Finite Group Databasebeta

← #25088210

Subgroups — #25088210

Order 11008.

668 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19147309C211yesmaximal, characteristic, complemented
2#19146831C211yesmaximal, characteristic
3SD16×D172C211yesmaximal, characteristic, complemented
4#19134197C211yesmaximal, characteristic, complemented
5#19134140C211yesmaximal, characteristic, complemented
6#19134137C211yesmaximal, characteristic, complemented
7#19133967C211yesmaximal, characteristic, complemented
8#19133936C211yesmaximal, characteristic, complemented
9#19133823C211yesmaximal, characteristic, complemented
10#19133773C211yesmaximal, characteristic, complemented
11#19133758C211yesmaximal, characteristic, complemented
12#19133705C211yesmaximal, characteristic, complemented
13#19133486C211yesmaximal, characteristic, complemented
14C8×#10621882C211yesmaximal, characteristic, complemented
15C43×#3340C211yesmaximal, characteristic, complemented
16C2×SD16×D43—21no—
17Q8×D172C2211yescharacteristic
18D4×D172C2211yescharacteristic
19#14583272C2211yescharacteristic
20#14583258—21no—
21#14583249C2211yescharacteristic, complemented
22#14583249C2211yescharacteristic, complemented
23#14583247C2211yescharacteristic
24#14583247C2211yescharacteristic
25#14583223—21no—
26#14583194—21no—
27#14583186C2211yescharacteristic
28#14583179C2211yescharacteristic, complemented
29#14583150—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
42#14582977C2211yescharacteristic
43C8×D172C2211yescharacteristic
44#14582903—21no—
45#14582651C2211yescharacteristic, complemented
46#14582650C2211yescharacteristic, complemented
47#14582648C2211yescharacteristic, complemented
48#14582646C2211yescharacteristic, complemented
49#14582645C2211yescharacteristic, complemented
50#14582642C2211yescharacteristic
51Q8⋊(C43⋊C8)C2211yescharacteristic, complemented
52D4⋊(C43⋊C8)C2211yescharacteristic
53D4⋊(C43⋊C8)C2211yescharacteristic, complemented
54#14582626C2211yescharacteristic
55C8×C43⋊C8C2211yescharacteristic, complemented
56C43×#556C2211yescharacteristic, complemented
57C43×#555C2211yescharacteristic
58C43×#498C2211yescharacteristic, complemented
59C43×#485—21no—
60C43×#482—21no—
61C43×#477C2211yescharacteristic, complemented
62C43×#474C2211yescharacteristic
63C43×#465—21no—
64C4×SD16×C43C2211yescharacteristic
65C8×D4×C43C2211yescharacteristic, complemented
66C43×#427—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
78C2×#10621877—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#13063258D411yescharacteristic
92#13063256—21no—
93C4⋊(C43⋊C8)—21no—
94C4⋊(C43⋊C8)C2311yescharacteristic
95C4⋊(C43⋊C8)C2311yescharacteristic
96C4⋊(C43⋊C8)C2311yescharacteristic
97#13063249—21no—
98#13063249—21no—
99#13063249—21no—
100#13063249—41no—
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×SD16×C43D411yescharacteristic
117C22×C8×C43—21no—
118C4⋊Q8×C43C2311yescharacteristic
119C4⋊Q8×C43C2311yescharacteristic
120#126×C43—41no—
121#124×C43—21no—
122#124×C43—41no—
123C22⋊Q8×C43—21no—
124#121×C43—21no—
125#121×C43—21no—
126C4×Q8×C43—21no—
127C4×Q8×C43C2311yescharacteristic
128C4×D4×C43C2311yescharacteristic
129C4×D4×C43C2311yescharacteristic
130#117×C43—21no—
131#117×C43—21no—
132#117×C43—21no—
133#107×C43—21no—
134#107×C43—21no—
135#106×C43—21no—
136#106×C43—21no—
137#106×C43D411yescharacteristic
138C4⋊C8×C43C2311yescharacteristic
139#103×C43—21no—
140#103×C43D411yescharacteristic
141#102×C43—21no—
142#102×C43D411yescharacteristic
143C22⋊C8×C43—21no—
144C4×C8×C43C2311yescharacteristic
145C23×D43—41no—
146C2×Q8×C43—21no—
147C2×Q8×C43—21no—
148C2×Q8×C43C2×D411yescharacteristic
149C2×D4×C43—41no—
150C2×D4×C43C2×D411yescharacteristic
151C2×D4×C43C2×D411yescharacteristic
152C22×C4×C43—21no—
153C22×C4×C43—21no—
154C2×C22⋊D43—41no—
155Q8×D43—42no—
156Q8×D43—81no—
157D4×D43—42no—
158D4×D43—81no—
159C2×D172—21no—
160C2×D172—21no—
161C2×D172—21no—
162C2×D172—41no—
163C2×D172—42no—
164C2×D172—81no—
165C2×D172C2×D411yescharacteristic
166C2×C4×D43—21no—
167C2×C4×D43—41no—
168C2×Dic86—21no—
169C2×Dic86—41no—
170SD16×C43—22no—
171C2×C8×C43—21no—
172C2×C8×C43—42no—
173C2×C8×C43C2×D411yescharacteristic
