Finite Group Databasebeta

← #25086425

Subgroups — #25086425

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19147309C211yesmaximal, characteristic, complemented
2#19146402C211yesmaximal, characteristic
3SD16×D172C211yesmaximal, characteristic, complemented
4#19134250C211yesmaximal, characteristic, complemented
5#19134165C211yesmaximal, characteristic, complemented
6#19134016C211yesmaximal, characteristic, complemented
7#19134015C211yesmaximal, characteristic, complemented
8#19133936C211yesmaximal, characteristic, complemented
9#19133842C211yesmaximal, characteristic, complemented
10#19133830C211yesmaximal, characteristic, complemented
11#19133773C211yesmaximal, characteristic, complemented
12#19133705C211yesmaximal, characteristic, complemented
13#19133432C211yesmaximal, characteristic, complemented
14#19133417C211yesmaximal, characteristic, complemented
15C43×#3302C211yesmaximal, characteristic, complemented
16C2×SD16×D43—21no—
17Q8×D172C2211yescharacteristic
18D4×D172C2211yescharacteristic
19#14583302C2211yescharacteristic, complemented
20#14583272C2211yescharacteristic
21#14583258—21no—
22#14583249C2211yescharacteristic, complemented
23#14583247C2211yescharacteristic
24#14583247C2211yescharacteristic
25#14583212—21no—
26#14583194—21no—
27#14583179C2211yescharacteristic, complemented
28#14583176C2211yescharacteristic
29#14583133—21no—
30#14583129—21no—
31#14583111—21no—
32#14583104C2211yescharacteristic
33#14583101C2211yescharacteristic
34#14583099C2211yescharacteristic
35#14583094C2211yescharacteristic
36#14583051—21no—
37#14583039—21no—
38#14583032—21no—
39#14583030C2211yescharacteristic, complemented
40#14582997—21no—
41C4×#10621882C2211yescharacteristic
42C8×D172C2211yescharacteristic
43D4×C43⋊C8C2211yescharacteristic
44C2×#13063245—21no—
45#14582650C2211yescharacteristic, complemented
46#14582649C2211yescharacteristic, complemented
47#14582646C2211yescharacteristic, complemented
48#14582645C2211yescharacteristic, complemented
49#14582645C2211yescharacteristic, complemented
50#14582642C2211yescharacteristic
51Q8⋊(C43⋊C8)C2211yescharacteristic, complemented
52D4⋊(C43⋊C8)C2211yescharacteristic
53D4⋊(C43⋊C8)C2211yescharacteristic, complemented
54#14582626C2211yescharacteristic
55#14582621C2211yescharacteristic, complemented
56C43×#556C2211yescharacteristic, complemented
57C43×#547C2211yescharacteristic
58C43×#487C2211yescharacteristic, complemented
59C43×#485—21no—
60C43×#474C2211yescharacteristic
61C43×#467—21no—
62C43×#462C2211yescharacteristic, complemented
63C43×#452—21no—
64C4×SD16×C43C2211yescharacteristic
65C8×D4×C43C2211yescharacteristic, complemented
66C43×#416—21no—
67C2×Q8×D43—21no—
68C22×D172—21no—
69C2×D4×D43—21no—
70C2×#10621873—21no—
71C2×#10621873—21no—
72C2×#10621883—21no—
73C2×#10621884—21no—
74C2×#10621884—21no—
75SD16×D43—44no—
76C2×#10621882—21no—
77C2×#10621882D411yescharacteristic
78C22×C43⋊C8—21no—
79C2×C8×D43—21no—
80#13063325—21no—
81#13063325C2311yescharacteristic
82#13063322C2311yescharacteristic
83#13063305—21no—
84#13063305—41no—
85#13063300—21no—
86#13063299—21no—
87#13063295—41no—
88C4×D172—21no—
89C4×D172C2311yescharacteristic
90#13063258—21no—
91#13063258—21no—
92#13063258—41no—
93#13063258D411yescharacteristic
94C4⋊(C43⋊C8)—21no—
95C4⋊(C43⋊C8)C2311yescharacteristic
96C4⋊(C43⋊C8)C2311yescharacteristic
