2413yes—
227C4⋊Dic47C2411yescharacteristic
228Dic47⋊C4—23no—
229Dic47⋊C4—21no—
230C4×Dic47—21no—
231C4×Dic47—23no—
232C4×Dic47—23no—
233#10625329—26no—
234#10625329—26no—
235C2×C47⋊C8—26no—
236C2×C47⋊C8—26no—
237C2×C47⋊C8C2411yescharacteristic
238C2×C47⋊C8C2413yes—
239C2×C47⋊C8C2413yes—
240C2×C47⋊C8C2411yescharacteristic
241C23×C47C22×D413yes—
242C22×D47—23no—
243C22×D47—212no—
244C22×D47—83no—
245C22×D47—83no—
246Q8×C47—83no—
247Q8×C47—83no—
248Q8×C47C22⋊Q814yes—
249D4×C47C22⋊Q8112yes—
250C2×C4×C47—21no—
251C2×C4×C47—21no—
252C2×C4×C47—23no—
253C2×C4×C47—23no—
254C2×C4×C47—43no—
255C2×C4×C47—43no—
256C2×C4×C47—83no—
257C2×C4×C47—83no—
258C2×C4×C47C22×D413yes—
259C2×C4×C47C22×D411yescharacteristic
260C2×C4×C47C22×D411yescharacteristic
261C2×C4×C47C22×D413yes—
262C2×C4×C47C22×D413yes—
263C2×C4×C47C22×Q816yes—
264C2×C4×C47C2×#4116yes—
265C2×C4×C47C2511yescharacteristic
266C22⋊D47—212no—
267C2×Dic47—21no—
268C2×Dic47—21no—
269C2×Dic47—23no—
270C2×Dic47—83no—
271D188—212no—
272D188—23no—
273D188C22×Q816yes—
274C4×D47—212no—
275C4×D47—46no—
276C4×D47—83no—
277C4×D47—86no—
278Dic94—21no—
279Dic94C22×Q812yes—
280C47⋊C8—21no—
281C47⋊C8—21no—
282C47⋊C8—23no—
283C47⋊C8—23no—
284#60496—471nomaximal, Sylow
285C22×C47—23no—
286C22×C47—23no—
287C22×C47C2×C22⋊Q816yes—
288C22×C47C23×D411yescharacteristic
289C2×D47—26no—
290C2×D47—43no—
291C2×D47—43no—
292C2×D47—812no—
293C4×C47—23no—
294C4×C47—21no—
295C4×C47—21no—
296C4×C47—23no—
297C4×C47—43no—
298C4×C47—41no—
299C4×C47—43no—
300C4×C47—41no—
301C4×C47C2×C22⋊Q816yes—
302C4×C47C23×Q811yescharacteristic
303C4×C47C22×#4111yescharacteristic
304Dic47—22no—
305Dic47—41no—
306Dic47—43no—
307#3504D4711yescharacteristic, complemented
308C4×H(5,𝔽2)—471no—
309#3084—473no—
310#3083—473no—
311#3081—473no—
312#3081—473no—
313#3080—473no—
314#3077—473no—
315#3076—473no—
316C2×#478—473no—
317#2913—471no—
318#2912—471no—
319#2906—471no—
320#2897—471no—
321#2897—471no—
322C2×C47—26no—
323C2×C47C22×C22⋊Q811yescharacteristic
324C2×C47#358411yescharacteristic
325C2×C47#358511yescharacteristic
326D47—46no—
327C2×H(5,𝔽2)—471no—
328C2×#131—471no—
329C2×#131—471no—
330#557—21no—
331#557—21no—
332#554—23no—
333#554—23no—
334#544C2×D4713yescomplemented
335#544C2×D4713yescomplemented
336#533C2×D4713yescomplemented
337C2×#125C2×D4713yescomplemented
338#520—471no—
339#520—473no—
340#520—941no—
341#520C2×D4711yescharacteristic
342#520C2×D4711yescharacteristic
343#518—473no—
344#518—943no—
345C4×#41—473no—
346C4×#41—941no—
347C4×#41C2×D4711yescharacteristic
348C2×C4×D4—473no—
349C2×C4×D4—943no—
350#479—471no—
351#479—473no—
352#479—476no—
353#478—473no—
354#478—476no—
355#478—476no—
356#478—476no—
357#478—473no—
358#478—476no—
359#477—473no—
360#476—473no—
361#476—476no—
362#475—473no—
363#474—473no—
364#474—476no—
365#428—473no—
366#428—473no—
367#428—473no—
368#428—473no—
369#427—473no—
370#427—473no—
371C2×#107—473no—
372C2×#106—473no—
373#424—473no—
374#424—473no—
375#423—473no—
376C2×C4⋊C8—473no—
377#418—473no—
378#418—473no—
379#417—473no—
380#417—473no—
381#416—473no—
382#416—473no—
383C2×#102—473no—
384C2×#102—473no—
385C47#6049611yescharacteristic, complemented, Sylow
386H(5,𝔽2)—478no—
387H(5,𝔽2)—944no—
388C2×#41—473no—
389C2×#41—941no—
390C2×#41C22×D4711yescharacteristic
391C22×D4—473no—
392C22×D4—943no—
393C23×C4—943no—
394#131—944no—
395#131—944no—
396C2×M16—473no—
397C2×M16—473no—
398C22×C8—473no—
399C22×C8—473no—
400C4⋊Q8—23no—
401C4⋊Q8—23no—
402C4⋊Q8C22×D4713yes—
403C4⋊Q8C22×D4713yes—
404#125—23no—
405#125—23no—
406#125—23no—
407#125—23no—
408#125C22×D4713yes—
409#125C22×D4716yes—
410#125C22×D4711yescharacteristic
411C22⋊Q8—23no—
412C22⋊Q8—23no—
413C22⋊Q8—26no—
414C22⋊Q8—26no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—23no—
418C4×Q8—23no—
419C4×Q8—471no—
420C4×Q8—473no—
421C4×Q8—941no—
422C4×Q8C22⋊D4711yescharacteristic
423C4×Q8C22⋊D4711yescharacteristic
