Finite Group Databasebeta

← #25253090

Subgroups — #25253090

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×#14583342C211yesmaximal, characteristic, complemented
2C22×#13063270C211yesmaximal, characteristic, complemented
3D4×Dic172C211yesmaximal, characteristic, complemented
4#19146282C211yesmaximal, characteristic
5#19136866C211yesmaximal, characteristic, complemented
6#19136708C211yesmaximal, characteristic, complemented
7#19136686C211yesmaximal, characteristic, complemented
8#19136658C211yesmaximal, characteristic, complemented
9#19136641C211yesmaximal, characteristic, complemented
10#19136537C211yesmaximal, characteristic, complemented
11#19136531C211yesmaximal, characteristic, complemented
12#19136526C211yesmaximal, characteristic, complemented
13#19136514C211yesmaximal, characteristic, complemented
14#19136461C211yesmaximal, characteristic, complemented
15C43×#2168C211yesmaximal, characteristic, complemented
16C2×#13063347C2211yescharacteristic
17C22×Dic172C2211yescharacteristic
18C22×Dic172C2211yescharacteristic, complemented
19C22×D344C2211yescharacteristic
20D4×Dic86—21no—
21#14583545—21no—
22#14583342—24no—
23C2×Dic344C2211yescharacteristic, complemented
24C2×Dic344C2211yescharacteristic, complemented
25C2×Dic344C2211yescharacteristic, complemented
26C2×#13063270—24no—
27C2×#13063270C2211yescharacteristic
28C2×#13063270C2211yescharacteristic
29C2×#13063270C2211yescharacteristic
30C2×#13063270C2211yescharacteristic
31C2×#13063270C2211yescharacteristic
32C2×#13063270C2211yescharacteristic
33#14583283C2211yescharacteristic
34#14583279C2211yescharacteristic, complemented
35#14583140C2211yescharacteristic
36#14583140C2211yescharacteristic
37#14583140C2211yescharacteristic, complemented
38#14583135C2211yescharacteristic
39#14583115—21no—
40#14583110—21no—
41#14583057—21no—
42#14583057—21no—
43#14583052—21no—
44#14583046—21no—
45C4×Dic172C2211yescharacteristic, complemented
46C4×D344C2211yescharacteristic
47#14582725C2211yescharacteristic, complemented
48#14582704C2211yescharacteristic, complemented
49#14582704C2211yescharacteristic, complemented
50#14582704C2211yescharacteristic, complemented
51#14582704C2211yescharacteristic, complemented
52#14582701C2211yescharacteristic
53#14582701C2211yescharacteristic
54C2×M32×C43C2211yescharacteristic, complemented
55C22×C16×C43C2211yescharacteristic
56C8×D4×C43C2211yescharacteristic
57C43×C4⋊C16C2211yescharacteristic, complemented
58C43×C22⋊C16C2211yescharacteristic
59C43×C22⋊C16C2211yescharacteristic, complemented
60C43×#346C2211yescharacteristic, complemented
61C2×#10621898—21no—
62C22×Dic86—21no—
63C22×Dic86—21no—
64C22×D172—21no—
65#13063347—22no—
66#13063347—41no—
67C2×Dic172—22no—
68C2×Dic172—22no—
69C2×Dic172—21no—
70C2×Dic172—21no—
71C2×Dic172—21no—
72C2×Dic172—41no—
73C2×Dic172—41no—
74C2×Dic172C2311yescharacteristic
75C2×Dic172C2311yescharacteristic
76C2×Dic172C2311yescharacteristic
77C2×Dic172C2311yescharacteristic
78C2×#10621873—21no—
79C2×#10621873—21no—
80C2×D344—22no—
81C2×D344—21no—
