Finite Group Databasebeta

← #27122810

Subgroups — #27122810

Order 12032.

2038 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20258157C212yesmaximal, complemented
2#20256540C214yesmaximal, complemented
3#20256534C214yesmaximal, complemented
4#20256533C214yesmaximal, complemented
5#20256528C214yesmaximal, complemented
6#20256524C211yesmaximal, characteristic, complemented
7#20256521C214yesmaximal, complemented
8C2×#15689553C212yesmaximal, complemented
9#508×D47C214yesmaximal, complemented
10C2×#15689541C211yesmaximal, characteristic, complemented
11C47×#3414C211yesmaximal, characteristic, complemented
12#15689959—22no—
13#15689937C2214yescomplemented
14#15689936C2214yescomplemented
15#15689930—24no—
16#15689929—24no—
17#15689925C2214yescomplemented
18#15689924C2214yescomplemented
19#15689921—22no—
20#15689920—24no—
21C2×#13094046C2212yescomplemented
22#135×D47C2218yescomplemented
23C2×#13094030C2211yescharacteristic
24#15689566C2214yescomplemented
25#15689565C2218yescomplemented
26#15689562C2218yescomplemented
27#15689561C2214yescomplemented
28#15689560C2218yescomplemented
29#15689557C2218yescomplemented
30#15689556C2214yescomplemented
31#15689555C2214yescomplemented
32#15689554C2218yescomplemented
33#15689553C2214yescomplemented
34#15689553C2214yescomplemented
35#15689553C2214yescomplemented
36#15689553C2214yescomplemented
37#15689552C2214yescomplemented
38C2×#13093970C2212yescomplemented
39Q32×D47C2214yescomplemented
40C2×#13093968C2212yescomplemented
41C2×#13093967C2212yescomplemented
42SD32×D47C2218yescomplemented
43C2×#13093965C2212yescomplemented
44D16×D47C2214yescomplemented
45#15689541C2214yescomplemented
46#15689541C2214yescomplemented
47C2×#13093963C2212yescomplemented
48C22×C47⋊C16C2211yescharacteristic, complemented
49C2×C16×D47C2212yescomplemented
50C2×#135×C47C2212yescomplemented
51C47×#510C2214yescomplemented
52C47×#509C2214yescomplemented
53C47×#508C2214yescomplemented
54C2×M32×C47C2211yescharacteristic, complemented
55#13094072—24no—
56#13094071—24no—
57#13094071—42no—
58#13094070—24no—
59#13094070—42no—
60#13094069—24no—
61#13094069—44no—
62C2×#10625350—22no—
63C2×#10625350—44no—
64C2×#10625350D411yescharacteristic
65#41×D47—28no—
66#41×D47—44no—
67#13094056—28no—
68#13094055—24no—
69#13094054—24no—
70#13094053—28no—
71#13094051—28no—
72#13094050—24no—
73#13094049—24no—
74#13094048—28no—
75#13094047C2318yes—
76#13094046—22no—
77#13094046—22no—
78#13094046C2314yes—
79#13094046C2314yes—
80#13094045—28no—
81#13094044—28no—
82#13094043C2318yes—
83C2×Dic188C2312yes—
84Q16×D47C2318yes—
85C2×#10625325—22no—
86SD16×D47—28no—
87C2×D376C2312yes—
88D8×D47C2318yes—
89#13094030D414yes—
90#13094030C2314yes—
91C2×#10625329D411yescharacteristic
92C2×#10625324D412yes—
93C2×C8×D47C2312yes—
94Dic376C2314yes—
95#13093970C2314yes—
96#13093970C2314yes—
97#13093969C2318yes—
98#13093968C2314yes—
99#13093968C2314yes—
100#13093967C2314yes—
101#13093967C2314yes—
102D752C2314yes—
103#13093965C2314yes—
104#13093965C2314yes—
105#13093963C2318yes—
106C2×C47⋊C16C2311yescharacteristic
107C2×C47⋊C16C2311yescharacteristic
108C2×C47⋊C16C2312yes—
109C2×C47⋊C16C2312yes—
110C16×D47C2314yes—
111C16×D47C2314yes—
112C2×#41×C47—22no—
113#135×C47—22no—
114#135×C47—22no—
