Finite Group Databasebeta

← #26217701

Subgroups — #26217701

Order 12032.

1926 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094049C211yesmaximal, characteristic
2C2×#15689119C211yesmaximal, characteristic, complemented
3C22×#13093944C211yesmaximal, characteristic, complemented
4#20245968C211yesmaximal, characteristic, complemented
5#20243743C214yesmaximal, complemented
6#20243736C214yesmaximal, complemented
7#20243107C211yesmaximal, characteristic, complemented
8#20243085C211yesmaximal, characteristic, complemented
9C2×C47×#520C211yesmaximal, characteristic, complemented
10C23×D188C2211yescharacteristic
11C2×#13094049—24no—
12C2×#13094049—24no—
13C2×#13094049C418yescomplemented
14C22×#10625336C2212yes—
15C22×#10625334C2212yes—
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19#15689119—24no—
20C2×#13093948C2212yescomplemented
21C2×#13093945C2212yescomplemented
22C2×#13093944—24no—
23C2×#13093944C2214yes—
24#15689025C2214yes—
25#15689024—21no—
26#15689024—21no—
27#15688978—22no—
28#15688885C2214yescomplemented
29#15688878C2214yescomplemented
30C22×#41×C47C2211yescharacteristic
31C47×#520—24no—
32C47×#520—24no—
33C2×C4×Q8×C47C2212yescomplemented
34C2×C4×D4×C47—22no—
35C2×C4×D4×C47C2212yescomplemented
36C2×#117×C47—21no—
37C2×#117×C47C2211yescharacteristic
38C22×C4⋊C4×C47—21no—
39C22×C4⋊C4×C47C2211yescharacteristic
40C24×D47—21no—
41C22×D188—22no—
42C22×D188—22no—
43C22×D188—22no—
44C22×D188—22no—
45C22×D188—21no—
46C22×D188—22no—
47C22×D188C2×C414yescomplemented
48C22×D188C2312yes—
49#13094049—232no—
50C2×#10625336—24no—
51C2×#10625336—24no—
52C2×#10625336C2×C418yescomplemented
53C2×#10625334—24no—
54C2×#10625334—24no—
55C2×#10625334C2×C418yescomplemented
56C2×#10625329—22no—
57C2×#10625329—22no—
58C2×#10625329C2×C414yescomplemented
59C22×C47⋊C8—22no—
60C22×C47⋊C8—22no—
61C22×C47⋊C8—22no—
62C22×C47⋊C8C2312yes—
63C22×C47⋊C8C2312yes—
64#13093983—22no—
65#13093983—22no—
66C2×C4⋊Dic47C2312yes—
67C2×C22⋊Dic47—21no—
68C2×(C2×D47)⋊C4—24no—
69C2×(C2×D47)⋊C4—24no—
70C2×C4×Dic47—21no—
71#13093948—24no—
72#13093945—24no—
73#13093944—28no—
74C2×#41×C47—22no—
75C2×#41×C47—22no—
76C2×#41×C47—22no—
77C2×#41×C47C2×C414yes—
78C2×#41×C47D414yes—
79C2×#41×C47D414yes—
80C22×Q8×C47C2311yescharacteristic
81C22×D4×C47—21no—
82C22×D4×C47C2311yescharacteristic
83C23×C4×C47—21no—
84C23×C4×C47—21no—
85C23×C4×C47—22no—
86C23×C4×C47C2311yescharacteristic
87C4×Q8×C47—24no—
88C4×Q8×C47—24no—
89C4×D4×C47—216no—
90C4×D4×C47—24no—
91C4×D4×C47—24no—
92#117×C47—28no—
93#117×C47—22no—
94#117×C47—22no—
95C2×C4⋊C4×C47—28no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47—22no—
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47D411yescharacteristic
101C2×C4⋊C4×C47D414yes—
102C2×C4⋊C4×C47C2312yes—
103C2×C22⋊C4×C47—22no—
104C2×C22⋊C4×C47D412yes—
105C2×C42×C47—22no—
106C2×C42×C47C2312yes—
107C24×C47—21no—
108C24×C47C2×D411yescharacteristic
109C23×D47—22no—
110C23×D47—24no—
111C23×D47—42no—
112C23×D47—42no—
113C23×D47—82no—
114C23×D47—82no—
115#41×C47—28no—
116#41×C47—28no—
117#41×C47—28no—
118#41×C47C22⋊C4116yescomplemented
119C2×Q8×C47—21no—
120C2×Q8×C47—21no—
121C2×Q8×C47—21no—
122C2×Q8×C47C22×C412yes—
123C2×Q8×C47C2×D412yes—
124C2×Q8×C47C2×D412yes—
125C2×D4×C47—24no—
126C2×D4×C47—24no—
127C2×D4×C47—24no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47C22×C412yes—
