Finite Group Databasebeta

← #26217812

Subgroups — #26217812

Order 12032.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256591C211yesmaximal, characteristic, complemented
2C2×#15689140C211yesmaximal, characteristic, complemented
3C2×#15689128C211yesmaximal, characteristic, complemented
4#20245970C211yesmaximal, characteristic, complemented
5#20243766C218yesmaximal, complemented
6#20243126C211yesmaximal, characteristic, complemented
7#20243099C211yesmaximal, characteristic, complemented
8C22×AGL(1,ℤ/8)×C47C211yesmaximal, characteristic
9C22×#10625329C2211yescharacteristic
10#15689606—22no—
11#15689606C2214yescomplemented
12#15689601—21no—
13#15689598—22no—
14#15689598—21no—
15#41×Dic47—22no—
16#41×Dic47—22no—
17C2×#13093983—22no—
18C2×#13093983C2211yescharacteristic
19C2×#13093983C2211yescharacteristic
20C2×#13093983C2211yescharacteristic
21#15689140—24no—
22C2×#13093951C2214yescomplemented
23#15689128—22no—
24#15689128C2214yescomplemented
25C2×#13093945—22no—
26#15689026—22no—
27#15689026C2214yescomplemented
28#15688979—22no—
29#15688891C2214yescomplemented
30#15688884C2214yescomplemented
31C22×#41×C47C2211yescharacteristic
32C23×D4×C47C2211yescharacteristic
33C2×AGL(1,ℤ/8)×C47—24no—
34C2×AGL(1,ℤ/8)×C47—22no—
35C2×AGL(1,ℤ/8)×C47C418yescomplemented
36C2×AGL(1,ℤ/8)×C47C2214yes—
37C22×SD16×C47—21no—
38C22×D8×C47—21no—
39C22×M16×C47C2211yescharacteristic
40C2×#10625329—21no—
41C2×#10625329—22no—
42C2×#10625329—22no—
43C2×#10625329C2312yes—
44C22×C47⋊C8—21no—
45Q8×Dic47—24no—
46Q8×Dic47—24no—
47D4×Dic47—28no—
48D4×Dic47—24no—
49D4×Dic47—24no—
50D4×Dic47—28no—
51#13093983—24no—
52#13093983—24no—
53#13093983—21no—
54#13093983—21no—
55#13093983—21no—
56#13093983—21no—
57#13093983—21no—
58#13093983—21no—
59#13093983—24no—
60#13093983—22no—
61#13093983—22no—
62#13093983D412yes—
63#13093983C2312yes—
64C2×C4⋊Dic47—24no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47D412yes—
67C2×C4⋊Dic47C2312yes—
68C2×C22⋊Dic47—24no—
69C2×C22⋊Dic47—21no—
70C2×C22⋊Dic47—21no—
71C2×C22⋊Dic47—28no—
72C2×C22⋊Dic47D412yes—
73C2×C4×Dic47—24no—
74C2×C4×Dic47—21no—
75C2×C4×Dic47D412yes—
76C2×C4×Dic47C2312yes—
77#13093951—28no—
78#13093945—28no—
79C25×C47—21no—
80C2×#41×C47—21no—
81C2×#41×C47—21no—
82C2×#41×C47—22no—
83C2×#41×C47—22no—
84C2×#41×C47C2×C414yes—
85C2×#41×C47D414yes—
86C2×#41×C47D412yes—
87C2×#41×C47C2312yes—
88C22×Q8×C47D411yescharacteristic
89C22×D4×C47—21no—
90C22×D4×C47—21no—
91C22×D4×C47—21no—
92C22×D4×C47—22no—
93C22×D4×C47—21no—
94C22×D4×C47—21no—
95C22×D4×C47—21no—
96C22×D4×C47—22no—
97C22×D4×C47—22no—
98C22×D4×C47C2×C414yescomplemented
99C22×D4×C47D411yescharacteristic
100C22×D4×C47C2312yes—
101C23×C4×C47—21no—
102C23×C4×C47C2311yescharacteristic
103AGL(1,ℤ/8)×C47—216no—
104AGL(1,ℤ/8)×C47—28no—
105AGL(1,ℤ/8)×C47C2×C4116yescomplemented
106C2×SD16×C47—28no—
107C2×SD16×C47—24no—
