Finite Group Databasebeta

← #26217009

Subgroups — #26217009

Order 12032.

1862 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D47C211yesmaximal, characteristic
2C2×#15689166C211yesmaximal, characteristic, complemented
3C2×#15689086C211yesmaximal, characteristic
4#20245767C211yesmaximal, characteristic
5#20243579C214yesmaximal, complemented
6#20243574C214yesmaximal, complemented
7#20243108C211yesmaximal, characteristic, complemented
8#20243107C211yesmaximal, characteristic, complemented
9C2×C47×#428C211yesmaximal, characteristic, complemented
10C23×C4×D47—21no—
11C23×C4×D47C2211yescharacteristic
12C22×#10625329C2211yescharacteristic
13C2×#13093983C2211yescharacteristic
14C22×C4⋊Dic47C2211yescharacteristic
15C22×Dic47⋊C4—21no—
16C2×C4⋊C4×D47—22no—
17C2×C4⋊C4×D47—24no—
18C2×C4⋊C4×D47—24no—
19C2×C4⋊C4×D47C2214yes—
20#15689166—24no—
21C2×#13093957C2212yescomplemented
22C2×#13093955C2212yescomplemented
23#15689086C418yescomplemented
24C2×(C2×D47)⋊C8C2214yes—
25#15688984—21no—
26#15688980C2212yes—
27#15688980C2212yes—
28#15688978—21no—
29C4×(C2×D47)⋊C4—22no—
30#15688886C2214yescomplemented
31#15688885C2214yescomplemented
32C22×M16×C47C2211yescharacteristic
33C2×#117×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2211yescharacteristic
35C47×#428—24no—
36C2×#107×C47C2212yescomplemented
37C2×#106×C47C2212yescomplemented
38C24×D47D411yescharacteristic
39C23×Dic47—21no—
40C23×Dic47D411yescharacteristic
41C22×C4×D47—22no—
42C22×C4×D47—22no—
43C22×C4×D47—22no—
44C22×C4×D47—22no—
45C22×C4×D47—28no—
46C22×C4×D47—24no—
47C22×C4×D47—48no—
48C22×C4×D47C2×C414yes—
49C22×C4×D47D412yes—
50C22×C4×D47Q814yes—
51C22×C4×D47C2312yes—
52C2×#10625329—22no—
53C2×#10625329—22no—
54C2×#10625329C2×C414yescomplemented
55C22×C47⋊C8C2312yes—
56#13093983—22no—
57#13093983—22no—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47—22no—
60C2×C4⋊Dic47D414yes—
61C2×C4⋊Dic47C2312yes—
62C2×C4⋊Dic47C2312yes—
63C2×Dic47⋊C4—28no—
64C2×Dic47⋊C4—24no—
65C4⋊C4×D47—216no—
66C4⋊C4×D47—28no—
67C4⋊C4×D47—28no—
68C2×C22⋊Dic47—21no—
69C2×(C2×D47)⋊C4—22no—
70C2×(C2×D47)⋊C4—22no—
71C2×C4×Dic47—21no—
72#13093957—24no—
73#13093955—24no—
74(C2×D47)⋊C8C2×C4116yescomplemented
75C47⋊#95—22no—
76C47⋊#95—22no—
77C23×C4×C47—21no—
78C23×C4×C47C2311yescharacteristic
79C2×M16×C47—22no—
80C2×M16×C47—22no—
81C2×M16×C47C2×C414yescomplemented
82C22×C8×C47C2312yes—
83#117×C47—22no—
84#117×C47—22no—
85C2×C4⋊C4×C47—21no—
86C2×C4⋊C4×C47—22no—
87C2×C4⋊C4×C47D414yes—
88C2×C4⋊C4×C47C2311yescharacteristic
89C2×C4⋊C4×C47C2311yescharacteristic
90C2×C4⋊C4×C47C2312yes—
91C2×C22⋊C4×C47—21no—
92C2×C42×C47—21no—
93#107×C47—24no—
94#106×C47—24no—
95C24×C47C2×D411yescharacteristic
96C23×D47—22no—
97C23×D47—22no—
98C23×D47—24no—
99C23×D47—24no—
100C23×D47C4⋊C414yes—
101C23×D47C2×D412yes—
102C22×C4×C47—24no—
103C22×C4×C47—22no—
104C22×C4×C47—21no—
105C22×C4×C47—21no—
106C22×C4×C47—44no—
107C22×C4×C47C22×C412yes—
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2×D412yes—
110C22×C4×C47C2×D412yes—
111C22×C4×C47C2×Q812yes—
112C22×C4×C47#4112yes—
