Finite Group Databasebeta

← #26217523

Subgroups — #26217523

Order 12032.

2278 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093995C211yesmaximal, characteristic
2C2×#117×D47C211yesmaximal, characteristic, complemented
3C2×#15689128C211yesmaximal, characteristic, complemented
4C2×#15689086C211yesmaximal, characteristic, complemented
5#20243707C214yesmaximal, complemented
6#20243702C214yesmaximal, complemented
7#20243109C211yesmaximal, characteristic, complemented
8#20243108C211yesmaximal, characteristic, complemented
9C2×C47×#418C211yesmaximal, characteristic, complemented
10C23×C22⋊D47—21no—
11C23×C4×D47C2211yescharacteristic
12C22×#10625329C2211yescharacteristic
13C2×#13093995—24no—
14C2×#13093995—21no—
15C2×#13093995—21no—
16C2×#13093995C418yescomplemented
17C2×#13093995C2212yes—
18C2×#13093995C2212yes—
19C2×#13093983C2211yescharacteristic
20C2×#13093982C2212yescomplemented
21C2×#13093981—21no—
22C2×#13093980—21no—
23#117×D47—24no—
24#117×D47—24no—
25C22×C4⋊Dic47C2211yescharacteristic
26C2×C4⋊C4×D47C2212yescomplemented
27C22×C22⋊Dic47—21no—
28C2×C22⋊C4×D47—21no—
29C2×C42×D47—21no—
30#15689128—24no—
31C2×#13093945C2214yescomplemented
32#15689086—24no—
33C2×(C2×D47)⋊C8C2212yescomplemented
34C2×(C2×D47)⋊C8C2212yescomplemented
35#15688886C2214yescomplemented
36#15688885C2214yescomplemented
37C23×D4×C47C2211yescharacteristic
38C22×M16×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C47×#418—24no—
41C2×#102×C47C2214yescomplemented
42C24×D47D411yescharacteristic
43C22×C22⋊D47—28no—
44C22×C22⋊D47—22no—
45C22×C22⋊D47—22no—
46C22×C22⋊D47—48no—
47C23×Dic47—21no—
48C23×Dic47D411yescharacteristic
49C22×C4×D47—22no—
50C22×C4×D47—22no—
51C22×C4×D47—22no—
52C22×C4×D47—22no—
53C22×C4×D47—22no—
54C22×C4×D47C2×C414yes—
55C22×C4×D47D411yescharacteristic
56C22×C4×D47D411yescharacteristic
57C22×C4×D47D414yes—
58C22×C4×D47C2311yescharacteristic
59C22×C4×D47C2311yescharacteristic
60C2×#10625329—22no—
61C2×#10625329—22no—
62C2×#10625329—22no—
63C22×C47⋊C8C2312yes—
64#13093995—28no—
65#13093995—28no—
66#13093995—24no—
67#13093995—24no—
68#13093995C2×C418yes—
69#13093995C2×C418yescomplemented
70#13093983—22no—
71#13093983D414yescomplemented
72#13093982—24no—
73#13093982—24no—
74#13093981—28no—
75#13093980—28no—
76C2×C4⋊Dic47—21no—
77C2×C4⋊Dic47—22no—
78C2×C4⋊Dic47C2×C414yescomplemented
79C2×C4⋊Dic47C2312yes—
80C2×C4⋊Dic47C2312yes—
81C2×Dic47⋊C4—22no—
82C4⋊C4×D47—24no—
83C4⋊C4×D47—24no—
84C2×C22⋊Dic47—21no—
85C2×C22⋊Dic47—28no—
86C2×C22⋊Dic47—22no—
87C2×C22⋊Dic47—22no—
88C2×(C2×D47)⋊C4—22no—
89C22⋊C4×D47—28no—
90C2×C4×Dic47—21no—
91C2×C4×Dic47—22no—
92C42×D47—28no—
93#13093945—28no—
94(C2×D47)⋊C8—24no—
95(C2×D47)⋊C8—24no—
96C25×C47—21no—
97C22×D4×C47—22no—
98C22×D4×C47—22no—
99C22×D4×C47—21no—
100C22×D4×C47—22no—
101C22×D4×C47—22no—
102C22×D4×C47—22no—
103C22×D4×C47C2×C414yescomplemented
104C22×D4×C47C2312yes—
105C23×C4×C47C2311yescharacteristic
106C2×M16×C47—22no—
107C2×M16×C47—22no—
108C2×M16×C47—22no—
109C22×C8×C47C2312yes—
110#117×C47—22no—
111#117×C47D414yescomplemented
112C2×C4⋊C4×C47C2312yes—
