Finite Group Databasebeta

← #26138167

Subgroups — #26138167

Order 12032.

2438 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093981C211yesmaximal, characteristic
2C22×C4⋊C4×D47C211yesmaximal, characteristic
3C2×#15688985C211yesmaximal, characteristic
4C2×#15688985C212yesmaximal
5#20241640C212yesmaximal, complemented
6#20241639C212yesmaximal, complemented
7#20241623C211yesmaximal, characteristic, complemented
8#20241621C211yesmaximal, characteristic, complemented
9#20241620C211yesmaximal, characteristic, complemented
10(C2×D47)⋊#95C212yesmaximal
11C47×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C23×C4×D47C2211yescharacteristic
14C23×C4×D47C2211yescharacteristic
15C2×#13093981—24no—
16C2×#13093981—28no—
17C22×C4⋊Dic47C2211yescharacteristic
18C22×Dic47⋊C4C2212yes—
19C22×Dic47⋊C4C2211yescharacteristic
20C22×Dic47⋊C4C2211yescharacteristic
21C2×C4⋊C4×D47—24no—
22C2×C4⋊C4×D47—24no—
23C2×C4⋊C4×D47C418yescomplemented
24C22×(C2×D47)⋊C4C2212yes—
25C22×(C2×D47)⋊C4C2212yes—
26C22×C4×Dic47C2211yescharacteristic
27#15688985—24no—
28#15688985—28no—
29C2×C47⋊#95C2211yescharacteristic
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C2×C47⋊#95C2212yes—
36C2×C47⋊#95C2212yes—
37C22×C4⋊C4×C47C2211yescharacteristic
38C22×C42×C47C2211yescharacteristic
39C2×#95×C47C2211yescharacteristic
40C2×#95×C47C2212yes—
41C2×#95×C47C2212yes—
42C24×D47C2311yescharacteristic
43C23×Dic47C2312yes—
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2312yes—
46C23×Dic47C2311yescharacteristic
47C23×Dic47C2311yescharacteristic
48C22×C4×D47—22no—
49C22×C4×D47—22no—
50C22×C4×D47—24no—
51C22×C4×D47—24no—
52C22×C4×D47—24no—
53C22×C4×D47—24no—
54C22×C4×D47—24no—
55C22×C4×D47—22no—
56C22×C4×D47—22no—
57C22×C4×D47—24no—
58C22×C4×D47C2×C414yes—
59C22×C4×D47C2×C414yes—
60C22×C4×D47C2×C414yes—
61C22×C4×D47D414yes—
62C22×C4×D47D414yes—
63#13093981—232no—
64C2×C4⋊Dic47—22no—
65C2×C4⋊Dic47—22no—
66C2×C4⋊Dic47C2×C414yes—
67C2×Dic47⋊C4—22no—
68C2×Dic47⋊C4—22no—
69C2×Dic47⋊C4—24no—
70C2×Dic47⋊C4—22no—
71C2×Dic47⋊C4—22no—
72C2×Dic47⋊C4—24no—
73C2×Dic47⋊C4—24no—
74C2×Dic47⋊C4C2×C414yes—
75C2×Dic47⋊C4C2×C414yes—
76C4⋊C4×D47—232no—
77C2×(C2×D47)⋊C4—24no—
78C2×(C2×D47)⋊C4—24no—
79C2×(C2×D47)⋊C4—24no—
80C2×(C2×D47)⋊C4—24no—
81C2×(C2×D47)⋊C4—24no—
82C2×(C2×D47)⋊C4D418yes—
83C2×C4×Dic47—22no—
84C2×C4×Dic47—24no—
85C47⋊#95—24no—
86C47⋊#95—22no—
87C47⋊#95—24no—
88C47⋊#95—24no—
89C47⋊#95—24no—
90C47⋊#95—24no—
91C47⋊#95—24no—
92C47⋊#95—24no—
93C23×C4×C47C2312yes—
94C23×C4×C47C2311yescharacteristic
95C23×C4×C47C2312yes—
96C23×C4×C47C2311yescharacteristic
97C23×C4×C47C2311yescharacteristic
98C2×C4⋊C4×C47—22no—
99C2×C4⋊C4×C47—22no—
100C2×C4⋊C4×C47C2×C414yes—
101C2×C42×C47—24no—
102C2×C42×C47—22no—
103#95×C47—24no—
104#95×C47—22no—
105#95×C47—24no—
106C24×C47C2411yescharacteristic
107C23×D47—22no—
108C23×D47—22no—
109C23×D47—22no—
110C23×D47—22no—
111C23×D47C22×C412yes—
112C23×D47C2×D412yes—
113C23×D47C2×D412yes—
114C23×D47C2×D414yes—
115C23×D47#4114yes—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47C22×C412yes—
139C22×C4×C47C22×C412yes—
140C22×C4×C47C22×C412yes—
141C22×C4×C47C2×D414yes—
142C22×C4×C47C2×D412yes—
143C22×C4×C47C2×D412yes—
144C22×C4×C47#4114yes—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47C22×C412yes—
