Finite Group Databasebeta

← #26219214

Subgroups — #26219214

Order 12032.

978 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
1C2×SD16×Dic47C211yesmaximal, characteristic
2#20245863C211yesmaximal, characteristic, complemented
3C4×D4×Dic47C211yesmaximal, characteristic
4#20244032C211yesmaximal, characteristic, complemented
5#20244026C211yesmaximal, characteristic, complemented
6#20243967C211yesmaximal, characteristic, complemented
7#20243965C211yesmaximal, characteristic, complemented
8#20243724C211yesmaximal, characteristic
9#20243635C211yesmaximal, characteristic
10#20243633C211yesmaximal, characteristic
11#20243280C211yesmaximal, characteristic, complemented
12#103×Dic47C211yesmaximal, characteristic, complemented
13#102×Dic47C211yesmaximal, characteristic
14#20243132C211yesmaximal, characteristic, complemented
15C47×#1877C211yesmaximal, characteristic
16C2×Q8×Dic47C2211yescharacteristic
17C2×D4×Dic47C2211yescharacteristic
18C2×D4×Dic47C2211yescharacteristic
19C22×C4×Dic47—21no—
20SD16×Dic47C418yescomplemented
21C2×#13093957C2211yescharacteristic
22C2×#13093957C2211yescharacteristic
23C2×#13093956C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093948C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29C2×C8×Dic47C2211yescharacteristic
30#15689030C2211yescharacteristic
31#15689021C2211yescharacteristic
32#15689019C2211yescharacteristic
33#15689008—21no—
34#15689006—21no—
35C4⋊C4×Dic47C2211yescharacteristic
36C4⋊C4×Dic47C2211yescharacteristic
37C4⋊C4×Dic47C2211yescharacteristic
38C4×C4⋊Dic47C2211yescharacteristic
39#15688974—21no—
40C22⋊C4×Dic47—21no—
41C22⋊C4×Dic47—21no—
42C4×C22⋊Dic47—21no—
43#15688968—21no—
44C42×Dic47C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688878C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C22×SD16×C47C2211yescharacteristic
54C2×C4×D4×C47C2211yescharacteristic
55C2×#103×C47C2211yescharacteristic
56C2×#102×C47C2211yescharacteristic
57C47×#391C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#336C2211yescharacteristic
60C23×Dic47—21no—
61C23×Dic47—21no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64Q8×Dic47—22no—
65Q8×Dic47C2×C414yescomplemented
66D4×Dic47—22no—
67D4×Dic47—22no—
68D4×Dic47—22no—
69D4×Dic47C2×C414yes—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—21no—
84C2×C4×Dic47—44no—
85C2×C4×Dic47D411yescharacteristic
86C2×C4×Dic47D411yescharacteristic
87C2×C4×Dic47D411yescharacteristic
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47C2311yescharacteristic
90C2×C4×Dic47C2311yescharacteristic
91#13093957—22no—
92#13093957—22no—
93#13093956C2×C414yescomplemented
94#13093948—22no—
95#13093948C2×C414yescomplemented
96#13093945—22no—
97#13093945C2×C414yes—
98C8×Dic47—22no—
99C8×Dic47C2×C414yescomplemented
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×Q8×C47C2311yescharacteristic
109C22×D4×C47C2311yescharacteristic
110C23×C4×C47—21no—
111C2×SD16×C47—22no—
112C2×SD16×C47—22no—
113C2×SD16×C47C2×C414yes—
114C22×C8×C47C2311yescharacteristic
115C22×C8×C47C2311yescharacteristic
116C4×D4×C47—22no—
117C4×D4×C47—22no—
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C22⋊C4×C47—21no—
121C2×C42×C47C2311yescharacteristic
122#103×C47—22no—
123#102×C47—22no—
124#95×C47—21no—
125#95×C47—21no—
