Finite Group Databasebeta

← #26370056

Subgroups — #26370056

Order 12032.

1150 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
1C2×#15689553C212yesmaximal, complemented
2#20252969C211yesmaximal, characteristic, complemented
3#20245328C212yesmaximal, complemented
4#20245322C211yesmaximal, characteristic, complemented
5#20245317C211yesmaximal, characteristic, complemented
6#20245311C212yesmaximal, complemented
7#20245305C211yesmaximal, characteristic, complemented
8#20245300C211yesmaximal, characteristic, complemented
9#20245095C212yesmaximal, complemented
10C2×C22⋊(C47⋊C16)C211yesmaximal, characteristic, complemented
11C22×#135×C47C211yesmaximal, characteristic, complemented
12C2×#13094046C2211yescharacteristic, complemented
13#15689634—21no—
14#15689553—28no—
15C2×#13093970C2212yescomplemented
16C2×#13093968C2212yescomplemented
17C2×#13093967C2212yescomplemented
18C2×#13093965C2212yescomplemented
19C22×C47⋊C16C2212yescomplemented
20#15689364C2211yescharacteristic
21#15689364C2211yescharacteristic, complemented
22#15689359C2211yescharacteristic
23#15689359C2211yescharacteristic, complemented
24#15689352—21no—
25#15689352—21no—
26#15689161C2211yescharacteristic, complemented
27#15689120—21no—
28#15688927C2212yescomplemented
29#15688927C2212yescomplemented
30#15688924C2212yescomplemented
31#15688924C2212yescomplemented
32C22⋊(C47⋊C16)C2211yescharacteristic
33C22⋊(C47⋊C16)C2211yescharacteristic
34C22⋊(C47⋊C16)C2212yescomplemented
35C22×#41×C47—21no—
36C2×#135×C47—28no—
37C2×#135×C47—21no—
38C2×#135×C47—21no—
39C2×#135×C47C2212yes—
40C2×#135×C47C2212yes—
41C22×Q16×C47C2211yescharacteristic
42C22×SD16×C47—21no—
43C22×D8×C47C2211yescharacteristic
44C23×C8×C47C2211yescharacteristic
45C2×#10625350—21no—
46#13094046—24no—
47C2×Dic188C2311yescharacteristic
48C2×#10625325—21no—
49C2×D376C2311yescharacteristic
50#13094007—41no—
51#13094003—21no—
52#13094003—21no—
53#13093998—21no—
54#13093998—21no—
55#13093993—41no—
56C4×C22⋊D47—41no—
57C4×C22⋊D47—41no—
58#13093983—21no—
59#13093970—24no—
60#13093968—24no—
61#13093967—24no—
62#13093965—24no—
63C2×C47⋊C16—24no—
64C2×C47⋊C16C2312yes—
65C2×C47⋊C16C2312yes—
66#13093958C2311yescharacteristic
67#13093958C2311yescharacteristic
68#13093956—21no—
69#13093949—21no—
70#13093949—21no—
71#13093946—21no—
72#13093946—21no—
73C2×#41×C47—28no—
74C2×#41×C47—22no—
75C2×#41×C47—22no—
76C2×#41×C47—44no—
77C2×#41×C47—41no—
78C2×#41×C47—41no—
79C22×Q8×C47—21no—
80C22×D4×C47—21no—
81C22×D4×C47—41no—
82C23×C4×C47—41no—
83C23×C4×C47D411yescharacteristic
84#135×C47—28no—
85#135×C47—28no—
86#135×C47—24no—
87#135×C47—24no—
88#135×C47—24no—
89#135×C47D418yescomplemented
90C2×Q16×C47—24no—
91C2×Q16×C47—21no—
92C2×Q16×C47C2312yes—
93C2×SD16×C47—28no—
94C2×SD16×C47—22no—
95C2×SD16×C47—22no—
96C2×D8×C47—24no—
97C2×D8×C47—21no—
98C2×D8×C47C2312yes—
99C22×C8×C47—24no—
100C22×C8×C47D411yescharacteristic
101C22×C8×C47D411yescharacteristic
102C22×C8×C47D411yescharacteristic
103C22×C8×C47C2311yescharacteristic
104C22×C8×C47C2311yescharacteristic
105C22×C8×C47C2311yescharacteristic
106C24×C47—41no—
107C24×C47D811yescharacteristic
108#41×C47—28no—
109#41×C47—28no—
110#41×C47—48no—
111#41×C47—44no—
112#41×C47—48no—
113#41×C47—44no—
114C2×Q8×C47—24no—
115C2×Q8×C47—22no—
116C2×Q8×C47—42no—
117C2×Q8×C47—41no—
118C2×D4×C47—24no—
119C2×D4×C47—22no—
120C2×D4×C47—42no—
121C2×D4×C47—48no—
122C2×D4×C47—42no—
123C2×D4×C47—42no—
124C2×D4×C47—41no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—48no—
128C22×C4×C47—41no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47D811yescharacteristic
132C22×C4×C47D811yescharacteristic
133C22×C4×C47D811yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C2×C22⋊D47—41no—
138C2×C22⋊D47—41no—
139C2×C22⋊D47—41no—
140#10625350—42no—
