Finite Group Databasebeta

← #26200072

Subgroups — #26200072

Order 12032.

690 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253970C211yesmaximal, characteristic, complemented
2#20253185C211yesmaximal, characteristic
3C2×D4×C47⋊C8C211yesmaximal, characteristic
4#20242841C212yesmaximal, complemented
5#20242837C212yesmaximal, complemented
6#20242785C212yesmaximal, complemented
7#20242751C212yesmaximal, complemented
8#20242146C211yesmaximal, characteristic, complemented
9#20242079C211yesmaximal, characteristic, complemented
10#20242078C211yesmaximal, characteristic, complemented
11C47×#3077C211yesmaximal, characteristic
12C23×C47⋊C8—21no—
13#15689680C2211yescharacteristic
14#15689526C2211yescharacteristic, complemented
15#15689502C2211yescharacteristic, complemented
16#15689475C2211yescharacteristic, complemented
17#15689473C2211yescharacteristic, complemented
18#15689423—22no—
19#15689413C2212yes—
20#15689403C2212yes—
21#15689362—21no—
22#15689362—21no—
23C4×#10625337—21no—
24C4×#10625337—21no—
25D4×C47⋊C8—21no—
26D4×C47⋊C8—21no—
27D4×C47⋊C8C2212yes—
28D4×C47⋊C8C2212yes—
29#15689163—21no—
30#15689147C2211yescharacteristic
31C2×C4⋊(C47⋊C8)C2211yescharacteristic
32#15689128—21no—
33C2×C22⋊(C47⋊C8)—21no—
34C2×C4×C47⋊C8C2211yescharacteristic
35#15688875C2212yescomplemented
36#15688875C2212yescomplemented
37#15688872C2212yescomplemented
38#15688871C2212yescomplemented
39#15688869C2212yescomplemented
40#15688868C2212yescomplemented
41C47×#533C2211yescharacteristic
42C2×C4×D4×C47C2211yescharacteristic
43C47×#478—22no—
44C47×#476C2212yes—
45C47×#474C2212yes—
46C47×#428—21no—
47C47×#423C2211yescharacteristic
48C47×#418—21no—
49C2×#10625337—21no—
50C2×#10625337—21no—
51C2×#10625329—21no—
52C22×C47⋊C8—21no—
53C22×C47⋊C8—22no—
54C22×C47⋊C8—22no—
55C22×C47⋊C8—44no—
56C22×C47⋊C8D411yescharacteristic
57C22×C47⋊C8D411yescharacteristic
58#13094024C2311yescharacteristic
59#13094024C2311yescharacteristic
60#13094016—21no—
61#13094003—21no—
62#13094003—21no—
63C4×Dic94—21no—
64C4×Dic94—21no—
65#13093983—21no—
66#13093957—22no—
67#13093957—21no—
68#13093957—21no—
69#13093955—22no—
70C4⋊(C47⋊C8)—21no—
71C4⋊(C47⋊C8)—21no—
72C4⋊(C47⋊C8)C2312yes—
73C4⋊(C47⋊C8)C2312yes—
74#13093948—21no—
75#13093948—21no—
76#13093947—21no—
77#13093947—21no—
78#13093945—22no—
79#13093945—22no—
80C22⋊(C47⋊C8)—22no—
81C22⋊(C47⋊C8)—22no—
82C4×C47⋊C8D411yescharacteristic
83C4×C47⋊C8D411yescharacteristic
84C4×C47⋊C8C2311yescharacteristic
85C4×C47⋊C8C2311yescharacteristic
86C22×D4×C47D411yescharacteristic
87C23×C4×C47—21no—
88C2×M16×C47—21no—
89C4⋊Q8×C47C2311yescharacteristic
90C4⋊Q8×C47C2311yescharacteristic
91#125×C47—21no—
92C22⋊Q8×C47—21no—
93C22⋊Q8×C47—21no—
94C4×Q8×C47—21no—
95C4×Q8×C47—21no—
96C4×D4×C47—21no—
97C4×D4×C47—21no—
98C4×D4×C47D412yes—
99C4×D4×C47C2312yes—
100#117×C47—21no—
101C2×C4⋊C4×C47D411yescharacteristic
102C2×C22⋊C4×C47—21no—
103C2×C42×C47C2311yescharacteristic
104#107×C47—22no—
105#106×C47—22no—
106C4⋊C8×C47—21no—
107C4⋊C8×C47C2312yes—
108#102×C47—22no—
109#102×C47—22no—
110C24×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×D4×C47—21no—
114C2×D4×C47—22no—
115C2×D4×C47Q1614yescomplemented
116C2×D4×C47C2×D412yes—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—42no—
121C22×C4×C47—42no—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47#4111yescharacteristic
125C2×Dic94—21no—
126C2×Dic94—21no—
127M16×C47—42no—
128C2×C8×C47—22no—
129C4⋊C4×C47—21no—
130C4⋊C4×C47—21no—
131C4⋊C4×C47—21no—
132C4⋊C4×C47—21no—
133C4⋊C4×C47—21no—
134C4⋊C4×C47—41no—
135C4⋊C4×C47—41no—
136C4⋊C4×C47C2×D412yes—
137C22⋊C4×C47—22no—
138C22⋊C4×C47—22no—
139C22⋊C4×C47—41no—
140C42×C47—41no—