113C22×C4×C47#4112yes—
114C22×C4×C47C2411yescharacteristic
115C22×Dic47—22no—
116C22×Dic47—22no—
117C22×Dic47—24no—
118C22×Dic47—22no—
119C22×Dic47—22no—
120C22×Dic47—44no—
121C22×Dic47C4⋊C414yes—
122C22×Dic47C2×D412yes—
123C2×C4×D47—28no—
124C2×C4×D47—28no—
125C2×C4×D47—24no—
126C2×C4×D47—28no—
127C2×C4×D47—28no—
128C2×C4×D47—24no—
129C2×C4×D47—22no—
130C2×C4×D47—22no—
131C2×C4×D47—22no—
132C2×C4×D47—22no—
133C2×C4×D47—216no—
134C2×C4×D47—432no—
135C2×C4×D47—48no—
136C2×C4×D47—48no—
137C2×C4×D47C4⋊C418yes—
138C2×C4×D47C22×C414yes—
139C2×C4×D47C2×Q814yes—
140M16×C47—28no—
141C2×C8×C47—22no—
142C2×C8×C47—22no—
143C2×C8×C47C22×C414yescomplemented
144C4⋊C4×C47—24no—
145C4⋊C4×C47—22no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47C2×D414yes—
151C22⋊C4×C47—42no—
152C42×C47—42no—
153C22⋊Dic47—42no—
154(C2×D47)⋊C4—44no—
155(C2×D47)⋊C4—44no—
156C4⋊Dic47—24no—
157C4⋊Dic47—22no—
158C4⋊Dic47—22no—
159C4⋊Dic47—22no—
160C4⋊Dic47C2×D414yes—
161Dic47⋊C4—216no—
162C4×Dic47—42no—
163#10625329—28no—
164C2×C47⋊C8—22no—
165C2×C47⋊C8—22no—
166C2×C47⋊C8C22×C414yescomplemented
167C23×C47—22no—
168C23×C47—22no—
169C23×C47C2×C4⋊C412yes—
170C23×C47#12012yes—
171C23×C47#12312yes—
172C23×C47C22×D411yescharacteristic
173C22×D47—28no—
174C22×D47—216no—
175C22×D47—28no—
176C22×D47—22no—
177C22×D47—22no—
178C22×D47—22no—
179C22×D47—22no—
180C22×D47—28no—
181C22×D47—216no—
182C22×D47C2×C4⋊C414yes—
183C22×D47AGL(1,ℤ/8)14yes—
184C22×D47#13714yes—
185C2×C4×C47—24no—
186C2×C4×C47—48no—
187C2×C4×C47—44no—
188C2×C4×C47—41no—
189C2×C4×C47—41no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47C2×C4⋊C412yes—
193C2×C4×C47C4×D414yes—
194C2×C4×C47C4×D414yes—
195C2×C4×C47#12012yes—
196C2×C4×C47C22⋊Q814yes—
197C2×C4×C47C22⋊Q814yes—
198C2×C4×C47#12312yes—
199C2×C4×C47C23×C411yescharacteristic
200C2×C4×C47C22×D411yescharacteristic
201C2×C4×C47C22×D411yescharacteristic
202C2×C4×C47C22×Q811yescharacteristic
203C2×C4×C47C2×#4111yescharacteristic
204C2×C4×C47C2×#4111yescharacteristic
205C2×Dic47—22no—
206C2×Dic47—22no—
207C2×Dic47—28no—
208C2×Dic47—24no—
209C2×Dic47—48no—
210C2×Dic47—44no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47—42no—
214C2×Dic47C2×C4⋊C414yes—
215C4×D47—24no—
216C4×D47—24no—
217C4×D47—24no—
218C4×D47—24no—
219C4×D47—216no—
220C4×D47—432no—
221C8×C47—24no—
222C47⋊C8—24no—
223#20844—471nomaximal, Sylow
224C22×C47—24no—
225C22×C47—24no—
226C22×C47—22no—
227C22×C47—22no—
228C22×C47#38514yes—
229C22×C47C22×C4⋊C411yescharacteristic
230C22×C47C2×#12011yescharacteristic
231C22×C47C2×#12311yescharacteristic
232C22×C47C2×AGL(1,ℤ/8)12yes—
233C22×C47C2×#13712yes—
234C2×D47—232no—
235C2×D47—24no—
236C2×D47—24no—
237C2×D47—24no—
238C2×D47—24no—
239C2×D47—24no—
240C2×D47—24no—
241C2×D47#42818yescomplemented
242C4×C47—44no—
243C4×C47—81no—
244C4×C47—81no—
245C4×C47#38514yes—
246C4×C47C2×C4×D411yescharacteristic
247C4×C47C2×C4×D411yescharacteristic
248C4×C47C2×C22⋊Q811yescharacteristic
249C4×C47C2×C22⋊Q811yescharacteristic
250Dic47—24no—
251Dic47—44no—
252Dic47—82no—
253C23×C4⋊C4—471no—
254C2×#428—471no—
