Finite Group Databasebeta

← #26198856

Subgroups — #26198856

Order 12032.

684 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253392C211yesmaximal, characteristic, complemented
2#20253384C211yesmaximal, characteristic
3C2×#15689202C211yesmaximal, characteristic, complemented
4#20242960C212yesmaximal, complemented
5#20242946C212yesmaximal, complemented
6#20242932C212yesmaximal, complemented
7#20242918C212yesmaximal, complemented
8#20242136C211yesmaximal, characteristic, complemented
9#20242045C211yesmaximal, characteristic, complemented
10#20242037C211yesmaximal, characteristic, complemented
11C47×#3131C211yesmaximal, characteristic, complemented
12C22×#10625324—21no—
13C2×C4×D188C2211yescharacteristic
14#15689451—21no—
15#15689434—21no—
16#15689410C2212yes—
17#15689400C2212yes—
18#15689325C2212yes—
19#15689315C2212yes—
20#15689273—21no—
21#15689256—21no—
22C4×#10625336—22no—
23C4×#10625334—22no—
24#15689202—21no—
25#15689202—21no—
26#15689202C2212yes—
27#15689202C2212yes—
28#15689176C2211yescharacteristic
29#15689176C2211yescharacteristic
30C2×C4⋊(C47⋊C8)C2211yescharacteristic
31C2×(C2×D47)⋊C8—21no—
32#15688876C2212yescomplemented
33#15688874C2212yescomplemented
34#15688872C2212yescomplemented
35#15688869C2212yescomplemented
36#15688869C2212yescomplemented
37#15688867C2212yescomplemented
38C47×#533C2211yescharacteristic
39C47×#532C2211yescharacteristic
40C47×#488—21no—
41C47×#488—21no—
42C47×#487C2211yescharacteristic, complemented
43C47×#487C2211yescharacteristic, complemented
44C47×#486C2211yescharacteristic, complemented
45C47×#486C2211yescharacteristic, complemented
46C47×#416—21no—
47C47×#416—21no—
48C2×C4×C8×C47C2211yescharacteristic
49C22×D188D411yescharacteristic
50C22×C4×D47—21no—
51C2×#10625336—22no—
52C2×#10625334—22no—
53C2×#10625324—22no—
54C2×#10625324—22no—
55C2×#10625324—44no—
56C22×C47⋊C8—21no—
57C4×D188—21no—
58C4×D188—21no—
59C4×D188D412yes—
60C4×D188C2312yes—
61C2×C4⋊Dic47D411yescharacteristic
62C2×(C2×D47)⋊C4—21no—
63#13093957—22no—
64#13093955—22no—
65C4⋊(C47⋊C8)—21no—
66C4⋊(C47⋊C8)C2312yes—
67C4⋊(C47⋊C8)C2312yes—
68C4⋊(C47⋊C8)C2312yes—
69#13093948—22no—
70#13093945—22no—
71#13093944—22no—
72#13093944—22no—
73C22⋊(C47⋊C8)—21no—
74C22⋊(C47⋊C8)—21no—
75(C2×D47)⋊C8—22no—
76(C2×D47)⋊C8—22no—
77C4×C47⋊C8—21no—
78C4×C47⋊C8—21no—
79C2×#41×C47—21no—
80C22×C8×C47D411yescharacteristic
81C22×C8×C47D411yescharacteristic
82C4⋊Q8×C47C2311yescharacteristic
83C4⋊Q8×C47C2311yescharacteristic
84C4⋊Q8×C47C2311yescharacteristic
85#127×C47C2311yescharacteristic
86#125×C47—21no—
87#124×C47—21no—
88C22⋊Q8×C47—21no—
89C22⋊Q8×C47—21no—
90#121×C47—21no—
91#121×C47—21no—
92C4×Q8×C47—21no—
93C4×Q8×C47—21no—
94C4×D4×C47—21no—
95C4×D4×C47—21no—
96#117×C47—21no—
97C2×C42×C47C2311yescharacteristic
98#103×C47—21no—
99#103×C47—21no—
