Finite Group Databasebeta

← #26217049

Subgroups — #26217049

Order 12032.

1462 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
1C2×#15689606C211yesmaximal, characteristic
2C2×#15689166C211yesmaximal, characteristic, complemented
3C2×#15689159C211yesmaximal, characteristic, complemented
4C2×#15689090C211yesmaximal, characteristic
5#20243585C212yesmaximal, complemented
6#20243585C212yesmaximal, complemented
7#20243580C212yesmaximal, complemented
8#20243575C212yesmaximal, complemented
9#20243107C211yesmaximal, characteristic, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C47×#2311C211yesmaximal, characteristic
12C22×#10625329C2211yescharacteristic
13C23×C47⋊C8C2211yescharacteristic
14#15689606—24no—
15#15689606—24no—
16C2×Q8×Dic47—21no—
17C2×D4×Dic47—22no—
18C2×D4×Dic47—21no—
19C2×#13093983C2212yes—
20C2×#13093983C2211yescharacteristic
21C22×C4⋊Dic47C2212yes—
22C22×C4⋊Dic47C2211yescharacteristic
23#15689166—24no—
24#15689159—24no—
25C2×#13093957C2211yescharacteristic
26C2×#13093957C2211yescharacteristic
27C2×#13093957C2212yes—
28C2×#13093955C2211yescharacteristic
29C2×#13093955C2211yescharacteristic
30C2×#13093955C2212yes—
31#15689090C418yescomplemented
32C2×C22⋊(C47⋊C8)C2212yes—
33C2×C22⋊(C47⋊C8)C2212yes—
34#15688885C2212yes—
35#15688885C2212yes—
36#15688885C2212yes—
37#15688885C2212yes—
38C22×#41×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C22×C4⋊C4×C47C2211yescharacteristic
41C47×#391—21no—
42C47×#390—21no—
43C47×#386—22no—
44C47×#380C2212yes—
45C47×#380C2212yes—
46C23×Dic47—22no—
47C23×Dic47—21no—
48C2×#10625329—22no—
49C2×#10625329—22no—
50C2×#10625329C2×C414yescomplemented
51C22×C47⋊C8—22no—
52C22×C47⋊C8—22no—
53C22×C47⋊C8C2×C414yescomplemented
54C22×C47⋊C8C2311yescharacteristic
55C22×C47⋊C8C2311yescharacteristic
56C22×C47⋊C8C2312yes—
57Q8×Dic47—28no—
58D4×Dic47—216no—
59D4×Dic47—28no—
60#13093983—22no—
61#13093983—24no—
62#13093983—24no—
63#13093983D414yes—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—22no—
67C2×C4⋊Dic47—24no—
68C2×C4⋊Dic47—24no—
69C2×C4⋊Dic47D414yes—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2312yes—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C22⋊Dic47—22no—
74C2×C22⋊Dic47—21no—
75C2×C4×Dic47—22no—
76C2×C4×Dic47—21no—
77#13093957—22no—
78#13093957—22no—
79#13093957—24no—
80#13093955—22no—
81#13093955—22no—
82#13093955—24no—
83C22⋊(C47⋊C8)C2×C418yescomplemented
84C22⋊(C47⋊C8)C2×C418yescomplemented
85C2×#41×C47—22no—
86C2×#41×C47—22no—
87C2×#41×C47—22no—
88C2×#41×C47—22no—
89C2×#41×C47C2×C414yes—
90C2×#41×C47Q814yes—
91C22×Q8×C47D411yescharacteristic
92C22×D4×C47D412yes—
93C22×D4×C47D411yescharacteristic
94C23×C4×C47—21no—
95C23×C4×C47C2312yes—
96C23×C4×C47C2311yescharacteristic
97#117×C47—22no—
98#117×C47—22no—
99C2×C4⋊C4×C47—22no—
100C2×C4⋊C4×C47—22no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47C2312yes—
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47—22no—
107C2×C22⋊C4×C47—22no—
108C2×C42×C47—21no—
109#95×C47—22no—
110#95×C47—22no—
111C24×C47C2×D412yes—
112C24×C47C2×D411yescharacteristic
113#41×C47—28no—
114#41×C47—28no—
