Finite Group Databasebeta

← #26217517

Subgroups — #26217517

Order 12032.

1934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D47C211yesmaximal, characteristic
2C2×#117×D47C211yesmaximal, characteristic
3C2×#15689130C212yesmaximal, complemented
4C2×#15689085C211yesmaximal, characteristic
5#20243701C211yesmaximal, characteristic, complemented
6#20243700C211yesmaximal, characteristic, complemented
7#20243698C211yesmaximal, characteristic, complemented
8#20243694C212yesmaximal, complemented
9#20243692C212yesmaximal, complemented
10#20243106C212yesmaximal, complemented
11C47×#1873C211yesmaximal, characteristic, complemented
12C22×Q8×D47C2211yescharacteristic
13C23×C4×D47C2211yescharacteristic
14C22×#10625324C2212yes—
15C2×#13094004C2211yescharacteristic
16C2×#13094003C2211yescharacteristic
17C2×#13094002—21no—
18C2×#13094001—21no—
19C2×#13094000C2211yescharacteristic
20C22⋊Q8×D47—24no—
21C22⋊Q8×D47C418yescomplemented
22C2×#13093983C2211yescharacteristic
23C2×#13093982C2212yes—
24C2×#13093981—21no—
25C2×#13093980—21no—
26#117×D47—28no—
27C2×C4⋊C4×D47—21no—
28C2×C4⋊C4×D47C2212yes—
29C2×C4⋊C4×D47C2211yescharacteristic
30C2×C22⋊C4×D47—21no—
31C2×C22⋊C4×D47—21no—
32C2×C42×D47—21no—
33#15689130—28no—
34C2×#13093949C2212yescomplemented
35C2×#13093948C2212yescomplemented
36C2×#13093947C2212yescomplemented
37#15689085—24no—
38C2×C22⋊(C47⋊C8)C2211yescharacteristic
39C2×(C2×D47)⋊C8C2212yes—
40#15688886C2212yescomplemented
41#15688885C2212yescomplemented
42#15688885C2212yescomplemented
43C2×C22⋊Q8×C47C2211yescharacteristic
44C2×#117×C47C2211yescharacteristic
45C2×#103×C47C2212yescomplemented
46C2×C22⋊C8×C47C2211yescharacteristic
47C47×#340C2212yescomplemented
48C24×D47D411yescharacteristic
49C22×Dic94—21no—
50C22×Dic94C2311yescharacteristic
51C2×Q8×D47—22no—
52C2×Q8×D47—24no—
53C2×Q8×D47—24no—
54C2×Q8×D47C2×C414yescomplemented
55C23×Dic47D411yescharacteristic
56C22×C4×D47—24no—
57C22×C4×D47—24no—
58C22×C4×D47—22no—
59C22×C4×D47—21no—
60C22×C4×D47—21no—
61C22×C4×D47C2×C414yes—
62C22×C4×D47D411yescharacteristic
63C22×C4×D47D411yescharacteristic
64C22×C4×D47D414yes—
65C22×C4×D47C2311yescharacteristic
66C22×C4×D47C2311yescharacteristic
67C2×#10625324—24no—
68C2×#10625324—24no—
69C2×#10625324—24no—
70C22×C47⋊C8C2312yes—
71#13094004—22no—
72#13094004C2×C414yes—
73#13094003—22no—
74#13094003C2×C414yescomplemented
75#13094002—28no—
76#13094001—28no—
77#13094000—22no—
78#13094000C2×C414yes—
79#13093983—24no—
80#13093982—24no—
81#13093982D418yes—
82#13093981—28no—
83#13093980—28no—
84C2×C4⋊Dic47—21no—
85C2×C4⋊Dic47C2311yescharacteristic
86C2×C4⋊Dic47C2312yes—
87C2×Dic47⋊C4—21no—
88C2×Dic47⋊C4—21no—
89C2×Dic47⋊C4—22no—
90C2×Dic47⋊C4—21no—
91C4⋊C4×D47—28no—
92C4⋊C4×D47—22no—
93C4⋊C4×D47—24no—
94C4⋊C4×D47—24no—
95C4⋊C4×D47C2×C414yescomplemented
96C2×C22⋊Dic47—21no—
97C2×C22⋊Dic47—21no—