126C24×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47C22×C412yes—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47C22×C412yes—
137C2×D4×C47#4112yes—
138C2×D4×C47#4112yes—
139C22×C4×C47—24no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—42no—
144C22×C4×C47—42no—
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2411yescharacteristic
148C22×Dic47—24no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—24no—
153C22×Dic47—21no—
154C22×Dic47—42no—
155C22×Dic47—42no—
156C22×Dic47—42no—
157C22×Dic47—42no—
158C22×Dic47C2×D411yescharacteristic
159C22×Dic47C2×D411yescharacteristic
160C22×Dic47C2×D411yescharacteristic
161C22×Dic47C2×D411yescharacteristic
162SD16×C47—28no—
163C2×C8×C47—21no—
164C2×C8×C47—21no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47C22×C412yes—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—21no—
172C4⋊C4×C47#4112yes—
173C22⋊C4×C47—24no—
174C42×C47—21no—
175C42×C47—21no—
176C22⋊Dic47—24no—
177C22⋊Dic47—24no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—24no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47C22×C412yes—
185C4⋊Dic47C22×C412yes—
186C4⋊Dic47#4112yes—
187Dic47⋊C4—42no—
188Dic47⋊C4—42no—
189C4×Dic47—22no—
190C4×Dic47—21no—
191C4×Dic47—21no—
192C4×Dic47—24no—
193C4×Dic47—22no—
194C4×Dic47—22no—
195C4×Dic47—22no—
196C4×Dic47—42no—
197C4×Dic47—42no—
198C4×Dic47—42no—
199C4×Dic47—42no—
200C4×Dic47C22×C412yes—
201C4×Dic47#4112yes—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8C22×C412yes—
208C23×C47—22no—
209C23×C47—22no—
210C23×C47—22no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47C22×D411yescharacteristic
216Q8×C47—22no—
217Q8×C47—22no—
218Q8×C47—41no—
219Q8×C47—41no—
220D4×C47—22no—
221D4×C47—22no—
222D4×C47—22no—
223D4×C47#11714yes—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—22no—
227C2×C4×C47—21no—
228C2×C4×C47—22no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—48no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
239C2×C4×C47#13512yes—
240C2×C4×C47C23×C411yescharacteristic
241C2×C4×C47C2×#4111yescharacteristic
242C2×C4×C47C2×#4111yescharacteristic
243C2×C4×C47C2×#4111yescharacteristic
244C2×C4×C47C2×#4111yescharacteristic
245C2×C4×C47C2×#4111yescharacteristic
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—21no—
253C2×Dic47—22no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—48no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—48no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47C4×D412yes—
268C2×Dic47C4×D412yes—
269C2×Dic47#13512yes—
270C2×Dic47AGL(1,ℤ/8)12yes—
271C2×Dic47#13712yes—
272C8×C47—22no—
273C8×C47—41no—
274C47⋊C8—22no—
275C47⋊C8—41no—
276#21114—471nomaximal, Sylow
277C22×C47—24no—
278C22×C47—42no—
279C22×C47—42no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—42no—
283C22×C47—42no—
284C22×C47C2×C4×D411yescharacteristic
285C22×C47#52511yescharacteristic
286C22×C47#52511yescharacteristic
287C22×C47C2×#13511yescharacteristic
288C22×C47C2×#13511yescharacteristic
289C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C47C2×#13711yescharacteristic
291C4×C47—22no—
292C4×C47—42no—
293C4×C47—41no—
294C4×C47—42no—
295C4×C47—81no—
296C4×C47C2×#11711yescharacteristic
297C4×C47C4×#4111yescharacteristic
298C4×C47#52911yescharacteristic
299C4×C47#52911yescharacteristic
