Finite Group Databasebeta

← #26217025

Subgroups — #26217025

Order 12032.

1630 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093982C211yesmaximal, characteristic
2C2×#15689162C212yesmaximal, complemented
3C2×#15689085C211yesmaximal, characteristic
4#380×D47C211yesmaximal, characteristic
5#20243590C212yesmaximal, complemented
6#20243586C212yesmaximal, complemented
7#20243581C211yesmaximal, characteristic, complemented
8#20243578C211yesmaximal, characteristic, complemented
9#20243575C211yesmaximal, characteristic, complemented
10#20243106C212yesmaximal, complemented
11C47×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C22×#10625324C2212yes—
14C2×#13093982—21no—
15C2×#13093982—21no—
16C2×#13093982—28no—
17C2×#13093982C2212yes—
18C2×#13093982C2212yes—
19C22×C4⋊Dic47C2211yescharacteristic
20C2×C4⋊C4×D47C2211yescharacteristic
21C2×C4⋊C4×D47C2211yescharacteristic
22C22×(C2×D47)⋊C4—21no—
23C2×C22⋊C4×D47—21no—
24C22×C4×Dic47—21no—
25#15689162—28no—
26C2×#13093958C2212yescomplemented
27C2×#13093957C2212yescomplemented
28C2×#13093955C2212yescomplemented
29#15689085C418yescomplemented
30C2×C22⋊(C47⋊C8)C2211yescharacteristic
31C2×(C2×D47)⋊C8C2212yes—
32#15688983C2211yescharacteristic
33#15688982—21no—
34#15688981—21no—
35#15688980C2211yescharacteristic
36#15688980C2211yescharacteristic
37#95×D47—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—21no—
52C22×C4×D47—21no—
53C22×C4×D47C2×C414yes—
54C22×C4×D47D411yescharacteristic
55C22×C4×D47D411yescharacteristic
56C22×C4×D47Q814yes—
57C22×C4×D47C2311yescharacteristic
58C22×C4×D47C2311yescharacteristic
59C2×#10625324—24no—
60C2×#10625324—24no—
61C2×#10625324C2×C418yescomplemented
62C22×C47⋊C8C2312yes—
63#13093982—28no—
64#13093982—28no—
65#13093982—24no—
66#13093982—24no—
67#13093982—24no—
68#13093982D418yes—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2312yes—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C4⋊C4×D47—24no—
77C4⋊C4×D47—24no—
78C2×C22⋊Dic47—21no—
79C2×(C2×D47)⋊C4—28no—
80C2×(C2×D47)⋊C4—22no—
81C2×(C2×D47)⋊C4—22no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C22⋊C4×D47—44no—
85C2×C4×Dic47—28no—
86C2×C4×Dic47—22no—
87C2×C4×Dic47—22no—
88#13093958—24no—
89#13093957—24no—
90#13093955—24no—
91C22⋊(C47⋊C8)C2×C414yescomplemented
92(C2×D47)⋊C8C2×C418yescomplemented
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C23×C4×C47—21no—
97C23×C4×C47C2311yescharacteristic
98C22×C8×C47C2312yes—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—24no—
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2312yes—
104C2×C22⋊C4×C47—21no—
105#106×C47—24no—
106C22⋊C8×C47C2×C414yescomplemented
107#95×C47—21no—
108C24×C47C2×D411yescharacteristic
109C23×D47—22no—
110C23×D47—22no—
111C23×D47—22no—
112C23×D47—22no—
113C23×D47—41no—
114C23×D47—41no—
115C23×D47C4⋊C414yes—
116C23×D47C2×D411yescharacteristic
117C23×D47C2×D411yescharacteristic
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—24no—
121C22×C4×C47—22no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—42no—
125C22×C4×C47—42no—
126C22×C4×C47C22×C412yes—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×Q812yes—
129C22×C4×C47C2411yescharacteristic
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—24no—
133C22×Dic47—22no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—42no—
137C22×Dic47—42no—
138C22×Dic47C4⋊C414yes—
