Finite Group Databasebeta

← #25253086

Subgroups — #25253086

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583340C211yesmaximal, characteristic, complemented
2C22×#13063268C211yesmaximal, characteristic, complemented
3#19146282C211yesmaximal, characteristic
4#19136864C211yesmaximal, characteristic, complemented
5#19136710C211yesmaximal, characteristic, complemented
6#19136684C211yesmaximal, characteristic, complemented
7#19136657C211yesmaximal, characteristic, complemented
8#19136637C211yesmaximal, characteristic, complemented
9#19136533C211yesmaximal, characteristic, complemented
10#19136529C211yesmaximal, characteristic, complemented
11#19136523C211yesmaximal, characteristic, complemented
12#19136512C211yesmaximal, characteristic, complemented
13#19136457C211yesmaximal, characteristic, complemented
14#19136421C211yesmaximal, characteristic, complemented
15D4×Q16×C43C211yesmaximal, characteristic, complemented
16C2×#13063347C2211yescharacteristic
17C22×D344C2211yescharacteristic
18#14583545—21no—
19#14583340—24no—
20C2×#13063271C2211yescharacteristic, complemented
21C2×#13063271C2211yescharacteristic, complemented
22C2×#13063271C2211yescharacteristic, complemented
23C2×#13063268—24no—
24C2×#13063268C2211yescharacteristic
25C2×#13063268C2211yescharacteristic
26C2×#13063268C2211yescharacteristic
27C2×#13063268C2211yescharacteristic
28C2×#13063268C2211yescharacteristic
29C2×#13063268C2211yescharacteristic
30C2×#13063265C2211yescharacteristic, complemented
31C22×C43⋊C16C2211yescharacteristic
32#14583283C2211yescharacteristic
33#14583140C2211yescharacteristic
34#14583135C2211yescharacteristic
35#14583115—21no—
36#14583052—21no—
37#14583046—21no—
38C4×D344C2211yescharacteristic
39#14582724C2211yescharacteristic, complemented
40C4⋊(C43⋊C16)C2211yescharacteristic, complemented
41#14582703C2211yescharacteristic, complemented
42#14582703C2211yescharacteristic, complemented
43#14582703C2211yescharacteristic, complemented
44#14582702C2211yescharacteristic, complemented
45#14582699C2211yescharacteristic
46#14582699C2211yescharacteristic
47C22⋊(C43⋊C16)C2211yescharacteristic
48C22⋊(C43⋊C16)C2211yescharacteristic, complemented
49#14582675C2211yescharacteristic, complemented
50C22×Q16×C43C2211yescharacteristic
51C22×Q16×C43C2211yescharacteristic, complemented
52D4×Q8×C43—21no—
53C43×#494C2211yescharacteristic, complemented
54C43×#467C2211yescharacteristic
55C43×#467C2211yescharacteristic, complemented
56C43×#462—21no—
57C43×#451—21no—
58C43×#451—21no—
59C4×Q16×C43C2211yescharacteristic, complemented
60C8×D4×C43C2211yescharacteristic
61C2×#10621898—21no—
62C22×D172—21no—
63#13063347—22no—
64#13063347—41no—
65C2×Dic172C2311yescharacteristic
66C2×#10621873—21no—
67C2×#10621873—21no—
68C2×D344—22no—
69C2×D344—21no—
70C2×D344—41no—
71C2×D344C2311yescharacteristic
72C2×D344C2311yescharacteristic
