Finite Group Databasebeta

← #26217704

Subgroups — #26217704

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×#13094050C211yesmaximal, characteristic
2C2×#15689603C211yesmaximal, characteristic, complemented
3C2×#15689117C211yesmaximal, characteristic, complemented
4C22×#13093945C211yesmaximal, characteristic, complemented
5#20243746C214yesmaximal, complemented
6#20243739C214yesmaximal, complemented
7#20243109C211yesmaximal, characteristic, complemented
8#20243085C211yesmaximal, characteristic, complemented
9C47×#2309C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C2×#13094050—24no—
12C2×#13094050—24no—
13C2×#13094050C418yescomplemented
14C22×#10625335C2212yes—
15C22×#10625334C2212yes—
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18#15689603—24no—
19#15689603—24no—
20C2×C4×Dic94C2212yescomplemented
21C2×C4×D188C2212yescomplemented
22C2×(C22⋊D47)⋊C4—22no—
23C2×#13093982—21no—
24C22×C4⋊Dic47C2211yescharacteristic
25C2×C4⋊C4×D47—21no—
26#15689117—24no—
27C2×#13093947C2212yescomplemented
28C2×#13093945—24no—
29C2×#13093945C2214yes—
30C2×#13093944C2212yescomplemented
31#15688885C2214yescomplemented
32#15688878C2214yescomplemented
33C23×D4×C47C2211yescharacteristic
34C2×#117×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C47×#390—21no—
37C47×#390—21no—
38C47×#390C2214yes—
39C47×#379—22no—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350C2×C414yes—
44C2×#10625350D414yes—
45C2×#10625350D414yes—
46C22×Dic94C2311yescharacteristic
47C22×D188C2311yescharacteristic
48C22×C22⋊D47—21no—
49C23×Dic47—21no—
50C22×C4×D47—21no—
51C22×C4×D47—22no—
52#13094050—232no—
53C2×#10625335—24no—
54C2×#10625335—24no—
55C2×#10625335C2×C418yescomplemented
56C2×#10625334—24no—
57C2×#10625334—24no—
58C2×#10625334C2×C418yescomplemented
59C2×#10625329—22no—
60C2×#10625329—22no—
61C2×#10625329C2×C414yescomplemented
62C22×C47⋊C8—22no—
63C22×C47⋊C8—22no—
64C22×C47⋊C8—22no—
65C22×C47⋊C8C2312yes—
66C22×C47⋊C8C2312yes—
67C4×Dic94—24no—
68C4×Dic94—24no—
69C4×D188—24no—
70C4×D188—24no—
71(C22⋊D47)⋊C4—216no—
72#13093982—28no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—22no—
75C2×C4⋊Dic47D414yes—
76C2×C4⋊Dic47C2312yes—
77C2×Dic47⋊C4—22no—
78C4⋊C4×D47—28no—
79C2×(C2×D47)⋊C4—22no—
80C2×C4×Dic47—22no—
81#13093947—24no—
82#13093945—28no—
83#13093944—24no—
84C25×C47—21no—
85C22×D4×C47—22no—
86C22×D4×C47—22no—
87C22×D4×C47—22no—
88C22×D4×C47—21no—
89C22×D4×C47—22no—
90C22×D4×C47—22no—
91C22×D4×C47C2×C414yescomplemented
92C22×D4×C47C2312yes—
93C23×C4×C47C2311yescharacteristic
94#117×C47—22no—
95#117×C47—22no—
96#117×C47—22no—
97#117×C47—22no—
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47C2312yes—
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—24no—
103C2×C22⋊C4×C47—24no—
104C2×C22⋊C4×C47D412yes—
105C2×C42×C47—21no—
106C2×C42×C47C2312yes—
107C24×C47—22no—
108C24×C47—24no—
109C24×C47—42no—
110C24×C47—42no—
111C24×C47—82no—
112C24×C47—82no—
113C24×C47C2×D411yescharacteristic
114C23×D47—21no—
115C2×D4×C47—28no—
116C2×D4×C47—28no—
117C2×D4×C47—24no—
118C2×D4×C47—22no—
119C2×D4×C47—22no—