97C4⋊(C43⋊C8)C2311yescharacteristic
98#13063249—21no—
99#13063249—21no—
100#13063249—21no—
101#13063248—21no—
102#13063246—21no—
103#13063246D411yescharacteristic
104#13063245—21no—
105#13063245—21no—
106#13063245—21no—
107#13063245—21no—
108#13063245—21no—
109#13063245—41no—
110#13063245D411yescharacteristic
111C22⋊(C43⋊C8)—21no—
112(C2×D43)⋊C8—21no—
113#13063236—21no—
114C4×C43⋊C8C2311yescharacteristic
115C4×C43⋊C8C2311yescharacteristic
116C2×#41×C43—21no—
117C2×Q16×C43—21no—
118C2×SD16×C43D411yescharacteristic
119C22×C8×C43—21no—
120C4⋊Q8×C43C2311yescharacteristic
121C4⋊Q8×C43C2311yescharacteristic
122#126×C43—41no—
123#124×C43—21no—
124#123×C43—41no—
125C22⋊Q8×C43—21no—
126C22⋊Q8×C43—21no—
127#121×C43—21no—
128C4×Q8×C43—21no—
129C4×Q8×C43C2311yescharacteristic
130C4×D4×C43C2311yescharacteristic
131C4×D4×C43C2311yescharacteristic
132#117×C43—21no—
133C2×C4⋊C4×C43—21no—
134#107×C43—21no—
135#106×C43D411yescharacteristic
136C4⋊C8×C43C2311yescharacteristic
137#103×C43—21no—
138#103×C43—21no—
139#103×C43—21no—
140#103×C43D411yescharacteristic
141#102×C43—21no—
142#102×C43D411yescharacteristic
143C22⋊C8×C43—21no—
144C4×C8×C43C2311yescharacteristic
145C23×D43—41no—
146#41×C43—42no—
147#41×C43—81no—
148C2×Q8×C43—21no—
149C2×Q8×C43—21no—
150C2×Q8×C43C2×D411yescharacteristic
151C2×D4×C43—41no—
152C2×D4×C43C2×D411yescharacteristic
153C2×D4×C43C2×D411yescharacteristic
154C22×C4×C43—21no—
155C22×C4×C43—21no—
156C22×C4×C43—41no—
157C2×C22⋊D43—41no—
158Q8×D43—42no—
159Q8×D43—81no—
160D4×D43—42no—
161D4×D43—81no—
162C2×D172—21no—
163C2×D172—21no—
164C2×D172—21no—
165C2×D172—41no—
166C2×D172—42no—
167C2×D172—81no—
168C2×D172C2×D411yescharacteristic
169C2×C4×D43—21no—
170C2×C4×D43—41no—
171C2×Dic86—21no—
172C2×Dic86—41no—
173Q16×C43—81no—
174SD16×C43—22no—
175C2×C8×C43—21no—
176C2×C8×C43—81no—
177C2×C8×C43C2×D411yescharacteristic
178C2×C8×C43C2×D411yescharacteristic
179C4⋊C4×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—21no—
184C4⋊C4×C43—41no—
185C4⋊C4×C43—41no—
186C4⋊C4×C43C2×D411yescharacteristic
187C4⋊C4×C43C2×D411yescharacteristic
188C4⋊C4×C43C2×D411yescharacteristic
189C22⋊C4×C43—21no—
190C22⋊C4×C43—21no—
191C22⋊C4×C43—41no—
192C22⋊C4×C43—41no—
193C22⋊C4×C43—41no—
194C42×C43—21no—
195C42×C43—41no—
196C42×C43C2411yescharacteristic
197#10621884—42no—
198#10621884—42no—
199#10621883—42no—
200#10621882—22no—
201#10621882—81no—
202(C2×D43)⋊C4—21no—
203(C2×D43)⋊C4—41no—
204(C2×D43)⋊C4—41no—
205C4⋊Dic43—21no—
206C4⋊Dic43—21no—
207C4⋊Dic43—41no—
208C4⋊Dic43C2×D411yescharacteristic
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8—21no—
212C2×C43⋊C8—21no—
213C2×C43⋊C8—81no—
214C2×C43⋊C8C2×D411yescharacteristic
215C2×C43⋊C8C2×D411yescharacteristic
216#10621873—42no—
217#10621873—81no—
218C8×D43—42no—
219C23×C43—21no—
220C23×C43—21no—
221C22×D43—21no—
222C22×D43—41no—
223C22×D43—82no—
224C22×D43—82no—
225C22×D43—161no—
226Q8×C43—21no—
227Q8×C43—21no—
228Q8×C43—41no—
229Q8×C43—41no—
230Q8×C43—41no—
231Q8×C43—41no—
232D4×C43—21no—
233D4×C43—21no—
234D4×C43—21no—
235D4×C43—82no—
236D4×C43AGL(1,ℤ/8)12yes—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43—82no—
248C2×C4×C43C22×D411yescharacteristic
249C2×C4×C43C22×D411yescharacteristic
250C2×C4×C43H(5,𝔽2)11yescharacteristic
251C22⋊D43—82no—
252C2×Dic43—21no—
253C2×Dic43—41no—
254D172—21no—
255D172—41no—
