Finite Group Databasebeta

← #25250068

Subgroups — #25250068

Order 11008.

668 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#509×D43C211yesmaximal, characteristic
2#19143509C211yesmaximal, characteristic, complemented
3#19143444C211yesmaximal, characteristic, complemented
4#19136496C211yesmaximal, characteristic, complemented
5#19136495C211yesmaximal, characteristic, complemented
6#19136494C211yesmaximal, characteristic, complemented
7#19136493C211yesmaximal, characteristic, complemented
8#19136492C211yesmaximal, characteristic, complemented
9#19136491C211yesmaximal, characteristic, complemented
10#19136490C211yesmaximal, characteristic, complemented
11#19136366C211yesmaximal, characteristic, complemented
12#361×D43C211yesmaximal, characteristic, complemented
13#19136329C211yesmaximal, characteristic, complemented
14#19136198C211yesmaximal, characteristic, complemented
15C43×AGL(1,ℤ/16)C211yesmaximal, characteristic, complemented
16#135×D43C2211yescharacteristic
17C2×D8×D43C2211yescharacteristic
18#14583664—21no—
19#14583371—21no—
20#14583337C2211yescharacteristic
21#14583336C412yescomplemented
22#14583335C2211yescharacteristic
23#14583334C2211yescharacteristic
24#14583333C412yescomplemented
25SD32×D43C412yescomplemented
26D16×D43C412yescomplemented
27M32×D43C2211yescharacteristic
28M32×D43C2211yescharacteristic
29#14583022C2211yescharacteristic, complemented
30#14583021C2211yescharacteristic, complemented
31#14583020C2211yescharacteristic, complemented
32#14583014—21no—
33#14583013—21no—
34#14583012C2211yescharacteristic, complemented
35#14583011C2211yescharacteristic, complemented
36#14583010C2211yescharacteristic, complemented
37C4×#10621882—21no—
38#14582979—21no—
39#108×D43C2211yescharacteristic
40#14582934C2211yescharacteristic
41#14582914—21no—
42#14582902—21no—
43#14582856C2211yescharacteristic
44#14582723C2211yescharacteristic, complemented
45#14582722C2211yescharacteristic, complemented
46#14582721C2211yescharacteristic, complemented
47#14582713C2211yescharacteristic, complemented
48#14582712C2211yescharacteristic, complemented
49#14582711C2211yescharacteristic, complemented
50#14582710C2211yescharacteristic, complemented
51#14582709C2211yescharacteristic, complemented
52#14582708C2211yescharacteristic, complemented
53#14582677C2211yescharacteristic, complemented
54#14582676C2211yescharacteristic, complemented
55#14582676C2211yescharacteristic, complemented
56C43×#509C2211yescharacteristic
57C43×#444C2211yescharacteristic, complemented
58C43×#442C2211yescharacteristic, complemented
59C43×#365C2211yescharacteristic, complemented
60C43×#361C2211yescharacteristic, complemented
61C43×#360C2211yescharacteristic, complemented
62C43×#347C2211yescharacteristic, complemented
63#41×D43—21no—
64C2×D4×D43—21no—
65#13063348C2×C411yescharacteristic, complemented
66#13063347C2311yescharacteristic
67#13063346—21no—
68#13063345—21no—
69#13063344C2×C411yescharacteristic, complemented
70Q16×D43C2×C411yescharacteristic, complemented
71SD16×D43—21no—
72C2×D344C2311yescharacteristic
73C2×#10621882—21no—
74D8×D43—21no—
75D8×D43—21no—
76D8×D43C2×C412yes—
77D8×D43C2×C412yes—
78D8×D43C2×C411yescharacteristic, complemented
79#13063333—21no—
80#13063333—41no—
81#13063332—21no—
82#13063331—21no—
83#13063331—41no—
84C2×#10621872—41no—
85C2×#10621872D411yescharacteristic
86M16×D43—21no—
87M16×D43—41no—
88C2×C8×D43C2311yescharacteristic
89C4×D172—21no—