120C2×D4×C47—22no—
121C2×D4×C47—22no—
122C2×D4×C47—22no—
123C2×D4×C47—42no—
124C2×D4×C47—42no—
125C2×D4×C47—44no—
126C2×D4×C47—44no—
127C2×D4×C47C22×C414yes—
128C22×C4×C47—22no—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D412yes—
136C22×C4×C47#4112yes—
137C22×C4×C47#4112yes—
138C22×C4×C47C2411yescharacteristic
139C2×C22⋊D47—24no—
140C2×C22⋊D47—24no—
141C2×C22⋊D47—24no—
142C22×Dic47—21no—
143C22×Dic47—22no—
144C22×Dic47—24no—
145C22×Dic47—82no—
146#10625350—28no—
147#10625350—28no—
148#10625350—28no—
149#10625350C22⋊C4116yescomplemented
150C2×D188—21no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188C22×C412yes—
154C2×D188C2×D412yes—
155C2×D188C2×D412yes—
156C2×C4×D47—24no—
157C2×C4×D47—24no—
158C2×C4×D47—24no—
159C2×C4×D47—28no—
160C2×C4×D47—84no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94C22×C412yes—
165C2×Dic94C2×D412yes—
166C2×Dic94C2×D412yes—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C22⋊C4×C47—24no—
172C22⋊C4×C47—42no—
173C22⋊C4×C47—84no—
174C22⋊C4×C47—84no—
175C42×C47—22no—
176C42×C47—22no—
177C42×C47—42no—
178#10625335—216no—
179#10625334—216no—
180(C2×D47)⋊C4—28no—
181C4⋊Dic47—24no—
182C4⋊Dic47—22no—
183C4⋊Dic47C2×D414yes—
184Dic47⋊C4—28no—
185C4×Dic47—28no—
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—24no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—44no—
200C23×C47—44no—
201C23×C47—48no—
202C23×C47—88no—
203C23×C47—84no—
204C23×C47—84no—
205C23×C47—88no—
206C23×C47C4×D412yes—
207C23×C47#12012yes—
208C23×C47#12112yes—
209C23×C47C22×D411yescharacteristic
210C22×D47—22no—
211C22×D47—22no—
212C22×D47—22no—
213C22×D47—42no—
214C22×D47—42no—
215D4×C47—24no—
216D4×C47—24no—
217D4×C47—42no—
218D4×C47—42no—
219D4×C47—48no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47C2×C22⋊C414yes—
228C2×C4×C47C4×D414yes—
229C2×C4×C47C4×D412yes—
230C2×C4×C47#12012yes—
231C2×C4×C47#12112yes—
232C2×C4×C47#12114yes—
233C2×C4×C47#12314yes—
234C2×C4×C47C23×C411yescharacteristic
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C22×D411yescharacteristic
237C2×C4×C47C22×D411yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C22⋊D47—216no—
241C2×Dic47—22no—
242C2×Dic47—22no—
243C2×Dic47—22no—
244C2×Dic47—24no—
245C2×Dic47—44no—
246C2×Dic47—84no—
247D188—22no—
248D188—22no—
249D188—22no—
250D188C2×C22⋊C414yes—
251C4×D47—216no—
252C4×D47—88no—
253Dic94—22no—
254Dic94—22no—
255Dic94—22no—
256Dic94C2×C22⋊C414yes—
257C47⋊C8—24no—
258C47⋊C8—42no—
259#20932—471nomaximal, Sylow
260C22×C47—24no—
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—24no—
264C22×C47—48no—
265C22×C47—44no—
266C22×C47—816no—
267C22×C47—82no—
268C22×C47—82no—
269C22×C47—84no—
270C22×C47#38614yes—
271C22×C47C2×C4×D411yescharacteristic
272C22×C47C2×#12011yescharacteristic
273C22×C47C2×#12111yescharacteristic
274C22×C47#57612yes—
275C22×C47#57712yes—
276C2×D47—24no—
277C2×D47—44no—
278C2×D47—42no—
279C2×D47—42no—
280C2×D47—44no—
281C4×C47—42no—
282C4×C47—42no—
283C4×C47—82no—
284C4×C47#38614yes—
285C4×C47C22×C22⋊C411yescharacteristic
286C4×C47C2×C4×D411yescharacteristic
