Finite Group Databasebeta

← #26373117

Subgroups — #26373117

Order 12032.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20255075C211yesmaximal, characteristic
2#20245622C211yesmaximal, characteristic, complemented
3#20245585C211yesmaximal, characteristic, complemented
4#20245579C211yesmaximal, characteristic, complemented
5#20245542C211yesmaximal, characteristic, complemented
6#20245486C211yesmaximal, characteristic, complemented
7#20245477C211yesmaximal, characteristic, complemented
8#20245450C211yesmaximal, characteristic, complemented
9#20245326C211yesmaximal, characteristic, complemented
10#20245316C211yesmaximal, characteristic, complemented
11C4×#13093967C211yesmaximal, characteristic, complemented
12D4×C47⋊C16C211yesmaximal, characteristic, complemented
13#20245093C211yesmaximal, characteristic, complemented
14C2×#15688923C211yesmaximal, characteristic, complemented
15C47×#3336C211yesmaximal, characteristic, complemented
16C2×#13094046C2211yescharacteristic
17C22×D376C2211yescharacteristic
18#15689769—21no—
19C2×#13093970C2211yescharacteristic, complemented
20C2×#13093967C2211yescharacteristic
21C2×#13093967C2211yescharacteristic
22C22×C47⋊C16C2211yescharacteristic
23C22×C47⋊C16C2211yescharacteristic, complemented
24#15689507C2211yescharacteristic
25#15689364C2211yescharacteristic
26#15689359C2211yescharacteristic
27#15689339—21no—
28#15689276—21no—
29#15689270—21no—
30C4×D376C2211yescharacteristic
31#15688950C2211yescharacteristic, complemented
32#15688950C2211yescharacteristic, complemented
33#15688948C2211yescharacteristic, complemented
34C4⋊(C47⋊C16)C2211yescharacteristic, complemented
35#15688927C2211yescharacteristic, complemented
36#15688926C2211yescharacteristic, complemented
37#15688926C2211yescharacteristic, complemented
38#15688926C2211yescharacteristic, complemented
39#15688923C2211yescharacteristic
40#15688923C2211yescharacteristic
41#15688923C2211yescharacteristic
42#15688923C2211yescharacteristic
43#15688923C2211yescharacteristic
44#15688923C2211yescharacteristic
45C22⋊(C47⋊C16)C2211yescharacteristic
46C22⋊(C47⋊C16)C2211yescharacteristic, complemented
47C4×C47⋊C16C2211yescharacteristic, complemented
48C47×#552—21no—
49C47×#499C2211yescharacteristic, complemented
50C47×#485—21no—
51C47×#483—21no—
52C47×#479—21no—
53C47×#467C2211yescharacteristic
54C47×#467C2211yescharacteristic, complemented
55C4×Q16×C47C2211yescharacteristic, complemented
56C8×D4×C47C2211yescharacteristic
57C47×#427C2211yescharacteristic, complemented
58C2×#107×C47C2211yescharacteristic
59C2×#10625350—21no—
60C22×D188—21no—
61#13094046—22no—
62#13094046—41no—
63C2×Dic188C2311yescharacteristic
64C2×#10625325—21no—
65C2×#10625325—21no—
66C2×D376—22no—
67C2×D376—21no—
68C2×D376—41no—
69C2×D376C2311yescharacteristic
70C2×D376C2311yescharacteristic
71#13094012—21no—
72#13094009—41no—
73#13094003—21no—
74#13093998—21no—
75#13093997—41no—
76#13093992—41no—
77C4×D188—21no—
78C22⋊C4×D47—41no—
79#13093970—41no—
80#13093967—22no—
81#13093967—41no—
82C2×C47⋊C16—41no—
83C2×C47⋊C16—41no—
84C2×C47⋊C16C2311yescharacteristic
85C2×C47⋊C16C2311yescharacteristic
86C2×C47⋊C16C2311yescharacteristic
87C2×C47⋊C16C2311yescharacteristic
88#13093958C2311yescharacteristic
89#13093949—21no—