90D4×Dic43—21no—
91#13063287—21no—
92C4×C22⋊D43—41no—
93(C22⋊D43)⋊C4—41no—
94#13063283—21no—
95#13063282D411yescharacteristic
96C2×(C2×D43)⋊C4—41no—
97C22⋊C4×D43—41no—
98#13063270C2×C412yescomplemented
99#13063269C2×C412yescomplemented
100#13063268C2×C412yescomplemented
101D688C2×C412yescomplemented
102#13063266C2×C412yescomplemented
103#13063266C2×C412yescomplemented
104#13063265C2311yescharacteristic
105#13063265C2311yescharacteristic
106#13063264—21no—
107#13063264C2×C412yescomplemented
108C16×D43—21no—
109C16×D43C2×C412yescomplemented
110#13063261C2311yescharacteristic
111#13063260—21no—
112#13063258—21no—
113#13063257C2311yescharacteristic
114#13063253—21no—
115#13063252—21no—
116#13063251—21no—
117#13063247—21no—
118#13063246—21no—
119#13063245—21no—
120#13063235C2311yescharacteristic
121C4×C43⋊C8D411yescharacteristic
122C8×Dic43D411yescharacteristic
123#135×C43C2311yescharacteristic
124C2×D8×C43C2311yescharacteristic
125#131×C43—21no—
126C4×D4×C43—21no—
127SD32×C43C2×C412yescomplemented
128D16×C43C2×C412yescomplemented
129M32×C43C2311yescharacteristic
130M32×C43C2311yescharacteristic
131#108×C43C2311yescharacteristic
132#106×C43C2311yescharacteristic
133#104×C43—21no—
134#102×C43—21no—
135#97×C43C2311yescharacteristic
136C23×D43—41no—
137#41×C43—21no—
138C2×D4×C43—21no—
139C22×C4×C43—41no—
140C2×C22⋊D43—41no—
141C22×Dic43—41no—
142#10621902—21no—
143Q8×D43—21no—
144#10621900—21no—
145#10621900—41no—
146D4×D43—22no—
147D4×D43—22no—
148D4×D43—21no—
149D4×D43—41no—
150D4×D43—41no—
151D4×D43—41no—
152#10621898—21no—
153#10621898—41no—
154C2×D172—21no—
155C2×C4×D43—41no—
156C2×C4×D43—41no—
157C2×C4×D43—41no—
158C2×C4×D43C2×D411yescharacteristic
159Q16×C43C22×C411yescharacteristic
160SD16×C43—21no—
161D8×C43—21no—
162D8×C43C22×C411yescharacteristic
163D8×C43C22×C412yes—
164M16×C43—21no—
165M16×C43—41no—
166C2×C8×C43—41no—
167C2×C8×C43C2×D411yescharacteristic
168C2×C8×C43C2411yescharacteristic
169C4⋊C4×C43—21no—
170C22⋊C4×C43—41no—
171C42×C43C2×D411yescharacteristic
172C22⋊Dic43—41no—
173#10621885—21no—
174#10621884—21no—
175#10621883—21no—
176#10621882—22no—
177#10621882—22no—
178#10621882—21no—
179(C2×D43)⋊C4—21no—
180(C2×D43)⋊C4—41no—
181(C2×D43)⋊C4—41no—
182(C2×D43)⋊C4—41no—
183C4⋊Dic43—21no—
184Dic43⋊C4—21no—
185C4×Dic43—41no—
186C4×Dic43C2×D411yescharacteristic
187#10621877—21no—
188#10621877—41no—
189C2×C43⋊C8—41no—
190C2×C43⋊C8C2×D411yescharacteristic
191C2×C43⋊C8C2×D411yescharacteristic
192Dic172C22×C411yescharacteristic
193D344—21no—
194D344C22×C411yescharacteristic
195D344C22×C412yes—
196#10621873—21no—
197#10621872—21no—
198#10621872—21no—
199#10621872—41no—
200#10621872—41no—
201#10621872—41no—
202C8×D43—41no—
203C8×D43—41no—
204C8×D43—41no—
205C8×D43C22×C411yescharacteristic
206C8×D43C22×C411yescharacteristic
207C8×D43#4111yescharacteristic
208C8×D43#4111yescharacteristic
209C16×C43—21no—
210C16×C43C22×C412yes—
211C43⋊C16—21no—
212C43⋊C16C22×C412yes—
213C23×C43—41no—
214C22×D43—41no—
215C22×D43—41no—
216C22×D43—42no—
217C22×D43—42no—
218C22×D43—161no—
219C22×D43—161no—
220C22×D43AGL(1,ℤ/8)11yescharacteristic
221Q8×C43—21no—
222D4×C43—22no—
223D4×C43—21no—
224D4×C43—41no—
225D4×C43—41no—
226C2×C4×C43—21no—
227C2×C4×C43—41no—
228C2×C4×C43—41no—
229C2×C4×C43—161no—
230C2×C4×C43C22×D411yescharacteristic
231C22⋊D43—41no—
232C22⋊D43—41no—
233C22⋊D43—41no—
234C22⋊D43—42no—
235C22⋊D43—42no—
236C2×Dic43—21no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—161no—