108C2×D8×C47—28no—
109C2×D8×C47—24no—
110C2×M16×C47—22no—
111C2×M16×C47—21no—
112C2×M16×C47C2×C414yescomplemented
113C2×M16×C47C2312yes—
114C22×C8×C47—21no—
115C24×C47—21no—
116C24×C47—22no—
117C24×C47—24no—
118C24×C47—41no—
119C24×C47—41no—
120C24×C47—42no—
121C24×C47—84no—
122C24×C47C2×D411yescharacteristic
123#41×C47—24no—
124#41×C47—24no—
125#41×C47—24no—
126#41×C47—24no—
127#41×C47—24no—
128#41×C47—22no—
129#41×C47—22no—
130#41×C47C22⋊C418yescomplemented
131#41×C47C22×C414yes—
132#41×C47C2×D414yes—
133C2×Q8×C47—21no—
134C2×Q8×C47—22no—
135C2×Q8×C47C22⋊C414yescomplemented
136C2×Q8×C47C2×D412yes—
137C2×D4×C47—24no—
138C2×D4×C47—22no—
139C2×D4×C47—24no—
140C2×D4×C47—22no—
141C2×D4×C47—24no—
142C2×D4×C47—24no—
143C2×D4×C47—24no—
144C2×D4×C47—24no—
145C2×D4×C47—24no—
146C2×D4×C47—21no—
147C2×D4×C47—22no—
148C2×D4×C47—22no—
149C2×D4×C47—22no—
150C2×D4×C47—22no—
151C2×D4×C47—22no—
152C2×D4×C47—22no—
153C2×D4×C47—41no—
154C2×D4×C47—41no—
155C2×D4×C47—42no—
156C2×D4×C47—42no—
157C2×D4×C47—42no—
158C2×D4×C47—42no—
159C2×D4×C47—42no—
160C2×D4×C47C22⋊C414yescomplemented
161C2×D4×C47C22×C414yes—
162C2×D4×C47C2×D412yes—
163C22×C4×C47—21no—
164C22×C4×C47—24no—
165C22×C4×C47—22no—
166C22×C4×C47—24no—
167C22×C4×C47—22no—
168C22×C4×C47C22×C412yes—
169C22×C4×C47C2×D411yescharacteristic
170C22×C4×C47C2×D411yescharacteristic
171C22×C4×C47C2×D412yes—
172C22×C4×C47C2×D411yescharacteristic
173C22×C4×C47C2×D411yescharacteristic
174C22×C4×C47C2×D411yescharacteristic
175C22×C4×C47#4112yes—
176C22×C4×C47#4112yes—
177C22×C4×C47C2411yescharacteristic
178C22×Dic47—28no—
179C22×Dic47—22no—
180C22×Dic47—22no—
181C22×Dic47C2×D414yes—
182SD16×C47—216no—
183D8×C47—216no—
184M16×C47—24no—
185M16×C47—22no—
186M16×C47C22×C414yes—
187C2×C8×C47—24no—
188C2×C8×C47—22no—
189C22⋊Dic47—28no—
190C22⋊Dic47—22no—
191C22⋊Dic47—22no—
192C22⋊Dic47—22no—
193C22⋊Dic47—22no—
194C22⋊Dic47—42no—
195C22⋊Dic47—88no—
196C4⋊Dic47—28no—
197C4⋊Dic47—22no—
198C4⋊Dic47—22no—
199C4⋊Dic47—22no—
200C4⋊Dic47—42no—
201C4⋊Dic47C2×D414yes—
202C4×Dic47—28no—
203C4×Dic47—22no—
204C4×Dic47—22no—
205C4×Dic47—22no—
206C4×Dic47—22no—
207C4×Dic47—22no—
208C4×Dic47—22no—
209#10625329—22no—
210#10625329—22no—
211#10625329—42no—
212C2×C47⋊C8—22no—
213C2×C47⋊C8—42no—
214C23×C47—22no—
215C23×C47—24no—
216C23×C47—24no—
217C23×C47—42no—
218C23×C47—48no—
219C23×C47—42no—
220C23×C47—42no—
221C23×C47—44no—
222C23×C47—44no—
223C23×C47—82no—
224C23×C47—84no—
225C23×C47—84no—
226C23×C47—88no—
227C23×C47—88no—
228C23×C47C4×D412yes—
229C23×C47#12011yescharacteristic
230C23×C47#12011yescharacteristic
231C23×C47#12112yes—
232C23×C47C22×D411yescharacteristic
233Q8×C47—22no—
234Q8×C47—24no—
235Q8×C47C2×C22⋊C414yes—
236D4×C47—24no—
237D4×C47—24no—
238D4×C47—28no—
239D4×C47—24no—
240D4×C47—24no—
241D4×C47—22no—
242D4×C47—24no—
243D4×C47—42no—
244D4×C47—42no—
