Finite Group Databasebeta

← #26217969

Subgroups — #26217969

Order 12032.

1914 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625324C211yesmaximal, characteristic, complemented
2C2×#15689630C211yesmaximal, characteristic
3C2×#15689119C211yesmaximal, characteristic, complemented
4#20245767C211yesmaximal, characteristic, complemented
5#20243823C214yesmaximal, complemented
6#20243818C214yesmaximal, complemented
7#20243105C211yesmaximal, characteristic, complemented
8#20243104C211yesmaximal, characteristic, complemented
9C2×C47×#416C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C22×#10625324—24no—
12C22×#10625324—24no—
13C22×#10625324—24no—
14C22×#10625324C2214yes—
15C23×C47⋊C8C2211yescharacteristic
16#15689630—24no—
17#15689630C418yescomplemented
18C2×#13094005—21no—
19C2×#13094000C2212yes—
20C2×#13093995C2212yes—
21C2×#13093992—21no—
22C2×C4×C22⋊D47—22no—
23C2×#13093983C2211yescharacteristic
24C2×#13093983C2211yescharacteristic
25#15689119—24no—
26C2×#13093948C2212yescomplemented
27C2×#13093945C2212yescomplemented
28#15688984—21no—
29#15688980C2214yes—
30#15688978—21no—
31C4×(C2×D47)⋊C4—22no—
32#15688886C2214yescomplemented
33#15688885C2214yescomplemented
34C22×#41×C47C2211yescharacteristic
35C23×C8×C47C2211yescharacteristic
36C2×#117×C47C2211yescharacteristic
37C47×#416—24no—
38C2×#103×C47C2212yescomplemented
39C2×#102×C47C2212yescomplemented
40C24×D47D411yescharacteristic
41C22×C22⋊D47—22no—
42C23×Dic47D411yescharacteristic
43C22×C4×D47—22no—
44C22×C4×D47—22no—
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47C2×C414yes—
48C22×C4×D47D412yes—
49C22×C4×D47D414yes—
50C22×C4×D47C2312yes—
51C2×#10625324—232no—
52C2×#10625324—28no—
53C2×#10625324—28no—
54C2×#10625324—28no—
55C22×C47⋊C8—22no—
56C22×C47⋊C8—22no—
57C22×C47⋊C8D414yescomplemented
58C22×C47⋊C8C2312yes—
59#13094005—28no—
60#13094000—24no—
61#13094000C2×C418yes—
62#13093995—24no—
63#13093995C2×C418yes—
64#13093992—28no—
65C4×C22⋊D47—216no—
66#13093983—22no—
67#13093983—22no—
68#13093983—22no—
69#13093983C2×C414yes—
70C2×C4⋊Dic47C2312yes—
71C2×C4⋊Dic47C2312yes—
72C2×Dic47⋊C4—22no—
73C2×C22⋊Dic47—21no—
74C2×C22⋊Dic47—22no—
75C2×C22⋊Dic47—21no—
76C2×(C2×D47)⋊C4—22no—
77C2×(C2×D47)⋊C4—24no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80#13093948—24no—
81#13093945—24no—
82C47⋊#95—24no—
83C2×#41×C47—22no—
84C2×#41×C47—22no—
85C2×#41×C47—22no—
86C2×#41×C47—22no—
87C2×#41×C47—22no—
88C2×#41×C47C2×C414yes—
89C22×Q8×C47C2311yescharacteristic
90C22×D4×C47—21no—
91C22×D4×C47C2311yescharacteristic
92C23×C4×C47—21no—
93C23×C4×C47C2311yescharacteristic
94C22×C8×C47—22no—
95C22×C8×C47—22no—
96C22×C8×C47D414yescomplemented
97C22×C8×C47C2312yes—
98#117×C47—22no—
99#117×C47—22no—
100C2×C4⋊C4×C47C2312yes—
101C2×C22⋊C4×C47—21no—
102C2×C42×C47—21no—
103#103×C47—24no—
104#102×C47—24no—
105C24×C47—21no—
106C24×C47C2×D411yescharacteristic
107C23×D47—22no—
108C23×D47—22no—
109C23×D47—44no—
110C23×D47C22⋊C414yes—
111C23×D47C2×D412yes—
112#41×C47—28no—
113#41×C47—28no—
114#41×C47—44no—
115#41×C47—44no—
116C2×Q8×C47—21no—
117C2×Q8×C47—21no—
118C2×Q8×C47—21no—
119C2×Q8×C47—21no—
120C2×Q8×C47—21no—
121C2×Q8×C47C22×C412yes—
122C2×D4×C47—24no—