256D172—41no—
257D172—41no—
258D172—41no—
259D172—81no—
260D172—82no—
261D172#13512yes—
262C4×D43—41no—
263C4×D43—41no—
264C4×D43—82no—
265Dic86—41no—
266Dic86—41no—
267Dic86—82no—
268C8×C43—21no—
269C8×C43—21no—
270C8×C43—41no—
271C43⋊C8—21no—
272C43⋊C8—21no—
273C43⋊C8—41no—
274C43⋊C8—41no—
275C43⋊C8—41no—
276C43⋊C8—41no—
277#19787—431nomaximal, Sylow
278C22×C43—22no—
279C22×C43—41no—
280C22×C43—41no—
281C22×C43—41no—
282C22×C43—41no—
283C22×C43—41no—
284C22×C43D4211yescharacteristic
285C2×D43—22no—
286C2×D43—41no—
287C2×D43—41no—
288C2×D43—81no—
289C2×D43—81no—
290C2×D43—81no—
291C2×D43—164no—
292C4×C43—21no—
293C4×C43—41no—
294C4×C43—41no—
295C4×C43—41no—
296C4×C43—41no—
297C4×C43—81no—
298C4×C43—81no—
299C4×C43C2×#13511yescharacteristic
300C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
301C4×C43#57611yescharacteristic
302C4×C43#57711yescharacteristic
303Dic43—41no—
304Dic43—81no—
305#3367—431no—
306#3302D4311yescharacteristic, complemented
307#3294—431no—
308D4×SD16—431no—
309#1719—431no—
310#1702—431no—
311#1679—431no—
312#1679—431no—
313#1665—431no—
314#1649—431no—
315#1647—431no—
316#1639—431no—
317#1628—431no—
318#1582—431no—
319#1580—431no—
320C2×C43—42no—
321C2×C43—81no—
322C2×C43#326411yescharacteristic
323C2×C43#328211yescharacteristic
324C2×C43#328911yescharacteristic
325D43—42no—
326D43—161no—
327C22×SD16—861no—
328#556C2×D4311yescharacteristic, complemented
329D4×Q8—431no—
330#547C2×D4311yescharacteristic
331D42—431no—
332#500—431no—
333#492—431no—
334#489—861no—
335#487—431no—
336#487C2×D4311yescharacteristic, complemented
337#486—431no—
338#486—431no—
339#485—21no—
340#480—861no—
341#477—861no—
342#474—431no—
343#474—431no—
344#474C2×D4311yescharacteristic
345#467—21no—
346#466—861no—
347#465—861no—
348#463—861no—
349#462C2×D4311yescharacteristic, complemented
350#461—431no—
351#461—431no—
352#460—431no—
353#459—431no—
354#452—21no—
355#450—861no—
356#448—861no—
357#447—861no—
358#446—431no—
359#440—861no—
360C4×SD16C2×D4311yescharacteristic
361C4×D8—431no—
362C8×D4—431no—
363C8×D4—431no—
364C8×D4C2×D4311yescharacteristic, complemented
365#416—21no—
366C2×#102—861no—
367#334—431no—
368#334—431no—
369#332—431no—
370#332—431no—
371#332—431no—
372#331—431no—
373Q8⋊C8—431no—
374D4⋊C8—431no—
375D4⋊C8—431no—
376D4⋊C8—431no—
377#322—431no—
378C43#1978711yescharacteristic, complemented, Sylow
379C2×#41—21no—
380C22×Q8—861no—
381C22×D4—861no—
382C22×D4—861no—
383C2×Q16—21no—
384C2×SD16—861no—
385C2×SD16—861no—
386C2×SD16—861no—
387C2×SD16—861no—
388C2×SD16—861no—
389C2×SD16—1724no—
390C2×SD16C22⋊D4311yescharacteristic
391C2×D8—431no—
392C2×D8—861no—
393C22×C8—21no—
394C22×C8—861no—
395C22×C8—861no—
396C4⋊Q8—431no—
397C4⋊Q8—861no—
398C4⋊Q8C22×D4311yescharacteristic
399C4⋊Q8C22×D4311yescharacteristic
400#127—431no—
401#126—41no—
402#124—21no—
403#123—41no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8—861no—
407C22⋊Q8—1721no—
408#121—21no—
409#121—861no—
410#121—861no—
411#120—1721no—
412C4×Q8—21no—
413C4×Q8C22×D4311yescharacteristic
414C4×D4—431no—
415C4×D4—861no—
416C4×D4C22×D4311yescharacteristic
417C4×D4C22×D4311yescharacteristic
418#117—21no—
419C2×C4⋊C4—21no—
420#107—21no—
421#107—431no—
422#107—861no—
423#107—861no—
424#107—1721no—
425#106C22⋊D4311yescharacteristic
426C4⋊C8—431no—
427C4⋊C8—431no—
428C4⋊C8—431no—
429C4⋊C8—861no—
