Finite Group Databasebeta

← #26201685

Subgroups — #26201685

Order 12032.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254329C211yesmaximal, characteristic, complemented
2#20253176C211yesmaximal, characteristic, complemented
3#20252499C211yesmaximal, characteristic, complemented
4#20242802C212yesmaximal, complemented
5#20242795C212yesmaximal, complemented
6#20242734C212yesmaximal, complemented
7#20242727C212yesmaximal, complemented
8#20242160C211yesmaximal, characteristic, complemented
9#20242079C211yesmaximal, characteristic, complemented
10C2×#15688867C211yesmaximal, characteristic, complemented
11C47×#3038C211yesmaximal, characteristic
12#15689680C2211yescharacteristic
13#15689495C2212yes—
14#15689469C2212yes—
15#15689430—21no—
16#15689423—21no—
17#15689418C2211yescharacteristic, complemented
18#15689413C2211yescharacteristic, complemented
19#15689408C2211yescharacteristic, complemented
20#15689403C2211yescharacteristic, complemented
21#15689320—21no—
22#15689320—21no—
23#15689274—22no—
24Q8×C47⋊C8C2212yescomplemented
25D4×C47⋊C8—22no—
26D4×C47⋊C8C2212yescomplemented
27#15689176—21no—
28#15689176C2211yescharacteristic
29#15689159—21no—
30C2×C4⋊(C47⋊C8)—21no—
31C2×C4⋊(C47⋊C8)C2211yescharacteristic
32C2×C4⋊(C47⋊C8)C2211yescharacteristic
33#15689119—21no—
34#15688873C2212yescomplemented
35#15688873C2212yescomplemented
36#15688871C2212yescomplemented
37#15688868C2212yescomplemented
38#15688867C2212yes—
39#15688867C2212yes—
40C2×AGL(1,ℤ/8)×C47—21no—
41C2×#127×C47C2211yescharacteristic
42C4×#41×C47C2211yescharacteristic
43C47×#460—21no—
44C47×#460C2212yes—
45C47×#459—21no—
46C47×#459C2212yes—
47C47×#423C2211yescharacteristic
48C47×#418—21no—
49C2×#10625335—22no—
50C2×#10625335—22no—
51C22×C47⋊C8—21no—
52C22×C47⋊C8—21no—
53C22×C47⋊C8—22no—
54#13094024C2311yescharacteristic
55#13094024C2311yescharacteristic
56#13094016—21no—
57#13094003—21no—
58#13094003—21no—
59C4×Dic94—21no—
60C4×Dic94—21no—
61#13093983—21no—
62#13093957—21no—
63#13093957—21no—
64#13093955—21no—
65#13093955—21no—
66C4⋊(C47⋊C8)—21no—
67C4⋊(C47⋊C8)—21no—
68C4⋊(C47⋊C8)—22no—
69C4⋊(C47⋊C8)D411yescharacteristic
70C4⋊(C47⋊C8)D411yescharacteristic
71C4⋊(C47⋊C8)C2312yes—
72C4⋊(C47⋊C8)C2311yescharacteristic
73C4⋊(C47⋊C8)C2311yescharacteristic
74#13093948—21no—
75#13093948—21no—
76#13093945—21no—
77#13093945—21no—
78#13093945—22no—
79#13093945—22no—
80C22⋊(C47⋊C8)—22no—
81C22⋊(C47⋊C8)D412yes—
82C4×C47⋊C8—22no—
83C4×C47⋊C8C2312yes—
84C2×#41×C47D411yescharacteristic
85C22×D4×C47—21no—
86C22×D4×C47—21no—
87C22×D4×C47—21no—
88AGL(1,ℤ/8)×C47—44no—
89C2×SD16×C47—22no—
90C2×D8×C47—22no—
91C2×M16×C47—21no—
92#127×C47—21no—
93#127×C47—21no—
94#127×C47—21no—
95#127×C47C2312yes—
96C4×Q8×C47D411yescharacteristic
97C4×Q8×C47C2311yescharacteristic
98C4×D4×C47—21no—
99C4×D4×C47—21no—
100C4×D4×C47D411yescharacteristic
