Finite Group Databasebeta

← #26243720

Subgroups — #26243720

Order 12032.

1164 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256793C211yesmaximal, characteristic, complemented
2C2×#15689097C211yesmaximal, characteristic
3C2×#15689097C211yesmaximal, characteristic
4#20244565C218yesmaximal, complemented
5#20243121C211yesmaximal, characteristic, complemented
6#20243120C211yesmaximal, characteristic, complemented
7#20243120C211yesmaximal, characteristic, complemented
8C2×C47×#410C211yesmaximal, characteristic
9#15689754—22no—
10#15689751—22no—
11#15689751—22no—
12#15689749C2214yes—
13#15689680—22no—
14#15689676—22no—
15#15689676—22no—
16C2×#13093983C2211yescharacteristic
17C2×#13093983C2211yescharacteristic
18C2×#13093983C2211yescharacteristic
19C2×#13093983C2211yescharacteristic
20#15689097—22no—
21#15689097—22no—
22#15689097—22no—
23#15689097—22no—
24C2×#13093939C2214yes—
25C2×#13093939C2214yes—
26#15688890C2214yes—
27#15688889C2214yes—
28#15688889C2214yes—
29C22×#41×C47C2211yescharacteristic
30C2×#117×C47C2211yescharacteristic
31C2×#117×C47C2211yescharacteristic
32C47×#410—22no—
33C47×#410—22no—
34C2×#99×C47C2214yes—
35#13094024—24no—
36#13094023—24no—
37#13094023—24no—
38#13094016—24no—
39#13094015—24no—
40#13094015—24no—
41#13094003—28no—
42#13094001—28no—
43#13094001—28no—
44C4×Dic94—28no—
45#13093983—21no—
46#13093983—21no—
47#13093983—21no—
48#13093983—21no—
49#13093983—21no—
50#13093983—21no—
51#13093983—21no—
52#13093983—21no—
53#13093983—21no—
54#13093983—21no—
55#13093983—21no—
56#13093983—21no—
57#13093983D412yes—
58#13093983D412yes—
59#13093983D412yes—
60#13093983D412yes—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C22⋊Dic47C2312yes—
66C2×C22⋊Dic47C2312yes—
67C2×C22⋊Dic47C2312yes—
68C2×C22⋊Dic47C2312yes—
69C2×C4×Dic47—21no—
70C2×C4×Dic47—21no—
71C2×C4×Dic47—21no—
72C2×C4×Dic47—21no—
73#13093939—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×C47—21no—
86C2×#41×C47—21no—
87C22×Q8×C47Q811yescharacteristic, complemented
88C22×D4×C47Q811yescharacteristic, complemented
89C22×D4×C47C2312yes—
90C23×C4×C47Q812yescomplemented
91C23×C4×C47C2311yescharacteristic
92#117×C47—21no—
93#117×C47—21no—
94#117×C47—21no—
95#117×C47—21no—
96#117×C47—21no—
97#117×C47—21no—
98#117×C47D412yes—
99#117×C47D412yes—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C22⋊C4×C47C2312yes—
103C2×C22⋊C4×C47C2312yes—
104C2×C42×C47—21no—
105C2×C42×C47—21no—
106#99×C47—28no—
107C24×C47C2×Q812yes—
108C24×C47C2411yescharacteristic
109#41×C47—41no—
110#41×C47—41no—
111#41×C47—41no—
112#41×C47—41no—
113#41×C47—41no—
114#41×C47—41no—
115#41×C47—41no—
116#41×C47—41no—
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×C47—21no—
131C2×D4×C47—22no—
132C2×D4×C47—22no—
133C2×D4×C47—22no—
134C2×D4×C47—22no—
135C2×D4×C47—22no—
136C2×D4×C47—22no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47—21no—
140C2×D4×C47—21no—
141C2×D4×C47—21no—
142C2×D4×C47—21no—
143C22×C4×C47—22no—
144C22×C4×C47—22no—
145C22×C4×C47—22no—
146C22×C4×C47—22no—
147C22×C4×C47—22no—
148C22×C4×C47—22no—
149C22×C4×C47—22no—
150C22×C4×C47—22no—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×Q812yes—
158C22×C4×C47C2×Q811yescharacteristic
159C22×C4×C47C2×Q811yescharacteristic
160C22×C4×C47#4111yescharacteristic
161C22×C4×C47#4111yescharacteristic
162C22×C4×C47#4111yescharacteristic
163C22×C4×C47#4111yescharacteristic
164C22×C4×C47#4111yescharacteristic
165C22×C4×C47#4111yescharacteristic