430C4⋊C8C22×D4311yescharacteristic
431#103—21no—
432#103—21no—
433#103—21no—
434#103—861no—
435#103—861no—
436#103—861no—
437#103—861no—
438#103C22⋊D4311yescharacteristic
439#102—21no—
440#102—431no—
441#102—431no—
442#102—861no—
443#102—861no—
444#102—861no—
445#102—861no—
446#102—861no—
447#102—861no—
448#102—1721no—
449#102C22⋊D4311yescharacteristic
450C22⋊C8—21no—
451C22⋊C8—861no—
452C22⋊C8—861no—
453#97—861no—
454C4×C8—431no—
455C4×C8—431no—
456C4×C8C22×D4311yescharacteristic
457C24—1721no—
458#41—42no—
459#41—81no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—861no—
463C2×Q8—3441no—
464C2×Q8—1721no—
465C2×Q8—1722no—
466C2×Q8C2×C22⋊D4311yescharacteristic
467C2×D4—41no—
468C2×D4—431no—
469C2×D4—861no—
470C2×D4—861no—
471C2×D4—861no—
472C2×D4—3441no—
473C2×D4—3441no—
474C2×D4—1721no—
475C2×D4—1722no—
476C2×D4—1721no—
477C2×D4—1722no—
478C2×D4D4×D4311yescharacteristic
479C2×D4C2×C22⋊D4311yescharacteristic
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483C22×C4—861no—
484C22×C4—1721no—
485Q16—81no—
486SD16—22no—
487SD16—3441no—
488SD16—1722no—
489SD16—1722no—
490SD16—1722no—
491SD16—1722no—
492D8—862no—
493D8—3441no—
494C2×C8—21no—
495C2×C8—81no—
496C2×C8—431no—
497C2×C8—431no—
498C2×C8—861no—
499C2×C8—861no—
500C2×C8—861no—
501C2×C8—861no—
502C2×C8—3441no—
503C2×C8—1722no—
504C2×C8C2×C22⋊D4311yescharacteristic
505C2×C8C2×C22⋊D4311yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—431no—
514C4⋊C4—861no—
515C4⋊C4—861no—
516C4⋊C4—1721no—
517C4⋊C4D4×D4311yescharacteristic
518C4⋊C4C2×C22⋊D4311yescharacteristic
519C4⋊C4C2×C22⋊D4311yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—861no—
526C22⋊C4—1721no—
527C22⋊C4—1721no—
528C42—21no—
529C42—41no—
530C42C23×D4311yescharacteristic
531C23—21no—
532C23—21no—
533C23—861no—
534C23—3442no—
535C23—3442no—
536C23—1721no—
537C23—6881no—
538Q8—21no—
539Q8—21no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—3442no—
545Q8—1721no—
546Q8—1721no—
547D4—21no—
548D4—21no—
549D4—21no—
550D4—82no—
551D4—432no—
552D4—861no—
553D4—3441no—
554D4—3442no—
555D4—3442no—
556D4—1721no—
557D4—1721no—
558D4—1721no—
559D4—1721no—
560D4#1306335012yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—82no—
572C2×C4—861no—
573C2×C4—3442no—
574C2×C4—1721no—
575C2×C4—1721no—
576C2×C4—1721no—
577C2×C4C2×D4×D4311yescharacteristic
578C2×C4C22×C22⋊D4311yescharacteristic
579C2×C4#1306337011yescharacteristic
580C8—21no—
581C8—21no—
582C8—41no—
583C8—861no—
584C8—861no—
585C8—1721no—
586C8—1721no—
587C8—1721no—
588C8—1721no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—862no—
596C22—3441no—
597C22—3441no—
598C22—3441no—
599C22—1721no—
600C22—1721no—
601C22—6884no—
602C22D4×C22⋊D4311yescharacteristic
603C4—21no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—3441no—
611C4—1721no—
612C4#135×D4311yescharacteristic
613C4C2×#1306335011yescharacteristic
614C4#1458370211yescharacteristic
615C4#1458370611yescharacteristic
616C2—42no—
617C2—81no—
618C2—1722no—
619C2—6881no—
620C2#1914607611yescharacteristic
621C2#1914629611yescharacteristic
622C2#1914636611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D43
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3302C2
socleC22×C43D42
prime radical#19146402C2
hypercenter#3302D43
hypocenterC43#19787
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ #3302