73#13063313—21no—
74#13063310—41no—
75#13063304—21no—
76#13063299—21no—
77#13063298—41no—
78#13063293—41no—
79C4×D172—21no—
80C22⋊C4×D43—41no—
81#13063271—22no—
82#13063271—22no—
83#13063271—41no—
84#13063268—22no—
85#13063268—22no—
86#13063268—22no—
87#13063268—22no—
88#13063268—41no—
89#13063268D414yescomplemented
90#13063265—22no—
91C2×C43⋊C16—22no—
92C2×C43⋊C16C2311yescharacteristic
93C2×C43⋊C16C2311yescharacteristic
94C2×C43⋊C16C2311yescharacteristic
95C2×C43⋊C16C2311yescharacteristic
96#13063259C2311yescharacteristic
97#13063250—21no—
98#13063247—21no—
99#13063247—21no—
100C22×Q8×C43—21no—
101C22×Q8×C43—21no—
102C2×Q16×C43—22no—
103C2×Q16×C43—22no—
104C2×Q16×C43—21no—
105C2×Q16×C43—21no—
106C2×Q16×C43—21no—
107C2×Q16×C43—41no—
108C2×Q16×C43—41no—
109C2×Q16×C43C2311yescharacteristic
110C2×Q16×C43C2311yescharacteristic
111C2×Q16×C43C2311yescharacteristic
112C22×C8×C43C2311yescharacteristic
113C22×C8×C43C2311yescharacteristic
114C4⋊Q8×C43—21no—
115C4⋊Q8×C43—41no—
116C22⋊Q8×C43—21no—
117C22⋊Q8×C43—21no—
118C22⋊Q8×C43—41no—
119C22⋊Q8×C43—41no—
120C4×Q8×C43—21no—
121C4×D4×C43—41no—
122C4×D4×C43D411yescharacteristic
123#107×C43C2311yescharacteristic
124C4⋊C8×C43D411yescharacteristic
125#103×C43—21no—
126#103×C43—21no—
127#103×C43—21no—
128C22⋊C8×C43D411yescharacteristic
129C22⋊C8×C43D411yescharacteristic
130C4×C8×C43C2311yescharacteristic
131C23×D43—41no—
132C2×Q8×C43—21no—
133C2×Q8×C43—21no—
134C2×Q8×C43—21no—
135C2×Q8×C43—42no—
136C2×Q8×C43—42no—
137C2×Q8×C43—41no—
138C2×Q8×C43—41no—
139C2×Q8×C43—81no—
140C2×Q8×C43—81no—
141C2×D4×C43D811yescharacteristic
142C22×C4×C43—41no—
143C22×C4×C43—41no—
144C22×C4×C43C2×D411yescharacteristic
145C22×C4×C43C2×D411yescharacteristic
146C2×C22⋊D43—41no—
147C2×C22⋊D43—41no—
148#10621898—42no—
149#10621898—81no—
150C2×D172—21no—
151C2×D172—21no—
152C2×D172—41no—
153C2×D172—42no—
154C2×D172—81no—
155C2×C4×D43—41no—
156C2×C4×D43—41no—
157C2×Dic86—21no—
158Q16×C43—21no—
159Q16×C43—21no—
160Q16×C43—21no—
161Q16×C43—21no—
162Q16×C43—21no—
163Q16×C43—42no—
164Q16×C43—42no—
165Q16×C43—81no—
166Q16×C43C2×D412yes—
167C2×C8×C43—21no—
168C2×C8×C43—21no—
169C2×C8×C43—21no—
170C2×C8×C43—21no—
171C2×C8×C43C2×D411yescharacteristic
172C2×C8×C43C2×D411yescharacteristic
173C2×C8×C43C2×D411yescharacteristic
174C2×C8×C43C2411yescharacteristic
175C4⋊C4×C43—21no—
176C4⋊C4×C43—41no—
177C4⋊C4×C43—41no—
178C4⋊C4×C43—41no—
179C4⋊C4×C43—41no—
180C4⋊C4×C43—41no—
181C4⋊C4×C43D811yescharacteristic
182C22⋊C4×C43—41no—
183C22⋊C4×C43—41no—
184C22⋊C4×C43D811yescharacteristic
185C22⋊C4×C43D811yescharacteristic
186C42×C43—41no—
187C42×C43C2×D411yescharacteristic
188C22⋊Dic43—41no—
189(C2×D43)⋊C4—41no—
190(C2×D43)⋊C4—41no—
191(C2×D43)⋊C4—41no—
192(C2×D43)⋊C4—41no—
193C4⋊Dic43—21no—
194Dic43⋊C4—41no—
195Dic172—21no—
196Dic172—21no—
197D344—21no—
198D344—21no—
199D344—21no—
200D344—42no—