115#41×C47—28no—
116#41×C47—28no—
117C2×Q8×C47—22no—
118C2×Q8×C47—22no—
119C2×Q8×C47C4⋊C414yes—
120C2×D4×C47—24no—
121C2×D4×C47—24no—
122C2×D4×C47—22no—
123C2×D4×C47—22no—
124C2×D4×C47C4⋊C418yes—
125C2×D4×C47C4⋊C414yes—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—22no—
133C22×C4×C47—42no—
134C22×C4×C47—81no—
135C22×C4×C47—81no—
136C22×C4×C47C22×C414yes—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×Q814yes—
143C22×C4×C47C2×Q812yes—
144C22×C4×C47#4112yes—
145C22×C4×C47#4112yes—
146C22×C4×C47C2411yescharacteristic
147C22×Dic47—21no—
148C22×Dic47—22no—
149C22×Dic47—21no—
150C22×Dic47—28no—
151C22×Dic47—24no—
152C22×Dic47—81no—
153C22×Dic47—82no—
154C22×Dic47—81no—
155C22×Dic47—82no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—22no—
162C4⋊C4×C47—42no—
163C22⋊C4×C47—42no—
164C22⋊C4×C47—82no—
165C22⋊C4×C47—82no—
166C42×C47—42no—
167C22⋊Dic47—28no—
168C22⋊Dic47—24no—
169C4⋊Dic47—28no—
170C4⋊Dic47—24no—
171C4⋊Dic47—22no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47C2×D412yes—
175C4⋊Dic47C2×D414yes—
176C4⋊Dic47C2×D412yes—
177C4×Dic47—28no—
178C4×Dic47—24no—
179#10625329—28no—
180C2×C47⋊C8—28no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8C22×C412yes—
188C2×C47⋊C8C22×C412yes—
189C2×C47⋊C8C22×C414yes—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—22no—
200C23×C47C2×C4⋊C414yes—
201C23×C47C2×C4⋊C412yes—
202C23×C47#12012yes—
203C23×C47#12312yes—
204C23×C47C22×D411yescharacteristic
205Q8×C47—28no—
206D4×C47—216no—
207D4×C47—28no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—24no—
213C2×C4×C47—24no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—42no—
217C2×C4×C47—41no—
218C2×C4×C47—42no—
219C2×C4×C47—41no—
220C2×C4×C47—42no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—84no—
224C2×C4×C47C2×C4⋊C414yes—
225C2×C4×C47C2×C4⋊C412yes—
226C2×C4×C47C4×D414yes—
227C2×C4×C47C4×D414yes—
228C2×C4×C47#12012yes—
229C2×C4×C47C22⋊Q814yes—
230C2×C4×C47C22⋊Q814yes—
231C2×C4×C47#12312yes—
232C2×C4×C47C23×C411yescharacteristic
233C2×C4×C47C22×D411yescharacteristic
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C22×Q811yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×Dic47—22no—
239C2×Dic47—24no—
240C2×Dic47—22no—
241C2×Dic47—42no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—42no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—84no—
248C2×Dic47—88no—
249C47⋊C8—22no—
250C47⋊C8—22no—
251C47⋊C8—24no—
252#20849—471nomaximal, Sylow
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—24no—
260C22×C47—24no—
261C22×C47—24no—
262C22×C47—42no—
263C22×C47—42no—
264C22×C47—42no—
265C22×C47—42no—
266C22×C47#38514yes—
267C22×C47C22×C4⋊C411yescharacteristic
268C22×C47C2×#12011yescharacteristic
269C22×C47C2×#12311yescharacteristic
270C22×C47#57611yescharacteristic
271C22×C47#57711yescharacteristic
272C22×C47#57711yescharacteristic
273C22×C47#57811yescharacteristic
274C4×C47—24no—
275C4×C47—82no—
276C4×C47—81no—
277C4×C47—81no—
278C4×C47#38514yes—
279C4×C47C2×C4×D411yescharacteristic
280C4×C47C2×C4×D411yescharacteristic
281C4×C47C2×C22⋊Q811yescharacteristic
