Finite Group Databasebeta

← #26376752

Subgroups — #26376752

Order 12032.

2094 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D47C211yesmaximal, characteristic
2#20246485C214yesmaximal, complemented
3#20246479C214yesmaximal, complemented
4#20246262C214yesmaximal
5#20246261C214yesmaximal
6#20246128C212yesmaximal, complemented
7#20246125C212yesmaximal, complemented
8#20246003C212yesmaximal
9#125×Dic47C211yesmaximal, characteristic, complemented
10C4×#13094016C211yesmaximal, characteristic, complemented
11#20245943C211yesmaximal, characteristic, complemented
12#20245794C211yesmaximal, characteristic
13#20245742C212yesmaximal
14C4×#13093981C211yesmaximal, characteristic
15C4×#125×C47C211yesmaximal, characteristic, complemented
16C2×#13094016C2211yescharacteristic
17C2×#13094015C2212yes—
18C2×#13094014C2214yes—
19#125×D47C4116yescomplemented
20C2×#13093982C2212yes—
21C2×#13093981C2211yescharacteristic
22C2×#13093981C2211yescharacteristic
23C2×C4⋊C4×D47C2212yes—
24C2×C4⋊C4×D47C2214yes—
25C2×C42×D47C2211yescharacteristic
26#15689022C2212yes—
27#15689021C2214yes—
28#15689021C2212yes—
29#15689020C2214yes—
30#15689020C2214yes—
31#15689019C2212yes—
32#15689019C2214yes—
33#15689018C2214yes—
34#15689018C2214yes—
35#15689009C2214yes—
36#15689008C2214yes—
37#15689007C2214yes—
38#15689007C2214yes—
39#15689006C2214yes—
40#15689005C2214yes—
41#15689005C2214yes—
42#15688992C2212yes—
43#15688989C2211yescharacteristic
44#15688982C2214yes—
45#15688982C2214yes—
46#15688980C2214yes—
47C4⋊C4×Dic47C2212yes—
48C4⋊C4×Dic47C2212yes—
49C4⋊C4×Dic47C2214yes—
50C4×Dic47⋊C4C2214yes—
51C4×Dic47⋊C4C2212yes—
52C4×Dic47⋊C4C2212yes—
53C4×Dic47⋊C4C2214yes—
54C4×C4⋊Dic47C2212yes—
55#15688974C2214yes—
56#15688974C2214yes—
57#15688974C2214yes—
58C4×(C2×D47)⋊C4C2212yes—
59C4×(C2×D47)⋊C4C2212yes—
60C4×(C2×D47)⋊C4C2214yes—
61#15688970C2214yes—
62#15688969C2211yescharacteristic
63#15688969C2211yescharacteristic
64C42×Dic47C2211yescharacteristic
65C2×#125×C47C2211yescharacteristic
66C47×#389C2212yes—
67C47×#387C2214yes—
68C47×#382C2211yescharacteristic
69C4×C4⋊C4×C47C2212yes—
70C4×C4⋊C4×C47C2214yes—
71C43×C47C2211yescharacteristic
72C22×C4×D47C2314yes—
73C22×C4×D47C2312yes—
74C22×C4×D47C2311yescharacteristic
75#13094016C2×C418yes—
76#13094015C2×C4116yes—
77#13094014C2×C4132yes—
78#13093982—28no—
79#13093981—24no—
80#13093981Q818yes—
81C2×C4⋊Dic47C2312yes—
82C2×C4⋊Dic47C2314yes—
83C2×C4⋊Dic47C2312yes—
84C2×C4⋊Dic47C2314yes—
85C2×Dic47⋊C4C2312yes—
86C2×Dic47⋊C4C2314yes—
