Finite Group Databasebeta

← #26215853

Subgroups — #26215853

Order 12032.

1518 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D47C211yesmaximal, characteristic
2C2×#15689160C211yesmaximal, characteristic, complemented
3C2×#15689158C211yesmaximal, characteristic, complemented
4#20245770C212yesmaximal
5#20243422C211yesmaximal, characteristic, complemented
6#20243421C211yesmaximal, characteristic, complemented
7#20243420C212yesmaximal, complemented
8#20243419C212yesmaximal, complemented
9#20243418C211yesmaximal, characteristic, complemented
10#20243103C212yesmaximal, complemented
11C47×#1809C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C22×#10625329C2211yescharacteristic
14C22×#10625324C2211yescharacteristic
15C22×#10625324C2211yescharacteristic
16C2×M16×D47—24no—
17C2×M16×D47—24no—
18C2×M16×D47C418yescomplemented
19C22×C8×D47C2211yescharacteristic
20C22×C8×D47C2211yescharacteristic
21C2×#13093982C2212yes—
22C2×#13093982C2212yes—
23#15689160—24no—
24#15689158—24no—
25C2×#13093958C2211yescharacteristic, complemented
26C2×#13093957C2212yescomplemented
27C2×#13093956C2211yescharacteristic, complemented
28C2×#13093955C2212yescomplemented
29#15688985—22no—
30#15688983C2212yes—
31#15688980C2212yes—
32#15688980C2212yes—
33#15688978—22no—
34C22⋊C4×Dic47—22no—
35#15688970—22no—
36#15688886C2212yescomplemented
37#15688885C2212yescomplemented
38#15688885C2212yescomplemented
39C22×M16×C47C2211yescharacteristic
40C2×#107×C47C2211yescharacteristic, complemented
41C2×#106×C47C2211yescharacteristic, complemented
42C47×#380C2212yes—
43C47×#340C2212yescomplemented
44C24×D47D411yescharacteristic
45C23×Dic47D411yescharacteristic
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47—22no—
49C22×C4×D47—22no—
50C22×C4×D47C2×C414yes—
51C22×C4×D47D411yescharacteristic
52C22×C4×D47D411yescharacteristic
53C22×C4×D47Q814yes—
54C22×C4×D47C2311yescharacteristic
55C22×C4×D47C2311yescharacteristic
56C2×#10625329—22no—
57C2×#10625329—22no—
58C2×#10625329C2×C414yes—
59C2×#10625324—22no—
60C2×#10625324—22no—
61C2×#10625324—22no—
62C2×#10625324—22no—
63C2×#10625324C2×C414yes—
64C2×#10625324C2×C414yes—
65M16×D47—232no—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C2×C8×D47—22no—
69C2×C8×D47—22no—
70C2×C8×D47C2×C414yes—
71C2×C8×D47C2×C414yes—
72C2×C8×D47D414yes—
73C2×C8×D47D414yes—
74#13093982—24no—
75#13093982—24no—
76#13093982—24no—
77#13093982—24no—
78C2×C4⋊Dic47C2312yes—
79C2×C4⋊Dic47C2312yes—
80C2×C22⋊Dic47—22no—
81C2×(C2×D47)⋊C4—22no—
82C2×(C2×D47)⋊C4—22no—
83C2×(C2×D47)⋊C4—22no—
84C2×(C2×D47)⋊C4—22no—
85C2×C4×Dic47—22no—
86C2×C4×Dic47—22no—
87#13093958—22no—
88#13093957—24no—
89#13093956—22no—
90#13093955—24no—
91C47⋊#95—22no—
92C47⋊#95—22no—
93C47⋊#95—22no—
94C23×C4×C47C2311yescharacteristic
95C2×M16×C47—22no—
96C2×M16×C47—22no—
97C2×M16×C47C2×C414yes—
98C22×C8×C47C2311yescharacteristic
99C22×C8×C47C2311yescharacteristic
100C2×C4⋊C4×C47C2312yes—
101C2×C4⋊C4×C47C2312yes—
102C2×C22⋊C4×C47—22no—
103#107×C47—22no—
104#106×C47—22no—
105#95×C47—22no—
106C24×C47C2×D411yescharacteristic
107C23×D47—22no—
108C23×D47—22no—
109C23×D47—42no—
110C23×D47—42no—
111C23×D47C4⋊C414yes—
112C23×D47C2×D411yescharacteristic
113C23×D47C2×D411yescharacteristic
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×Q812yes—
