Finite Group Databasebeta

← #26243656

Subgroups — #26243656

Order 12032.

1214 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689751C211yesmaximal, characteristic, complemented
2#20251907C211yesmaximal, characteristic
3#20245980C211yesmaximal, characteristic
4#20244547C214yesmaximal, complemented
5#20244542C214yesmaximal, complemented
6C2×#15688891C211yesmaximal, characteristic, complemented
7#20243099C211yesmaximal, characteristic, complemented
8#20243097C211yesmaximal, characteristic, complemented
9C47×#2896C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11#15689751—22no—
12#15689751—22no—
13#15689751—22no—
14#15689751C2214yes—
15C2×#13094024—21no—
16C2×#13094023—21no—
17C2×#13094016—21no—
18C2×#13094015—21no—
19C2×#13094001—22no—
20C2×#13093983C2211yescharacteristic
21C2×#13093983C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23#15689132C2212yes—
24#15689128C2212yes—
25#15689124—21no—
26#15689124—21no—
27#15689119—22no—
28#15689030C2212yes—
29#15689026C2212yes—
30#15689003—22no—
31#15688983—22no—
32#15688891C2214yes—
33#15688884C2214yes—
34#15688882C2214yes—
35C22×#41×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C2×#117×C47C2211yescharacteristic
38C47×#419C2212yes—
39C47×#418C2212yes—
40C47×#417—21no—
41C47×#417—21no—
42C47×#416—22no—
43C22×Dic94—22no—
44C23×Dic47D412yes—
45C2×#10625329—21no—
46C2×#10625329—22no—
47C2×#10625329—22no—
48C2×#10625329C2312yes—
49C22×C47⋊C8—21no—
50#13094024—24no—
51#13094024—24no—
52#13094023—24no—
53#13094023—24no—
54#13094016—24no—
55#13094016—24no—
56#13094015—24no—
57#13094015—24no—
58#13094001—28no—
59#13094001—28no—
60#13093983—21no—
61#13093983—21no—
62#13093983—22no—
63#13093983—22no—
64#13093983D412yes—
65#13093983C2312yes—
66C2×C4⋊Dic47—22no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47D412yes—
73C2×C4⋊Dic47D411yescharacteristic
74C2×C4⋊Dic47D411yescharacteristic
75C2×C4⋊Dic47C2312yes—
76C2×Dic47⋊C4—22no—
77C2×Dic47⋊C4—22no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×C22⋊Dic47—24no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47C2312yes—
83#13093948—22no—
84#13093948—22no—
85#13093945—22no—
86#13093945—22no—
87C47⋊#95—24no—
88C2×#41×C47—22no—
89C2×#41×C47—21no—
90C2×#41×C47—21no—
91C2×#41×C47—21no—
92C2×#41×C47—21no—
93C2×#41×C47—22no—
94C2×#41×C47—22no—
95C2×#41×C47Q814yescomplemented
96C22×Q8×C47C2311yescharacteristic
97C22×D4×C47—21no—
98C22×D4×C47C2311yescharacteristic
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101C2×M16×C47—21no—
102C2×M16×C47—22no—
103C2×M16×C47—22no—
104C2×M16×C47C2312yes—
105C22×C8×C47—21no—
106#117×C47—21no—
107#117×C47—21no—
108#117×C47D412yes—
109#117×C47C2312yes—
110C2×C4⋊C4×C47D411yescharacteristic
111C2×C4⋊C4×C47D411yescharacteristic
112C2×C22⋊C4×C47—21no—
113C2×C42×C47—21no—
114#103×C47—22no—
115#103×C47—22no—
116#102×C47—22no—
117#102×C47—22no—
118C24×C47—21no—
119C24×C47C2×D411yescharacteristic
120#41×C47—24no—
121#41×C47—24no—
122#41×C47—24no—
123#41×C47—24no—
124#41×C47—42no—
125#41×C47—42no—
126#41×C47—42no—
127#41×C47—42no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47—21no—
133C2×Q8×C47C2×Q812yescomplemented
134C2×D4×C47—24no—
135C2×D4×C47—21no—
136C2×D4×C47—21no—