90#13093946—21no—
91#13093946—21no—
92C2×Q16×C47C2311yescharacteristic
93C22×C8×C47C2311yescharacteristic
94C22×C8×C47C2311yescharacteristic
95#126×C47—41no—
96#125×C47—21no—
97#125×C47—41no—
98#123×C47—41no—
99C22⋊Q8×C47—21no—
100C22⋊Q8×C47—21no—
101C4×Q8×C47—21no—
102C4×D4×C47D411yescharacteristic
103#117×C47—21no—
104#117×C47—41no—
105C2×C4⋊C4×C47—21no—
106#107×C47—21no—
107#107×C47C2311yescharacteristic
108#107×C47C2311yescharacteristic
109#107×C47C2311yescharacteristic
110#106×C47—21no—
111#106×C47—21no—
112C4⋊C8×C47D411yescharacteristic
113#103×C47—21no—
114#103×C47—21no—
115#103×C47—21no—
116C22⋊C8×C47D411yescharacteristic
117C22⋊C8×C47D411yescharacteristic
118C4×C8×C47C2311yescharacteristic
119C23×D47—41no—
120C2×Q8×C47—21no—
121C2×D4×C47D811yescharacteristic
122C22×C4×C47—41no—
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D411yescharacteristic
125C2×C22⋊D47—41no—
126C2×C22⋊D47—41no—
127#10625350—42no—
128#10625350—81no—
129C2×D188—21no—
130C2×D188—21no—
131C2×D188—42no—
132C2×D188—41no—
133C2×D188—81no—
134C2×C4×D47—41no—
135C2×C4×D47—41no—
136C2×Dic94—21no—
137Q16×C47—21no—
138Q16×C47—21no—
139C2×C8×C47—21no—
140C2×C8×C47—21no—
141C2×C8×C47—21no—
142C2×C8×C47—21no—
143C2×C8×C47C2×D411yescharacteristic
144C2×C8×C47C2×D411yescharacteristic
145C2×C8×C47C2×D411yescharacteristic
146C2×C8×C47C2411yescharacteristic
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—41no—
151C4⋊C4×C47—41no—
152C4⋊C4×C47—41no—
153C4⋊C4×C47—41no—
154C4⋊C4×C47—41no—
155C4⋊C4×C47—41no—
156C4⋊C4×C47D811yescharacteristic
157C22⋊C4×C47—41no—
158C22⋊C4×C47—41no—
159C22⋊C4×C47—41no—
160C22⋊C4×C47D811yescharacteristic
161C22⋊C4×C47D811yescharacteristic
162C42×C47—41no—
163C42×C47—41no—
164C42×C47C2×D411yescharacteristic
165C22⋊Dic47—41no—
166(C2×D47)⋊C4—41no—
167(C2×D47)⋊C4—41no—
168(C2×D47)⋊C4—41no—
169(C2×D47)⋊C4—41no—
170C4⋊Dic47—21no—
171Dic47⋊C4—41no—
172Dic188—21no—
173Dic188—21no—
174D376—21no—
175D376—21no—
176D376—21no—
177D376—42no—
178D376#4112yes—
179#10625325—42no—
180#10625325—81no—
181C47⋊C16—21no—
182C47⋊C16—21no—
183C47⋊C16—21no—
184C47⋊C16—21no—
185C23×C47C2×D811yescharacteristic
186C23×C47C2×D811yescharacteristic
187C22×D47—41no—
188C22×D47—41no—
189C22×D47—82no—
190C22×D47—82no—
191C22×D47—161no—
192Q8×C47—41no—
193Q8×C47—41no—
194D4×C47—22no—
195C2×C4×C47—21no—
196C2×C4×C47—21no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—82no—
204C2×C4×C47C2×D811yescharacteristic
205C2×C4×C47C2×D811yescharacteristic
206C2×C4×C47C2×D811yescharacteristic
207C2×C4×C47C2×D811yescharacteristic
208C2×C4×C47C22×D411yescharacteristic
209C22⋊D47—82no—
210C22⋊D47—161no—
211C2×Dic47—41no—
212C2×Dic47—41no—
213D188—22no—
214D188—41no—
215D188—41no—
216D188—41no—
217D188—82no—
218C4×D47—82no—
219C4×D47—161no—
220Dic94—41no—
221Dic94—41no—
222C8×C47—21no—
223C8×C47—21no—
224C8×C47—21no—
225C8×C47C2×#4111yescharacteristic
226C8×C47H(5,𝔽2)11yescharacteristic
227#33863—471nomaximal, Sylow
228C22×C47—21no—
229C22×C47—21no—
230C22×C47—21no—
231C22×C47—21no—
232C22×C47—21no—
233C22×C47#50812yes—
234C22×C47C22×D811yescharacteristic
235C2×D47—42no—
236C2×D47—81no—
237C2×D47—81no—
238C2×D47—81no—
239C2×D47—81no—
240C2×D47—81no—
241C2×D47—164no—
242C4×C47—21no—
243C4×C47—21no—
244C4×C47—21no—