282C4×C47C2×C22⋊Q811yescharacteristic
283Dic47—44no—
284Dic47—42no—
285Dic47—42no—
286C2×#520—471no—
287C2×#428—471no—
288C2×#427—471no—
289C2×#408—471no—
290#2311D4711yescharacteristic
291#1854—472no—
292#1854—472no—
293#1853—472no—
294#1852—472no—
295#1736—471no—
296C2×#340—471no—
297C2×C47—24no—
298C2×C47—44no—
299C2×C47C2×#38511yescharacteristic
300C2×C47#291211yescharacteristic
301C2×C47#291311yescharacteristic
302C2×C47#301211yescharacteristic
303C2×C47#301311yescharacteristic
304C2×C47#309611yescharacteristic
305C2×C47#309711yescharacteristic
306C22×#41C2×D4711yescharacteristic
307C22×M16—471no—
308C23×C8—471no—
309#520—944no—
310#520—944no—
311C2×C4×Q8—941no—
312C2×C4×D4—942no—
313C2×C4×D4—941no—
314C2×#117—471no—
315C2×#117—472no—
316C2×#117C2×D4711yescharacteristic
317C22×C4⋊C4—471no—
318C22×C4⋊C4—472no—
319C22×C4⋊C4C2×D4711yescharacteristic
320#428—944no—
321#427—944no—
322C2×#107—471no—
323C2×#107—472no—
324C2×#107—471no—
325C2×#106—471no—
326C2×#106—472no—
327C2×#106—471no—
328#408—478no—
329C2×C22⋊C8—472no—
330C2×C22⋊C8—472no—
331#391—21no—
332#390—21no—
333#386—22no—
334#380C2×D4712yes—
335#380C2×D4712yes—
336#340—472no—
337#340—472no—
338#340—472no—
339#340—472no—
340C47#2084911yescharacteristic, complemented, Sylow
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41Dic9414yes—
346C2×#41C4×D4714yes—
347C22×Q8C22⋊D4711yescharacteristic
348C22×D4C22⋊D4712yes—
349C22×D4C22⋊D4711yescharacteristic
350C23×C4—21no—
351C23×C4—942no—
352C23×C4—941no—
353C23×C4C22×D4712yes—
354C23×C4C22×D4711yescharacteristic
355C2×M16—474no—
356C2×M16—942no—
357C2×M16—942no—
358C22×C8—474no—
359C22×C8—471no—
360C22×C8—471no—
361C22×C8—472no—
362C22×C8—942no—
363C22×C8—942no—
364C4×Q8—948no—
365C4×D4—9416no—
366C4×D4—948no—
367#117—22no—
368#117—22no—
369#117—474no—
370#117—944no—
371#117—944no—
372#117—942no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—474no—
377C2×C4⋊C4—472no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—944no—
381C2×C4⋊C4—944no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4C22×D4712yes—
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—942no—
392C2×C22⋊C4—941no—
393C2×C42—21no—
394C2×C42—942no—
395C2×C42—941no—
396#107—942no—
397#107—942no—
398#107—944no—
399#106—942no—
400#106—942no—
401#106—944no—
402C22⋊C8—478no—
403C22⋊C8—478no—
404#95—22no—
405#95—22no—
406C24C2×C22⋊D4712yes—
407C24C2×C22⋊D4711yescharacteristic
408#41—28no—
409#41—28no—
410#41—28no—
411#41—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8Dic47⋊C414yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4Dic47⋊C418yes—
420C2×D4Dic47⋊C414yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—948no—
432C22×C4—944no—
433C22×C4—941no—
434C22×C4—942no—
435C22×C4—941no—
436C22×C4—3761no—
437C22×C4—3761no—
438C22×C4—3762no—
439C22×C4—3762no—
440C22×C4C2×Dic9414yes—
441C22×C4C2×Dic9412yes—
442C22×C4C2×C4×D4714yes—
443C22×C4C2×C4×D4712yes—
444C22×C4D4×D4712yes—
445C22×C4D4×D4712yes—
446C22×C4#1062535212yes—
447C22×C4#1062535212yes—
448C22×C4C2×C22⋊D4712yes—
449C22×C4C2×C22⋊D4711yescharacteristic
