Finite Group Databasebeta

← #450×Dic47 #26221859

Subgroups — #450×Dic47 #26221859

Order 12032.

1880 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic47C211yesmaximal, characteristic
2C2×SD16×Dic47C212yesmaximal, complemented
3C22⋊Q8×Dic47C211yesmaximal, characteristic, complemented
4#20243724C212yesmaximal, complemented
5#20243696C211yesmaximal, characteristic, complemented
6#20243651C212yesmaximal, complemented
7#20243583C211yesmaximal, characteristic, complemented
8#20243319C211yesmaximal, characteristic
9#102×Dic47C212yesmaximal, complemented
10C22⋊C8×Dic47C211yesmaximal, characteristic, complemented
11C2×C47×#450C211yesmaximal, characteristic
12C24×Dic47—21no—
13C2×Q8×Dic47C2211yescharacteristic, complemented
14C2×D4×Dic47—28no—
15C2×D4×Dic47—21no—
16C2×D4×Dic47—21no—
17C2×D4×Dic47C2212yes—
18C2×D4×Dic47C2212yes—
19C22×C4⋊Dic47C2211yescharacteristic
20C22×C22⋊Dic47—21no—
21C22×C4×Dic47C2211yescharacteristic
22SD16×Dic47—28no—
23C2×#13093956C2212yescomplemented
24C2×#13093948C2212yescomplemented
25C2×#13093945C2212yes—
26C2×#13093945C2212yes—
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×C8×Dic47C2212yescomplemented
29#15689030C2211yescharacteristic, complemented
30#15689022C2211yescharacteristic, complemented
31#15689009—21no—
32#15688998—21no—
33#15688983C2211yescharacteristic
34C4⋊C4×Dic47—21no—
35C4⋊C4×Dic47C2211yescharacteristic, complemented
36C22⋊C4×Dic47—21no—
37#15688886C2212yescomplemented
38#15688885C2212yescomplemented
39#15688877C2212yescomplemented
40C23×D4×C47C2211yescharacteristic
41C22×SD16×C47C2212yes—
42C2×C22⋊Q8×C47C2211yescharacteristic
43C47×#450C418yescomplemented
44C2×#102×C47C2212yes—
45C2×C22⋊C8×C47C2211yescharacteristic
46C23×Dic47—28no—
47C23×Dic47—21no—
48C23×Dic47—22no—
49C23×Dic47—22no—
50C23×Dic47—44no—
51C23×Dic47—44no—
52C23×Dic47D411yescharacteristic
53C23×Dic47D411yescharacteristic
54C22×C47⋊C8C2312yes—
55Q8×Dic47—24no—
56D4×Dic47—28no—
57D4×Dic47—28no—
58D4×Dic47—24no—
59D4×Dic47—24no—
60D4×Dic47—24no—
61D4×Dic47D418yes—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—24no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2312yes—
69C2×C22⋊Dic47—28no—
70C2×C22⋊Dic47—22no—
71C2×C22⋊Dic47—22no—
72C2×C22⋊Dic47—21no—
73C2×C4×Dic47—24no—
74C2×C4×Dic47—21no—
75C2×C4×Dic47—21no—
76C2×C4×Dic47D411yescharacteristic
77C2×C4×Dic47D411yescharacteristic
78C2×C4×Dic47C2311yescharacteristic
79C2×C4×Dic47C2311yescharacteristic
80#13093956—24no—
81#13093948—24no—
82#13093945—24no—
83#13093945—24no—
84C22⋊(C47⋊C8)—22no—
85C8×Dic47—24no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C25×C47—21no—
92C22×Q8×C47C2311yescharacteristic
93C22×D4×C47—24no—
94C22×D4×C47—24no—
95C22×D4×C47—21no—
96C22×D4×C47—22no—
97C22×D4×C47C2×C414yes—
98C22×D4×C47C2312yes—
99C23×C4×C47C2311yescharacteristic
100C2×SD16×C47—24no—
101C2×SD16×C47—24no—
102C2×SD16×C47C2×C418yescomplemented
103C22×C8×C47C2312yes—
104C22⋊Q8×C47—22no—
105C22⋊Q8×C47C2×C414yescomplemented
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C22⋊C4×C47—21no—
109#102×C47C2×C418yescomplemented
110C22⋊C8×C47C2×C414yescomplemented
111C24×C47—24no—
112C24×C47—24no—
113C24×C47—24no—
114C24×C47—22no—
115C24×C47—42no—
116C24×C47—42no—
117C24×C47—42no—
118C24×C47—42no—
119C24×C47C2×D411yescharacteristic
120C2×Q8×C47—21no—
121C2×Q8×C47—22no—
122C2×Q8×C47—22no—
123C2×Q8×C47C22×C412yescomplemented
124C2×D4×C47—24no—
125C2×D4×C47—216no—
126C2×D4×C47—24no—
127C2×D4×C47—216no—
128C2×D4×C47—24no—
129C2×D4×C47—22no—
130C2×D4×C47—22no—
131C2×D4×C47—22no—
132C2×D4×C47C22×C414yes—