141#10625350—42no—
142C2×D188—21no—
143C2×C4×D47—41no—
144C2×Dic94—21no—
145Q16×C47—24no—
146Q16×C47—22no—
147Q16×C47C2×D414yes—
148SD16×C47—28no—
149SD16×C47—28no—
150D8×C47—24no—
151D8×C47—22no—
152D8×C47C2×D414yes—
153C2×C8×C47—22no—
154C2×C8×C47—22no—
155C2×C8×C47—22no—
156C2×C8×C47—22no—
157C2×C8×C47—22no—
158C2×C8×C47C2×D411yescharacteristic
159C2×C8×C47C2×D411yescharacteristic
160C2×C8×C47C2×D411yescharacteristic
161C2×C8×C47C2×D414yes—
162C2×C8×C47C2411yescharacteristic
163C22⋊Dic47—41no—
164C22⋊Dic47—41no—
165C22⋊Dic47—41no—
166(C2×D47)⋊C4—41no—
167(C2×D47)⋊C4—41no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170Dic47⋊C4—41no—
171Dic47⋊C4—41no—
172C4×Dic47—41no—
173Dic188—22no—
174D376—22no—
175#10625325—42no—
176C47⋊C16—22no—
177C47⋊C16—22no—
178C23×C47—22no—
179C23×C47—22no—
180C23×C47—41no—
181C23×C47—44no—
182C23×C47—42no—
183C23×C47—82no—
184C23×C47—82no—
185C23×C47C2×D811yescharacteristic
186C23×C47C2×D811yescharacteristic
187C23×C47C2×D811yescharacteristic
188C22×D47—41no—
189Q8×C47—24no—
190Q8×C47—44no—
191Q8×C47—42no—
192D4×C47—24no—
193D4×C47—44no—
194D4×C47—48no—
195D4×C47—48no—
196D4×C47—42no—
197C2×C4×C47—22no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—48no—
205C2×C4×C47—44no—
206C2×C4×C47—42no—
207C2×C4×C47—48no—
208C2×C4×C47#12012yes—
209C2×C4×C47C2×D811yescharacteristic
210C2×C4×C47C2×D811yescharacteristic
211C2×C4×C47C2×D811yescharacteristic
212C2×C4×C47AGL(1,ℤ/8)12yes—
213C2×C4×C47C22×D411yescharacteristic
214C22⋊D47—82no—
215C22⋊D47—82no—
216C22⋊D47—82no—
217C2×Dic47—41no—
218C2×Dic47—41no—
219C2×Dic47—41no—
220D188—41no—
221D188—41no—
222C4×D47—82no—
223Dic94—41no—
224Dic94—41no—
225C8×C47—21no—
226C8×C47—21no—
227C8×C47#12012yes—
228C8×C47C22×D412yes—
229#33516—471nomaximal, Sylow
230C22×C47—21no—
231C22×C47—21no—
232C22×C47—21no—
233C22×C47—21no—
234C22×C47—21no—
235C22×C47—21no—
236C22×C47—21no—
237C22×C47—44no—
238C22×C47—44no—
239C22×C47—88no—
240C22×C47—82no—
241C22×C47—82no—
242C22×C47#44712yes—
243C22×C47#50812yescomplemented
244C22×C47C22×D811yescharacteristic
245C2×D47—81no—
246C2×D47—81no—
247C2×D47—81no—
248C4×C47—21no—
249C4×C47—21no—
250C4×C47—44no—
251C4×C47#44712yes—
252C4×C47C2×#12011yescharacteristic
253C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
254Dic47—81no—
255Dic47—81no—
256Dic47—81no—
257C22×#135D4711yescharacteristic, complemented
258C2×#508—472no—
259#3050—471no—
260#2189—472no—
261#2188—471no—
262#2187—471no—
263#2186—472no—
264#2185—471no—
265#2184—471no—
266#2138—472no—
267C2×C22⋊C16—471no—
268C2×C47—22no—
269C2×C47—22no—
270C2×C47—41no—
271C2×C47—84no—
272C2×C47C2×#44711yescharacteristic
273C2×C47C2×#50811yescharacteristic
274C2×C47#341411yescharacteristic
275D47—161no—
276C22×#41—21no—
277C2×#135—28no—
278C2×#135—21no—
279C2×#135—21no—
280C2×#135—471no—
281C2×#135C2×D4712yes—
282C2×#135C2×D4712yes—
283C22×Q16C2×D4711yescharacteristic
284C22×SD16—21no—
285C22×D8C2×D4711yescharacteristic
286C23×C8C2×D4711yescharacteristic
287#525—941no—
288#508—948no—
289C2×Q32—472no—
290C2×SD32—472no—
291C2×SD32—472no—
292C2×D16—472no—
293C22×C16—472no—
294#467—471no—
295#467—471no—
296#466—471no—
297#466—471no—
298#465—941no—
299#465—941no—
300#427—471no—
301#416—941no—
302#358—472no—
303#358—472no—
304#357—472no—
305#357—472no—
306C22⋊C16—471no—
307C22⋊C16—471no—
308C22⋊C16—472no—
309C47#3351611yescharacteristic, complemented, Sylow
310C2×#41—28no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—44no—
314C2×#41—41no—
315C2×#41—41no—
316C2×#41—941no—
317C22×Q8—21no—
318C22×D4—21no—
319C22×D4—41no—
320C23×C4—41no—
321C23×C4C22⋊D4711yescharacteristic
322#135—28no—
323#135—28no—
324#135—24no—
325#135—24no—
326#135—24no—
327#135—944no—
328#135C22⋊D4718yescomplemented
329C2×Q16—24no—
330C2×Q16—21no—
331C2×Q16—471no—