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C2×C4×D47—22no—
142C2×C4×D47—22no—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—216no—
148C2×C4×D47—216no—
149C2×C4×D47—42no—
150C2×C4×D47—42no—
151C2×C4×D47—42no—
152C2×C4×D47—42no—
153C2×C4×D47—42no—
154C2×C4×D47—42no—
155C2×C4×D47C4⋊C414yes—
156C2×C4×D47C4⋊C414yes—
157C2×C4×D47C22×C412yes—
158C2×C4×D47C22×C412yes—
159C2×C4×D47C2×D414yes—
160C2×C4×D47C2×Q812yes—
161C2×C4×D47C2×Q812yes—
162C2×C4×D47#4114yes—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C2×C8×C47C22×C414yes—
166C4⋊C4×C47—24no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C4⋊C4×C47—22no—
170C4⋊C4×C47C2×D414yes—
171C22⋊C4×C47—42no—
172C22⋊Dic47—42no—
173(C2×D47)⋊C4—28no—
174(C2×D47)⋊C4—28no—
175(C2×D47)⋊C4—42no—
176(C2×D47)⋊C4—42no—
177C4⋊Dic47—24no—
178C4⋊Dic47—22no—
179C4⋊Dic47—22no—
180C4⋊Dic47—22no—
181C4⋊Dic47C2×D414yes—
182Dic47⋊C4—42no—
183Dic47⋊C4—42no—
184C4×Dic47—28no—
185C4×Dic47—28no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8C22×C414yes—
189#10625324—216no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47C2×C4⋊C412yes—
195C23×C47AGL(1,ℤ/8)12yes—
196C23×C47#13712yes—
197C23×C47C22×D411yescharacteristic
198C22×D47—21no—
199C22×D47—21no—
200C22×D47—21no—
201C22×D47—21no—
202C22×D47—21no—
203C22×D47—21no—
204C22×D47—21no—
205C22×D47—21no—
206C22×D47—21no—
207C22×D47—21no—
208C22×D47—28no—
209C22×D47—28no—
210C22×D47—44no—
211C22×D47—44no—
212C22×D47—44no—
213C22×D47—44no—
214C22×D47—44no—
215C22×D47C2×C4⋊C412yes—
216C22×D47C2×C4⋊C412yes—
217C22×D47#12012yes—
218C22×D47#12312yes—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—24no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—48no—
225C2×C4×C47—42no—
226C2×C4×C47—42no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47C2×C4⋊C412yes—
234C2×C4×C47C23×C411yescharacteristic
235C2×C4×C47C22×D412yes—
236C2×C4×C47C22×Q811yescharacteristic
237C2×C4×C47C2×#4112yes—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—28no—
241C2×Dic47—24no—
242C2×Dic47—48no—
243C2×Dic47—42no—
244C2×Dic47—42no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47C2×C4⋊C412yes—
252C2×Dic47C2×C4⋊C412yes—
253C2×Dic47#12012yes—
254C2×Dic47#12312yes—
255C4×D47—28no—
256C4×D47—24no—
257C4×D47—24no—
258C4×D47—28no—
259C4×D47—84no—
260C4×D47—84no—
261C4×D47C4×D418yes—
262C4×D47C22⋊Q818yes—
263C8×C47—24no—
264C47⋊C8—24no—
265#20846—471nomaximal, Sylow
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47#42814yescomplemented
273C22×C47C22×C4⋊C411yescharacteristic
274C22×C47C2×#12011yescharacteristic
275C22×C47C2×#12311yescharacteristic
276C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C47C2×#13711yescharacteristic
278C22×C47#57511yescharacteristic
279C22×C47#57511yescharacteristic
280C2×D47—22no—
281C2×D47—22no—
282C2×D47—22no—
283C2×D47—22no—
284C2×D47—22no—
285C2×D47—22no—
286C2×D47—22no—
287C2×D47—22no—
288C2×D47—22no—
289C2×D47—416no—
290C2×D47—41no—
291C2×D47—41no—
292C2×D47—41no—
293C2×D47—41no—
294C2×D47#38514yes—
295C4×C47—22no—
296C4×C47—44no—
297C4×C47—81no—
298C4×C47—81no—
299C4×C47C2×C4×D412yes—
300C4×C47C2×C22⋊Q812yes—
301Dic47—22no—
302Dic47—44no—
303Dic47—81no—
304Dic47—81no—
305Dic47#38514yes—
306C22×#117—471no—
307C2×#428—472no—
308C2×#407—471no—
309C2×#380—471no—
310#1854—472no—
311#1854—472no—
312#1853—471no—
313#1853D4711yescharacteristic, complemented
314#1852—471no—
