Finite Group Databasebeta

← #26222205

Subgroups — #26222205

Order 12032.

1422 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094069C211yesmaximal, characteristic
2C2×#15689300C211yesmaximal, characteristic
3C4×#13094055C212yesmaximal, complemented
4#20243777C211yesmaximal, characteristic, complemented
5#20243760C212yesmaximal, complemented
6#20243613C212yesmaximal, complemented
7#20243597C211yesmaximal, characteristic, complemented
8#20243341C211yesmaximal, characteristic
9#20243199C212yesmaximal, complemented
10C4×#13093941C211yesmaximal, characteristic, complemented
11C47×#2326C211yesmaximal, characteristic, complemented
12C2×#13094069C2211yescharacteristic
13C2×#13094055C2212yes—
14#15689604C2212yes—
15#15689602—22no—
16#15689598C2211yescharacteristic
17#15689596—22no—
18#15689595—22no—
19C4×#10625350C2212yes—
20C4×#10625352—22no—
21C2×C4×C22⋊D47—22no—
22C4×D4×D47—22no—
23#15689300C418yescomplemented
24#15689228—22no—
25#15689221—22no—
26C4×#10625337—22no—
27C4×#10625336—22no—
28#15689163C2212yescomplemented
29C2×#13093952C2212yes—
30C2×#13093952C2212yes—
31#15689132C2212yescomplemented
32#15689122C2212yes—
33C2×#13093941C2211yescharacteristic
34C2×#13093941C2211yescharacteristic
35C4×#10625329C2212yescomplemented
36#15688892C2212yescomplemented
37#15688883C2212yescomplemented
38#15688883C2212yescomplemented
39C2×#124×C47C2211yescharacteristic
40C2×C4×Q8×C47C2211yescharacteristic, complemented
41C2×C4×D4×C47C2211yescharacteristic
42C47×#391C2211yescharacteristic, complemented
43C47×#390C2211yescharacteristic
44C47×#388—22no—
45C47×#385—22no—
46C47×#384C2211yescharacteristic, complemented
47C47×#376C2211yescharacteristic, complemented
48#13094069—22no—
49#13094069—24no—
50#13094069C2×C414yes—
51C2×#10625350C2312yes—
52C2×#10625352—22no—
53C22×C22⋊D47—22no—
54C2×D4×D47—22no—
55C22×C4×D47—22no—
56#13094055—24no—
57#13094055C2×C418yescomplemented
58C2×#10625337—22no—
59C2×#10625336—22no—
60C2×#10625329C2312yes—
61C2×#10625329C2312yes—
62C4×Dic94D412yes—
63C4×Dic94D412yes—
64#13093989—22no—
65C4×D188D412yes—
66C4×D188D412yes—
67D4×Dic47—22no—
68D4×Dic47—22no—
69#13093986—22no—
70C4×C22⋊D47—24no—
71C4×C22⋊D47—24no—
72C4×C22⋊D47—22no—
73C4×C22⋊D47—24no—
74C4×C22⋊D47—22no—
75(C22⋊D47)⋊C4—24no—
76(C22⋊D47)⋊C4—24no—
77#13093983C2312yes—
78#13093982—22no—
79#13093981—22no—
80#13093980—24no—
81C2×Dic47⋊C4—22no—
82C4⋊C4×D47—22no—
83C2×C22⋊Dic47—22no—
84C2×(C2×D47)⋊C4—22no—
85C22⋊C4×D47—24no—
86C2×C4×Dic47—22no—
87C42×D47—22no—
88#13093957—22no—
89#13093955—22no—
90#13093952—24no—
91#13093952C2×C418yescomplemented
92#13093948—22no—
93#13093948—22no—
94#13093947—22no—
95#13093944—22no—
96#13093941—22no—
97#13093941C2×C414yescomplemented
98#13093935—22no—
99C4×C47⋊C8—22no—
100C22×Q8×C47C2311yescharacteristic
101C22×D4×C47C2311yescharacteristic
102C23×C4×C47D412yes—
103#124×C47—22no—
104#124×C47C2×C414yescomplemented
105C4×Q8×C47—22no—
106C4×Q8×C47—22no—
107C4×D4×C47—22no—
108C4×D4×C47—22no—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—22no—
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C22⋊C4×C47—22no—
114C2×C22⋊C4×C47—22no—
115C2×C22⋊C4×C47D412yes—
116C2×C42×C47—21no—
117C2×C42×C47C2311yescharacteristic
118C2×C42×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—22no—
123C24×C47C2×D412yes—
124C23×D47—22no—
125C2×Q8×C47—21no—
126C2×Q8×C47—22no—
127C2×Q8×C47—22no—
128C2×Q8×C47C22×C412yescomplemented
129C2×D4×C47—22no—
130C2×D4×C47—22no—
131C2×D4×C47C22×C412yes—
132C2×D4×C47#4112yes—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47—82no—
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C2×C22⋊D47—24no—
145C2×C22⋊D47—28no—
146C2×C22⋊D47—28no—
147C2×C22⋊D47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151#10625352—28no—
152#10625352—28no—
153D4×D47—28no—
154D4×D47—28no—
155#10625350—22no—
156#10625350—22no—
157#10625350C22×C414yes—
158#10625350#4114yes—
159C2×D188C2×D412yes—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×C4×D47—24no—
