Finite Group Databasebeta

← #26375424

Subgroups — #26375424

Order 12032.

2070 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689736C211yesmaximal, characteristic
2#20246191C214yesmaximal, complemented
3#20246162C214yesmaximal
4#20246107C211yesmaximal, characteristic, complemented
5#20246071C212yesmaximal
6#20246035C212yesmaximal, complemented
7#20245996C211yesmaximal, characteristic
8#20245955C211yesmaximal, characteristic, complemented
9#125×Dic47C211yesmaximal, characteristic, complemented
10#20245947C214yesmaximal, complemented
11#20245938C214yesmaximal
12#20245918C212yesmaximal
13C4×#13093989C212yesmaximal, complemented
14#20245727C211yesmaximal, characteristic
15C47×#2330C211yesmaximal, characteristic, complemented
16#15689736C4116yescomplemented
17C2×#13094024C2211yescharacteristic
18C2×#13094019C2214yes—
19C2×#13094015C2214yes—
20C2×#13094004C2212yes—
21C2×C4×Dic94C2212yes—
22C2×#13093989C2212yes—
23C2×#13093989C2212yes—
24C2×#13093981C2211yescharacteristic
25C2×#13093981C2211yescharacteristic
26C2×C42×D47C2211yescharacteristic
27#15689029C2212yes—
28#15689028C2212yes—
29#15689022C2212yes—
30#15689021C2214yes—
31#15689019C2214yes—
32#15689015C2214yes—
33#15689014C2214yes—
34#15689013C2214yes—
35#15689010C2214yes—
36#15689009C2214yes—
37#15689008C2214yes—
38#15689008C2214yes—
39#15689008C2214yes—
40#15689006C2214yes—
41#15689004C2214yes—
42#15689004C2214yes—
43#15688994C2214yes—
44#15688990C2214yes—
45#15688989C2211yescharacteristic
46#15688988C2214yes—
47#15688980C2212yes—
48C4⋊C4×Dic47C2212yes—
49C4⋊C4×Dic47C2214yes—
50C4⋊C4×Dic47C2212yes—
51C4⋊C4×Dic47C2214yes—
52C4⋊C4×Dic47C2214yes—
53C4×Dic47⋊C4C2214yes—
54#15688974C2214yes—
55#15688974C2212yes—
56#15688974C2214yes—
57#15688974C2212yes—
58#15688970C2212yes—
59#15688970C2214yes—
60#15688970C2212yes—
61#15688969C2211yescharacteristic
62C42×Dic47C2214yes—
63C42×Dic47C2211yescharacteristic
64C42×Dic47C2211yescharacteristic
65C2×#125×C47C2211yescharacteristic
66C47×#389C2212yes—
67C47×#389C2214yes—
68C47×#387C2214yes—
69C47×#384C2211yescharacteristic
70C4×C4⋊C4×C47C2212yes—
71C47×#376C2211yescharacteristic
72C22×Dic94C2312yes—
73C22×C4×D47C2311yescharacteristic
74C22×C4×D47C2311yescharacteristic
75C22×C4×D47C2311yescharacteristic
76#13094024C2×C418yescomplemented
77#13094019C2×C4132yes—
78#13094015C2×C4132yescomplemented
79#13094004C2×C4116yes—
80C4×Dic94—28no—
81#13093989—28no—
82#13093989C2×C4116yescomplemented
83#13093981—24no—
84#13093981C2×C418yes—
85C2×C4⋊Dic47C2312yes—
86C2×C4⋊Dic47C2314yes—
87C2×C4⋊Dic47C2314yes—
88C2×C4⋊Dic47C2314yes—
89C2×Dic47⋊C4C2314yes—
90C2×Dic47⋊C4C2312yes—
91C2×Dic47⋊C4C2314yes—
92C2×Dic47⋊C4C2312yes—
93C2×Dic47⋊C4C2314yes—
94C2×(C2×D47)⋊C4C2312yes—
95C2×(C2×D47)⋊C4C2314yes—
96C2×(C2×D47)⋊C4C2312yes—
97C2×(C2×D47)⋊C4C2314yes—
98C2×C4×Dic47C2312yes—
99C2×C4×Dic47C2311yescharacteristic
100C2×C4×Dic47C2314yes—
101C2×C4×Dic47C2312yes—
102C2×C4×Dic47C2314yes—
103C2×C4×Dic47C2314yes—
104C2×C4×Dic47C2311yescharacteristic
105C2×C4×Dic47C2311yescharacteristic
106C2×C4×Dic47C2312yes—
107C2×C4×Dic47C2312yes—
108C2×C4×Dic47C2314yes—
109C2×C4×Dic47C2311yescharacteristic