255C2×#428D4711yescharacteristic, complemented
256C2×#407—471no—
257#2280—471no—
258#1853—474no—
259#1852—474no—
260#1736—471no—
261#1736—471no—
262C2×C47—24no—
263C2×C47C2×#38511yescharacteristic
264C2×C47C2×#42812yes—
265C2×C47#300311yescharacteristic
266C2×C47#300411yescharacteristic
267C2×C47#309212yes—
268D47—28no—
269C24×C4—471no—
270C24×C4—941no—
271C22×M16—471no—
272C22×M16C2×D4711yescharacteristic
273C2×#117—471no—
274C2×#117C2×D4711yescharacteristic
275C22×C4⋊C4—471no—
276C22×C4⋊C4—474no—
277C22×C4⋊C4—944no—
278C22×C4⋊C4—942no—
279C22×C4⋊C4—941no—
280C22×C4⋊C4—944no—
281C22×C4⋊C4C2×D4711yescharacteristic
282#428—24no—
283#428—944no—
284C2×#107—472no—
285C2×#107C2×D4712yescomplemented
286C2×#106—472no—
287C2×#106C2×D4712yescomplemented
288#407—478no—
289C2×C22⋊C8—474no—
290#381—941no—
291#380—472no—
292#380—472no—
293#379—941no—
294C4×C22⋊C4—942no—
295#340—474no—
296#340—474no—
297C47#2084411yescharacteristic, complemented, Sylow
298C25—471no—
299C23×C4—21no—
300C23×C4—471no—
301C23×C4—474no—
302C23×C4—472no—
303C23×C4—474no—
304C23×C4—472no—
305C23×C4—944no—
306C23×C4—948no—
307C23×C4—942no—
308C23×C4—942no—
309C23×C4—942no—
310C23×C4—942no—
311C23×C4—941no—
312C23×C4—1888no—
313C23×C4C22×D4711yescharacteristic
314C2×M16—22no—
315C2×M16—22no—
316C2×M16—474no—
317C2×M16—942no—
318C2×M16—942no—
319C2×M16C4×D4714yescomplemented
320C22×C8—472no—
321C22×C8C22×D4712yes—
322#117—22no—
323#117—22no—
324#117—942no—
325#117—942no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—474no—
329C2×C4⋊C4—472no—
330C2×C4⋊C4—472no—
331C2×C4⋊C4—948no—
332C2×C4⋊C4—9416no—
333C2×C4⋊C4—944no—
334C2×C4⋊C4—948no—
335C2×C4⋊C4—941no—
336C2×C4⋊C4—942no—
337C2×C4⋊C4—948no—
338C2×C4⋊C4C22⋊D4714yes—
339C2×C4⋊C4C22×D4711yescharacteristic
340C2×C4⋊C4C22×D4711yescharacteristic
341C2×C4⋊C4C22×D4712yes—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—942no—
344C2×C22⋊C4—942no—
345C2×C22⋊C4—941no—
346C2×C42—21no—
347C2×C42—941no—
348#107—24no—
349#107—944no—
350#106—24no—
351#106—944no—
352C22⋊C8—4716no—
353#95—942no—
354#95—942no—
355C24—472no—
356C24—474no—
357C24—944no—
358C24—944no—
359C24—942no—
360C24—942no—
361C24D4×D4711yescharacteristic
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—44no—
367C22×C4—474no—
368C22×C4—472no—
369C22×C4—478no—
370C22×C4—474no—
371C22×C4—474no—
372C22×C4—9416no—
373C22×C4—942no—
374C22×C4—942no—
375C22×C4—948no—
376C22×C4—948no—
377C22×C4—944no—
378C22×C4—948no—
379C22×C4—948no—
380C22×C4—944no—
381C22×C4—942no—
382C22×C4—942no—
383C22×C4—942no—
384C22×C4—942no—
385C22×C4—944no—
386C22×C4—942no—
387C22×C4—942no—
388C22×C4—1888no—
389C22×C4—18832no—
390C22×C4—1888no—
391C22×C4—1884no—
392C22×C4C2×C4×D4712yes—
393C22×C4#1062535012yes—
394C22×C4D4×D4711yescharacteristic
395C22×C4D4×D4712yes—
396C22×C4Q8×D4712yes—
397C22×C4#1062535412yes—
398C22×C4C2×C22⋊D4712yes—
399C22×C4C23×D4711yescharacteristic
400M16—28no—
401M16—948no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—474no—
405C2×C8—942no—
406C2×C8—942no—
407C2×C8C2×C4×D4714yescomplemented