240C2×Dic43AGL(1,ℤ/8)11yescharacteristic
241D172—21no—
242D172—22no—
243D172—41no—
244D172—41no—
245C4×D43—41no—
246C4×D43—41no—
247C4×D43—41no—
248C4×D43—161no—
249C4×D43—161no—
250C4×D43C4×D411yescharacteristic
251C4×D43C4×D411yescharacteristic
252C4×D43AGL(1,ℤ/8)11yescharacteristic
253C4×D43AGL(1,ℤ/8)11yescharacteristic
254Dic86—21no—
255Dic86—41no—
256C8×C43—21no—
257C8×C43—41no—
258C8×C43C23×C411yescharacteristic
259C8×C43C2×#4111yescharacteristic
260C43⋊C8—21no—
261C43⋊C8—41no—
262C43⋊C8C4×D411yescharacteristic
263C43⋊C8C4×D411yescharacteristic
264#33471—431nomaximal, Sylow
265C22×C43—42no—
266C22×C43—41no—
267C22×C43—81no—
268C22×C43—81no—
269C22×C43C2×AGL(1,ℤ/8)11yescharacteristic
270C2×D43—41no—
271C2×D43—42no—
272C2×D43—41no—
273C2×D43—81no—
274C2×D43—81no—
275C2×D43—161no—
276C2×D43—162no—
277C2×D43—162no—
278C2×D43#44211yescharacteristic
279C2×D43#44211yescharacteristic
280C4×C43—41no—
281C4×C43—41no—
282C4×C43—81no—
283C4×C43C2×C4×D411yescharacteristic
284C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
285Dic43—41no—
286Dic43—41no—
287Dic43—81no—
288Dic43#44211yescharacteristic
289Dic43#44211yescharacteristic
290C2×#509—431no—
291#2957—431no—
292#2948—431no—
293AGL(1,ℤ/16)—431no—
294AGL(1,ℤ/16)—431no—
295AGL(1,ℤ/16)—431no—
296AGL(1,ℤ/16)—431no—
297AGL(1,ℤ/16)—431no—
298AGL(1,ℤ/16)—431no—
299AGL(1,ℤ/16)—431no—
300AGL(1,ℤ/16)D4311yescharacteristic, complemented
301#2155—431no—
302C2×#361—431no—
303#2145—431no—
304#2110—431no—
305C2×C43—21no—
306C2×C43—82no—
307C2×C43—81no—
308C2×C43C2×#44211yescharacteristic
309D43—21no—
310D43—21no—
311D43—82no—
312D43—81no—
313C2×#135—431no—
314C22×D8—431no—
315#568—861no—
316#518—861no—
317#509—431no—
318#509—431no—
319#509—432no—
320#509—432no—
321#509—431no—
322#509C2×D4311yescharacteristic
323C2×SD32—432no—
324C2×D16—432no—
325C2×M32—431no—
326C2×M32—431no—
327#444—431no—
328#444—431no—
329#444—431no—
330#444C2×D4311yescharacteristic, complemented
331#443—861no—
332#443—861no—
333#442—431no—
334#442—431no—
335#442—431no—
336#442C2×D4311yescharacteristic, complemented
337C4×D8—861no—
338C4×D8—861no—
339C2×#108—431no—
340#427—431no—
341#421—861no—
342#418—861no—
343#405—431no—
344#365—431no—
345#365—431no—
346#365—431no—
347#365C2×D4311yescharacteristic, complemented
348#361—431no—
349#361—431no—
350#361—431no—
351#361C2×D4311yescharacteristic, complemented
352#360—431no—
353#360—431no—
354#360—431no—
355#360C2×D4311yescharacteristic, complemented
356#347—431no—
357#347—431no—
358#347—431no—
359#347C2×D4311yescharacteristic, complemented
360C43#3347111yescharacteristic, complemented, Sylow
361C2×#41—861no—
362C22×D4—861no—
363#135—431no—
364#135—431no—
365#135—431no—
366#135—861no—
367#135—861no—
368#135C22×D4311yescharacteristic
369C2×Q16—431no—
370C2×SD16—861no—
371C2×D8—432no—
372C2×D8—432no—
373C2×D8—431no—
374C2×D8—431no—
375C2×D8—861no—
376C2×D8—861no—
377C2×D8—861no—
378C2×D8C22×D4311yescharacteristic
379#131—21no—
380#131—861no—
381#131—861no—
382#131—861no—
383#131—1721no—
384#131—1721no—
385C2×M16—431no—
386C2×M16—861no—
387C2×M16—1721no—
388C2×M16—1721no—
389C22×C8—431no—
390C4×D4—21no—
391C4×D4—861no—
392C4×D4—861no—
393C4×D4—861no—
394C4×D4—1721no—
395C4×D4—1721no—
396#117—431no—
397#117—861no—
398C2×C22⋊C4—1721no—
399C2×C22⋊C4—1721no—
400SD32—432no—
401SD32—432no—
402SD32—432no—
403SD32C4×D4312yescomplemented
404D16—432no—
405D16—432no—
406D16—432no—
