Finite Group Databasebeta

← #26204144

Subgroups — #26204144

Order 12032.

848 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254326C211yesmaximal, characteristic, complemented
2#20254313C211yesmaximal, characteristic, complemented
3#20253948C211yesmaximal, characteristic, complemented
4#20253840C211yesmaximal, characteristic
5C2×#15689471C211yesmaximal, characteristic, complemented
6#20243017C212yesmaximal, complemented
7#20243011C212yesmaximal, complemented
8#20242990C212yesmaximal, complemented
9#20242981C212yesmaximal, complemented
10M16×C47⋊C8C211yesmaximal, characteristic, complemented
11C47×#3165C211yesmaximal, characteristic, complemented
12C2×#13094052—22no—
13#15689673C2211yescharacteristic
14C2×#13094021C2211yescharacteristic
15#15689519C2211yescharacteristic, complemented
16#15689512—22no—
17#15689500C2212yes—
18#15689497C2212yes—
19#15689497C2212yescomplemented
20#15689497C2211yescharacteristic, complemented
21#15689496C2212yes—
22#15689473C2211yescharacteristic, complemented
23#15689473C2212yescomplemented
24#15689471—22no—
25#15689471C2212yes—
26#15689471C2212yes—
27#15689471C2212yes—
28#15689471C2211yescharacteristic, complemented
29#15689354—22no—
30#15689325—22no—
31#15689323—22no—
32#15689273—22no—
33#15689262—22no—
34C4×#10625336—22no—
35#15689176C2211yescharacteristic
36C2×#13093944—22no—
37C2×C4×C47⋊C8C2211yescharacteristic
38#15688844C2212yescomplemented
39C8×C47⋊C8C2212yescomplemented
40C47×#533C2211yescharacteristic
41C2×C4⋊Q8×C47C2211yescharacteristic
42C47×#501—22no—
43C47×#500C2212yescomplemented
44C47×#498C2212yescomplemented
45C47×#428—22no—
46C4×M16×C47C2211yescharacteristic
47C2×#10625350—21no—
48C22×D188—21no—
49C22×D188—21no—
50C22×D188—21no—
51#13094052—48no—
52C2×#10625325—24no—
53C2×#10625336—22no—
54C2×#10625336—24no—
55C2×#10625336—24no—
56C2×D376—24no—
57C22×C47⋊C8D412yes—
58#13094024C2311yescharacteristic
59#13094021—21no—
60#13094021—21no—
61#13094021—21no—
62#13094021C2312yes—
63#13094021C2311yescharacteristic
64#13094012—21no—
65#13093998—21no—
66#13093998—21no—
67C4×D188—21no—
68C4×D188—21no—
69#13093955—22no—
70C4⋊(C47⋊C8)D412yes—
71#13093948—24no—
72#13093948—22no—
73#13093944—24no—
74#13093944—22no—
75#13093944—24no—
76#13093944—24no—
77C22⋊(C47⋊C8)D412yes—
78C4×C47⋊C8D412yes—
79C4×C47⋊C8C2312yes—
80C4×C47⋊C8C2311yescharacteristic
81C4×C47⋊C8C2311yescharacteristic
82C22×Q8×C47—21no—
83C2×M16×C47D412yes—
84C4⋊Q8×C47—21no—
85C4⋊Q8×C47—21no—
86C4⋊Q8×C47—21no—
87C4⋊Q8×C47C2311yescharacteristic
88C4⋊Q8×C47C2312yes—
89C4⋊Q8×C47C2311yescharacteristic
90#125×C47—21no—
91C22⋊Q8×C47—21no—
92C22⋊Q8×C47—21no—
93C4×Q8×C47—21no—
94C4×Q8×C47—21no—
95#117×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—21no—
98C2×C42×C47C2311yescharacteristic
99#107×C47—24no—
100#106×C47—24no—
101#97×C47D412yes—
102C4×C8×C47C2312yes—
103C23×D47—41no—
104C2×Q8×C47—21no—
105C2×Q8×C47—21no—
106C2×Q8×C47—22no—
107C2×Q8×C47—41no—
108C2×Q8×C47—82no—
109C22×C4×C47—41no—
110C22×C4×C47C2×D411yescharacteristic
111C22×C4×C47C2×D411yescharacteristic
112C22×C4×C47C2×D411yescharacteristic
113C2×C22⋊D47—41no—
114#10625350—44no—
115C2×D188—22no—
116C2×D188—21no—
117C2×D188—22no—
118C2×D188—21no—
119C2×D188—21no—
120C2×D188—22no—
121C2×D188—44no—
122C2×D188—44no—
123C2×D188—41no—
124C2×D188—41no—
125C2×D188—41no—
126C2×D188—82no—
127C2×C4×D47—41no—
128C2×Dic94—21no—
129M16×C47SD1618yescomplemented
130C2×C8×C47C2×D414yes—
131C4⋊C4×C47—21no—
132C4⋊C4×C47—22no—
133C4⋊C4×C47—21no—
134C4⋊C4×C47—22no—
135C4⋊C4×C47—21no—
136C4⋊C4×C47—21no—
137C4⋊C4×C47—41no—