123C2×D4×C47—21no—
124C2×D4×C47—21no—
125C2×D4×C47—21no—
126C2×D4×C47—21no—
127C2×D4×C47—21no—
128C2×D4×C47—42no—
129C2×D4×C47—42no—
130C2×D4×C47C22×C412yes—
131C22×C4×C47—21no—
132C22×C4×C47—24no—
133C22×C4×C47—22no—
134C22×C4×C47—42no—
135C22×C4×C47—42no—
136C22×C4×C47C22×C412yes—
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×D412yes—
141C22×C4×C47#4112yes—
142C22×C4×C47#4112yes—
143C22×C4×C47C2411yescharacteristic
144C2×C22⋊D47—28no—
145C2×C22⋊D47—48no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47C22⋊C414yes—
151C22×Dic47C2×D412yes—
152C2×C4×D47—28no—
153C2×C4×D47—28no—
154C2×C4×D47—28no—
155C2×C4×D47—24no—
156C2×C4×D47—24no—
157C2×C4×D47—28no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×C4×D47C22⋊C418yes—
163C2×C4×D47C22×C414yes—
164C2×C4×D47C2×D414yes—
165C2×C8×C47—28no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47C2×D414yes—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C22⋊C4×C47—42no—
172C42×C47—42no—
173C22⋊Dic47—24no—
174C22⋊Dic47—28no—
175C22⋊Dic47—42no—
176(C2×D47)⋊C4—28no—
177(C2×D47)⋊C4—48no—
178C4⋊Dic47—22no—
179C4⋊Dic47—22no—
180C4⋊Dic47—22no—
181C4⋊Dic47C22×C414yes—
182Dic47⋊C4—28no—
183C4×Dic47—24no—
184C4×Dic47—42no—
185C2×C47⋊C8—28no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8C2×D414yes—
189#10625324—232no—
190C23×C47—24no—
191C23×C47—22no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—44no—
195C23×C47C2×C22⋊C412yes—
196C23×C47#12112yes—
197C23×C47#12112yes—
198C23×C47C22×D411yescharacteristic
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—24no—
202C22×D47—24no—
203C22×D47—28no—
204C22×D47—48no—
205C22×D47—48no—
206C22×D47—48no—
207C22×D47C2×C22⋊C414yes—
208Q8×C47—22no—
209Q8×C47—22no—
210Q8×C47—41no—
211Q8×C47—41no—
212D4×C47—22no—
213D4×C47—22no—
214D4×C47—48no—
215D4×C47—41no—
216D4×C47—41no—
217C2×C4×C47—22no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—48no—
221C2×C4×C47—42no—
222C2×C4×C47—44no—
223C2×C4×C47C2×C22⋊C412yes—
224C2×C4×C47C4×D414yes—
225C2×C4×C47C4×D414yes—
226C2×C4×C47#12014yes—
227C2×C4×C47#12112yes—
228C2×C4×C47#12112yes—
229C2×C4×C47#12314yes—
230C2×C4×C47C23×C411yescharacteristic
231C2×C4×C47C22×D411yescharacteristic
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C22⋊D47—416no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—28no—
240C2×Dic47—24no—
241C2×Dic47—44no—
242C2×Dic47—42no—
243C2×Dic47—44no—
244C2×Dic47C2×C22⋊C414yes—
245C4×D47—24no—
246C4×D47—24no—
247C4×D47—24no—
248C4×D47—24no—
249C4×D47—216no—
250C8×C47—24no—
251C47⋊C8—24no—
252#20970—471nomaximal, Sylow
253C22×C47—24no—
254C22×C47—24no—
255C22×C47—24no—
256C22×C47—44no—
257C22×C47—42no—
258C22×C47—82no—
259C22×C47—82no—
260C22×C47#38614yes—
261C22×C47C22×C22⋊C411yescharacteristic
262C22×C47C2×#12111yescharacteristic
263C22×C47C2×#12111yescharacteristic
264C22×C47#57514yes—
265C2×D47—24no—
266C2×D47—24no—
267C2×D47—24no—
268C2×D47—416no—
269C2×D47—44no—
270C2×D47—44no—
271C4×C47—42no—
272C4×C47—82no—
273C4×C47#38614yes—
274C4×C47C2×C4×D411yescharacteristic
275C4×C47C2×C4×D411yescharacteristic
276C4×C47C2×#12011yescharacteristic
277C4×C47C2×#12311yescharacteristic
278Dic47—24no—
279Dic47—44no—
280Dic47—82no—
