Finite Group Databasebeta

← #26370054

Subgroups — #26370054

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
1C22×#13094046C211yesmaximal, characteristic, complemented
2C2×#15689553C212yesmaximal, complemented
3#20245326C212yesmaximal, complemented
4#20245321C211yesmaximal, characteristic, complemented
5#20245316C211yesmaximal, characteristic, complemented
6#20245309C212yesmaximal, complemented
7#20245304C211yesmaximal, characteristic, complemented
8#20245299C211yesmaximal, characteristic, complemented
9#20245093C212yesmaximal, complemented
10C2×C22⋊(C47⋊C16)C211yesmaximal, characteristic, complemented
11C47×#3050C211yesmaximal, characteristic, complemented
12C22×#10625350—21no—
13C2×#13094046—28no—
14C2×#13094046—21no—
15C2×#13094046—21no—
16C2×#13094046C2212yes—
17C2×#13094046C2212yes—
18C22×Dic188C2211yescharacteristic
19C22×#10625325—21no—
20C22×D376C2211yescharacteristic
21#15689553—28no—
22C2×#13093970C2212yescomplemented
23C2×#13093968C2212yescomplemented
24C2×#13093967C2212yescomplemented
25C2×#13093965C2212yescomplemented
26C22×C47⋊C16C2212yescomplemented
27#15688926C2212yescomplemented
28#15688926C2212yescomplemented
29#15688923C2212yescomplemented
30#15688923C2212yescomplemented
31C22⋊(C47⋊C16)C2211yescharacteristic
32C22⋊(C47⋊C16)C2211yescharacteristic
33C22⋊(C47⋊C16)C2212yescomplemented
34C2×#135×C47C2211yescharacteristic, complemented
35C23×C8×C47C2211yescharacteristic
36C47×#525—21no—
37C47×#467C2211yescharacteristic
38C47×#467C2211yescharacteristic, complemented
39C47×#466C2211yescharacteristic
40C47×#466C2211yescharacteristic, complemented
41C47×#465—21no—
42C47×#465—21no—
43C47×#427C2211yescharacteristic, complemented
44C47×#416—21no—
45C2×#10625350—28no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—44no—
49C2×#10625350—41no—
50C2×#10625350—41no—
51C22×Dic94—21no—
52C22×D188—21no—
53C22×C22⋊D47—41no—
54C22×C4×D47—41no—
55#13094046—28no—
56#13094046—28no—
57#13094046—24no—
58#13094046—24no—
59#13094046—24no—
60#13094046D418yescomplemented
61C2×Dic188—21no—
62C2×Dic188—24no—
63C2×Dic188C2312yes—
64C2×#10625325—28no—
65C2×#10625325—22no—
66C2×#10625325—22no—
67C2×D376—21no—
68C2×D376—24no—
69C2×D376C2312yes—
70#13093970—24no—
71#13093968—24no—
72#13093967—24no—
73#13093965—24no—
74C2×C47⋊C16—24no—
75C2×C47⋊C16C2312yes—
76C2×C47⋊C16C2312yes—
77C2×#41×C47—21no—
78C23×C4×C47D411yescharacteristic
79#135×C47—24no—
80C2×Q16×C47C2311yescharacteristic
81C2×SD16×C47—21no—
82C2×D8×C47C2311yescharacteristic
83C22×C8×C47—24no—
84C22×C8×C47D411yescharacteristic
85C22×C8×C47D411yescharacteristic
86C22×C8×C47D411yescharacteristic
87C22×C8×C47C2311yescharacteristic
88C22×C8×C47C2311yescharacteristic
89C22×C8×C47C2311yescharacteristic
90#123×C47—41no—
91C22⋊Q8×C47—21no—
92C22⋊Q8×C47—21no—
93#121×C47—21no—
94#121×C47—21no—
95#120×C47—41no—
96C4×D4×C47—41no—
97C4×D4×C47—41no—
98#117×C47—21no—
99#107×C47C2311yescharacteristic
100#107×C47C2311yescharacteristic
101#106×C47—21no—
102#103×C47—21no—