201D344C2×D412yes—
202#10621873—42no—
203#10621873—81no—
204C43⋊C16—21no—
205C43⋊C16—21no—
206C43⋊C16—21no—
207C43⋊C16C2×D412yes—
208C23×C43C2×D811yescharacteristic
209C23×C43C2×D811yescharacteristic
210C22×D43—41no—
211C22×D43—41no—
212C22×D43—82no—
213C22×D43—82no—
214C22×D43—161no—
215Q8×C43—22no—
216Q8×C43—41no—
217Q8×C43—41no—
218Q8×C43—41no—
219Q8×C43—41no—
220Q8×C43—41no—
221Q8×C43—82no—
222Q8×C43—82no—
223D4×C43—22no—
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—21no—
228C2×C4×C43—41no—
229C2×C4×C43—41no—
230C2×C4×C43—41no—
231C2×C4×C43—41no—
232C2×C4×C43—82no—
233C2×C4×C43—82no—
234C2×C4×C43C2×D811yescharacteristic
235C2×C4×C43C2×D811yescharacteristic
236C2×C4×C43C2×D811yescharacteristic
237C2×C4×C43C2×D811yescharacteristic
238C2×C4×C43C22×D411yescharacteristic
239C22⋊D43—82no—
240C22⋊D43—161no—
241C2×Dic43—41no—
242C2×Dic43—41no—
243D172—22no—
244D172—41no—
245D172—41no—
246D172—41no—
247D172—82no—
248C4×D43—82no—
249C4×D43—161no—
250Dic86—41no—
251Dic86—41no—
252C8×C43—21no—
253C8×C43—21no—
254C8×C43—21no—
255C8×C43C22×D411yescharacteristic
256C8×C43H(5,𝔽2)11yescharacteristic
257#33832—431nomaximal, Sylow
258C22×C43—21no—
259C22×C43—21no—
260C22×C43—21no—
261C22×C43—21no—
262C22×C43—21no—
263C22×C43#51012yescomplemented
264C22×C43C22×D811yescharacteristic
265C2×D43—42no—
266C2×D43—81no—
267C2×D43—81no—
268C2×D43—81no—
269C2×D43—81no—
270C2×D43—81no—
271C2×D43—164no—
272C4×C43—21no—
273C4×C43—21no—
274C4×C43—21no—
275C4×C43—21no—
276C4×C43—42no—
277C4×C43—81no—
278C4×C43—81no—
279C4×C43—81no—
280C4×C43D4211yescharacteristic
281C4×C43#57611yescharacteristic
282Dic43—81no—
283Dic43—81no—
284C2×#510—431no—
285C22×SD32—431no—
286D4×Q16D4311yescharacteristic, complemented
287#3280—431no—
288#2251—431no—
289#2220—431no—
290#2216—431no—
291#2211—431no—
292#2208—431no—
293#2189—431no—
294#2188—431no—
295#2187—431no—
296#2185—431no—
297#2175—431no—
298#2168—431no—
299C2×C43—22no—
300C2×C43—41no—
301C2×C43D4×D811yescharacteristic
302C2×C43C2×#51011yescharacteristic
303C2×C43#341611yescharacteristic
304D43—82no—
305D43—161no—
306C2×#135—431no—
307C22×Q16C2×D4311yescharacteristic
308C22×Q16C2×D4311yescharacteristic, complemented
309C22×D8—431no—
310D4×Q8—21no—
311#546—861no—
312#510—864no—
313C2×Q32—431no—
314C2×Q32—431no—
315C2×Q32—431no—
316C2×SD32—431no—
317C2×SD32—431no—
318C2×SD32—431no—
319C2×SD32—431no—
320C2×SD32—431no—
321C2×SD32—431no—
322C2×SD32—864no—
323C2×M32—431no—
324C22×C16—431no—
325#495—431no—
326#494C2×D4311yescharacteristic, complemented
327#467—431no—
328#467C2×D4311yescharacteristic
329#467C2×D4311yescharacteristic, complemented
330#466—431no—
331#463—861no—
332#462—21no—
333#451—21no—
334#451—21no—
335#450—861no—
336#449—861no—
337C4×Q16C2×D4311yescharacteristic, complemented
338C4×D8—431no—
339C8×D4C2×D4311yescharacteristic
340#366—431no—
341C4⋊C16—431no—
342#358—431no—
