Finite Group Databasebeta

← #26201679

Subgroups — #26201679

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#20254326C211yesmaximal, characteristic, complemented
2#20252868C211yesmaximal, characteristic
3#20252499C211yesmaximal, characteristic, complemented
4#20242806C212yesmaximal, complemented
5#20242799C212yesmaximal, complemented
6#20242738C212yesmaximal, complemented
7#20242731C212yesmaximal, complemented
8#20242158C211yesmaximal, characteristic, complemented
9#20242078C211yesmaximal, characteristic, complemented
10C2×#15688866C211yesmaximal, characteristic, complemented
11C47×#3076C211yesmaximal, characteristic, complemented
12C2×#13094049—21no—
13C2×#13094021C2211yescharacteristic
14#15689497C2212yes—
15#15689471C2212yes—
16#15689323—21no—
17#15689323—21no—
18#15689323—21no—
19#15689323C2212yes—
20#15689318—21no—
21#15689318C2212yes—
22#15689273—22no—
23Q8×C47⋊C8C2212yescomplemented
24D4×C47⋊C8—22no—
25D4×C47⋊C8C2212yescomplemented
26#15689176—21no—
27#15689176C2211yescharacteristic
28#15689147C2211yescharacteristic
29C2×C4⋊(C47⋊C8)—21no—
30C2×C4⋊(C47⋊C8)C2211yescharacteristic
31#15689127—21no—
32#15688873C2212yescomplemented
33#15688873C2212yescomplemented
34#15688872C2212yescomplemented
35#15688869C2212yescomplemented
36#15688866C2212yes—
37#15688866C2212yes—
38C47×#533C2211yescharacteristic
39C4×#41×C47C2211yescharacteristic
40C47×#479—21no—
41C47×#478—21no—
42C47×#477C2211yescharacteristic, complemented
43C47×#476C2211yescharacteristic, complemented
44C47×#475C2211yescharacteristic, complemented
45C47×#474C2211yescharacteristic, complemented
46C47×#427—21no—
47C2×C4⋊C8×C47C2211yescharacteristic
48C47×#416—21no—
49C22×D188—21no—
50C22×D188—21no—
51C22×D188—21no—
52#13094049—44no—
53C2×#10625336—22no—
54C2×#10625336—22no—
55C2×#10625336—22no—
56C2×#10625334—22no—
57C2×#10625329—21no—
58C22×C47⋊C8—21no—
59C22×C47⋊C8—22no—
60#13094021—21no—
61#13094021—21no—
62#13094021—21no—
63#13094021C2312yes—
64C4⋊(C47⋊C8)—21no—
65C4⋊(C47⋊C8)—21no—
66C4⋊(C47⋊C8)—22no—
67C4⋊(C47⋊C8)D411yescharacteristic
68C4⋊(C47⋊C8)D411yescharacteristic
69C4⋊(C47⋊C8)C2312yes—
70C4⋊(C47⋊C8)C2312yes—
71#13093944—22no—
72#13093944—22no—
73#13093944—22no—
74#13093944—22no—
75C22⋊(C47⋊C8)—22no—
76C22⋊(C47⋊C8)D412yes—
77C4×C47⋊C8—22no—
78C4×C47⋊C8C2312yes—
79C2×#41×C47D411yescharacteristic
80C22×C8×C47—21no—
81C4⋊Q8×C47C2311yescharacteristic
82C4⋊Q8×C47C2311yescharacteristic
83#125×C47—21no—
84C22⋊Q8×C47—21no—
85C22⋊Q8×C47—21no—
86C4×Q8×C47—21no—
87C4×Q8×C47—21no—
88C4×Q8×C47D411yescharacteristic
89C4×Q8×C47C2311yescharacteristic
90C4×D4×C47—21no—
91C4×D4×C47—21no—
92C4×D4×C47D411yescharacteristic
93C4×D4×C47C2311yescharacteristic
94#117×C47—21no—
95#117×C47—21no—
96#117×C47D411yescharacteristic
97C2×C42×C47—21no—
98C2×C42×C47C2311yescharacteristic
99#107×C47—21no—
100#107×C47—21no—
101#106×C47—21no—
102#106×C47—21no—
103C4⋊C8×C47—21no—
104C4⋊C8×C47C2311yescharacteristic
105C4⋊C8×C47C2311yescharacteristic
106#103×C47—21no—
107#103×C47—21no—
108#102×C47—21no—
109#102×C47—21no—
110C23×D47—41no—
111#41×C47—22no—
112#41×C47SD1614yescomplemented
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47C2×D411yescharacteristic
116C2×D4×C47—21no—
117C2×D4×C47C2×D411yescharacteristic
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47#4111yescharacteristic
124C2×D188—22no—
125C2×D188—22no—
126C2×D188—22no—
127C2×D188—42no—
128C2×D188—42no—
129C2×D188—41no—
130C2×D188—41no—
131C2×D188—41no—
132C2×D188—81no—
133C2×D188—81no—
134C2×D188—82no—
135C2×C8×C47—21no—
136C2×C8×C47—21no—
137C2×C8×C47—42no—
138C4⋊C4×C47—21no—
139C4⋊C4×C47—21no—
140C4⋊C4×C47—21no—