123C22×C4×C47C2411yescharacteristic
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47C4⋊C414yes—
129C22×Dic47C2×D411yescharacteristic
130C22×Dic47C2×D411yescharacteristic
131C2×C4×D47—22no—
132C2×C4×D47—22no—
133C2×C4×D47—21no—
134C2×C4×D47—21no—
135C2×C4×D47—21no—
136C2×C4×D47—21no—
137C2×C4×D47—22no—
138C2×C4×D47—22no—
139C2×C4×D47—28no—
140C2×C4×D47—28no—
141C2×C4×D47—28no—
142C2×C4×D47—28no—
143C2×C4×D47C4⋊C414yes—
144C2×C4×D47C4⋊C414yes—
145C2×C4×D47C22×C412yes—
146C2×C4×D47C22×C412yes—
147C2×C4×D47C2×D412yes—
148C2×C4×D47C2×D412yes—
149C2×C4×D47C2×Q812yes—
150C2×C4×D47C2×Q812yes—
151C2×C4×D47#4112yes—
152C2×C4×D47#4112yes—
153M16×C47—28no—
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47C22×C412yes—
157C2×C8×C47C22×C412yes—
158C2×C8×C47C2×D412yes—
159C2×C8×C47C2×D412yes—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C22⋊C4×C47—44no—
165C22⋊Dic47—44no—
166(C2×D47)⋊C4—44no—
167(C2×D47)⋊C4—44no—
168(C2×D47)⋊C4—44no—
169(C2×D47)⋊C4—44no—
170C4⋊Dic47—22no—
171C4⋊Dic47—22no—
172C4⋊Dic47—22no—
173C4⋊Dic47—22no—
174C4×Dic47—44no—
175C4×Dic47—44no—
176#10625329—28no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8C22×C412yes—
180C2×C47⋊C8C22×C412yes—
181C2×C47⋊C8C2×D412yes—
182C2×C47⋊C8C2×D412yes—
183#10625324—28no—
184#10625324—28no—
185C8×D47—28no—
186C8×D47C22⋊C4116yescomplemented
187C23×C47—21no—
188C23×C47—21no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47C2×C4⋊C412yes—
192C23×C47C22×D411yescharacteristic
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—21no—
196C22×D47—21no—
197C22×D47—22no—
198C22×D47—22no—
199C22×D47—28no—
200C22×D47—48no—
201C22×D47—48no—
202C22×D47—44no—
203C22×D47—44no—
204C22×D47C2×C4⋊C412yes—
205C22×D47C2×C4⋊C412yes—
206C22×D47#12112yes—
207C22×D47#12112yes—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47C2×C4⋊C412yes—
221C2×C4×C47C23×C411yescharacteristic
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C22×D411yescharacteristic
224C2×C4×C47C22×Q811yescharacteristic
225C2×C4×C47C2×#4111yescharacteristic
226C2×C4×C47C2×#4111yescharacteristic
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—28no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47C2×C4⋊C412yes—
237C2×Dic47C2×C4⋊C412yes—
238C2×Dic47#12112yes—
239C2×Dic47#12112yes—
240C4×D47—28no—
241C4×D47—22no—
242C4×D47—22no—
243C4×D47—22no—
244C4×D47—22no—
245C4×D47—28no—
246C4×D47C2×C22⋊C414yes—
247C4×D47#11714yes—
248C4×D47C22⋊Q814yes—
249C4×D47C22⋊Q814yes—
250C8×C47—22no—
251C8×C47C2×C22⋊C414yes—
252C47⋊C8—22no—
253C47⋊C8C2×C22⋊C414yes—
254#20660—471nomaximal, Sylow
255C22×C47—24no—
256C22×C47—22no—
257C22×C47—42no—
258C22×C47—41no—
259C22×C47—41no—
260C22×C47C22×C4⋊C411yescharacteristic
261C22×C47C2×#12111yescharacteristic
262C22×C47C2×#12111yescharacteristic
263C22×C47#57512yes—
264C22×C47#57512yes—
265C2×D47—24no—
266C2×D47—24no—
267C2×D47—22no—
268C2×D47—22no—
269C2×D47—24no—
270C2×D47—22no—
271C2×D47—416no—
272C2×D47—42no—
273C2×D47—41no—
274C2×D47—41no—
275C2×D47#38014yes—
276C4×C47—22no—
277C4×C47—82no—
278C4×C47—82no—
279C4×C47C22×C22⋊C411yescharacteristic
280C4×C47C2×#11711yescharacteristic
281C4×C47C2×C22⋊Q811yescharacteristic
282C4×C47C2×C22⋊Q811yescharacteristic
283Dic47—22no—
284Dic47—82no—
285Dic47—82no—