132C2×D4×C47C2×D412yes—
133C2×D4×C47C2×D412yes—
134C22×C4×C47—21no—
135C22×C4×C47—24no—
136C22×C4×C47—24no—
137C22×C4×C47—24no—
138C22×C4×C47—22no—
139C22×C4×C47—28no—
140C22×C4×C47—24no—
141C22×C4×C47—84no—
142C22×C4×C47—82no—
143C22×C4×C47C22×C412yes—
144C22×C4×C47C2×D412yes—
145C22×C4×C47C2×D412yes—
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D412yes—
150C22×C4×C47#4112yes—
151C22×C4×C47#4112yes—
152C22×C4×C47C2411yescharacteristic
153C22×Dic47—22no—
154C2×D188—28no—
155C2×D188—28no—
156C2×D188—22no—
157C2×D188—22no—
158C2×D188—22no—
159C2×D188—22no—
160C2×D188—24no—
161C2×D188—22no—
162C2×D188—42no—
163C2×D188—44no—
164C2×D188—42no—
165C2×D188—44no—
166C2×D188C22×C414yes—
167C4⋊C4×C47—28no—
168C4⋊C4×C47—22no—
169C4⋊C4×C47—24no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47C2×D414yes—
173C22⋊C4×C47—28no—
174C22⋊C4×C47—24no—
175C42×C47—28no—
176C42×C47—22no—
177C42×C47—22no—
178C22⋊Dic47—42no—
179#10625336—216no—
180#10625334—216no—
181(C2×D47)⋊C4—84no—
182(C2×D47)⋊C4—84no—
183C4⋊Dic47—22no—
184C4⋊Dic47—22no—
185C4×Dic47—42no—
186#10625329—28no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—44no—
193C2×C47⋊C8C22×C414yes—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47C4×D412yes—
202C23×C47#12012yes—
203C23×C47#12112yes—
204C23×C47C22×D411yescharacteristic
205C22×D47—24no—
206C22×D47—42no—
207C22×D47—42no—
208C22×D47—44no—
209C22×D47—44no—
210C22×D47—48no—
211C22×D47—88no—
212C22×D47—84no—
213C22×D47—84no—
214C22×D47—88no—
215Q8×C47—22no—
216Q8×C47—22no—
217Q8×C47—22no—
218Q8×C47C2×C22⋊C414yes—
219D4×C47—216no—
220D4×C47—22no—
221D4×C47—22no—
222D4×C47—22no—
223D4×C47C2×C22⋊C414yes—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—216no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—44no—
234C2×C4×C47—84no—
235C2×C4×C47—88no—
236C2×C4×C47C2×C22⋊C414yes—
237C2×C4×C47C4×D414yes—
238C2×C4×C47C4×D412yes—
239C2×C4×C47#12012yes—
240C2×C4×C47#12114yes—
241C2×C4×C47#12112yes—
242C2×C4×C47#12314yes—
243C2×C4×C47C23×C411yescharacteristic
244C2×C4×C47C22×D411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C2×#4111yescharacteristic
248C2×C4×C47C2×#4111yescharacteristic
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252D188—24no—
253D188—24no—
254D188—42no—
255D188—42no—
256D188—48no—
257C47⋊C8—24no—
258C47⋊C8—42no—
259#20932—471nomaximal, Sylow
260C22×C47—24no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—24no—
264C22×C47—24no—
265C22×C47—44no—
266C22×C47—42no—
267C22×C47—42no—
268C22×C47—44no—
269C22×C47#38614yes—
270C22×C47C2×C4×D411yescharacteristic
271C22×C47C2×#12011yescharacteristic
272C22×C47C2×#12111yescharacteristic
273C22×C47#57612yes—
274C22×C47#57712yes—
275C2×D47—48no—
276C2×D47—44no—
277C2×D47—816no—
278C2×D47—82no—
279C2×D47—82no—
280C2×D47—84no—
281C4×C47—24no—
282C4×C47—42no—
283C4×C47—44no—
284C4×C47—42no—
285C4×C47#38614yes—
286C4×C47C22×C22⋊C411yescharacteristic
287C4×C47C2×C4×D411yescharacteristic
288C4×C47C2×#12111yescharacteristic
289C4×C47C2×#12311yescharacteristic
290Dic47—82no—
291C22×AGL(1,ℤ/8)—471no—
292C2×#520D4711yescharacteristic, complemented
293C2×#416—471no—
294C22×#102—471no—
295#2309—471no—
296#1880—474no—
297#1879—474no—
298#1736—471no—
299#1729—471no—
300C2×C47—24no—
301C2×C47—44no—