113C2×C22⋊C4×C47—21no—
114C2×C42×C47—21no—
115#102×C47—28no—
116C24×C47—24no—
117C24×C47—22no—
118C24×C47—44no—
119C24×C47—44no—
120C24×C47—42no—
121C24×C47—42no—
122C24×C47C2×D411yescharacteristic
123C23×D47—22no—
124C23×D47—22no—
125C23×D47—24no—
126C23×D47—42no—
127C23×D47C22⋊C414yes—
128C23×D47C2×D411yescharacteristic
129C23×D47C2×D411yescharacteristic
130C2×D4×C47—24no—
131C2×D4×C47—28no—
132C2×D4×C47—28no—
133C2×D4×C47—22no—
134C2×D4×C47—22no—
135C2×D4×C47—22no—
136C2×D4×C47—22no—
137C2×D4×C47—22no—
138C2×D4×C47—42no—
139C2×D4×C47—44no—
140C2×D4×C47—44no—
141C2×D4×C47—42no—
142C2×D4×C47C22×C414yes—
143C22×C4×C47—22no—
144C22×C4×C47C22×C412yes—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D412yes—
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×D412yes—
149C22×C4×C47#4112yes—
150C22×C4×C47#4112yes—
151C22×C4×C47C2411yescharacteristic
152C2×C22⋊D47—28no—
153C2×C22⋊D47—28no—
154C2×C22⋊D47—432no—
155C2×C22⋊D47—48no—
156C2×C22⋊D47—48no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—24no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—44no—
163C22×Dic47C22⋊C414yes—
164C22×Dic47C2×D411yescharacteristic
165C22×Dic47C2×D411yescharacteristic
166C2×C4×D47—28no—
167C2×C4×D47—28no—
168C2×C4×D47—22no—
169C2×C4×D47—22no—
170C2×C4×D47—28no—
171C2×C4×D47—28no—
172C2×C4×D47—22no—
173C2×C4×D47—22no—
174C2×C4×D47—21no—
175C2×C4×D47—21no—
176C2×C4×D47—21no—
177C2×C4×D47—21no—
178C2×C4×D47—21no—
179C2×C4×D47—21no—
180C2×C4×D47—21no—
181C2×C4×D47—21no—
182C2×C4×D47—28no—
183C2×C4×D47—44no—
184C2×C4×D47—44no—
185C2×C4×D47C22⋊C414yes—
186C2×C4×D47C22⋊C414yes—
187C2×C4×D47C22×C412yes—
188C2×C4×D47C22×C412yes—
189C2×C4×D47C2×D412yes—
190C2×C4×D47C2×D412yes—
191M16×C47—44no—
192C2×C8×C47—22no—
193C2×C8×C47—22no—
194C2×C8×C47—22no—
195C4⋊C4×C47—22no—
196C4⋊C4×C47C2×D414yescomplemented
197C22⋊C4×C47—24no—
198C42×C47—24no—
199C22⋊Dic47—24no—
200C22⋊Dic47—28no—
201C22⋊Dic47—28no—
202(C2×D47)⋊C4—28no—
203C4⋊Dic47—24no—
204C4⋊Dic47—22no—
205C4⋊Dic47—22no—
206C4⋊Dic47C22×C414yes—
207C4⋊Dic47C2×D414yescomplemented
208Dic47⋊C4—28no—
209C4×Dic47—28no—
210C4×Dic47—24no—
211#10625329—44no—
212C2×C47⋊C8—22no—
213C2×C47⋊C8—22no—
214C2×C47⋊C8—22no—
215C23×C47—22no—
216C23×C47—22no—
217C23×C47—24no—
218C23×C47—48no—
219C23×C47—48no—
220C23×C47—42no—
221C23×C47—42no—
222C23×C47—44no—
223C23×C47—44no—
224C23×C47—48no—
225C23×C47—88no—
226C23×C47—88no—
227C23×C47C2×C22⋊C412yes—
228C23×C47#12012yes—
229C23×C47#12312yes—
230C23×C47C22×D411yescharacteristic
231C22×D47—21no—
232C22×D47—21no—
233C22×D47—22no—
234C22×D47—21no—
235C22×D47—21no—
236C22×D47—22no—
237C22×D47—22no—
238C22×D47—28no—
239C22×D47—28no—
240C22×D47—44no—
241C22×D47—48no—
242C22×D47—48no—
243C22×D47C2×C22⋊C412yes—
244C22×D47C2×C22⋊C412yes—
245C22×D47AGL(1,ℤ/8)14yes—
246D4×C47—24no—
247D4×C47—24no—
248D4×C47—42no—
249D4×C47—42no—
250D4×C47—48no—
251C2×C4×C47—24no—
252C2×C4×C47—42no—
253C2×C4×C47—42no—
254C2×C4×C47C2×C22⋊C412yes—
255C2×C4×C47C4×D414yes—
256C2×C4×C47C4×D414yes—
257C2×C4×C47#12012yes—
258C2×C4×C47#12114yes—