281C23×M16—471no—
282C2×#525—471no—
283C2×#416—471no—
284C2×#416D4711yescharacteristic, complemented
285#2280—471no—
286#1895—474no—
287#1894—474no—
288#1735—471no—
289#1735—471no—
290C2×C47—24no—
291C2×C47—82no—
292C2×C47C2×#38611yescharacteristic
293C2×C47#289612yes—
294C2×C47#305412yes—
295C2×C47#305412yes—
296D47—44no—
297C22×#41C2×D4711yescharacteristic
298C24×C4—471no—
299C22×M16—474no—
300C22×M16—944no—
301C22×M16—944no—
302C22×M16—944no—
303C23×C8—471no—
304C23×C8C2×D4711yescharacteristic
305#525—478no—
306#525—944no—
307C2×#123—941no—
308C2×C22⋊Q8—472no—
309C2×#121—472no—
310C2×#120—941no—
311C2×C4×D4—942no—
312C2×#117—471no—
313C2×#117—471no—
314C2×#117C2×D4711yescharacteristic
315#416—24no—
316#416—944no—
317C2×#103—472no—
318C2×#103C2×D4712yescomplemented
319C2×#102—472no—
320C2×#102C2×D4712yescomplemented
321#381—941no—
322#380—474no—
323#379—941no—
324C4×C22⋊C4—942no—
325#340—474no—
326#340—474no—
327C47#2097011yescharacteristic, complemented, Sylow
328C25—471no—
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41C4×D4714yes—
335C22×Q8C22×D4711yescharacteristic
336C22×D4—21no—
337C22×D4—942no—
338C22×D4C22×D4711yescharacteristic
339C23×C4—21no—
340C23×C4—471no—
341C23×C4—474no—
342C23×C4—474no—
343C23×C4—472no—
344C23×C4—472no—
345C23×C4—942no—
346C23×C4—942no—
347C23×C4—942no—
348C23×C4—942no—
349C23×C4C22×D4711yescharacteristic
350C2×M16—948no—
351C2×M16—948no—
352C2×M16—9432no—
353C2×M16—948no—
354C22×C8—22no—
355C22×C8—22no—
356C22×C8—474no—
357C22×C8—472no—
358C22×C8—942no—
359C22×C8—942no—
360C22×C8C22⋊D4714yescomplemented
361C22×C8C22×D4712yes—
362#123—948no—
363C22⋊Q8—478no—
364C22⋊Q8—944no—
365#121—478no—
366#121—944no—
367#120—948no—
368C4×D4—9416no—
369#117—22no—
370#117—22no—
371#117—474no—
372#117—942no—
373#117—942no—
374#117—942no—
375C2×C4⋊C4—472no—
376C2×C4⋊C4—472no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4C22×D4712yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—942no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—941no—
383C2×C22⋊C4—944no—
384C2×C22⋊C4—941no—
385C2×C42—21no—
386C2×C42—941no—
387C2×C42—941no—
388#103—24no—
389#103—944no—
390#102—24no—
391#102—944no—
392#95—944no—
393C24—21no—
394C24—472no—
395C24—474no—
396C24—942no—
397C24—942no—
398C24—1884no—
399C24D4×D4711yescharacteristic
400#41—28no—
401#41—28no—
402#41—44no—
403#41—44no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8C2×C4×D4712yes—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—948no—
419C2×D4—1888no—
420C2×D4C2×C4×D4712yes—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—474no—
427C22×C4—472no—
428C22×C4—478no—
429C22×C4—474no—
430C22×C4—474no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—948no—
434C22×C4—948no—
435C22×C4—948no—
436C22×C4—944no—
437C22×C4—944no—
438C22×C4—948no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4C2×C4×D4712yes—
446C22×C4#1062535012yes—
447C22×C4D4×D4712yes—
448C22×C4D4×D4712yes—
449C22×C4D4×D4711yescharacteristic
450C22×C4#1062535212yes—
451C22×C4C2×C22⋊D4712yes—
452C22×C4C23×D4711yescharacteristic
453M16—9432no—
454C2×C8—28no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—474no—
458C2×C8—948no—