163C2×C4×D47—22no—
164C2×C4×D47—84no—
165C2×Dic94C2×D412yes—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—41no—
169C4⋊C4×C47—41no—
170C4⋊C4×C47—82no—
171C22⋊C4×C47—28no—
172C22⋊C4×C47—24no—
173C22⋊C4×C47—82no—
174C42×C47—21no—
175C42×C47—22no—
176C42×C47—42no—
177C42×C47C22×C412yescomplemented
178C42×C47C2×D412yes—
179C42×C47C2×D412yes—
180C22⋊Dic47—24no—
181C22⋊Dic47—22no—
182#10625337—28no—
183#10625336—28no—
184(C2×D47)⋊C4—24no—
185(C2×D47)⋊C4—22no—
186(C2×D47)⋊C4—22no—
187C4⋊Dic47C2×D412yes—
188C4⋊Dic47C2×D412yes—
189Dic47⋊C4—24no—
190Dic47⋊C4—22no—
191Dic47⋊C4—22no—
192C4×Dic47—24no—
193C4×Dic47—22no—
194C4×Dic47—22no—
195C4×Dic47—22no—
196#10625329—22no—
197#10625329—22no—
198#10625329—22no—
199#10625329C22×C414yescomplemented
200C2×C47⋊C8—22no—
201C2×C47⋊C8—22no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—42no—
205C23×C47—42no—
206C23×C47C4×D414yes—
207C23×C47C22×D411yescharacteristic
208C22×D47—22no—
209C22×D47—28no—
210C22×D47—82no—
211C22×D47—82no—
212Q8×C47—24no—
213Q8×C47—42no—
214D4×C47—24no—
215D4×C47—24no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—44no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—42no—
227C2×C4×C47—41no—
228C2×C4×C47—42no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—82no—
233C2×C4×C47#12011yescharacteristic
234C2×C4×C47#12012yes—
235C2×C4×C47#12011yescharacteristic
236C2×C4×C47C23×C411yescharacteristic
237C2×C4×C47C22×D412yes—
238C2×C4×C47C2×#4112yes—
239C2×C4×C47C2×#4111yescharacteristic
240C22⋊D47—216no—
241C22⋊D47—24no—
242C22⋊D47—24no—
243C2×Dic47—28no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—82no—
248C2×Dic47—82no—
249D188—22no—
250D188C4×D414yes—
251C4×D47—24no—
252C4×D47—24no—
253C4×D47—44no—
254C4×D47—82no—
255C4×D47—84no—
256Dic94—22no—
257Dic94C4×D414yes—
258C47⋊C8—24no—
259C47⋊C8—42no—
260#21474—471nomaximal, Sylow
261C22×C47—21no—
262C22×C47—21no—
263C22×C47—24no—
264C22×C47—44no—
265C22×C47—42no—
266C22×C47—42no—
267C22×C47—82no—
268C22×C47C2×C4×D411yescharacteristic
269C22×C47C2×#12011yescharacteristic
270C22×C47#52511yescharacteristic
271C2×D47—24no—
272C2×D47—42no—
273C2×D47—42no—
274C2×D47—88no—
275C4×C47—21no—
276C4×C47—21no—
277C4×C47—22no—
278C4×C47—42no—
279C4×C47—42no—
280C4×C47—42no—
281C4×C47—81no—
282C4×C47—81no—
283C4×C47C2×C4×D412yes—
284C4×C47#52512yes—
285Dic47—24no—
286Dic47—42no—
287Dic47—42no—
288C4×H(5,𝔽2)—471no—
289C2×#455—471no—
290C4×#137—472no—
291#2326D4711yescharacteristic, complemented
292#1885—471no—
293#1882—472no—
294#1859—472no—
295#1856—471no—
296#1792—471no—
297#1759—472no—
298C4×#100—471no—
299C2×C47—21no—
300C2×C47—21no—
301C2×C47—44no—
302C2×C47C4×#12011yescharacteristic
303C2×C47#301911yescharacteristic
304C2×C47#302211yescharacteristic
305D47—44no—
306C2×H(5,𝔽2)—471no—
307C2×#137—472no—
308C2×#124C2×D4711yescharacteristic
309#520—472no—
310#520—942no—
311#518—471no—
312#518—942no—
313#518—942no—
314C4×#41—472no—
315C4×#41—942no—
316C2×C4×Q8C2×D4711yescharacteristic, complemented
317C2×C4×D4—942no—
318C2×C4×D4—942no—
319C2×C4×D4C2×D4711yescharacteristic
320#455—478no—
321#441—942no—
322#440—942no—
323C4×Q16—942no—
324C4×SD16—942no—
325#428—472no—
326C2×#104—472no—
327C2×#104—472no—
328#419—472no—
329#417—472no—
330C2×#100—471no—
331C2×#100—471no—
332C4×M16—472no—
333#391C2×D4711yescharacteristic, complemented
334#390C2×D4711yescharacteristic
335#388—22no—
336#385—22no—
337#384C2×D4711yescharacteristic, complemented
338#376C2×D4711yescharacteristic, complemented
339#343—472no—
340#339—472no—
341#339—472no—
342C47#2147411yescharacteristic, complemented, Sylow
343H(5,𝔽2)—474no—
344H(5,𝔽2)—944no—
345H(5,𝔽2)—942no—
346C2×#41—472no—
347C2×#41—942no—
348C22×Q8C22×D4711yescharacteristic
349C22×D4—942no—
350C22×D4—942no—
351C22×D4C22×D4711yescharacteristic
352C23×C4—942no—
353C23×C4C22⋊D4712yes—
354#137—478no—
355#137—944no—
356C2×Q16—942no—
357C2×SD16—942no—
358C2×M16—472no—