259C2×C4×C47#12114yes—
260C2×C4×C47#12312yes—
261C2×C4×C47C23×C411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47C2×#4111yescharacteristic
267C22⋊D47—416no—
268C22⋊D47—416no—
269C2×Dic47—21no—
270C2×Dic47—21no—
271C2×Dic47—21no—
272C2×Dic47—21no—
273C2×Dic47—28no—
274C2×Dic47—24no—
275C2×Dic47—24no—
276C2×Dic47—48no—
277C2×Dic47—42no—
278C2×Dic47—44no—
279C2×Dic47—42no—
280C2×Dic47C2×C22⋊C412yes—
281C2×Dic47C2×C22⋊C412yes—
282C4×D47—22no—
283C4×D47—22no—
284C4×D47—22no—
285C4×D47—22no—
286C4×D47—22no—
287C4×D47—22no—
288C4×D47—22no—
289C4×D47—22no—
290C4×D47—28no—
291C4×D47—28no—
292C4×D47—416no—
293C8×C47—42no—
294C47⋊C8—42no—
295#20912—471nomaximal, Sylow
296C22×C47—24no—
297C22×C47—22no—
298C22×C47—22no—
299C22×C47—24no—
300C22×C47—44no—
301C22×C47—44no—
302C22×C47—44no—
303C22×C47—816no—
304C22×C47—84no—
305C22×C47—84no—
306C22×C47#38614yes—
307C22×C47C22×C22⋊C411yescharacteristic
308C22×C47C2×#12011yescharacteristic
309C22×C47C2×#12311yescharacteristic
310C22×C47C2×AGL(1,ℤ/8)12yes—
311C22×C47#57512yes—
312C2×D47—24no—
313C2×D47—22no—
314C2×D47—24no—
315C2×D47—22no—
316C2×D47—22no—
317C2×D47—22no—
318C2×D47—22no—
319C2×D47—416no—
320C2×D47—42no—
321C2×D47—42no—
322C2×D47#41814yescomplemented
323C4×C47—44no—
324C4×C47#38614yes—
325C4×C47C2×C4×D411yescharacteristic
326C4×C47C2×C4×D411yescharacteristic
327C4×C47C2×#12111yescharacteristic
328C4×C47C2×#12111yescharacteristic
329Dic47—22no—
330Dic47—22no—
331Dic47—44no—
332Dic47—44no—
333C22×#121—471no—
334C22×#117—471no—
335C2×#418—471no—
336C2×#418D4711yescharacteristic, complemented
337C2×#407—471no—
338#1875—474no—
339#1874—474no—
340#1736—471no—
341#1736—471no—
342C2×C47—24no—
343C2×C47—84no—
344C2×C47C2×#38611yescharacteristic
345C2×C47C2×#41811yescharacteristic
346C2×C47#289611yescharacteristic
347C2×C47#300612yes—
348C2×C47#309412yes—
349D47—24no—
350D47—42no—
351C23×D4—941no—
352C23×D4C2×D4711yescharacteristic
353C24×C4—471no—
354C22×M16—471no—
355C22×M16C2×D4711yescharacteristic
356C2×#121—478no—
357C2×#121—472no—
358C2×#121—472no—
359C2×#121—941no—
360C2×#121—944no—
361C2×#121—941no—
362C2×#117—471no—
363C2×#117—472no—
364C2×#117—941no—
365C2×#117—941no—
366C2×#117—944no—
367C2×#117—944no—
368C2×#117C2×D4711yescharacteristic
369C22×C4⋊C4—471no—
370C22×C4⋊C4—472no—
371C22×C22⋊C4—941no—
372C22×C22⋊C4—941no—
373C22×C42—941no—
374#418—24no—
375#418—944no—
376C2×#102—474no—
377C2×#102C2×D4714yescomplemented
378#407—944no—
379C2×C22⋊C8—472no—
380C2×C22⋊C8—472no—
381#340—474no—
382#340—474no—
383C47#2091211yescharacteristic, complemented, Sylow
384C25—21no—
385C25—471no—
386C22×D4—22no—
387C22×D4—22no—
388C22×D4—21no—
389C22×D4—22no—
390C22×D4—22no—
391C22×D4—22no—
392C22×D4—948no—
393C22×D4—942no—
394C22×D4—942no—
395C22×D4—1888no—
396C22×D4C4×D4714yescomplemented
397C22×D4C22×D4712yes—
398C23×C4—471no—
399C23×C4—474no—
400C23×C4—471no—
401C23×C4—474no—
402C23×C4—471no—
403C23×C4—471no—
404C23×C4—471no—
405C23×C4—942no—
406C23×C4—942no—
407C23×C4—942no—
408C23×C4—942no—
409C23×C4—942no—
410C23×C4—941no—
