Finite Group Databasebeta

← #26216088

Subgroups — #26216088

Order 12032.

1542 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D47C211yesmaximal, characteristic
2C2×#15689085C211yesmaximal, characteristic
3#380×D47C211yesmaximal, characteristic
4#20243484C211yesmaximal, characteristic, complemented
5#20243483C211yesmaximal, characteristic, complemented
6#20243480C211yesmaximal, characteristic, complemented
7#20243477C212yesmaximal, complemented
8#20243473C212yesmaximal, complemented
9#20243106C212yesmaximal, complemented
10#20243106C212yesmaximal, complemented
11C47×#1825C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C22×#10625324C2212yes—
14C2×#13093983C2211yescharacteristic
15C2×#13093982C2211yescharacteristic
16C2×#13093982C2211yescharacteristic
17C2×#13093981—21no—
18C2×#13093980—21no—
19#117×D47—24no—
20#117×D47C418yescomplemented
21C2×C4⋊C4×D47C2212yes—
22C2×C4⋊C4×D47C2211yescharacteristic
23C2×C4⋊C4×D47C2211yescharacteristic
24C2×C22⋊C4×D47—21no—
25C2×C22⋊C4×D47—21no—
26C2×C42×D47—21no—
27#15689085—24no—
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×(C2×D47)⋊C8C2212yes—
30#15688983C2211yescharacteristic
31#15688982—21no—
32#15688981—21no—
33#15688980C2212yes—
34#95×D47—21no—
35#15688886C2212yescomplemented
36#15688886C2212yescomplemented
37#15688885C2212yescomplemented
38#15688885C2212yescomplemented
39#15688885C2212yescomplemented
40#15688885C2212yescomplemented
41C2×#117×C47C2211yescharacteristic
42C2×C22⋊C8×C47C2211yescharacteristic
43C47×#380C2211yescharacteristic
44C47×#340C2212yescomplemented
45C47×#340C2212yescomplemented
46C24×D47D411yescharacteristic
47C23×Dic47D411yescharacteristic
48C22×C4×D47—28no—
49C22×C4×D47—21no—
50C22×C4×D47—21no—
51C22×C4×D47—22no—
52C22×C4×D47C2×C414yes—
53C22×C4×D47D411yescharacteristic
54C22×C4×D47D411yescharacteristic
55C22×C4×D47D414yes—
56C22×C4×D47C2311yescharacteristic
57C22×C4×D47C2311yescharacteristic
58C2×#10625324—24no—
59C2×#10625324—28no—
60C22×C47⋊C8C2312yes—
61#13093983—22no—
62#13093983C2×C414yescomplemented
63#13093982—22no—
64#13093982—22no—
65#13093982C2×C414yes—
66#13093982C2×C414yes—
67#13093981—28no—
68#13093980—28no—
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—22no—
75C4⋊C4×D47—22no—
76C4⋊C4×D47—22no—
77C4⋊C4×D47—28no—
78C4⋊C4×D47C2×C414yescomplemented
79C4⋊C4×D47C2×C414yescomplemented
80C2×C22⋊Dic47—21no—
81C2×C22⋊Dic47—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—22no—
85C22⋊C4×D47—28no—
86C22⋊C4×D47—44no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90C42×D47—28no—
91C22⋊(C47⋊C8)—22no—
92(C2×D47)⋊C8—24no—
93C47⋊#95—21no—
94C47⋊#95—22no—
95C23×C4×C47C2311yescharacteristic
96C22×C8×C47C2312yes—
97#117×C47—22no—
98#117×C47C2×C414yescomplemented
99C2×C4⋊C4×C47C2312yes—
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C22⋊C4×C47—21no—
103C2×C22⋊C4×C47—21no—
104C2×C42×C47—21no—
105C22⋊C8×C47—22no—
106#95×C47—21no—
107C24×C47C2×D411yescharacteristic
108C23×D47—24no—
109C23×D47—22no—
110C23×D47—22no—
111C23×D47—42no—
112C23×D47C22⋊C414yes—
113C23×D47C2×D411yescharacteristic
114C23×D47C2×D411yescharacteristic