359C2×M16—472no—
360#124—22no—
361#124C4×D4714yescomplemented
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—472no—
365C4×Q8—472no—
366C4×Q8—942no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—472no—
370C4×D4—472no—
371C4×D4—944no—
372C4×D4—944no—
373C4×D4—942no—
374C4×D4—942no—
375C4×D4—942no—
376C4×D4—944no—
377C4×D4—944no—
378C4×D4—942no—
379C4×D4—944no—
380C4×D4—942no—
381#117—472no—
382#117—944no—
383#117—942no—
384#117—942no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4—942no—
389C2×C4⋊C4C22×D4711yescharacteristic
390C2×C4⋊C4C22×D4711yescharacteristic
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—944no—
394C2×C22⋊C4—942no—
395C2×C22⋊C4—942no—
396C2×C22⋊C4C22⋊D4712yes—
397C2×C42—21no—
398C2×C42—942no—
399C2×C42—942no—
400C2×C42C22×D4711yescharacteristic
401C2×C42C22×D4711yescharacteristic
402C2×C42C22×D4711yescharacteristic
403#107—942no—
404#106—942no—
405#104—478no—
406#104—944no—
407#103—942no—
408#103—942no—
409#103—942no—
410#102—942no—
411#100—474no—
412#100—942no—
413#97—942no—
414C4×C8—942no—
415#95—21no—
416#95—21no—
417#95—22no—
418C24—942no—
419C24C2×C22⋊D4712yes—
420#41—474no—
421#41—474no—
422#41—948no—
423#41—948no—
424#41—942no—
425#41—942no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—472no—
430C2×Q8C2×C4×D4712yescomplemented
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—472no—
434C2×D4—948no—
435C2×D4—948no—
436C2×D4—944no—
437C2×D4—948no—
438C2×D4—948no—
439C2×D4—942no—
440C2×D4C2×C4×D4712yes—
441C2×D4#1062535012yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—82no—
449C22×C4—944no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—3764no—
457C22×C4C2×C22⋊D4712yes—
458C22×C4C2×C22⋊D4711yescharacteristic
459C22×C4C2×C22⋊D4711yescharacteristic
460C22×C4C23×D4711yescharacteristic
461Q16—948no—
462SD16—948no—
463M16—474no—
464M16—942no—
465M16—942no—
466M16—942no—
467C2×C8—942no—
468C2×C8—942no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—82no—
474C4⋊C4—472no—
475C4⋊C4—472no—
476C4⋊C4—944no—
477C4⋊C4—942no—
478C4⋊C4—942no—
479C22⋊C4—28no—
480C22⋊C4—24no—
481C22⋊C4—82no—
482C22⋊C4—944no—
483C22⋊C4—944no—
484C22⋊C4—942no—
485C22⋊C4—942no—
486C22⋊C4—942no—
487C42—21no—
488C42—22no—
489C42—42no—
490C42—944no—
491C42—942no—
492C42—942no—
493C42—942no—
494C42C2×C4×D4712yescomplemented
495C42D4×D4712yes—
496C42D4×D4712yes—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—948no—
502C23—942no—
503C23—3762no—
504C23—3762no—
505C23C4×C22⋊D4714yes—
506C23C22×C22⋊D4711yescharacteristic
507Q8—24no—
508Q8—42no—
509Q8—474no—
510Q8—942no—
511D4—24no—
512D4—24no—
513D4—474no—
514D4—9416no—
515D4—944no—
516D4—944no—
517D4—942no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—82no—
535C2×C4—948no—
536C2×C4—942no—
537C2×C4—944no—
538C2×C4—944no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—3762no—
542C2×C4—3764no—
543C2×C4—3762no—
544C2×C4—3762no—
545C2×C4—1884no—
546C2×C4#1309399212yes—
547C2×C4#1309399211yescharacteristic
548C2×C4#1309399211yescharacteristic
549C2×C4C22×C4×D4711yescharacteristic
550C2×C4C2×D4×D4712yes—
551C2×C4#41×D4712yes—
552C2×C4C2×#1062535011yescharacteristic
553C8—944no—
554C8—1882no—
555C22—21no—
556C22—21no—
557C22—24no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22—944no—
563C22—3768no—
564C22—1882no—
565C22—1882no—
566C22C2×C4×C22⋊D4711yescharacteristic
567C22C2×#1309399211yescharacteristic
568C22#1568963011yescharacteristic
569C4—21no—
570C4—21no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—944no—
578C4—1882no—
579C4—1882no—
580C4C4×D4×D4712yes—
581C4#1568963212yes—
582C2—21no—
583C2—21no—
584C2—44no—
585C2—1884no—
586C2C4×#1309399211yescharacteristic
587C2#2025274411yescharacteristic
588C2#2025276611yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2326C2
socleC22×C47C2×#120
prime radicalC2×#13094069C22
cosocleC22×C4×C47C24
hypercenter#2326D47
hypocenterC47#21474
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2326

