Finite Group Databasebeta

← #26201681

Subgroups — #26201681

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#20254331C211yesmaximal, characteristic, complemented
2#20253176C211yesmaximal, characteristic, complemented
3#20252868C211yesmaximal, characteristic
4#20242804C212yesmaximal, complemented
5#20242797C212yesmaximal, complemented
6#20242736C212yesmaximal, complemented
7#20242729C212yesmaximal, complemented
8#20242159C211yesmaximal, characteristic, complemented
9#20242076C211yesmaximal, characteristic, complemented
10C2×#15688866C211yesmaximal, characteristic, complemented
11C47×#2988C211yesmaximal, characteristic, complemented
12C2×#13094049—21no—
13#15689680C2211yescharacteristic
14C2×#13094021C2211yescharacteristic
15#15689496C2212yes—
16#15689470C2212yes—
17#15689430—21no—
18#15689423—21no—
19#15689418C2211yescharacteristic, complemented
20#15689413C2211yescharacteristic, complemented
21#15689408C2211yescharacteristic, complemented
22#15689403C2211yescharacteristic, complemented
23#15689324—21no—
24#15689324—21no—
25#15689323—21no—
26#15689323C2212yes—
27#15689318—21no—
28#15689318C2212yes—
29#15689276—22no—
30#15689159—21no—
31#15689147C2211yescharacteristic
32C2×C4⋊(C47⋊C8)C2211yescharacteristic
33#15689127—21no—
34#15689119—21no—
35#15688874C2212yescomplemented
36#15688874C2212yescomplemented
37#15688871C2212yescomplemented
38#15688870C2212yescomplemented
39#15688866C2212yes—
40#15688866C2212yes—
41C4×#41×C47C2211yescharacteristic
42C8×Q8×C47C2212yescomplemented
43C8×D4×C47—22no—
44C8×D4×C47C2212yescomplemented
45C47×#431—21no—
46C47×#431C2211yescharacteristic
47C2×C4⋊C8×C47—21no—
48C2×C4⋊C8×C47C2211yescharacteristic
49C22×D188—21no—
50C22×D188—21no—
51C22×D188—21no—
52#13094049—44no—
53C2×#10625325—22no—
54C2×#10625325—22no—
55C2×#10625336—22no—
56C2×#10625334—22no—
57C2×#10625329—21no—
58C22×C47⋊C8—21no—
59#13094024C2311yescharacteristic
60#13094024C2311yescharacteristic
61#13094021—21no—
62#13094021—21no—
63#13094021—21no—
64#13094021C2312yes—
65#13094016—21no—
66#13094003—21no—
67#13094003—21no—
68C4×Dic94—21no—
69C4×Dic94—21no—
70#13093983—21no—
71#13093957—21no—
72#13093957—21no—
73#13093955—21no—
74#13093955—21no—
75C4⋊(C47⋊C8)—21no—
76C4⋊(C47⋊C8)—21no—
77C4⋊(C47⋊C8)C2312yes—
78C4⋊(C47⋊C8)C2311yescharacteristic
79C4⋊(C47⋊C8)C2311yescharacteristic
80#13093948—21no—
81#13093948—21no—
82#13093946—22no—
83#13093946—22no—
84#13093945—21no—
85#13093945—21no—
86#13093944—22no—
87#13093944—22no—
88C2×#41×C47D411yescharacteristic
89C22×C8×C47—21no—
90C22×C8×C47—22no—
91C4×Q8×C47D411yescharacteristic
92C4×Q8×C47C2311yescharacteristic
93C4×D4×C47—21no—
94C4×D4×C47—21no—
95C4×D4×C47D411yescharacteristic
96C4×D4×C47C2311yescharacteristic
97#117×C47—21no—
98#117×C47D411yescharacteristic
99C2×C42×C47—21no—
100C2×C42×C47C2311yescharacteristic
101C4⋊C8×C47—22no—
102C4⋊C8×C47—21no—
103C4⋊C8×C47D411yescharacteristic
104C4⋊C8×C47D411yescharacteristic
105C4⋊C8×C47C2312yes—
106C22⋊C8×C47—22no—
107C22⋊C8×C47D412yes—
108C4×C8×C47—22no—
109C4×C8×C47C2312yes—
110C23×D47—41no—
111#41×C47—22no—
112#41×C47SD1614yescomplemented
113C2×Q8×C47C2×D411yescharacteristic
114C2×D4×C47—21no—
115C2×D4×C47C2×D411yescharacteristic
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47#4111yescharacteristic
122C2×D188—22no—
123C2×D188—22no—
124C2×D188—22no—
125C2×D188—42no—
126C2×D188—42no—
127C2×D188—41no—
128C2×D188—41no—
129C2×D188—41no—
130C2×D188—81no—
131C2×D188—81no—
132C2×D188—82no—
133C2×Dic94—21no—
134C2×Dic94—21no—
135C2×C8×C47—22no—
136C2×C8×C47—82no—
137C2×C8×C47—81no—
138C2×C8×C47C2×D412yes—
139C2×C8×C47C2×D412yes—
140C4⋊C4×C47—21no—
141C4⋊C4×C47—21no—