286Dic47#38014yes—
287C23×M16—471no—
288C2×#427—471no—
289C2×#427—471no—
290#2280—472no—
291#1809—471no—
292#1809—471no—
293#1809—472no—
294#1809—472no—
295#1809—471no—
296#1809D4711yescharacteristic, complemented
297#1735—472no—
298C2×C47—42no—
299C2×C47C2×#38011yescharacteristic
300C2×C47#291112yes—
301C2×C47#305412yes—
302C2×C47#305411yescharacteristic
303C2×C47#305411yescharacteristic
304D47—24no—
305D47—42no—
306C24×C4—471no—
307C22×M16—471no—
308C22×M16—478no—
309C22×M16—471no—
310C22×M16—471no—
311C22×M16—944no—
312C22×M16—944no—
313C22×M16C2×D4711yescharacteristic
314C23×C8—471no—
315C23×C8—471no—
316C2×#117—472no—
317C2×#117—472no—
318#427—944no—
319#427—944no—
320C2×#107—471no—
321C2×#107—472no—
322C2×#107C2×D4711yescharacteristic, complemented
323C2×#106—471no—
324C2×#106—472no—
325C2×#106C2×D4711yescharacteristic, complemented
326#381—942no—
327#380—472no—
328#380—472no—
329#380—472no—
330#380C2×D4712yes—
331#379—942no—
332C4×C22⋊C4—942no—
333C4×C22⋊C4—942no—
334#340—472no—
335#340—472no—
336#340—472no—
337#340C2×D4712yescomplemented
338C47#2066011yescharacteristic, complemented, Sylow
339C25—471no—
340C23×C4—471no—
341C23×C4—471no—
342C23×C4—474no—
343C23×C4—471no—
344C23×C4—471no—
345C23×C4—474no—
346C23×C4—471no—
347C23×C4—942no—
348C23×C4—942no—
349C23×C4—942no—
350C23×C4—942no—
351C23×C4C22×D4711yescharacteristic
352C2×M16—22no—
353C2×M16—22no—
354C2×M16—474no—
355C2×M16—474no—
356C2×M16—474no—
357C2×M16—942no—
358C2×M16—942no—
359C2×M16—9432no—
360C2×M16—942no—
361C2×M16—942no—
362C2×M16—942no—
363C2×M16—942no—
364C2×M16C4×D4714yes—
365C22×C8—471no—
366C22×C8—474no—
367C22×C8—474no—
368C22×C8—474no—
369C22×C8—471no—
370C22×C8—474no—
371C22×C8—942no—
372C22×C8—942no—
373C22×C8C22×D4711yescharacteristic
374C22×C8C22×D4711yescharacteristic
375#117—944no—
376#117—944no—
377#117—944no—
378#117—944no—
379C2×C4⋊C4—472no—
380C2×C4⋊C4—472no—
381C2×C4⋊C4C22×D4712yes—
382C2×C4⋊C4C22×D4712yes—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—942no—
385C2×C22⋊C4—942no—
386C2×C22⋊C4—942no—
387C2×C22⋊C4—942no—
388C2×C22⋊C4—942no—
389C2×C42—942no—
390C2×C42—942no—
391#107—22no—
392#107—944no—
393#107—942no—
394#106—22no—
395#106—944no—
396#106—942no—
397#95—22no—
398#95—942no—
399#95—942no—
400#95—942no—
401C24—471no—
402C24—471no—
403C24—474no—
404C24—942no—
405C24—942no—
406C24—1882no—
407C24—1882no—
408C24D4×D4711yescharacteristic
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—471no—
416C22×C4—472no—
417C22×C4—472no—
418C22×C4—472no—
419C22×C4—472no—
420C22×C4—472no—
421C22×C4—472no—
422C22×C4—474no—
423C22×C4—471no—
424C22×C4—474no—
425C22×C4—474no—
426C22×C4—472no—
427C22×C4—472no—
428C22×C4—948no—
429C22×C4—942no—
430C22×C4—942no—
431C22×C4—948no—
432C22×C4—948no—
433C22×C4—948no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—941no—
439C22×C4—941no—
440C22×C4—941no—
441C22×C4—941no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4C2×C4×D4712yes—
445C22×C4D4×D4711yescharacteristic
446C22×C4Q8×D4712yes—
447C22×C4C23×D4711yescharacteristic
448M16—28no—
449M16—948no—
450M16—948no—
451M16—948no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—472no—
455C2×C8—472no—
456C2×C8—472no—
457C2×C8—4716no—
458C2×C8—472no—
459C2×C8—948no—
460C2×C8—941no—
461C2×C8—941no—