245C4×C47—21no—
246C4×C47—81no—
247C4×C47—81no—
248C4×C47—81no—
249C4×C47—81no—
250C4×C47#54611yescharacteristic
251C4×C47#57711yescharacteristic
252Dic47—81no—
253Dic47—81no—
254#3336D4711yescharacteristic, complemented
255#3280—471no—
256#2242—471no—
257#2235—471no—
258#2234—471no—
259#2228—471no—
260#2217—471no—
261#2216—471no—
262#2211—471no—
263#2189—471no—
264#2187—471no—
265C4×SD32—471no—
266D4×C16—471no—
267#2138—471no—
268C2×#357—471no—
269C2×C47—22no—
270C2×C47—41no—
271C2×C47#329411yescharacteristic
272C2×C47C2×#50811yescharacteristic
273C2×C47#341611yescharacteristic
274D47—82no—
275D47—161no—
276C2×#135—471no—
277C22×D8—471no—
278#552—21no—
279#546—941no—
280C2×Q32—471no—
281C2×SD32—471no—
282C2×SD32—471no—
283C22×C16—471no—
284C22×C16—471no—
285#499C2×D4711yescharacteristic, complemented
286#495—471no—
287#485—21no—
288#483—21no—
289#479—21no—
290#467—471no—
291#467C2×D4711yescharacteristic
292#467C2×D4711yescharacteristic, complemented
293#466—471no—
294#463—941no—
295#450—941no—
296#449—941no—
297C4×Q16C2×D4711yescharacteristic, complemented
298C4×D8—471no—
299C8×D4C2×D4711yescharacteristic
300#427C2×D4711yescharacteristic, complemented
301C2×#107C2×D4711yescharacteristic
302#367—471no—
303#367—471no—
304#366—471no—
305C4⋊C16—471no—
306#358—471no—
307#358—471no—
308#358—471no—
309#358—471no—
310#357—471no—
311#357—471no—
312#357—471no—
313#357—471no—
314#357—471no—
315#357—471no—
316C22⋊C16—471no—
317C22⋊C16—471no—
318C4×C16—471no—
319C47#3386311yescharacteristic, complemented, Sylow
320C2×#41—941no—
321C22×D4—941no—
322#135—942no—
323#135—1881no—
324C2×Q16—471no—
325C2×Q16C22×D4711yescharacteristic
326C2×SD16—941no—
327C2×SD16—941no—
328C2×D8—471no—
329C2×D8—471no—
330C2×D8—942no—
331C2×D8—941no—
332C2×D8—1881no—
333C22×C8C22×D4711yescharacteristic
334C22×C8C22×D4711yescharacteristic
335#126—41no—
336#125—21no—
337#125—41no—
338#124—941no—
339#123—41no—
340#123—1881no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—941no—
344#121—941no—
345#121—1881no—
346#120—1881no—
347C4×Q8—21no—
348C4×D4—941no—
349C4×D4C22⋊D4711yescharacteristic
350#117—21no—
351#117—41no—
352C2×C4⋊C4—21no—
353C2×C22⋊C4—1881no—
354Q32—1881no—
355SD32—942no—
356SD32—1881no—
357C2×C16—471no—
358C2×C16—471no—
359C2×C16—471no—
360C2×C16—471no—
361C2×C16—1881no—
362C2×C16—1881no—
363#107—21no—
364#107—471no—
365#107C22×D4711yescharacteristic
366#107C22×D4711yescharacteristic
367#107C22×D4711yescharacteristic
368#106—21no—
369#106—21no—
370C4⋊C8C22⋊D4711yescharacteristic
371#103—21no—
372#103—21no—
373#103—21no—
374#103—941no—
375#102—941no—
376#102—941no—
377C22⋊C8C22⋊D4711yescharacteristic
378C22⋊C8C22⋊D4711yescharacteristic
379C4×C8C22×D4711yescharacteristic
380C24—1881no—
381#41—3761no—
382#41—1882no—
383C2×Q8—21no—
384C2×Q8—941no—
385C2×D4—941no—
386C2×D4—941no—
387C2×D4—3761no—
388C2×D4—1882no—
389C2×D4—1881no—
390C2×D4—1881no—
391C2×D4—1881no—
392C2×D4#1062533411yescharacteristic
393C22×C4—41no—
394C22×C4—1881no—
395C22×C4—1881no—
396C22×C4C2×C22⋊D4711yescharacteristic
397C22×C4C2×C22⋊D4711yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—941no—
401Q16—941no—
402SD16—3761no—
403SD16—1882no—
404D8—472no—
405D8—941no—
406D8—941no—
407D8—941no—
