Finite Group Databasebeta

← #26243768

Subgroups — #26243768

Order 12032.

1902 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093995C211yesmaximal, characteristic
2C2×#15689166C211yesmaximal, characteristic, complemented
3C2×#15689128C211yesmaximal, characteristic, complemented
4C2×#15689086C211yesmaximal, characteristic, complemented
5#20245767C211yesmaximal, characteristic, complemented
6#20244575C214yesmaximal, complemented
7#20244568C214yesmaximal, complemented
8#20243108C211yesmaximal, characteristic, complemented
9C2×C47×#418C211yesmaximal, characteristic, complemented
10C23×C22⋊D47—21no—
11C23×C4×D47C2211yescharacteristic
12C22×#10625329C2211yescharacteristic
13C2×#13093995—22no—
14C2×#13093995—24no—
15C2×#13093995—24no—
16C2×#13093995C2214yes—
17C2×#13093983C2211yescharacteristic
18C22×C4⋊Dic47C2211yescharacteristic
19C22×C22⋊Dic47—21no—
20#15689166—24no—
21C2×#13093957C2212yescomplemented
22C2×#13093955C2212yescomplemented
23#15689128—24no—
24C2×#13093945C2214yescomplemented
25#15689086—24no—
26C2×(C2×D47)⋊C8C2214yescomplemented
27#15688984—21no—
28#15688980C2212yescomplemented
29#15688980C2212yescomplemented
30#15688978—21no—
31C4×(C2×D47)⋊C4—22no—
32#15688886C2214yescomplemented
33C23×D4×C47C2211yescharacteristic
34C22×M16×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C47×#418—24no—
37C2×#102×C47C2212yescomplemented
38C2×#102×C47C2212yescomplemented
39C24×D47D411yescharacteristic
40C22×C22⋊D47—28no—
41C22×C22⋊D47—24no—
42C22×C22⋊D47—48no—
43C23×Dic47—21no—
44C23×Dic47D411yescharacteristic
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47—22no—
49C22×C4×D47—22no—
50C22×C4×D47—22no—
51C22×C4×D47D412yes—
52C22×C4×D47C2312yes—
53C2×#10625329—22no—
54C2×#10625329—22no—
55C2×#10625329D414yescomplemented
56C22×C47⋊C8C2312yes—
57#13093995—216no—
58#13093995—28no—
59#13093995—28no—
60#13093983—22no—
61#13093983—22no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—22no—
64C2×C4⋊Dic47D414yes—
65C2×C4⋊Dic47C2312yes—
66C2×C4⋊Dic47C2312yes—
67C2×C22⋊Dic47—28no—
68C2×C22⋊Dic47—24no—
69C2×C22⋊Dic47—21no—
70C2×(C2×D47)⋊C4—22no—
71C2×(C2×D47)⋊C4—22no—
72C2×C4×Dic47—21no—
73#13093957—24no—
74#13093955—24no—
75#13093945—28no—
76(C2×D47)⋊C8—28no—
77C47⋊#95—22no—
78C47⋊#95—22no—
79C25×C47—21no—
80C22×D4×C47—22no—
81C22×D4×C47—21no—
82C22×D4×C47—22no—
83C22×D4×C47—22no—
84C22×D4×C47—22no—
85C22×D4×C47—22no—
86C22×D4×C47D414yes—
87C22×D4×C47C2312yes—
88C23×C4×C47C2311yescharacteristic
89C2×M16×C47—22no—
90C2×M16×C47—22no—
91C2×M16×C47—22no—
92C22×C8×C47C2312yes—
93#117×C47—22no—
94#117×C47—22no—
95C2×C4⋊C4×C47C2311yescharacteristic
96C2×C4⋊C4×C47C2311yescharacteristic
97C2×C22⋊C4×C47—21no—
98C2×C42×C47—21no—
99#102×C47—24no—
100#102×C47—24no—
101C24×C47—24no—
102C24×C47—22no—
103C24×C47—44no—
104C24×C47—44no—
105C24×C47—42no—
106C24×C47—42no—
107C24×C47C2×D411yescharacteristic
108C23×D47—22no—
109C23×D47—24no—
110C23×D47—22no—
111C23×D47—22no—
112C23×D47—42no—
113C23×D47C2×D412yes—
114C2×D4×C47—24no—
115C2×D4×C47—28no—
116C2×D4×C47—28no—
117C2×D4×C47—22no—
118C2×D4×C47—22no—
119C2×D4×C47—22no—
120C2×D4×C47—22no—
121C2×D4×C47—22no—
122C2×D4×C47—44no—
123C2×D4×C47—42no—
124C2×D4×C47—42no—
125C2×D4×C47—44no—