87C2×Dic47⋊C4C2314yes—
88C2×Dic47⋊C4C2314yes—
89C2×Dic47⋊C4C2314yes—
90C2×Dic47⋊C4C2314yes—
91C2×Dic47⋊C4C2314yes—
92C2×Dic47⋊C4C2314yes—
93C2×Dic47⋊C4C2311yescharacteristic
94C2×Dic47⋊C4C2311yescharacteristic
95C4⋊C4×D47C2×C4116yes—
96C4⋊C4×D47C2×C4132yes—
97C2×(C2×D47)⋊C4C2314yes—
98C2×(C2×D47)⋊C4C2312yes—
99C2×(C2×D47)⋊C4C2311yescharacteristic
100C2×(C2×D47)⋊C4C2311yescharacteristic
101C2×C4×Dic47C2311yescharacteristic
102C2×C4×Dic47C2314yes—
103C2×C4×Dic47C2312yes—
104C2×C4×Dic47C2312yes—
105C2×C4×Dic47C2314yes—
106C2×C4×Dic47C2314yes—
107C2×C4×Dic47C2314yes—
108C2×C4×Dic47C2311yescharacteristic
109C2×C4×Dic47C2312yes—
110C2×C4×Dic47C2311yescharacteristic
111C42×D47C2×C418yes—
112C47⋊#95C2312yes—
113C47⋊#95C2312yes—
114C47⋊#95C2312yes—
115C47⋊#95C2312yes—
116C47⋊#95C2314yes—
117C47⋊#95C2314yes—
118C47⋊#95C2314yes—
119C47⋊#95C2314yes—
120C47⋊#95C2314yes—
121C47⋊#95C2314yes—
122C47⋊#95C2314yes—
123#125×C47C2×C418yes—
124C2×C4⋊C4×C47C2312yes—
125C2×C4⋊C4×C47C2314yes—
126C2×C4⋊C4×C47C2314yes—
127C2×C4⋊C4×C47C2312yes—
128C2×C42×C47C2311yescharacteristic
129C2×C42×C47C2312yes—
130C2×C42×C47C2312yes—
131C2×C42×C47C2314yes—
132C2×C42×C47C2311yescharacteristic
133C2×C42×C47C2311yescharacteristic
134#95×C47C2314yes—
135#95×C47C2314yes—
136#95×C47C2314yes—
137C23×D47C2411yescharacteristic
138C22×C4×C47C2412yes—
139C22×C4×C47C2414yes—
140C22×C4×C47C2412yes—
141C22×C4×C47C2414yes—
142C22×C4×C47C2411yescharacteristic
143C22×C4×C47C2411yescharacteristic
144C22×C4×C47C2411yescharacteristic
145C22×Dic47C2412yes—
146C22×Dic47C2414yes—
147C22×Dic47C2414yes—
148C22×Dic47C2412yes—
149C22×Dic47C2411yescharacteristic
150C22×Dic47C2411yescharacteristic
151C22×Dic47C2411yescharacteristic
152C2×C4×D47—44no—
153C2×C4×D47—44no—
154C2×C4×D47—42no—
155C2×C4×D47—42no—
156C2×C4×D47C22×C418yes—
157C2×C4×D47C22×C4116yes—
158C2×C4×D47C22×C414yes—
159C2×C4×D47C2×Q814yes—
160C2×C4×D47#4114yes—
161C4⋊C4×C47—24no—
162C4⋊C4×C47—44no—
163C4⋊C4×C47C22×C4116yes—
164C4⋊C4×C47C22×C418yes—
165C42×C47—22no—
166C42×C47—42no—
167C42×C47—42no—
168C42×C47—44no—
169C42×C47C22×C414yes—
170C42×C47C2×Q814yes—
171(C2×D47)⋊C4—24no—
172(C2×D47)⋊C4—28no—
173(C2×D47)⋊C4C2×Q814yes—
174(C2×D47)⋊C4C2×Q814yes—
175C4⋊Dic47—24no—
176C4⋊Dic47—44no—
177C4⋊Dic47C22×C4116yes—
178C4⋊Dic47C22×C418yes—
179Dic47⋊C4—24no—
180Dic47⋊C4—44no—
181Dic47⋊C4—44no—
182Dic47⋊C4—44no—
183Dic47⋊C4—44no—
184Dic47⋊C4C22×C4116yes—
185Dic47⋊C4C22×C4116yes—
186Dic47⋊C4C22×C4116yes—
187Dic47⋊C4C2×Q814yes—