302C2×C47C2×#38611yescharacteristic
303C2×C47#294612yes—
304C2×C47#300811yescharacteristic
305C2×C47#301011yescharacteristic
306C2×C47#304512yes—
307D47—84no—
308C22×#41C2×D4711yescharacteristic
309C23×D4—471no—
310C2×AGL(1,ℤ/8)—478no—
311C2×AGL(1,ℤ/8)—944no—
312C2×AGL(1,ℤ/8)—944no—
313C22×SD16—472no—
314C22×D8—472no—
315C22×M16—471no—
316C23×C8—471no—
317#520—24no—
318#520—24no—
319C2×C4×Q8C2×D4712yescomplemented
320C2×C4×D4—22no—
321C2×C4×D4C2×D4712yescomplemented
322C2×#117—21no—
323C2×#117—471no—
324C2×#117C2×D4711yescharacteristic
325C22×C4⋊C4—21no—
326C22×C4⋊C4C2×D4711yescharacteristic
327#416—944no—
328C2×#103—472no—
329C2×#102—472no—
330C2×#102—474no—
331C2×#102—944no—
332#390—474no—
333#390—941no—
334#390—941no—
335#379—942no—
336#340—474no—
337#336—474no—
338C47#2093211yescharacteristic, complemented, Sylow
339C25—941no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41C4×D4714yes—
344C2×#41D18814yes—
345C2×#41C22⋊D4714yes—
346C22×Q8C22×D4711yescharacteristic
347C22×D4—21no—
348C22×D4—474no—
349C22×D4—472no—
350C22×D4—942no—
351C22×D4—942no—
352C22×D4—942no—
353C22×D4—942no—
354C22×D4—942no—
355C22×D4—941no—
356C22×D4C22×D4711yescharacteristic
357C23×C4—21no—
358C23×C4—21no—
359C23×C4—22no—
360C23×C4C22×D4711yescharacteristic
361AGL(1,ℤ/8)—9432no—
362C2×SD16—478no—
363C2×SD16—944no—
364C2×SD16—944no—
365C2×D8—478no—
366C2×D8—944no—
367C2×D8—944no—
368C2×M16—474no—
369C2×M16—942no—
370C2×M16—942no—
371C22×C8—472no—
372C22×C8—472no—
373C22×C8—942no—
374C22×C8—942no—
375C22×C8—942no—
376C4×Q8—24no—
377C4×Q8—24no—
378C4×D4—216no—
379C4×D4—24no—
380C4×D4—24no—
381#117—28no—
382#117—22no—
383#117—22no—
384#117—942no—
385#117—942no—
386C2×C4⋊C4—28no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—472no—
391C2×C4⋊C4C22⋊D4711yescharacteristic
392C2×C4⋊C4C22⋊D4711yescharacteristic
393C2×C4⋊C4C22⋊D4714yes—
394C2×C4⋊C4C22×D4712yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—944no—
397C2×C22⋊C4—944no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4C22⋊D4712yes—
400C2×C42—22no—
401C2×C42—941no—
402C2×C42C22×D4712yes—
403#103—944no—
404#102—944no—
405#102—948no—
406C24—21no—
407C24—944no—
408C24—942no—
409C24—3762no—
410C24—3762no—
411C24—1882no—
412C24—1882no—
413C24C2×C22⋊D4711yescharacteristic
414#41—28no—
415#41—28no—
416#41—28no—
417#41(C2×D47)⋊C4116yescomplemented
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8C2×C4×D4712yes—
422C2×Q8C2×D18812yes—
423C2×Q8C2×C22⋊D4712yes—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—474no—
431C2×D4—948no—
432C2×D4—948no—
433C2×D4—944no—
434C2×D4—942no—
435C2×D4—942no—
436C2×D4—942no—
437C2×D4—942no—
438C2×D4—942no—
439C2×D4—1882no—
440C2×D4—1882no—
441C2×D4—1884no—
442C2×D4—1884no—
443C2×D4C2×C4×D4712yes—
444C2×D4C2×D18812yes—
445C2×D4C2×C22⋊D4712yes—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—84no—
454C22×C4—82no—
455C22×C4—942no—
456C22×C4C2×C4×D4712yes—
457C22×C4C2×D18812yes—
458C22×C4#1062535012yes—
459C22×C4#1062535012yes—
460C22×C4C2×C22⋊D4712yes—
461C22×C4C2×C22⋊D4712yes—
462C22×C4C2×C22⋊D4712yes—
463C22×C4C2×C22⋊D4711yescharacteristic
464C22×C4C2×C22⋊D4712yes—
465C22×C4C23×D4711yescharacteristic
466SD16—9416no—
467D8—9416no—
468M16—948no—
469C2×C8—474no—
470C2×C8—942no—
471C2×C8—942no—
472C2×C8—942no—
473C2×C8—942no—
474C2×C8—942no—