141C42×C47C2×D411yescharacteristic
142C42×C47C2×D411yescharacteristic
143C42×C47#4111yescharacteristic
144C42×C47C2411yescharacteristic
145C22⋊Dic47—41no—
146#10625337—42no—
147#10625337—42no—
148C4⋊Dic47—21no—
149C4⋊Dic47—21no—
150C4⋊Dic47—21no—
151C4⋊Dic47—21no—
152Dic47⋊C4—41no—
153Dic47⋊C4—41no—
154C4×Dic47—41no—
155#10625329—42no—
156C2×C47⋊C8—22no—
157C2×C47⋊C8—22no—
158C2×C47⋊C8—22no—
159C2×C47⋊C8—41no—
160C2×C47⋊C8—44no—
161C2×C47⋊C8—42no—
162C2×C47⋊C8—44no—
163C2×C47⋊C8C2×D411yescharacteristic
164C2×C47⋊C8C2×D411yescharacteristic
165C2×C47⋊C8C2×D411yescharacteristic
166C2×C47⋊C8C2×D411yescharacteristic
167C23×C47—22no—
168C23×C47—24no—
169C23×C47—42no—
170C23×C47—42no—
171C23×C47#12111yescharacteristic
172Q8×C47—41no—
173Q8×C47—41no—
174Q8×C47—81no—
175D4×C47—22no—
176D4×C47—24no—
177D4×C47C2×Q1614yes—
178C2×C4×C47—21no—
179C2×C4×C47—21no—
180C2×C4×C47—22no—
181C2×C4×C47—41no—
182C2×C4×C47—44no—
183C2×C4×C47—42no—
184C2×C4×C47—42no—
185C2×C4×C47—42no—
186C2×C4×C47—41no—
187C2×C4×C47—41no—
188C2×C4×C47#12111yescharacteristic
189C2×C4×C47#12111yescharacteristic
190C2×C4×C47#12111yescharacteristic
191C2×C4×C47C2×Q1612yes—
192C2×C4×C47#13512yes—
193C2×C4×C47C22×D411yescharacteristic
194C2×C4×C47C22×D411yescharacteristic
195C2×C4×C47C2×#4111yescharacteristic
196C2×Dic47—41no—
197C2×Dic47—41no—
198Dic94—41no—
199Dic94—41no—
200Dic94—81no—
201C8×C47—42no—
202C47⋊C8—21no—
203C47⋊C8—21no—
204C47⋊C8—21no—
205C47⋊C8—42no—
206C47⋊C8—42no—
207C47⋊C8AGL(1,ℤ/8)12yes—
208#19256—471nomaximal, Sylow
209C22×C47—21no—
210C22×C47—21no—
211C22×C47—24no—
212C22×C47—48no—
213C22×C47—42no—
214C22×C47—42no—
215C22×C47#46712yes—
216C22×C47C2×#12111yescharacteristic
217C4×C47—21no—
218C4×C47—21no—
219C4×C47—21no—
220C4×C47—42no—
221C4×C47—81no—
222C4×C47—81no—
223C4×C47#46712yes—
224C4×C47C22×Q1611yescharacteristic
225C4×C47C2×#13511yescharacteristic
226C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
227C4×C47#57511yescharacteristic
228Dic47—81no—
229Dic47—81no—
230#3170—471no—
231#3077—471no—
232#3077D4711yescharacteristic
233C2×C8×D4—471no—
234#1683—472no—
235#1683—472no—
236#1675—472no—
237#1669—472no—
238#1567—471no—
239#1550—471no—
240#1550—471no—
241C2×C47—22no—
242C2×C47—44no—
243C2×C47#303611yescharacteristic
244C2×C47C2×#46711yescharacteristic
245C2×C47#306711yescharacteristic
246C23×C8—941no—
247#533—471no—
248#533C2×D4711yescharacteristic
249C2×C4×D4C2×D4711yescharacteristic
250#500—471no—
251#494—471no—
252#487—471no—
253#487—471no—
254#478—22no—
255#478—942no—
256#476—472no—
257#476C2×D4712yes—
258#474—472no—
259#474C2×D4712yes—
260#467—941no—
261#467—941no—
262C4×Q16—941no—
263C4×Q16—941no—
264C8×D4—472no—
265C8×D4—472no—
266C8×D4—941no—
267C8×D4—941no—
268#428—21no—
269#428—941no—
270#423—471no—
271#423C2×D4711yescharacteristic
272C2×C4⋊C8—471no—
273#418—21no—
274#418—941no—
275C2×C22⋊C8—941no—
276C2×C4×C8—471no—
277#335—472no—
278#335—472no—
279#333—472no—
280#333—472no—
281#332—472no—
282#332—472no—
283C47#1925611yescharacteristic, complemented, Sylow
284C22×D4C22⋊D4711yescharacteristic
285C23×C4—21no—
286C2×Q16—941no—
287C2×Q16—941no—
288C2×M16—21no—
289C2×M16—941no—
290C22×C8—471no—
291C22×C8—471no—
292C22×C8—942no—
293C22×C8—942no—
294C22×C8—941no—
295C22×C8—1884no—
296C4⋊Q8—471no—
297C4⋊Q8—471no—
298C4⋊Q8C22×D4711yescharacteristic
299C4⋊Q8C22×D4711yescharacteristic
300#125—21no—
301#125—941no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304C22⋊Q8—941no—
305C22⋊Q8—941no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—941no—
309C4×Q8—941no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4C22⋊D4712yes—
313C4×D4C22×D4712yes—
314#117—21no—
315#117—941no—