110C2×C4×Dic47C2314yes—
111C2×C4×Dic47C2311yescharacteristic
112C42×D47—24no—
113C47⋊#95C2312yes—
114C47⋊#95C2312yes—
115C47⋊#95C2314yes—
116C47⋊#95C2314yes—
117C47⋊#95C2314yes—
118C47⋊#95C2314yes—
119C47⋊#95C2314yes—
120C47⋊#95C2314yes—
121C47⋊#95C2314yes—
122C47⋊#95C2314yes—
123C47⋊#95C2314yes—
124#125×C47C2×C418yescomplemented
125C2×C4⋊C4×C47C2312yes—
126C2×C4⋊C4×C47C2314yes—
127C2×C4⋊C4×C47C2314yes—
128C2×C4⋊C4×C47C2314yes—
129C2×C4⋊C4×C47C2312yes—
130C2×C42×C47C2311yescharacteristic
131C2×C42×C47C2314yes—
132C2×C42×C47C2311yescharacteristic
133C2×C42×C47C2311yescharacteristic
134#95×C47C2312yes—
135#95×C47C2312yes—
136#95×C47C2314yes—
137#95×C47C2314yes—
138C23×D47C2411yescharacteristic
139C22×C4×C47C2411yescharacteristic
140C22×C4×C47C2414yes—
141C22×C4×C47C2411yescharacteristic
142C22×C4×C47C2412yes—
143C22×C4×C47C2414yes—
144C22×C4×C47C2412yes—
145C22×C4×C47C2411yescharacteristic
146C22×Dic47C2414yes—
147C22×Dic47C2412yes—
148C22×Dic47C2412yes—
149C22×Dic47C2414yes—
150C22×Dic47C2411yescharacteristic
151C22×Dic47C2411yescharacteristic
152C22×Dic47C2411yescharacteristic
153C2×C4×D47—22no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—24no—
158C2×C4×D47—42no—
159C2×C4×D47C22×C414yes—
160C2×Dic94—24no—
161C2×Dic94—24no—
162C2×Dic94C22×C418yes—
163C4⋊C4×C47—24no—
164C4⋊C4×C47—44no—
165C4⋊C4×C47—44no—
166C4⋊C4×C47C22×C418yes—
167C4⋊C4×C47C22×C4116yes—
168C42×C47—22no—
169C42×C47—44no—
170C42×C47C22×C414yes—
171C42×C47#4114yes—
172(C2×D47)⋊C4—24no—
173(C2×D47)⋊C4—44no—
174(C2×D47)⋊C4C22×C418yes—
175(C2×D47)⋊C4C22×C4116yes—
176C4⋊Dic47—24no—
177C4⋊Dic47—44no—
178C4⋊Dic47C22×C4116yes—
179C4⋊Dic47C22×C4116yes—
180Dic47⋊C4—24no—
181Dic47⋊C4—28no—
182Dic47⋊C4—28no—
183Dic47⋊C4C22×C418yes—
184Dic47⋊C4C22×C4116yes—
185C4×Dic47—22no—
186C4×Dic47—28no—
187C4×Dic47—22no—
188C4×Dic47—22no—
189C4×Dic47—28no—
190C4×Dic47—24no—
191C4×Dic47—28no—
192C4×Dic47—24no—
193C4×Dic47—42no—
194C4×Dic47—44no—
195C4×Dic47—42no—
196C4×Dic47C22×C414yes—
197C4×Dic47C22×C4116yes—
198C4×Dic47#4114yes—
199C23×C47C2511yescharacteristic
200C22×D47—21no—
201C22×D47—21no—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47C23×C412yes—
205C2×C4×C47—22no—
206C2×C4×C47—24no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—22no—
215C2×C4×C47—24no—
216C2×C4×C47—24no—
217C2×C4×C47—22no—
218C2×C4×C47C23×C418yes—
219C2×C4×C47C23×C414yes—
220C2×C4×C47C23×C412yes—
221C2×C4×C47C2×#4112yes—
222C2×C4×C47C2×#4114yes—
223C2×C4×C47C2×#4112yes—
224C2×Dic47—21no—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—21no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—24no—
232C2×Dic47—22no—
233C2×Dic47—24no—
234C2×Dic47—22no—
235C2×Dic47—22no—
236C2×Dic47—24no—
237C2×Dic47C23×C414yes—
238C2×Dic47C23×C418yes—
239C2×Dic47C23×C412yes—
240C2×Dic47C2×#4112yes—
241C2×Dic47C2×#4112yes—
242C2×Dic472−1+414yes—
243C4×D47—28no—
244C4×D47—44no—
245C4×D47—44no—
246Dic94—216no—
247C4×#541—471nomaximal, Sylow
248C22×C47C24×C411yescharacteristic
249C22×C47C22×#4111yescharacteristic
250C22×C47C22×#4111yescharacteristic
251C22×C47C2×2−1+412yes—
252C22×C47#58512yes—
253C2×D47—22no—
254C2×D47—24no—
255C2×D47—41no—
