Finite Group Databasebeta

← #26243710

Subgroups — #26243710

Order 12032.

1148 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256793C211yesmaximal, characteristic, complemented
2C2×#15689140C211yesmaximal, characteristic
3C2×#15689097C211yesmaximal, characteristic
4#20244560C218yesmaximal, complemented
5C2×#15688895C211yesmaximal, characteristic, complemented
6#20243098C211yesmaximal, characteristic, complemented
7C2×#15688880C211yesmaximal, characteristic, complemented
8C2×C47×#421C211yesmaximal, characteristic
9C22×#10625329C2211yescharacteristic
10#15689754—22no—
11#15689751—22no—
12#15689751—22no—
13#15689749—22no—
14#15689680C2214yescomplemented
15#15689676—22no—
16#15689676—22no—
17C2×#13093983C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19C2×#13093983C2211yescharacteristic
20#15689140—22no—
21#15689140C2214yes—
22C2×#13093951—22no—
23#15689097—22no—
24#15689097C2214yes—
25C2×#13093939—22no—
26#15688895C2214yescomplemented
27#15688883C2214yescomplemented
28#15688880C2214yescomplemented
29C22×#41×C47C2211yescharacteristic
30C22×M16×C47C2211yescharacteristic
31C2×#117×C47C2211yescharacteristic
32C47×#421—22no—
33C47×#421—22no—
34C2×#104×C47C2214yes—
35C2×#10625329—21no—
36C2×#10625329—21no—
37C2×#10625329—21no—
38C2×#10625329—21no—
39C2×#10625329—21no—
40C2×#10625329C2312yes—
41C22×C47⋊C8—21no—
42#13094024—24no—
43#13094023—24no—
44#13094023—24no—
45#13094016—24no—
46#13094015—24no—
47#13094015—24no—
48#13094003—28no—
49#13094001—28no—
50#13094001—28no—
51C4×Dic94—28no—
52#13093983—21no—
53#13093983—21no—
54#13093983—21no—
55#13093983—21no—
56#13093983—21no—
57#13093983—21no—
58#13093983—21no—
59#13093983D412yes—
60#13093983D412yes—
61#13093983C2312yes—
62#13093983C2312yes—
63#13093983C2312yes—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—21no—
66C2×C4⋊Dic47—21no—
67C2×C22⋊Dic47—21no—
68C2×C22⋊Dic47—21no—
69C2×C22⋊Dic47D412yes—
70C2×C4×Dic47—21no—
71C2×C4×Dic47—21no—
72C2×C4×Dic47D412yes—
73#13093951—28no—
74#13093939—28no—
75C2×#41×C47—21no—
76C2×#41×C47—21no—
77C2×#41×C47—21no—
78C2×#41×C47—21no—
79C2×#41×C47—21no—
80C2×#41×C47—21no—
81C2×#41×C47—21no—
82C2×#41×C47—21no—
83C2×#41×C47—21no—
84C2×#41×C47—21no—
85C2×#41×C47Q812yescomplemented
86C2×#41×C47C2312yes—
87C22×Q8×C47Q811yescharacteristic, complemented
88C22×D4×C47—21no—
89C22×D4×C47Q811yescharacteristic, complemented
90C23×C4×C47—21no—
91C23×C4×C47C2311yescharacteristic
92C2×M16×C47—21no—
93C2×M16×C47—21no—
94C2×M16×C47—21no—
95C2×M16×C47—21no—
96C2×M16×C47—21no—
97C2×M16×C47C2312yes—
98C22×C8×C47—21no—
99#117×C47—21no—
100#117×C47—21no—
101#117×C47D412yes—
102#117×C47D412yes—
103C2×C4⋊C4×C47—21no—
104C2×C22⋊C4×C47—21no—
105C2×C42×C47C2312yes—
106#104×C47—28no—
107C24×C47—21no—
108C24×C47C2×D411yescharacteristic
109#41×C47—22no—
110#41×C47—22no—
111#41×C47—22no—
112#41×C47—22no—
113#41×C47—22no—
114#41×C47—22no—
115#41×C47—22no—
116#41×C47—22no—
117#41×C47—41no—
118#41×C47—41no—
119#41×C47—41no—
120#41×C47—41no—
121#41×C47—41no—
122#41×C47—41no—
123#41×C47—41no—
124#41×C47—41no—
125C2×Q8×C47—21no—
126C2×Q8×C47—21no—
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47C2×Q812yes—
131C2×D4×C47—22no—
132C2×D4×C47—22no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47—41no—
139C2×D4×C47—41no—
140C2×D4×C47—41no—
141C2×D4×C47—41no—
142C2×D4×C47C2×Q812yes—
143C22×C4×C47—21no—
144C22×C4×C47—22no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—41no—
148C22×C4×C47—41no—
149C22×C4×C47—41no—
150C22×C4×C47—41no—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×Q811yescharacteristic