101C4×D4×C47C2311yescharacteristic
102#117×C47—21no—
103#117×C47D411yescharacteristic
104C2×C42×C47—21no—
105C2×C42×C47C2311yescharacteristic
106C4⋊C8×C47—21no—
107C4⋊C8×C47C2312yes—
108#102×C47—22no—
109#102×C47—22no—
110C24×C47—41no—
111#41×C47—22no—
112#41×C47SD1614yescomplemented
113C2×Q8×C47C2×D411yescharacteristic
114C2×D4×C47—21no—
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C2×D4×C47—41no—
119C2×D4×C47—41no—
120C2×D4×C47—41no—
121C2×D4×C47—42no—
122C2×D4×C47—42no—
123C2×D4×C47—81no—
124C2×D4×C47—81no—
125C2×D4×C47—82no—
126C2×D4×C47C2×D411yescharacteristic
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47#4111yescharacteristic
133C2×Dic94—21no—
134C2×Dic94—21no—
135SD16×C47—44no—
136D8×C47—44no—
137M16×C47—42no—
138C2×C8×C47—22no—
139C4⋊C4×C47—21no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47C2×D411yescharacteristic
142C4⋊C4×C47C2×D411yescharacteristic
143C22⋊C4×C47—21no—
144C22⋊C4×C47—21no—
145C22⋊C4×C47—21no—
146C42×C47—21no—
147C42×C47—21no—
148C42×C47—21no—
149C42×C47C2×D411yescharacteristic
150C42×C47C2×D411yescharacteristic
151C42×C47#4111yescharacteristic
152C42×C47C2411yescharacteristic
153C22⋊Dic47—41no—
154#10625335—82no—
155#10625335—82no—
156C4⋊Dic47—21no—
157C4⋊Dic47—21no—
158C4⋊Dic47—21no—
159C4⋊Dic47—21no—
160Dic47⋊C4—41no—
161Dic47⋊C4—41no—
162C4×Dic47—41no—
163C2×C47⋊C8—22no—
164C2×C47⋊C8—21no—
165C2×C47⋊C8—21no—
166C2×C47⋊C8—42no—
167C2×C47⋊C8—82no—
168C2×C47⋊C8—81no—
169C2×C47⋊C8C2×D412yes—
170C2×C47⋊C8C2×D412yes—
171C23×C47—21no—
172C23×C47—42no—
173C23×C47—82no—
174C23×C47—82no—
175C23×C47—82no—
176C23×C47#12111yescharacteristic
177Q8×C47—21no—
178Q8×C47C2×SD1612yes—
179D4×C47—24no—
180D4×C47—21no—
181D4×C47—42no—
182D4×C47—42no—
183D4×C47—42no—
184D4×C47—42no—
185D4×C47—82no—
186D4×C47—82no—
187D4×C47—82no—
188D4×C47—82no—
189D4×C47C2×SD1612yes—
190C2×C4×C47—21no—
191C2×C4×C47—21no—
192C2×C4×C47—21no—
193C2×C4×C47—24no—
194C2×C4×C47—21no—
195C2×C4×C47—21no—
196C2×C4×C47—41no—
197C2×C4×C47—41no—
198C2×C4×C47—41no—
199C2×C4×C47—81no—
200C2×C4×C47#12111yescharacteristic
201C2×C4×C47#12111yescharacteristic
202C2×C4×C47#12111yescharacteristic
203C2×C4×C47C2×SD1612yes—
204C2×C4×C47#13512yes—
205C2×C4×C47C22×D411yescharacteristic
206C2×C4×C47C22×D411yescharacteristic
207C2×C4×C47C2×#4111yescharacteristic
208C2×Dic47—41no—
209C2×Dic47—41no—
210Dic94—41no—
211Dic94—41no—
212Dic94—81no—
213C8×C47—42no—
214C47⋊C8—22no—
215C47⋊C8—22no—
216C47⋊C8—41no—
217C47⋊C8—41no—
218C47⋊C8—42no—
219#19409—471nomaximal, Sylow
220C22×C47—21no—
221C22×C47—21no—
222C22×C47—22no—
223C22×C47—41no—
224C22×C47—41no—
225C22×C47—82no—
226C22×C47—82no—
227C22×C47—82no—
228C22×C47—164no—
229C22×C47#46512yes—
230C22×C47C2×#12111yescharacteristic
231C4×C47—21no—
232C4×C47—22no—
233C4×C47—21no—
234C4×C47—21no—
235C4×C47—41no—
236C4×C47—41no—
237C4×C47#46512yes—