407D16C4×D4312yescomplemented
408M32—431no—
409M32—432no—
410M32—431no—
411M32—861no—
412M32C22×D4311yescharacteristic
413M32C22×D4311yescharacteristic
414C2×C16—432no—
415C2×C16—861no—
416#108—431no—
417#108—861no—
418#108C22×D4311yescharacteristic
419#107—861no—
420#106—431no—
421#106C22×D4311yescharacteristic
422#104—21no—
423#104—861no—
424#104—861no—
425#104—861no—
426#102—21no—
427#102—861no—
428#102—861no—
429#102—861no—
430#97—431no—
431#97C22×D4311yescharacteristic
432C4×C8—431no—
433C4×C8—431no—
434C24—1721no—
435#41—21no—
436#41—861no—
437#41—861no—
438#41—861no—
439#41—1721no—
440#41—1721no—
441C2×Q8—861no—
442C2×D4—21no—
443C2×D4—862no—
444C2×D4—862no—
445C2×D4—861no—
446C2×D4—861no—
447C2×D4—1721no—
448C2×D4—1721no—
449C2×D4—1721no—
450C2×D4—1721no—
451C22×C4—41no—
452C22×C4—431no—
453C22×C4—1721no—
454C22×C4—1721no—
455C22×C4—1721no—
456C22×C4—1721no—
457Q16—431no—
458Q16—861no—
459Q16C2×C4×D4311yescharacteristic
460SD16—21no—
461SD16—861no—
462SD16—861no—
463SD16—861no—
464D8—21no—
465D8—431no—
466D8—432no—
467D8—862no—
468D8—862no—
469D8—861no—
470D8—861no—
471D8C2×C4×D4311yescharacteristic
472D8C2×C4×D4312yes—
473M16—21no—
474M16—41no—
475M16—861no—
476M16—861no—
477M16—861no—
478M16—1721no—
479M16—1721no—
480M16—1721no—
481M16—1721no—
482C2×C8—41no—
483C2×C8—431no—
484C2×C8—431no—
485C2×C8—431no—
486C2×C8—431no—
487C2×C8—431no—
488C2×C8—431no—
489C2×C8—1721no—
490C2×C8—1721no—
491C2×C8—1721no—
492C2×C8—1721no—
493C2×C8D4×D4311yescharacteristic
494C2×C8C23×D4311yescharacteristic
495C4⋊C4—21no—
496C4⋊C4—861no—
497C4⋊C4—861no—
498C22⋊C4—41no—
499C22⋊C4—861no—
500C22⋊C4—1721no—
501C22⋊C4—1721no—
502C22⋊C4—1721no—
503C22⋊C4—1721no—
504C42—431no—
505C42—1721no—
506C42D4×D4311yescharacteristic
507C16—21no—
508C16—432no—
509C16—861no—
510C16C2×C4×D4312yes—
511C23—41no—
512C23—431no—
513C23—1721no—
514C23—1722no—
515C23—1722no—
516C23—1721no—
517C23—6881no—
518C23—6881no—
519Q8—21no—
520Q8—861no—
521Q8—1721no—
522D4—22no—
523D4—21no—
524D4—41no—
525D4—41no—
526D4—862no—
527D4—861no—
528D4—1721no—
529D4—1722no—
530D4—1722no—
531D4—1721no—
532D4—1721no—
533D4—1721no—
534D4—1721no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—161no—
539C2×C4—431no—
540C2×C4—431no—
541C2×C4—431no—
542C2×C4—431no—
543C2×C4—431no—
544C2×C4—861no—
545C2×C4—1721no—
546C2×C4—1721no—
547C2×C4—1721no—
548C2×C4—1721no—
549C2×C4—1721no—
550C2×C4—6881no—
551C2×C4—6881no—
552C2×C4—6881no—
553C2×C4C2×D4×D4311yescharacteristic
554C8—21no—
555C8—41no—
556C8—431no—
557C8—431no—
558C8—861no—
559C8—1721no—
560C8C22×C4×D4311yescharacteristic
561C8#41×D4311yescharacteristic
562C22—41no—
563C22—42no—
564C22—81no—
565C22—81no—
566C22—431no—
567C22—431no—
568C22—3441no—
569C22—3441no—
570C22—1721no—
571C22—1721no—
572C22—1722no—
573C22—6882no—
574C22—6882no—
575C22—6881no—
576C22AGL(1,ℤ/8)×D4311yescharacteristic
577C4—41no—
578C4—41no—
579C4—81no—
580C4—431no—
581C4—431no—
582C4—3441no—
583C4—1721no—
584C4—1721no—
585C4C4×D4×D4311yescharacteristic
586C4AGL(1,ℤ/8)×D4311yescharacteristic
587C2—21no—
588C2—82no—
589C2—81no—
590C2—861no—
591C2—861no—
592C2—3442no—
593C2—3441no—
594C2#442×D4311yescharacteristic
subgroupG/H
centerC2#442×D43
commutatorC8×C43C23×C4
FrattiniC2×C8C23×D43
FittingC43×AGL(1,ℤ/16)C2
socleC2×C43C2×#442
prime radical#509×D43C2
cosocleC2×C8×C43C24
hypercenterAGL(1,ℤ/16)D43
hypocenterC43#33471
LCSG ▷ C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ AGL(1,ℤ/16)