315#1852—471no—
316#1736—472no—
317C2×C47—24no—
318C2×C47C2×#38511yescharacteristic
319C2×C47C2×#42811yescharacteristic
320C2×C47#291111yescharacteristic
321C2×C47#300611yescharacteristic
322C2×C47#300711yescharacteristic
323C2×C47#309411yescharacteristic
324C2×C47#309511yescharacteristic
325D47—24no—
326D47—42no—
327C24×C4—471no—
328C22×M16—472no—
329C2×#117—472no—
330C2×#117—472no—
331C2×#117—948no—
332C2×#117—941no—
333C2×#117—941no—
334C22×C4⋊C4—471no—
335C22×C4⋊C4—471no—
336C22×C4⋊C4—471no—
337C22×C4⋊C4C2×D4711yescharacteristic
338C22×C22⋊C4—941no—
339C22×C22⋊C4—941no—
340C22×C42—941no—
341#428—948no—
342C2×#107—472no—
343C2×#107—472no—
344C2×#106—472no—
345C2×#106C2×D4712yescomplemented
346#407—478no—
347C2×C22⋊C8—471no—
348C2×C22⋊C8—472no—
349C2×C22⋊C8C2×D4711yescharacteristic
350#380—471no—
351#380—471no—
352#380—471no—
353#380—941no—
354#380—941no—
355#380C2×D4711yescharacteristic
356C2×#95—941no—
357#340—472no—
358#340—472no—
359#340—472no—
360#340C2×D4712yescomplemented
361C47#2084611yescharacteristic, complemented, Sylow
362C25—471no—
363C23×C4—21no—
364C23×C4—471no—
365C23×C4—471no—
366C23×C4—474no—
367C23×C4—471no—
368C23×C4—471no—
369C23×C4—474no—
370C23×C4—471no—
371C23×C4—944no—
372C23×C4—944no—
373C23×C4—941no—
374C23×C4—941no—
375C23×C4—941no—
376C23×C4C22×D4711yescharacteristic
377C2×M16—478no—
378C2×M16—944no—
379C2×M16—944no—
380C22×C8—472no—
381C22×C8C22×D4712yes—
382#117—478no—
383#117—944no—
384#117—944no—
385#117—948no—
386#117—948no—
387#117—944no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—472no—
391C2×C4⋊C4—471no—
392C2×C4⋊C4—471no—
393C2×C4⋊C4—944no—
394C2×C4⋊C4—944no—
395C2×C4⋊C4—944no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4—941no—
399C2×C4⋊C4C22×D4711yescharacteristic
400C2×C4⋊C4C22×D4711yescharacteristic
401C2×C4⋊C4C22×D4712yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—942no—
404C2×C22⋊C4—942no—
405C2×C22⋊C4—948no—
406C2×C22⋊C4—941no—
407C2×C22⋊C4—941no—
408C2×C22⋊C4—941no—
409C2×C22⋊C4—1884no—
410C2×C42—942no—
411C2×C42—942no—
412C2×C42—948no—
413#107—944no—
414#107—944no—
415#106—24no—
416#106—944no—
417C22⋊C8—474no—
418C22⋊C8—478no—
419C22⋊C8C4×D4714yescomplemented
420#95—21no—
421#95—941no—
422#95—941no—
423#95—941no—
424C24—471no—
425C24—471no—
426C24—474no—
427C24—942no—
428C24—942no—
429C24—942no—
430C24—942no—
431C24—1881no—
432C24—1881no—
433C24D4×D4711yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—471no—
443C22×C4—472no—
444C22×C4—474no—
445C22×C4—472no—
446C22×C4—474no—
447C22×C4—474no—
448C22×C4—471no—
449C22×C4—474no—
450C22×C4—474no—
451C22×C4—472no—
452C22×C4—472no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—9416no—
456C22×C4—9416no—
457C22×C4—942no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—944no—
464C22×C4—942no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—1882no—
468C22×C4—1882no—
469C22×C4—1882no—
470C22×C4—1882no—
471C22×C4—1882no—
472C22×C4—1882no—
473C22×C4—1882no—
474C22×C4—1882no—
475C22×C4C2×C4×D4712yes—
476C22×C4D4×D4711yescharacteristic
477C22×C4Q8×D4712yes—
478C22×C4C23×D4711yescharacteristic
479M16—9416no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—474no—
483C2×C8—942no—
484C2×C8—942no—
485C2×C8C2×C4×D4714yes—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—474no—
491C4⋊C4—944no—
492C4⋊C4—942no—
493C4⋊C4—942no—
494C4⋊C4—942no—