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—22no—
170C2×Dic94—28no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—41no—
174C4⋊C4×C47—41no—
175C22⋊C4×C47—22no—
176C22⋊C4×C47—22no—
177C22⋊C4×C47—22no—
178C22⋊C4×C47C2×D414yes—
179C42×C47—22no—
180C42×C47—22no—
181C42×C47—41no—
182C42×C47—41no—
183C22⋊Dic47—22no—
184C22⋊Dic47—22no—
185C22⋊Dic47—22no—
186C22⋊Dic47—22no—
187C22⋊Dic47—22no—
188C22⋊Dic47—22no—
189C22⋊Dic47C2×D414yes—
190C22⋊Dic47C2×D414yes—
191C4⋊Dic47—28no—
192C4⋊Dic47—22no—
193C4⋊Dic47—22no—
194C4⋊Dic47—22no—
195C4⋊Dic47—22no—
196C4⋊Dic47—41no—
197C4⋊Dic47—41no—
198C4⋊Dic47—41no—
199C4⋊Dic47—41no—
200Dic47⋊C4—28no—
201Dic47⋊C4—28no—
202C4×Dic47—22no—
203C4×Dic47—22no—
204C4×Dic47—22no—
205C4×Dic47—22no—
206C4×Dic47—41no—
207C4×Dic47—41no—
208C4×Dic47—41no—
209C4×Dic47—41no—
210C23×C47—22no—
211C23×C47—22no—
212C23×C47—22no—
213C23×C47—81no—
214C23×C47—82no—
215C23×C47C22×D411yescharacteristic
216C23×C47C22×D411yescharacteristic
217C23×C47C22×D411yescharacteristic
218C23×C47C22×Q811yescharacteristic
219C23×C47C2×#4111yescharacteristic
220C23×C47C2×#4111yescharacteristic
221C23×C47C2×#4111yescharacteristic
222Q8×C47—41no—
223Q8×C47—41no—
224Q8×C47—41no—
225Q8×C47—41no—
226D4×C47—42no—
227D4×C47—42no—
228D4×C47—42no—
229D4×C47—42no—
230D4×C47—41no—
231D4×C47—41no—
232D4×C47—41no—
233D4×C47—41no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—82no—
244C2×C4×C47—82no—
245C2×C4×C47#12011yescharacteristic
246C2×C4×C47#12011yescharacteristic
247C2×C4×C47#12011yescharacteristic
248C2×C4×C47#12011yescharacteristic
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×C47C22⋊Q811yescharacteristic
256C2×C4×C47C22⋊Q811yescharacteristic
257C2×C4×C47C22⋊Q811yescharacteristic
258C2×C4×C47C22⋊Q811yescharacteristic
259C2×C4×C47C22⋊Q811yescharacteristic
260C2×C4×C47C22⋊Q811yescharacteristic
261C2×C4×C47#12411yescharacteristic
262C2×C4×C47#12411yescharacteristic
263C2×C4×C47#12411yescharacteristic
264C2×C4×C47#12411yescharacteristic
265C2×C4×C47#12411yescharacteristic
266C2×C4×C47#12411yescharacteristic
267C2×C4×C47#12411yescharacteristic
268C2×C4×C47#12411yescharacteristic
269C2×C4×C47#12411yescharacteristic
270C2×C4×C47#12411yescharacteristic
271C2×C4×C47#12411yescharacteristic
272C2×C4×C47#12411yescharacteristic
273C2×Dic47—24no—
274C2×Dic47—24no—
275C2×Dic47—42no—
276C2×Dic47—42no—
277C2×Dic47—82no—
278C2×Dic47—82no—
279C2×Dic47—82no—
280C2×Dic47—82no—
281Dic94—88no—
282#23474—471nomaximal, Sylow
283C22×C47—41no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47—41no—
290C22×C47—42no—
291C22×C47—82no—
292C22×C47—82no—
293C22×C47—82no—
294C22×C47C2×#12011yescharacteristic
295C22×C47C2×C22⋊Q811yescharacteristic
296C22×C47C2×C22⋊Q811yescharacteristic
297C22×C47C2×C22⋊Q811yescharacteristic
298C22×C47C2×#12411yescharacteristic
299C22×C47C2×#12411yescharacteristic
300C22×C47C2×#12411yescharacteristic
301C4×C47—44no—
302C4×C47—42no—
303C4×C47—82no—
304C4×C47#39311yescharacteristic
305C4×C47#39311yescharacteristic
306C4×C47#39311yescharacteristic
307C4×C47#39311yescharacteristic
308C4×C47#39311yescharacteristic
309C4×C47#39311yescharacteristic
310C4×C47#39311yescharacteristic
311C4×C47#39311yescharacteristic
312Dic47—44no—
313Dic47—44no—
314Dic47—82no—
315Dic47—82no—
316#3458—471no—
317C2×#410—471no—
318C2×#410—471no—
319C2×#410D4711yescharacteristic
320#2032—478no—
321#1740—471no—
322#1740—471no—
323#1740—471no—