103#103×C47—21no—
104#102×C47—21no—
105#102×C47—21no—
106C24×C47D811yescharacteristic
107C23×D47—41no—
108#41×C47—42no—
109#41×C47—42no—
110C2×Q8×C47—21no—
111C2×D4×C47—21no—
112C2×D4×C47—41no—
113C2×D4×C47—41no—
114C2×D4×C47—41no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—41no—
118C22×C4×C47D811yescharacteristic
119C22×C4×C47D811yescharacteristic
120C22×C4×C47D811yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C2×C22⋊D47—48no—
125C2×C22⋊D47—42no—
126C2×C22⋊D47—42no—
127C22×Dic47—41no—
128#10625350—28no—
129#10625350—28no—
130#10625350—48no—
131#10625350—48no—
132#10625350—44no—
133#10625350—44no—
134C2×D188—24no—
135C2×D188—22no—
136C2×D188—42no—
137C2×D188—41no—
138C2×C4×D47—48no—
139C2×C4×D47—42no—
140C2×C4×D47—42no—
141C2×Dic94—24no—
142C2×Dic94—22no—
143C2×Dic94—42no—
144C2×Dic94—41no—
145Q16×C47—22no—
146SD16×C47—42no—
147D8×C47—22no—
148C2×C8×C47—22no—
149C2×C8×C47—22no—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C2×C8×C47—22no—
153C2×C8×C47C2×D411yescharacteristic
154C2×C8×C47C2×D411yescharacteristic
155C2×C8×C47C2×D411yescharacteristic
156C2×C8×C47C2×D414yes—
157C2×C8×C47C2411yescharacteristic
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—41no—
161C4⋊C4×C47—41no—
162C22⋊C4×C47—41no—
163C22⋊C4×C47—41no—
164C22⋊C4×C47—41no—
165C22⋊C4×C47—41no—
166C22⋊C4×C47—41no—
167C42×C47—41no—
168Dic188—24no—
169Dic188—22no—
170Dic188C2×D414yes—
171D376—24no—
172D376—22no—
173D376C2×D414yes—
174#10625325—28no—
175#10625325—28no—
176C47⋊C16—22no—
177C47⋊C16—22no—
178C23×C47—22no—
179C23×C47—22no—
180C23×C47—41no—
181C23×C47—41no—
182C23×C47C2×D811yescharacteristic
183C23×C47C2×D811yescharacteristic
184C23×C47C2×D811yescharacteristic
185C22×D47—44no—
186C22×D47—42no—
187C22×D47—82no—
188C22×D47—82no—
189Q8×C47—41no—
190Q8×C47—41no—
191D4×C47—41no—
192D4×C47—41no—
193D4×C47—82no—
194D4×C47—82no—
195D4×C47—82no—
196C2×C4×C47—22no—
197C2×C4×C47—21no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—82no—
207C2×C4×C47#12012yes—
208C2×C4×C47C2×D811yescharacteristic
209C2×C4×C47C2×D811yescharacteristic
210C2×C4×C47C2×D811yescharacteristic
211C2×C4×C47AGL(1,ℤ/8)12yes—
212C2×C4×C47C22×D411yescharacteristic
213C22⋊D47—48no—
214C22⋊D47—48no—
215C2×Dic47—44no—
216C2×Dic47—42no—
217D188—24no—
218D188—44no—
219D188—42no—
220C4×D47—48no—
221C4×D47—48no—
222Dic94—24no—
223Dic94—44no—
224Dic94—42no—
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—81no—
239C22×C47—81no—
240C22×C47—81no—
241C22×C47#44712yes—
242C22×C47#50812yescomplemented
243C22×C47C22×D811yescharacteristic
244C2×D47—44no—
245C2×D47—88no—
246C2×D47—82no—
247C2×D47—82no—
248C4×C47—21no—
249C4×C47—21no—
250C4×C47—81no—
251C4×C47—81no—
252C4×C47—81no—
253C4×C47#44712yes—
254C4×C47C2×#12011yescharacteristic
255C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
256Dic47—44no—
257C22×#135—471no—
258C2×#508—472no—
259#3050D4711yescharacteristic, complemented
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—161no—
272C2×C47C2×#44711yescharacteristic
