Finite Group Databasebeta

← #26217012

Subgroups — #26217012

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
1C22×C4⋊C4×D47C211yesmaximal, characteristic
2C2×#15689162C212yesmaximal, complemented
3C2×#15689085C211yesmaximal, characteristic
4#20245770C211yesmaximal, characteristic
5#20243580C211yesmaximal, characteristic, complemented
6#20243579C212yesmaximal, complemented
7#20243578C211yesmaximal, characteristic, complemented
8#20243577C212yesmaximal, complemented
9#20243576C211yesmaximal, characteristic, complemented
10#20243106C212yesmaximal, complemented
11C47×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D47—21no—
13C23×C4×D47C2211yescharacteristic
14C22×#10625324C2212yes—
15C2×#13093982C2211yescharacteristic
16C2×#13093982C2211yescharacteristic
17C22×C4⋊Dic47C2211yescharacteristic
18C22×Dic47⋊C4—21no—
19C2×C4⋊C4×D47—21no—
20C2×C4⋊C4×D47—21no—
21C2×C4⋊C4×D47—28no—
22C2×C4⋊C4×D47C2212yes—
23C2×C4⋊C4×D47C2212yes—
24#15689162—28no—
25C2×#13093958C2212yescomplemented
26C2×#13093957C2212yescomplemented
27C2×#13093955C2212yescomplemented
28#15689085C418yescomplemented
29C2×C22⋊(C47⋊C8)C2211yescharacteristic
30C2×(C2×D47)⋊C8C2212yes—
31#15688985—21no—
32#15688983C2211yescharacteristic
33#15688980C2211yescharacteristic
34#15688980C2211yescharacteristic
35#15688978—21no—
36C22⋊C4×Dic47—21no—
37#15688970—21no—
38#15688886C2212yescomplemented
39#15688885C2212yescomplemented
40#15688885C2212yescomplemented
41C22×C4⋊C4×C47C2211yescharacteristic
42C2×#106×C47C2212yescomplemented
43C2×C22⋊C8×C47C2211yescharacteristic
44C47×#380C2211yescharacteristic
45C47×#340C2212yescomplemented
46C24×D47D411yescharacteristic
47C23×Dic47—21no—
48C23×Dic47D411yescharacteristic
49C22×C4×D47—24no—
50C22×C4×D47—24no—
51C22×C4×D47—28no—
52C22×C4×D47—22no—
53C22×C4×D47—22no—
54C22×C4×D47—44no—
55C22×C4×D47—44no—
56C22×C4×D47C2×C414yes—
57C22×C4×D47D411yescharacteristic
58C22×C4×D47D411yescharacteristic
59C22×C4×D47Q814yes—
60C22×C4×D47C2311yescharacteristic
61C22×C4×D47C2311yescharacteristic
62C2×#10625324—24no—
63C2×#10625324—24no—
64C2×#10625324C2×C418yescomplemented
65C22×C47⋊C8C2312yes—
66#13093982—24no—
67#13093982—24no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—24no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2312yes—
73C2×Dic47⋊C4—28no—
74C2×Dic47⋊C4—22no—
75C2×Dic47⋊C4—22no—
76C4⋊C4×D47—28no—
77C4⋊C4×D47—28no—
78C4⋊C4×D47—24no—
79C4⋊C4×D47—24no—
80C4⋊C4×D47—24no—
81C4⋊C4×D47D418yes—
82C2×C22⋊Dic47—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89#13093958—24no—
90#13093957—24no—
91#13093955—24no—
92C22⋊(C47⋊C8)C2×C414yescomplemented
93(C2×D47)⋊C8C2×C418yescomplemented
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C23×C4×C47—21no—
98C23×C4×C47C2311yescharacteristic
99C22×C8×C47C2312yes—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—24no—
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2312yes—
105C2×C22⋊C4×C47—21no—
106#106×C47—24no—
107C22⋊C8×C47C2×C414yescomplemented
108#95×C47—21no—
109C24×C47C2×D411yescharacteristic
110C23×D47—22no—
111C23×D47—22no—
112C23×D47—22no—
113C23×D47—22no—
114C23×D47—22no—
115C23×D47—22no—
116C23×D47C4⋊C414yes—
117C23×D47C2×D411yescharacteristic
118C23×D47C2×D411yescharacteristic
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—24no—
122C22×C4×C47—22no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—42no—
126C22×C4×C47—42no—
127C22×C4×C47C22×C412yes—
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×Q812yes—