462C2×C8C2×C4×D4712yes—
463C2×C8C2×C4×D4712yes—
464C2×C8D4×D4712yes—
465C2×C8D4×D4712yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—942no—
471C4⋊C4—942no—
472C4⋊C4—942no—
473C4⋊C4—942no—
474C22⋊C4—44no—
475C22⋊C4—1884no—
476C22⋊C4—1884no—
477C22⋊C4—1884no—
478C22⋊C4—1884no—
479C22⋊C4—1884no—
480C42—1884no—
481C42—1884no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23—472no—
487C23—472no—
488C23—472no—
489C23—472no—
490C23—948no—
491C23—942no—
492C23—942no—
493C23—941no—
494C23—941no—
495C23—942no—
496C23—942no—
497C23—1888no—
498C23—1888no—
499C23—1884no—
500C23—1884no—
501C23C4⋊C4×D4712yes—
502C23C2×D4×D4711yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—472no—
516C2×C4—472no—
517C2×C4—472no—
518C2×C4—474no—
519C2×C4—474no—
520C2×C4—474no—
521C2×C4—474no—
522C2×C4—472no—
523C2×C4—948no—
524C2×C4—948no—
525C2×C4—948no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—942no—
529C2×C4—942no—
530C2×C4—942no—
531C2×C4—942no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—1882no—
535C2×C4—1882no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4C4⋊C4×D4712yes—
539C2×C4C22×C4×D4711yescharacteristic
540C2×C4C2×D4×D4711yescharacteristic
541C2×C4C2×D4×D4711yescharacteristic
542C2×C4C2×Q8×D4711yescharacteristic
543C2×C4#41×D4711yescharacteristic
544C2×C4#41×D4711yescharacteristic
545C8—22no—
546C8—474no—
547C8—942no—
548C8C22⋊C4×D4714yes—
549C22—24no—
550C22—22no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—474no—
555C22—944no—
556C22—942no—
557C22—944no—
558C22—942no—
559C22—944no—
560C22—942no—
561C22—18816no—
562C22—1882no—
563C22—1881no—
564C22—1881no—
565C22C2×C4⋊C4×D4711yescharacteristic
566C22#121×D4711yescharacteristic
567C22#121×D4711yescharacteristic
568C22#1568991912yes—
569C22#1568992212yes—
570C4—22no—
571C4—82no—
572C4—82no—
573C4—474no—
574C4—942no—
575C4—3762no—
576C4—3762no—
577C4C2×C22⋊C4×D4711yescharacteristic
578C4#117×D4711yescharacteristic
579C4C22⋊Q8×D4711yescharacteristic
580C4C22⋊Q8×D4711yescharacteristic
581C2—42no—
582C2—944no—
583C2—1882no—
584C2#380×D4711yescharacteristic
585C2#2025197512yes—
586C2#2025299212yes—
587C2#2025299511yescharacteristic
588C2#2025299911yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1809C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1809D47
hypocenterC47#20660
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
2. C2×#15689160C2semidirect
3. C2×#15689158C2semidirect
5. #20243422C2semidirect
6. #20243421C2semidirect
7. #20243420C2semidirect
8. #20243419C2semidirect
9. #20243418C2semidirect
10. #20243103C2semidirect
11. C47×#1809C2semidirect
18. C2×M16×D47C4semidirect
25. C2×#13093958C22semidirect
26. C2×#13093957C22semidirect
27. C2×#13093956C22semidirect
28. C2×#13093955C22semidirect
36. #15688886C22semidirect
37. #15688885C22semidirect
38. #15688885C22semidirect
40. C2×#107×C47C22semidirect
41. C2×#106×C47C22semidirect
43. C47×#340C22semidirect
186. C8×D47C22⋊C4semidirect
296. #1809D47semidirect
322. C2×#107C2×D47semidirect
325. C2×#106C2×D47semidirect
337. #340C2×D47semidirect
338. C47#20660semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D472
C2×#156891602
C2×#156891582
#202457702
#202434222
#202434212
#202434202
#202434192
#202434182
#202431032
C47×#18092
#2066047