287C4×C47C2×#12111yescharacteristic
288C4×C47C2×#12311yescharacteristic
289Dic47—24no—
290Dic47—44no—
291C22×AGL(1,ℤ/8)—471no—
292C2×#520—471no—
293C2×#416—471no—
294C22×#102—471no—
295#2309D4711yescharacteristic, complemented
296#1880—474no—
297#1879—474no—
298#1736—471no—
299#1729—471no—
300C2×C47—24no—
301C2×C47—84no—
302C2×C47C2×#38611yescharacteristic
303C2×C47#294612yes—
304C2×C47#300811yescharacteristic
305C2×C47#301011yescharacteristic
306C2×C47#304512yes—
307D47—44no—
308C22×#41—471no—
309C23×D4C2×D4711yescharacteristic
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—944no—
318#520—944no—
319C2×C4×Q8—472no—
320C2×C4×D4—472no—
321C2×C4×D4—942no—
322C2×#117—941no—
323C2×#117C2×D4711yescharacteristic
324C2×#117C2×D4711yescharacteristic
325C22×C4⋊C4—471no—
326C22×C4⋊C4—941no—
327#416—944no—
328C2×#103—472no—
329C2×#102—474no—
330C2×#102—472no—
331C2×#102—944no—
332#390—21no—
333#390—21no—
334#390C2×D4714yes—
335#379—22no—
336#340—474no—
337#336—474no—
338C47#2093211yescharacteristic, complemented, Sylow
339C25—21no—
340C2×#41—474no—
341C2×#41—474no—
342C2×#41—474no—
343C2×#41—942no—
344C2×#41—942no—
345C2×#41—942no—
346C22×Q8—471no—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—21no—
353C22×D4—471no—
354C22×D4—941no—
355C22×D4C4×D4714yescomplemented
356C22×D4C22×D4712yes—
357C23×C4—941no—
358C23×C4—941no—
359C23×C4—942no—
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—944no—
377C4×Q8—944no—
378C4×D4—944no—
379C4×D4—944no—
380C4×D4—9416no—
381#117—22no—
382#117—22no—
383#117—22no—
384#117—22no—
385#117—948no—
386C2×C4⋊C4—474no—
387C2×C4⋊C4—472no—
388C2×C4⋊C4—948no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—942no—
392C2×C4⋊C4C22⋊D4711yescharacteristic
393C2×C4⋊C4C22⋊D4711yescharacteristic
394C2×C4⋊C4C22×D4712yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—942no—
399C2×C22⋊C4C22⋊D4712yes—
400C2×C42—21no—
401C2×C42—942no—
402C2×C42C22×D4712yes—
403#103—944no—
404#102—944no—
405#102—948no—
406C24—22no—
407C24—24no—
408C24—42no—
409C24—42no—
410C24—82no—
411C24—82no—
412C24—941no—
413C24C2×C22⋊D4711yescharacteristic
414#41—4716no—
415#41—948no—
416#41—948no—
417#41—948no—
418C2×Q8—472no—
419C2×Q8—472no—
420C2×Q8—472no—
421C2×Q8—941no—
422C2×Q8—941no—
423C2×Q8—941no—
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—472no—
437C2×D4—472no—
438C2×D4—472no—
439C2×D4—944no—
440C2×D4—944no—
441C2×D4—944no—
442C2×D4—941no—
443C2×D4—941no—
444C2×D4—941no—
445C2×D4C2×C4×D4714yes—
446C22×C4—22no—
447C22×C4—948no—
448C22×C4—941no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—944no—
453C22×C4—942no—
454C22×C4—3764no—
455C22×C4—3762no—
456C22×C4C2×C4×D4712yes—
457C22×C4#1062535012yes—
458C22×C4D4×D4712yes—
459C22×C4D4×D4712yes—
460C22×C4D4×D4712yes—
461C22×C4#1062535212yes—
462C22×C4C2×C22⋊D4712yes—
463C22×C4C2×C22⋊D4712yes—
464C22×C4C2×C22⋊D4711yescharacteristic
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—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—474no—
481C4⋊C4—948no—
482C4⋊C4—944no—