238C4×C47C22×SD1611yescharacteristic
239C4×C47C2×#13511yescharacteristic
240C4×C47#57611yescharacteristic
241C4×C47#57711yescharacteristic
242Dic47—81no—
243Dic47—81no—
244#3208—471no—
245#3076—471no—
246#3038D4711yescharacteristic
247#2988—471no—
248#1678—472no—
249#1677—472no—
250#1666—472no—
251#1665—472no—
252#1570—471no—
253#1550—471no—
254C2×#331—471no—
255C2×C47—22no—
256C2×C47—42no—
257C2×C47—162no—
258C2×C47#304511yescharacteristic
259C2×C47C2×#46511yescharacteristic
260C2×C47#306811yescharacteristic
261C2×AGL(1,ℤ/8)—21no—
262#533—471no—
263C2×#127C2×D4711yescharacteristic
264C4×#41C2×D4711yescharacteristic
265#492—472no—
266#486—472no—
267#479—941no—
268#478—941no—
269#477—471no—
270#476—471no—
271#475—471no—
272#474—471no—
273#460—21no—
274#460—941no—
275#460—941no—
276#460C2×D4712yes—
277#459—21no—
278#459C2×D4712yes—
279#450—942no—
280C8×Q8—472no—
281C8×D4—472no—
282C8×D4—942no—
283#431—471no—
284#431—941no—
285#427—941no—
286#423C2×D4711yescharacteristic
287C2×C4⋊C8—471no—
288C2×C4⋊C8—471no—
289C2×C4⋊C8—941no—
290#418—21no—
291#416—941no—
292#334—472no—
293#334—472no—
294#333—472no—
295#332—472no—
296#331—472no—
297#331—472no—
298C47#1940911yescharacteristic, complemented, Sylow
299C2×#41C22⋊D4711yescharacteristic
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303AGL(1,ℤ/8)—44no—
304C2×SD16—22no—
305C2×SD16—942no—
306C2×SD16—942no—
307C2×D8—22no—
308C2×M16—21no—
309C22×C8—942no—
310C22×C8—941no—
311C22×C8—941no—
312C4⋊Q8—471no—
313C4⋊Q8—471no—
314#127—21no—
315#127—21no—
316#127—21no—
317#127C22×D4712yes—
318#125—941no—
319C22⋊Q8—941no—
320C22⋊Q8—941no—
321C4×Q8—941no—
322C4×Q8—941no—
323C4×Q8C22⋊D4711yescharacteristic
324C4×Q8C22×D4711yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D4711yescharacteristic
328C4×D4C22×D4711yescharacteristic
329#117—21no—
330#117—941no—
331#117C22⋊D4711yescharacteristic
332C2×C42—21no—
333C2×C42C22×D4711yescharacteristic
334#107—941no—
335#107—941no—
336#106—941no—
337#106—941no—
338C4⋊C8—21no—
339C4⋊C8—471no—
340C4⋊C8—471no—
341C4⋊C8—472no—
342C4⋊C8—471no—
343C4⋊C8—471no—
344C4⋊C8—942no—
345C4⋊C8—941no—
346C4⋊C8—941no—
347C4⋊C8C22×D4712yes—
348#103—941no—
349#103—941no—
350#102—22no—
351#102—22no—
352#102—942no—
353#102—942no—
354#102—941no—
355#102—941no—
356C22⋊C8—472no—
357C22⋊C8—942no—
358C4×C8—472no—
359C4×C8—942no—
360C24—41no—
361#41—22no—
362#41#1062533514yescomplemented
363C2×Q8—941no—
364C2×Q8—941no—
365C2×Q8C2×C22⋊D4711yescharacteristic
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—82no—
378C2×D4C2×C22⋊D4711yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4D4×D4711yescharacteristic
383C22×C4#1062535211yescharacteristic
384C22×C4C2×C22⋊D4711yescharacteristic
385SD16—44no—
386SD16—3762no—
387SD16—3762no—
388D8—44no—
389M16—42no—
390C2×C8—22no—
391C2×C8—472no—
392C2×C8—472no—
393C2×C8—942no—
