Finite Group Databasebeta

← #26217026

Subgroups — #26217026

Order 12032.

1630 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689166C212yesmaximal, complemented
2C2×#15689087C211yesmaximal, characteristic
3C2×#15688983C211yesmaximal, characteristic
4#20243587C214yesmaximal, complemented
5#20243581C212yesmaximal, complemented
6#20243576C212yesmaximal, complemented
7#20243109C212yesmaximal, complemented
8C22×#117×C47C211yesmaximal, characteristic
9C22×#10625329C2212yes—
10C22×C4⋊Dic47C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×C22⋊Dic47—21no—
13#15689166—28no—
14C2×#13093957C2214yescomplemented
15C2×#13093955C2214yescomplemented
16#15689087C418yescomplemented
17C2×C22⋊(C47⋊C8)C2212yes—
18C2×C22⋊(C47⋊C8)C2212yes—
19#15688983—21no—
20#15688983—21no—
21#15688983C2212yes—
22#15688983C2212yes—
23C2×C47⋊#95—21no—
24#15688885C2214yescomplemented
25#15688885C2214yescomplemented
26C24×C4×C47C2211yescharacteristic
27C2×#117×C47—21no—
28C2×#117×C47—21no—
29C2×#117×C47—28no—
30C2×#117×C47C2212yes—
31C2×#117×C47C2212yes—
32C22×C4⋊C4×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C22⋊C4×C47—21no—
35C22×C42×C47—21no—
36C23×Dic47—21no—
37C23×Dic47—21no—
38C2×#10625329—24no—
39C2×#10625329—24no—
40C2×#10625329C2×C418yescomplemented
41C22×C47⋊C8C2314yes—
42C2×C4⋊Dic47—24no—
43C2×C4⋊Dic47—24no—
44C2×C4⋊Dic47—21no—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47C2312yes—
47C2×C4⋊Dic47C2312yes—
48C2×C22⋊Dic47—22no—
49C2×C22⋊Dic47—22no—
50C2×C22⋊Dic47—44no—
51#13093957—28no—
52#13093955—28no—
53C22⋊(C47⋊C8)C2×C418yescomplemented
54C22⋊(C47⋊C8)C2×C418yescomplemented
55C47⋊#95—22no—
56C47⋊#95—22no—
57C25×C47D411yescharacteristic
58C23×C4×C47—24no—
59C23×C4×C47—24no—
60C23×C4×C47—21no—
61C23×C4×C47—21no—
62C23×C4×C47C2×C414yes—
63C23×C4×C47D411yescharacteristic
64C23×C4×C47D411yescharacteristic
65C23×C4×C47D411yescharacteristic
66C23×C4×C47Q814yes—
67C23×C4×C47C2311yescharacteristic
68C23×C4×C47C2311yescharacteristic
69C23×C4×C47C2311yescharacteristic
70#117×C47—28no—
71#117×C47—28no—
72#117×C47—24no—
73#117×C47—24no—
74#117×C47—24no—
75#117×C47D418yes—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—21no—
78C2×C4⋊C4×C47—24no—
79C2×C4⋊C4×C47—24no—
80C2×C4⋊C4×C47C2312yes—
81C2×C4⋊C4×C47C2312yes—
82C2×C22⋊C4×C47—28no—
83C2×C22⋊C4×C47—22no—
84C2×C22⋊C4×C47—22no—
85C2×C42×C47—28no—
86C2×C42×C47—22no—
87C2×C42×C47—22no—
88C24×C47—22no—
89C24×C47—22no—
90C24×C47—22no—
91C24×C47—22no—
92C24×C47—41no—
93C24×C47—41no—
94C24×C47C4⋊C414yes—
95C24×C47C2×D411yescharacteristic
96C24×C47C2×D411yescharacteristic
97C24×C47C2×D411yescharacteristic
98C22×C4×C47—22no—
99C22×C4×C47—22no—
100C22×C4×C47—216no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—216no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—24no—
111C22×C4×C47—24no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—42no—
115C22×C4×C47—42no—
116C22×C4×C47—42no—
117C22×C4×C47—42no—
118C22×C4×C47C4⋊C414yes—