137C2×D4×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47—21no—
140C2×D4×C47—41no—
141C2×D4×C47—41no—
142C2×D4×C47—42no—
143C2×D4×C47C2×Q812yescomplemented
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—24no—
148C22×C4×C47—41no—
149C22×C4×C47—41no—
150C22×C4×C47—42no—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×Q812yes—
157C22×C4×C47#4112yes—
158C22×C4×C47#4112yes—
159C22×C4×C47#4112yes—
160C22×C4×C47C2411yescharacteristic
161C22×Dic47—21no—
162C22×Dic47—21no—
163C22×Dic47—22no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—82no—
167C22×Dic47C2×D414yes—
168C2×Dic94—28no—
169C2×Dic94—84no—
170M16×C47—22no—
171M16×C47—22no—
172M16×C47—42no—
173C2×C8×C47—22no—
174C2×C8×C47—42no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47C2×D412yes—
178C4⋊C4×C47C2×D412yes—
179C22⋊C4×C47—22no—
180C22⋊C4×C47—22no—
181C42×C47—22no—
182C42×C47—22no—
183C22⋊Dic47—22no—
184C22⋊Dic47—22no—
185C22⋊Dic47—42no—
186C22⋊Dic47—84no—
187C4⋊Dic47—28no—
188C4⋊Dic47—22no—
189C4⋊Dic47—22no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—22no—
193C4⋊Dic47—22no—
194C4⋊Dic47—42no—
195C4⋊Dic47C2×D412yes—
196C4⋊Dic47C2×D412yes—
197Dic47⋊C4—28no—
198Dic47⋊C4—28no—
199C4×Dic47—22no—
200C4×Dic47—22no—
201C4×Dic47—22no—
202C4×Dic47—22no—
203#10625329—22no—
204#10625329—22no—
205#10625329—42no—
206C2×C47⋊C8—22no—
207C2×C47⋊C8—42no—
208C23×C47—22no—
209C23×C47—41no—
210C23×C47—41no—
211C23×C47—42no—
212C23×C47—42no—
213C23×C47—41no—
214C23×C47—41no—
215C23×C47#12011yescharacteristic
216C23×C47#12011yescharacteristic
217C23×C47C22⋊Q812yes—
218C23×C47#12412yes—
219C23×C47C22×D411yescharacteristic
220Q8×C47—22no—
221Q8×C47—22no—
222Q8×C47—41no—
223Q8×C47—41no—
224D4×C47—22no—
225D4×C47—22no—
226D4×C47—44no—
227D4×C47—44no—
228D4×C47—41no—
229D4×C47—41no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—44no—
238C2×C4×C47—44no—
239C2×C4×C47#12011yescharacteristic
240C2×C4×C47#12011yescharacteristic
241C2×C4×C47C22⋊Q812yes—
242C2×C4×C47C22⋊Q812yes—
243C2×C4×C47C22⋊Q812yes—
244C2×C4×C47C22⋊Q812yes—
245C2×C4×C47C22⋊Q812yes—
246C2×C4×C47#12412yes—
247C2×C4×C47#12412yes—
248C2×C4×C47#12412yes—
249C2×C4×C47#12412yes—
250C2×C4×C47#12412yes—
251C2×C4×C47C22×D411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C2×C4×C47C22×Q811yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—42no—
261C2×Dic47—44no—
262C2×Dic47—42no—
263C2×Dic47—44no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—82no—
268Dic94—88no—
269C8×C47—42no—
270C47⋊C8—42no—
271#23462—471nomaximal, Sylow
272C22×C47—21no—
273C22×C47—21no—
274C22×C47—21no—
275C22×C47—21no—
276C22×C47—21no—
277C22×C47—21no—
278C22×C47—42no—
279C22×C47—42no—
280C22×C47—44no—
281C22×C47—42no—
282C22×C47—82no—
283C22×C47—81no—
284C22×C47—81no—
285C22×C47#39312yes—
286C22×C47#39312yes—
287C22×C47C2×#12011yescharacteristic
288C22×C47C2×C22⋊Q811yescharacteristic
289C22×C47C2×#12411yescharacteristic
290C4×C47—42no—
291C4×C47—42no—
292C4×C47—42no—
293C4×C47#39312yes—
294C4×C47#39312yes—
295C4×C47C2×C22⋊Q811yescharacteristic
296C4×C47C2×C22⋊Q811yescharacteristic
297C4×C47C2×#12411yescharacteristic
298C4×C47C2×#12411yescharacteristic
299Dic47—42no—
300Dic47—44no—
301Dic47—42no—
302Dic47—82no—
303C2×#544—471no—
304#2896—471no—