459C2×C8—942no—
460C2×C8—942no—
461C2×C8C2×C22⋊D4714yes—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—474no—
465C4⋊C4—948no—
466C4⋊C4—942no—
467C4⋊C4—942no—
468C4⋊C4—942no—
469C22⋊C4—42no—
470C22⋊C4—948no—
471C22⋊C4—948no—
472C22⋊C4—944no—
473C22⋊C4—1888no—
474C22⋊C4—1882no—
475C42—42no—
476C42—944no—
477C42—1882no—
478C23—24no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—44no—
483C23—474no—
484C23—948no—
485C23—942no—
486C23—942no—
487C23—944no—
488C23—944no—
489C23—1888no—
490C23—1888no—
491C23—1888no—
492C23C22⋊C4×D4712yes—
493C23#1309399512yes—
494C23#1309399712yes—
495C23C2×D4×D4711yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—41no—
499Q8—41no—
500D4—22no—
501D4—22no—
502D4—48no—
503D4—41no—
504D4—41no—
505D4—18816no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—474no—
513C2×C4—948no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—9416no—
517C2×C4—944no—
518C2×C4—944no—
519C2×C4—944no—
520C2×C4—944no—
521C2×C4—944no—
522C2×C4—1882no—
523C2×C4—1884no—
524C2×C4—1884no—
525C2×C4C22⋊C4×D4712yes—
526C2×C4(C22⋊D47)⋊C414yes—
527C2×C4C4×C22⋊D4714yes—
528C2×C4#1309399214yes—
529C2×C4#1309399512yes—
530C2×C4#1309399712yes—
531C2×C4#1309400614yes—
532C2×C4C22×C4×D4711yescharacteristic
533C2×C4C2×D4×D4711yescharacteristic
534C2×C4C2×D4×D4711yescharacteristic
535C2×C4C22×C22⋊D4711yescharacteristic
536C2×C4C2×#1062535211yescharacteristic
537C2×C4C2×#1062535011yescharacteristic
538C8—24no—
539C8—944no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—42no—
544C22—44no—
545C22—82no—
546C22—82no—
547C22—944no—
548C22—944no—
549C22—944no—
550C22—18816no—
551C22—1884no—
552C22—1884no—
553C22#1568899914yes—
554C22C2×C22⋊C4×D4711yescharacteristic
555C22C2×#1309399511yescharacteristic
556C22C2×#1309399711yescharacteristic
557C22#1568992014yes—
558C4—42no—
559C4—82no—
560C4—944no—
561C4—3762no—
562C4—1884no—
563C4#1568899914yes—
564C4C2×(C22⋊D47)⋊C411yescharacteristic
565C4C2×C4×C22⋊D4711yescharacteristic
566C4C2×#1309399211yescharacteristic
567C4C2×#1309400611yescharacteristic
568C2—24no—
569C2—82no—
570C2—1884no—
571C2C2×#1568899911yescharacteristic
572C2#2025190312yes—
573C2#2025299312yes—
574C2#2025299612yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#416C2
socleC23×C47C22×D4
prime radicalC2×#15689630C2
cosocleC22×C4×C47C24
hypercenterC2×#416D47
hypocenterC47#20970
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
1. C23×#10625324C2semidirect
3. C2×#15689119C2semidirect
4. #20245767C2semidirect
5. #20243823C2semidirect
6. #20243818C2semidirect
7. #20243105C2semidirect
8. #20243104C2semidirect
9. C2×C47×#416C2semidirect
17. #15689630C4semidirect
26. C2×#13093948C22semidirect
27. C2×#13093945C22semidirect
32. #15688886C22semidirect
33. #15688885C22semidirect
38. C2×#103×C47C22semidirect
39. C2×#102×C47C22semidirect
57. C22×C47⋊C8D4semidirect
96. C22×C8×C47D4semidirect
284. C2×#416D47semidirect
318. C2×#103C2×D47semidirect
320. C2×#102C2×D47semidirect
327. C47#20970semidirect
360. C22×C8C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C23×#106253242
C2×#156896302
C2×#156891192
#202457672
#202438232
#202438182
#202431052
#202431042
C2×C47×#4162
#2097047