Finite Group Databasebeta

← #447×Dic47 #26221848

Subgroups — #447×Dic47 #26221848

Order 12032.

1964 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic47C211yesmaximal, characteristic, complemented
2C2×D8×Dic47C212yesmaximal, complemented
3#121×Dic47C211yesmaximal, characteristic, complemented
4#20243738C212yesmaximal, complemented
5#20243706C211yesmaximal, characteristic, complemented
6#20243637C212yesmaximal, complemented
7#20243578C211yesmaximal, characteristic, complemented
8#20243319C211yesmaximal, characteristic, complemented
9#102×Dic47C212yesmaximal, complemented
10C22⋊C8×Dic47C211yesmaximal, characteristic, complemented
11C2×C47×#447C211yesmaximal, characteristic
12C24×Dic47—21no—
13C2×D4×Dic47—28no—
14C2×D4×Dic47—21no—
15C2×D4×Dic47—21no—
16C2×D4×Dic47—21no—
17C2×D4×Dic47C2212yes—
18C2×D4×Dic47C2212yes—
19C2×D4×Dic47C2211yescharacteristic, complemented
20C22×C4⋊Dic47C2211yescharacteristic
21C22×C22⋊Dic47—21no—
22C22×C4×Dic47C2211yescharacteristic
23D8×Dic47—28no—
24C2×#13093958C2212yescomplemented
25C2×#13093945C2212yes—
26C2×#13093945C2212yes—
27C2×#13093945C2212yescomplemented
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×C8×Dic47C2212yescomplemented
30#15689026C2211yescharacteristic, complemented
31#15689022C2211yescharacteristic, complemented
32#15689016—21no—
33#15689003—21no—
34#15688983C2211yescharacteristic
35C4⋊C4×Dic47C2211yescharacteristic, complemented
36C22⋊C4×Dic47—21no—
37#15688886C2212yescomplemented
38#15688885C2212yescomplemented
39#15688877C2212yescomplemented
40C23×D4×C47C2211yescharacteristic
41C22×D8×C47C2212yes—
42C2×#121×C47C2211yescharacteristic
43C47×#447C418yescomplemented
44C2×#102×C47C2212yes—
45C2×C22⋊C8×C47C2211yescharacteristic
46C23×Dic47—28no—
47C23×Dic47—21no—
48C23×Dic47—22no—
49C23×Dic47—22no—
50C23×Dic47—21no—
51C23×Dic47—44no—
52C23×Dic47—44no—
53C23×Dic47D411yescharacteristic
54C23×Dic47D411yescharacteristic
55C22×C47⋊C8C2312yes—
56D4×Dic47—28no—
57D4×Dic47—28no—
58D4×Dic47—24no—
59D4×Dic47—24no—
60D4×Dic47—24no—
61D4×Dic47—24no—
62D4×Dic47—44no—
63D4×Dic47D418yescomplemented
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—24no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2312yes—
70C2×C22⋊Dic47—28no—
71C2×C22⋊Dic47—22no—
72C2×C22⋊Dic47—22no—
73C2×C22⋊Dic47—21no—
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77C2×C4×Dic47—24no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47D411yescharacteristic
80C2×C4×Dic47D411yescharacteristic
81C2×C4×Dic47C2311yescharacteristic
82C2×C4×Dic47C2311yescharacteristic
83#13093958—24no—
84#13093945—24no—
85#13093945—24no—
86#13093945—24no—
87C22⋊(C47⋊C8)—22no—
88C8×Dic47—24no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C25×C47—21no—
93C22×D4×C47—24no—
94C22×D4×C47—24no—
95C22×D4×C47—21no—
96C22×D4×C47—21no—
97C22×D4×C47—22no—
98C22×D4×C47C2×C414yescomplemented
99C22×D4×C47C2311yescharacteristic
100C22×D4×C47C2312yes—
101C23×C4×C47C2311yescharacteristic
102C2×D8×C47—24no—
103C2×D8×C47—24no—