168C22×Dic47C22×C412yes—
169C22×Dic47C22×C412yes—
170C22×Dic47C2×D414yes—
171C22×Dic47C2×D412yes—
172C22×Dic47C2×D412yes—
173C22×Dic47#4114yes—
174C2×C4×D47—216no—
175C2×C4×D47—216no—
176C2×C4×D47—28no—
177C2×C4×D47—216no—
178C2×C4×D47—28no—
179C2×C4×D47—216no—
180C2×C4×D47—28no—
181C2×C4×D47—28no—
182C2×C4×D47—28no—
183C2×C4×D47—28no—
184C2×C4×D47—216no—
185C2×C4×D47—44no—
186C2×C4×D47—48no—
187C2×C4×D47—44no—
188C2×C4×D47C22⋊C4116yes—
189C4⋊C4×C47—28no—
190C42×C47—28no—
191(C2×D47)⋊C4—216no—
192(C2×D47)⋊C4—48no—
193C4⋊Dic47—28no—
194Dic47⋊C4—216no—
195Dic47⋊C4—28no—
196Dic47⋊C4—28no—
197C4×Dic47—28no—
198C23×C47C23×C411yescharacteristic
199C23×C47C22×D411yescharacteristic
200C23×C47C22×D411yescharacteristic
201C23×C47C22×D412yes—
202C23×C47C2×#4112yes—
203C23×C47C2×#4112yes—
204C23×C47C2×#4112yes—
205C23×C472−1+412yes—
206C23×C472−1+412yes—
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—24no—
210C22×D47—24no—
211C22×D47—24no—
212C22×D47—22no—
213C22×D47—24no—
214C22×D47—22no—
215C22×D47—24no—
216C22×D47—24no—
217C22×D47—24no—
218C22×D47—24no—
219C22×D47—24no—
220C22×D47—24no—
221C22×D47—24no—
222C22×D47—24no—
223C22×D47C2×C22⋊C414yes—
224C22×D47C4×D418yes—
225C22×D47#12014yes—
226C22×D47#12118yes—
227C22×D47#12314yes—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—24no—
231C2×C4×C47—22no—
232C2×C4×C47—24no—
233C2×C4×C47—22no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—22no—
237C2×C4×C47—24no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—24no—
241C2×C4×C47—41no—
242C2×C4×C47—42no—
243C2×C4×C47—41no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47—42no—
254C2×C4×C47—42no—
255C2×C4×C47C2×C22⋊C414yes—
256C2×C4×C47#11714yescomplemented
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—22no—
261C2×Dic47—24no—
262C2×Dic47—22no—
263C2×Dic47—24no—
264C2×Dic47—24no—
265C2×Dic47—22no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—24no—
269C2×Dic47—24no—
270C2×Dic47—41no—
271C2×Dic47—42no—
272C2×Dic47—41no—
273C2×Dic47—42no—
274C2×Dic47—42no—
275C2×Dic47—42no—
276C2×Dic47—42no—
277C2×Dic47—42no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—42no—
282C2×Dic47—42no—
283C2×Dic47—42no—
284C2×Dic47C2×C22⋊C414yes—
285C2×Dic47#11714yescomplemented
286C4×D47—232no—
287C4×D47—232no—
288C4×D47—416no—
289#13870—471nomaximal, Sylow
290C22×C47C22×C22⋊C411yescharacteristic
291C22×C47C2×#11711yescharacteristic
292C22×C47C2×#11711yescharacteristic
293C22×C47C2×C4×D412yes—
294C22×C47#51912yes—
295C22×C47C2×#12011yescharacteristic
296C22×C47C2×#12112yes—
297C22×C47C2×#12311yescharacteristic
298C22×C47#52512yes—
299C22×C47#52512yes—
300C22×C47#53611yescharacteristic
301C22×C47#53612yes—
302C22×C47#53611yescharacteristic
303C22×C47#54712yes—
304C22×C47#54712yes—
305C22×C47#54712yes—
306C22×C47#54712yes—
307C22×C47#55212yes—
308C22×C47#55212yes—
309C22×C47#55212yes—
310C22×C47#55212yes—
311C2×D47—24no—
312C2×D47—28no—
313C2×D47—28no—
314C2×D47—28no—
315C2×D47—28no—
316C2×D47—28no—
317C2×D47—28no—
318C2×D47—24no—
319C2×D47#38618yes—
320C4×C47—24no—
321C4×C47—24no—
322C4×C47—44no—
323C4×C47—42no—
324C4×C47—82no—
325Dic47—24no—
326Dic47—24no—
327Dic47—44no—
328Dic47—42no—
329Dic47—82no—
330C22×#117—471no—