126C2×D4×C47C2×D414yes—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47#4112yes—
134C22×C4×C47#4112yes—
135C22×C4×C47#4112yes—
136C22×C4×C47C2411yescharacteristic
137C2×C22⋊D47—216no—
138C2×C22⋊D47—432no—
139C2×C22⋊D47—48no—
140C2×C22⋊D47—48no—
141C22×Dic47—22no—
142C22×Dic47—24no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—44no—
148C22×Dic47C2×D412yes—
149C2×C4×D47—24no—
150C2×C4×D47—24no—
151C2×C4×D47—24no—
152C2×C4×D47—22no—
153C2×C4×D47—22no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—44no—
159C2×C4×D47—44no—
160C2×C4×D47—44no—
161C2×C4×D47—44no—
162M16×C47—44no—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C2×C8×C47—22no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C22⋊C4×C47—42no—
171C42×C47—42no—
172C22⋊Dic47—216no—
173C22⋊Dic47—42no—
174(C2×D47)⋊C4—44no—
175(C2×D47)⋊C4—44no—
176C4⋊Dic47—24no—
177C4⋊Dic47—22no—
178C4⋊Dic47—22no—
179C4⋊Dic47—22no—
180C4⋊Dic47C2×D414yes—
181C4×Dic47—42no—
182#10625329—28no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8C2×D414yescomplemented
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—24no—
189C23×C47—48no—
190C23×C47—48no—
191C23×C47—42no—
192C23×C47—42no—
193C23×C47—44no—
194C23×C47—44no—
195C23×C47—48no—
196C23×C47—88no—
197C23×C47—88no—
198C23×C47#12012yes—
199C23×C47#12112yes—
200C23×C47#12312yes—
201C23×C47C22×D411yescharacteristic
202C22×D47—28no—
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—48no—
210C22×D47—48no—
211C22×D47—44no—
212C22×D47—44no—
213C22×D47AGL(1,ℤ/8)14yes—
214D4×C47—24no—
215D4×C47—24no—
216D4×C47—42no—
217D4×C47—42no—
218D4×C47—48no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47#12012yes—
224C2×C4×C47#12114yes—
225C2×C4×C47#12114yes—
226C2×C4×C47#12112yes—
227C2×C4×C47#12312yes—
228C2×C4×C47#12314yes—
229C2×C4×C47#12414yes—
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C22×D411yescharacteristic
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C22⋊D47—432no—
237C2×Dic47—24no—
238C2×Dic47—22no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—48no—
242C2×Dic47—44no—
243C2×Dic47—44no—
244C2×Dic47—42no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C4×D47—42no—
248C4×D47—42no—
249C4×D47—42no—
250C4×D47—42no—
251C4×D47—48no—
252C8×C47—42no—
253C47⋊C8—24no—
254#23480—471nomaximal, Sylow
255C22×C47—24no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—24no—
259C22×C47—44no—
260C22×C47—44no—
261C22×C47—44no—
262C22×C47—816no—
263C22×C47—84no—
264C22×C47—84no—
265C22×C47#39414yes—
266C22×C47C2×#12011yescharacteristic
267C22×C47C2×#12111yescharacteristic
268C22×C47C2×#12311yescharacteristic
269C22×C47C2×AGL(1,ℤ/8)12yes—
270C22×C47#57512yes—
271C2×D47—24no—
272C2×D47—24no—
273C2×D47—24no—
274C2×D47—416no—
275C2×D47—42no—
276C2×D47—42no—
277C2×D47—42no—
278C2×D47—42no—
279C4×C47—81no—
280C4×C47—81no—
281C4×C47#39414yes—
282C4×C47C2×#12111yescharacteristic
283C4×C47C2×#12111yescharacteristic
284C4×C47C2×#12311yescharacteristic
285C4×C47C2×#12411yescharacteristic
286Dic47—42no—
287Dic47—44no—
288Dic47—82no—
289C22×#121—471no—
290C2×#428—471no—
291C2×#418—471no—
292C2×#418D4711yescharacteristic, complemented
293C2×#407—471no—
294#2280—471no—
295#2034—474no—
296#2033—474no—
297#1736—471no—