343#358—431no—
344#358—431no—
345#358—431no—
346#357—431no—
347#357—431no—
348C22⋊C16—431no—
349C22⋊C16—431no—
350#346—431no—
351C43#3383211yescharacteristic, complemented, Sylow
352C2×#41—861no—
353C22×Q8—21no—
354C22×Q8—21no—
355C22×D4—861no—
356#135—862no—
357#135—1721no—
358C2×Q16—22no—
359C2×Q16—22no—
360C2×Q16—21no—
361C2×Q16—21no—
362C2×Q16—21no—
363C2×Q16—41no—
364C2×Q16—41no—
365C2×Q16—431no—
366C2×Q16C22×D4311yescharacteristic
367C2×Q16C22×D4311yescharacteristic
368C2×Q16C22×D4311yescharacteristic
369C2×SD16—861no—
370C2×SD16—861no—
371C2×D8—431no—
372C2×D8—431no—
373C2×D8—862no—
374C2×D8—861no—
375C2×D8—1721no—
376C22×C8C22×D4311yescharacteristic
377C22×C8C22×D4311yescharacteristic
378C4⋊Q8—21no—
379C4⋊Q8—41no—
380#124—861no—
381#123—1721no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—41no—
385C22⋊Q8—41no—
386C22⋊Q8—861no—
387#121—861no—
388#121—1721no—
389#120—1721no—
390C4×Q8—21no—
391C4×D4—41no—
392C4×D4—861no—
393C4×D4C22⋊D4311yescharacteristic
394C2×C22⋊C4—1721no—
395Q32—862no—
396Q32—862no—
397Q32—1721no—
398SD32—434no—
399SD32—862no—
400SD32—862no—
401SD32—862no—
402SD32—862no—
403SD32—1721no—
404M32—862no—
405C2×C16—431no—
406C2×C16—431no—
407C2×C16—431no—
408C2×C16—431no—
409C2×C16—862no—
410#107—431no—
411#107C22×D4311yescharacteristic
412C4⋊C8C22⋊D4311yescharacteristic
413#103—21no—
414#103—21no—
415#103—21no—
416#103—861no—
417#102—861no—
418#102—861no—
419C22⋊C8C22⋊D4311yescharacteristic
420C22⋊C8C22⋊D4311yescharacteristic
421C4×C8C22×D4311yescharacteristic
422C24—1721no—
423#41—3441no—
424#41—1722no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—81no—
433C2×Q8—81no—
434C2×Q8—861no—
435C2×D4—861no—
436C2×D4—861no—
437C2×D4—3441no—
438C2×D4—1722no—
439C2×D4—1721no—
440C2×D4—1721no—
441C2×D4—1721no—
442C2×D4#1062188211yescharacteristic
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—1721no—
446C22×C4—1721no—
447C22×C4C2×C22⋊D4311yescharacteristic
448C22×C4C2×C22⋊D4311yescharacteristic
449Q16—21no—
450Q16—21no—
451Q16—21no—
452Q16—21no—
453Q16—21no—
454Q16—42no—
455Q16—42no—
456Q16—81no—
457Q16—861no—
458Q16—861no—
459Q16D4×D4312yes—
460SD16—3441no—
461SD16—1722no—
462D8—432no—
463D8—861no—
464D8—861no—
465D8—861no—
466D8—1722no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8C2×C22⋊D4311yescharacteristic
472C2×C8C2×C22⋊D4311yescharacteristic
473C2×C8C2×C22⋊D4311yescharacteristic
474C2×C8C23×D4311yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—861no—
482C4⋊C4—1721no—
483C4⋊C4#1062188211yescharacteristic
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—1721no—
487C22⋊C4—1721no—
488C22⋊C4—1721no—
489C22⋊C4—1721no—
490C22⋊C4—1721no—
491C22⋊C4#1062188211yescharacteristic
492C22⋊C4#1062188211yescharacteristic
493C42—41no—
494C42C2×C22⋊D4311yescharacteristic
495C16—432no—
496C16—861no—
497C16—861no—
498C16—861no—
499C23—3442no—