305#2896D4711yescharacteristic
306#2311—471no—
307#2028—474no—
308#2027—474no—
309C2×#343—471no—
310#1733—471no—
311#1733—471no—
312C2×C47—21no—
313C2×C47—21no—
314C2×C47—22no—
315C2×C47—22no—
316C2×C47—82no—
317C2×C47C2×#39311yescharacteristic
318C2×C47#301911yescharacteristic
319C2×C47#302011yescharacteristic
320C22×#41C2×D4711yescharacteristic
321C22×M16—471no—
322C22×M16C2×D4711yescharacteristic
323#544—474no—
324#544—942no—
325#544—942no—
326#544—942no—
327C2×C4⋊Q8—941no—
328C2×C4⋊Q8—941no—
329C2×#125—941no—
330C2×#125—941no—
331C2×C22⋊Q8—942no—
332C2×#117—471no—
333C2×#117—471no—
334C2×#117C2×D4711yescharacteristic
335C22×C4⋊C4—471no—
336#419—472no—
337#419C2×D4712yes—
338#418—472no—
339#418C2×D4712yes—
340#417—21no—
341#417—21no—
342#417—941no—
343#417—941no—
344#416—22no—
345#416—942no—
346#391—472no—
347#390—472no—
348#386—942no—
349#380—942no—
350#343—474no—
351#339—474no—
352#339—474no—
353C47#2346211yescharacteristic, complemented, Sylow
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—22no—
359C2×#41—21no—
360C2×#41—21no—
361C2×#41Dic9414yescomplemented
362C22×Q8—942no—
363C22×Q8C22×D4711yescharacteristic
364C22×D4—21no—
365C22×D4C22×D4711yescharacteristic
366C23×C4—21no—
367C23×C4—472no—
368C23×C4C22×D4711yescharacteristic
369C2×M16—21no—
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—472no—
373C2×M16—941no—
374C2×M16—942no—
375C2×M16—942no—
376C2×M16C22×D4712yes—
377C22×C8—21no—
378C22×C8—941no—
379C4⋊Q8—944no—
380C4⋊Q8—944no—
381C4⋊Q8—944no—
382C4⋊Q8—944no—
383#125—944no—
384#125—944no—
385#125—944no—
386#125—944no—
387C22⋊Q8—948no—
388C22⋊Q8—948no—
389#117—21no—
390#117—21no—
391#117—472no—
392#117—472no—
393#117—941no—
394#117—941no—
395#117—942no—
396#117—942no—
397#117C22⋊D4712yes—
398#117C22×D4712yes—
399C2×C4⋊C4—471no—
400C2×C4⋊C4—471no—
401C2×C4⋊C4—472no—
402C2×C4⋊C4—472no—
403C2×C4⋊C4—942no—
404C2×C4⋊C4—942no—
405C2×C4⋊C4—942no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4—941no—
409C2×C4⋊C4—941no—
410C2×C4⋊C4—941no—
411C2×C4⋊C4C22⋊D4711yescharacteristic
412C2×C4⋊C4C22⋊D4711yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—941no—
415C2×C22⋊C4—944no—
416C2×C22⋊C4—941no—
417C2×C42—21no—
418C2×C42—472no—
419C2×C42—941no—
420#103—22no—
421#103—22no—
422#103—942no—
423#103—942no—
424#102—22no—
425#102—22no—
426#102—942no—
427#102—942no—
428#95—944no—
429C24—21no—
430C24D4×D4711yescharacteristic
431#41—24no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—42no—
436#41—42no—
437#41—42no—
438#41—42no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—948no—
445C2×Q8—3764no—
446C2×Q8C2×Dic9412yescomplemented
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4—42no—
456C2×D4C2×Dic9412yescomplemented
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—474no—
465C22×C4—942no—
466C22×C4—941no—
467C22×C4—941no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—3762no—
471C22×C4C2×Dic9412yes—
472C22×C4#1062535012yes—
473C22×C4D4×D4711yescharacteristic
474C22×C4D4×D4711yescharacteristic
475C22×C4D4×D4711yescharacteristic
476C22×C4#1062535212yes—
477C22×C4#1062535212yes—
478C22×C4C2×C22⋊D4711yescharacteristic
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C23×D4711yescharacteristic