411C23×C4C22×D4711yescharacteristic
412C2×M16—22no—
413C2×M16—22no—
414C2×M16—22no—
415C2×M16—942no—
416C2×M16—942no—
417C2×M16—942no—
418C22×C8—472no—
419C22×C8C22×D4712yes—
420#121—478no—
421#121—478no—
422#121—948no—
423#121—948no—
424#121—944no—
425#121—944no—
426#117—22no—
427#117—474no—
428#117—948no—
429#117—948no—
430#117—944no—
431#117—942no—
432#117—944no—
433#117C22⋊D4714yescomplemented
434C2×C4⋊C4—474no—
435C2×C4⋊C4—472no—
436C2×C4⋊C4—472no—
437C2×C4⋊C4—944no—
438C2×C4⋊C4—942no—
439C2×C4⋊C4—941no—
440C2×C4⋊C4—942no—
441C2×C4⋊C4—944no—
442C2×C4⋊C4C22×D4712yes—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—948no—
445C2×C22⋊C4—942no—
446C2×C22⋊C4—948no—
447C2×C22⋊C4—942no—
448C2×C22⋊C4—942no—
449C2×C22⋊C4—941no—
450C2×C42—21no—
451C2×C42—948no—
452C2×C42—942no—
453C2×C42—941no—
454#102—28no—
455#102—948no—
456C22⋊C8—944no—
457C22⋊C8—944no—
458C24—24no—
459C24—22no—
460C24—44no—
461C24—44no—
462C24—42no—
463C24—42no—
464C24—471no—
465C24—471no—
466C24—474no—
467C24—944no—
468C24—942no—
469C24—942no—
470C24—1882no—
471C24D4×D4711yescharacteristic
472C2×D4—24no—
473C2×D4—28no—
474C2×D4—28no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—42no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—42no—
484C2×D4—948no—
485C2×D4—948no—
486C2×D4—1888no—
487C2×D4—18832no—
488C2×D4—1888no—
489C2×D4C2×C4×D4714yes—
490C22×C4—22no—
491C22×C4—474no—
492C22×C4—471no—
493C22×C4—471no—
494C22×C4—474no—
495C22×C4—474no—
496C22×C4—472no—
497C22×C4—472no—
498C22×C4—472no—
499C22×C4—472no—
500C22×C4—948no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4—948no—
504C22×C4—948no—
505C22×C4—942no—
506C22×C4—942no—
507C22×C4—948no—
508C22×C4—948no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—941no—
512C22×C4—941no—
513C22×C4—941no—
514C22×C4—941no—
515C22×C4—941no—
516C22×C4—941no—
517C22×C4—941no—
518C22×C4—941no—
519C22×C4—944no—
520C22×C4—942no—
521C22×C4—942no—
522C22×C4—1884no—
523C22×C4—1884no—
524C22×C4—1884no—
525C22×C4C2×C4×D4712yes—
526C22×C4#1062535012yes—
527C22×C4D4×D4711yescharacteristic
528C22×C4D4×D4712yes—
529C22×C4D4×D4712yes—
530C22×C4#1062535212yes—
531C22×C4C2×C22⋊D4712yes—
532C22×C4C23×D4711yescharacteristic
533M16—44no—
534M16—1884no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—942no—
539C2×C8—942no—
540C2×C8—942no—
541C4⋊C4—22no—
542C4⋊C4—474no—
543C4⋊C4—474no—
544C4⋊C4—948no—
545C4⋊C4—944no—
546C4⋊C4—942no—
547C4⋊C4—942no—
548C4⋊C4C2×C22⋊D4714yescomplemented
549C22⋊C4—24no—
550C22⋊C4—948no—
551C22⋊C4—948no—
552C22⋊C4—948no—
553C22⋊C4—944no—
554C42—24no—
555C42—948no—
556C42—944no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—88no—
568C23—88no—
569C23—472no—
570C23—474no—
571C23—472no—
572C23—948no—
573C23—941no—
574C23—941no—
575C23—942no—
576C23—948no—
577C23—941no—
578C23—941no—
579C23—942no—
580C23—942no—
581C23—1884no—
582C23—1888no—
583C23—1888no—
584C23C22⋊C4×D4712yes—
585C23#1309399212yes—
586C23#1309400612yes—
587C23C2×D4×D4711yescharacteristic
588D4—24no—
589D4—24no—
590D4—48no—
591D4—42no—
592D4—42no—
593D4—18816no—
594D4—18816no—