188Dic47⋊C4C2×Q814yes—
189C4×Dic47—22no—
190C4×Dic47—28no—
191C4×Dic47—44no—
192C4×Dic47—44no—
193C4×Dic47—42no—
194C4×Dic47—42no—
195C4×Dic47—44no—
196C4×Dic47C22×C418yes—
197C4×Dic47C22×C414yes—
198C4×Dic47C2×Q814yes—
199C23×C47C2511yescharacteristic
200C22×D47—22no—
201C22×D47—22no—
202C22×D47C23×C412yes—
203C22×D47C22×Q812yes—
204C22×D47C2×#4112yes—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—24no—
209C2×C4×C47—24no—
210C2×C4×C47—22no—
211C2×C4×C47—24no—
212C2×C4×C47—22no—
213C2×C4×C47—24no—
214C2×C4×C47—22no—
215C2×C4×C47—24no—
216C2×C4×C47—22no—
217C2×C4×C47C23×C414yes—
218C2×C4×C47C23×C418yes—
219C2×C4×C47C23×C412yes—
220C2×C4×C47C22×Q812yes—
221C2×C4×C47C22×Q812yes—
222C2×C4×C47C22×Q812yes—
223C2×C4×C47C2×#4112yes—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—24no—
227C2×Dic47—24no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—24no—
236C2×Dic47C23×C414yes—
237C2×Dic47C23×C418yes—
238C2×Dic47C23×C412yes—
239C2×Dic47C22×Q812yes—
240C2×Dic47C22×Q812yes—
241C2×Dic47C22×Q812yes—
242C2×Dic47C2×#4112yes—
243C4×D47—48no—
244C4×D47—416no—
245C4×D47C4×Q8116yes—
246#34217—471nomaximal, Sylow
247C22×C47C24×C411yescharacteristic
248C22×C47C23×Q811yescharacteristic
249C22×C47C22×#4111yescharacteristic
250C22×C47#58512yes—
251C22×C47#58512yes—
252C2×D47—24no—
253C2×D47—22no—
254C2×D47—22no—
255C2×D47—24no—
256C2×D47C2×C4×Q814yes—
257C4×C47—24no—
258C4×C47—28no—
259C4×C47—22no—
260C4×C47—22no—
261C4×C47—44no—
262C4×C47—42no—
263C4×C47C2×C4×Q814yes—
264Dic47—28no—
265Dic47—24no—
266Dic47—22no—
267Dic47—22no—
268Dic47—42no—
269Dic47—44no—
270Dic47C2×C4×Q814yes—
271C22×#125—471no—
272#2370—474no—
273#2370—474no—
274#2345—474no—
275#2345—474no—
276#2330—472no—
277#2330—472no—
278#2316—472no—
279C4×#125—471no—
280C4×#125—471no—
281C4×#125—471no—
282C4×#125D4711yescharacteristic, complemented
283#2285—471no—
284#2274—472no—
285C4×#117—471no—
286C2×C47C22×C4×Q811yescharacteristic
287C2×C47#342812yes—
288C2×C47#350411yescharacteristic
289C2×C47#350411yescharacteristic
290C2×C47#351212yes—
291D47—24no—
292C2×#125—4716no—
293C2×#125—472no—
294C2×#125—474no—
295C2×#125—471no—
296C2×#125C2×D4711yescharacteristic
297C2×#117—472no—
298C2×#117—471no—
299C2×#117—471no—
300C22×C4⋊C4—472no—
301C22×C4⋊C4—474no—
302C22×C42—471no—
303#389—472no—
304#389—472no—
305#389—474no—
306#389—474no—
307#389—474no—
308#389—472no—
309#389—474no—
310#389—474no—
311#389—474no—
312#389C2×D4712yes—
313#387—474no—
314#387—474no—
315#387—474no—