115C22×C4×C47—24no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D412yes—
122C22×C4×C47C2411yescharacteristic
123C22×Dic47—24no—
124C22×Dic47—21no—
125C22×Dic47—21no—
126C22×Dic47—22no—
127C22×Dic47C22⋊C414yes—
128C22×Dic47C2×D411yescharacteristic
129C22×Dic47C2×D411yescharacteristic
130C2×C4×D47—24no—
131C2×C4×D47—24no—
132C2×C4×D47—24no—
133C2×C4×D47—216no—
134C2×C4×D47—216no—
135C2×C4×D47—24no—
136C2×C4×D47—24no—
137C2×C4×D47—44no—
138C2×C4×D47—42no—
139C2×C4×D47—42no—
140C2×C4×D47—44no—
141C2×C4×D47—44no—
142C2×C4×D47—44no—
143C2×C4×D47C22⋊C414yes—
144C2×C4×D47C22⋊C414yes—
145C2×C4×D47C22×C412yes—
146C2×C4×D47C22×C412yes—
147C2×C4×D47C2×D412yes—
148C2×C4×D47C2×D412yes—
149C2×C4×D47#4118yes—
150C2×C8×C47—22no—
151C2×C8×C47—24no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—24no—
155C4⋊C4×C47C22×C412yes—
156C4⋊C4×C47C22×C412yes—
157C22⋊C4×C47—24no—
158C22⋊C4×C47—42no—
159C42×C47—24no—
160C22⋊Dic47—24no—
161C22⋊Dic47—42no—
162(C2×D47)⋊C4—24no—
163(C2×D47)⋊C4—24no—
164(C2×D47)⋊C4—44no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167C4⋊Dic47—24no—
168C4⋊Dic47C22×C412yes—
169C4⋊Dic47C22×C412yes—
170Dic47⋊C4—24no—
171Dic47⋊C4—24no—
172Dic47⋊C4—44no—
173C4×Dic47—24no—
174C4×Dic47—24no—
175C4×Dic47—24no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—24no—
178#10625324—48no—
179C23×C47—22no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47—22no—
183C23×C47C2×C22⋊C412yes—
184C23×C47C22×D411yescharacteristic
185C22×D47—22no—
186C22×D47—21no—
187C22×D47—21no—
188C22×D47—22no—
189C22×D47—22no—
190C22×D47—28no—
191C22×D47—28no—
192C22×D47—44no—
193C22×D47—48no—
194C22×D47—48no—
195C22×D47C2×C22⋊C412yes—
196C22×D47C2×C22⋊C412yes—
197C22×D47#12314yes—
198C22×D47AGL(1,ℤ/8)12yes—
199C22×D47#13712yes—
200C2×C4×C47—22no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47C2×C22⋊C412yes—
212C2×C4×C47C23×C411yescharacteristic
213C2×C4×C47C22×D411yescharacteristic
214C2×C4×C47C2×#4114yes—
215C2×Dic47—22no—
216C2×Dic47—28no—
217C2×Dic47—22no—
218C2×Dic47—22no—
219C2×Dic47—41no—
220C2×Dic47—41no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—42no—
225C2×Dic47C2×C22⋊C412yes—
226C2×Dic47C2×C22⋊C412yes—
227C2×Dic47#12314yes—
228C4×D47—28no—
229C4×D47—24no—
230C4×D47—24no—
231C4×D47—28no—
232C4×D47—48no—
233C4×D47—48no—
234C4×D47—88no—
235C4×D47#11718yes—
236C4×D47#12318yes—
237C8×C47—42no—
238C47⋊C8—42no—
239#20702—471nomaximal, Sylow
240C22×C47—22no—
241C22×C47—22no—
242C22×C47—22no—
243C22×C47—42no—
244C22×C47—42no—
245C22×C47C22×C22⋊C411yescharacteristic
246C22×C47C2×#12312yes—
247C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C47C2×#13711yescharacteristic
249C22×C47#57512yes—
250C2×D47—24no—
251C2×D47—24no—
252C2×D47—24no—
253C2×D47—24no—
254C2×D47—22no—
255C2×D47—22no—
256C2×D47—22no—
257C2×D47—22no—
258C2×D47—416no—
259C2×D47#38114yes—
260C2×D47#41714yes—
261C4×C47—22no—
262C4×C47—42no—
263C4×C47—42no—
264C4×C47—82no—
265C4×C47C2×#11712yes—
266C4×C47C2×#12312yes—
267Dic47—22no—
268Dic47—42no—
269Dic47—42no—
270Dic47—82no—
271Dic47#38114yes—
272C22×#117—471no—
273C2×#407—471no—