394C2×C8—941no—
395C2×C8—941no—
396C2×C8—3762no—
397C2×C8—3761no—
398C2×C8—1882no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—941no—
402C4⋊C4—941no—
403C4⋊C4—941no—
404C4⋊C4—941no—
405C4⋊C4—1881no—
406C4⋊C4—1881no—
407C4⋊C4C2×C22⋊D4711yescharacteristic
408C4⋊C4C2×C22⋊D4711yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—1881no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—1881no—
417C42D4×D4711yescharacteristic
418C42#1062535211yescharacteristic
419C42C2×C22⋊D4711yescharacteristic
420C42C23×D4711yescharacteristic
421C23—21no—
422C23—42no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23#1309399611yescharacteristic
427Q8—21no—
428Q8—3761no—
429Q8—1881no—
430Q8—1881no—
431Q8C2×#1062533512yes—
432D4—24no—
433D4—21no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—82no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4C2×#1062533512yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—1881no—
454C2×C4—1881no—
455C2×C4#1309399611yescharacteristic
456C2×C4#1309399611yescharacteristic
457C2×C4#1309399611yescharacteristic
458C2×C4C2×#1062533512yes—
459C2×C4#1309404412yes—
460C2×C4C2×D4×D4711yescharacteristic
461C2×C4C22×C22⋊D4711yescharacteristic
462C2×C4C2×#1062535211yescharacteristic
463C8—42no—
464C8—942no—
465C8—942no—
466C8—1881no—
467C8—1881no—
468C8—1882no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—164no—
478C22#1568935012yes—
479C22C2×#1309399611yescharacteristic
480C4—21no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—3761no—
487C4—3761no—
488C4#1568935012yes—
489C4C22×#1062533511yescharacteristic
490C4C2×#1309404411yescharacteristic
491C4#1568992711yescharacteristic
492C4#1568993311yescharacteristic
493C2—22no—
494C2—42no—
495C2—162no—
496C2#2025293111yescharacteristic
497C2C2×#1568935011yescharacteristic
498C2#2025311311yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
Fitting, prime radicalC47×#3038C2
socleC22×C47C2×#121
hypercenter#3038D47
hypocenterC47#19409
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #20254329C2semidirect
2. #20253176C2semidirect
3. #20252499C2semidirect
4. #20242802C2semidirect
5. #20242795C2semidirect
6. #20242734C2semidirect
7. #20242727C2semidirect
8. #20242160C2semidirect
9. #20242079C2semidirect
10. C2×#15688867C2semidirect
17. #15689418C22semidirect
18. #15689413C22semidirect
19. #15689408C22semidirect
20. #15689403C22semidirect
24. Q8×C47⋊C8C22semidirect
26. D4×C47⋊C8C22semidirect
34. #15688873C22semidirect
35. #15688873C22semidirect
36. #15688871C22semidirect
37. #15688868C22semidirect
112. #41×C47SD16semidirect
298. C47#19409semidirect
362. #41#10625335semidirect

Maximal subgroups

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

subgroupindex
#202543292
#202531762
#202524992
#202428022
#202427952
#202427342
#202427272
#202421602
#202420792
C2×#156888672
C47×#30382
#1940947