300Dic47—22no—
301Dic47—22no—
302Dic47—42no—
303Dic47—41no—
304Dic47—42no—
305Dic47—42no—
306Dic47—81no—
307Dic47#44014yes—
308C2×C4×SD16—471no—
309#2295—471no—
310C42×D4—471no—
311#1928—471no—
312#1928—471no—
313#1919—471no—
314#1919—471no—
315#1877—471no—
316#1877D4711yescharacteristic
317#1862—471no—
318#1862—471no—
319C4×#107—471no—
320C4×#103—471no—
321C4×#102—471no—
322#1744—471no—
323C2×C47—44no—
324C2×C47#229011yescharacteristic
325C2×C47C4×#13511yescharacteristic
326C2×C47C2×#44011yescharacteristic
327C2×C47#310511yescharacteristic
328C2×C47#310511yescharacteristic
329C2×C47#310711yescharacteristic
330C2×C47#310711yescharacteristic
331C22×SD16C2×D4711yescharacteristic
332C2×C4×Q8—471no—
333C2×C4×D4—471no—
334C2×C4×D4—471no—
335C2×C4×D4C2×D4711yescharacteristic
336C22×C42—941no—
337C4×SD16—478no—
338C2×#107—471no—
339C2×#107—471no—
340C2×#106—471no—
341C2×#103—471no—
342C2×#103—471no—
343C2×#103C2×D4711yescharacteristic
344C2×#102—471no—
345C2×#102—471no—
346C2×#102C2×D4711yescharacteristic
347C2×C4×C8—471no—
348C2×C4×C8—471no—
349#391—471no—
350#391C2×D4711yescharacteristic
351#389—471no—
352#389—471no—
353#387—941no—
354#387—941no—
355C4×C4⋊C4—471no—
356C4×C4⋊C4—471no—
357C4×C4⋊C4—471no—
358C4×C4⋊C4—471no—
359C4×C4⋊C4—941no—
360C4×C22⋊C4—941no—
361C4×C22⋊C4—941no—
362C4×C22⋊C4—941no—
363#376—941no—
364C43—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#340—471no—
370#340C2×D4711yescharacteristic
371#336—471no—
372#336—471no—
373#336—471no—
374#336C2×D4711yescharacteristic
375C47#2111411yescharacteristic, complemented, Sylow
376C22×Q8C22×D4711yescharacteristic
377C22×D4C22×D4711yescharacteristic
378C23×C4—21no—
379C23×C4—941no—
380C23×C4—941no—
381C2×SD16—22no—
382C2×SD16—22no—
383C2×SD16C4×D4714yes—
384C22×C8—471no—
385C22×C8—471no—
386C22×C8C22×D4711yescharacteristic
387C22×C8C22×D4711yescharacteristic
388C4×Q8—474no—
389C4×Q8—942no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—474no—
393C4×D4—942no—
394C4×D4—942no—
395C4×D4—942no—
396C2×C4⋊C4—471no—
397C2×C4⋊C4—471no—
398C2×C4⋊C4—471no—
399C2×C4⋊C4—471no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4C22×D4711yescharacteristic
404C2×C4⋊C4C22×D4711yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—941no—
407C2×C22⋊C4—941no—
408C2×C42—471no—
409C2×C42—471no—
410C2×C42—471no—
411C2×C42—471no—
412C2×C42—471no—
413C2×C42—471no—
414C2×C42—941no—
415C2×C42—941no—
416C2×C42—941no—
417C2×C42—941no—
418C2×C42—941no—
419C2×C42—1884no—
420C2×C42C22×D4711yescharacteristic
421#107—942no—
422#107—942no—
423#106—474no—
424#103—22no—
425#103—474no—
426#103—942no—
427#102—22no—
428#102—474no—
429#102—942no—
430C4×C8—474no—
431C4×C8—942no—
432#95—21no—
433#95—21no—
434#95—941no—
435#95—941no—
436#95—941no—
437#95—941no—
438#95—941no—
439#95—941no—
440#95—941no—
441#95—941no—
442C24—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×C4×D4712yes—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D4712yes—
453C2×D4#1062535212yes—
454C2×D4#1062535212yes—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—471no—
462C22×C4—471no—
463C22×C4—471no—
464C22×C4—471no—
465C22×C4—944no—
466C22×C4—944no—
467C22×C4—941no—
468C22×C4—941no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4—1882no—
472C22×C4—1882no—