104C2×D8×C47C2×C418yescomplemented
105C22×C8×C47C2312yes—
106#121×C47—22no—
107#121×C47C2×C414yescomplemented
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C22⋊C4×C47—21no—
110#102×C47C2×C418yescomplemented
111C22⋊C8×C47C2×C414yescomplemented
112C24×C47—24no—
113C24×C47—24no—
114C24×C47—24no—
115C24×C47—22no—
116C24×C47—21no—
117C24×C47—42no—
118C24×C47—42no—
119C24×C47—42no—
120C24×C47—42no—
121C24×C47C2×D411yescharacteristic
122C2×D4×C47—24no—
123C2×D4×C47—216no—
124C2×D4×C47—24no—
125C2×D4×C47—216no—
126C2×D4×C47—24no—
127C2×D4×C47—24no—
128C2×D4×C47—22no—
129C2×D4×C47—22no—
130C2×D4×C47—22no—
131C2×D4×C47—22no—
132C2×D4×C47—22no—
133C2×D4×C47—21no—
134C2×D4×C47—42no—
135C2×D4×C47—42no—
136C2×D4×C47C22×C414yes—
137C2×D4×C47C22×C412yescomplemented
138C2×D4×C47C2×D414yes—
139C2×D4×C47#4114yes—
140C22×C4×C47—22no—
141C22×C4×C47—22no—
142C22×C4×C47—21no—
143C22×C4×C47C22×C412yes—
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47#4112yes—
146C22×C4×C47C2411yescharacteristic
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—24no—
150C22×Dic47—28no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—28no—
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—41no—
158C22×Dic47—44no—
159C22×Dic47—432no—
160C22×Dic47—42no—
161C22×Dic47—42no—
162C22×Dic47—44no—
163C22×Dic47—44no—
164C22×Dic47—44no—
165C22×Dic47—42no—
166C22×Dic47—42no—
167C22×Dic47—42no—
168C22×Dic47D814yescomplemented
169C22×Dic47C2×D411yescharacteristic
170C22×Dic47C2×D411yescharacteristic
171C22×Dic47C2×D411yescharacteristic
172C22×Dic47C2×D411yescharacteristic
173D8×C47—216no—
174C2×C8×C47—22no—
175C2×C8×C47—22no—
176C2×C8×C47C22×C414yescomplemented
177C4⋊C4×C47—21no—
178C4⋊C4×C47C22×C412yescomplemented
179C22⋊C4×C47—24no—
180C22⋊Dic47—28no—
181C22⋊Dic47—28no—
182C22⋊Dic47—42no—
183C22⋊Dic47—42no—
184C22⋊Dic47—42no—
185C22⋊Dic47—42no—
186C4⋊Dic47—24no—
187C4⋊Dic47—22no—
188C4⋊Dic47—22no—
189C4⋊Dic47—22no—
190C4⋊Dic47—42no—
191C4⋊Dic47C2×D414yes—
192C4×Dic47—22no—
193C4×Dic47—22no—
194C4×Dic47—22no—
195C4×Dic47—42no—
196C4×Dic47C2×D414yes—
197C2×C47⋊C8—22no—
198C2×C47⋊C8—22no—
199C2×C47⋊C8—22no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47—216no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—24no—
208C23×C47—22no—
209C23×C47—48no—
210C23×C47—48no—
211C23×C47—48no—
212C23×C47—48no—
213C23×C47—48no—
214C23×C47—42no—
215C23×C47—42no—
216C23×C47—42no—
217C23×C47—42no—
218C23×C47—48no—
219C23×C47—41no—
220C23×C47—41no—
221C23×C47—41no—
222C23×C47—41no—
223C23×C47—41no—
224C23×C47—41no—
225C23×C47C4×D412yes—
226C23×C47C2×D812yes—
227C23×C47#13512yes—
228C23×C47C22×D411yescharacteristic
229D4×C47—216no—
230D4×C47—24no—
231D4×C47—24no—
232D4×C47—24no—
233D4×C47—24no—
234D4×C47—48no—
235D4×C47—42no—
236D4×C47C4×D418yescomplemented
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—24no—
240C2×C4×C47—24no—
241C2×C4×C47—22no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47C4×D412yes—
245C2×C4×C47C23×C411yescharacteristic
246C2×C4×C47C22×D412yes—