100#103×C47—21no—
101#103×C47—21no—
102#102×C47—21no—
103#102×C47—21no—
104#102×C47—21no—
105#102×C47—21no—
106C4×C8×C47D411yescharacteristic
107C4×C8×C47D411yescharacteristic
108C4×C8×C47C2311yescharacteristic
109C4×C8×C47C2311yescharacteristic
110C23×D47—21no—
111#41×C47—42no—
112#41×C47—42no—
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—41no—
120C22×C4×C47—41no—
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47#4111yescharacteristic
124C22×Dic47—21no—
125C2×D188—21no—
126C2×D188—42no—
127C2×D188C2×D412yes—
128C2×C4×D47—22no—
129C2×C4×D47—22no—
130C2×C4×D47—42no—
131C2×C4×D47—42no—
132C2×C8×C47—22no—
133C2×C8×C47—22no—
134C2×C8×C47C2×D411yescharacteristic
135C2×C8×C47C2×D411yescharacteristic
136C2×C8×C47C2×D411yescharacteristic
137C2×C8×C47C2×D411yescharacteristic
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—
146C22⋊C4×C47—41no—
147C22⋊C4×C47—41no—
148C22⋊C4×C47—41no—
149C42×C47—41no—
150C42×C47C2×D411yescharacteristic
151C42×C47C2×D411yescharacteristic
152C42×C47#4111yescharacteristic
153C42×C47C2411yescharacteristic
154#10625336—44no—
155#10625334—44no—
156(C2×D47)⋊C4—22no—
157(C2×D47)⋊C4—22no—
158C4⋊Dic47—21no—
159C4⋊Dic47C2×D412yes—
160C2×C47⋊C8—22no—
161C2×C47⋊C8—22no—
162C2×C47⋊C8—22no—
163C2×C47⋊C8—42no—
164#10625324—44no—
165#10625324—44no—
166C23×C47—41no—
167C23×C47#12111yescharacteristic
168C22×D47—22no—
169C22×D47—42no—
170C22×D47—42no—
171C22×D47—42no—
172Q8×C47—41no—
173Q8×C47—41no—
174Q8×C47—42no—
175Q8×C47—81no—
176D4×C47—41no—
177D4×C47—41no—
178D4×C47—42no—
179D4×C47—81no—
180D4×C47—82no—
181C2×C4×C47—22no—
182C2×C4×C47—41no—
183C2×C4×C47—41no—
184C2×C4×C47—41no—
185C2×C4×C47—41no—
186C2×C4×C47—82no—
187C2×C4×C47#12111yescharacteristic
188C2×C4×C47#12111yescharacteristic
189C2×C4×C47#12111yescharacteristic
190C2×C4×C47#13512yes—
191C2×C4×C47AGL(1,ℤ/8)12yes—
192C2×C4×C47C22×D411yescharacteristic
193C2×C4×C47C22×D411yescharacteristic
194C2×C4×C47C2×#4111yescharacteristic
195C2×Dic47—22no—
196C2×Dic47—42no—
197D188—22no—
198D188—22no—
199D188—42no—
200C4×D47—42no—
201C4×D47—42no—
202C4×D47—44no—
203C8×C47—21no—
204C8×C47—21no—
205C8×C47—21no—
206C8×C47—21no—
207C47⋊C8—42no—
208C47⋊C8—42no—
209C47⋊C8—42no—
210#19142—471nomaximal, Sylow
211C22×C47—21no—
212C22×C47—21no—
213C22×C47—81no—
214C22×C47—81no—
215C22×C47—81no—
216C22×C47#46312yes—
217C22×C47C2×#12111yescharacteristic
218C2×D47—42no—
219C2×D47—42no—
220C2×D47—42no—
221C2×D47—84no—
222C4×C47—21no—
223C4×C47—21no—
224C4×C47—21no—
225C4×C47—81no—
226C4×C47—81no—
227C4×C47—81no—
228C4×C47#46312yes—
229C4×C47C2×#13511yescharacteristic
230C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
231C4×C47#57512yes—
232Dic47—42no—
233#3131D4711yescharacteristic, complemented
234#3101—471no—
235#3100—471no—
236C2×#436—471no—
237#1703—472no—