298C2×C47—24no—
299C2×C47—84no—
300C2×C47C2×#39411yescharacteristic
301C2×C47#300311yescharacteristic
302C2×C47#300511yescharacteristic
303C2×C47#305512yes—
304C2×C47#309412yes—
305D47—44no—
306C23×D4—941no—
307C23×D4C2×D4711yescharacteristic
308C24×C4—471no—
309C22×M16—471no—
310C22×M16C2×D4711yescharacteristic
311C2×#121—474no—
312C2×#121—944no—
313C2×#121—942no—
314C2×#121—944no—
315C2×#117—471no—
316C2×#117C2×D4711yescharacteristic
317C22×C4⋊C4—471no—
318C22×C22⋊C4—941no—
319#428—944no—
320C2×#107—472no—
321C2×#106—472no—
322#418—24no—
323#418—944no—
324C2×#102—474no—
325C2×#102C2×D4712yescomplemented
326C2×#102C2×D4712yescomplemented
327#407—944no—
328C2×C22⋊C8—474no—
329#381—941no—
330#380—472no—
331#380—472no—
332#379—941no—
333C4×C22⋊C4—942no—
334#340—474no—
335C47#2348011yescharacteristic, complemented, Sylow
336C25—21no—
337C25—471no—
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—22no—
341C22×D4—21no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—944no—
345C22×D4—948no—
346C22×D4—1888no—
347C22×D4C22⋊D4714yes—
348C22×D4C22×D4712yes—
349C23×C4—471no—
350C23×C4—472no—
351C23×C4—472no—
352C23×C4—941no—
353C23×C4—942no—
354C23×C4—942no—
355C23×C4—942no—
356C23×C4—942no—
357C23×C4—942no—
358C23×C4—942no—
359C23×C4C22×D4711yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—22no—
363C2×M16—474no—
364C2×M16—942no—
365C2×M16—942no—
366C22×C8—472no—
367C22×C8C22×D4712yes—
368#121—948no—
369#121—9416no—
370#121—948no—
371#117—22no—
372#117—22no—
373#117—942no—
374#117—942no—
375C2×C4⋊C4—474no—
376C2×C4⋊C4—472no—
377C2×C4⋊C4—472no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—942no—
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C4⋊C4C22×D4711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—944no—
384C2×C22⋊C4—948no—
385C2×C22⋊C4—942no—
386C2×C22⋊C4—942no—
387C2×C22⋊C4—941no—
388C2×C42—21no—
389C2×C42—941no—
390#107—944no—
391#106—944no—
392#102—24no—
393#102—24no—
394#102—948no—
395C22⋊C8—948no—
396#95—942no—
397#95—942no—
398C24—24no—
399C24—22no—
400C24—44no—
401C24—44no—
402C24—42no—
403C24—42no—
404C24—472no—
405C24—944no—
406C24—942no—
407C24—942no—
408C24—942no—
409C24—1882no—
410C24D4×D4711yescharacteristic
411C2×D4—24no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—9416no—
424C2×D4—1888no—
425C2×D4—18832no—
426C2×D4—1888no—
427C2×D4C2×C22⋊D4714yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—472no—
431C22×C4—942no—
432C22×C4—944no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—944no—
437C22×C4—944no—
438C22×C4—944no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—1884no—
447C22×C4—1884no—
448C22×C4—1884no—
449C22×C4—1884no—
450C22×C4—1884no—
451C22×C4#1062535012yes—
452C22×C4#1062535012yes—
453C22×C4D4×D4711yescharacteristic
454C22×C4D4×D4712yes—
455C22×C4D4×D4712yes—
456C22×C4#1062535212yes—
457C22×C4C2×C22⋊D4712yes—
458C22×C4C23×D4711yescharacteristic
459M16—44no—
460M16—948no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—474no—
465C2×C8—942no—
466C2×C8—942no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—474no—
472C4⋊C4—944no—
473C4⋊C4—942no—
474C4⋊C4—942no—
475C4⋊C4—942no—
476C22⋊C4—42no—
477C22⋊C4—9416no—
478C22⋊C4—1884no—
479C22⋊C4—1884no—