82C2×D344—41no—
83C2×D344C2311yescharacteristic
84C2×D344C2311yescharacteristic
85#13063325—21no—
86#13063324—41no—
87#13063313—21no—
88#13063310—41no—
89#13063304—21no—
90#13063304—21no—
91#13063304—21no—
92#13063302—41no—
93#13063302—41no—
94#13063299—21no—
95#13063298—41no—
96#13063293—41no—
97C4×Dic86—21no—
98C4×D172—21no—
99D4×Dic43—41no—
100C22⋊C4×D43—41no—
101Dic344—22no—
102Dic344—22no—
103Dic344—41no—
104#13063270—22no—
105#13063270—22no—
106#13063270—22no—
107#13063270—22no—
108#13063270—41no—
109#13063270D414yescomplemented
110#13063259C2311yescharacteristic
111#13063259C2311yescharacteristic
112#13063250—21no—
113#13063250—21no—
114#13063250—21no—
115#13063250—21no—
116#13063247—21no—
117#13063247—21no—
118C22×C8×C43C2311yescharacteristic
119C22×C8×C43C2311yescharacteristic
120C4×D4×C43D411yescharacteristic
121M32×C43—22no—
122C2×C16×C43—22no—
123C2×C16×C43C2311yescharacteristic
124C2×C16×C43C2311yescharacteristic
125C2×C16×C43C2311yescharacteristic
126C2×C16×C43C2311yescharacteristic
127C4⋊C8×C43D411yescharacteristic
128C22⋊C8×C43D411yescharacteristic
129C22⋊C8×C43D411yescharacteristic
130C4×C8×C43C2311yescharacteristic
131C23×D43—41no—
132C2×D4×C43D811yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2×D411yescharacteristic
135C2×C22⋊D43—41no—
136C2×C22⋊D43—41no—
137C22×Dic43—41no—
138C22×Dic43—41no—
139#10621898—42no—
140#10621898—81no—
141C2×D172—21no—
142C2×D172—21no—
143C2×D172—41no—
144C2×D172—42no—
145C2×D172—81no—
146C2×C4×D43—41no—
147C2×C4×D43—41no—
148C2×Dic86—21no—
149C2×Dic86—21no—
150C2×Dic86—21no—
151C2×Dic86—21no—
152C2×Dic86—42no—
153C2×Dic86—42no—
154C2×Dic86—41no—
155C2×Dic86—41no—
156C2×Dic86—81no—
157C2×Dic86—81no—
158C2×C8×C43—21no—
159C2×C8×C43—21no—
160C2×C8×C43—21no—
161C2×C8×C43—21no—
162C2×C8×C43C2×D411yescharacteristic
163C2×C8×C43C2×D411yescharacteristic
164C2×C8×C43C2×D411yescharacteristic
165C2×C8×C43C2411yescharacteristic
166C4⋊C4×C43D811yescharacteristic
167C22⋊C4×C43D811yescharacteristic
168C22⋊C4×C43D811yescharacteristic
169C42×C43C2×D411yescharacteristic
170C22⋊Dic43—41no—
171C22⋊Dic43—41no—
172C22⋊Dic43—41no—
173(C2×D43)⋊C4—41no—
174(C2×D43)⋊C4—41no—
175(C2×D43)⋊C4—41no—
176(C2×D43)⋊C4—41no—
177C4⋊Dic43—21no—
178C4⋊Dic43—21no—
179C4⋊Dic43—41no—
180C4⋊Dic43—41no—
181Dic43⋊C4—41no—
182Dic43⋊C4—41no—
183Dic43⋊C4—41no—
184Dic43⋊C4—41no—
185C4×Dic43—41no—
186Dic172—21no—
187Dic172—21no—
188Dic172—21no—
189Dic172—21no—
190Dic172—21no—
191Dic172—21no—
192Dic172—21no—
193Dic172—42no—
194Dic172—42no—
195Dic172—81no—
196Dic172C2×D412yes—
197D344—21no—
198D344—21no—
199D344—21no—
200D344—42no—
201D344C2×D412yes—
202#10621873—42no—
203#10621873—81no—
204C16×C43—21no—
205C16×C43—21no—
206C16×C43—21no—
207C16×C43C2×D412yes—
208C23×C43C2×D811yescharacteristic
209C23×C43C2×D811yescharacteristic
210C22×D43—41no—
211C22×D43—41no—
212C22×D43—82no—
213C22×D43—82no—
214C22×D43—161no—
215D4×C43—22no—
216C2×C4×C43—21no—
217C2×C4×C43—21no—
218C2×C4×C43—21no—
219C2×C4×C43—21no—
220C2×C4×C43C2×D811yescharacteristic