316C2×C4⋊C4C22⋊D4711yescharacteristic
317C2×C22⋊C4—21no—
318C2×C42C22×D4711yescharacteristic
319#107—22no—
320#107—941no—
321#107—941no—
322#107—942no—
323#106—22no—
324#106—942no—
325C4⋊C8—21no—
326C4⋊C8—472no—
327C4⋊C8—472no—
328C4⋊C8—941no—
329C4⋊C8—941no—
330C4⋊C8C22×D4712yes—
331#103—941no—
332#103—941no—
333#103—941no—
334#103—941no—
335#102—22no—
336#102—22no—
337#102—942no—
338#102—942no—
339C22⋊C8—942no—
340C22⋊C8—942no—
341C4×C8—471no—
342C4×C8—471no—
343C4×C8—471no—
344C4×C8—471no—
345C24—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—941no—
349C2×Q8—941no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4#1062533714yescomplemented
353C2×D4C2×C22⋊D4712yes—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4D4×D4711yescharacteristic
360C22×C4#1062535211yescharacteristic
361C22×C4C2×C22⋊D4711yescharacteristic
362Q16—1882no—
363Q16—1882no—
364M16—42no—
365M16—1882no—
366C2×C8—22no—
367C2×C8—471no—
368C2×C8—471no—
369C2×C8—471no—
370C2×C8—471no—
371C2×C8—942no—
372C2×C8—942no—
373C2×C8—942no—
374C2×C8—1884no—
375C2×C8—1881no—
376C2×C8—1884no—
377C2×C8—1882no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—941no—
386C4⋊C4—941no—
387C4⋊C4—941no—
388C4⋊C4—941no—
389C4⋊C4—1881no—
390C4⋊C4—1881no—
391C4⋊C4C2×C22⋊D4712yes—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—41no—
395C22⋊C4—1881no—
396C42—41no—
397C42—1881no—
398C42D4×D4711yescharacteristic
399C42#1062535211yescharacteristic
400C42C2×C22⋊D4711yescharacteristic
401C42C23×D4711yescharacteristic
402C23—22no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23#1309399611yescharacteristic
407Q8—41no—
408Q8—41no—
409Q8—81no—
410Q8—3761no—
411Q8—1881no—
412Q8—1881no—
413D4—24no—
414D4—22no—
415D4C2×#1062533714yes—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—1881no—
427C2×C4—1881no—
428C2×C4#1309399611yescharacteristic
429C2×C4#1309399611yescharacteristic
430C2×C4#1309399611yescharacteristic
431C2×C4C2×#1062533712yes—
432C2×C4#1309404412yes—
433C2×C4C2×D4×D4711yescharacteristic
434C2×C4C22×C22⋊D4711yescharacteristic
435C2×C4C2×#1062535211yescharacteristic
436C8—42no—
437C8—472no—
438C8—941no—
439C8—941no—
440C8—941no—
441C8—1882no—
442C8—1882no—
443C22—21no—
444C22—21no—
445C22—24no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22#1568936212yes—
450C22C2×#1309399611yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4—3761no—
458C4—3761no—
459C4#1568936212yes—
460C4C22×#1062533711yescharacteristic
461C4C2×#1309404411yescharacteristic
462C4AGL(1,ℤ/8)×D4711yescharacteristic
463C4#1568991911yescharacteristic
464C2—22no—
465C2—44no—
466C2#2025284911yescharacteristic
467C2C2×#1568936211yescharacteristic
468C2#2025310511yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3077C2
socleC22×C47C2×#121
prime radicalC22×D4×C47D4
hypercenter#3077D47
hypocenterC47#19256
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #20253970C2semidirect
4. #20242841C2semidirect
5. #20242837C2semidirect
6. #20242785C2semidirect
7. #20242751C2semidirect
8. #20242146C2semidirect
9. #20242079C2semidirect
10. #20242078C2semidirect
14. #15689526C22semidirect
15. #15689502C22semidirect
16. #15689475C22semidirect
17. #15689473C22semidirect
35. #15688875C22semidirect
36. #15688875C22semidirect
37. #15688872C22semidirect
38. #15688871C22semidirect
39. #15688869C22semidirect
40. #15688868C22semidirect
115. C2×D4×C47Q16semidirect
283. C47#19256semidirect
352. C2×D4#10625337semidirect

Maximal subgroups

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

subgroupindex
#202539702
#202531852
C2×D4×C47⋊C82
#202428412
#202428372
#202427852
#202427512
#202421462
#202420792
#202420782
C47×#30772
#1925647