480C22⋊C4—1882no—
481C42—42no—
482C42—1882no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—48no—
487C23—48no—
488C23—48no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—88no—
494C23—88no—
495C23—474no—
496C23—948no—
497C23—942no—
498C23—942no—
499C23—942no—
500C23—942no—
501C23—942no—
502C23—942no—
503C23—1888no—
504C23—1888no—
505C23—1884no—
506C23—1884no—
507C23#1309399212yes—
508C23#1309399712yes—
509C23#1309400612yes—
510C23C2×D4×D4711yescharacteristic
511D4—24no—
512D4—24no—
513D4—48no—
514D4—42no—
515D4—42no—
516D4—18832no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—942no—
522C2×C4—944no—
523C2×C4—942no—
524C2×C4—942no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1888no—
529C2×C4—1884no—
530C2×C4—1884no—
531C2×C4—1888no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—1882no—
535C2×C4—1882no—
536C2×C4#1309399212yes—
537C2×C4#1309399614yes—
538C2×C4#1309399712yes—
539C2×C4#1309399714yes—
540C2×C4#1309400612yes—
541C2×C4#1309400714yes—
542C2×C4#1309401014yes—
543C2×C4C2×D4×D4711yescharacteristic
544C2×C4C2×D4×D4711yescharacteristic
545C2×C4C22×C22⋊D4711yescharacteristic
546C2×C4C2×#1062535211yescharacteristic
547C2×C4C2×#1062535011yescharacteristic
548C2×C4C2×#1062535011yescharacteristic
549C8—42no—
550C8—944no—
551C22—24no—
552C22—24no—
553C22—22no—
554C22—22no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—816no—
559C22—84no—
560C22—84no—
561C22—944no—
562C22—944no—
563C22—944no—
564C22—18816no—
565C22—1882no—
566C22—1882no—
567C22—1882no—
568C22—1882no—
569C22#1568903814yes—
570C22C2×#1309399211yescharacteristic
571C22C2×#1309399711yescharacteristic
572C22C2×#1309400611yescharacteristic
573C22AGL(1,ℤ/8)×D4712yes—
574C22#1568991912yes—
575C4—81no—
576C4—81no—
577C4—3762no—
578C4—1882no—
579C4—1884no—
580C4#1568903814yes—
581C4C2×#1309399611yescharacteristic
582C4C2×#1309399711yescharacteristic
583C4C2×#1309400711yescharacteristic
584C4C2×#1309401011yescharacteristic
585C2—24no—
586C2—84no—
587C2—1884no—
588C2C2×#1568903811yescharacteristic
589C2#2025261311yescharacteristic
590C2#2025262711yescharacteristic
591C2#2025300512yes—
592C2#2025332612yes—
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#418C2
socleC23×C47C22×D4
prime radicalC22×#13093995C2
hypercenterC2×#418D47
hypocenterC47#23480
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×#15689166C2semidirect
3. C2×#15689128C2semidirect
4. C2×#15689086C2semidirect
5. #20245767C2semidirect
6. #20244575C2semidirect
7. #20244568C2semidirect
8. #20243108C2semidirect
9. C2×C47×#418C2semidirect
21. C2×#13093957C22semidirect
22. C2×#13093955C22semidirect
24. C2×#13093945C22semidirect
26. C2×(C2×D47)⋊C8C22semidirect
28. #15688980C22semidirect
29. #15688980C22semidirect
32. #15688886C22semidirect
37. C2×#102×C47C22semidirect
38. C2×#102×C47C22semidirect
55. C2×#10625329D4semidirect
185. C2×C47⋊C8C2×D4semidirect
292. C2×#418D47semidirect
325. C2×#102C2×D47semidirect
326. C2×#102C2×D47semidirect
335. C47#23480semidirect

Maximal subgroups

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

subgroupindex
C22×#130939952
C2×#156891662
C2×#156891282
C2×#156890862
#202457672
#202445752
#202445682
#202431082
C2×C47×#4182
#2348047