495C4⋊C4—1882no—
496C4⋊C4—1882no—
497C4⋊C4D4×D4714yes—
498C22⋊C4—42no—
499C22⋊C4—948no—
500C22⋊C4—948no—
501C22⋊C4—1882no—
502C22⋊C4—1882no—
503C22⋊C4—1882no—
504C42—948no—
505C42—948no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—472no—
511C23—472no—
512C23—472no—
513C23—472no—
514C23—948no—
515C23—941no—
516C23—941no—
517C23—941no—
518C23—941no—
519C23—941no—
520C23—941no—
521C23—941no—
522C23—941no—
523C23—941no—
524C23—941no—
525C23—948no—
526C23—1884no—
527C23—1884no—
528C23—1884no—
529C23—1884no—
530C23—1884no—
531C23C4⋊C4×D4712yes—
532C23#1309405112yes—
533C23#1309405312yes—
534C23C2×D4×D4711yescharacteristic
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—472no—
550C2×C4—472no—
551C2×C4—472no—
552C2×C4—478no—
553C2×C4—478no—
554C2×C4—472no—
555C2×C4—948no—
556C2×C4—948no—
557C2×C4—948no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—944no—
561C2×C4—944no—
562C2×C4—944no—
563C2×C4—3764no—
564C2×C4—3764no—
565C2×C4—1881no—
566C2×C4—1881no—
567C2×C4—1881no—
568C2×C4—1881no—
569C2×C4—1881no—
570C2×C4—1881no—
571C2×C4—1888no—
572C2×C4—1882no—
573C2×C4—1882no—
574C2×C4C4⋊C4×D4712yes—
575C2×C4C22×C4×D4711yescharacteristic
576C2×C4C2×D4×D4712yes—
577C2×C4C2×Q8×D4711yescharacteristic
578C2×C4#41×D4712yes—
579C8—24no—
580C8—944no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—474no—
588C22—942no—
589C22—942no—
590C22—942no—
591C22—942no—
592C22—942no—
593C22—942no—
594C22—942no—
595C22—942no—
596C22—942no—
597C22—18816no—
598C22—1881no—
599C22—1881no—
600C22—1881no—
601C22—1881no—
602C22#1568916214yescomplemented
603C22C2×C4⋊C4×D4711yescharacteristic
604C22#120×D4711yescharacteristic
605C22#123×D4711yescharacteristic
606C22C2×#1309405111yescharacteristic
607C22C2×#1309405311yescharacteristic
608C22#1568991911yescharacteristic
609C22#1568992211yescharacteristic
610C4—22no—
611C4—44no—
612C4—81no—
613C4—81no—
614C4—474no—
615C4—942no—
616C4—3761no—
617C4—3761no—
618C4—1884no—
619C4C4×D4×D4712yes—
620C4C22⋊Q8×D4712yes—
621C2—24no—
622C2—944no—
623C2—1882no—
624C2#385×D4711yescharacteristic
625C2C2×#1568916211yescharacteristic
626C2#2025197511yescharacteristic
627C2#2025263911yescharacteristic
628C2#2025264311yescharacteristic
629C2#2025332511yescharacteristic
630C2#2025333111yescharacteristic
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#20846
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. #20243590C2semidirect
6. #20243586C2semidirect
7. #20243581C2semidirect
8. #20243578C2semidirect
9. #20243575C2semidirect
10. #20243106C2semidirect
11. C47×#1853C2semidirect
26. C2×#13093958C22semidirect
27. C2×#13093957C22semidirect
28. C2×#13093955C22semidirect
29. #15689085C4semidirect
38. #15688886C22semidirect
39. #15688885C22semidirect
40. #15688885C22semidirect
42. C2×#106×C47C22semidirect
45. C47×#340C22semidirect
61. C2×#10625324C2×C4semidirect
91. C22⋊(C47⋊C8)C2×C4semidirect
92. (C2×D47)⋊C8C2×C4semidirect
106. C22⋊C8×C47C2×C4semidirect
272. C22×C47#428semidirect
313. #1853D47semidirect
345. C2×#106C2×D47semidirect
360. #340C2×D47semidirect
361. C47#20846semidirect
419. C22⋊C8C4×D47semidirect
602. C22#15689162semidirect

Maximal subgroups

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

subgroupindex
C22×#130939822
C2×#156891622
C2×#156890852
#380×D472
#202435902
#202435862
#202435812
#202435782
#202435752
#202431062
C47×#18532
#2084647