174C2×C8×C43C2×D411yescharacteristic
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—21no—
179C4⋊C4×C43—21no—
180C4⋊C4×C43—41no—
181C4⋊C4×C43C2×D411yescharacteristic
182C4⋊C4×C43C2×D411yescharacteristic
183C4⋊C4×C43C2×D411yescharacteristic
184C22⋊C4×C43—21no—
185C22⋊C4×C43—21no—
186C22⋊C4×C43—41no—
187C22⋊C4×C43—41no—
188C22⋊C4×C43—41no—
189C22⋊C4×C43—41no—
190C42×C43—21no—
191C42×C43—41no—
192C42×C43—41no—
193C42×C43C2411yescharacteristic
194#10621884—42no—
195#10621884—42no—
196#10621883—42no—
197#10621882—22no—
198#10621882—81no—
199(C2×D43)⋊C4—21no—
200(C2×D43)⋊C4—41no—
201(C2×D43)⋊C4—41no—
202C4⋊Dic43—21no—
203C4⋊Dic43—21no—
204C4⋊Dic43—41no—
205C4⋊Dic43C2×D411yescharacteristic
206#10621877—81no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—21no—
209C2×C43⋊C8—21no—
210C2×C43⋊C8—21no—
211C2×C43⋊C8C2×D411yescharacteristic
212C2×C43⋊C8C2×D411yescharacteristic
213#10621873—42no—
214#10621873—81no—
215C8×D43—42no—
216C23×C43—21no—
217C23×C43—21no—
218C22×D43—21no—
219C22×D43—41no—
220C22×D43—82no—
221C22×D43—82no—
222C22×D43—161no—
223Q8×C43—21no—
224Q8×C43—21no—
225Q8×C43—41no—
226Q8×C43—41no—
227Q8×C43—81no—
228D4×C43—21no—
229D4×C43—21no—
230D4×C43—21no—
231D4×C43—21no—
232D4×C43—161no—
233C2×C4×C43—21no—
234C2×C4×C43—21no—
235C2×C4×C43—21no—
236C2×C4×C43—21no—
237C2×C4×C43—41no—
238C2×C4×C43—41no—
239C2×C4×C43—41no—
240C2×C4×C43—41no—
241C2×C4×C43—41no—
242C2×C4×C43—41no—
243C2×C4×C43C22×D411yescharacteristic
244C2×C4×C43C22×D411yescharacteristic
245C2×C4×C43H(5,𝔽2)11yescharacteristic
246C22⋊D43—82no—
247C2×Dic43—21no—
248C2×Dic43—41no—
249D172—21no—
250D172—41no—
251D172—41no—
252D172—41no—
253D172—41no—
254D172—81no—
255D172—82no—
256D172#13512yes—
257C4×D43—41no—
258C4×D43—41no—
259C4×D43—82no—
260Dic86—41no—
261Dic86—41no—
262Dic86—82no—
263C8×C43—21no—
264C8×C43—41no—
265C8×C43AGL(1,ℤ/8)12yes—
266C43⋊C8—21no—
267C43⋊C8—21no—
268C43⋊C8—41no—
269C43⋊C8—41no—
270C43⋊C8—41no—
271C43⋊C8—41no—
272#19910—431nomaximal, Sylow
273C22×C43—41no—
274C22×C43—41no—
275C22×C43—41no—
276C22×C43—41no—
277C22×C43—41no—
278C22×C43—41no—
279C22×C43D4211yescharacteristic
280C2×D43—22no—
281C2×D43—41no—
282C2×D43—41no—
283C2×D43—81no—
284C2×D43—81no—
285C2×D43—81no—
286C2×D43—164no—
287C4×C43—21no—
288C4×C43—41no—
289C4×C43—41no—
290C4×C43—41no—
291C4×C43—41no—
292C4×C43—81no—
293C4×C43—81no—
294C4×C43C2×#13511yescharacteristic
295C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
296C4×C43#57711yescharacteristic
297C4×C43#57711yescharacteristic
298Dic43—41no—
299Dic43—81no—
300#3367—431no—
301#3340D4311yescharacteristic, complemented
302#3325—431no—
303D4×SD16—431no—
304#1708—431no—
305#1699—431no—
306#1698—431no—
307#1671—431no—
308#1665—431no—
309#1646—431no—
310#1639—431no—
311#1637—431no—
312#1628—431no—
313#1590—431no—
314C8×D8—431no—
315C2×C43—81no—
316C2×C43—81no—
317C2×C43#327211yescharacteristic
318C2×C43#328211yescharacteristic
319C2×C43#329011yescharacteristic
320D43—42no—
321D43—161no—
322C22×SD16—861no—
323#556C2×D4311yescharacteristic, complemented
324#555C2×D4311yescharacteristic
325D4×Q8—431no—
326D42—431no—
327#498C2×D4311yescharacteristic, complemented
328#492—431no—
329#489—861no—
330#487—431no—
331#487—431no—
332#486—431no—
333#486—431no—
334#485—21no—
335#482—21no—
336#482—861no—
337#477—861no—
338#477C2×D4311yescharacteristic, complemented
339#476—431no—
340#474—431no—
341#474C2×D4311yescharacteristic
342#469—861no—
343#465—21no—
344#465—861no—
345#463—861no—
346#461—431no—
347#461—431no—
348#460—431no—
349#459—431no—
350#450—861no—
351#448—861no—
352#447—861no—
353#446—431no—
354#440—861no—
355C4×SD16C2×D4311yescharacteristic
356C4×D8—431no—
357#436—431no—
358C8×D4—431no—
359C8×D4C2×D4311yescharacteristic, complemented
360#427—21no—
361#418—861no—
362#335—431no—
363#334—431no—
364#333—431no—
365#332—431no—
366#332—431no—
367#331—431no—
368Q8⋊C8—431no—
369D4⋊C8—431no—
370D4⋊C8—431no—
371D4⋊C8—431no—
372C82—431no—
373C43#1991011yescharacteristic, complemented, Sylow
374C22×Q8—861no—
375C22×D4—861no—
376C22×D4—861no—
377C2×SD16—861no—
378C2×SD16—861no—
379C2×SD16—861no—
380C2×SD16—861no—