332C2×Q16C22×D4712yes—
333C2×SD16—28no—
334C2×SD16—22no—
335C2×SD16—22no—
336C2×SD16—941no—
337C2×D8—24no—
338C2×D8—21no—
339C2×D8—471no—
340C2×D8C22×D4712yes—
341C22×C8—24no—
342C22×C8C22⋊D4711yescharacteristic
343C22×C8C22⋊D4711yescharacteristic
344C22×C8C22⋊D4711yescharacteristic
345C22×C8C22×D4711yescharacteristic
346C22×C8C22×D4711yescharacteristic
347C22×C8C22×D4711yescharacteristic
348#123—1881no—
349C22⋊Q8—941no—
350C22⋊Q8—941no—
351#121—941no—
352#121—941no—
353#120—1881no—
354C4×D4—1881no—
355C4×D4—1881no—
356#117—941no—
357Q32—944no—
358SD32—944no—
359SD32—944no—
360D16—944no—
361C2×C16—472no—
362C2×C16—472no—
363C2×C16—944no—
364#107—471no—
365#107—471no—
366#106—941no—
367#103—941no—
368#103—941no—
369#102—941no—
370#102—941no—
371C24—41no—
372C24#1062533411yescharacteristic
373#41—28no—
374#41—28no—
375#41—48no—
376#41—48no—
377#41—44no—
378#41—44no—
379#41—1882no—
380#41—1882no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—42no—
384C2×Q8—41no—
385C2×Q8—941no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—42no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—41no—
393C2×D4—941no—
394C2×D4—1881no—
395C2×D4—1881no—
396C2×D4—1881no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—48no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—1881no—
404C22×C4#1062533411yescharacteristic
405C22×C4#1062533411yescharacteristic
406C22×C4#1062533411yescharacteristic
407C22×C4C2×C22⋊D4711yescharacteristic
408C22×C4C2×C22⋊D4711yescharacteristic
409C22×C4C2×C22⋊D4711yescharacteristic
410Q16—24no—
411Q16—22no—
412Q16—942no—
413Q16C2×C22⋊D4714yes—
414SD16—28no—
415SD16—28no—
416SD16—1882no—
417D8—24no—
418D8—22no—
419D8—942no—
420D8C2×C22⋊D4714yes—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8C2×C22⋊D4711yescharacteristic
427C2×C8C2×C22⋊D4711yescharacteristic
428C2×C8C2×C22⋊D4711yescharacteristic
429C2×C8C2×C22⋊D4714yes—
430C2×C8C23×D4711yescharacteristic
431C4⋊C4—941no—
432C4⋊C4—941no—
433C4⋊C4—1881no—
434C4⋊C4—1881no—
435C22⋊C4—1881no—
436C22⋊C4—1881no—
437C22⋊C4—1881no—
438C22⋊C4—1881no—
439C22⋊C4—1881no—
440C42—1881no—
441C16—942no—
442C16—942no—
443C23—22no—
444C23—22no—
445C23—41no—
446C23—44no—
447C23—42no—
448C23—82no—
449C23—82no—
450C23—1881no—
451C23C2×#1062533411yescharacteristic
452C23C2×#1062533411yescharacteristic
453C23C2×#1062533411yescharacteristic
454Q8—24no—
455Q8—44no—
456Q8—42no—
457Q8—1881no—
458Q8—1881no—
459D4—24no—
460D4—44no—
461D4—48no—
462D4—48no—
463D4—42no—
464D4—3762no—
465D4—3762no—
466D4—3762no—
467D4—1881no—
468D4—1881no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—48no—
480C2×C4—3762no—
481C2×C4—1881no—
482C2×C4—1881no—
483C2×C4—1881no—
484C2×C4#1309399312yes—
485C2×C4C2×#1062533411yescharacteristic
486C2×C4C2×#1062533411yescharacteristic
487C2×C4C2×#1062533411yescharacteristic
488C2×C4#1309405012yes—
489C2×C4C22×C22⋊D4711yescharacteristic
490C8—21no—
491C8—21no—
492C8#1309399312yes—
493C8C22×C22⋊D4712yes—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—44no—
502C22—44no—
503C22—88no—
504C22—82no—
505C22—82no—
506C22—3761no—
507C22—3761no—
508C22—3761no—
509C22#1568925712yes—
510C22#1568955312yescomplemented
511C22C22×#1062533411yescharacteristic
512C4—21no—
513C4—21no—
514C4—44no—
515C4—3761no—
516C4—3761no—
517C4—3761no—
518C4#1568925712yes—
519C4C2×#1309399311yescharacteristic
520C4C2×#1309405011yescharacteristic
521C2—22no—
522C2—22no—
523C2—41no—
524C2—84no—
525C2—7521no—
526C2C2×#1568925711yescharacteristic
527C2C2×#1568955311yescharacteristic
528C2#2025652211yescharacteristic
subgroupG/H
centerC2×C4C2×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC22×#135×C47C2
socleC22×C47C22×D8
hypercenterC22×#135D47
hypocenterC47#33516
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C22×#135

