Finite Group Databasebeta

← #26201668

Subgroups — #26201668

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254310C211yesmaximal, characteristic, complemented
2#20253203C211yesmaximal, characteristic, complemented
3#20252499C211yesmaximal, characteristic, complemented
4#20242820C212yesmaximal, complemented
5#20242813C212yesmaximal, complemented
6#20242734C212yesmaximal, complemented
7#20242727C212yesmaximal, complemented
8#20242158C211yesmaximal, characteristic, complemented
9#20242065C211yesmaximal, characteristic, complemented
10C2×#15688868C211yesmaximal, characteristic, complemented
11C47×#3035C211yesmaximal, characteristic
12C2×#13094024C2211yescharacteristic
13#15689495C2212yes—
14#15689475C2212yes—
15#15689418—21no—
16#15689413—21no—
17#15689408C2212yes—
18#15689403C2212yes—
19#15689328—21no—
20#15689328—21no—
21#15689261—22no—
22Q8×C47⋊C8C2212yes—
23D4×C47⋊C8—22no—
24D4×C47⋊C8C2212yes—
25#15689176—21no—
26#15689176C2211yescharacteristic
27#15689166—21no—
28#15689147C2211yescharacteristic
29C2×C4⋊(C47⋊C8)—21no—
30C2×C4⋊(C47⋊C8)C2211yescharacteristic
31#15689124—21no—
32#15688873C2212yes—
33#15688873C2212yes—
34#15688869C2212yes—
35#15688868C2212yes—
36#15688868C2212yes—
37#15688867C2212yes—
38C2×#135×C47—21no—
39C47×#532C2211yescharacteristic
40C4×#41×C47C2211yescharacteristic
41C47×#464—21no—
42C47×#463—21no—
43C47×#462C2211yescharacteristic
44C47×#461C2211yescharacteristic
45C47×#460C2211yescharacteristic
46C47×#459C2211yescharacteristic
47C2×C4⋊C8×C47C2211yescharacteristic
48C47×#416—21no—
49C22×Dic94—21no—
50C2×#10625335—22no—
51C2×#10625335—22no—
52C2×#10625329—21no—
53C22×C47⋊C8—21no—
54C22×C47⋊C8—22no—
55#13094024—21no—
56#13094024—21no—
57#13094024—21no—
58#13094024C2312yes—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—21no—
61#13093957—22no—
62#13093955—22no—
63C4⋊(C47⋊C8)—21no—
64C4⋊(C47⋊C8)—21no—
65C4⋊(C47⋊C8)—22no—
66C4⋊(C47⋊C8)D411yescharacteristic
67C4⋊(C47⋊C8)D411yescharacteristic
68C4⋊(C47⋊C8)C2312yes—
69C4⋊(C47⋊C8)C2312yes—
70#13093948—22no—
71#13093947—22no—
72#13093947—22no—
73#13093945—22no—
74C22⋊(C47⋊C8)—22no—
75C22⋊(C47⋊C8)D412yes—
76C4×C47⋊C8—22no—
77C4×C47⋊C8C2312yes—
78C2×#41×C47—21no—
79C2×#41×C47D411yescharacteristic
80#135×C47—44no—
81C2×Q16×C47—21no—
82C2×SD16×C47—21no—
83C2×SD16×C47—21no—
84C2×D8×C47—21no—
85C22×C8×C47—21no—
86C4⋊Q8×C47C2311yescharacteristic
87#127×C47C2311yescharacteristic
88#124×C47—21no—
89#121×C47—21no—
90#121×C47—21no—
91C4×Q8×C47D411yescharacteristic
92C4×Q8×C47C2311yescharacteristic
93C4×D4×C47—21no—
94C4×D4×C47—21no—
95C4×D4×C47—21no—
96C4×D4×C47—21no—
97C4×D4×C47D411yescharacteristic
98C4×D4×C47C2311yescharacteristic
99#117×C47—21no—
100#117×C47D411yescharacteristic
101C2×C42×C47—21no—
102C2×C42×C47C2311yescharacteristic
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—
110#41×C47—22no—
111#41×C47—42no—
112#41×C47—42no—
113#41×C47SD1614yescomplemented
114C2×Q8×C47—21no—
115C2×Q8×C47C2×D411yescharacteristic
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—21no—
120C2×D4×C47—41no—
121C2×D4×C47C2×D411yescharacteristic
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—41no—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47#4111yescharacteristic
129C22×Dic47—41no—
130C2×Dic94—22no—
131C2×Dic94—41no—
132C2×Dic94—81no—
133C2×Dic94—81no—
134Q16×C47—42no—
135SD16×C47—42no—
136SD16×C47—42no—
137D8×C47—42no—
138C2×C8×C47—21no—
139C2×C8×C47—21no—
140C2×C8×C47—42no—
141C4⋊C4×C47—21no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47C2×D411yescharacteristic