473C22×C4—1882no—
474C22×C4—1882no—
475C22×C4D4×D4711yescharacteristic
476C22×C4D4×D4711yescharacteristic
477C22×C4C23×D4711yescharacteristic
478SD16—28no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—472no—
485C2×C8—941no—
486C2×C8—941no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8C2×C4×D4712yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—472no—
495C4⋊C4—472no—
496C4⋊C4—472no—
497C4⋊C4—944no—
498C4⋊C4—941no—
499C4⋊C4—941no—
500C4⋊C4—941no—
501C4⋊C4—941no—
502C4⋊C4—941no—
503C4⋊C4—1882no—
504C4⋊C4—1882no—
505C4⋊C4#1062535212yes—
506C22⋊C4—24no—
507C22⋊C4—944no—
508C22⋊C4—944no—
509C42—21no—
510C42—21no—
511C42—472no—
512C42—472no—
513C42—942no—
514C42—942no—
515C42—942no—
516C42—944no—
517C42—942no—
518C42—941no—
519C42—941no—
520C42—1882no—
521C42—1882no—
522C42—1882no—
523C42—1882no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D4711yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1309398014yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—472no—
556C2×C4—472no—
557C2×C4—472no—
558C2×C4—472no—
559C2×C4—472no—
560C2×C4—941no—
561C2×C4—941no—
562C2×C4—941no—
563C2×C4—941no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—941no—
567C2×C4—942no—
568C2×C4—941no—
569C2×C4—941no—
570C2×C4—1888no—
571C2×C4—1888no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4#1309404512yes—
582C2×C4C22×C4×D4711yescharacteristic
583C2×C4#41×D4711yescharacteristic
584C2×C4#41×D4711yescharacteristic
585C2×C4C2×#1062535211yescharacteristic
586C2×C4C2×#1062535211yescharacteristic
587C2×C4C2×#1062535211yescharacteristic
588C8—22no—
589C8—41no—
590C8—942no—
591C8—1881no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C4×D4×D4711yescharacteristic
600C22#1568962911yescharacteristic
601C22#1568963311yescharacteristic
602C22#135×D4711yescharacteristic
603C22C2×#1309404511yescharacteristic
604C22AGL(1,ℤ/8)×D4711yescharacteristic
605C22#137×D4711yescharacteristic
606C4—22no—
607C4—41no—
608C4—42no—
609C4—42no—
610C4—81no—
611C4—474no—
612C4—942no—
613C4—942no—
614C4—3761no—
615C4—1882no—
616C4—1881no—
617C4—1882no—
618C4—1882no—
619C4C2×#1309398011yescharacteristic
620C4C4×#1062535211yescharacteristic
621C4#1568965011yescharacteristic
622C4#1568965411yescharacteristic
623C2—44no—
624C2#2024582811yescharacteristic
625C2C4×#1309404511yescharacteristic
626C2#440×D4711yescharacteristic
627C2#2025343411yescharacteristic
628C2#2025343611yescharacteristic
629C2#2025345211yescharacteristic
630C2#2025345611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1877C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#1877D47
hypocenterC47#21114
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #20245863C2semidirect
4. #20244032C2semidirect
5. #20244026C2semidirect
6. #20243967C2semidirect
7. #20243965C2semidirect
11. #20243280C2semidirect
12. #103×Dic47C2semidirect
14. #20243132C2semidirect
20. SD16×Dic47C4semidirect
65. Q8×Dic47C2×C4semidirect
93. #13093956C2×C4semidirect
95. #13093948C2×C4semidirect
99. C8×Dic47C2×C4semidirect
375. C47#21114semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic472
#202458632
C4×D4×Dic472
#202440322
#202440262
#202439672
#202439652
#202437242
#202436352
#202436332
#202432802
#103×Dic472
#102×Dic472
#202431322
C47×#18772
#2111447