273C2×C47C2×#50811yescharacteristic
274C2×C47#341411yescharacteristic
275D47—84no—
276C22×#41—941no—
277C2×#135—472no—
278C2×#135—472no—
279C2×#135—948no—
280C2×#135—941no—
281C2×#135—941no—
282C2×#135C2×D4711yescharacteristic, complemented
283C22×Q16—471no—
284C22×SD16—941no—
285C22×D8—471no—
286C23×C8C2×D4711yescharacteristic
287#525—21no—
288#508—948no—
289C2×Q32—472no—
290C2×SD32—472no—
291C2×SD32—472no—
292C2×D16—472no—
293C22×C16—472no—
294#467C2×D4711yescharacteristic
295#467C2×D4711yescharacteristic, complemented
296#466C2×D4711yescharacteristic
297#466C2×D4711yescharacteristic, complemented
298#465—21no—
299#465—21no—
300#427C2×D4711yescharacteristic, complemented
301#416—21no—
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—21no—
311C2×#41—948no—
312C2×#41—942no—
313C2×#41—942no—
314C2×#41—1884no—
315C2×#41—1881no—
316C2×#41—1881no—
317C22×Q8—941no—
318C22×D4—941no—
319C22×D4—1881no—
320C23×C4—1881no—
321C23×C4C22⋊D4711yescharacteristic
322#135—24no—
323#135—478no—
324#135—944no—
325#135—944no—
326#135—948no—
327#135—948no—
328#135—944no—
329C2×Q16—472no—
330C2×Q16—944no—
331C2×Q16—941no—
332C2×Q16C22×D4711yescharacteristic
333C2×SD16—21no—
334C2×SD16—948no—
335C2×SD16—942no—
336C2×SD16—942no—
337C2×D8—472no—
338C2×D8—944no—
339C2×D8—941no—
340C2×D8C22×D4711yescharacteristic
341C22×C8—24no—
342C22×C8C22⋊D4711yescharacteristic
343C22×C8C22⋊D4711yescharacteristic
344C22×C8C22⋊D4711yescharacteristic
345C22×C8C22×D4711yescharacteristic
346C22×C8C22×D4711yescharacteristic
347C22×C8C22×D4711yescharacteristic
348#123—41no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351#121—21no—
352#121—21no—
353#120—41no—
354C4×D4—41no—
355C4×D4—41no—
356#117—21no—
357Q32—944no—
358SD32—944no—
359SD32—944no—
360D16—944no—
361C2×C16—472no—
362C2×C16—472no—
363C2×C16—944no—
364#107C22×D4711yescharacteristic
365#107C22×D4711yescharacteristic
366#106—21no—
367#103—21no—
368#103—21no—
369#102—21no—
370#102—21no—
371C24—1881no—
372C24#1062533411yescharacteristic
373#41—42no—
374#41—42no—
375#41—948no—
376#41—948no—
377#41—1888no—
378#41—1888no—
379#41—1884no—
380#41—1884no—
381C2×Q8—21no—
382C2×Q8—944no—
383C2×Q8—942no—
384C2×Q8—1882no—
385C2×Q8—1881no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—944no—
391C2×D4—942no—
392C2×D4—1882no—
393C2×D4—1888no—
394C2×D4—1882no—
395C2×D4—1882no—
396C2×D4—1881no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—1888no—
401C22×C4—1881no—
402C22×C4—1882no—
403C22×C4—1882no—
404C22×C4#1062533411yescharacteristic
405C22×C4#1062533411yescharacteristic
406C22×C4#1062533411yescharacteristic
407C22×C4C2×C22⋊D4711yescharacteristic
408C22×C4C2×C22⋊D4711yescharacteristic
409C22×C4C2×C22⋊D4711yescharacteristic
410Q16—22no—
411Q16—474no—
412Q16—944no—
413Q16—942no—
414SD16—42no—
415SD16—948no—
416SD16—948no—
417D8—22no—
418D8—474no—
419D8—944no—
420D8—942no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8D4×D4714yes—
427C2×C8C2×C22⋊D4711yescharacteristic