408D8—1882no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C22⋊D4711yescharacteristic
414C2×C8C2×C22⋊D4711yescharacteristic
415C2×C8C2×C22⋊D4711yescharacteristic
416C2×C8C23×D4711yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—941no—
427C4⋊C4—1881no—
428C4⋊C4#1062533411yescharacteristic
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—1881no—
433C22⋊C4—1881no—
434C22⋊C4—1881no—
435C22⋊C4—1881no—
436C22⋊C4—1881no—
437C22⋊C4#1062533411yescharacteristic
438C22⋊C4#1062533411yescharacteristic
439C42—41no—
440C42—41no—
441C42C2×C22⋊D4711yescharacteristic
442C16—941no—
443C16—941no—
444C16—941no—
445C16—941no—
446C23—3762no—
447C23—3762no—
448C23—1881no—
449C23—1881no—
450C23—7521no—
451C23C2×#1062533411yescharacteristic
452C23C2×#1062533411yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—1881no—
456Q8—1881no—
457D4—22no—
458D4—942no—
459D4—3762no—
460D4—3762no—
461D4—1881no—
462D4—1881no—
463D4—1881no—
464D4—7521no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—3762no—
475C2×C4—1881no—
476C2×C4—1881no—
477C2×C4—7521no—
478C2×C4C2×#1062533411yescharacteristic
479C2×C4C2×#1062533411yescharacteristic
480C2×C4C2×#1062533411yescharacteristic
481C2×C4C2×#1062533411yescharacteristic
482C2×C4C22×C22⋊D4711yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D4711yescharacteristic
487C8#1309407011yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—3761no—
494C22—3761no—
495C22—3761no—
496C22—3761no—
497C22—3761no—
498C22—1882no—
499C22—7524no—
500C22#1568955312yes—
501C22C22×#1062533411yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—3761no—
511C4—3761no—
512C4#1568976111yescharacteristic
513C4#1568993111yescharacteristic
514C2—22no—
515C2—41no—
516C2—3762no—
517C2—7521no—
518C2#2025519511yescharacteristic
519C2C2×#1568955311yescharacteristic
520C2#2025653211yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3336C2
socleC22×C47C22×D8
prime radical#20255075C2
hypercenter#3336D47
hypocenterC47#33863
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #20245622C2semidirect
3. #20245585C2semidirect
4. #20245579C2semidirect
5. #20245542C2semidirect
6. #20245486C2semidirect
7. #20245477C2semidirect
8. #20245450C2semidirect
9. #20245326C2semidirect
10. #20245316C2semidirect
11. C4×#13093967C2semidirect
12. D4×C47⋊C16C2semidirect
13. #20245093C2semidirect
14. C2×#15688923C2semidirect
15. C47×#3336C2semidirect
19. C2×#13093970C22semidirect
23. C22×C47⋊C16C22semidirect
31. #15688950C22semidirect
32. #15688950C22semidirect
33. #15688948C22semidirect
34. C4⋊(C47⋊C16)C22semidirect
35. #15688927C22semidirect
36. #15688926C22semidirect
37. #15688926C22semidirect
38. #15688926C22semidirect
46. C22⋊(C47⋊C16)C22semidirect
47. C4×C47⋊C16C22semidirect
49. C47×#499C22semidirect
54. C47×#467C22semidirect
55. C4×Q16×C47C22semidirect
57. C47×#427C22semidirect
254. #3336D47semidirect
285. #499C2×D47semidirect
292. #467C2×D47semidirect
297. C4×Q16C2×D47semidirect
300. #427C2×D47semidirect
319. C47#33863semidirect

Maximal subgroups

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

subgroupindex
#202550752
#202456222
#202455852
#202455792
#202455422
#202454862
#202454772
#202454502
#202453262
#202453162
C4×#130939672
D4×C47⋊C162
#202450932
C2×#156889232
C47×#33362
#3386347