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—474no—
415C4⋊C4—9416no—
416C4⋊C4—944no—
417C4⋊C4—942no—
418C4⋊C4—942no—
419C4⋊C4—942no—
420C4⋊C4C2×C22⋊D4714yes—
421C22⋊C4—42no—
422C22⋊C4—1884no—
423C22⋊C4—1884no—
424C22⋊C4—1882no—
425C42—42no—
426C42—1882no—
427C23—22no—
428C23—22no—
429C23—474no—
430C23—474no—
431C23—474no—
432C23—9416no—
433C23—948no—
434C23—9416no—
435C23—948no—
436C23—948no—
437C23—942no—
438C23—942no—
439C23—942no—
440C23—942no—
441C23C4⋊C4×D4712yes—
442C23#1309399212yes—
443C23#1309400912yes—
444C23C2×D4×D4711yescharacteristic
445C2×C4—24no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—474no—
453C2×C4—948no—
454C2×C4—942no—
455C2×C4—942no—
456C2×C4—9416no—
457C2×C4—944no—
458C2×C4—944no—
459C2×C4—944no—
460C2×C4—944no—
461C2×C4—944no—
462C2×C4—18832no—
463C2×C4—1882no—
464C2×C4—1882no—
465C2×C4—1882no—
466C2×C4—1888no—
467C2×C4—1884no—
468C2×C4C4⋊C4×D4712yes—
469C2×C4C4×C22⋊D4714yes—
470C2×C4#1309398614yes—
471C2×C4#1309399212yes—
472C2×C4#1309400014yes—
473C2×C4#1309400214yes—
474C2×C4#1309400912yes—
475C2×C4C22×C4×D4711yescharacteristic
476C2×C4C2×D4×D4711yescharacteristic
477C2×C4C22×C22⋊D4711yescharacteristic
478C2×C4C2×Q8×D4711yescharacteristic
479C2×C4C2×#1062535011yescharacteristic
480C2×C4C2×#1062535411yescharacteristic
481C8—24no—
482C8—944no—
483C22—24no—
484C22—24no—
485C22—22no—
486C22—22no—
487C22—478no—
488C22—9432no—
489C22—944no—
490C22—944no—
491C22—944no—
492C22—944no—
493C22—944no—
494C22—944no—
495C22#1568899414yes—
496C22C2×C4⋊C4×D4711yescharacteristic
497C22C2×#1309399211yescharacteristic
498C22C2×#1309400911yescharacteristic
499C22AGL(1,ℤ/8)×D4712yes—
500C22#137×D4712yes—
501C4—44no—
502C4—81no—
503C4—81no—
504C4—944no—
505C4—3762no—
506C4—1884no—
507C4#1568899414yes—
508C4C2×C4×C22⋊D4711yescharacteristic
509C4C2×#1309398611yescharacteristic
510C4C2×#1309400011yescharacteristic
511C4C2×#1309400211yescharacteristic
512C2—24no—
513C2—948no—
514C2C2×#1568899411yescharacteristic
515C2#428×D4712yes—
516C2#2025261311yescharacteristic
517C2#2025262011yescharacteristic
518C2#2025331412yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenterC2×#428D47
hypocenterC47#20844
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#15689166C2semidirect
5. #20243579C2semidirect
6. #20243574C2semidirect
7. #20243108C2semidirect
8. #20243107C2semidirect
9. C2×C47×#428C2semidirect
21. C2×#13093957C22semidirect
22. C2×#13093955C22semidirect
23. #15689086C4semidirect
30. #15688886C22semidirect
31. #15688885C22semidirect
36. C2×#107×C47C22semidirect
37. C2×#106×C47C22semidirect
54. C2×#10625329C2×C4semidirect
74. (C2×D47)⋊C8C2×C4semidirect
81. C2×M16×C47C2×C4semidirect
143. C2×C8×C47C22×C4semidirect
166. C2×C47⋊C8C22×C4semidirect
241. C2×D47#428semidirect
255. C2×#428D47semidirect
285. C2×#107C2×D47semidirect
287. C2×#106C2×D47semidirect
297. C47#20844semidirect
319. C2×M16C4×D47semidirect
407. C2×C8C2×C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D472
C2×#156891662
C2×#156890862
#202457672
#202435792
#202435742
#202431082
#202431072
C2×C47×#4282
#2084447