157C22×C4×C47C2×Q811yescharacteristic
158C22×C4×C47#4111yescharacteristic
159C22×C4×C47#4111yescharacteristic
160C22×C4×C47#4111yescharacteristic
161C22×C4×C47#4111yescharacteristic
162C22×C4×C47#4111yescharacteristic
163C22×C4×C47#4111yescharacteristic
164C22×C4×C47C2411yescharacteristic
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47—41no—
168C2×Dic94—28no—
169M16×C47—22no—
170M16×C47—22no—
171M16×C47—41no—
172M16×C47—41no—
173C2×C8×C47—22no—
174C2×C8×C47—41no—
175C2×C8×C47—41no—
176C4⋊C4×C47—22no—
177C4⋊C4×C47—22no—
178C22⋊C4×C47—22no—
179C22⋊C4×C47—22no—
180C42×C47—22no—
181C42×C47C2×D414yes—
182C22⋊Dic47—22no—
183C22⋊Dic47—22no—
184C22⋊Dic47—22no—
185C22⋊Dic47—22no—
186C22⋊Dic47—41no—
187C22⋊Dic47C2×D414yes—
188C4⋊Dic47—28no—
189C4⋊Dic47—22no—
190C4⋊Dic47—22no—
191C4⋊Dic47—22no—
192C4⋊Dic47—22no—
193C4⋊Dic47—22no—
194C4⋊Dic47—41no—
195Dic47⋊C4—28no—
196Dic47⋊C4—28no—
197C4×Dic47—22no—
198C4×Dic47—22no—
199C4×Dic47—22no—
200C4×Dic47—22no—
201C4×Dic47—41no—
202C4×Dic47C2×D414yes—
203#10625329—22no—
204#10625329—22no—
205#10625329—41no—
206#10625329—41no—
207C2×C47⋊C8—22no—
208C2×C47⋊C8—41no—
209C2×C47⋊C8—41no—
210C23×C47—22no—
211C23×C47—41no—
212C23×C47—41no—
213C23×C47—81no—
214C23×C47—82no—
215C23×C47#12011yescharacteristic
216C23×C47#12011yescharacteristic
217C23×C47C22⋊Q811yescharacteristic
218C23×C47C22⋊Q811yescharacteristic
219C23×C47#12411yescharacteristic
220C23×C47#12411yescharacteristic
221C23×C47C22×D411yescharacteristic
222Q8×C47—22no—
223Q8×C47—22no—
224Q8×C47—41no—
225Q8×C47—41no—
226D4×C47—22no—
227D4×C47—22no—
228D4×C47—42no—
229D4×C47—42no—
230D4×C47—42no—
231D4×C47—42no—
232D4×C47—41no—
233D4×C47—41no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—82no—
243C2×C4×C47#12011yescharacteristic
244C2×C4×C47#12011yescharacteristic
245C2×C4×C47C22⋊Q811yescharacteristic
246C2×C4×C47C22⋊Q811yescharacteristic
247C2×C4×C47C22⋊Q811yescharacteristic
248C2×C4×C47C22⋊Q811yescharacteristic
249C2×C4×C47C22⋊Q811yescharacteristic
250C2×C4×C47C22⋊Q811yescharacteristic
251C2×C4×C47C22⋊Q811yescharacteristic
252C2×C4×C47C22⋊Q811yescharacteristic
253C2×C4×C47C22⋊Q811yescharacteristic
254C2×C4×C47C22⋊Q811yescharacteristic
255C2×C4×C47#12411yescharacteristic
256C2×C4×C47#12411yescharacteristic
257C2×C4×C47#12411yescharacteristic
258C2×C4×C47#12411yescharacteristic
259C2×C4×C47#12411yescharacteristic
260C2×C4×C47#12411yescharacteristic
261C2×C4×C47#12411yescharacteristic
262C2×C4×C47#12411yescharacteristic
263C2×C4×C47#12411yescharacteristic
264C2×C4×C47#12411yescharacteristic
265C2×C4×C47C22×D411yescharacteristic
266C2×C4×C47C22×D411yescharacteristic
267C2×C4×C47C22×Q811yescharacteristic
268C2×C4×C47C2×#4111yescharacteristic
269C2×C4×C47C2×#4111yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—42no—
274C2×Dic47—82no—
275C2×Dic47—82no—
276C2×Dic47—82no—
277Dic94—88no—
278C8×C47—42no—
279C47⋊C8—42no—
280#23472—471nomaximal, Sylow
281C22×C47—41no—
282C22×C47—41no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—42no—
289C22×C47—82no—
290C22×C47—82no—
291C22×C47—82no—
292C22×C47#39311yescharacteristic
293C22×C47#39311yescharacteristic
294C22×C47#39311yescharacteristic
295C22×C47#39311yescharacteristic
296C22×C47C2×#12011yescharacteristic
297C22×C47C2×C22⋊Q811yescharacteristic
298C22×C47C2×#12411yescharacteristic
299C4×C47—44no—
300C4×C47—42no—
301C4×C47#39311yescharacteristic
302C4×C47#39311yescharacteristic
303C4×C47#39311yescharacteristic
304C4×C47#39311yescharacteristic
305C4×C47C2×C22⋊Q811yescharacteristic
306C4×C47C2×C22⋊Q811yescharacteristic
307C4×C47C2×#12411yescharacteristic