274C2×#380—471no—
275#1825—471no—
276#1825—471no—
277#1825—471no—
278#1825D4711yescharacteristic, complemented
279#1824—472no—
280#1823—472no—
281#1736—472no—
282#1736—472no—
283C2×C47—42no—
284C2×C47C2×#38111yescharacteristic
285C2×C47C2×#41711yescharacteristic
286C2×C47#289611yescharacteristic
287C2×C47#309412yes—
288C2×C47#309512yes—
289D47—24no—
290D47—24no—
291C24×C4—471no—
292C22×M16—472no—
293C2×#117—471no—
294C2×#117—478no—
295C2×#117—471no—
296C2×#117—471no—
297C2×#117—941no—
298C2×#117—941no—
299C2×#117—944no—
300C2×#117C2×D4711yescharacteristic
301C22×C4⋊C4—471no—
302C22×C4⋊C4—471no—
303C22×C4⋊C4—472no—
304C22×C22⋊C4—941no—
305C22×C22⋊C4—941no—
306C22×C42—941no—
307#407—944no—
308C2×C22⋊C8—471no—
309C2×C22⋊C8—472no—
310C2×C22⋊C8C2×D4711yescharacteristic
311#380—471no—
312#380—472no—
313#380—941no—
314#380—941no—
315#380C2×D4711yescharacteristic
316C2×#95—941no—
317#340—472no—
318#340—472no—
319#340—472no—
320#340—472no—
321#340—472no—
322#340—472no—
323#340C2×D4712yescomplemented
324#340C2×D4712yescomplemented
325C47#2070211yescharacteristic, complemented, Sylow
326C25—471no—
327C23×C4—471no—
328C23×C4—471no—
329C23×C4—474no—
330C23×C4—471no—
331C23×C4—474no—
332C23×C4—471no—
333C23×C4—471no—
334C23×C4—948no—
335C23×C4—941no—
336C23×C4—941no—
337C23×C4—942no—
338C23×C4C22×D4711yescharacteristic
339C2×M16—948no—
340C2×M16—944no—
341C22×C8—472no—
342C22×C8C22×D4712yes—
343#117—22no—
344#117—474no—
345#117—474no—
346#117—474no—
347#117—948no—
348#117—948no—
349#117—942no—
350#117—942no—
351#117—942no—
352#117C4×D4714yescomplemented
353C2×C4⋊C4—471no—
354C2×C4⋊C4—471no—
355C2×C4⋊C4—474no—
356C2×C4⋊C4—474no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—942no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4—948no—
361C2×C4⋊C4—941no—
362C2×C4⋊C4—941no—
363C2×C4⋊C4—942no—
364C2×C4⋊C4C22×D4712yes—
365C2×C4⋊C4C22×D4711yescharacteristic
366C2×C4⋊C4C22×D4711yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—948no—
370C2×C22⋊C4—941no—
371C2×C22⋊C4—941no—
372C2×C22⋊C4—941no—
373C2×C22⋊C4—942no—
374C2×C22⋊C4—941no—
375C2×C22⋊C4—1884no—
376C2×C42—21no—
377C2×C42—948no—
378C2×C42—941no—
379C2×C42—941no—
380C2×C42—941no—
381C22⋊C8—22no—
382C22⋊C8—944no—
383C22⋊C8—942no—
384#95—21no—
385#95—942no—
386#95—941no—
387C24—471no—
388C24—471no—
389C24—474no—
390C24—942no—
391C24—944no—
392C24—942no—
393C24—1882no—
394C24D4×D4711yescharacteristic
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—471no—
400C22×C4—472no—
401C22×C4—472no—
402C22×C4—478no—
403C22×C4—474no—
404C22×C4—471no—
405C22×C4—474no—
406C22×C4—474no—
407C22×C4—472no—
408C22×C4—472no—
409C22×C4—944no—
410C22×C4—9416no—
411C22×C4—9416no—
412C22×C4—944no—
413C22×C4—944no—
414C22×C4—944no—
415C22×C4—944no—
416C22×C4—944no—
417C22×C4—941no—
418C22×C4—941no—
419C22×C4—942no—
420C22×C4—1884no—
421C22×C4—1884no—
422C22×C4—1884no—
423C22×C4—1882no—
424C22×C4—1882no—
425C22×C4—1884no—
426C22×C4C2×C4×D4712yes—
427C22×C4D4×D4711yescharacteristic
428C22×C4D4×D4712yes—
429C22×C4C23×D4711yescharacteristic
430M16—1888no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—942no—