146C4⋊C4×C47C2×D411yescharacteristic
147C22⋊C4×C47—21no—
148C22⋊C4×C47—21no—
149C22⋊C4×C47—21no—
150C22⋊C4×C47—41no—
151C22⋊C4×C47—41no—
152C42×C47—21no—
153C42×C47—21no—
154C42×C47—21no—
155C42×C47C2×D411yescharacteristic
156C42×C47C2×D411yescharacteristic
157C42×C47#4111yescharacteristic
158C42×C47C2411yescharacteristic
159#10625335—82no—
160#10625335—82no—
161C4⋊Dic47—22no—
162C4⋊Dic47—22no—
163C4⋊Dic47—41no—
164C4⋊Dic47—41no—
165#10625329—42no—
166C2×C47⋊C8—22no—
167C2×C47⋊C8—22no—
168C2×C47⋊C8—82no—
169C2×C47⋊C8—81no—
170C2×C47⋊C8C2×D412yes—
171C2×C47⋊C8C2×D412yes—
172C23×C47—21no—
173C23×C47—41no—
174C23×C47#12111yescharacteristic
175Q8×C47—21no—
176Q8×C47—41no—
177Q8×C47—41no—
178Q8×C47C2×SD1612yes—
179D4×C47—24no—
180D4×C47—21no—
181D4×C47—41no—
182D4×C47—41no—
183D4×C47—41no—
184D4×C47—41no—
185D4×C47—81no—
186D4×C47—82no—
187D4×C47C2×SD1612yes—
188C2×C4×C47—21no—
189C2×C4×C47—21no—
190C2×C4×C47—21no—
191C2×C4×C47—24no—
192C2×C4×C47—21no—
193C2×C4×C47—21no—
194C2×C4×C47—41no—
195C2×C4×C47—41no—
196C2×C4×C47—41no—
197C2×C4×C47—41no—
198C2×C4×C47—81no—
199C2×C4×C47—82no—
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—42no—
209C2×Dic47—82no—
210Dic94—42no—
211Dic94—42no—
212Dic94—82no—
213C8×C47—41no—
214C8×C47—41no—
215C47⋊C8—22no—
216C47⋊C8—22no—
217C47⋊C8—42no—
218C47⋊C8—42no—
219#19408—471nomaximal, Sylow
220C22×C47—21no—
221C22×C47—21no—
222C22×C47—22no—
223C22×C47—41no—
224C22×C47—41no—
225C22×C47—81no—
226C22×C47—81no—
227C22×C47—81no—
228C22×C47#46512yes—
229C22×C47C2×#12111yescharacteristic
230C4×C47—21no—
231C4×C47—22no—
232C4×C47—21no—
233C4×C47—21no—
234C4×C47—41no—
235C4×C47—41no—
236C4×C47—81no—
237C4×C47#46512yes—
238C4×C47C22×SD1611yescharacteristic
239C4×C47C2×#13511yescharacteristic
240C4×C47#57611yescharacteristic
241C4×C47#57711yescharacteristic
242Dic47—82no—
243#3206—471no—
244#3079—471no—
245#3035D4711yescharacteristic
246#2988—471no—
247#1680—472no—
248#1679—472no—
249#1666—472no—
250#1665—472no—
251#1570—471no—
252#1547—471no—
253C2×#332—471no—
254C2×C47—22no—
255C2×C47—42no—
256C2×C47—161no—
257C2×C47#304511yescharacteristic
258C2×C47C2×#46511yescharacteristic
259C2×C47#306911yescharacteristic
260C2×#135—21no—
261#532C2×D4711yescharacteristic
262C2×C4⋊Q8—471no—
263C4×#41C2×D4711yescharacteristic
264#492—472no—
265#487—472no—
266#477—941no—
267#476—941no—
268#475—472no—
269#474—472no—
270#464—21no—
271#463—21no—
272#462C2×D4711yescharacteristic
273#461—941no—
274#461—941no—
275#461C2×D4711yescharacteristic
276#460C2×D4711yescharacteristic
277#459C2×D4711yescharacteristic
278#448—942no—
279C8×Q8—472no—
280C8×D4—472no—
281C8×D4—942no—
282#431—471no—
283#431—941no—
284#428—941no—
285#423—471no—
286C2×C4⋊C8—471no—
287C2×C4⋊C8—941no—
288C2×C4⋊C8C2×D4711yescharacteristic
289#417—941no—
290#416—21no—
291#334—472no—
292#334—472no—
293#332—472no—
294#332—472no—
295#332—472no—
296#331—472no—
297C47#1940811yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C2×#41C22⋊D4711yescharacteristic
300C22×Q8—941no—
301#135—44no—
302C2×Q16—21no—
303C2×SD16—21no—
304C2×SD16—21no—
305C2×SD16—942no—
306C2×SD16—942no—
307C2×D8—21no—
308C2×M16—941no—
309C22×C8—21no—
310C22×C8—942no—
311C22×C8—941no—
312C4⋊Q8—472no—
313C4⋊Q8—941no—
314C4⋊Q8—941no—
315C4⋊Q8—941no—
316C4⋊Q8C22×D4711yescharacteristic
317#127C22×D4711yescharacteristic
318#124—21no—
319#121—21no—
320#121—21no—
321C4×Q8C22⋊D4711yescharacteristic
322C4×Q8C22×D4711yescharacteristic