256C2×D47—41no—
257C2×D47—42no—
258C4×C47—24no—
259C4×C47—28no—
260C4×C47—24no—
261C4×C47—41no—
262C4×C47—41no—
263C4×C47—44no—
264C4×C47C4×#4114yes—
265Dic47—28no—
266Dic47—22no—
267Dic47—24no—
268Dic47—22no—
269Dic47—44no—
270Dic47—42no—
271Dic47#54114yescomplemented
272C2×#541—471no—
273#2336—474no—
274#2334—474no—
275#2330D4711yescharacteristic, complemented
276#2327—471no—
277#2323—472no—
278#2320—472no—
279#2314—471no—
280C4×C4⋊Q8—471no—
281C4×#125—474no—
282C4×#125—471no—
283C4×#126—474no—
284C4×C22⋊Q8—472no—
285C42×Q8—472no—
286C4×#117—471no—
287C2×C47C2×C4×#4111yescharacteristic
288C2×C47C4×2−1+412yes—
289C2×C47C2×#54111yescharacteristic
290C2×C47#345511yescharacteristic
291C2×C47#346112yes—
292D47—42no—
293#541—4716no—
294C2×C4⋊Q8—471no—
295C2×#126—474no—
296C2×#125—474no—
297C2×#125C2×D4711yescharacteristic
298C2×C22⋊Q8—472no—
299C2×C4×Q8—472no—
300C2×C4×Q8—472no—
301C2×C4×Q8—472no—
302C2×#117—471no—
303C2×#117—471no—
304C22×C42—471no—
305#391—472no—
306#391—472no—
307#389—472no—
308#389—474no—
309#389—474no—
310#389C2×D4714yes—
311#389C2×D4712yes—
312#388—474no—
313#388—474no—
314#388—474no—
315#387—474no—
316#387—474no—
317#387—474no—
318#387—474no—
319#387—474no—
320#387—474no—
321#387—474no—
322#387—474no—
323#387C2×D4714yes—
324#385—474no—
325#384C2×D4711yescharacteristic
326#383—474no—
327#382—474no—
328#382—471no—
329#380—472no—
330C4×C4⋊C4—474no—
331C4×C4⋊C4—474no—
332C4×C4⋊C4—474no—
333C4×C4⋊C4—472no—
334C4×C4⋊C4—472no—
335C4×C4⋊C4—474no—
336C4×C4⋊C4—474no—
337C4×C4⋊C4—472no—
338C4×C4⋊C4—474no—
339C4×C4⋊C4—472no—
340C4×C4⋊C4C2×D4712yes—
341C4×C22⋊C4—472no—
342C4×C22⋊C4—474no—
343C4×C22⋊C4—472no—
344#376—471no—
345#376C2×D4711yescharacteristic
346C43—474no—
347C43—471no—
348C43—471no—
349C47C4×#54111yescharacteristic, complemented, Sylow
350C22×Q8—472no—
351C23×C4—471no—
352C23×C4—471no—
353C23×C4—471no—
354C4⋊Q8—478no—
355#126—4732no—
356#125—4732no—
357#125C4×D4718yescomplemented
358C22⋊Q8—4716no—
359C4×Q8—4716no—
360C4×Q8—948no—
361C4×Q8—948no—
362#117—478no—
363#117—944no—
364C2×C4⋊C4—472no—
365C2×C4⋊C4—474no—
366C2×C4⋊C4—474no—
367C2×C4⋊C4—474no—
368C2×C4⋊C4—472no—
369C2×C4⋊C4—474no—
370C2×C4⋊C4—474no—
371C2×C4⋊C4—472no—
372C2×C4⋊C4—474no—
373C2×C4⋊C4C22×D4712yes—
374C2×C4⋊C4C22×D4714yes—
375C2×C4⋊C4C22×D4714yes—
376C2×C4⋊C4C22×D4712yes—
377C2×C4⋊C4C22×D4714yes—
378C2×C22⋊C4—472no—
379C2×C22⋊C4—474no—
380C2×C22⋊C4—472no—
381C2×C22⋊C4—474no—
382C2×C42—471no—
383C2×C42—471no—
384C2×C42—474no—
385C2×C42—471no—
386C2×C42—474no—
387C2×C42—474no—
388C2×C42—472no—
389C2×C42—474no—
390C2×C42—472no—
391C2×C42—471no—
392C2×C42—471no—
393C2×C42—472no—
394C2×C42—472no—
395C2×C42—474no—
396C2×C42—944no—
397C2×C42C22×D4711yescharacteristic
398C2×C42C22×D4714yes—
399C2×C42C22×D4711yescharacteristic
400C2×C42C22×D4711yescharacteristic
401#95—472no—
402#95—472no—
403#95—474no—
404#95—474no—
405#95—474no—
406#95—474no—
407#95—474no—
408#95—474no—
409#95—474no—
410#95—474no—
411#95—474no—
412#95C22×D4712yes—
413#95C22×D4712yes—
414#95C22×D4714yes—
415#95C22×D4714yes—
416C24—471no—
417C2×Q8—478no—
418C2×Q8—944no—
419C2×Q8—944no—
420C22×C4—471no—
421C22×C4—472no—