119C22×C4×C47C4⋊C414yes—
120C22×C4×C47C4⋊C414yes—
121C22×C4×C47C22×C412yes—
122C22×C4×C47C22×C412yes—
123C22×C4×C47C22×C412yes—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D414yes—
128C22×C4×C47C2×Q812yes—
129C22×C4×C47C2×Q812yes—
130C22×C4×C47C2×Q812yes—
131C22×C4×C47#4114yes—
132C22×C4×C47C2411yescharacteristic
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—42no—
136C22×Dic47—42no—
137C22×Dic47—42no—
138C22×Dic47—42no—
139C22×Dic47—42no—
140C22×Dic47—42no—
141C4⋊C4×C47—24no—
142C4⋊C4×C47—24no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47C2×D414yes—
146C4⋊C4×C47C2×D414yes—
147C22⋊C4×C47—28no—
148C22⋊C4×C47—28no—
149C42×C47—28no—
150C42×C47—28no—
151C22⋊Dic47—44no—
152C22⋊Dic47—44no—
153C4⋊Dic47—24no—
154C4⋊Dic47—24no—
155C4⋊Dic47—42no—
156C4⋊Dic47—42no—
157#10625329—216no—
158C2×C47⋊C8—24no—
159C2×C47⋊C8—24no—
160C2×C47⋊C8C22×C418yes—
161C23×C47—21no—
162C23×C47—21no—
163C23×C47—21no—
164C23×C47—21no—
165C23×C47—21no—
166C23×C47—21no—
167C23×C47—21no—
168C23×C47—21no—
169C23×C47—21no—
170C23×C47—21no—
171C23×C47—21no—
172C23×C47—21no—
173C23×C47—21no—
174C23×C47—21no—
175C23×C47—28no—
176C23×C47—28no—
177C23×C47—44no—
178C23×C47—44no—
179C23×C47—44no—
180C23×C47—44no—
181C23×C47—44no—
182C23×C47C2×C4⋊C412yes—
183C23×C47C2×C4⋊C412yes—
184C23×C47C2×C4⋊C412yes—
185C23×C47#12012yes—
186C23×C47#12312yes—
187C23×C47AGL(1,ℤ/8)12yes—
188C23×C47#13712yes—
189C23×C47C22×D411yescharacteristic
190C2×C4×C47—21no—
191C2×C4×C47—21no—
192C2×C4×C47—21no—
193C2×C4×C47—21no—
194C2×C4×C47—28no—
195C2×C4×C47—24no—
196C2×C4×C47—24no—
197C2×C4×C47—24no—
198C2×C4×C47—24no—
199C2×C4×C47—28no—
200C2×C4×C47—28no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—48no—
204C2×C4×C47—48no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47C2×C4⋊C412yes—
210C2×C4×C47C2×C4⋊C412yes—
211C2×C4×C47C2×C4⋊C412yes—
212C2×C4×C47C4×D418yes—
213C2×C4×C47#12012yes—
214C2×C4×C47C22⋊Q818yes—
215C2×C4×C47#12312yes—
216C2×C4×C47C23×C411yescharacteristic
217C2×C4×C47C22×D412yes—
218C2×C4×C47C22×Q811yescharacteristic
219C2×C4×C47C2×#4112yes—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—84no—
227C2×Dic47—84no—
228C47⋊C8—28no—
229#20846—471nomaximal, Sylow
230C22×C47—22no—
231C22×C47—22no—
232C22×C47—22no—
233C22×C47—22no—
234C22×C47—22no—
235C22×C47—22no—
236C22×C47—22no—
237C22×C47—22no—
238C22×C47—22no—
239C22×C47—22no—
240C22×C47—22no—
241C22×C47—22no—
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—416no—
246C22×C47—41no—
247C22×C47—41no—
248C22×C47—41no—
249C22×C47—41no—
250C22×C47#38514yes—
251C22×C47#42814yescomplemented
252C22×C47C22×C4⋊C411yescharacteristic
253C22×C47C2×#12011yescharacteristic
254C22×C47C2×#12311yescharacteristic
255C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C47C2×#13711yescharacteristic
257C22×C47#57511yescharacteristic
258C22×C47#57511yescharacteristic
259C4×C47—22no—
260C4×C47—22no—
261C4×C47—44no—
262C4×C47—44no—
263C4×C47#38514yes—
264C4×C47C2×C4×D412yes—