247C2×C4×C47C2×#4112yes—
248C2×C4×C47C2×#4111yescharacteristic
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—28no—
254C2×Dic47—21no—
255C2×Dic47—24no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—44no—
259C2×Dic47—416no—
260C2×Dic47—48no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—416no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—84no—
271C2×Dic47#12012yes—
272C2×Dic47#12012yes—
273C2×Dic47C2×D812yes—
274C2×Dic47C2×D812yes—
275C2×Dic47AGL(1,ℤ/8)12yes—
276C8×C47—24no—
277C47⋊C8—42no—
278C4×#447—471nomaximal, Sylow
279C22×C47—22no—
280C22×C47—22no—
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—22no—
285C22×C47—28no—
286C22×C47—432no—
287C22×C47—44no—
288C22×C47—44no—
289C22×C47—44no—
290C22×C47—44no—
291C22×C47—44no—
292C22×C47—44no—
293C22×C47—42no—
294C22×C47—42no—
295C22×C47—42no—
296C22×C47—81no—
297C22×C47—81no—
298C22×C47—81no—
299C22×C47—81no—
300C22×C47C4×D814yescomplemented
301C22×C47C2×C4×D411yescharacteristic
302C22×C47C2×#12011yescharacteristic
303C22×C47#52511yescharacteristic
304C22×C47C22×D811yescharacteristic
305C22×C47C2×#13511yescharacteristic
306C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
307C22×C47#57511yescharacteristic
308C4×C47—22no—
309C4×C47—42no—
310C4×C47C2×C4×D412yes—
311C4×C47#52512yes—
312Dic47—22no—
313Dic47—22no—
314Dic47—44no—
315Dic47—41no—
316Dic47—81no—
317Dic47—81no—
318Dic47#44714yesdirect product factor
319C22×C4×D4—471no—
320C2×#447D4711yescharacteristic
321C2×C4×D8—472no—
322C4×#121—471no—
323#1879—472no—
324#1874—471no—
325#1862—472no—
326#1852—471no—
327#1786—471no—
328C4×#102—472no—
329C4×C22⋊C8—471no—
330C2×C47—24no—
331C2×C47—48no—
332C2×C47—82no—
333C2×C47C4×#12011yescharacteristic
334C2×C47C2×C4×D811yescharacteristic
335C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
336C2×C47C2×#44711yescharacteristic
337C2×C47#300211yescharacteristic
338C2×C47#310011yescharacteristic
339C2×C47#310511yescharacteristic
340C23×D4C2×D4711yescharacteristic
341C24×C4—941no—
342C22×D8C2×D4712yes—
343C2×#121C2×D4711yescharacteristic
344C2×C4×D4—472no—
345C2×C4×D4—472no—
346C2×C4×D4—471no—
347C2×C4×D4—941no—
348C2×C4×D4—941no—
349C2×C4×D4—948no—
350C2×C4×D4—941no—
351C22×C4⋊C4—471no—
352C22×C22⋊C4—941no—
353C22×C42—471no—
354#447Dic4718yesdirect product factor
355C4×D8—948no—
356C2×#107—472no—
357C2×#102—472no—
358C2×#102—472no—
359C2×#102—472no—
360C2×#102C2×D4712yes—
361C2×C22⋊C8—471no—
362C2×C22⋊C8C2×D4711yescharacteristic
363C2×C4×C8—472no—
364#390—471no—
365#389—471no—
366#388—941no—
367#386—941no—
368#380—471no—
369C4×C4⋊C4—471no—
370C4×C22⋊C4—941no—
371#340—472no—
372#340—472no—
373#336—472no—
374C47C4×#44711yescharacteristic, complemented, Sylow
375C25—21no—
376C22×D4—24no—
377C22×D4—24no—
378C22×D4—21no—
379C22×D4—21no—
380C22×D4—22no—
381C22×D4C2×Dic4714yescomplemented
382C22×D4C22×D4711yescharacteristic
383C22×D4C22×D4712yes—
384C23×C4—471no—
385C23×C4—471no—
386C23×C4—942no—
387C23×C4—942no—
388C23×C4—941no—
389C23×C4—948no—