424C4×D4—23no—
425C4×D4—23no—
426C4×D4—473no—
427C4×D4—476no—
428C4×D4—473no—
429C4×D4—943no—
430C4×D4—943no—
431C4×D4—943no—
432C4×D4—943no—
433C4×D4—943no—
434C4×D4—946no—
435C4×D4—943no—
436C4×D4—943no—
437C4×D4C22⋊D4713yes—
438C4×D4C22⋊D4713yes—
439#117—473no—
440#117—943no—
441#117—943no—
442#117C22×D4713yes—
443#117C22×D4713yes—
444#117C22×D4713yes—
445C2×C4⋊C4—23no—
446C2×C4⋊C4—23no—
447C2×C4⋊C4—473no—
448C2×C4⋊C4—943no—
449C2×C4⋊C4C22×D4713yes—
450C2×C4⋊C4C22×D4713yes—
451C2×C22⋊C4—943no—
452C2×C22⋊C4—946no—
453C2×C42—943no—
454C2×C42C22×D4713yes—
455#107—473no—
456#107—476no—
457#107—471no—
458#107—473no—
459#107—473no—
460#106—473no—
461#106—476no—
462#106—471no—
463#106—473no—
464#106—473no—
465C4⋊C8—471no—
466C4⋊C8—473no—
467C4⋊C8—473no—
468C4⋊C8—473no—
469C4⋊C8—476no—
470#103—471no—
471#103—471no—
472#103—473no—
473#103—473no—
474#102—473no—
475#102—473no—
476#102—476no—
477#102—476no—
478#102—473no—
479#102—473no—
480C24—943no—
481#41—4712no—
482#41—948no—
483#41—946no—
484#41Q8×D4714yes—
485#41#1062535414yes—
486C2×Q8—23no—
487C2×Q8—23no—
488C2×Q8—471no—
489C2×Q8C2×C22⋊D4711yescharacteristic
490C2×D4—4712no—
491C2×D4—473no—
492C2×D4—943no—
493C2×D4—9424no—
494C2×D4—943no—
495C2×D4—946no—
496C2×D4C2×C22⋊D4713yes—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—943no—
500C22×C4—943no—
501C22×C4—943no—
502C22×C4—946no—
503C22×C4—943no—
504C22×C4—3766no—
505C22×C4C2×C22⋊D4713yes—
506C22×C4C2×C22⋊D4713yes—
507C22×C4C23×D4713yes—
508M16—946no—
509M16—946no—
510C2×C8—471no—
511C2×C8—473no—
512C2×C8—473no—
513C2×C8—471no—
514C2×C8—946no—
515C2×C8—946no—
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—26no—
519C4⋊C4—26no—
520C4⋊C4—23no—
521C4⋊C4—23no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—23no—
525C4⋊C4—23no—
526C4⋊C4—473no—
527C4⋊C4—471no—
528C4⋊C4—943no—
529C4⋊C4—941no—
530C4⋊C4—943no—
531C4⋊C4C2×C22⋊D4713yes—
532C4⋊C4C2×C22⋊D4713yes—
533C4⋊C4C23×D4713yes—
534C4⋊C4C23×D4711yescharacteristic
535C4⋊C4C23×D4713yes—
536C4⋊C4C23×D4711yescharacteristic
537C22⋊C4—23no—
538C22⋊C4—23no—
539C22⋊C4—943no—
540C22⋊C4—943no—
541C22⋊C4—946no—
542C22⋊C4—943no—
543C22⋊C4C2×C22⋊D4716yes—
544C42—23no—
545C42—23no—
546C42—941no—
547C42—943no—
548C42—943no—
549C42C2×C22⋊D4713yes—
550C42C2×C22⋊D4713yes—
551C42C23×D4711yescharacteristic
552C42C23×D4713yes—
553C23—9412no—
554C23—943no—
555C23—3763no—
556C23—3763no—
557C23C22×C22⋊D4713yes—
558Q8—83no—
559Q8—83no—
560Q8—472no—
561Q8—941no—
562Q8#1309400014yes—
563D4—476no—
564D4—9412no—
565D4—9412no—
566D4—943no—
567D4#13094000112yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—23no—
571C2×C4—23no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—83no—
575C2×C4—83no—
576C2×C4—941no—
577C2×C4—9412no—
578C2×C4—941no—
579C2×C4—943no—
580C2×C4—3763no—
581C2×C4—3766no—
582C2×C4—3763no—
583C2×C4—1886no—
584C2×C4C22×C22⋊D4711yescharacteristic
585C2×C4C22×C22⋊D4713yes—
586C2×C4C22×C22⋊D4711yescharacteristic
587C2×C4C22×C22⋊D4713yes—
588C2×C4C22×C22⋊D4713yes—
589C2×C4C2×Q8×D4716yes—
590C2×C4C2×#1062535416yes—
591C2×C4C24×D4711yescharacteristic
592C8—941no—
593C8—941no—
594C8—943no—
595C8—943no—
596C22—23no—
597C22—23no—
598C22—946no—
599C22—37612no—
600C22—1883no—
601C22—1883no—
602C22C2×#1309400016yes—
603C22C23×C22⋊D4711yescharacteristic
604C4—21no—
605C4—23no—
606C4—21no—
607C4—23no—
608C4—43no—
609C4—41no—
610C4—43no—
611C4—41no—
612C4—942no—
613C4—1881no—
614C4—1883no—
615C4C2×#1309400016yes—
616C4C22×Q8×D4711yescharacteristic
617C4C22×#1062535411yescharacteristic
618C2—26no—
619C2—1886no—
620C2C22×#1309400011yescharacteristic
621C2#2025815911yescharacteristic
622C2#2025816711yescharacteristic
subgroupG/H
centerC22C23×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#3504C2
socleC22×C47C23×D4
prime radicalC2×#13094070C22
hypercenter#3504D47
hypocenterC47#60496
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3504