98C2×(C2×D47)⋊C4—22no—
99C2×(C2×D47)⋊C4—21no—
100C2×(C2×D47)⋊C4—21no—
101C22⋊C4×D47—28no—
102C22⋊C4×D47—28no—
103C2×C4×Dic47—21no—
104C2×C4×Dic47—22no—
105C42×D47—28no—
106#13093949—24no—
107#13093948—24no—
108#13093947—24no—
109C22⋊(C47⋊C8)—22no—
110(C2×D47)⋊C8—24no—
111C22×Q8×C47C2311yescharacteristic
112C23×C4×C47C2311yescharacteristic
113C22×C8×C47C2312yes—
114C22⋊Q8×C47—22no—
115C22⋊Q8×C47C2×C414yescomplemented
116#117×C47—24no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2312yes—
120C2×C22⋊C4×C47—21no—
121C2×C22⋊C4×C47—21no—
122C2×C42×C47—21no—
123#103×C47—24no—
124C22⋊C8×C47—22no—
125C24×C47C2×D411yescharacteristic
126C23×D47—22no—
127C23×D47—22no—
128C23×D47—24no—
129C23×D47—42no—
130C23×D47C22⋊C414yes—
131C23×D47C2×D411yescharacteristic
132C23×D47C2×D411yescharacteristic
133C2×Q8×C47—21no—
134C2×Q8×C47—22no—
135C2×Q8×C47—22no—
136C2×Q8×C47C22×C412yes—
137C22×C4×C47—22no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47C22×C412yes—
143C22×C4×C47C2×D411yescharacteristic
144C22×C4×C47C2×D412yes—
145C22×C4×C47C2411yescharacteristic
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47C22⋊C414yes—
152C22×Dic47C2×D411yescharacteristic
153C22×Dic47C2×D411yescharacteristic
154Q8×D47—216no—
155Q8×D47—48no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×C4×D47—216no—
163C2×C4×D47—216no—
164C2×C4×D47—28no—
165C2×C4×D47—24no—
166C2×C4×D47—24no—
167C2×C4×D47—44no—
168C2×C4×D47—44no—
169C2×C4×D47—42no—
170C2×C4×D47—42no—
171C2×C4×D47—44no—
172C2×C4×D47C22⋊C414yes—
173C2×C4×D47C22⋊C414yes—
174C2×C4×D47C22×C412yes—
175C2×C4×D47C22×C412yes—
176C2×C4×D47C2×D412yes—
177C2×C4×D47C2×D414yes—
178C2×C4×D47C2×D412yes—
179C2×C4×D47#4114yes—
180C2×Dic94—22no—
181C2×Dic94—22no—
182C2×Dic94—21no—
183C2×Dic94—24no—
184C2×Dic94—42no—
185C2×Dic94—42no—
186C2×Dic94C22×C412yes—
187C2×C8×C47—22no—
188C2×C8×C47—22no—
189C2×C8×C47—22no—
190C4⋊C4×C47—24no—
191C4⋊C4×C47—22no—
192C4⋊C4×C47—21no—
193C4⋊C4×C47C22×C412yes—
194C4⋊C4×C47C2×D414yes—
195C22⋊C4×C47—24no—
196C22⋊C4×C47—24no—
197C42×C47—24no—
198C22⋊Dic47—24no—
199C22⋊Dic47—24no—
200(C2×D47)⋊C4—24no—
201(C2×D47)⋊C4—24no—
202(C2×D47)⋊C4—28no—
203C4⋊Dic47—22no—
204C4⋊Dic47—21no—
205C4⋊Dic47—24no—
206C4⋊Dic47C22×C412yes—
207C4⋊Dic47C2×D414yes—
208Dic47⋊C4—28no—
209Dic47⋊C4—24no—
210Dic47⋊C4—24no—
211Dic47⋊C4—24no—
212C4×Dic47—24no—
213C4×Dic47—28no—
214C2×C47⋊C8—22no—
215C2×C47⋊C8—22no—
216C2×C47⋊C8—22no—
217#10625324—48no—
218C23×C47—21no—
219C23×C47—21no—
220C23×C47—22no—
221C23×C47—22no—
222C23×C47C2×C22⋊C412yes—
223C23×C47C22×D411yescharacteristic
224C22×D47—21no—
225C22×D47—21no—
226C22×D47—22no—
227C22×D47—22no—
228C22×D47—22no—
229C22×D47—28no—
230C22×D47—28no—
231C22×D47—44no—
232C22×D47—48no—
233C22×D47—48no—
234C22×D47C2×C22⋊C412yes—
235C22×D47C2×C22⋊C412yes—
236C22×D47#12012yes—
237C22×D47#12312yes—