133C2×D4×C47C2×D414yes—
134C2×D4×C47#4114yes—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—21no—
138C22×C4×C47—21no—
139C22×C4×C47C22×C412yes—
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47#4112yes—
142C22×C4×C47C2411yescharacteristic
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—24no—
146C22×Dic47—28no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—28no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—41no—
154C22×Dic47—44no—
155C22×Dic47—432no—
156C22×Dic47—42no—
157C22×Dic47—42no—
158C22×Dic47—44no—
159C22×Dic47—44no—
160C22×Dic47—44no—
161C22×Dic47SD1614yescomplemented
162C22×Dic47C2×D411yescharacteristic
163C22×Dic47C2×D411yescharacteristic
164C22×Dic47C2×D411yescharacteristic
165C22×Dic47C2×D411yescharacteristic
166SD16×C47—216no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C2×C8×C47C22×C414yescomplemented
170C4⋊C4×C47—24no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47C22×C412yescomplemented
173C22⋊C4×C47—24no—
174C22⋊Dic47—28no—
175C22⋊Dic47—28no—
176C22⋊Dic47—42no—
177C4⋊Dic47—24no—
178C4⋊Dic47—22no—
179C4⋊Dic47—22no—
180C4⋊Dic47—22no—
181C4⋊Dic47—42no—
182C4⋊Dic47—42no—
183C4⋊Dic47C2×D414yes—
184C4×Dic47—22no—
185C4×Dic47—22no—
186C4×Dic47—22no—
187C4×Dic47—42no—
188C4×Dic47—42no—
189C4×Dic47C2×D414yes—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8—22no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—216no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47—24no—
201C23×C47—48no—
202C23×C47—48no—
203C23×C47—48no—
204C23×C47—48no—
205C23×C47—48no—
206C23×C47—42no—
207C23×C47—42no—
208C23×C47—42no—
209C23×C47—42no—
210C23×C47—48no—
211C23×C47C4×D412yes—
212C23×C47C2×SD1612yes—
213C23×C47#13512yes—
214C23×C47C22×D411yescharacteristic
215Q8×C47—24no—
216Q8×C47—42no—
217D4×C47—216no—
218D4×C47—24no—
219D4×C47—24no—
220D4×C47—24no—
221D4×C47C4×D418yes—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—24no—
225C2×C4×C47—24no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47C4×D412yes—
233C2×C4×C47C23×C411yescharacteristic
234C2×C4×C47C22×D412yes—
235C2×C4×C47C2×#4112yes—
236C2×C4×C47C2×#4111yescharacteristic
237C2×Dic47—21no—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—28no—
242C2×Dic47—21no—
243C2×Dic47—24no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—44no—
247C2×Dic47—416no—
248C2×Dic47—48no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—416no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47#12012yes—
259C2×Dic47#12012yes—
260C2×Dic47C2×SD1612yes—
261C2×Dic47C2×SD1612yes—
262C2×Dic47AGL(1,ℤ/8)12yes—
263C8×C47—24no—
264C47⋊C8—42no—
265C4×#450—471nomaximal, Sylow
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—28no—
273C22×C47—432no—
274C22×C47—44no—
275C22×C47—44no—
276C22×C47—44no—
277C22×C47—44no—
278C22×C47—44no—
279C22×C47—44no—
280C22×C47C4×SD1614yescomplemented
281C22×C47C2×C4×D411yescharacteristic
282C22×C47C2×#12011yescharacteristic
283C22×C47#52511yescharacteristic
284C22×C47C22×SD1611yescharacteristic
285C22×C47C2×#13511yescharacteristic
286C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C47#57511yescharacteristic
288C4×C47—22no—
289C4×C47—42no—
290C4×C47—42no—
291C4×C47C2×C4×D412yes—
292C4×C47#52512yes—
293Dic47—22no—
294Dic47—22no—
295Dic47—44no—
296Dic47—41no—
297Dic47—81no—
298Dic47—81no—
299Dic47#45014yesdirect product factor
300C22×C4×D4—471no—
301C2×#450D4711yescharacteristic
302C2×C4×SD16—472no—
303C4×C22⋊Q8—471no—
304#1877—472no—
305#1872—471no—
306#1864—472no—
307#1853—471no—
308#1786—471no—