381C2×SD16—861no—
382C2×SD16—1724no—
383C2×SD16C22⋊D4311yescharacteristic
384C2×D8—431no—
385C2×D8—861no—
386C2×M16—861no—
387C22×C8—21no—
388C22×C8—861no—
389C4⋊Q8—431no—
390C4⋊Q8—861no—
391C4⋊Q8C22×D4311yescharacteristic
392C4⋊Q8C22×D4311yescharacteristic
393#127—431no—
394#126—41no—
395#124—21no—
396#124—41no—
397C22⋊Q8—21no—
398C22⋊Q8—861no—
399C22⋊Q8—1721no—
400#121—21no—
401#121—21no—
402#121—861no—
403#121—861no—
404#120—1721no—
405C4×Q8—21no—
406C4×Q8C22×D4311yescharacteristic
407C4×D4—431no—
408C4×D4—861no—
409C4×D4C22×D4311yescharacteristic
410C4×D4C22×D4311yescharacteristic
411#117—21no—
412#117—21no—
413#117—21no—
414#107—21no—
415#107—21no—
416#107—431no—
417#107—861no—
418#106—21no—
419#106—21no—
420#106—861no—
421#106C22⋊D4311yescharacteristic
422C4⋊C8—431no—
423C4⋊C8—431no—
424C4⋊C8—431no—
425C4⋊C8—861no—
426C4⋊C8C22×D4311yescharacteristic
427#103—21no—
428#103—861no—
429#103—861no—
430#103—861no—
431#103—861no—
432#103—1721no—
433#103C22⋊D4311yescharacteristic
434#102—21no—
435#102—431no—
436#102—431no—
437#102—861no—
438#102—861no—
439#102—861no—
440#102—861no—
441#102—861no—
442#102—861no—
443#102—1721no—
444#102C22⋊D4311yescharacteristic
445C22⋊C8—21no—
446C22⋊C8—861no—
447C22⋊C8—861no—
448#97—861no—
449C4×C8—431no—
450C4×C8—431no—
451C4×C8C22×D4311yescharacteristic
452C24—1721no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—861no—
456C2×Q8—3441no—
457C2×Q8—1721no—
458C2×Q8—1722no—
459C2×Q8C2×C22⋊D4311yescharacteristic
460C2×D4—41no—
461C2×D4—431no—
462C2×D4—861no—
463C2×D4—861no—
464C2×D4—861no—
465C2×D4—3441no—
466C2×D4—3441no—
467C2×D4—1721no—
468C2×D4—1722no—
469C2×D4—1721no—
470C2×D4—1722no—
471C2×D4D4×D4311yescharacteristic
472C2×D4C2×C22⋊D4311yescharacteristic
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—861no—
476C22×C4—1721no—
477SD16—22no—
478SD16—3441no—
479SD16—1722no—
480SD16—1722no—
481SD16—1722no—
482SD16—1722no—
483D8—862no—
484D8—3441no—
485M16—3441no—
486C2×C8—21no—
487C2×C8—42no—
488C2×C8—431no—
489C2×C8—431no—
490C2×C8—861no—
491C2×C8—861no—
492C2×C8—861no—
493C2×C8—861no—
494C2×C8—1722no—
495C2×C8C2×C22⋊D4311yescharacteristic
496C2×C8C2×C22⋊D4311yescharacteristic
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—41no—
503C4⋊C4—431no—
504C4⋊C4—861no—
505C4⋊C4—861no—
506C4⋊C4—1721no—
507C4⋊C4D4×D4311yescharacteristic
508C4⋊C4C2×C22⋊D4311yescharacteristic
509C4⋊C4C2×C22⋊D4311yescharacteristic
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—861no—
517C22⋊C4—1721no—
518C22⋊C4—1721no—
519C42—21no—
520C42—41no—
521C42—41no—
522C42C23×D4311yescharacteristic
523C23—21no—
524C23—21no—
525C23—861no—
526C23—3442no—
527C23—3442no—
528C23—1721no—
529C23—6881no—
530Q8—21no—
531Q8—21no—
532Q8—41no—
533Q8—41no—
534Q8—81no—
535Q8—3442no—
536Q8—1721no—
537Q8—1721no—
538D4—21no—
539D4—21no—
540D4—21no—
541D4—21no—
542D4—161no—
543D4—432no—
544D4—861no—
545D4—3441no—
546D4—3442no—
547D4—3442no—
548D4—1721no—
549D4—1721no—
550D4—1721no—
551D4—1721no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—861no—
563C2×C4—3442no—
564C2×C4—1721no—
565C2×C4—1721no—
566C2×C4—1721no—
567C2×C4C2×D4×D4311yescharacteristic
568C2×C4C22×C22⋊D4311yescharacteristic
569C2×C4#1306337011yescharacteristic
570C8—21no—
571C8—41no—
572C8—861no—
573C8—861no—
574C8—1721no—
575C8—1721no—
576C8—1721no—
577C8—1721no—
578C8#1306335012yes—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—862no—
586C22—3441no—
587C22—3441no—
588C22—3441no—
589C22—1721no—
590C22—1721no—
591C22—6884no—
592C22D4×C22⋊D4311yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—3441no—
601C4—1721no—
602C4#135×D4311yescharacteristic
603C4C2×#1306335011yescharacteristic
604C4#1458370611yescharacteristic
605C4#1458370711yescharacteristic
606C2—81no—
607C2—81no—
608C2—1722no—
609C2—6881no—
610C2#1914619511yescharacteristic
611C2#1914629611yescharacteristic
612C2#1914637511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D43
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3340C2
socleC22×C43D42
prime radical#19146831C2
hypercenter#3340D43
hypocenterC43#19910
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ #3340