238C22×D47#13714yes—
239Q8×C47—24no—
240Q8×C47—42no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—24no—
244C2×C4×C47—24no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—24no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47C2×C22⋊C412yes—
254C2×C4×C47C23×C411yescharacteristic
255C2×C4×C47C22×D412yes—
256C2×C4×C47C22×D411yescharacteristic
257C2×C4×C47C2×#4112yes—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—28no—
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—24no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47C2×C22⋊C412yes—
270C2×Dic47C2×C22⋊C412yes—
271C2×Dic47#12012yes—
272C2×Dic47#12312yes—
273C4×D47—28no—
274C4×D47—24no—
275C4×D47—24no—
276C4×D47—28no—
277C4×D47—48no—
278C4×D47—416no—
279C4×D47—48no—
280C4×D47C4×D418yes—
281C4×D47#12118yes—
282Dic94—24no—
283Dic94—42no—
284Dic94—48no—
285C8×C47—42no—
286C47⋊C8—42no—
287#20911—471nomaximal, Sylow
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47C22×C22⋊C411yescharacteristic
294C22×C47C2×#12011yescharacteristic
295C22×C47C2×#12311yescharacteristic
296C22×C47C2×#13712yes—
297C22×C47#57512yes—
298C2×D47—24no—
299C2×D47—24no—
300C2×D47—22no—
301C2×D47—22no—
302C2×D47—22no—
303C2×D47—24no—
304C2×D47—24no—
305C2×D47—22no—
306C2×D47—416no—
307C2×D47#38614yes—
308C2×D47#41914yescomplemented
309C4×C47—22no—
310C4×C47—42no—
311C4×C47—44no—
312C4×C47—42no—
313C4×C47C2×C4×D412yes—
314C4×C47C2×#12112yes—
315Dic47—22no—
316Dic47—42no—
317Dic47—44no—
318Dic47—42no—
319Dic47#38614yes—
320C22×C22⋊Q8—471no—
321C22×#117—471no—
322C2×#419—472no—
323C2×#407—471no—
324#1873—471no—
325#1873—471no—
326#1873—471no—
327#1873D4711yescharacteristic, complemented
328#1872—472no—
329#1872—472no—
330#1736—472no—
331C2×C47—42no—
332C2×C47C2×#38611yescharacteristic
333C2×C47C2×#41911yescharacteristic
334C2×C47#289611yescharacteristic
335C2×C47#300712yes—
336C2×C47#309512yes—
337D47—24no—
338D47—24no—
339C23×Q8—471no—
340C24×C4—471no—
341C22×M16—472no—
342C2×C22⋊Q8—471no—
343C2×C22⋊Q8—471no—
344C2×C22⋊Q8—478no—
345C2×C22⋊Q8—471no—
346C2×C22⋊Q8—941no—
347C2×C22⋊Q8—941no—
348C2×C22⋊Q8—944no—
349C2×C22⋊Q8C2×D4711yescharacteristic
350C2×#117—471no—
351C2×#117—472no—
352C2×#117—941no—
353C2×#117—941no—
354C2×#117—948no—
355C2×#117C2×D4711yescharacteristic
356C22×C4⋊C4—471no—
357C22×C4⋊C4—472no—
358C22×C4⋊C4—941no—
359C22×C22⋊C4—941no—
360C22×C22⋊C4—941no—
361C22×C42—941no—
362#419—948no—
363C2×#103—472no—
364C2×#103—472no—
365C2×#103—472no—
366C2×#103C2×D4712yescomplemented
367#407—944no—
368C2×C22⋊C8—471no—
369C2×C22⋊C8—472no—
370C2×C22⋊C8C2×D4711yescharacteristic
371#340—472no—
372#340—472no—
373#340—472no—
374#340C2×D4712yescomplemented
375C47#2091111yescharacteristic, complemented, Sylow
376C25—471no—
377C22×Q8—474no—
378C22×Q8—471no—
379C22×Q8—941no—
380C22×Q8—942no—
381C22×Q8—944no—
382C22×Q8—944no—