316#387—474no—
317#387—474no—
318#387—474no—
319#387—474no—
320#387C2×D4714yes—
321#384—472no—
322#382—471no—
323#382C2×D4711yescharacteristic
324#380—474no—
325#380—474no—
326#380—474no—
327C4×C4⋊C4—472no—
328C4×C4⋊C4—472no—
329C4×C4⋊C4—474no—
330C4×C4⋊C4—472no—
331C4×C4⋊C4—474no—
332C4×C4⋊C4—472no—
333C4×C4⋊C4—472no—
334C4×C4⋊C4—474no—
335C4×C4⋊C4—474no—
336C4×C4⋊C4—474no—
337C4×C4⋊C4—474no—
338C4×C4⋊C4C2×D4712yes—
339C4×C4⋊C4C2×D4714yes—
340C4×C22⋊C4—472no—
341C4×C22⋊C4—472no—
342C4×C22⋊C4—474no—
343C4×C22⋊C4—474no—
344#376—471no—
345#376—471no—
346C43—471no—
347C43C2×D4711yescharacteristic
348C47#3421711yescharacteristic, complemented, Sylow
349C23×C4—471no—
350C23×C4—474no—
351C23×C4—472no—
352#125—4732no—
353#125—4716no—
354#125—478no—
355#125C4×D4718yes—
356#117—478no—
357#117—944no—
358#117—948no—
359C2×C4⋊C4—474no—
360C2×C4⋊C4—474no—
361C2×C4⋊C4—474no—
362C2×C4⋊C4—472no—
363C2×C4⋊C4—474no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—4716no—
367C2×C4⋊C4—4732no—
368C2×C4⋊C4—472no—
369C2×C4⋊C4—472no—
370C2×C4⋊C4—474no—
371C2×C4⋊C4—474no—
372C2×C4⋊C4—474no—
373C2×C4⋊C4—474no—
374C2×C4⋊C4—474no—
375C2×C4⋊C4C22×D4712yes—
376C2×C4⋊C4C22×D4714yes—
377C2×C4⋊C4C22×D4714yes—
378C2×C4⋊C4C22×D4712yes—
379C2×C22⋊C4—471no—
380C2×C22⋊C4—471no—
381C2×C22⋊C4—472no—
382C2×C22⋊C4—474no—
383C2×C42—471no—
384C2×C42—471no—
385C2×C42—472no—
386C2×C42—478no—
387C2×C42—471no—
388C2×C42—474no—
389C2×C42—472no—
390C2×C42—472no—
391C2×C42—474no—
392C2×C42—474no—
393C2×C42—474no—
394C2×C42C22×D4711yescharacteristic
395C2×C42C22×D4712yes—
396C2×C42C22×D4712yes—
397C2×C42C22×D4714yes—
398C2×C42C22×D4711yescharacteristic
399C2×C42C22×D4711yescharacteristic
400#95—472no—
401#95—472no—
402#95—472no—
403#95—472no—
404#95—474no—
405#95—474no—
406#95—474no—
407#95—474no—
408#95—474no—
409#95—474no—
410#95—474no—
411#95C22×D4714yes—
412#95C22×D4714yes—
413#95C22×D4714yes—
414C24—471no—
415C22×C4—471no—
416C22×C4—474no—
417C22×C4—474no—
418C22×C4—474no—
419C22×C4—471no—
420C22×C4—471no—
421C22×C4—472no—
422C22×C4—474no—
423C22×C4—478no—
424C22×C4—4716no—
425C22×C4—474no—
426C22×C4—472no—
427C22×C4—1882no—
428C22×C4—1882no—
429C22×C4—1884no—
430C22×C4—1884no—
431C22×C4C23×D4712yes—
432C22×C4C23×D4714yes—
433C22×C4C23×D4712yes—
434C22×C4C23×D4714yes—
435C22×C4C23×D4711yescharacteristic
436C22×C4C23×D4711yescharacteristic
437C22×C4C23×D4711yescharacteristic
438C4⋊C4—24no—
439C4⋊C4—44no—
440C4⋊C4—4716no—
441C4⋊C4—4716no—
442C4⋊C4—4716no—
443C4⋊C4—4716no—