422C22×C4—471no—
423C22×C4—474no—
424C22×C4—471no—
425C22×C4—474no—
426C22×C4—472no—
427C22×C4—474no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—944no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—1882no—
434C22×C4C23×D4711yescharacteristic
435C22×C4C23×D4711yescharacteristic
436C22×C4C23×D4714yes—
437C22×C4C23×D4714yes—
438C22×C4C23×D4712yes—
439C22×C4C23×D4712yes—
440C22×C4C23×D4711yescharacteristic
441C4⋊C4—24no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—478no—
445C4⋊C4—4716no—
446C4⋊C4—4716no—
447C4⋊C4—4716no—
448C4⋊C4—948no—
449C4⋊C4—944no—
450C4⋊C4—948no—
451C4⋊C4—944no—
452C4⋊C4—1884no—
453C4⋊C4C2×C4×D4718yes—
454C4⋊C4C2×C4×D47116yes—
455C22⋊C4—478no—
456C22⋊C4—4716no—
457C22⋊C4—944no—
458C22⋊C4—1884no—
459C42—22no—
460C42—44no—
461C42—474no—
462C42—4716no—
463C42—474no—
464C42—944no—
465C42—948no—
466C42—944no—
467C42—942no—
468C42—942no—
469C42—948no—
470C42—942no—
471C42—948no—
472C42—1882no—
473C42—1882no—
474C42—1884no—
475C42C2×C4×D4714yes—
476C42#1062535414yes—
477C23—472no—
478C23—941no—
479C23—941no—
480C23—942no—
481C23—942no—
482C23C24×D4711yescharacteristic
483Q8—9416no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—472no—
498C2×C4—474no—
499C2×C4—474no—
500C2×C4—478no—
501C2×C4—472no—
502C2×C4—472no—
503C2×C4—941no—
504C2×C4—944no—
505C2×C4—944no—
506C2×C4—941no—
507C2×C4—942no—
508C2×C4—942no—
509C2×C4—942no—
510C2×C4—948no—
511C2×C4—942no—
512C2×C4—944no—
513C2×C4—942no—
514C2×C4—944no—
515C2×C4—944no—
516C2×C4—942no—
517C2×C4—1884no—
518C2×C4—1884no—
519C2×C4C22×C4×D4718yes—
520C2×C4C22×C4×D4714yes—
521C2×C4C22×C4×D4712yes—
522C2×C4#41×D4712yes—
523C2×C4C2×#1062535414yes—
524C2×C4C2×#1062535412yes—
525C22—942no—
526C22—944no—
527C22—1881no—
528C22—1882no—
529C22—1881no—
530C22C23×C4×D4711yescharacteristic
531C22C2×#41×D4711yescharacteristic
532C22C22×#1062535411yescharacteristic
533C222−1+4×D4712yes—
534C22#1568995812yes—
535C4—24no—
536C4—28no—
537C4—24no—
538C4—41no—
539C4—44no—
540C4—41no—
541C4—474no—
542C4—942no—
543C4—942no—
544C4—948no—
545C4—944no—
546C4—1882no—
547C4—1884no—
548C4C4×#1062535414yes—
549C2—1882no—
550C2C2×C4×#1062535411yescharacteristic
551C2#2025659912yes—
552C2#541×D4711yescharacteristic
553C2#2025677211yescharacteristic
554C2#2025682212yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2330C2
socle, cosocleC23×C47C25
prime radicalC23×D47C24
hypercenter#2330D47
hypocenterC47C4×#541
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #20246191C2semidirect
4. #20246107C2semidirect
6. #20246035C2semidirect
8. #20245955C2semidirect
9. #125×Dic47C2semidirect
10. #20245947C2semidirect
13. C4×#13093989C2semidirect
15. C47×#2330C2semidirect
16. #15689736C4semidirect
76. #13094024C2×C4semidirect
78. #13094015C2×C4semidirect
82. #13093989C2×C4semidirect
124. #125×C47C2×C4semidirect
271. Dic47#541semidirect
275. #2330D47semidirect
349. C47C4×#541semidirect
357. #125C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156897362
#202461912
#202461622
#202461072
#202460712
#202460352
#202459962
#202459552
#125×Dic472
#202459472
#202459382
#202459182
C4×#130939892
#202457272
C47×#23302
C4×#54147