383C22×Q8C22×D4711yescharacteristic
384C23×C4—471no—
385C23×C4—471no—
386C23×C4—474no—
387C23×C4—471no—
388C23×C4—474no—
389C23×C4—471no—
390C23×C4—471no—
391C23×C4—944no—
392C23×C4—944no—
393C23×C4—941no—
394C23×C4—942no—
395C23×C4—941no—
396C23×C4C22×D4711yescharacteristic
397C2×M16—944no—
398C2×M16—944no—
399C2×M16—944no—
400C22×C8—472no—
401C22×C8C22×D4712yes—
402C22⋊Q8—22no—
403C22⋊Q8—474no—
404C22⋊Q8—474no—
405C22⋊Q8—474no—
406C22⋊Q8—948no—
407C22⋊Q8—948no—
408C22⋊Q8—942no—
409C22⋊Q8—942no—
410C22⋊Q8—942no—
411C22⋊Q8C4×D4714yescomplemented
412#117—24no—
413#117—478no—
414#117—948no—
415#117—948no—
416#117—944no—
417#117—944no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—472no—
420C2×C4⋊C4—471no—
421C2×C4⋊C4—474no—
422C2×C4⋊C4—944no—
423C2×C4⋊C4—942no—
424C2×C4⋊C4—948no—
425C2×C4⋊C4—942no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4—944no—
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C4⋊C4C22×D4712yes—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—948no—
436C2×C22⋊C4—948no—
437C2×C22⋊C4—942no—
438C2×C22⋊C4—941no—
439C2×C22⋊C4—941no—
440C2×C22⋊C4—941no—
441C2×C22⋊C4—941no—
442C2×C42—21no—
443C2×C42—948no—
444C2×C42—942no—
445C2×C42—941no—
446#103—24no—
447#103—944no—
448#103—944no—
449#103—944no—
450C22⋊C8—22no—
451C22⋊C8—944no—
452C22⋊C8—942no—
453C24—471no—
454C24—471no—
455C24—474no—
456C24—942no—
457C24—942no—
458C24—944no—
459C24—1882no—
460C24D4×D4711yescharacteristic
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—472no—
465C2×Q8—944no—
466C2×Q8—9416no—
467C2×Q8—941no—
468C2×Q8—942no—
469C2×Q8—942no—
470C2×Q8—1882no—
471C2×Q8—1882no—
472C2×Q8—1888no—
473C2×Q8C2×C4×D4712yes—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—471no—
480C22×C4—472no—
481C22×C4—472no—
482C22×C4—474no—
483C22×C4—474no—
484C22×C4—474no—
485C22×C4—471no—
486C22×C4—474no—
487C22×C4—474no—
488C22×C4—472no—
489C22×C4—472no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—9416no—
493C22×C4—9416no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—944no—
497C22×C4—944no—
498C22×C4—948no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—941no—
505C22×C4—941no—
506C22×C4—1884no—
507C22×C4—1882no—
508C22×C4—1882no—
509C22×C4—1884no—
510C22×C4—1884no—
511C22×C4C2×C4×D4712yes—
512C22×C4D4×D4711yescharacteristic
513C22×C4D4×D4712yes—
514C22×C4C23×D4711yescharacteristic
515M16—1888no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—942no—
520C2×C8—942no—
521C2×C8—942no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—472no—
526C4⋊C4—474no—
527C4⋊C4—944no—
528C4⋊C4—944no—
529C4⋊C4—944no—
530C4⋊C4—948no—
531C4⋊C4—944no—
532C4⋊C4—942no—
533C4⋊C4—941no—
534C4⋊C4C2×C4×D4712yes—
535C4⋊C4D4×D4714yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—948no—
539C22⋊C4—944no—
540C22⋊C4—944no—
541C22⋊C4—944no—
542C22⋊C4—944no—
543C42—24no—