331C23×C4⋊C4—471no—
332C2×#381—472no—
333C2×#381—471no—
334#1441—472no—
335#1441—472no—
336#1436—471no—
337#1436—471no—
338#1436—471no—
339#1436D4711yescharacteristic, complemented
340C22⋊#95—472no—
341C2×C47#228011yescharacteristic
342C2×C47C2×#38611yescharacteristic
343C2×C47#231011yescharacteristic
344C2×C47#234412yes—
345C2×C47#234412yes—
346C2×C47#240612yes—
347C2×C47#241312yes—
348C2×C47#241312yes—
349C2×C47#248412yes—
350D47—28no—
351C24×C4—471no—
352C24×C4—471no—
353C24×C4—471no—
354C2×#117—948no—
355C2×#117—944no—
356C22×C4⋊C4—471no—
357C22×C4⋊C4—471no—
358C22×C4⋊C4—471no—
359C22×C4⋊C4—478no—
360C22×C4⋊C4—472no—
361C22×C4⋊C4—944no—
362C22×C4⋊C4—944no—
363C22×C4⋊C4C2×D4711yescharacteristic
364C22×C22⋊C4—472no—
365C22×C22⋊C4—472no—
366C22×C42—471no—
367C22×C42C2×D4711yescharacteristic
368#381—948no—
369#381—944no—
370C2×#95—472no—
371C2×#95—472no—
372C2×#95—472no—
373C2×#95—471no—
374C2×#95—472no—
375C2×#95—472no—
376C2×#95—472no—
377C2×#95—472no—
378C2×#95C2×D4712yes—
379C2×#95C2×D4711yescharacteristic
380C2×#95C2×D4712yes—
381C47#1387011yescharacteristic, complemented, Sylow
382C25—471no—
383C23×C4—471no—
384C23×C4—471no—
385C23×C4—474no—
386C23×C4—474no—
387C23×C4—474no—
388C23×C4—472no—
389C23×C4—471no—
390C23×C4—474no—
391C23×C4—474no—
392C23×C4—472no—
393C23×C4—944no—
394C23×C4—942no—
395C23×C4—942no—
396C23×C4—944no—
397C23×C4—944no—
398C23×C4—942no—
399C23×C4—942no—
400C23×C4—944no—
401C23×C4—944no—
402C23×C4—944no—
403C23×C4C22×D4712yes—
404C23×C4C22×D4711yescharacteristic
405C23×C4C22×D4712yes—
406C23×C4C22×D4711yescharacteristic
407C23×C4C22×D4711yescharacteristic
408#117—9432no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—474no—
412C2×C4⋊C4—474no—
413C2×C4⋊C4—474no—
414C2×C4⋊C4—9432no—
415C2×C4⋊C4—942no—
416C2×C4⋊C4—942no—
417C2×C4⋊C4—942no—
418C2×C4⋊C4—942no—
419C2×C4⋊C4—944no—
420C2×C4⋊C4—944no—
421C2×C4⋊C4—942no—
422C2×C4⋊C4—942no—
423C2×C4⋊C4—944no—
424C2×C4⋊C4C4×D4714yes—
425C2×C22⋊C4—478no—
426C2×C22⋊C4—944no—
427C2×C22⋊C4—944no—
428C2×C22⋊C4—944no—
429C2×C22⋊C4—944no—
430C2×C22⋊C4—944no—
431C2×C42—24no—
432C2×C42—22no—
433C2×C42—944no—
434C2×C42—942no—
435#95—24no—
436#95—24no—
437#95—22no—
438#95—944no—
439#95—944no—
440#95—944no—
441#95—944no—
442#95—944no—
443#95—944no—
444#95—944no—
445#95—942no—
446C24—472no—
447C24—474no—
448C24—472no—
449C24—472no—
450C24—474no—
451C24—942no—
452C24—942no—
453C24—942no—
454C24—942no—
455C24C23×D4711yescharacteristic
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—472no—
479C22×C4—472no—
480C22×C4—472no—
481C22×C4—472no—
482C22×C4—474no—
483C22×C4—4716no—
484C22×C4—472no—
485C22×C4—474no—
486C22×C4—948no—
487C22×C4—948no—
488C22×C4—948no—
489C22×C4—9416no—
490C22×C4—9416no—
491C22×C4—948no—
492C22×C4—9416no—
493C22×C4—9416no—
494C22×C4—948no—
495C22×C4—9416no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—948no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—941no—
506C22×C4—941no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—942no—
512C22×C4—942no—
513C22×C4—942no—
514C22×C4—942no—
515C22×C4—942no—
516C22×C4—942no—
517C22×C4—942no—
518C22×C4—942no—
519C22×C4—1884no—
520C22×C4—1888no—
521C22×C4—1884no—
522C22×C4C2×C4×D4712yes—
523C22×C4C2×C4×D4712yes—
524C22×C4C2×C4×D4712yes—