Complements

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

NH ≅ G/Ntype
1. #19147309C2semidirect
3. SD16×D172C2semidirect
4. #19134197C2semidirect
5. #19134140C2semidirect
6. #19134137C2semidirect
7. #19133967C2semidirect
8. #19133936C2semidirect
9. #19133823C2semidirect
10. #19133773C2semidirect
11. #19133758C2semidirect
12. #19133705C2semidirect
13. #19133486C2semidirect
14. C8×#10621882C2semidirect
15. C43×#3340C2semidirect
21. #14583249C22semidirect
22. #14583249C22semidirect
28. #14583179C22semidirect
39. #14583030C22semidirect
45. #14582651C22semidirect
46. #14582650C22semidirect
47. #14582648C22semidirect
48. #14582646C22semidirect
49. #14582645C22semidirect
51. Q8⋊(C43⋊C8)C22semidirect
53. D4⋊(C43⋊C8)C22semidirect
55. C8×C43⋊C8C22semidirect
56. C43×#556C22semidirect
58. C43×#498C22semidirect
61. C43×#477C22semidirect
65. C8×D4×C43C22semidirect
301. #3340D43semidirect
323. #556C2×D43semidirect
327. #498C2×D43semidirect
338. #477C2×D43semidirect
359. C8×D4C2×D43semidirect
373. C43#19910semidirect

Maximal subgroups

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

subgroupindex
#191473092
#191468312
SD16×D1722
#191341972
#191341402
#191341372
#191339672
#191339362
#191338232
#191337732
#191337582
#191337052
#191334862
C8×#106218822
C43×#33402
#1991043