544C42—948no—
545C42—944no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—472no—
551C23—472no—
552C23—472no—
553C23—472no—
554C23—474no—
555C23—948no—
556C23—948no—
557C23—941no—
558C23—941no—
559C23—942no—
560C23—942no—
561C23—942no—
562C23—1884no—
563C23—1888no—
564C23—1888no—
565C23C22⋊C4×D4712yes—
566C23C2×D4×D4711yescharacteristic
567Q8—24no—
568Q8—42no—
569Q8—944no—
570Q8—1888no—
571Q8—1882no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—472no—
585C2×C4—472no—
586C2×C4—472no—
587C2×C4—478no—
588C2×C4—478no—
589C2×C4—472no—
590C2×C4—948no—
591C2×C4—948no—
592C2×C4—948no—
593C2×C4—941no—
594C2×C4—941no—
595C2×C4—944no—
596C2×C4—944no—
597C2×C4—942no—
598C2×C4—942no—
599C2×C4—944no—
600C2×C4—1888no—
601C2×C4—18816no—
602C2×C4—1888no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4—1882no—
608C2×C4C22⋊C4×D4712yes—
609C2×C4C22×C4×D4711yescharacteristic
610C2×C4C2×D4×D4712yes—
611C2×C4C2×D4×D4711yescharacteristic
612C2×C4#41×D4712yes—
613C8—42no—
614C8—1882no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—42no—
619C22—42no—
620C22—474no—
621C22—474no—
622C22—942no—
623C22—944no—
624C22—944no—
625C22—944no—
626C22—944no—
627C22—942no—
628C22—942no—
629C22—942no—
630C22—18816no—
631C22C2×C22⋊C4×D4711yescharacteristic
632C22#120×D4711yescharacteristic
633C22#123×D4711yescharacteristic
634C22#137×D4712yes—
635C22#1568992012yes—
636C4—22no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—474no—
641C4—942no—
642C4—1882no—
643C4—1884no—
644C4—1882no—
645C4C4×D4×D4712yes—
646C4#121×D4712yes—
647C2—42no—
648C2—944no—
649C2—944no—
650C2#386×D4711yescharacteristic
651C2#419×D4711yescharacteristic
652C2#2025190311yescharacteristic
653C2#2025264512yes—
654C2#2025333512yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1873C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1873D47
hypocenterC47#20911
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
3. C2×#15689130C2semidirect
5. #20243701C2semidirect
6. #20243700C2semidirect
7. #20243698C2semidirect
8. #20243694C2semidirect
9. #20243692C2semidirect
10. #20243106C2semidirect
11. C47×#1873C2semidirect
21. C22⋊Q8×D47C4semidirect
34. C2×#13093949C22semidirect
35. C2×#13093948C22semidirect
36. C2×#13093947C22semidirect
40. #15688886C22semidirect
41. #15688885C22semidirect
42. #15688885C22semidirect
45. C2×#103×C47C22semidirect
47. C47×#340C22semidirect
54. C2×Q8×D47C2×C4semidirect
74. #13094003C2×C4semidirect
95. C4⋊C4×D47C2×C4semidirect
115. C22⋊Q8×C47C2×C4semidirect
308. C2×D47#419semidirect
327. #1873D47semidirect
366. C2×#103C2×D47semidirect
374. #340C2×D47semidirect
375. C47#20911semidirect
411. C22⋊Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D472
C2×#117×D472
C2×#156891302
C2×#156890852
#202437012
#202437002
#202436982
#202436942
#202436922
#202431062
C47×#18732
#2091147