245D4×C47—44no—
246D4×C47—44no—
247D4×C47C2×C22⋊C414yes—
248C2×C4×C47—22no—
249C2×C4×C47—24no—
250C2×C4×C47—24no—
251C2×C4×C47—24no—
252C2×C4×C47—28no—
253C2×C4×C47C2×C22⋊C412yes—
254C2×C4×C47C2×C22⋊C412yes—
255C2×C4×C47C4×D412yes—
256C2×C4×C47C4×D412yes—
257C2×C4×C47C4×D412yes—
258C2×C4×C47#12011yescharacteristic
259C2×C4×C47#12011yescharacteristic
260C2×C4×C47#12112yes—
261C2×C4×C47#12112yes—
262C2×C4×C47#12112yes—
263C2×C4×C47#12312yes—
264C2×C4×C47#12312yes—
265C2×C4×C47C23×C411yescharacteristic
266C2×C4×C47C22×D411yescharacteristic
267C2×C4×C47C22×D411yescharacteristic
268C2×C4×C47C22×D411yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—44no—
274C2×Dic47—44no—
275C2×Dic47—42no—
276C2×Dic47—82no—
277C2×Dic47—88no—
278C8×C47—24no—
279C47⋊C8—42no—
280#20945—471nomaximal, Sylow
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—24no—
285C22×C47—42no—
286C22×C47—42no—
287C22×C47—48no—
288C22×C47—44no—
289C22×C47—44no—
290C22×C47—84no—
291C22×C47—84no—
292C22×C47—88no—
293C22×C47—88no—
294C22×C47—84no—
295C22×C47#38612yes—
296C22×C47#38612yes—
297C22×C47C2×C4×D411yescharacteristic
298C22×C47C2×#12011yescharacteristic
299C22×C47C2×#12111yescharacteristic
300C4×C47—24no—
301C4×C47#38612yes—
302C4×C47#38612yes—
303C4×C47C22×C22⋊C411yescharacteristic
304C4×C47C2×C4×D411yescharacteristic
305C4×C47C2×#12111yescharacteristic
306C4×C47C2×#12311yescharacteristic
307Dic47—44no—
308Dic47—44no—
309Dic47—82no—
310C22×AGL(1,ℤ/8)D4711yescharacteristic
311#3428—471no—
312C2×#421—471no—
313C2×#418—471no—
314#2309—471no—
315#1883—478no—
316#1742—471no—
317#1733—471no—
318C2×C47—21no—
319C2×C47—21no—
320C2×C47—22no—
321C2×C47—22no—
322C2×C47—44no—
323C2×C47—84no—
324C2×C47C2×#38611yescharacteristic
325C2×C47#302112yes—
326C22×#41C2×D4711yescharacteristic
327C23×D4C2×D4711yescharacteristic
328C2×AGL(1,ℤ/8)—24no—
329C2×AGL(1,ℤ/8)—22no—
330C2×AGL(1,ℤ/8)Dic4718yescomplemented
331C2×AGL(1,ℤ/8)C2×D4714yes—
332C22×SD16—21no—
333C22×D8—21no—
334C22×M16—471no—
335C22×M16C2×D4711yescharacteristic
336#520—474no—
337#520—942no—
338#519—941no—
339#518—942no—
340#518—941no—
341C4×#41—942no—
342C4×#41—942no—
343C2×#117—471no—
344C2×#117—471no—
345C2×#117—471no—
346C2×#117—942no—
347#421—944no—
348C2×#104—474no—
349#418—474no—
350#418—942no—
351C2×#102—942no—
352#390—474no—
353#390—942no—
354#379—942no—
355#343—474no—
356#339—474no—
357C47#2094511yescharacteristic, complemented, Sylow
358C25—21no—
359C2×#41—21no—
360C2×#41—21no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41C2×Dic4714yes—
364C2×#41C22⋊D4714yes—
365C2×#41C22⋊D4712yes—
366C2×#41C22×D4712yes—
367C22×Q8C22⋊D4711yescharacteristic
368C22×D4—21no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4—21no—
372C22×D4—22no—
373C22×D4—22no—
374C22×D4—21no—
375C22×D4—21no—
376C22×D4—22no—
377C22×D4C2×Dic4714yescomplemented