115#135×C47C2314yes—
116#135×C47C2314yes—
117C2×Q16×C47C2312yes—
118C2×SD16×C47—22no—
119C2×D8×C47C2312yes—
120C22×C8×C47C2311yescharacteristic
121Q32×C47C2314yes—
122SD32×C47C2318yes—
123D16×C47C2314yes—
124M32×C47C2314yes—
125C2×C16×C47C2312yes—
126#41×C47—24no—
127#41×C47—24no—
128#41×C47—42no—
129#41×C47—42no—
130C2×Q8×C47—22no—
131C2×D4×C47—22no—
132C2×D4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47C2×D411yescharacteristic
135C2×C22⋊D47—42no—
136C2×C22⋊D47—48no—
137C2×C22⋊D47D812yes—
138#10625354—28no—
139#10625354—44no—
140Q8×D47—28no—
141Q8×D47—44no—
142#10625352—28no—
143#10625352—48no—
144#10625352—48no—
145#10625352—44no—
146D4×D47—28no—
147D4×D47—48no—
148D4×D47—48no—
149D4×D47—44no—
150#10625350—24no—
151#10625350—24no—
152#10625350—42no—
153#10625350—42no—
154#10625350—48no—
155#10625350—44no—
156#10625350—48no—
157#10625350—44no—
158#10625350D814yes—
159#10625350C2×D414yes—
160C2×D188—22no—
161C2×D188—44no—
162C2×D188D811yescharacteristic
163C2×C4×D47—42no—
164C2×C4×D47—48no—
165C2×C4×D47C2×D412yes—
166C2×Dic94—22no—
167C2×Dic94—44no—
168C2×Dic94D811yescharacteristic
169Q16×C47—22no—
170Q16×C47C2414yes—
171SD16×C47—24no—
172SD16×C47—24no—
173D8×C47—22no—
174D8×C47C2414yes—
175C2×C8×C47C2×D412yes—
176C2×C8×C47C2×D411yescharacteristic
177C2×C8×C47C2411yescharacteristic
178C2×C8×C47C2412yes—
179#10625337—28no—
180#10625336—28no—
181#10625335—28no—
182#10625334—28no—
183#10625329C2×D414yes—
184C2×C47⋊C8C2×D412yes—
185Dic188—22no—
186Dic188C2414yes—
187D376—22no—
188D376C2414yes—
189#10625325—24no—
190#10625325—24no—
191#10625324C2×D418yes—
192C8×D47C2×D414yes—
193C8×D47C2414yes—
194C16×C47C2414yes—
195C47⋊C16C2411yescharacteristic
196C47⋊C16C2411yescharacteristic
197C47⋊C16C2412yes—
198C23×C47—42no—
199C23×C47C2×D811yescharacteristic
200C22×D47—42no—
201C22×D47—48no—
202C22×D47C2×D812yes—
203Q8×C47—24no—
204Q8×C47—42no—
205D4×C47—24no—
206D4×C47—44no—
207D4×C47—44no—
208D4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—44no—
211C2×C4×C47—44no—
212C2×C4×C47C2×D811yescharacteristic
213C2×C4×C47C2×D812yes—
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C22×D412yes—
216C22⋊D47—44no—
217C22⋊D47—44no—
218C22⋊D47—48no—
219C22⋊D47—48no—
220C22⋊D47C2×D818yes—
221C2×Dic47—42no—
222C2×Dic47—48no—
223C2×Dic47C2×D812yes—
224D188—24no—
225D188—48no—
226D188—42no—
227D188C2×D814yes—
228C4×D47—48no—
229C4×D47—44no—
230C4×D47—44no—
231C4×D47—48no—
232C4×D47C2×D814yes—
233C4×D47C22×D414yes—
234Dic94—24no—
235Dic94—42no—
236Dic94—48no—
237Dic94C2×D814yes—
238C8×C47C22×D411yescharacteristic
239C8×C47C22×D412yes—
240C8×C47C2511yescharacteristic
241C47⋊C8C22×D414yes—
242#62635—471nomaximal, Sylow
243C22×C47—41no—
244C22×C47—44no—
245C22×C47—162no—
246C22×C47C22×D812yes—
247C22×C47C22×D811yescharacteristic
248C2×D47—42no—
249C2×D47—44no—
250C2×D47—168no—
251C2×D47—162no—
252C2×D47C22×D814yes—
253C4×C47—44no—
254C4×C47C22×D812yes—
255C4×C47C22×D811yescharacteristic
256C4×C47C23×D411yescharacteristic
257Dic47—44no—
258Dic47C22×D814yes—
259#3584—472no—
260#3417—474no—
261#3416—474no—
262#3416—474no—
263#3415—474no—
264#3414—474no—
265#3414—471no—
266#3414D4711yescharacteristic, complemented
267C2×#508—474no—
268C2×#508—472no—