434C2×C8—944no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—472no—
439C4⋊C4—472no—
440C4⋊C4—944no—
441C4⋊C4—944no—
442C4⋊C4—941no—
443C4⋊C4—941no—
444C4⋊C4—944no—
445C4⋊C4—1884no—
446C4⋊C4C2×C4×D4712yes—
447C4⋊C4C2×C4×D4712yes—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—944no—
451C22⋊C4—944no—
452C22⋊C4—944no—
453C22⋊C4—1884no—
454C22⋊C4—1882no—
455C42—24no—
456C42—944no—
457C42—944no—
458C42—944no—
459C23—22no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—472no—
464C23—472no—
465C23—474no—
466C23—472no—
467C23—472no—
468C23—948no—
469C23—948no—
470C23—942no—
471C23—941no—
472C23—941no—
473C23—942no—
474C23—942no—
475C23—1884no—
476C23—1888no—
477C23—1888no—
478C23C22⋊C4×D4712yes—
479C23C2×D4×D4711yescharacteristic
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—472no—
492C2×C4—474no—
493C2×C4—478no—
494C2×C4—478no—
495C2×C4—472no—
496C2×C4—948no—
497C2×C4—948no—
498C2×C4—948no—
499C2×C4—942no—
500C2×C4—944no—
501C2×C4—944no—
502C2×C4—942no—
503C2×C4—942no—
504C2×C4—3768no—
505C2×C4—1888no—
506C2×C4—1888no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4—1882no—
510C2×C4—1881no—
511C2×C4—1881no—
512C2×C4—1882no—
513C2×C4C22⋊C4×D4712yes—
514C2×C4C22×C4×D4711yescharacteristic
515C2×C4C2×D4×D4711yescharacteristic
516C2×C4#41×D4714yes—
517C8—42no—
518C8—1882no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—42no—
523C22—42no—
524C22—474no—
525C22—474no—
526C22—942no—
527C22—944no—
528C22—944no—
529C22—944no—
530C22—944no—
531C22—942no—
532C22—942no—
533C22—942no—
534C22—18816no—
535C22C2×C22⋊C4×D4711yescharacteristic
536C22#123×D4712yes—
537C22AGL(1,ℤ/8)×D4711yescharacteristic
538C22#137×D4711yescharacteristic
539C22#1568992012yes—
540C4—22no—
541C4—42no—
542C4—42no—
543C4—82no—
544C4—474no—
545C4—942no—
546C4—3762no—
547C4—1882no—
548C4—1882no—
549C4#117×D4712yes—
550C4#123×D4712yes—
551C2—42no—
552C2—944no—
553C2—944no—
554C2#381×D4711yescharacteristic
555C2#417×D4711yescharacteristic
556C2#2025190311yescharacteristic
557C2#2025332712yes—
558C2#2025333512yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1825C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1825D47
hypocenterC47#20702
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
4. #20243484C2semidirect
5. #20243483C2semidirect
6. #20243480C2semidirect
7. #20243477C2semidirect
8. #20243473C2semidirect
9. #20243106C2semidirect
10. #20243106C2semidirect
11. C47×#1825C2semidirect
20. #117×D47C4semidirect
35. #15688886C22semidirect
36. #15688886C22semidirect
37. #15688885C22semidirect
38. #15688885C22semidirect
39. #15688885C22semidirect
40. #15688885C22semidirect
44. C47×#340C22semidirect
45. C47×#340C22semidirect
62. #13093983C2×C4semidirect
78. C4⋊C4×D47C2×C4semidirect
79. C4⋊C4×D47C2×C4semidirect
98. #117×C47C2×C4semidirect
278. #1825D47semidirect
323. #340C2×D47semidirect
324. #340C2×D47semidirect
325. C47#20702semidirect
352. #117C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D472
C2×#156890852
#380×D472
#202434842
#202434832
#202434802
#202434772
#202434732
#202431062
C47×#18252
#2070247