309C4×#102—472no—
310C4×C22⋊C8—471no—
311C2×C47—24no—
312C2×C47—48no—
313C2×C47C4×#12011yescharacteristic
314C2×C47C2×C4×SD1611yescharacteristic
315C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C47C2×#45011yescharacteristic
317C2×C47#300211yescharacteristic
318C2×C47#310111yescharacteristic
319C2×C47#310511yescharacteristic
320C23×D4C2×D4711yescharacteristic
321C24×C4—941no—
322C22×SD16C2×D4712yes—
323C2×C22⋊Q8C2×D4711yescharacteristic
324C2×C4×Q8—471no—
325C2×C4×D4—472no—
326C2×C4×D4—472no—
327C2×C4×D4—941no—
328C2×C4×D4—941no—
329C2×C4×D4—948no—
330C22×C4⋊C4—471no—
331C22×C22⋊C4—941no—
332C22×C42—471no—
333#450Dic4718yesdirect product factor
334C4×SD16—948no—
335C2×#106—472no—
336C2×#103—472no—
337C2×#102—472no—
338C2×#102—472no—
339C2×#102C2×D4712yes—
340C2×C22⋊C8—471no—
341C2×C22⋊C8C2×D4711yescharacteristic
342C2×C4×C8—472no—
343#391—471no—
344#389—471no—
345#387—941no—
346#385—941no—
347#380—471no—
348C4×C4⋊C4—471no—
349C4×C4⋊C4—941no—
350C4×C22⋊C4—941no—
351#340—472no—
352#340—472no—
353#336—472no—
354C47C4×#45011yescharacteristic, complemented, Sylow
355C25—21no—
356C22×Q8C22×D4711yescharacteristic
357C22×D4—24no—
358C22×D4—24no—
359C22×D4—21no—
360C22×D4—22no—
361C22×D4C2×Dic4714yes—
362C22×D4C22×D4712yes—
363C23×C4—471no—
364C23×C4—471no—
365C23×C4—942no—
366C23×C4—942no—
367C23×C4—948no—
368C23×C4—941no—
369C23×C4—1884no—
370C23×C4—1884no—
371C23×C4C22×D4711yescharacteristic
372C2×SD16—24no—
373C2×SD16—24no—
374C2×SD16C2×Dic4718yescomplemented
375C22×C8—472no—
376C22×C8C22×D4712yes—
377C22⋊Q8—22no—
378C22⋊Q8C2×Dic4714yescomplemented
379C4×Q8—944no—
380C4×D4—478no—
381C4×D4—944no—
382C4×D4—944no—
383C4×D4—948no—
384C4×D4—948no—
385C4×D4—944no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—472no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—944no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4C22×D4711yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—942no—
397C2×C22⋊C4—942no—
398C2×C22⋊C4—948no—
399C2×C22⋊C4—941no—
400C2×C42—471no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—471no—
404C2×C42—944no—
405C2×C42—941no—
406C2×C42—941no—
407#106—944no—
408#103—944no—
409#102—944no—
410#102—944no—
411#102C2×Dic4718yescomplemented
412C22⋊C8—942no—
413C22⋊C8C2×Dic4714yescomplemented
414C4×C8—944no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420C24—24no—
421C24—24no—
422C24—24no—
423C24—22no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24D4×D4711yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C22×Dic4712yescomplemented
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—216no—
436C2×D4—216no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4D4×D4714yes—
442C2×D4#1062535214yes—
443C2×D4C22×Dic4714yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—471no—
449C22×C4—474no—
450C22×C4—471no—
451C22×C4—471no—
452C22×C4—471no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—948no—
457C22×C4—944no—
458C22×C4—948no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—941no—
462C22×C4—941no—
463C22×C4—1884no—
464C22×C4—1884no—
465C22×C4—1884no—
466C22×C4—1881no—
467C22×C4—18832no—
468C22×C4—1884no—
469C22×C4—1882no—
470C22×C4—1882no—
471C22×C4D4×D4711yescharacteristic
472C22×C4#1062535212yes—
473C22×C4C22×Dic4712yes—
474C22×C4C23×D4711yescharacteristic
475SD16—216no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—942no—
479C2×C8—942no—
480C2×C8—942no—
481C2×C8C22×Dic4714yescomplemented
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—474no—
485C4⋊C4—944no—
486C4⋊C4—942no—
487C4⋊C4—942no—