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D4711yescharacteristic
328C4×D4C22×D4711yescharacteristic
329#117—21no—
330#117C22⋊D4711yescharacteristic
331C2×C4⋊C4—941no—
332C2×C4⋊C4—941no—
333C2×C42—21no—
334C2×C42C22×D4711yescharacteristic
335#107—942no—
336#106—942no—
337C4⋊C8—21no—
338C4⋊C8—471no—
339C4⋊C8—471no—
340C4⋊C8—472no—
341C4⋊C8—472no—
342C4⋊C8—942no—
343C4⋊C8—941no—
344C4⋊C8—941no—
345C4⋊C8C22×D4711yescharacteristic
346C4⋊C8C22×D4711yescharacteristic
347#103—21no—
348#103—21no—
349#103—942no—
350#103—942no—
351#103—942no—
352#102—21no—
353#102—21no—
354#102—942no—
355C22⋊C8—472no—
356C22⋊C8—942no—
357C4×C8—472no—
358C4×C8—942no—
359#41—22no—
360#41—42no—
361#41—42no—
362#41#1062533514yescomplemented
363C2×Q8—21no—
364C2×Q8—942no—
365C2×Q8—3761no—
366C2×Q8—3761no—
367C2×Q8—1881no—
368C2×Q8C2×C22⋊D4711yescharacteristic
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4C2×C22⋊D4711yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4—1881no—
380C22×C4D4×D4711yescharacteristic
381C22×C4#1062535211yescharacteristic
382C22×C4C2×C22⋊D4711yescharacteristic
383Q16—42no—
384SD16—42no—
385SD16—42no—
386SD16—3762no—
387SD16—3762no—
388D8—42no—
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—942no—
404C4⋊C4—942no—
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—41no—
413C22⋊C4—41no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42D4×D4711yescharacteristic
418C42#1062535211yescharacteristic
419C42C2×C22⋊D4711yescharacteristic
420C42C23×D4711yescharacteristic
421C23—21no—
422C23—41no—
423C23#1309399611yescharacteristic
424Q8—21no—
425Q8—41no—
426Q8—41no—
427Q8—3762no—
428Q8—1882no—
429Q8—1882no—
430Q8C2×#1062533512yes—
431D4—24no—
432D4—21no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—82no—
439D4C2×#1062533512yes—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—3762no—
453C2×C4—1882no—
454C2×C4#1309399611yescharacteristic
455C2×C4#1309399611yescharacteristic
456C2×C4#1309399611yescharacteristic
457C2×C4C2×#1062533512yes—
458C2×C4#1309404412yes—
459C2×C4C2×D4×D4711yescharacteristic
460C2×C4C22×C22⋊D4711yescharacteristic
461C2×C4C2×#1062535211yescharacteristic
462C8—41no—
463C8—41no—
464C8—942no—
465C8—942no—
466C8—1882no—
467C8—1882no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—41no—
472C22—41no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22#1568935012yes—
477C22C2×#1309399611yescharacteristic
478C4—22no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—81no—
485C4—3762no—
486C4#1568935012yes—
487C4C22×#1062533511yescharacteristic
488C4C2×#1309404411yescharacteristic
489C4#1568992511yescharacteristic
490C4#1568993011yescharacteristic
491C2—22no—
492C2—42no—
493C2—161no—
494C2#2025292511yescharacteristic
495C2C2×#1568935011yescharacteristic
496C2#2025312711yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
Fitting, prime radicalC47×#3035C2
socleC22×C47C2×#121
hypercenter#3035D47
hypocenterC47#19408
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #20254310C2semidirect
2. #20253203C2semidirect
3. #20252499C2semidirect
4. #20242820C2semidirect
5. #20242813C2semidirect
6. #20242734C2semidirect
7. #20242727C2semidirect
8. #20242158C2semidirect
9. #20242065C2semidirect
10. C2×#15688868C2semidirect
113. #41×C47SD16semidirect
297. C47#19408semidirect
362. #41#10625335semidirect

Maximal subgroups

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

subgroupindex
#202543102
#202532032
#202524992
#202428202
#202428132
#202427342
#202427272
#202421582
#202420652
C2×#156888682
C47×#30352
#1940847