390C23×C4—941no—
391C23×C4—1884no—
392C23×C4—1884no—
393C23×C4C22×D4711yescharacteristic
394C2×D8—24no—
395C2×D8—24no—
396C2×D8C2×Dic4718yescomplemented
397C22×C8—472no—
398C22×C8C22×D4712yes—
399#121—22no—
400#121C2×Dic4714yescomplemented
401C4×D4—478no—
402C4×D4—944no—
403C4×D4—944no—
404C4×D4—944no—
405C4×D4—948no—
406C4×D4—948no—
407C4×D4—944no—
408C4×D4—1884no—
409C2×C4⋊C4—472no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—471no—
412C2×C4⋊C4—944no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4C22×D4711yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—942no—
418C2×C22⋊C4—942no—
419C2×C22⋊C4—948no—
420C2×C22⋊C4—941no—
421C2×C22⋊C4—941no—
422C2×C22⋊C4—941no—
423C2×C22⋊C4—941no—
424C2×C42—471no—
425C2×C42—471no—
426C2×C42—471no—
427C2×C42—471no—
428C2×C42—944no—
429C2×C42—941no—
430#107—944no—
431#102—944no—
432#102—944no—
433#102—944no—
434#102C2×Dic4718yescomplemented
435C22⋊C8—942no—
436C22⋊C8C2×Dic4714yescomplemented
437C4×C8—944no—
438#95—941no—
439#95—941no—
440#95—941no—
441C24—24no—
442C24—24no—
443C24—24no—
444C24—22no—
445C24—21no—
446C24—42no—
447C24—42no—
448C24—42no—
449C24—42no—
450C24D4×D4711yescharacteristic
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—216no—
454C2×D4—216no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4D4×D4714yes—
466C2×D4#1062535214yes—
467C2×D4C22×Dic4714yes—
468C2×D4C22×Dic4712yescomplemented
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—471no—
473C22×C4—474no—
474C22×C4—471no—
475C22×C4—471no—
476C22×C4—471no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—948no—
481C22×C4—944no—
482C22×C4—948no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4—1882no—
488C22×C4—1882no—
489C22×C4—1882no—
490C22×C4—1884no—
491C22×C4—1884no—
492C22×C4—1884no—
493C22×C4—1881no—
494C22×C4—18832no—
495C22×C4—1884no—
496C22×C4—1882no—
497C22×C4—1882no—
498C22×C4D4×D4711yescharacteristic
499C22×C4#1062535212yes—
500C22×C4C22×Dic4712yes—
501C22×C4C23×D4711yescharacteristic
502D8—216no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—942no—
506C2×C8—942no—
507C2×C8—942no—
508C2×C8C22×Dic4714yescomplemented
509C4⋊C4—21no—
510C4⋊C4—474no—
511C4⋊C4—944no—
512C4⋊C4—942no—
513C4⋊C4—942no—
514C4⋊C4—942no—
515C4⋊C4—1882no—
516C4⋊C4C22×Dic4712yescomplemented
517C22⋊C4—24no—
518C22⋊C4—948no—
519C22⋊C4—948no—
520C22⋊C4—1882no—
521C22⋊C4—1882no—
522C22⋊C4—1882no—
523C22⋊C4—1882no—
524C42—474no—
525C42—942no—
526C42—942no—
527C42—942no—
528C42—1882no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—216no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—48no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23D4×Dic4712yes—
555C23D8×D4712yes—
556C23#1309404312yes—
557C23C2×D4×D4711yescharacteristic
558D4—216no—
559D4—24no—
560D4—24no—
561D4—24no—
562D4—24no—
563D4—48no—
564D4—42no—
565D4D4×Dic4718yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—472no—
574C2×C4—472no—
575C2×C4—472no—
576C2×C4—472no—
577C2×C4—472no—
578C2×C4—948no—
579C2×C4—941no—
580C2×C4—941no—