269C2×#504—471no—
270C2×C47—21no—
271C2×C47—22no—
272C2×C47—84no—
273C2×C47C23×D811yescharacteristic
274D47—24no—
275D47—84no—
276#585—942no—
277#578—474no—
278#578—474no—
279#577—944no—
280#577—944no—
281#576—474no—
282#576—474no—
283#575—944no—
284#575—942no—
285C2×#135—478no—
286C2×#135—472no—
287C2×#135C2×D4712yescomplemented
288C2×#131—471no—
289#510—478no—
290#510—478no—
291#510—474no—
292#510C2×D4714yescomplemented
293#509—478no—
294#509—478no—
295#509—474no—
296#509C2×D4714yescomplemented
297#508—474no—
298#508—474no—
299#508—474no—
300#508—474no—
301#508—474no—
302#508—478no—
303#508—474no—
304#508—474no—
305#508C2×D4714yescomplemented
306C2×Q32—472no—
307C2×Q32—474no—
308C2×SD32—472no—
309C2×SD32—472no—
310C2×SD32—478no—
311C2×D16—472no—
312C2×D16—474no—
313#504—474no—
314#504—474no—
315C2×M32—472no—
316C2×M32C2×D4711yescharacteristic, complemented
317C22×C16—471no—
318C22×C16—472no—
319C47#6263511yescharacteristic, complemented, Sylow
3202−1+4—944no—
3212−1+4—944no—
3222−1+4—1882no—
323H(5,𝔽2)—944no—
324H(5,𝔽2)—944no—
325H(5,𝔽2)—1884no—
326H(5,𝔽2)—1882no—
327C2×#41—22no—
328C2×#41—471no—
329C2×#41—948no—
330C2×#41—942no—
331C2×#41—1884no—
332C2×#41—1884no—
333#137—948no—
334#137—948no—
335#137—944no—
336#137—944no—
337AGL(1,ℤ/8)—948no—
338AGL(1,ℤ/8)—948no—
339AGL(1,ℤ/8)—944no—
340AGL(1,ℤ/8)—944no—
341#135—22no—
342#135—22no—
343#135—478no—
344#135—478no—
345#135—474no—
346#135—474no—
347#135—948no—
348#135—948no—
349#135—942no—
350#135—942no—
351#135C22×D4714yes—
352#135C22×D4714yes—
353C2×Q16—478no—
354C2×Q16—472no—
355C2×Q16C22×D4712yes—
356C2×SD16—22no—
357C2×SD16—948no—
358C2×SD16—942no—
359C2×D8—478no—
360C2×D8—472no—
361C2×D8C22×D4712yes—
362#131—474no—
363#131—474no—
364C2×M16—471no—
365C2×M16—472no—
366C22×C8—472no—
367C22×C8C22×D4711yescharacteristic
368Q32—474no—
369Q32—474no—
370Q32—474no—
371Q32C22×D4714yes—
372SD32—474no—
373SD32—474no—
374SD32—474no—
375SD32—474no—
376SD32—478no—
377SD32C22×D4718yes—
378D16—474no—
379D16—474no—
380D16—474no—
381D16C22×D4714yes—
382M32—478no—
383M32C22×D4714yes—
384C2×C16—471no—
385C2×C16—471no—
386C2×C16—472no—
387C2×C16—472no—
388C2×C16—474no—
389C2×C16—474no—
390C2×C16C22×D4712yes—
391#41—24no—
392#41—24no—
393#41—42no—
394#41—42no—
395#41—474no—
396#41—474no—
397#41—948no—
398#41—948no—
399#41—944no—
400#41—944no—
401#41—1888no—
402#41—1888no—
403#41—1884no—
404#41—1884no—
405#41—1888no—
406#41—1888no—
407#41—1884no—
408#41—1884no—
409#41—1882no—
410#41—1882no—
411C2×Q8—22no—
412C2×Q8—471no—
413C2×Q8—948no—
414C2×Q8—942no—
415C2×Q8—1884no—
416C2×Q8—1884no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—472no—
420C2×D4—471no—
421C2×D4—948no—
422C2×D4—942no—
423C2×D4—1884no—
424C2×D4—1888no—
425C2×D4—1884no—
426C2×D4—1888no—
427C2×D4—1888no—
428C2×D4—1882no—
429C22×C4—42no—
430C22×C4—472no—
431C22×C4—1888no—
432C22×C4—1882no—
433C22×C4D4×D4711yescharacteristic
434Q16—22no—
435Q16—474no—
436Q16—948no—
437Q16—942no—
438Q16C23×D4714yes—
439SD16—24no—
440SD16—24no—
441SD16—948no—
442SD16—948no—
443SD16—944no—
444SD16—944no—
445D8—22no—
446D8—474no—
447D8—948no—
448D8—942no—
449D8C23×D4714yes—
450M16—474no—
451M16—478no—
452C2×C8—472no—
453C2×C8—474no—
454C2×C8—474no—
455C2×C8D4×D4711yescharacteristic