238#1701—472no—
239#1699—472no—
240#1697—472no—
241#1564—471no—
242#1542—471no—
243#1540—471no—
244C2×C47—22no—
245C2×C47—161no—
246C2×C47C2×#46311yescharacteristic
247C2×C47#305411yescharacteristic
248C2×C47#306711yescharacteristic
249D47—82no—
250C22×M16—941no—
251#533C2×D4711yescharacteristic
252#532C2×D4711yescharacteristic
253C2×C4×D4—471no—
254#488—21no—
255#488—21no—
256#487C2×D4711yescharacteristic, complemented
257#487C2×D4711yescharacteristic, complemented
258#486C2×D4711yescharacteristic, complemented
259#486C2×D4711yescharacteristic, complemented
260#483—941no—
261#480—941no—
262#476—472no—
263#474—472no—
264#461—472no—
265#459—472no—
266#450—941no—
267#447—941no—
268C4×SD16—942no—
269C4×D8—942no—
270#436—472no—
271#436—472no—
272#436—941no—
273#436—941no—
274#431—471no—
275#431—471no—
276C2×C4⋊C8—471no—
277#416—21no—
278#416—21no—
279C2×C22⋊C8—941no—
280C2×C4×C8C2×D4711yescharacteristic
281#335—472no—
282#334—472no—
283#333—472no—
284#332—472no—
285#332—472no—
286#331—472no—
287C47#1914211yescharacteristic, complemented, Sylow
288C2×#41—21no—
289C22×D4—471no—
290C23×C4—941no—
291C2×SD16—942no—
292C2×D8—942no—
293C2×M16—942no—
294C2×M16—942no—
295C2×M16—1884no—
296C22×C8—941no—
297C22×C8C22⋊D4711yescharacteristic
298C22×C8C22⋊D4711yescharacteristic
299C4⋊Q8C22×D4711yescharacteristic
300C4⋊Q8C22×D4711yescharacteristic
301C4⋊Q8C22×D4711yescharacteristic
302#127C22×D4711yescharacteristic
303#125—21no—
304#124—21no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307#121—21no—
308#121—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—472no—
314C4×D4—472no—
315C4×D4—941no—
316C4×D4—941no—
317#117—21no—
318C2×C4⋊C4—471no—
319C2×C22⋊C4—941no—
320C2×C42C22×D4711yescharacteristic
321#107—942no—
322#106—942no—
323C4⋊C8—472no—
324C4⋊C8—472no—
325C4⋊C8—472no—
326C4⋊C8—941no—
327#103—21no—
328#103—21no—
329#103—21no—
330#103—21no—
331#103—942no—
332#102—21no—
333#102—21no—
334#102—21no—
335#102—21no—
336#102—942no—
337#102—942no—
338#102—942no—
339C22⋊C8—941no—
340C22⋊C8—941no—
341C22⋊C8—942no—
342C22⋊C8—942no—
343C4×C8—941no—
344C4×C8—941no—
345C4×C8C22⋊D4711yescharacteristic
346C4×C8C22⋊D4711yescharacteristic
347C4×C8C22×D4711yescharacteristic
348C4×C8C22×D4711yescharacteristic
349C24—941no—
350#41—42no—
351#41—42no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—41no—
359C2×D4—472no—
360C2×D4—941no—
361C2×D4—1882no—
362C22×C4—41no—
363C22×C4—941no—
364C22×C4—942no—
365C22×C4—942no—
366C22×C4—1882no—
367C22×C4—1882no—
368C22×C4D4×D4711yescharacteristic
369C22×C4#1062535211yescharacteristic
370C22×C4C2×C22⋊D4711yescharacteristic
371SD16—1884no—
372D8—1884no—
373M16—1884no—
374M16—1884no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—942no—
378C2×C8—942no—
379C2×C8—942no—
380C2×C8—1882no—
381C2×C8C2×C22⋊D4711yescharacteristic