581C2×C4—941no—
582C2×C4—941no—
583C2×C4—941no—
584C2×C4—944no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—3764no—
588C2×C4—18816no—
589C2×C4—1884no—
590C2×C4—18816no—
591C2×C4—1881no—
592C2×C4—1881no—
593C2×C4—1881no—
594C2×C4—1881no—
595C2×C4—1881no—
596C2×C4—1881no—
597C2×C4—1888no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4D4×Dic4712yes—
601C2×C4C23×Dic4711yescharacteristic
602C2×C4C2×D4×D4712yes—
603C2×C4C2×#1062535212yes—
604C2×C4C2×#1062535211yescharacteristic
605C8—24no—
606C8—1882no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—432no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22D8×Dic4714yescomplemented
629C22C2×D4×Dic4711yescharacteristic
630C22#120×D4711yescharacteristic
631C22#1568962911yescharacteristic
632C22C2×D8×D4711yescharacteristic
633C22C2×#1309404311yescharacteristic
634C22AGL(1,ℤ/8)×D4711yescharacteristic
635C22#1568991911yescharacteristic
636C4—22no—
637C4—42no—
638C4—474no—
639C4—942no—
640C4—942no—
641C4—3761no—
642C4—3761no—
643C4—1884no—
644C4—1881no—
645C4C2×D4×Dic4712yes—
646C4#1568962912yes—
647C2—24no—
648C2—48no—
649C2—82no—
650C2#120×Dic4711yescharacteristic
651C2C2×D8×Dic4711yescharacteristic
652C2AGL(1,ℤ/8)×Dic4711yescharacteristic
653C2#447×D4711yescharacteristic
654C2#2025260111yescharacteristic
655C2#2025338311yescharacteristic
656C2#2025343011yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#447C2
socleC23×C47C22×D4
cosocleC22×C4×C47C24
hypercenterC2×#447D47
hypocenterC47C4×#447
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C22×D4×Dic47C2semidirect
2. C2×D8×Dic47C2semidirect
3. #121×Dic47C2semidirect
4. #20243738C2semidirect
5. #20243706C2semidirect
6. #20243637C2semidirect
7. #20243578C2semidirect
8. #20243319C2semidirect
9. #102×Dic47C2semidirect
10. C22⋊C8×Dic47C2semidirect
19. C2×D4×Dic47C22semidirect
24. C2×#13093958C22semidirect
27. C2×#13093945C22semidirect
29. C2×C8×Dic47C22semidirect
30. #15689026C22semidirect
31. #15689022C22semidirect
35. C4⋊C4×Dic47C22semidirect
37. #15688886C22semidirect
38. #15688885C22semidirect
39. #15688877C22semidirect
43. C47×#447C4semidirect
63. D4×Dic47D4semidirect
98. C22×D4×C47C2×C4semidirect
104. C2×D8×C47C2×C4semidirect
107. #121×C47C2×C4semidirect
110. #102×C47C2×C4semidirect
111. C22⋊C8×C47C2×C4semidirect
137. C2×D4×C47C22×C4semidirect
168. C22×Dic47D8semidirect
176. C2×C8×C47C22×C4semidirect
178. C4⋊C4×C47C22×C4semidirect
236. D4×C47C4×D4semidirect
300. C22×C47C4×D8semidirect
318. Dic47#447direct (N × H)
354. #447Dic47direct (N × H)
374. C47C4×#447semidirect
381. C22×D4C2×Dic47semidirect
396. C2×D8C2×Dic47semidirect
400. #121C2×Dic47semidirect
434. #102C2×Dic47semidirect
436. C22⋊C8C2×Dic47semidirect
468. C2×D4C22×Dic47semidirect
508. C2×C8C22×Dic47semidirect
516. C4⋊C4C22×Dic47semidirect
565. D4D4×Dic47semidirect
628. C22D8×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×D8×Dic472
#121×Dic472
#202437382
#202437062
#202436372
#202435782
#202433192
#102×Dic472
C22⋊C8×Dic472
C2×C47×#4472
C4×#44747