525C22×C4#1062535014yes—
526C22×C4D4×D4714yes—
527C22×C4D4×D4712yes—
528C22×C4D4×D4712yes—
529C4⋊C4—28no—
530C4⋊C4—948no—
531C4⋊C4—948no—
532C4⋊C4—9416no—
533C4⋊C4—948no—
534C22⋊C4—9416no—
535C22⋊C4—1888no—
536C42—28no—
537C42—948no—
538C23—474no—
539C23—478no—
540C23—474no—
541C23—478no—
542C23—474no—
543C23—944no—
544C23—944no—
545C23—944no—
546C23—944no—
547C23—944no—
548C23—944no—
549C23—944no—
550C23—944no—
551C23—942no—
552C23—942no—
553C23—944no—
554C23—944no—
555C23—944no—
556C23—942no—
557C23—944no—
558C23—942no—
559C23C22×C4×D4711yescharacteristic
560C23C2×D4×D4712yes—
561C23C2×D4×D4711yescharacteristic
562C23C2×D4×D4711yescharacteristic
563C23#41×D4712yes—
564C23#41×D4712yes—
565C23C2×#1062535012yes—
566C23#1309407112yes—
567C23#1309407212yes—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—474no—
596C2×C4—474no—
597C2×C4—9432no—
598C2×C4—9432no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—944no—
602C2×C4—942no—
603C2×C4—944no—
604C2×C4—942no—
605C2×C4—944no—
606C2×C4—944no—
607C2×C4—944no—
608C2×C4—944no—
609C2×C4—942no—
610C2×C4—944no—
611C2×C4—944no—
612C2×C4—18816no—
613C2×C4—1882no—
614C2×C4—1882no—
615C2×C4—1882no—
616C2×C4—1882no—
617C2×C4—1882no—
618C2×C4—1882no—
619C2×C4—1882no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4—1881no—
623C2×C4—1882no—
624C2×C4—1881no—
625C2×C4—1882no—
626C2×C4—1882no—
627C2×C4C22⋊C4×D4714yes—
628C2×C4#1309398114yescomplemented
629C22—478no—
630C22—948no—
631C22—944no—
632C22—948no—
633C22—948no—
634C22—948no—
635C22—944no—
636C22—948no—
637C22—948no—
638C22C2×C22⋊C4×D4711yescharacteristic
639C22#117×D4711yescharacteristic
640C22C2×#1309398111yescharacteristic
641C22C4×D4×D4712yes—
642C22#1568959912yes—
643C22#120×D4711yescharacteristic
644C22#121×D4712yes—
645C22#123×D4711yescharacteristic
646C22#1568963212yes—
647C22#1568963212yes—
648C22#1568969311yescharacteristic
649C22#1568969612yes—
650C22#1568969811yescharacteristic
651C22#1568977112yes—
652C22#1568977412yes—
653C22#1568977512yes—
654C22#1568977812yes—
655C22#1568980512yes—
656C22#1568980912yes—
657C22#1568981212yes—
658C22#1568981412yes—
659C4—24no—
660C4—24no—
661C4—42no—
662C4—44no—
663C4—82no—
664C4—944no—
665C4—944no—
666C4—3762no—
667C4—1882no—
668C4—1884no—
669C2—948no—
670C2#2024576911yescharacteristic
671C2#386×D4711yescharacteristic
672C2#2024597111yescharacteristic
673C2#2024625312yes—
674C2#2024625412yes—
675C2#2024672112yes—
676C2#2024679212yes—
677C2#2024679412yes—
678C2#2024767312yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1436C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenter#1436D47
hypocenterC47#13870
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
5. #20241640C2semidirect
6. #20241639C2semidirect
7. #20241623C2semidirect
8. #20241621C2semidirect
9. #20241620C2semidirect
11. C47×#1436C2semidirect
23. C2×C4⋊C4×D47C4semidirect
256. C2×C4×C47#117semidirect
285. C2×Dic47#117semidirect
339. #1436D47semidirect
381. C47#13870semidirect
628. C2×C4#13093981semidirect

Maximal subgroups

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

subgroupindex
C22×#130939812
C22×C4⋊C4×D472
C2×#156889852
#202416402
#202416392
#202416232
#202416212
#202416202
(C2×D47)⋊#952
C47×#14362
#1387047