Finite Group Databasebeta

← #26199059

Subgroups — #26199059

Order 12032.

760 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253848C211yesmaximal, characteristic, complemented
2C2×#15689403C211yesmaximal, characteristic, complemented
3D4×#10625329C211yesmaximal, characteristic, complemented
4#20242875C212yesmaximal, complemented
5#20242851C212yesmaximal, complemented
6#20242751C212yesmaximal, complemented
7#20242727C212yesmaximal, complemented
8#20242141C211yesmaximal, characteristic, complemented
9#20242065C211yesmaximal, characteristic, complemented
10#20242059C211yesmaximal, characteristic, complemented
11C47×#3100C211yesmaximal, characteristic
12C2×#13094054—21no—
13C2×#13094054—21no—
14C22×#10625329—21no—
15C2×#13094024C2211yescharacteristic
16#15689518—22no—
17#15689502C2212yescomplemented
18#15689495C2212yescomplemented
19#15689403—21no—
20#15689403—21no—
21#15689403C2212yes—
22#15689403C2212yes—
23#15689201C2212yes—
24#15689195—22no—
25D4×C47⋊C8C2212yes—
26C2×#13093957—21no—
27#15689147C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×#13093945—21no—
30#15689087—21no—
31C4×#10625329C2211yescharacteristic
32#15688875C2212yescomplemented
33#15688873C2212yescomplemented
34#15688869C2212yescomplemented
35#15688868C2212yescomplemented
36#15688868C2212yescomplemented
37#15688867C2212yescomplemented
38C47×#532C2211yescharacteristic
39C2×C4×D4×C47C2211yescharacteristic
40C47×#480—21no—
41C47×#474C2211yescharacteristic
42C47×#474C2211yescharacteristic
43C47×#459C2211yescharacteristic
44C47×#459C2211yescharacteristic
45C47×#447—21no—
46C4×D8×C47—21no—
47C4×D8×C47—21no—
48C47×#431C2211yescharacteristic
49C22×Dic94—21no—
50#13094054—44no—
51#13094054—44no—
52C2×#10625337—22no—
53C2×#10625337—22no—
54C2×#10625335—22no—
55C2×#10625335—22no—
56C2×#10625329—21no—
57C2×#10625329—22no—
58C2×#10625329—44no—
59C2×#10625329D411yescharacteristic
60C2×#10625329D411yescharacteristic
61C22×C47⋊C8—21no—
62C22×C47⋊C8—22no—
63#13094024—21no—
64#13094024—21no—
65#13094024—21no—
66#13094024C2312yes—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69#13093957—22no—
70#13093957—22no—
71C4⋊(C47⋊C8)—21no—
72C4⋊(C47⋊C8)—21no—
73C4⋊(C47⋊C8)C2312yes—
74C4⋊(C47⋊C8)C2312yes—
75#13093945—22no—
76#13093945—22no—
77C22⋊(C47⋊C8)—22no—
78C22⋊(C47⋊C8)—22no—
79#13093935D412yes—
80C4×C47⋊C8C2312yes—
81C2×#41×C47—21no—
82C22×D4×C47D411yescharacteristic
83C23×C4×C47—21no—
84C2×D8×C47—21no—
85C2×D8×C47—21no—
86C4⋊Q8×C47C2311yescharacteristic
87#127×C47C2311yescharacteristic
88#124×C47—21no—
89#121×C47—21no—
90#121×C47—21no—
91C4×D4×C47—21no—
92C4×D4×C47—21no—
93C4×D4×C47—21no—
94C4×D4×C47—21no—
95C4×D4×C47D411yescharacteristic
96C4×D4×C47D411yescharacteristic
97C4×D4×C47C2311yescharacteristic
98C4×D4×C47C2311yescharacteristic
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C22⋊C4×C47—21no—
101C2×C42×C47C2311yescharacteristic
102#107×C47—21no—
103#107×C47—21no—
104C4⋊C8×C47C2311yescharacteristic
105C4⋊C8×C47C2311yescharacteristic
106#102×C47—21no—
107#102×C47—21no—
108#102×C47—21no—
109#102×C47—21no—
110C22⋊C8×C47—21no—
111C4×C8×C47—21no—
112C24×C47—21no—
113#41×C47—42no—
114#41×C47—42no—
115C2×Q8×C47—21no—
116C2×D4×C47—22no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—21no—
121C2×D4×C47—41no—
122C2×D4×C47—41no—
123C2×D4×C47C2×D411yescharacteristic
124C2×D4×C47C2×D411yescharacteristic
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47—41no—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47#4111yescharacteristic
136C22×Dic47—41no—
137C2×Dic94—22no—
138C2×Dic94—42no—
139C2×Dic94—42no—
140C2×Dic94—41no—
141D8×C47—42no—
142D8×C47—42no—
143C2×C8×C47—21no—
144C2×C8×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47C2×D411yescharacteristic
149C4⋊C4×C47C2×D411yescharacteristic
150C22⋊C4×C47—21no—
151C22⋊C4×C47—21no—
152C22⋊C4×C47—21no—
153C22⋊C4×C47—21no—
154C22⋊C4×C47—41no—