444C4⋊C4—478no—
445C4⋊C4—474no—
446C4⋊C4—474no—
447C4⋊C4—944no—
448C4⋊C4—944no—
449C4⋊C4—1884no—
450C4⋊C4—1884no—
451C4⋊C4—1884no—
452C4⋊C4—1884no—
453C4⋊C4—1884no—
454C4⋊C4C2×C4×D47116yes—
455C4⋊C4C2×C4×D4718yes—
456C22⋊C4—474no—
457C22⋊C4—474no—
458C22⋊C4—944no—
459C22⋊C4—948no—
460C42—22no—
461C42—42no—
462C42—44no—
463C42—42no—
464C42—478no—
465C42—474no—
466C42—474no—
467C42—948no—
468C42—942no—
469C42—1884no—
470C42—1882no—
471C42—1882no—
472C42—1884no—
473C42—1884no—
474C42C2×C4×D4714yes—
475C42Q8×D4714yes—
476C23—472no—
477C23—472no—
478C23—472no—
479C23—942no—
480C23—942no—
481C23C24×D4711yescharacteristic
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—472no—
495C2×C4—472no—
496C2×C4—472no—
497C2×C4—4716no—
498C2×C4—474no—
499C2×C4—478no—
500C2×C4—472no—
501C2×C4—472no—
502C2×C4—942no—
503C2×C4—942no—
504C2×C4—944no—
505C2×C4—944no—
506C2×C4—942no—
507C2×C4—942no—
508C2×C4—942no—
509C2×C4—944no—
510C2×C4—944no—
511C2×C4—942no—
512C2×C4—942no—
513C2×C4—944no—
514C2×C4—1888no—
515C2×C4—18816no—
516C2×C4C22×C4×D4714yes—
517C2×C4C22×C4×D4718yes—
518C2×C4C22×C4×D4712yes—
519C2×C4C2×Q8×D4712yes—
520C2×C4C2×Q8×D4712yes—
521C2×C4C2×Q8×D4712yes—
522C2×C4#41×D4712yes—
523C22—474no—
524C22—944no—
525C22—944no—
526C22—942no—
527C22—942no—
528C22C23×C4×D4711yescharacteristic
529C22C22×Q8×D4711yescharacteristic
530C22C2×#41×D4711yescharacteristic
531C22#1568995912yes—
532C22#1568995912yes—
533C4—24no—
534C4—28no—
535C4—22no—
536C4—22no—
537C4—42no—
538C4—44no—
539C4—474no—
540C4—944no—
541C4—948no—
542C4—942no—
543C4—942no—
544C4—1884no—
545C4—1882no—
546C4C4×Q8×D4714yes—
547C2—944no—
548C2C2×C4×Q8×D4711yescharacteristic
549C2#2025658712yes—
550C2#2025720911yescharacteristic
551C2#2025721211yescharacteristic
552C2#2025730312yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC4×#125×C47C2
socle, cosocleC23×C47C25
prime radicalC23×D47C24
hypercenterC4×#125D47
hypocenterC47#34217
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
2. #20246485C2semidirect
3. #20246479C2semidirect
6. #20246128C2semidirect
7. #20246125C2semidirect
9. #125×Dic47C2semidirect
10. C4×#13094016C2semidirect
11. #20245943C2semidirect
15. C4×#125×C47C2semidirect
19. #125×D47C4semidirect
282. C4×#125D47semidirect
348. C47#34217semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D472
#202464852
#202464792
#202462622
#202462612
#202461282
#202461252
#202460032
#125×Dic472
C4×#130940162
#202459432
#202457942
#202457422
C4×#130939812
C4×#125×C472
#3421747