130C22×C4×C47C2411yescharacteristic
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—24no—
134C22×Dic47—22no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—42no—
138C22×Dic47—42no—
139C22×Dic47C4⋊C414yes—
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C2×C4×D47—22no—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—216no—
149C2×C4×D47—216no—
150C2×C4×D47—28no—
151C2×C4×D47—28no—
152C2×C4×D47—432no—
153C2×C4×D47—44no—
154C2×C4×D47—44no—
155C2×C4×D47—44no—
156C2×C4×D47—44no—
157C2×C4×D47C4⋊C414yes—
158C2×C4×D47C4⋊C414yes—
159C2×C4×D47C22×C412yes—
160C2×C4×D47C22×C412yes—
161C2×C4×D47C2×D414yes—
162C2×C4×D47C2×Q812yes—
163C2×C4×D47C2×Q812yes—
164C2×C4×D47#4114yes—
165C2×C8×C47—22no—
166C2×C8×C47—22no—
167C2×C8×C47C22×C414yescomplemented
168C4⋊C4×C47—24no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47C2×D414yes—
173C22⋊C4×C47—42no—
174C22⋊Dic47—42no—
175(C2×D47)⋊C4—42no—
176(C2×D47)⋊C4—42no—
177(C2×D47)⋊C4—42no—
178(C2×D47)⋊C4—42no—
179C4⋊Dic47—24no—
180C4⋊Dic47—22no—
181C4⋊Dic47—22no—
182C4⋊Dic47—22no—
183C4⋊Dic47C2×D414yes—
184Dic47⋊C4—28no—
185Dic47⋊C4—28no—
186C4×Dic47—42no—
187C4×Dic47—42no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8C22×C414yescomplemented
191#10625324—216no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47C2×C4⋊C412yes—
197C23×C47AGL(1,ℤ/8)12yes—
198C23×C47#13712yes—
199C23×C47C22×D411yescharacteristic
200C22×D47—28no—
201C22×D47—28no—
202C22×D47—28no—
203C22×D47—28no—
204C22×D47—21no—
205C22×D47—21no—
206C22×D47—21no—
207C22×D47—21no—
208C22×D47—21no—
209C22×D47—21no—
210C22×D47—21no—
211C22×D47—21no—
212C22×D47—28no—
213C22×D47—28no—
214C22×D47—28no—
215C22×D47C2×C4⋊C412yes—
216C22×D47C2×C4⋊C412yes—
217C22×D47#12012yes—
218C22×D47#12312yes—
219C22×D47AGL(1,ℤ/8)12yes—
220C22×D47#13712yes—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—24no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—48no—
227C2×C4×C47—42no—
228C2×C4×C47—42no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47C2×C4⋊C412yes—
236C2×C4×C47C23×C411yescharacteristic
237C2×C4×C47C22×D412yes—
238C2×C4×C47C22×Q811yescharacteristic
239C2×C4×C47C2×#4112yes—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—28no—
243C2×Dic47—24no—
244C2×Dic47—48no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47C2×C4⋊C412yes—
254C2×Dic47C2×C4⋊C412yes—
255C2×Dic47#12012yes—
256C2×Dic47#12312yes—
257C4×D47—28no—
258C4×D47—24no—
259C4×D47—24no—
260C4×D47—28no—
261C4×D47—416no—
262C4×D47—416no—
263C4×D47C4×D418yes—
264C4×D47C22⋊Q818yes—
265C8×C47—24no—
266C47⋊C8—24no—
267#20844—471nomaximal, Sylow
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—22no—
273C22×C47—22no—
274C22×C47#42814yescomplemented
275C22×C47C22×C4⋊C411yescharacteristic
276C22×C47C2×#12011yescharacteristic
277C22×C47C2×#12311yescharacteristic
278C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C47C2×#13711yescharacteristic
281C22×C47C2×#13711yescharacteristic
282C2×D47—232no—
283C2×D47—22no—
284C2×D47—22no—
285C2×D47—22no—
286C2×D47—22no—
287C2×D47—22no—
288C2×D47—22no—
289C2×D47—22no—
290C2×D47—22no—
291C2×D47—22no—
292C2×D47—22no—
293C2×D47—22no—
294C2×D47—22no—
295C2×D47#38514yes—
296C2×D47#42814yescomplemented
297C4×C47—22no—
298C4×C47—44no—
299C4×C47—81no—
300C4×C47—81no—
301C4×C47C2×C4×D412yes—
302C4×C47C2×C22⋊Q812yes—
303Dic47—22no—
304Dic47—44no—
305Dic47—81no—
306Dic47—81no—
307Dic47#38514yes—
308C23×C4⋊C4—471no—
309C2×#428—472no—
310C2×#407—471no—
311#2280—471no—
312#1853—472no—
313#1853—471no—