155C22⋊C4×C47—41no—
156C42×C47C2×D411yescharacteristic
157C42×C47C2×D411yescharacteristic
158C42×C47#4111yescharacteristic
159C42×C47C2411yescharacteristic
160#10625337—44no—
161#10625337—44no—
162#10625335—44no—
163#10625335—44no—
164C4⋊Dic47—22no—
165C4⋊Dic47—22no—
166C4⋊Dic47—41no—
167C4⋊Dic47—41no—
168#10625329—22no—
169#10625329—42no—
170#10625329—44no—
171#10625329D814yescomplemented
172C2×C47⋊C8—22no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—42no—
175C2×C47⋊C8—44no—
176C2×C47⋊C8C2×D412yes—
177C2×C47⋊C8C2×D412yes—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—42no—
181C23×C47—42no—
182C23×C47—41no—
183C23×C47—81no—
184C23×C47#12111yescharacteristic
185Q8×C47—41no—
186Q8×C47—41no—
187D4×C47—21no—
188D4×C47—21no—
189D4×C47—21no—
190D4×C47—42no—
191D4×C47—41no—
192D4×C47—41no—
193D4×C47—41no—
194D4×C47—41no—
195D4×C47—81no—
196D4×C47—82no—
197D4×C47#13712yescomplemented
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—41no—
203C2×C4×C47—42no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—41no—
211C2×C4×C47—82no—
212C2×C4×C47#12111yescharacteristic
213C2×C4×C47#12111yescharacteristic
214C2×C4×C47#12111yescharacteristic
215C2×C4×C47C2×D812yes—
216C2×C4×C47AGL(1,ℤ/8)12yes—
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×Dic47—42no—
221C2×Dic47—82no—
222Dic94—42no—
223Dic94—42no—
224Dic94—82no—
225C8×C47—41no—
226C8×C47—41no—
227C47⋊C8—22no—
228C47⋊C8—42no—
229C47⋊C8—42no—
230C47⋊C8C2×D814yes—
231#19163—471nomaximal, Sylow
232C22×C47—21no—
233C22×C47—21no—
234C22×C47—22no—
235C22×C47—41no—
236C22×C47—41no—
237C22×C47—41no—
238C22×C47—41no—
239C22×C47—41no—
240C22×C47—84no—
241C22×C47—81no—
242C22×C47—81no—
243C22×C47—81no—
244C22×C47#45912yes—
245C22×C47C2×#12111yescharacteristic
246C4×C47—21no—
247C4×C47—21no—
248C4×C47—21no—
249C4×C47—41no—
250C4×C47—41no—
251C4×C47—81no—
252C4×C47#45912yes—
253C4×C47C22×D811yescharacteristic
254C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C47C2×#13711yescharacteristic
256C4×C47#57511yescharacteristic
257Dic47—82no—
258#3152—471no—
259#3100D4711yescharacteristic
260C2×#474—471no—
261D4×M16—471no—
262#1689—472no—
263#1685—472no—
264#1669—472no—
265#1665—472no—
266#1565—471no—
267#1547—471no—
268#1546—471no—
269C2×C47—22no—
270C2×C47—42no—
271C2×C47—81no—
272C2×C47—161no—
273C2×C47C2×#45911yescharacteristic
274C2×C47#305611yescharacteristic
275C2×C47#306911yescharacteristic
276C2×#137—941no—
277C2×#137—941no—
278C22×M16—941no—
279#532C2×D4711yescharacteristic
280C2×C4⋊Q8—471no—
281C2×C4×D4C2×D4711yescharacteristic
282#497—942no—
283#494—472no—
284#492—472no—
285#480—21no—
286#474—472no—
287#474—472no—
288#474—941no—
289#474—941no—
290#474C2×D4711yescharacteristic
291#474C2×D4711yescharacteristic
292#459C2×D4711yescharacteristic
293#459C2×D4711yescharacteristic
294#447—21no—
295C4×D8—21no—
296C4×D8—21no—
297#436—472no—
298#435—942no—
299C8×D4—472no—
300#431C2×D4711yescharacteristic
301C2×#107—941no—
302#423—471no—
303C2×C4⋊C8—471no—
304C2×#102—941no—
305#407—941no—
306C4×M16—471no—
307#335—472no—
308#334—472no—
309#332—472no—
310#332—472no—
311#332—472no—
312#331—472no—
313C47#1916311yescharacteristic, complemented, Sylow
314C2×#41—21no—
315C22×Q8—941no—
316C22×D4C22⋊D4711yescharacteristic
317C23×C4—21no—
318#137—1884no—
319#137—1884no—
320C2×Q16—942no—
321C2×Q16—942no—
322C2×SD16—942no—
323C2×SD16—942no—
324C2×D8—21no—
325C2×D8—21no—
326C2×M16—471no—
327C2×M16—471no—
328C2×M16—942no—
329C2×M16—941no—
330C2×M16—1884no—
331C22×C8—942no—
332C22×C8—941no—
333C4⋊Q8—472no—
334C4⋊Q8—941no—
335C4⋊Q8—941no—
336C4⋊Q8—941no—
337C4⋊Q8C22×D4711yescharacteristic
338#127C22×D4711yescharacteristic
339#124—21no—
340#121—21no—
341#121—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4C22⋊D4711yescharacteristic
347C4×D4C22⋊D4711yescharacteristic
348C4×D4C22×D4711yescharacteristic
349C4×D4C22×D4711yescharacteristic
350C2×C4⋊C4—941no—