138C4⋊C4×C47—41no—
139C4⋊C4×C47—41no—
140C4⋊C4×C47—41no—
141C22⋊C4×C47—41no—
142C42×C47—41no—
143C42×C47C2×D411yescharacteristic
144C42×C47C2×D411yescharacteristic
145C42×C47C2×D411yescharacteristic
146C42×C47C2411yescharacteristic
147#10625336—44no—
148#10625336—84no—
149#10625336—84no—
150(C2×D47)⋊C4—41no—
151(C2×D47)⋊C4—41no—
152C4⋊Dic47—21no—
153C4⋊Dic47—21no—
154C2×C47⋊C8—42no—
155C2×C47⋊C8C2×D412yes—
156C2×C47⋊C8C2×D412yes—
157C2×C47⋊C8C2×D414yes—
158D376—48no—
159#10625325—48no—
160C23×C47#12711yescharacteristic
161C22×D47—41no—
162C22×D47—42no—
163C22×D47—82no—
164C22×D47—82no—
165C22×D47—82no—
166Q8×C47—42no—
167Q8×C47—42no—
168Q8×C47—44no—
169Q8×C47—82no—
170C2×C4×C47—22no—
171C2×C4×C47—22no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47—42no—
175C2×C4×C47—82no—
176C2×C4×C47#12711yescharacteristic
177C2×C4×C47#12711yescharacteristic
178C2×C4×C47#12711yescharacteristic
179C2×C4×C47C2×SD1614yes—
180C2×C4×C47C22×D411yescharacteristic
181C2×C4×C47C22×D411yescharacteristic
182C2×C4×C47C22×D411yescharacteristic
183C22⋊D47—82no—
184C2×Dic47—41no—
185D188—44no—
186D188—42no—
187D188—42no—
188D188—44no—
189D188—42no—
190D188—44no—
191D188—82no—
192D188—82no—
193D188—82no—
194C4×D47—82no—
195Dic94—42no—
196C8×C47C2×SD1618yes—
197C47⋊C8—22no—
198C47⋊C8—22no—
199C47⋊C8—24no—
200#19623—471nomaximal, Sylow
201C22×C47—21no—
202C22×C47—21no—
203C22×C47#49212yes—
204C22×C47C2×#12711yescharacteristic
205C2×D47—81no—
206C2×D47—82no—
207C2×D47—82no—
208C2×D47—81no—
209C2×D47—82no—
210C2×D47—81no—
211C2×D47—164no—
212C4×C47—21no—
213C4×C47—21no—
214C4×C47—81no—
215C4×C47—82no—
216C4×C47—81no—
217C4×C47#49212yes—
218C4×C47C22×SD1612yes—
219C4×C47#57512yes—
220C4×C47#57612yes—
221Dic47—81no—
222#3208—471no—
223#3206—471no—
224#3167—471no—
225#3165D4711yescharacteristic, complemented
226#3151—471no—
227C2×#486—471no—
228#1715—472no—
229#1714—472no—
230#1708—472no—
231#1706—472no—
232C8×M16—471no—
233C2×C47—22no—
234C2×C47C2×#49211yescharacteristic
235C2×C47#315311yescharacteristic
236C2×C47#315511yescharacteristic
237D47—161no—
238D47—162no—
239C2×AGL(1,ℤ/8)—942no—
240#533C2×D4711yescharacteristic
241#532—471no—
242C2×C4⋊Q8C2×D4711yescharacteristic
243C2×#127—471no—
244#501—22no—
245#500C2×D4712yescomplemented
246#498—471no—
247#498C2×D4712yescomplemented
248#496—942no—
249#493—472no—
250#492—472no—
251#492—471no—
252#492—472no—
253#492—472no—
254#487—471no—
255#487—472no—
256#486—472no—
257#486—472no—
258#486—471no—
259#486—472no—
260#486—942no—
261#465—942no—
262#461—942no—
263#460—942no—
264#450—942no—
265#448—942no—
266C4×SD16—942no—
267#431—471no—
268#428—22no—
269C2×#102—942no—
270C4×M16C2×D4711yescharacteristic
271C2×C4×C8—471no—
272#322—472no—
273C82—472no—
274C47#1962311yescharacteristic, complemented, Sylow
275C2×#41—941no—
276C22×Q8—21no—
277C22×D4—941no—
278C22×D4—941no—
279C22×D4—941no—
280AGL(1,ℤ/8)—1888no—
281C2×SD16—944no—
282C2×SD16—944no—
283C2×SD16—942no—
284C2×SD16—944no—
285C2×D8—944no—
286C2×M16C22⋊D4712yes—
287C22×C8—472no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—471no—
292C4⋊Q8C22×D4712yes—
293C4⋊Q8C22×D4711yescharacteristic
294C4⋊Q8C22×D4711yescharacteristic
295#127—472no—
296#127—471no—
297#127—941no—
298#127—941no—
299#127—941no—
300#125—21no—
301#124—941no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—941no—
305#121—941no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×D4—941no—
309C4×D4—941no—
310#117—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C42C22×D4711yescharacteristic
314#107—24no—
315#106—24no—
316#106—942no—