142C4⋊C4×C47C2×D411yescharacteristic
143C4⋊C4×C47C2×D411yescharacteristic
144C22⋊C4×C47—21no—
145C22⋊C4×C47—21no—
146C22⋊C4×C47—21no—
147C42×C47—21no—
148C42×C47—21no—
149C42×C47—21no—
150C42×C47C2×D411yescharacteristic
151C42×C47C2×D411yescharacteristic
152C42×C47#4111yescharacteristic
153C42×C47C2411yescharacteristic
154C22⋊Dic47—41no—
155#10625336—44no—
156#10625334—44no—
157C4⋊Dic47—21no—
158C4⋊Dic47—21no—
159C4⋊Dic47—21no—
160C4⋊Dic47—21no—
161Dic47⋊C4—41no—
162Dic47⋊C4—41no—
163C4×Dic47—41no—
164#10625329—42no—
165C2×C47⋊C8—22no—
166C2×C47⋊C8—21no—
167C2×C47⋊C8—21no—
168C2×C47⋊C8—42no—
169#10625325—82no—
170#10625325—82no—
171C23×C47—21no—
172C23×C47#12111yescharacteristic
173C22×D47—42no—
174C22×D47—82no—
175C22×D47—82no—
176C22×D47—82no—
177Q8×C47—21no—
178Q8×C47C2×SD1612yes—
179D4×C47—24no—
180D4×C47—21no—
181D4×C47C2×SD1612yes—
182C2×C4×C47—21no—
183C2×C4×C47—21no—
184C2×C4×C47—21no—
185C2×C4×C47—24no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—41no—
189C2×C4×C47—41no—
190C2×C4×C47—41no—
191C2×C4×C47—81no—
192C2×C4×C47#12111yescharacteristic
193C2×C4×C47#12111yescharacteristic
194C2×C4×C47#12111yescharacteristic
195C2×C4×C47C2×SD1612yes—
196C2×C4×C47#13512yes—
197C2×C4×C47C22×D411yescharacteristic
198C2×C4×C47C22×D411yescharacteristic
199C2×C4×C47C2×#4111yescharacteristic
200C2×Dic47—41no—
201C2×Dic47—41no—
202D188—42no—
203D188—42no—
204D188—42no—
205D188—42no—
206D188—82no—
207D188—82no—
208D188—82no—
209D188—82no—
210Dic94—41no—
211Dic94—41no—
212Dic94—81no—
213C8×C47—22no—
214C8×C47—22no—
215C8×C47—42no—
216C47⋊C8—42no—
217C47⋊C8—41no—
218C47⋊C8—41no—
219#19409—471nomaximal, Sylow
220C22×C47—21no—
221C22×C47—21no—
222C22×C47—22no—
223C22×C47—41no—
224C22×C47—41no—
225C22×C47#46512yes—
226C22×C47C2×#12111yescharacteristic
227C2×D47—82no—
228C2×D47—82no—
229C2×D47—82no—
230C2×D47—164no—
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#3038—471no—
247#2988D4711yescharacteristic, complemented
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#304511yescharacteristic
258C2×C47C2×#46511yescharacteristic
259C2×C47#306811yescharacteristic
260D47—162no—
261C2×AGL(1,ℤ/8)—941no—
262#533—471no—
263C2×#127—471no—
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—472no—
274#460—941no—
275#460—941no—
276#460—941no—
277#459—472no—
278#459—941no—
279#450—942no—
280C8×Q8C2×D4712yescomplemented
281C8×D4—22no—
282C8×D4C2×D4712yescomplemented
283#431—21no—
284#431C2×D4711yescharacteristic
285#427—941no—
286#423—471no—
287C2×C4⋊C8—21no—
288C2×C4⋊C8—471no—
289C2×C4⋊C8C2×D4711yescharacteristic
290#418—941no—
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×#41D18811yescharacteristic
300C22×D4—941no—
301C22×D4—941no—
302C22×D4—941no—
303AGL(1,ℤ/8)—1884no—
304C2×SD16—942no—
305C2×SD16—942no—
306C2×SD16—942no—
307C2×D8—942no—
308C2×M16—941no—
309C22×C8—21no—
310C22×C8—22no—
311C22×C8—941no—
312C4⋊Q8—471no—
313C4⋊Q8—471no—
314#127—472no—
315#127—941no—
316#127—941no—
317#127—941no—
318#125—941no—
319C22⋊Q8—941no—
320C22⋊Q8—941no—
321C4×Q8—941no—
322C4×Q8—941no—
323C4×Q8D18811yescharacteristic
324C4×Q8C22×D4711yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4D18811yescharacteristic
328C4×D4C22×D4711yescharacteristic
329#117—21no—
330#117—941no—
331#117D18811yescharacteristic
332C2×C42—21no—
333C2×C42C22×D4711yescharacteristic
334#107—941no—
335#107—941no—
336#106—941no—
337#106—941no—
338C4⋊C8—22no—
339C4⋊C8—21no—
340C4⋊C8—471no—
341C4⋊C8—471no—
342C4⋊C8—472no—
343C4⋊C8—941no—
344C4⋊C8—941no—
345C4⋊C8C22⋊D4711yescharacteristic
346C4⋊C8C22⋊D4711yescharacteristic
347C4⋊C8C22×D4712yes—
348#103—941no—
349#103—941no—
350#102—941no—