Complements

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

NH ≅ G/Ntype
2. #19143509C2semidirect
3. #19143444C2semidirect
4. #19136496C2semidirect
5. #19136495C2semidirect
6. #19136494C2semidirect
7. #19136493C2semidirect
8. #19136492C2semidirect
9. #19136491C2semidirect
10. #19136490C2semidirect
11. #19136366C2semidirect
12. #361×D43C2semidirect
13. #19136329C2semidirect
14. #19136198C2semidirect
15. C43×AGL(1,ℤ/16)C2semidirect
21. #14583336C4semidirect
24. #14583333C4semidirect
25. SD32×D43C4semidirect
26. D16×D43C4semidirect
29. #14583022C22semidirect
30. #14583021C22semidirect
31. #14583020C22semidirect
34. #14583012C22semidirect
35. #14583011C22semidirect
36. #14583010C22semidirect
44. #14582723C22semidirect
45. #14582722C22semidirect
46. #14582721C22semidirect
47. #14582713C22semidirect
48. #14582712C22semidirect
49. #14582711C22semidirect
50. #14582710C22semidirect
51. #14582709C22semidirect
52. #14582708C22semidirect
53. #14582677C22semidirect
54. #14582676C22semidirect
55. #14582676C22semidirect
57. C43×#444C22semidirect
58. C43×#442C22semidirect
59. C43×#365C22semidirect
60. C43×#361C22semidirect
61. C43×#360C22semidirect
62. C43×#347C22semidirect
65. #13063348C2×C4semidirect
69. #13063344C2×C4semidirect
70. Q16×D43C2×C4semidirect
78. D8×D43C2×C4semidirect
98. #13063270C2×C4semidirect
99. #13063269C2×C4semidirect
100. #13063268C2×C4semidirect
101. D688C2×C4semidirect
102. #13063266C2×C4semidirect
103. #13063266C2×C4semidirect
107. #13063264C2×C4semidirect
109. C16×D43C2×C4semidirect
127. SD32×C43C2×C4semidirect
128. D16×C43C2×C4semidirect
300. AGL(1,ℤ/16)D43semidirect
330. #444C2×D43semidirect
336. #442C2×D43semidirect
347. #365C2×D43semidirect
351. #361C2×D43semidirect
355. #360C2×D43semidirect
359. #347C2×D43semidirect
360. C43#33471semidirect
403. SD32C4×D43semidirect
407. D16C4×D43semidirect

Maximal subgroups

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

subgroupindex
#509×D432
#191435092
#191434442
#191364962
#191364952
#191364942
#191364932
#191364922
#191364912
#191364902
#191363662
#361×D432
#191363292
#191361982
C43×AGL(1,ℤ/16)2
#3347143