317C4⋊C8—472no—
318#103—942no—
319#103—944no—
320#102—944no—
321#102—944no—
322#102—942no—
323#102—944no—
324C22⋊C8—472no—
325#97C22⋊D4712yes—
326C4×C8—472no—
327C4×C8—471no—
328C4×C8—472no—
329C4×C8—471no—
330C4×C8C22×D4712yes—
331C24—1881no—
332#41—1884no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—21no—
336C2×Q8—41no—
337C2×Q8—82no—
338C2×Q8—941no—
339C2×D4—942no—
340C2×D4—941no—
341C2×D4—942no—
342C2×D4—941no—
343C2×D4—941no—
344C2×D4—942no—
345C2×D4—3762no—
346C2×D4—1884no—
347C2×D4—1881no—
348C2×D4—1881no—
349C2×D4—1881no—
350C2×D4—1881no—
351C2×D4—1884no—
352C22×C4—41no—
353C22×C4—1881no—
354C22×C4D4×D4711yescharacteristic
355C22×C4D4×D4711yescharacteristic
356C22×C4C2×C22⋊D4711yescharacteristic
357SD16—3764no—
358SD16—3764no—
359SD16—1884no—
360SD16—1888no—
361D8—1888no—
362M16#1062533618yescomplemented
363C2×C8—472no—
364C2×C8—472no—
365C2×C8—474no—
366C2×C8—1882no—
367C2×C8C2×C22⋊D4714yes—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—941no—
379C4⋊C4—941no—
380C22⋊C4—41no—
381C22⋊C4—1881no—
382C22⋊C4—1881no—
383C42—41no—
384C42D4×D4711yescharacteristic
385C42D4×D4711yescharacteristic
386C42C2×C22⋊D4711yescharacteristic
387C42C23×D4711yescharacteristic
388C23—3762no—
389C23—3762no—
390C23—3762no—
391C23—1881no—
392C23—1882no—
393C23#1309402011yescharacteristic
394Q8—42no—
395Q8—42no—
396Q8—44no—
397Q8—82no—
398Q8—1882no—
399D4—3762no—
400D4—3762no—
401D4—3762no—
402D4—3762no—
403D4—1882no—
404D4—1884no—
405D4—1882no—
406D4—1884no—
407D4—1882no—
408D4—1884no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—3762no—
416C2×C4—1881no—
417C2×C4#1309402011yescharacteristic
418C2×C4#1309402011yescharacteristic
419C2×C4#1309402011yescharacteristic
420C2×C4C2×#1062533614yes—
421C2×C4C2×D4×D4711yescharacteristic
422C2×C4C2×D4×D4711yescharacteristic
423C2×C4C22×C22⋊D4711yescharacteristic
424C8—942no—
425C8—942no—
426C8—944no—
427C8C2×#1062533618yes—
428C22—21no—
429C22—21no—
430C22—3761no—
431C22—3762no—
432C22—3762no—
433C22—3761no—
434C22—3762no—
435C22—3761no—
436C22—7524no—
437C22#1568949712yes—
438C22C2×#1309402011yescharacteristic
439C4—21no—
440C4—21no—
441C4—81no—
442C4—82no—
443C4—81no—
444C4—3761no—
445C4#1568949712yes—
446C4C22×#1062533612yes—
447C4#1568992212yes—
448C4#1568992712yes—
449C2—22no—
450C2—7522no—
451C2—7521no—
452C2C2×#1568949711yescharacteristic
453C2#2025385511yescharacteristic
454C2#2025387311yescharacteristic
subgroupG/H
centerC22C2×#13094020
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3165C2
socleC22×C47C2×#127
prime radical#20253840C2
hypercenter#3165D47
hypocenterC47#19623
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
1. #20254326C2semidirect
2. #20254313C2semidirect
3. #20253948C2semidirect
5. C2×#15689471C2semidirect
6. #20243017C2semidirect
7. #20243011C2semidirect
8. #20242990C2semidirect
9. #20242981C2semidirect
10. M16×C47⋊C8C2semidirect
11. C47×#3165C2semidirect
15. #15689519C22semidirect
19. #15689497C22semidirect
20. #15689497C22semidirect
22. #15689473C22semidirect
23. #15689473C22semidirect
28. #15689471C22semidirect
38. #15688844C22semidirect
39. C8×C47⋊C8C22semidirect
43. C47×#500C22semidirect
44. C47×#498C22semidirect
129. M16×C47SD16semidirect
225. #3165D47semidirect
245. #500C2×D47semidirect
247. #498C2×D47semidirect
274. C47#19623semidirect
362. M16#10625336semidirect

Maximal subgroups

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

subgroupindex
#202543262
#202543132
#202539482
#202538402
C2×#156894712
#202430172
#202430112
#202429902
#202429812
M16×C47⋊C82
C47×#31652
#1962347