Complements

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

NH ≅ G/Ntype
1. #19147309C2semidirect
3. SD16×D172C2semidirect
4. #19134250C2semidirect
5. #19134165C2semidirect
6. #19134016C2semidirect
7. #19134015C2semidirect
8. #19133936C2semidirect
9. #19133842C2semidirect
10. #19133830C2semidirect
11. #19133773C2semidirect
12. #19133705C2semidirect
13. #19133432C2semidirect
14. #19133417C2semidirect
15. C43×#3302C2semidirect
19. #14583302C22semidirect
22. #14583249C22semidirect
27. #14583179C22semidirect
39. #14583030C22semidirect
45. #14582650C22semidirect
46. #14582649C22semidirect
47. #14582646C22semidirect
48. #14582645C22semidirect
49. #14582645C22semidirect
51. Q8⋊(C43⋊C8)C22semidirect
53. D4⋊(C43⋊C8)C22semidirect
55. #14582621C22semidirect
56. C43×#556C22semidirect
58. C43×#487C22semidirect
62. C43×#462C22semidirect
65. C8×D4×C43C22semidirect
306. #3302D43semidirect
328. #556C2×D43semidirect
336. #487C2×D43semidirect
349. #462C2×D43semidirect
364. C8×D4C2×D43semidirect
378. C43#19787semidirect

Maximal subgroups

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

subgroupindex
#191473092
#191464022
SD16×D1722
#191342502
#191341652
#191340162
#191340152
#191339362
#191338422
#191338302
#191337732
#191337052
#191334322
#191334172
C43×#33022
#1978743