308C4×C47C2×#12411yescharacteristic
309Dic47—44no—
310Dic47—44no—
311Dic47—82no—
312#3458—471no—
313C2×#421—471no—
314C2×#421D4711yescharacteristic
315C2×#410—471no—
316#2031—478no—
317C2×#344—471no—
318#1733—471no—
319C2×H(3,ℤ/4)—471no—
320C2×C47—21no—
321C2×C47—21no—
322C2×C47—21no—
323C2×C47—21no—
324C2×C47—21no—
325C2×C47—21no—
326C2×C47—21no—
327C2×C47—82no—
328C2×C47C2×#39311yescharacteristic
329C22×#41C2×D4711yescharacteristic
330C22×M16—471no—
331C22×M16C2×D4711yescharacteristic
332#544—942no—
333#544—942no—
334#544—942no—
335#543—942no—
336#533—474no—
337#533—942no—
338#533—942no—
339C2×#117—471no—
340C2×#117—471no—
341C2×#117—471no—
342C2×#117C2×D4711yescharacteristic
343#421—22no—
344#421—22no—
345#421—474no—
346#421—942no—
347C2×#104—942no—
348C2×#104C2×D4714yes—
349#410—474no—
350#410—942no—
351C2×#99—942no—
352#344—474no—
353#339—474no—
354H(3,ℤ/4)—474no—
355C47#2347211yescharacteristic, complemented, Sylow
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—21no—
359C2×#41—21no—
360C2×#41—21no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41Dic9412yescomplemented
367C2×#41C22×D4712yes—
368C22×Q8Dic9411yescharacteristic, complemented
369C22×D4—21no—
370C22×D4Dic9411yescharacteristic, complemented
371C23×C4—21no—
372C23×C4C22×D4711yescharacteristic
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—472no—
379C2×M16—941no—
380C2×M16—941no—
381C2×M16—941no—
382C2×M16—941no—
383C2×M16—941no—
384C2×M16C22×D4712yes—
385C22×C8—21no—
386C22×C8—941no—
387C4⋊Q8—944no—
388C4⋊Q8—944no—
389C4⋊Q8—944no—
390#125—944no—
391#125—944no—
392#125—944no—
393C22⋊Q8—948no—
394C22⋊Q8—948no—
395C22⋊Q8—948no—
396C4×Q8—948no—
397#117—21no—
398#117—21no—
399#117—472no—
400#117—472no—
401#117—472no—
402#117—472no—
403#117—472no—
404#117—941no—
405#117—941no—
406#117—941no—
407#117—941no—
408#117—941no—
409#117—941no—
410#117—941no—
411#117C22⋊D4712yes—
412#117C22⋊D4712yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—941no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—472no—
419C2×C22⋊C4—941no—
420C2×C22⋊C4—941no—
421C2×C42—472no—
422C2×C42—941no—
423C2×C42—941no—
424C2×C42C22×D4712yes—
425#104—28no—
426#104—948no—
427#99—948no—
428C24—21no—
429C24D4×D4711yescharacteristic
430#41—22no—
431#41—22no—
432#41—22no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—41no—
439#41—41no—
440#41—41no—
441#41—41no—
442#41—41no—
443#41—41no—
444#41—41no—
445#41—41no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—948no—
452C2×Q8C2×Dic9412yes—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4C2×Dic9412yes—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—1881no—
476C22×C4C2×Dic9411yescharacteristic
477C22×C4C2×Dic9411yescharacteristic
478C22×C4#1062535011yescharacteristic
479C22×C4#1062535011yescharacteristic
480C22×C4D4×D4711yescharacteristic
481C22×C4D4×D4711yescharacteristic
482C22×C4D4×D4711yescharacteristic
483C22×C4#1062535211yescharacteristic
484C22×C4#1062535211yescharacteristic
485C22×C4#1062535211yescharacteristic
486C22×C4#1062535211yescharacteristic
487C22×C4C2×C22⋊D4711yescharacteristic
488C22×C4C2×C22⋊D4711yescharacteristic
489C22×C4C23×D4711yescharacteristic
490M16—22no—
491M16—22no—
492M16—41no—
493M16—41no—
494M16—942no—
495M16—942no—
496M16—1881no—
497M16—1881no—
498C2×C8—22no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—942no—
502C2×C8—1881no—
503C2×C8—1881no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—948no—
507C4⋊C4—948no—
508C4⋊C4—948no—
509C4⋊C4—942no—
510C4⋊C4—942no—
511C4⋊C4—942no—
512C4⋊C4—942no—
513C4⋊C4—942no—