481M16—22no—
482M16—22no—
483M16—42no—
484M16—942no—
485M16—942no—
486M16—1882no—
487C2×C8—22no—
488C2×C8—42no—
489C2×C8—942no—
490C2×C8—1882no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—472no—
494C4⋊C4—472no—
495C4⋊C4—948no—
496C4⋊C4—948no—
497C4⋊C4—948no—
498C4⋊C4—942no—
499C4⋊C4—942no—
500C4⋊C4—941no—
501C4⋊C4—941no—
502C4⋊C4—942no—
503C4⋊C4—942no—
504C4⋊C4—1882no—
505C4⋊C4C2×C22⋊D4712yes—
506C4⋊C4C2×C22⋊D4712yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—942no—
510C22⋊C4—942no—
511C22⋊C4—3764no—
512C22⋊C4—1882no—
513C42—22no—
514C42—22no—
515C42—942no—
516C42—942no—
517C42—942no—
518C42—942no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23#1309399211yescharacteristic
527C23#1309399211yescharacteristic
528C23#1309400112yes—
529C23#1309401012yes—
530C23C2×D4×D4711yescharacteristic
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535Q8—3768no—
536D4—22no—
537D4—22no—
538D4—44no—
539D4—44no—
540D4—41no—
541D4—41no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—942no—
552C2×C4—942no—
553C2×C4—944no—
554C2×C4—3762no—
555C2×C4—1882no—
556C2×C4—1882no—
557C2×C4—1882no—
558C2×C4—1882no—
559C2×C4—1884no—
560C2×C4—1882no—
561C2×C4—1884no—
562C2×C4#1309399211yescharacteristic
563C2×C4#1309399211yescharacteristic
564C2×C4#1309400112yes—
565C2×C4#1309400112yes—
566C2×C4#1309400112yes—
567C2×C4#1309400312yes—
568C2×C4#1309400312yes—
569C2×C4#1309401012yes—
570C2×C4#1309401012yes—
571C2×C4#1309401012yes—
572C2×C4#1309401112yes—
573C2×C4#1309401112yes—
574C2×C4C2×D4×D4711yescharacteristic
575C2×C4C22×C22⋊D4711yescharacteristic
576C2×C4C22×Dic9411yescharacteristic
577C2×C4C2×#1062535211yescharacteristic
578C2×C4C2×#1062535211yescharacteristic
579C2×C4C2×#1062535011yescharacteristic
580C8—42no—
581C8—1882no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—42no—
589C22—42no—
590C22—44no—
591C22—42no—
592C22—82no—
593C22—81no—
594C22—81no—
595C22#1568903512yes—
596C22#1568903512yes—
597C22C2×#1309399211yescharacteristic
598C22C2×#1309400111yescharacteristic
599C22C2×#1309401011yescharacteristic
600C4—42no—
601C4—42no—
602C4—42no—
603C4—3762no—
604C4—1884no—
605C4—1882no—
606C4—1882no—
607C4#1568903512yes—
608C4#1568903512yes—
609C4C2×#1309400111yescharacteristic
610C4C2×#1309400311yescharacteristic
611C4C2×#1309401011yescharacteristic
612C4C2×#1309401111yescharacteristic
613C2—21no—
614C2—21no—
615C2—22no—
616C2—22no—
617C2—82no—
618C2C2×#1568903511yescharacteristic
619C2#2025274311yescharacteristic
620C2#2025275011yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2896C2
socleC22×C47C2×#120
prime radicalC22×#41×C47C22
hypercenter#2896D47
hypocenterC47#23462
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
1. C2×#15689751C2semidirect
4. #20244547C2semidirect
5. #20244542C2semidirect
6. C2×#15688891C2semidirect
7. #20243099C2semidirect
8. #20243097C2semidirect
95. C2×#41×C47Q8semidirect
133. C2×Q8×C47C2×Q8semidirect
143. C2×D4×C47C2×Q8semidirect
353. C47#23462semidirect
361. C2×#41Dic94semidirect
446. C2×Q8C2×Dic94semidirect
456. C2×D4C2×Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156897512
#202519072
#202459802
#202445472
#202445422
C2×#156888912
#202430992
#202430972
C47×#28962
#2346247