324C2×C47—21no—
325C2×C47—21no—
326C2×C47—21no—
327C2×C47—21no—
328C2×C47—21no—
329C2×C47—21no—
330C2×C47—21no—
331C2×C47—82no—
332C2×C47C2×#39311yescharacteristic
333C22×#41C2×D4711yescharacteristic
334#544—942no—
335#544—942no—
336#544—942no—
337#543—474no—
338#533—942no—
339#533—942no—
340#533—942no—
341C2×#117—471no—
342C2×#117—471no—
343C2×#117—471no—
344C2×#117—471no—
345C2×#117C2×D4711yescharacteristic
346C2×#117C2×D4711yescharacteristic
347#410—22no—
348#410—22no—
349#410—942no—
350#410—942no—
351#410—942no—
352#410—942no—
353C2×#99—474no—
354C2×#99—474no—
355C2×#99C2×D4714yes—
356#342—474no—
357#342—474no—
358#342—474no—
359C47#2347411yescharacteristic, complemented, Sylow
360C2×#41—21no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—21no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—21no—
370C2×#41—21no—
371C2×#41—21no—
372C22×Q8Dic9411yescharacteristic, complemented
373C22×D4Dic9411yescharacteristic, complemented
374C22×D4C22×D4712yes—
375C23×C4Dic9412yescomplemented
376C23×C4C22×D4711yescharacteristic
377C4⋊Q8—944no—
378C4⋊Q8—944no—
379C4⋊Q8—944no—
380#125—944no—
381#125—944no—
382#125—944no—
383C22⋊Q8—948no—
384C22⋊Q8—948no—
385C22⋊Q8—948no—
386C4×Q8—948no—
387#117—21no—
388#117—21no—
389#117—21no—
390#117—21no—
391#117—21no—
392#117—21no—
393#117—472no—
394#117—472no—
395#117—472no—
396#117—472no—
397#117—941no—
398#117—941no—
399#117—941no—
400#117—941no—
401#117—941no—
402#117—941no—
403#117—941no—
404#117—941no—
405#117—941no—
406#117—941no—
407#117—941no—
408#117—941no—
409#117C22⋊D4712yes—
410#117C22⋊D4712yes—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—941no—
414C2×C4⋊C4—941no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—941no—
417C2×C22⋊C4—472no—
418C2×C22⋊C4—472no—
419C2×C22⋊C4—472no—
420C2×C22⋊C4—472no—
421C2×C22⋊C4C22×D4712yes—
422C2×C22⋊C4C22×D4712yes—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—941no—
426C2×C42—941no—
427C2×C42—941no—
428C2×C42—941no—
429#99—28no—
430#99—948no—
431#99—948no—
432C24C2×Dic9412yes—
433C24C23×D4711yescharacteristic
434#41—41no—
435#41—41no—
436#41—41no—
437#41—41no—
438#41—41no—
439#41—41no—
440#41—41no—
441#41—41no—
442#41—41no—
443#41—41no—
444#41—41no—
445#41—41no—
446#41—41no—
447#41—41no—
448#41—41no—
449#41—41no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—948no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
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—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4C2×Dic9412yes—
482C22×C4C2×Dic9411yescharacteristic
483C22×C4C2×Dic9411yescharacteristic
484C22×C4#1062535011yescharacteristic
485C22×C4#1062535011yescharacteristic
486C22×C4D4×D4711yescharacteristic
487C22×C4D4×D4711yescharacteristic
488C22×C4D4×D4711yescharacteristic
489C22×C4D4×D4711yescharacteristic
490C22×C4#1062535211yescharacteristic
491C22×C4#1062535211yescharacteristic
492C22×C4#1062535211yescharacteristic
493C22×C4#1062535211yescharacteristic
494C22×C4C2×C22⋊D4711yescharacteristic
495C22×C4C2×C22⋊D4711yescharacteristic
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—948no—
501C4⋊C4—948no—
502C4⋊C4—948no—
503C4⋊C4—942no—
504C4⋊C4—942no—
505C4⋊C4—942no—
506C4⋊C4—942no—
507C4⋊C4—1881no—
508C4⋊C4—1881no—
509C4⋊C4—1881no—
510C4⋊C4—1881no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—474no—
515C22⋊C4—474no—
516C22⋊C4—942no—
517C22⋊C4—942no—
518C22⋊C4—942no—
519C22⋊C4—942no—
520C22⋊C4—942no—
521C22⋊C4—942no—