500C23—3442no—
501C23—1721no—
502C23—1721no—
503C23—6881no—
504C23C2×#1062188211yescharacteristic
505C23C2×#1062188211yescharacteristic
506Q8—22no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—82no—
513Q8—82no—
514Q8—1721no—
515Q8—1721no—
516D4—22no—
517D4—862no—
518D4—3442no—
519D4—3442no—
520D4—1721no—
521D4—1721no—
522D4—1721no—
523D4—6881no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—3442no—
535C2×C4—1721no—
536C2×C4—1721no—
537C2×C4—6881no—
538C2×C4C2×#1062188211yescharacteristic
539C2×C4C2×#1062188211yescharacteristic
540C2×C4C2×#1062188211yescharacteristic
541C2×C4C2×#1062188211yescharacteristic
542C2×C4C22×C22⋊D4311yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8C2×D4×D4311yescharacteristic
547C8#1306337011yescharacteristic
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—3441no—
554C22—3441no—
555C22—3441no—
556C22—3441no—
557C22—3441no—
558C22—1722no—
559C22—6884no—
560C22#1458334012yescomplemented
561C22C22×#1062188211yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—3441no—
571C4—3441no—
572C4D4×C22⋊D4311yescharacteristic
573C4#1458370211yescharacteristic
574C2—22no—
575C2—41no—
576C2—3442no—
577C2—6881no—
578C2D4×#1062188211yescharacteristic
579C2C2×#1458334011yescharacteristic
580C2#1914773911yescharacteristic
subgroupG/H
centerC22C22×#10621882
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingD4×Q16×C43C2
socleC22×C43C22×D8
prime radical#19146282C2
hypercenterD4×Q16D43
hypocenterC43#33832
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#14583340C2semidirect
2. C22×#13063268C2semidirect
4. #19136864C2semidirect
5. #19136710C2semidirect
6. #19136684C2semidirect
7. #19136657C2semidirect
8. #19136637C2semidirect
9. #19136533C2semidirect
10. #19136529C2semidirect
11. #19136523C2semidirect
12. #19136512C2semidirect
13. #19136457C2semidirect
14. #19136421C2semidirect
15. D4×Q16×C43C2semidirect
20. C2×#13063271C22semidirect
21. C2×#13063271C22semidirect
22. C2×#13063271C22semidirect
30. C2×#13063265C22semidirect
39. #14582724C22semidirect
40. C4⋊(C43⋊C16)C22semidirect
41. #14582703C22semidirect
42. #14582703C22semidirect
43. #14582703C22semidirect
44. #14582702C22semidirect
48. C22⋊(C43⋊C16)C22semidirect
49. #14582675C22semidirect
51. C22×Q16×C43C22semidirect
53. C43×#494C22semidirect
55. C43×#467C22semidirect
59. C4×Q16×C43C22semidirect
89. #13063268D4semidirect
263. C22×C43#510semidirect
286. D4×Q16D43semidirect
308. C22×Q16C2×D43semidirect
326. #494C2×D43semidirect
329. #467C2×D43semidirect
337. C4×Q16C2×D43semidirect
351. C43#33832semidirect
560. C22#14583340semidirect

Maximal subgroups

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

subgroupindex
C2×#145833402
C22×#130632682
#191462822
#191368642
#191367102
#191366842
#191366572
#191366372
#191365332
#191365292
#191365232
#191365122
#191364572
#191364212
D4×Q16×C432
#3383243