Complements

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

NH ≅ G/Ntype
3. C4×#13094055C2semidirect
4. #20243777C2semidirect
5. #20243760C2semidirect
6. #20243613C2semidirect
7. #20243597C2semidirect
9. #20243199C2semidirect
10. C4×#13093941C2semidirect
11. C47×#2326C2semidirect
23. #15689300C4semidirect
28. #15689163C22semidirect
31. #15689132C22semidirect
35. C4×#10625329C22semidirect
36. #15688892C22semidirect
37. #15688883C22semidirect
38. #15688883C22semidirect
40. C2×C4×Q8×C47C22semidirect
42. C47×#391C22semidirect
46. C47×#384C22semidirect
47. C47×#376C22semidirect
57. #13094055C2×C4semidirect
91. #13093952C2×C4semidirect
97. #13093941C2×C4semidirect
104. #124×C47C2×C4semidirect
128. C2×Q8×C47C22×C4semidirect
177. C42×C47C22×C4semidirect
199. #10625329C22×C4semidirect
291. #2326D47semidirect
316. C2×C4×Q8C2×D47semidirect
333. #391C2×D47semidirect
337. #384C2×D47semidirect
338. #376C2×D47semidirect
342. C47#21474semidirect
361. #124C4×D47semidirect
430. C2×Q8C2×C4×D47semidirect
494. C42C2×C4×D47semidirect

Maximal subgroups

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

subgroupindex
C4×#130940692
C2×#156893002
C4×#130940552
#202437772
#202437602
#202436132
#202435972
#202433412
#202431992
C4×#130939412
C47×#23262
#2147447