265C4×C47C2×C22⋊Q812yes—
266Dic47—82no—
267Dic47—82no—
268C22×#117D4711yescharacteristic
269C2×#428—472no—
270C2×#407—471no—
271C2×#380—471no—
272#1854—474no—
273#1853—472no—
274#1852—472no—
275#1736—472no—
276C2×C47—24no—
277C2×C47—24no—
278C2×C47—42no—
279C2×C47C2×#38511yescharacteristic
280C2×C47C2×#42811yescharacteristic
281C2×C47#291111yescharacteristic
282C2×C47#300611yescharacteristic
283C2×C47#300711yescharacteristic
284C2×C47#309411yescharacteristic
285C2×C47#309511yescharacteristic
286C24×C4C2×D4711yescharacteristic
287C22×M16—472no—
288C2×#117—21no—
289C2×#117—21no—
290C2×#117—28no—
291C2×#117C2×D4712yes—
292C2×#117C2×D4712yes—
293C22×C4⋊C4—471no—
294C22×C4⋊C4—471no—
295C22×C4⋊C4C2×D4711yescharacteristic
296C22×C4⋊C4C2×D4711yescharacteristic
297C22×C22⋊C4—21no—
298C22×C22⋊C4—941no—
299C22×C42—21no—
300#428—948no—
301C2×#107—474no—
302C2×#106—474no—
303#407—478no—
304C2×C22⋊C8—472no—
305C2×C22⋊C8—472no—
306#380—472no—
307#380—472no—
308#380—941no—
309#380—941no—
310C2×#95—941no—
311#340—474no—
312#340—474no—
313C47#2084611yescharacteristic, complemented, Sylow
314C25C22⋊D4711yescharacteristic
315C23×C4—24no—
316C23×C4—24no—
317C23×C4—21no—
318C23×C4—21no—
319C23×C4—941no—
320C23×C4—941no—
321C23×C4Dic9414yes—
322C23×C4C4×D4714yes—
323C23×C4C22⋊D4711yescharacteristic
324C23×C4C22⋊D4711yescharacteristic
325C23×C4C22⋊D4711yescharacteristic
326C23×C4C22×D4711yescharacteristic
327C23×C4C22×D4711yescharacteristic
328C23×C4C22×D4711yescharacteristic
329C2×M16—478no—
330C2×M16—944no—
331C2×M16—944no—
332C22×C8—474no—
333#117—28no—
334#117—28no—
335#117—24no—
336#117—24no—
337#117—24no—
338#117C22⋊D4718yes—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—472no—
344C2×C4⋊C4—472no—
345C2×C4⋊C4—944no—
346C2×C4⋊C4—944no—
347C2×C4⋊C4—941no—
348C2×C4⋊C4—941no—
349C2×C4⋊C4C22×D4712yes—
350C2×C4⋊C4C22×D4712yes—
351C2×C22⋊C4—28no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—942no—
355C2×C22⋊C4—942no—
356C2×C22⋊C4—1884no—
357C2×C42—28no—
358C2×C42—22no—
359C2×C42—22no—
360#107—948no—
361#106—948no—
362C22⋊C8—478no—
363C22⋊C8—478no—
364#95—942no—
365#95—942no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—41no—
371C24—41no—
372C24Dic47⋊C414yes—
373C24C2×C22⋊D4711yescharacteristic
374C24C2×C22⋊D4711yescharacteristic
375C24C2×C22⋊D4711yescharacteristic
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—216no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—942no—
397C22×C4—942no—
398C22×C4—1882no—
399C22×C4—1882no—
400C22×C4—1882no—
401C22×C4—1882no—
402C22×C4—1882no—
403C22×C4—1882no—
404C22×C4Dic47⋊C414yes—
405C22×C4Dic47⋊C414yes—
406C22×C4Dic47⋊C414yes—
407C22×C4C2×Dic9412yes—
408C22×C4C2×Dic9412yes—
409C22×C4C2×Dic9412yes—
410C22×C4C2×C4×D4712yes—
411C22×C4C2×C4×D4712yes—
412C22×C4C2×C4×D4712yes—
413C22×C4#1062535014yes—
414C22×C4C2×C22⋊D4711yescharacteristic
415C22×C4C2×C22⋊D4711yescharacteristic
416C22×C4C2×C22⋊D4711yescharacteristic
417C22×C4C2×C22⋊D4714yes—