314#1853D4711yescharacteristic, complemented
315#1852—471no—
316#1852—472no—
317#1852—471no—
318#1736—472no—
319C2×C47—24no—
320C2×C47C2×#38511yescharacteristic
321C2×C47C2×#42811yescharacteristic
322C2×C47C2×#42811yescharacteristic
323C2×C47#300311yescharacteristic
324C2×C47#300411yescharacteristic
325C2×C47#309211yescharacteristic
326C2×C47#309211yescharacteristic
327D47—24no—
328D47—24no—
329C24×C4—471no—
330C24×C4—941no—
331C22×M16—472no—
332C2×#117—471no—
333C2×#117—471no—
334C22×C4⋊C4—472no—
335C22×C4⋊C4—471no—
336C22×C4⋊C4—472no—
337C22×C4⋊C4—948no—
338C22×C4⋊C4—941no—
339C22×C4⋊C4—941no—
340C22×C4⋊C4—941no—
341C22×C4⋊C4C2×D4711yescharacteristic
342#428—948no—
343C2×#107—472no—
344C2×#107—472no—
345C2×#106—472no—
346C2×#106C2×D4712yescomplemented
347#407—478no—
348C2×C22⋊C8—471no—
349C2×C22⋊C8—472no—
350C2×C22⋊C8C2×D4711yescharacteristic
351#381—941no—
352#380—471no—
353#380—471no—
354#380—471no—
355#380C2×D4711yescharacteristic
356#379—941no—
357C4×C22⋊C4—941no—
358C4×C22⋊C4—941no—
359#340—472no—
360#340—472no—
361#340—472no—
362#340C2×D4712yescomplemented
363C47#2084411yescharacteristic, complemented, Sylow
364C25—471no—
365C23×C4—21no—
366C23×C4—471no—
367C23×C4—471no—
368C23×C4—474no—
369C23×C4—471no—
370C23×C4—471no—
371C23×C4—474no—
372C23×C4—471no—
373C23×C4—944no—
374C23×C4—944no—
375C23×C4—942no—
376C23×C4—942no—
377C23×C4—948no—
378C23×C4—941no—
379C23×C4—1884no—
380C23×C4—1884no—
381C23×C4C22×D4711yescharacteristic
382C2×M16—478no—
383C2×M16—944no—
384C2×M16—944no—
385C22×C8—472no—
386C22×C8C22×D4712yes—
387#117—944no—
388#117—944no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—472no—
392C2×C4⋊C4—478no—
393C2×C4⋊C4—471no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—944no—
396C2×C4⋊C4—944no—
397C2×C4⋊C4—948no—
398C2×C4⋊C4—948no—
399C2×C4⋊C4—944no—
400C2×C4⋊C4—942no—
401C2×C4⋊C4—942no—
402C2×C4⋊C4—948no—
403C2×C4⋊C4—941no—
404C2×C4⋊C4—944no—
405C2×C4⋊C4C22×D4711yescharacteristic
406C2×C4⋊C4C22×D4711yescharacteristic
407C2×C4⋊C4C22×D4712yes—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—941no—
410C2×C22⋊C4—941no—
411C2×C22⋊C4—941no—
412C2×C22⋊C4—941no—
413C2×C22⋊C4—941no—
414C2×C42—941no—
415C2×C42—941no—
416#107—944no—
417#107—944no—
418#106—24no—
419#106—944no—
420C22⋊C8—474no—
421C22⋊C8—478no—
422C22⋊C8C4×D4714yescomplemented
423#95—21no—
424#95—941no—
425#95—941no—
426#95—941no—
427C24—471no—
428C24—471no—
429C24—474no—
430C24—942no—
431C24—942no—
432C24—942no—
433C24—942no—
434C24—942no—
435C24—942no—
436C24D4×D4711yescharacteristic
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—471no—
446C22×C4—472no—
447C22×C4—474no—
448C22×C4—472no—
449C22×C4—474no—
450C22×C4—474no—
451C22×C4—471no—
452C22×C4—474no—
453C22×C4—474no—
454C22×C4—472no—
455C22×C4—472no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—9416no—
459C22×C4—9416no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—948no—
463C22×C4—948no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—944no—
469C22×C4—942no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4—1884no—
473C22×C4—18832no—
474C22×C4—1884no—
475C22×C4—1884no—
476C22×C4—1884no—
477C22×C4—1882no—
478C22×C4—1882no—
479C22×C4C2×C4×D4712yes—
480C22×C4D4×D4711yescharacteristic
481C22×C4Q8×D4712yes—
482C22×C4C23×D4711yescharacteristic
483M16—9416no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—474no—
487C2×C8—942no—
488C2×C8—942no—
489C2×C8C2×C4×D4714yescomplemented
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—474no—
495C4⋊C4—948no—