351#102—941no—
352#102—942no—
353#102—942no—
354#102—942no—
355#102—942no—
356C22⋊C8—22no—
357C22⋊C8C22⋊D4712yes—
358C4×C8—22no—
359C4×C8C22×D4712yes—
360C24—1881no—
361#41—22no—
362#41#1062532514yescomplemented
363C2×Q8—941no—
364C2×Q8—941no—
365C2×Q8C2×D18811yescharacteristic
366C2×D4—21no—
367C2×D4—942no—
368C2×D4—942no—
369C2×D4—942no—
370C2×D4—3761no—
371C2×D4—3761no—
372C2×D4—3762no—
373C2×D4—1881no—
374C2×D4—1881no—
375C2×D4—1881no—
376C2×D4—1882no—
377C2×D4—1882no—
378C2×D4C2×D18811yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4C2×D18811yescharacteristic
383C22×C4#1062535011yescharacteristic
384C22×C4C2×C22⋊D4711yescharacteristic
385SD16—3762no—
386SD16—3762no—
387SD16—1884no—
388D8—1884no—
389M16—1882no—
390C2×C8—22no—
391C2×C8—82no—
392C2×C8—81no—
393C2×C8—942no—
394C2×C8—941no—
395C2×C8—941no—
396C2×C8—1882no—
397C2×C8C2×C22⋊D4712yes—
398C2×C8C2×C22⋊D4712yes—
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×D18811yescharacteristic
408C4⋊C4C2×D18811yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—1881no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—1881no—
417C42C2×D18811yescharacteristic
418C42#1062535011yescharacteristic
419C42C2×C22⋊D4711yescharacteristic
420C42C23×D4711yescharacteristic
421C23—21no—
422C23—3762no—
423C23—3762no—
424C23—3762no—
425C23—1882no—
426C23#1309399811yescharacteristic
427Q8—21no—
428Q8—3761no—
429Q8—1881no—
430Q8—1881no—
431Q8C2×#1062532512yes—
432D4—24no—
433D4—21no—
434D4—3762no—
435D4—3762no—
436D4—3762no—
437D4—3762no—
438D4—1882no—
439D4—1882no—
440D4—1882no—
441D4—1882no—
442D4C2×#1062532512yes—
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#1309399811yescharacteristic
456C2×C4#1309399811yescharacteristic
457C2×C4#1309399811yescharacteristic
458C2×C4C2×#1062532512yes—
459C2×C4#1309404612yes—
460C2×C4C22×C22⋊D4711yescharacteristic
461C2×C4C22×D18811yescharacteristic
462C2×C4C2×#1062535011yescharacteristic
463C8—22no—
464C8—22no—
465C8—42no—
466C8—1882no—
467C8—1881no—
468C8—1881no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—3762no—
475C22—3762no—
476C22—3762no—
477C22—7524no—
478C22#1568935212yes—
479C22C2×#1309399811yescharacteristic
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—3761no—
487C4—3761no—
488C4#1568935212yes—
489C4C22×#1062532511yescharacteristic
490C4C2×#1309404611yescharacteristic
491C4#1568992611yescharacteristic
492C4#1568993111yescharacteristic
493C2—22no—
494C2—42no—
495C2—7522no—
496C2#2025292711yescharacteristic
497C2C2×#1568935211yescharacteristic
498C2#2025311611yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2988C2
socleC22×C47C2×#121
prime radical#20252868C2
hypercenter#2988D47
hypocenterC47#19409
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #20254331C2semidirect
2. #20253176C2semidirect
4. #20242804C2semidirect
5. #20242797C2semidirect
6. #20242736C2semidirect
7. #20242729C2semidirect
8. #20242159C2semidirect
9. #20242076C2semidirect
10. C2×#15688866C2semidirect
11. C47×#2988C2semidirect
19. #15689418C22semidirect
20. #15689413C22semidirect
21. #15689408C22semidirect
22. #15689403C22semidirect
35. #15688874C22semidirect
36. #15688874C22semidirect
37. #15688871C22semidirect
38. #15688870C22semidirect
42. C8×Q8×C47C22semidirect
44. C8×D4×C47C22semidirect
112. #41×C47SD16semidirect
247. #2988D47semidirect
280. C8×Q8C2×D47semidirect
282. C8×D4C2×D47semidirect
298. C47#19409semidirect
362. #41#10625325semidirect

Maximal subgroups

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

subgroupindex
#202543312
#202531762
#202528682
#202428042
#202427972
#202427362
#202427292
#202421592
#202420762
C2×#156888662
C47×#29882
#1940947