141C4⋊C4×C47—21no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—41no—
145C4⋊C4×C47—41no—
146C4⋊C4×C47C2×D411yescharacteristic
147C4⋊C4×C47C2×D411yescharacteristic
148C22⋊C4×C47—21no—
149C22⋊C4×C47—21no—
150C22⋊C4×C47—21no—
151C22⋊C4×C47—41no—
152C42×C47—21no—
153C42×C47—21no—
154C42×C47—21no—
155C42×C47—41no—
156C42×C47C2×D411yescharacteristic
157C42×C47C2×D411yescharacteristic
158C42×C47#4111yescharacteristic
159C42×C47C2411yescharacteristic
160#10625336—44no—
161#10625336—82no—
162#10625336—82no—
163#10625334—44no—
164#10625329—42no—
165C2×C47⋊C8—22no—
166C2×C47⋊C8—22no—
167C2×C47⋊C8—82no—
168C2×C47⋊C8—81no—
169C2×C47⋊C8C2×D412yes—
170C2×C47⋊C8C2×D412yes—
171C23×C47—21no—
172C23×C47#12111yescharacteristic
173C22×D47—42no—
174C22×D47—82no—
175C22×D47—82no—
176C22×D47—82no—
177Q8×C47—21no—
178Q8×C47—41no—
179Q8×C47—41no—
180Q8×C47—81no—
181Q8×C47C2×SD1612yes—
182D4×C47—24no—
183D4×C47—21no—
184D4×C47C2×SD1612yes—
185C2×C4×C47—21no—
186C2×C4×C47—21no—
187C2×C4×C47—21no—
188C2×C4×C47—24no—
189C2×C4×C47—21no—
190C2×C4×C47—21no—
191C2×C4×C47—41no—
192C2×C4×C47—41no—
193C2×C4×C47—41no—
194C2×C4×C47—41no—
195C2×C4×C47—41no—
196C2×C4×C47—81no—
197C2×C4×C47#12111yescharacteristic
198C2×C4×C47#12111yescharacteristic
199C2×C4×C47#12111yescharacteristic
200C2×C4×C47C2×SD1612yes—
201C2×C4×C47#13512yes—
202C2×C4×C47C22×D411yescharacteristic
203C2×C4×C47C22×D411yescharacteristic
204C2×C4×C47C2×#4111yescharacteristic
205D188—42no—
206D188—42no—
207D188—42no—
208D188—42no—
209D188—82no—
210D188—82no—
211D188—82no—
212D188—82no—
213C8×C47—41no—
214C8×C47—41no—
215C47⋊C8—22no—
216C47⋊C8—22no—
217C47⋊C8—42no—
218C47⋊C8—42no—
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—81no—
238C4×C47—81no—
239C4×C47#46512yes—
240C4×C47C22×SD1611yescharacteristic
241C4×C47C2×#13511yescharacteristic
242C4×C47#57611yescharacteristic
243C4×C47#57711yescharacteristic
244#3208—471no—
245#3076D4711yescharacteristic, complemented
246#3038—471no—
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#304511yescharacteristic
258C2×C47C2×#46511yescharacteristic
259C2×C47#306811yescharacteristic
260D47—162no—
261C2×AGL(1,ℤ/8)—941no—
262#533C2×D4711yescharacteristic
263C2×#127—471no—
264C4×#41C2×D4711yescharacteristic
265#492—472no—
266#486—472no—
267#479—21no—
268#478—21no—
269#477C2×D4711yescharacteristic, complemented
270#476C2×D4711yescharacteristic, complemented
271#475C2×D4711yescharacteristic, complemented
272#474C2×D4711yescharacteristic, complemented
273#460—472no—
274#460—941no—
275#460—941no—
276#460—941no—
277#459—472no—
278#459—941no—
279#450—942no—
280C8×Q8—472no—
281C8×D4—472no—
282C8×D4—942no—
283#431—471no—
284#431—941no—
285#427—21no—
286#423—471no—
287C2×C4⋊C8—471no—
288C2×C4⋊C8—941no—
289C2×C4⋊C8C2×D4711yescharacteristic
290#418—941no—
291#416—21no—
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—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—942no—
311C22×C8—941no—
312C4⋊Q8C22×D4711yescharacteristic
313C4⋊Q8C22×D4711yescharacteristic
314#127—472no—
315#127—941no—
316#127—941no—
317#127—941no—
318#125—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8C22⋊D4711yescharacteristic
324C4×Q8C22×D4711yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D4711yescharacteristic
328C4×D4C22×D4711yescharacteristic
329#117—21no—
330#117—21no—
331#117C22⋊D4711yescharacteristic
332C2×C42—21no—
333C2×C42C22×D4711yescharacteristic
334#107—21no—
335#107—21no—
336#106—21no—
337#106—21no—
338C4⋊C8—21no—
339C4⋊C8—471no—
340C4⋊C8—471no—
341C4⋊C8—472no—
342C4⋊C8—472no—
343C4⋊C8—942no—
344C4⋊C8—941no—
345C4⋊C8—941no—
346C4⋊C8C22×D4711yescharacteristic