514C4⋊C4—1881no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—474no—
518C22⋊C4—942no—
519C22⋊C4—942no—
520C22⋊C4—942no—
521C22⋊C4—942no—
522C22⋊C4—1881no—
523C42—22no—
524C42—474no—
525C42—942no—
526C42—942no—
527C42—942no—
528C42—942no—
529C42—1881no—
530C42C2×C22⋊D4714yes—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—81no—
535C23—82no—
536C23#1309399211yescharacteristic
537C23#1309399211yescharacteristic
538C23#1309400111yescharacteristic
539C23#1309400111yescharacteristic
540C23#1309401011yescharacteristic
541C23#1309401011yescharacteristic
542C23C2×D4×D4711yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547Q8—3768no—
548D4—22no—
549D4—22no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—41no—
555D4—41no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—82no—
565C2×C4—944no—
566C2×C4—944no—
567C2×C4—3762no—
568C2×C4—3762no—
569C2×C4—3762no—
570C2×C4—1882no—
571C2×C4#1309399211yescharacteristic
572C2×C4#1309399211yescharacteristic
573C2×C4#1309400111yescharacteristic
574C2×C4#1309400111yescharacteristic
575C2×C4#1309400111yescharacteristic
576C2×C4#1309400111yescharacteristic
577C2×C4#1309400111yescharacteristic
578C2×C4#1309400111yescharacteristic
579C2×C4#1309400311yescharacteristic
580C2×C4#1309400311yescharacteristic
581C2×C4#1309400311yescharacteristic
582C2×C4#1309400311yescharacteristic
583C2×C4#1309401011yescharacteristic
584C2×C4#1309401011yescharacteristic
585C2×C4#1309401011yescharacteristic
586C2×C4#1309401011yescharacteristic
587C2×C4#1309401011yescharacteristic
588C2×C4#1309401011yescharacteristic
589C2×C4#1309401111yescharacteristic
590C2×C4#1309401111yescharacteristic
591C2×C4#1309401111yescharacteristic
592C2×C4#1309401111yescharacteristic
593C2×C4C2×D4×D4711yescharacteristic
594C2×C4C22×C22⋊D4711yescharacteristic
595C2×C4C22×Dic9411yescharacteristic
596C2×C4C2×#1062535211yescharacteristic
597C2×C4C2×#1062535211yescharacteristic
598C2×C4C2×#1062535011yescharacteristic
599C8—42no—
600C8—1882no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22#1568903511yescharacteristic
613C22#1568903511yescharacteristic
614C22#1568903511yescharacteristic
615C22#1568903511yescharacteristic
616C22C2×#1309399211yescharacteristic
617C22C2×#1309400111yescharacteristic
618C22C2×#1309401011yescharacteristic
619C4—44no—
620C4—42no—
621C4—3762no—
622C4—1884no—
623C4—1884no—
624C4#1568903511yescharacteristic
625C4#1568903511yescharacteristic
626C4#1568903511yescharacteristic
627C4#1568903511yescharacteristic
628C4C2×#1309400111yescharacteristic
629C4C2×#1309400311yescharacteristic
630C4C2×#1309401011yescharacteristic
631C4C2×#1309401111yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—82no—
640C2C2×#1568903511yescharacteristic
subgroupG/H
centerC4C2×#13094010
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#421C2
socleC2×C47C2×#393
prime radicalC22×#41×C47C22
hypercenterC2×#421D47
hypocenterC47#23472
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
1. #20256793C2semidirect
4. #20244560C2semidirect
5. C2×#15688895C2semidirect
6. #20243098C2semidirect
7. C2×#15688880C2semidirect
14. #15689680C22semidirect
26. #15688895C22semidirect
27. #15688883C22semidirect
28. #15688880C22semidirect
85. C2×#41×C47Q8semidirect
87. C22×Q8×C47Q8semidirect
89. C22×D4×C47Q8semidirect
355. C47#23472semidirect
366. C2×#41Dic94semidirect
368. C22×Q8Dic94semidirect
370. C22×D4Dic94semidirect

Maximal subgroups

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

subgroupindex
#202567932
C2×#156891402
C2×#156890972
#202445602
C2×#156888952
#202430982
C2×#156888802
C2×C47×#4212
#2347247