595C2×C4—24no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—472no—
599C2×C4—472no—
600C2×C4—948no—
601C2×C4—941no—
602C2×C4—941no—
603C2×C4—941no—
604C2×C4—941no—
605C2×C4—948no—
606C2×C4—948no—
607C2×C4—942no—
608C2×C4—942no—
609C2×C4—942no—
610C2×C4—942no—
611C2×C4—942no—
612C2×C4—942no—
613C2×C4—942no—
614C2×C4—942no—
615C2×C4—944no—
616C2×C4—944no—
617C2×C4—18816no—
618C2×C4—1888no—
619C2×C4—1882no—
620C2×C4—1884no—
621C2×C4—1882no—
622C2×C4C22⋊C4×D4712yes—
623C2×C4(C22⋊D47)⋊C414yes—
624C2×C4C4×C22⋊D4714yes—
625C2×C4#1309399212yes—
626C2×C4#1309399514yes—
627C2×C4#1309399714yes—
628C2×C4#1309400612yes—
629C2×C4C22×C4×D4711yescharacteristic
630C2×C4C2×D4×D4711yescharacteristic
631C2×C4C2×D4×D4711yescharacteristic
632C2×C4C22×C22⋊D4711yescharacteristic
633C2×C4C2×#1062535211yescharacteristic
634C2×C4C2×#1062535011yescharacteristic
635C8—42no—
636C8—1882no—
637C22—24no—
638C22—24no—
639C22—22no—
640C22—22no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—816no—
645C22—84no—
646C22—84no—
647C22—474no—
648C22—942no—
649C22—944no—
650C22—944no—
651C22—942no—
652C22—942no—
653C22—942no—
654C22—942no—
655C22—18816no—
656C22—1882no—
657C22—1882no—
658C22#1568899914yes—
659C22C2×C22⋊C4×D4711yescharacteristic
660C22C2×#1309399211yescharacteristic
661C22C2×#1309400611yescharacteristic
662C22AGL(1,ℤ/8)×D4712yes—
663C22#1568991912yes—
664C4—44no—
665C4—942no—
666C4—942no—
667C4—1884no—
668C4—1884no—
669C4#1568899914yes—
670C4C2×(C22⋊D47)⋊C411yescharacteristic
671C4C2×C4×C22⋊D4711yescharacteristic
672C4C2×#1309399511yescharacteristic
673C4C2×#1309399711yescharacteristic
674C2—24no—
675C2—84no—
676C2—944no—
677C2—1882no—
678C2C2×#1568899911yescharacteristic
679C2#418×D4711yescharacteristic
680C2#2025190211yescharacteristic
681C2#2025263612yes—
682C2#2025332612yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#418C2
socleC23×C47C22×D4
prime radicalC22×#13093995C2
cosocleC22×C4×C47C24
hypercenterC2×#418D47
hypocenterC47#20912
LCSG ▷ C2×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×#117×D47C2semidirect
3. C2×#15689128C2semidirect
4. C2×#15689086C2semidirect
5. #20243707C2semidirect
6. #20243702C2semidirect
7. #20243109C2semidirect
8. #20243108C2semidirect
9. C2×C47×#418C2semidirect
16. C2×#13093995C4semidirect
20. C2×#13093982C22semidirect
26. C2×C4⋊C4×D47C22semidirect
31. C2×#13093945C22semidirect
33. C2×(C2×D47)⋊C8C22semidirect
34. C2×(C2×D47)⋊C8C22semidirect
35. #15688886C22semidirect
36. #15688885C22semidirect
41. C2×#102×C47C22semidirect
69. #13093995C2×C4semidirect
71. #13093983D4semidirect
78. C2×C4⋊Dic47C2×C4semidirect
103. C22×D4×C47C2×C4semidirect
111. #117×C47D4semidirect
196. C4⋊C4×C47C2×D4semidirect
207. C4⋊Dic47C2×D4semidirect
322. C2×D47#418semidirect
336. C2×#418D47semidirect
377. C2×#102C2×D47semidirect
383. C47#20912semidirect
396. C22×D4C4×D47semidirect
433. #117C22⋊D47semidirect
548. C4⋊C4C2×C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130939952
C2×#117×D472
C2×#156891282
C2×#156890862
#202437072
#202437022
#202431092
#202431082
C2×C47×#4182
#2091247