522C22⋊C4C2×C22⋊D4714yes—
523C42—22no—
524C42—22no—
525C42—41no—
526C42—41no—
527C42—942no—
528C42—942no—
529C42—942no—
530C42—942no—
531C42—1881no—
532C42—1881no—
533C42—1881no—
534C42—1881no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—81no—
539C23—82no—
540C23C2×D4×D4711yescharacteristic
541C23C2×D4×D4711yescharacteristic
542C23C22×C22⋊D4711yescharacteristic
543C23C22×Dic9411yescharacteristic
544C23C2×#1062535211yescharacteristic
545C23C2×#1062535211yescharacteristic
546C23C2×#1062535011yescharacteristic
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—3768no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—41no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—944no—
572C2×C4—944no—
573C2×C4—3762no—
574C2×C4—3762no—
575C2×C4—3762no—
576C2×C4—3762no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4#1309399211yescharacteristic
580C2×C4#1309399211yescharacteristic
581C2×C4#1309399211yescharacteristic
582C2×C4#1309399211yescharacteristic
583C2×C4#1309400111yescharacteristic
584C2×C4#1309400111yescharacteristic
585C2×C4#1309400111yescharacteristic
586C2×C4#1309400111yescharacteristic
587C2×C4#1309400111yescharacteristic
588C2×C4#1309400111yescharacteristic
589C2×C4#1309400111yescharacteristic
590C2×C4#1309400111yescharacteristic
591C2×C4#1309400311yescharacteristic
592C2×C4#1309400311yescharacteristic
593C2×C4#1309400311yescharacteristic
594C2×C4#1309400311yescharacteristic
595C2×C4#1309401011yescharacteristic
596C2×C4#1309401011yescharacteristic
597C2×C4#1309401011yescharacteristic
598C2×C4#1309401011yescharacteristic
599C2×C4#1309401011yescharacteristic
600C2×C4#1309401011yescharacteristic
601C2×C4#1309401011yescharacteristic
602C2×C4#1309401011yescharacteristic
603C2×C4#1309401111yescharacteristic
604C2×C4#1309401111yescharacteristic
605C2×C4#1309401111yescharacteristic
606C2×C4#1309401111yescharacteristic
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C2×#1309399211yescharacteristic
619C22C2×#1309400111yescharacteristic
620C22C2×#1309400111yescharacteristic
621C22C2×#1309400311yescharacteristic
622C22C2×#1309401011yescharacteristic
623C22C2×#1309401011yescharacteristic
624C22C2×#1309401111yescharacteristic
625C4—44no—
626C4—42no—
627C4—82no—
628C4—3762no—
629C4—3762no—
630C4—1884no—
631C4—1884no—
632C4#1568903511yescharacteristic
633C4#1568903511yescharacteristic
634C4#1568903511yescharacteristic
635C4#1568903511yescharacteristic
636C4#1568903511yescharacteristic
637C4#1568903511yescharacteristic
638C4#1568903511yescharacteristic
639C4#1568903511yescharacteristic
640C2—21no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—21no—
646C2—21no—
647C2—82no—
648C2C2×#1568903511yescharacteristic
subgroupG/H
centerC4#15689035
commutator, cosocleC24×C47C24
FrattiniC24C23×D47
FittingC2×C47×#410C2
socleC2×C47C2×#393
prime radicalC22×#41×C47C22
hypercenterC2×#410D47
hypocenterC47#23474
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
1. #20256793C2semidirect
4. #20244565C2semidirect
5. #20243121C2semidirect
6. #20243120C2semidirect
7. #20243120C2semidirect
87. C22×Q8×C47Q8semidirect
88. C22×D4×C47Q8semidirect
90. C23×C4×C47Q8semidirect
359. C47#23474semidirect
372. C22×Q8Dic94semidirect
373. C22×D4Dic94semidirect
375. C23×C4Dic94semidirect

Maximal subgroups

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

subgroupindex
#202567932
C2×#156890972
#202445652
#202431212
#202431202
C2×C47×#4102
#2347447