488C4⋊C4—942no—
489C4⋊C4—1882no—
490C4⋊C4—1882no—
491C4⋊C4C22×Dic4712yescomplemented
492C22⋊C4—24no—
493C22⋊C4—948no—
494C22⋊C4—948no—
495C22⋊C4—1882no—
496C42—474no—
497C42—942no—
498C42—942no—
499C42—942no—
500C42—1882no—
501C42—1882no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—216no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—48no—
520C23D4×Dic4712yes—
521C23SD16×D4712yes—
522C23#1309404512yes—
523C23C2×D4×D4711yescharacteristic
524Q8—24no—
525Q8—42no—
526D4—216no—
527D4—24no—
528D4—24no—
529D4—24no—
530D4D4×Dic4718yes—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—472no—
542C2×C4—472no—
543C2×C4—472no—
544C2×C4—472no—
545C2×C4—472no—
546C2×C4—948no—
547C2×C4—941no—
548C2×C4—941no—
549C2×C4—941no—
550C2×C4—941no—
551C2×C4—941no—
552C2×C4—944no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—18816no—
556C2×C4—1884no—
557C2×C4—18816no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4—1888no—
565C2×C4—1882no—
566C2×C4—1882no—
567C2×C4D4×Dic4712yes—
568C2×C4C23×Dic4711yescharacteristic
569C2×C4C2×D4×D4712yes—
570C2×C4C2×#1062535212yes—
571C2×C4C2×#1062535211yescharacteristic
572C8—24no—
573C8—1882no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—28no—
581C22—432no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22SD16×Dic4714yescomplemented
589C22C2×D4×Dic4711yescharacteristic
590C22#120×D4711yescharacteristic
591C22#1568962911yescharacteristic
592C22C2×SD16×D4711yescharacteristic
593C22C2×#1309404511yescharacteristic
594C22AGL(1,ℤ/8)×D4711yescharacteristic
595C22#1568991911yescharacteristic
596C4—22no—
597C4—42no—
598C4—42no—
599C4—474no—
600C4—942no—
601C4—942no—
602C4—3761no—
603C4—3761no—
604C4—1884no—
605C4—1881no—
606C4C2×D4×Dic4712yes—
607C4#1568962912yes—
608C2—24no—
609C2—48no—
610C2#120×Dic4711yescharacteristic
611C2C2×SD16×Dic4711yescharacteristic
612C2AGL(1,ℤ/8)×Dic4711yescharacteristic
613C2#450×D4711yescharacteristic
614C2#2025260711yescharacteristic
615C2#2025339111yescharacteristic
616C2#2025343611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#450C2
socleC23×C47C22×D4
prime radicalC23×D4×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#450D47
hypocenterC47C4×#450
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×Dic47C2semidirect
3. C22⋊Q8×Dic47C2semidirect
4. #20243724C2semidirect
5. #20243696C2semidirect
6. #20243651C2semidirect
7. #20243583C2semidirect
9. #102×Dic47C2semidirect
10. C22⋊C8×Dic47C2semidirect
13. C2×Q8×Dic47C22semidirect
23. C2×#13093956C22semidirect
24. C2×#13093948C22semidirect
28. C2×C8×Dic47C22semidirect
29. #15689030C22semidirect
30. #15689022C22semidirect
35. C4⋊C4×Dic47C22semidirect
37. #15688886C22semidirect
38. #15688885C22semidirect
39. #15688877C22semidirect
43. C47×#450C4semidirect
102. C2×SD16×C47C2×C4semidirect
105. C22⋊Q8×C47C2×C4semidirect
109. #102×C47C2×C4semidirect
110. C22⋊C8×C47C2×C4semidirect
123. C2×Q8×C47C22×C4semidirect
161. C22×Dic47SD16semidirect
169. C2×C8×C47C22×C4semidirect
172. C4⋊C4×C47C22×C4semidirect
280. C22×C47C4×SD16semidirect
299. Dic47#450direct (N × H)
333. #450Dic47direct (N × H)
354. C47C4×#450semidirect
374. C2×SD16C2×Dic47semidirect
378. C22⋊Q8C2×Dic47semidirect
411. #102C2×Dic47semidirect
413. C22⋊C8C2×Dic47semidirect
432. C2×Q8C22×Dic47semidirect
481. C2×C8C22×Dic47semidirect
491. C4⋊C4C22×Dic47semidirect
588. C22SD16×Dic47semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic472
C2×SD16×Dic472
C22⋊Q8×Dic472
#202437242
#202436962
#202436512
#202435832
#202433192
#102×Dic472
C22⋊C8×Dic472
C2×C47×#4502
C4×#45047