Complements

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

NH ≅ G/Ntype
1. C2×#15689553C2semidirect
2. #20252969C2semidirect
3. #20245328C2semidirect
4. #20245322C2semidirect
5. #20245317C2semidirect
6. #20245311C2semidirect
7. #20245305C2semidirect
8. #20245300C2semidirect
9. #20245095C2semidirect
10. C2×C22⋊(C47⋊C16)C2semidirect
11. C22×#135×C47C2semidirect
12. C2×#13094046C22semidirect
15. C2×#13093970C22semidirect
16. C2×#13093968C22semidirect
17. C2×#13093967C22semidirect
18. C2×#13093965C22semidirect
19. C22×C47⋊C16C22semidirect
21. #15689364C22semidirect
23. #15689359C22semidirect
26. #15689161C22semidirect
28. #15688927C22semidirect
29. #15688927C22semidirect
30. #15688924C22semidirect
31. #15688924C22semidirect
34. C22⋊(C47⋊C16)C22semidirect
89. #135×C47D4semidirect
243. C22×C47#508semidirect
257. C22×#135D47semidirect
309. C47#33516semidirect
328. #135C22⋊D47semidirect
510. C22#15689553semidirect

Maximal subgroups

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

subgroupindex
C2×#156895532
#202529692
#202453282
#202453222
#202453172
#202453112
#202453052
#202453002
#202450952
C2×C22⋊(C47⋊C16)2
C22×#135×C472
#3351647