378C22×D4C22⋊D4711yescharacteristic
379C22×D4C22×D4712yes—
380C23×C4—21no—
381C23×C4C22×D4711yescharacteristic
382AGL(1,ℤ/8)—216no—
383AGL(1,ℤ/8)—28no—
384AGL(1,ℤ/8)C2×Dic47116yescomplemented
385C2×SD16—28no—
386C2×SD16—24no—
387C2×D8—28no—
388C2×D8—24no—
389C2×M16—22no—
390C2×M16—21no—
391C2×M16—472no—
392C2×M16—941no—
393C2×M16—942no—
394C2×M16—942no—
395C2×M16C2×Dic4714yescomplemented
396C2×M16C22×D4712yes—
397C22×C8—21no—
398C22×C8—941no—
399C4×Q8—944no—
400C4×Q8—944no—
401C4×D4—948no—
402C4×D4—948no—
403C4×D4—944no—
404C4×D4—944no—
405#117—472no—
406#117—472no—
407#117—944no—
408#117—944no—
409#117—944no—
410#117—941no—
411#117—941no—
412#117—941no—
413#117—941no—
414#117—941no—
415#117—941no—
416#117—942no—
417#117—942no—
418C2×C4⋊C4—472no—
419C2×C4⋊C4—472no—
420C2×C4⋊C4—944no—
421C2×C4⋊C4—941no—
422C2×C22⋊C4—472no—
423C2×C22⋊C4—944no—
424C2×C22⋊C4—941no—
425C2×C22⋊C4—948no—
426C2×C22⋊C4—941no—
427C2×C42—472no—
428C2×C42—472no—
429C2×C42—944no—
430C2×C42—941no—
431#104—948no—
432#102—948no—
433C24—21no—
434C24—22no—
435C24—24no—
436C24—41no—
437C24—41no—
438C24—42no—
439C24—84no—
440C24D4×D4711yescharacteristic
441#41—24no—
442#41—24no—
443#41—24no—
444#41—24no—
445#41—24no—
446#41—22no—
447#41—22no—
448#41C22⋊Dic4718yescomplemented
449#41C22×Dic4714yes—
450#41C2×C22⋊D4714yes—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8C22⋊Dic4714yescomplemented
454C2×Q8C2×C22⋊D4712yes—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4C22⋊Dic4714yescomplemented
479C2×D4C22×Dic4714yes—
480C2×D4C2×C22⋊D4712yes—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—474no—
487C22×C4—948no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4D4×D4711yescharacteristic
491C22×C4D4×D4711yescharacteristic
492C22×C4D4×D4711yescharacteristic
493C22×C4#1062535212yes—
494C22×C4#1062535212yes—
495C22×C4C22×Dic4712yes—
496C22×C4C2×C22⋊D4711yescharacteristic
497C22×C4C2×C22⋊D4711yescharacteristic
498C22×C4C2×C22⋊D4712yes—
499C22×C4C23×D4711yescharacteristic
500SD16—216no—
501D8—216no—
502M16—24no—
503M16—22no—
504M16—942no—
505M16—942no—
506M16—1882no—
507M16C22×Dic4714yes—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—942no—
511C2×C8—1882no—
512C4⋊C4—474no—
513C4⋊C4—948no—
514C4⋊C4—942no—
515C4⋊C4—942no—
516C4⋊C4—942no—
517C4⋊C4—1882no—
518C22⋊C4—948no—
519C22⋊C4—942no—
520C22⋊C4—942no—
521C22⋊C4—942no—
522C22⋊C4—942no—
523C22⋊C4—3768no—
524C22⋊C4—1882no—
525C42—948no—
526C42—942no—
527C42—942no—
528C42—942no—
529C42—942no—
530C42—942no—
531C42—942no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—42no—
536C23—48no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—82no—
542C23—84no—
543C23—84no—
544C23—88no—
545C23—88no—
546C23D4×Dic4712yes—
547C23#1309399211yescharacteristic
548C23#1309399211yescharacteristic
549C23#1309399612yes—
550C23C2×D4×D4711yescharacteristic