456C2×C8D4×D4712yes—
457C2×C8C23×D4711yescharacteristic
458C2×C8C23×D4712yes—
459C16—471no—
460C16—471no—
461C16—472no—
462C16C23×D4714yes—
463C23—42no—
464C23—472no—
465C23—1888no—
466C23—1882no—
467C23D8×D4711yescharacteristic
468Q8—24no—
469Q8—42no—
470Q8—474no—
471Q8—944no—
472Q8—1888no—
473Q8—1882no—
474D4—24no—
475D4—44no—
476D4—44no—
477D4—42no—
478D4—478no—
479D4—474no—
480D4—944no—
481D4—1888no—
482D4—1888no—
483D4—1888no—
484D4—1884no—
485D4—1884no—
486D4—1882no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—472no—
491C2×C4—474no—
492C2×C4—474no—
493C2×C4—1888no—
494C2×C4—1888no—
495C2×C4—1888no—
496C2×C4—1882no—
497C2×C4—1884no—
498C2×C4—1884no—
499C2×C4D8×D4711yescharacteristic
500C2×C4D8×D4712yes—
501C2×C4C2×D4×D4712yes—
502C2×C4C2×D4×D4711yescharacteristic
503C8—474no—
504C8C2×D4×D4711yescharacteristic
505C8C2×D4×D4712yes—
506C8C24×D4711yescharacteristic
507C22—41no—
508C22—44no—
509C22—162no—
510C22—474no—
511C22—1882no—
512C22—1884no—
513C22—7522no—
514C22—7528no—
515C22C2×D8×D4712yes—
516C22C2×D8×D4711yescharacteristic
517C4—44no—
518C4—474no—
519C4—1884no—
520C4C2×D8×D4712yes—
521C4C2×D8×D4711yescharacteristic
522C4C22×D4×D4711yescharacteristic
523C2—22no—
524C2—21no—
525C2—84no—
526C2—944no—
527C2—3764no—
528C2C22×D8×D4711yescharacteristic
subgroupG/H
centerC4C2×D8×D47
commutator, cosocleC8×C47C25
FrattiniC8C24×D47
FittingC47×#3414C2
socleC2×C47C23×D8
hypercenter#3414D47
hypocenterC47#62635
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #20258157C2semidirect
2. #20256540C2semidirect
3. #20256534C2semidirect
4. #20256533C2semidirect
5. #20256528C2semidirect
6. #20256524C2semidirect
7. #20256521C2semidirect
8. C2×#15689553C2semidirect
9. #508×D47C2semidirect
10. C2×#15689541C2semidirect
11. C47×#3414C2semidirect
13. #15689937C22semidirect
14. #15689936C22semidirect
17. #15689925C22semidirect
18. #15689924C22semidirect
21. C2×#13094046C22semidirect
22. #135×D47C22semidirect
24. #15689566C22semidirect
25. #15689565C22semidirect
26. #15689562C22semidirect
27. #15689561C22semidirect
28. #15689560C22semidirect
29. #15689557C22semidirect
30. #15689556C22semidirect
31. #15689555C22semidirect
32. #15689554C22semidirect
33. #15689553C22semidirect
34. #15689553C22semidirect
35. #15689553C22semidirect
36. #15689553C22semidirect
37. #15689552C22semidirect
38. C2×#13093970C22semidirect
39. Q32×D47C22semidirect
40. C2×#13093968C22semidirect
41. C2×#13093967C22semidirect
42. SD32×D47C22semidirect
43. C2×#13093965C22semidirect
44. D16×D47C22semidirect
45. #15689541C22semidirect
46. #15689541C22semidirect
47. C2×#13093963C22semidirect
48. C22×C47⋊C16C22semidirect
49. C2×C16×D47C22semidirect
50. C2×#135×C47C22semidirect
51. C47×#510C22semidirect
52. C47×#509C22semidirect
53. C47×#508C22semidirect
54. C2×M32×C47C22semidirect
266. #3414D47semidirect
287. C2×#135C2×D47semidirect
292. #510C2×D47semidirect
296. #509C2×D47semidirect
305. #508C2×D47semidirect
316. C2×M32C2×D47semidirect
319. C47#62635semidirect

Maximal subgroups

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

subgroupindex
#202581572
#202565402
#202565342
#202565332
#202565282
#202565242
#202565212
C2×#156895532
#508×D472
C2×#156895412
C47×#34142
#6263547