428C2×C8C2×C22⋊D4711yescharacteristic
429C2×C8C2×C22⋊D4711yescharacteristic
430C2×C8C23×D4711yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C42—41no—
441C16—942no—
442C16—942no—
443C23—22no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—3762no—
448C23—3762no—
449C23—1884no—
450C23—1882no—
451C23C2×#1062533411yescharacteristic
452C23C2×#1062533411yescharacteristic
453C23C2×#1062533411yescharacteristic
454Q8—41no—
455Q8—41no—
456Q8—944no—
457Q8—1884no—
458Q8—1882no—
459D4—41no—
460D4—41no—
461D4—82no—
462D4—82no—
463D4—82no—
464D4—944no—
465D4—1884no—
466D4—1888no—
467D4—1888no—
468D4—1882no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—82no—
480C2×C4—1888no—
481C2×C4—1884no—
482C2×C4—1882no—
483C2×C4—1888no—
484C2×C4#1309399212yes—
485C2×C4C2×#1062533411yescharacteristic
486C2×C4C2×#1062533411yescharacteristic
487C2×C4C2×#1062533411yescharacteristic
488C2×C4#1309404912yes—
489C2×C4C22×C22⋊D4711yescharacteristic
490C8—21no—
491C8—21no—
492C8#1309399212yes—
493C8C2×D4×D4712yes—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—44no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—3768no—
506C22—3762no—
507C22—3762no—
508C22—1884no—
509C22#1568925612yes—
510C22#1568955312yescomplemented
511C22C22×#1062533411yescharacteristic
512C4—21no—
513C4—21no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—1884no—
518C4#1568925612yes—
519C4C2×#1309399211yescharacteristic
520C4C2×#1309404911yescharacteristic
521C2—22no—
522C2—22no—
523C2—41no—
524C2—161no—
525C2—3764no—
526C2C2×#1568925611yescharacteristic
527C2C2×#1568955311yescharacteristic
528C2#2025652211yescharacteristic
subgroupG/H
centerC2×C4C2×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3050C2
socleC22×C47C22×D8
hypercenter#3050D47
hypocenterC47#33516
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C22×#13094046C2semidirect
2. C2×#15689553C2semidirect
3. #20245326C2semidirect
4. #20245321C2semidirect
5. #20245316C2semidirect
6. #20245309C2semidirect
7. #20245304C2semidirect
8. #20245299C2semidirect
9. #20245093C2semidirect
10. C2×C22⋊(C47⋊C16)C2semidirect
11. C47×#3050C2semidirect
22. C2×#13093970C22semidirect
23. C2×#13093968C22semidirect
24. C2×#13093967C22semidirect
25. C2×#13093965C22semidirect
26. C22×C47⋊C16C22semidirect
27. #15688926C22semidirect
28. #15688926C22semidirect
29. #15688923C22semidirect
30. #15688923C22semidirect
33. C22⋊(C47⋊C16)C22semidirect
34. C2×#135×C47C22semidirect
38. C47×#467C22semidirect
40. C47×#466C22semidirect
43. C47×#427C22semidirect
60. #13094046D4semidirect
242. C22×C47#508semidirect
259. #3050D47semidirect
282. C2×#135C2×D47semidirect
295. #467C2×D47semidirect
297. #466C2×D47semidirect
300. #427C2×D47semidirect
309. C47#33516semidirect
510. C22#15689553semidirect

Maximal subgroups

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

subgroupindex
C22×#130940462
C2×#156895532
#202453262
#202453212
#202453162
#202453092
#202453042
#202452992
#202450932
C2×C22⋊(C47⋊C16)2
C47×#30502
#3351647