347C4⋊C8C22×D4711yescharacteristic
348#103—21no—
349#103—21no—
350#102—21no—
351#102—21no—
352#102—942no—
353#102—942no—
354#102—942no—
355#102—942no—
356C22⋊C8—472no—
357C22⋊C8—942no—
358C4×C8—472no—
359C4×C8—942no—
360C24—1881no—
361#41—22no—
362#41#1062533614yescomplemented
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8C2×C22⋊D4711yescharacteristic
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×C22⋊D4711yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4D4×D4711yescharacteristic
383C22×C4#1062535211yescharacteristic
384C22×C4C2×C22⋊D4711yescharacteristic
385SD16—3762no—
386SD16—3762no—
387SD16—1884no—
388D8—1884no—
389M16—1882no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—42no—
393C2×C8—472no—
394C2×C8—472no—
395C2×C8—942no—
396C2×C8—942no—
397C2×C8—3762no—
398C2×C8—3761no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4C2×C22⋊D4711yescharacteristic
408C4⋊C4C2×C22⋊D4711yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—41no—
417C42D4×D4711yescharacteristic
418C42#1062535211yescharacteristic
419C42C2×C22⋊D4711yescharacteristic
420C42C23×D4711yescharacteristic
421C23—21no—
422C23—3762no—
423C23—3762no—
424C23—3762no—
425C23—1882no—
426C23#1309399611yescharacteristic
427Q8—21no—
428Q8—41no—
429Q8—41no—
430Q8—81no—
431Q8C2×#1062533612yes—
432D4—24no—
433D4—21no—
434D4—3762no—
435D4—3762no—
436D4—3762no—
437D4—3762no—
438D4—1882no—
439D4—1882no—
440D4—1882no—
441D4—1882no—
442D4C2×#1062533612yes—
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—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4#1309399611yescharacteristic
456C2×C4#1309399611yescharacteristic
457C2×C4#1309399611yescharacteristic
458C2×C4C2×#1062533612yes—
459C2×C4#1309404412yes—
460C2×C4C2×D4×D4711yescharacteristic
461C2×C4C22×C22⋊D4711yescharacteristic
462C2×C4C2×#1062535211yescharacteristic
463C8—41no—
464C8—41no—
465C8—942no—
466C8—942no—
467C8—1882no—
468C8—1882no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—3762no—
475C22—3762no—
476C22—3762no—
477C22—7524no—
478C22#1568935412yes—
479C22C2×#1309399611yescharacteristic
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—81no—
488C4#1568935412yes—
489C4C22×#1062533611yescharacteristic
490C4C2×#1309404411yescharacteristic
491C4#1568992511yescharacteristic
492C4#1568993011yescharacteristic
493C2—22no—
494C2—42no—
495C2—7522no—
496C2#2025292511yescharacteristic
497C2C2×#1568935411yescharacteristic
498C2#2025311811yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3076C2
socleC22×C47C2×#121
prime radical#20252868C2
hypercenter#3076D47
hypocenterC47#19409
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #20254326C2semidirect
3. #20252499C2semidirect
4. #20242806C2semidirect
5. #20242799C2semidirect
6. #20242738C2semidirect
7. #20242731C2semidirect
8. #20242158C2semidirect
9. #20242078C2semidirect
10. C2×#15688866C2semidirect
11. C47×#3076C2semidirect
23. Q8×C47⋊C8C22semidirect
25. D4×C47⋊C8C22semidirect
32. #15688873C22semidirect
33. #15688873C22semidirect
34. #15688872C22semidirect
35. #15688869C22semidirect
42. C47×#477C22semidirect
43. C47×#476C22semidirect
44. C47×#475C22semidirect
45. C47×#474C22semidirect
112. #41×C47SD16semidirect
245. #3076D47semidirect
269. #477C2×D47semidirect
270. #476C2×D47semidirect
271. #475C2×D47semidirect
272. #474C2×D47semidirect
298. C47#19409semidirect
362. #41#10625336semidirect

Maximal subgroups

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

subgroupindex
#202543262
#202528682
#202524992
#202428062
#202427992
#202427382
#202427312
#202421582
#202420782
C2×#156888662
C47×#30762
#1940947