351C2×C4⋊C4—941no—
352C2×C4⋊C4C22⋊D4711yescharacteristic
353C2×C22⋊C4—21no—
354C2×C42C22×D4711yescharacteristic
355#107—21no—
356#107—21no—
357#107—942no—
358#107—942no—
359C4⋊C8—472no—
360C4⋊C8—472no—
361C4⋊C8—941no—
362C4⋊C8—941no—
363C4⋊C8C22×D4711yescharacteristic
364C4⋊C8C22×D4711yescharacteristic
365#102—21no—
366#102—21no—
367#102—21no—
368#102—21no—
369#102—942no—
370#102—942no—
371C22⋊C8—21no—
372C22⋊C8—942no—
373C22⋊C8—942no—
374#97—472no—
375C4×C8—21no—
376C4×C8—472no—
377C24—21no—
378#41—42no—
379#41—42no—
380C2×Q8—21no—
381C2×Q8—942no—
382C2×Q8—1882no—
383C2×Q8—1882no—
384C2×Q8—1881no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4C2×C22⋊D4711yescharacteristic
393C2×D4C2×C22⋊D4711yescharacteristic
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—1881no—
403C22×C4D4×D4711yescharacteristic
404C22×C4#1062535211yescharacteristic
405C22×C4C2×C22⋊D4711yescharacteristic
406Q16—1884no—
407Q16—1884no—
408SD16—1884no—
409SD16—1884no—
410D8—42no—
411D8—42no—
412M16—474no—
413M16—942no—
414M16—1884no—
415M16—1882no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—472no—
419C2×C8—472no—
420C2×C8—942no—
421C2×C8—942no—
422C2×C8—1884no—
423C2×C8—1882no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—942no—
428C4⋊C4—942no—
429C4⋊C4—1881no—
430C4⋊C4—1881no—
431C4⋊C4C2×C22⋊D4711yescharacteristic
432C4⋊C4C2×C22⋊D4711yescharacteristic
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C42D4×D4711yescharacteristic
440C42#1062535211yescharacteristic
441C42C2×C22⋊D4711yescharacteristic
442C42C23×D4711yescharacteristic
443C23—21no—
444C23—21no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—81no—
449C23#1309399611yescharacteristic
450Q8—41no—
451Q8—41no—
452Q8—3762no—
453Q8—1882no—
454Q8—1882no—
455D4—21no—
456D4—21no—
457D4—21no—
458D4—42no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—81no—
464D4—82no—
465D4#1309405412yescomplemented
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—82no—
480C2×C4—3762no—
481C2×C4—1882no—
482C2×C4#1309399611yescharacteristic
483C2×C4#1309399611yescharacteristic
484C2×C4#1309399611yescharacteristic
485C2×C4D8×D4712yes—
486C2×C4#1309404812yes—
487C2×C4C2×D4×D4711yescharacteristic
488C2×C4C22×C22⋊D4711yescharacteristic
489C2×C4C2×#1062535211yescharacteristic
490C8—41no—
491C8—41no—
492C8—474no—
493C8—942no—
494C8—1882no—
495C8—1882no—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—84no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22#1568931612yes—
509C22C2×#1309399611yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—3762no—
517C4#1568931612yes—
518C4C2×D8×D4711yescharacteristic
519C4C2×#1309404811yescharacteristic
520C4C2×#1309405411yescharacteristic
521C4#1568992111yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—161no—
526C2C2×#1568931611yescharacteristic
527C2#2025302011yescharacteristic
528C2#2025312711yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
Fitting, prime radicalC47×#3100C2
socleC22×C47C2×#121
hypercenter#3100D47
hypocenterC47#19163
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #20253848C2semidirect
2. C2×#15689403C2semidirect
3. D4×#10625329C2semidirect
4. #20242875C2semidirect
5. #20242851C2semidirect
6. #20242751C2semidirect
7. #20242727C2semidirect
8. #20242141C2semidirect
9. #20242065C2semidirect
10. #20242059C2semidirect
17. #15689502C22semidirect
18. #15689495C22semidirect
32. #15688875C22semidirect
33. #15688873C22semidirect
34. #15688869C22semidirect
35. #15688868C22semidirect
36. #15688868C22semidirect
37. #15688867C22semidirect
171. #10625329D8semidirect
197. D4×C47#137semidirect
313. C47#19163semidirect
465. D4#13094054semidirect

Maximal subgroups

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

subgroupindex
#202538482
C2×#156894032
D4×#106253292
#202428752
#202428512
#202427512
#202427272
#202421412
#202420652
#202420592
C47×#31002
#1916347