483C4⋊C4—942no—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—948no—
489C42—22no—
490C42—22no—
491C42—42no—
492C42—948no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—48no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—88no—
502C23—84no—
503C23—84no—
504C23—88no—
505C23—942no—
506C23—942no—
507C23—942no—
508C23—1882no—
509C23—1882no—
510C23C4×C22⋊D4712yes—
511C23#1309399212yes—
512C23#1309399512yes—
513C23C22×C22⋊D4711yescharacteristic
514Q8—474no—
515Q8—942no—
516Q8—942no—
517Q8—942no—
518D4—24no—
519D4—24no—
520D4—48no—
521D4—42no—
522D4—42no—
523D4—474no—
524D4—9416no—
525D4—942no—
526D4—942no—
527D4—942no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—942no—
536C2×C4—942no—
537C2×C4—942no—
538C2×C4—9416no—
539C2×C4—944no—
540C2×C4—3768no—
541C2×C4—3764no—
542C2×C4—1884no—
543C2×C4C22⋊C4×D4714yes—
544C2×C4(C22⋊D47)⋊C414yes—
545C2×C4C4×C22⋊D4712yes—
546C2×C4#1309399212yes—
547C2×C4#1309399512yes—
548C2×C4#1309399714yes—
549C2×C4#1309400614yes—
550C2×C4C22×C4×D4711yescharacteristic
551C2×C4C2×D4×D4711yescharacteristic
552C2×C4C2×D4×D4711yescharacteristic
553C2×C4C2×D4×D4711yescharacteristic
554C2×C4C2×#1062535211yescharacteristic
555C2×C4C2×#1062535011yescharacteristic
556C8—944no—
557C8—1882no—
558C22—22no—
559C22—24no—
560C22—22no—
561C22—24no—
562C22—44no—
563C22—48no—
564C22—816no—
565C22—82no—
566C22—82no—
567C22—84no—
568C22—944no—
569C22—1882no—
570C22—1884no—
571C22—1882no—
572C22—1884no—
573C22#1568899914yes—
574C22C2×C4×C22⋊D4711yescharacteristic
575C22C2×#1309399211yescharacteristic
576C22C2×#1309399511yescharacteristic
577C22#1568992612yes—
578C22#1568993412yes—
579C4—42no—
580C4—42no—
581C4—82no—
582C4—944no—
583C4—1884no—
584C4#1568899914yes—
585C4C2×C22⋊C4×D4711yescharacteristic
586C4C2×(C22⋊D47)⋊C411yescharacteristic
587C4C2×#1309399711yescharacteristic
588C4C2×#1309400611yescharacteristic
589C2—24no—
590C2—84no—
591C2—1884no—
592C2C2×#1568899911yescharacteristic
593C2#2025222012yes—
594C2#2025265411yescharacteristic
595C2#2025267211yescharacteristic
596C2#2025292312yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2309C2
socleC23×C47C22×D4
prime radicalC22×#13094050C2
cosocleC22×C4×C47C24
hypercenter#2309D47
hypocenterC47#20932
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. C2×#15689603C2semidirect
3. C2×#15689117C2semidirect
4. C22×#13093945C2semidirect
5. #20243746C2semidirect
6. #20243739C2semidirect
7. #20243109C2semidirect
8. #20243085C2semidirect
9. C47×#2309C2semidirect
13. C2×#13094050C4semidirect
20. C2×C4×Dic94C22semidirect
21. C2×C4×D188C22semidirect
27. C2×#13093947C22semidirect
30. C2×#13093944C22semidirect
31. #15688885C22semidirect
32. #15688878C22semidirect
55. C2×#10625335C2×C4semidirect
58. C2×#10625334C2×C4semidirect
61. C2×#10625329C2×C4semidirect
91. C22×D4×C47C2×C4semidirect
149. #10625350C22⋊C4semidirect
295. #2309D47semidirect
338. C47#20932semidirect
355. C22×D4C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940502
C2×#156896032
C2×#156891172
C22×#130939452
#202437462
#202437392
#202431092
#202430852
C47×#23092
#2093247