450C22×C4C23×D4711yescharacteristic
451M16—948no—
452C2×C8—472no—
453C2×C8—472no—
454C2×C8—474no—
455C2×C8—948no—
456C2×C8—941no—
457C2×C8—942no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—941no—
461C2×C8—942no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—474no—
470C4⋊C4—472no—
471C4⋊C4—472no—
472C4⋊C4—948no—
473C4⋊C4—944no—
474C4⋊C4—941no—
475C4⋊C4—942no—
476C4⋊C4—941no—
477C22⋊C4—42no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—948no—
481C22⋊C4—944no—
482C42—42no—
483C42—948no—
484C42—944no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23C2×Dic47⋊C414yes—
496C23C2×Dic47⋊C412yes—
497C23#1309399212yes—
498C23#1309400512yes—
499C23C22×C22⋊D4711yescharacteristic
500Q8—28no—
501D4—216no—
502D4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—84no—
519C2×C4—942no—
520C2×C4—944no—
521C2×C4—942no—
522C2×C4—3764no—
523C2×C4—3768no—
524C2×C4—1882no—
525C2×C4—1881no—
526C2×C4—1882no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4C2×Dic47⋊C414yes—
531C2×C4C2×Dic47⋊C412yes—
532C2×C4(C22⋊D47)⋊C414yes—
533C2×C4(C22⋊D47)⋊C414yes—
534C2×C4#1309399212yes—
535C2×C4#1309400114yes—
536C2×C4#1309400114yes—
537C2×C4#1309400512yes—
538C2×C4C22×C4×D4711yescharacteristic
539C2×C4C2×D4×D4711yescharacteristic
540C2×C4C2×D4×D4711yescharacteristic
541C2×C4C22×Dic9411yescharacteristic
542C2×C4C2×#1062535211yescharacteristic
543C2×C4C2×#1062535211yescharacteristic
544C8—942no—
545C8—942no—
546C8—944no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22#1568899514yes—
561C22C22×Dic47⋊C411yescharacteristic
562C22C2×#1309399211yescharacteristic
563C22C2×#1309400511yescharacteristic
564C22#1568992611yescharacteristic
565C22#1568993111yescharacteristic
566C22#1568993411yescharacteristic
567C22#1568993811yescharacteristic
568C4—24no—
569C4—81no—
570C4—81no—
571C4—82no—
572C4—1884no—
573C4—1882no—
574C4—1882no—
575C4#1568899514yes—
576C4C2×(C22⋊D47)⋊C411yescharacteristic
577C4C2×(C22⋊D47)⋊C411yescharacteristic
578C4C2×#1309400111yescharacteristic
579C4C2×#1309400111yescharacteristic
580C2—24no—
581C2—44no—
582C2C2×#1568899511yescharacteristic
583C2#2025198111yescharacteristic
584C2#2025198611yescharacteristic
585C2#2025268911yescharacteristic
586C2#2025270011yescharacteristic
587C2#2025334011yescharacteristic
588C2#2025335111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2311C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenter#2311D47
hypocenterC47#20849
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#15689166C2semidirect
3. C2×#15689159C2semidirect
5. #20243585C2semidirect
6. #20243585C2semidirect
7. #20243580C2semidirect
8. #20243575C2semidirect
9. #20243107C2semidirect
10. C2×#15688885C2semidirect
31. #15689090C4semidirect
50. C2×#10625329C2×C4semidirect
53. C22×C47⋊C8C2×C4semidirect
83. C22⋊(C47⋊C8)C2×C4semidirect
84. C22⋊(C47⋊C8)C2×C4semidirect
340. C47#20849semidirect

Maximal subgroups

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

subgroupindex
C2×#156896062
C2×#156891662
C2×#156891592
C2×#156890902
#202435852
#202435802
#202435752
#202431072
C2×#156888852
C47×#23112
#2084947