418C22×C4C23×D4711yescharacteristic
419M16—9416no—
420C2×C8—478no—
421C2×C8—944no—
422C2×C8—944no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—944no—
428C4⋊C4—944no—
429C4⋊C4—1882no—
430C4⋊C4—1882no—
431C4⋊C4C2×C22⋊D4714yes—
432C4⋊C4C2×C22⋊D4714yes—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—1884no—
436C22⋊C4—1884no—
437C42—28no—
438C42—28no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—28no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—28no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23C2×Dic47⋊C412yes—
461C23C2×Dic47⋊C412yes—
462C23C2×Dic47⋊C412yes—
463C23#1309399312yes—
464C23#1309400712yes—
465C23#1309404912yes—
466C23#1309405412yes—
467C23C22×C22⋊D4711yescharacteristic
468C2×C4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—3764no—
488C2×C4—3764no—
489C2×C4—1882no—
490C2×C4—1882no—
491C2×C4—1882no—
492C2×C4—1882no—
493C2×C4—1882no—
494C2×C4—1882no—
495C2×C4C2×Dic47⋊C412yes—
496C2×C4C2×Dic47⋊C412yes—
497C2×C4C2×Dic47⋊C412yes—
498C2×C4C4×C22⋊D4718yes—
499C2×C4#1309399312yes—
500C2×C4#1309400318yes—
501C2×C4#1309400712yes—
502C2×C4C22×C4×D4711yescharacteristic
503C2×C4C22×C22⋊D4712yes—
504C2×C4C22×Dic9411yescharacteristic
505C2×C4C2×#1062535012yes—
506C8—948no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—416no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22#1568899614yes—
528C22#1568916614yescomplemented
529C22C22×Dic47⋊C411yescharacteristic
530C22C2×#1309399311yescharacteristic
531C22C2×#1309400711yescharacteristic
532C22C2×#1309404911yescharacteristic
533C22C2×#1309405411yescharacteristic
534C22#1568992111yescharacteristic
535C22#1568992111yescharacteristic
536C4—22no—
537C4—22no—
538C4—44no—
539C4—44no—
540C4—3762no—
541C4—3762no—
542C4#1568899614yes—
543C4C2×C4×C22⋊D4712yes—
544C4C2×#1309400312yes—
545C2—24no—
546C2—24no—
547C2—42no—
548C2C2×#1568899611yescharacteristic
549C2C2×#1568916611yescharacteristic
550C2#2025197711yescharacteristic
551C2#2025264011yescharacteristic
552C2#2025264811yescharacteristic
553C2#2025332811yescharacteristic
554C2#2025333611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#117×C47C2
socleC23×C47C22×D4
prime radicalC25×C47D4
cosocleC22×C4×C47C24
hypercenterC22×#117D47
hypocenterC47#20846
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C2×#15689166C2semidirect
4. #20243587C2semidirect
5. #20243581C2semidirect
6. #20243576C2semidirect
7. #20243109C2semidirect
14. C2×#13093957C22semidirect
15. C2×#13093955C22semidirect
16. #15689087C4semidirect
24. #15688885C22semidirect
25. #15688885C22semidirect
40. C2×#10625329C2×C4semidirect
53. C22⋊(C47⋊C8)C2×C4semidirect
54. C22⋊(C47⋊C8)C2×C4semidirect
251. C22×C47#428semidirect
313. C47#20846semidirect
528. C22#15689166semidirect

Maximal subgroups

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

subgroupindex
C2×#156891662
C2×#156890872
C2×#156889832
#202435872
#202435812
#202435762
#202431092
C22×#117×C472
#2084647