551Q8—24no—
552Q8—22no—
553Q8C2×C22⋊Dic4714yes—
554D4—28no—
555D4—24no—
556D4—24no—
557D4—24no—
558D4—24no—
559D4—24no—
560D4—22no—
561D4—44no—
562D4—44no—
563D4—42no—
564D4—42no—
565D4C2×C22⋊Dic4714yes—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—944no—
572C2×C4—944no—
573C2×C4—3762no—
574C2×C4—3768no—
575C2×C4—1884no—
576C2×C4—1882no—
577C2×C4—1884no—
578C2×C4C2×C22⋊Dic4712yes—
579C2×C4C2×C22⋊Dic4712yes—
580C2×C4D4×Dic4712yes—
581C2×C4D4×Dic4712yes—
582C2×C4D4×Dic4712yes—
583C2×C4#1309399211yescharacteristic
584C2×C4#1309399211yescharacteristic
585C2×C4#1309399612yes—
586C2×C4#1309399612yes—
587C2×C4#1309399612yes—
588C2×C4#1309400512yes—
589C2×C4#1309400512yes—
590C2×C4C23×Dic4711yescharacteristic
591C2×C4C2×D4×D4711yescharacteristic
592C2×C4C22×C22⋊D4711yescharacteristic
593C2×C4C22×C22⋊D4711yescharacteristic
594C2×C4C2×#1062535211yescharacteristic
595C2×C4C2×#1062535211yescharacteristic
596C8—24no—
597C8—1882no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—48no—
605C22—44no—
606C22—44no—
607C22—84no—
608C22—84no—
609C22—88no—
610C22—88no—
611C22—84no—
612C22#1568900312yes—
613C22#1568900312yes—
614C22C2×D4×Dic4711yescharacteristic
615C22C2×#1309399211yescharacteristic
616C22C2×#1309399611yescharacteristic
617C4—24no—
618C4—3762no—
619C4—1884no—
620C4—1884no—
621C4#1568900312yes—
622C4#1568900312yes—
623C4C22×C22⋊Dic4711yescharacteristic
624C4C2×D4×Dic4711yescharacteristic
625C4C2×#1309399611yescharacteristic
626C4C2×#1309400511yescharacteristic
627C2—21no—
628C2—21no—
629C2—22no—
630C2—22no—
631C2—44no—
632C2—84no—
633C2C2×#1568900311yescharacteristic
634C2#2025275712yes—
subgroupG/H
centerC22C2×#13093992
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC22×AGL(1,ℤ/8)×C47C2
socleC22×C47C2×#120
cosocleC22×C4×C47C24
hypercenterC22×AGL(1,ℤ/8)D47
hypocenterC47#20945
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20256591C2semidirect
2. C2×#15689140C2semidirect
3. C2×#15689128C2semidirect
4. #20245970C2semidirect
5. #20243766C2semidirect
6. #20243126C2semidirect
7. #20243099C2semidirect
11. #15689606C22semidirect
22. C2×#13093951C22semidirect
24. #15689128C22semidirect
27. #15689026C22semidirect
29. #15688891C22semidirect
30. #15688884C22semidirect
35. C2×AGL(1,ℤ/8)×C47C4semidirect
98. C22×D4×C47C2×C4semidirect
105. AGL(1,ℤ/8)×C47C2×C4semidirect
112. C2×M16×C47C2×C4semidirect
130. #41×C47C22⋊C4semidirect
135. C2×Q8×C47C22⋊C4semidirect
160. C2×D4×C47C22⋊C4semidirect
330. C2×AGL(1,ℤ/8)Dic47semidirect
357. C47#20945semidirect
377. C22×D4C2×Dic47semidirect
384. AGL(1,ℤ/8)C2×Dic47semidirect
395. C2×M16C2×Dic47semidirect
448. #41C22⋊Dic47semidirect
453. C2×Q8C22⋊Dic47semidirect
478. C2×D4C22⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
#202565912
C2×#156891402
C2×#156891282
#202459702
#202437662
#202431262
#202430992
C22×AGL(1,ℤ/8)×C472
#2094547