496C4⋊C4—948no—
497C4⋊C4—944no—
498C4⋊C4—942no—
499C4⋊C4—942no—
500C4⋊C4—942no—
501C4⋊C4D4×D4714yes—
502C22⋊C4—42no—
503C22⋊C4—1882no—
504C22⋊C4—1882no—
505C22⋊C4—1882no—
506C22⋊C4—1882no—
507C22⋊C4—1882no—
508C42—1882no—
509C42—1882no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—472no—
515C23—472no—
516C23—472no—
517C23—472no—
518C23—472no—
519C23—472no—
520C23—948no—
521C23—948no—
522C23—948no—
523C23—948no—
524C23—948no—
525C23—948no—
526C23—941no—
527C23—941no—
528C23—941no—
529C23—941no—
530C23—941no—
531C23—941no—
532C23—941no—
533C23—941no—
534C23—948no—
535C23C4⋊C4×D4712yes—
536C23#1309405112yes—
537C23#1309405312yes—
538C23C2×D4×D4711yescharacteristic
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—472no—
554C2×C4—472no—
555C2×C4—472no—
556C2×C4—478no—
557C2×C4—478no—
558C2×C4—472no—
559C2×C4—948no—
560C2×C4—948no—
561C2×C4—948no—
562C2×C4—941no—
563C2×C4—941no—
564C2×C4—944no—
565C2×C4—944no—
566C2×C4—944no—
567C2×C4—18816no—
568C2×C4—18816no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1888no—
576C2×C4—1882no—
577C2×C4—1882no—
578C2×C4C4⋊C4×D4712yes—
579C2×C4C22×C4×D4711yescharacteristic
580C2×C4C2×D4×D4712yes—
581C2×C4C2×Q8×D4711yescharacteristic
582C2×C4#41×D4712yes—
583C8—24no—
584C8—944no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—474no—
592C22—474no—
593C22—9432no—
594C22—942no—
595C22—942no—
596C22—942no—
597C22—942no—
598C22—942no—
599C22—942no—
600C22—942no—
601C22—942no—
602C22—942no—
603C22—942no—
604C22—942no—
605C22—942no—
606C22#1568916214yescomplemented
607C22C2×C4⋊C4×D4711yescharacteristic
608C22#120×D4711yescharacteristic
609C22#123×D4711yescharacteristic
610C22AGL(1,ℤ/8)×D4711yescharacteristic
611C22C2×#1309405111yescharacteristic
612C22#137×D4711yescharacteristic
613C22C2×#1309405311yescharacteristic
614C4—22no—
615C4—44no—
616C4—81no—
617C4—81no—
618C4—474no—
619C4—942no—
620C4—3761no—
621C4—3761no—
622C4—1884no—
623C4C4×D4×D4712yes—
624C4C22⋊Q8×D4712yes—
625C2—24no—
626C2—944no—
627C2—944no—
628C2#385×D4711yescharacteristic
629C2#428×D4711yescharacteristic
630C2C2×#1568916211yescharacteristic
631C2#2025261611yescharacteristic
632C2#2025261911yescharacteristic
633C2#2025331011yescharacteristic
634C2#2025331211yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1853C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1853D47
hypocenterC47#20844
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#15689162C2semidirect
5. #20243580C2semidirect
6. #20243579C2semidirect
7. #20243578C2semidirect
8. #20243577C2semidirect
9. #20243576C2semidirect
10. #20243106C2semidirect
11. C47×#1853C2semidirect
25. C2×#13093958C22semidirect
26. C2×#13093957C22semidirect
27. C2×#13093955C22semidirect
28. #15689085C4semidirect
38. #15688886C22semidirect
39. #15688885C22semidirect
40. #15688885C22semidirect
42. C2×#106×C47C22semidirect
45. C47×#340C22semidirect
64. C2×#10625324C2×C4semidirect
92. C22⋊(C47⋊C8)C2×C4semidirect
93. (C2×D47)⋊C8C2×C4semidirect
107. C22⋊C8×C47C2×C4semidirect
167. C2×C8×C47C22×C4semidirect
190. C2×C47⋊C8C22×C4semidirect
274. C22×C47#428semidirect
296. C2×D47#428semidirect
314. #1853D47semidirect
346. C2×#106C2×D47semidirect
362. #340C2×D47semidirect
363. C47#20844semidirect
422. C22⋊C8C4×D47semidirect
489. C2×C8C2×C4×D47semidirect
606. C22#15689162semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D472
C2×#156891622
C2×#156890852
#202457702
#202435802
#202435792
#202435782
#202435772
#202435762
#202431062
C47×#18532
#2084447