475C2×C8—1884no—
476C4⋊C4—28no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—942no—
482C4⋊C4—942no—
483C4⋊C4C2×C22⋊D4714yes—
484C22⋊C4—28no—
485C22⋊C4—24no—
486C22⋊C4—3764no—
487C22⋊C4—3764no—
488C22⋊C4—1882no—
489C42—28no—
490C42—22no—
491C42—22no—
492C42—1882no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23—944no—
501C23—3764no—
502C23—3764no—
503C23—3768no—
504C23—3768no—
505C23—1888no—
506C23—1882no—
507C23—1882no—
508C23—1884no—
509C23—1884no—
510C23C4×C22⋊D4712yes—
511C23#1309399312yes—
512C23#1309399812yes—
513C23C22×C22⋊D4711yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—22no—
517Q8C2×(C2×D47)⋊C414yes—
518D4—216no—
519D4—22no—
520D4—22no—
521D4—22no—
522D4—944no—
523D4—944no—
524D4—1888no—
525D4—1882no—
526D4—1882no—
527D4C2×(C2×D47)⋊C414yes—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—216no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—44no—
538C2×C4—84no—
539C2×C4—88no—
540C2×C4—1882no—
541C2×C4—1882no—
542C2×C4—1882no—
543C2×C4C2×(C2×D47)⋊C414yes—
544C2×C4C4×C22⋊D4712yes—
545C2×C4C4×C22⋊D4714yes—
546C2×C4#1309399312yes—
547C2×C4#1309399812yes—
548C2×C4#1309399814yes—
549C2×C4#1309400714yes—
550C2×C4C22×C4×D4711yescharacteristic
551C2×C4C22×C22⋊D4711yescharacteristic
552C2×C4C22×C22⋊D4711yescharacteristic
553C2×C4C22×D18811yescharacteristic
554C2×C4C2×#1062535011yescharacteristic
555C2×C4C2×#1062535011yescharacteristic
556C8—944no—
557C8—1882no—
558C22—22no—
559C22—24no—
560C22—22no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22—37616no—
568C22—3762no—
569C22—3762no—
570C22—3764no—
571C22—1884no—
572C22—1888no—
573C22#1568900214yes—
574C22C2×C4×C22⋊D4711yescharacteristic
575C22C2×#1309399311yescharacteristic
576C22C2×#1309399811yescharacteristic
577C22#1568992612yes—
578C22#1568993112yes—
579C4—24no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—3762no—
584C4#1568900214yes—
585C4C22×(C2×D47)⋊C411yescharacteristic
586C4C2×C4×C22⋊D4711yescharacteristic
587C4C2×#1309399811yescharacteristic
588C4C2×#1309400711yescharacteristic
589C2—24no—
590C2—44no—
591C2—3764no—
592C2C2×#1568900211yescharacteristic
593C2#2025221612yes—
594C2#2025265611yescharacteristic
595C2#2025267411yescharacteristic
596C2#2025292712yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#520C2
socleC23×C47C22×D4
prime radicalC22×#13094049C2
cosocleC22×C4×C47C24
hypercenterC2×#520D47
hypocenterC47#20932
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
2. C2×#15689119C2semidirect
3. C22×#13093944C2semidirect
4. #20245968C2semidirect
5. #20243743C2semidirect
6. #20243736C2semidirect
7. #20243107C2semidirect
8. #20243085C2semidirect
9. C2×C47×#520C2semidirect
13. C2×#13094049C4semidirect
20. C2×#13093948C22semidirect
21. C2×#13093945C22semidirect
28. #15688885C22semidirect
29. #15688878C22semidirect
33. C2×C4×Q8×C47C22semidirect
35. C2×C4×D4×C47C22semidirect
47. C22×D188C2×C4semidirect
52. C2×#10625336C2×C4semidirect
55. C2×#10625334C2×C4semidirect
58. C2×#10625329C2×C4semidirect
118. #41×C47C22⋊C4semidirect
292. C2×#520D47semidirect
319. C2×C4×Q8C2×D47semidirect
321. C2×C4×D4C2×D47semidirect
338. C47#20932semidirect
417. #41(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130940492
C2×#156891192
C22×#130939442
#202459682
#202437432
#202437362
#202431072
#202430852
C2×C47×#5202
#2093247