382C2×C8C2×C22⋊D4711yescharacteristic
383C2×C8C2×C22⋊D4711yescharacteristic
384C2×C8C2×C22⋊D4711yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—472no—
394C4⋊C4—941no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—942no—
399C22⋊C4—942no—
400C42—41no—
401C42D4×D4711yescharacteristic
402C42#1062535211yescharacteristic
403C42C2×C22⋊D4711yescharacteristic
404C42C23×D4711yescharacteristic
405C23—41no—
406C23—942no—
407C23—1882no—
408C23—1882no—
409C23—1882no—
410C23#1309399511yescharacteristic
411Q8—41no—
412Q8—41no—
413Q8—42no—
414Q8—81no—
415D4—41no—
416D4—41no—
417D4—42no—
418D4—81no—
419D4—82no—
420D4—942no—
421D4—942no—
422D4—1882no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—82no—
429C2×C4—942no—
430C2×C4—1882no—
431C2×C4—1884no—
432C2×C4—1882no—
433C2×C4—1882no—
434C2×C4#1309399511yescharacteristic
435C2×C4#1309399511yescharacteristic
436C2×C4#1309399511yescharacteristic
437C2×C4#1309404412yes—
438C2×C4#1309404912yes—
439C2×C4C2×D4×D4711yescharacteristic
440C2×C4C22×C22⋊D4711yescharacteristic
441C2×C4C2×#1062535211yescharacteristic
442C8—21no—
443C8—21no—
444C8—21no—
445C8—21no—
446C8—1882no—
447C8—1882no—
448C8—1882no—
449C22—21no—
450C22—21no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—3764no—
455C22—1882no—
456C22—1882no—
457C22—1882no—
458C22#1568933512yes—
459C22C2×#1309399511yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—1882no—
467C4#1568933512yes—
468C4C2×#1309404411yescharacteristic
469C4C2×#1309404911yescharacteristic
470C4#1568992012yes—
471C2—22no—
472C2—161no—
473C2—3762no—
474C2C2×#1568933511yescharacteristic
475C2#2025299311yescharacteristic
476C2#2025310411yescharacteristic
subgroupG/H
centerC2×C4#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3131C2
socleC22×C47C2×#121
prime radical#20253384C2
hypercenter#3131D47
hypocenterC47#19142
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #20253392C2semidirect
3. C2×#15689202C2semidirect
4. #20242960C2semidirect
5. #20242946C2semidirect
6. #20242932C2semidirect
7. #20242918C2semidirect
8. #20242136C2semidirect
9. #20242045C2semidirect
10. #20242037C2semidirect
11. C47×#3131C2semidirect
32. #15688876C22semidirect
33. #15688874C22semidirect
34. #15688872C22semidirect
35. #15688869C22semidirect
36. #15688869C22semidirect
37. #15688867C22semidirect
42. C47×#487C22semidirect
43. C47×#487C22semidirect
44. C47×#486C22semidirect
45. C47×#486C22semidirect
233. #3131D47semidirect
256. #487C2×D47semidirect
257. #487C2×D47semidirect
258. #486C2×D47semidirect
259. #486C2×D47semidirect
287. C47#19142semidirect

Maximal subgroups

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

subgroupindex
#202533922
#202533842
C2×#156892022
#202429602
#202429462
#202429322
#202429182
#202421362
#202420452
#202420372
C47×#31312
#1914247