Complements

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

NH ≅ G/Ntype
2. #20253246C2semidirect
3. #20253235C2semidirect
4. #20253213C2semidirect
5. #20253213C2semidirect
6. #20253206C2semidirect
7. #20253181C2semidirect
8. #20253170C2semidirect
9. C2×#15689420C2semidirect
10. #20251986C2semidirect
11. #20251981C2semidirect
12. #20251956C2semidirect
15. C47×#3504C2semidirect
28. #15689427C22semidirect
29. #15689427C22semidirect
30. #15689427C22semidirect
31. #15689420C22semidirect
32. #15689420C22semidirect
33. #15689420C22semidirect
34. #15689420C22semidirect
35. #15689420C22semidirect
36. #15689420C22semidirect
37. #15689415C22semidirect
38. #15689410C22semidirect
39. #15689410C22semidirect
40. #15689405C22semidirect
41. #15689400C22semidirect
42. #15689400C22semidirect
43. #15689166C22semidirect
44. #15689166C22semidirect
45. #15689163C22semidirect
46. #15689163C22semidirect
47. #15689159C22semidirect
48. #15689159C22semidirect
49. C2×#13093957C22semidirect
50. C2×#13093955C22semidirect
51. #15689150C22semidirect
52. #15689150C22semidirect
53. #15689147C22semidirect
54. C2×C4⋊(C47⋊C8)C22semidirect
67. C47×#544C22semidirect
68. C47×#544C22semidirect
69. C47×#533C22semidirect
70. C2×#125×C47C22semidirect
75. #13094070Q8semidirect
307. #3504D47semidirect
334. #544C2×D47semidirect
335. #544C2×D47semidirect
336. #533C2×D47semidirect
337. C2×#125C2×D47semidirect
385. C47#60496semidirect

Maximal subgroups

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

subgroupindex
C4×#130940702
#202532462
#202532352
#202532132
#202532062
#202531812
#202531702
C2×#156894202
#202519862
#202519812
#202519562
#202519112
C47×#35042
#6049647