221C2×C4×C43C2×D811yescharacteristic
222C2×C4×C43C2×D811yescharacteristic
223C2×C4×C43C2×D811yescharacteristic
224C2×C4×C43C22×D411yescharacteristic
225C22⋊D43—82no—
226C22⋊D43—161no—
227C2×Dic43—41no—
228C2×Dic43—41no—
229C2×Dic43—41no—
230C2×Dic43—41no—
231C2×Dic43—41no—
232C2×Dic43—41no—
233C2×Dic43—82no—
234C2×Dic43—82no—
235D172—22no—
236D172—41no—
237D172—41no—
238D172—41no—
239D172—82no—
240C4×D43—82no—
241C4×D43—161no—
242Dic86—22no—
243Dic86—41no—
244Dic86—41no—
245Dic86—41no—
246Dic86—41no—
247Dic86—41no—
248Dic86—41no—
249Dic86—41no—
250Dic86—82no—
251Dic86—82no—
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×C43D4211yescharacteristic
277C4×C43#57611yescharacteristic
278Dic43—42no—
279Dic43—81no—
280Dic43—81no—
281Dic43—81no—
282Dic43—81no—
283Dic43—81no—
284C2×#510—431no—
285C22×SD32—431no—
286D4×Q16—431no—
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#2168D4311yescharacteristic, complemented
299C2×C43—22no—
300C2×C43—41no—
301C2×C43D4×D811yescharacteristic
302C2×C43C2×#51011yescharacteristic
303C2×C43#341611yescharacteristic
304D43—82no—
305D43—161no—
306C2×#135—431no—
307C22×Q16—431no—
308C22×Q16—431no—
309C22×D8—431no—
310D4×Q8—861no—
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×M32C2×D4311yescharacteristic, complemented
324C22×C16C2×D4311yescharacteristic
325#495—431no—
326#494—431no—
327#467—431no—
328#467—431no—
329#467—431no—
330#466—431no—
331#463—861no—
332#462—861no—
333#451—861no—
334#451—861no—
335#450—861no—
336#449—861no—
337C4×Q16—431no—
338C4×D8—431no—
339C8×D4C2×D4311yescharacteristic
340#366—431no—
341C4⋊C16C2×D4311yescharacteristic, complemented
342#358—431no—
343#358—431no—
344#358—431no—
345#358—431no—
346#357—431no—
347#357—431no—
348C22⋊C16C2×D4311yescharacteristic
349C22⋊C16C2×D4311yescharacteristic, complemented
350#346C2×D4311yescharacteristic, complemented
351C43#3383211yescharacteristic, complemented, Sylow
352C2×#41—861no—
353C22×Q8—861no—
354C22×Q8—861no—
355C22×D4—861no—
356#135—862no—
357#135—1721no—
358C2×Q16—431no—
359C2×Q16—431no—
360C2×Q16—431no—
361C2×Q16—431no—
362C2×Q16—862no—
363C2×Q16—862no—
364C2×Q16—861no—
365C2×Q16—861no—
366C2×Q16—861no—
367C2×Q16—1721no—
368C2×Q16—1721no—
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—861no—
379C4⋊Q8—1721no—
380#124—861no—
381#123—1721no—
382C22⋊Q8—861no—
383C22⋊Q8—861no—
384C22⋊Q8—861no—
385C22⋊Q8—1721no—
386C22⋊Q8—1721no—
387#121—861no—
388#121—1721no—
389#120—1721no—
390C4×Q8—861no—
391C4×D4—861no—
392C4×D4—1721no—
393C4×D4D17211yescharacteristic
394C2×C22⋊C4—1721no—
395Q32—862no—
396Q32—862no—
397Q32—1721no—
398SD32—434no—
399SD32—862no—
400SD32—862no—
401SD32—862no—
402SD32—862no—
403SD32—1721no—
404M32—22no—
405C2×C16—22no—
406C2×C16C22×D4311yescharacteristic
407C2×C16C22×D4311yescharacteristic
408C2×C16C22×D4311yescharacteristic
409C2×C16C22×D4311yescharacteristic
410#107—431no—
411#107—431no—
412C4⋊C8D17211yescharacteristic
413#103—861no—
414#103—861no—
415#103—861no—
416#103—861no—
417#102—861no—
418#102—861no—
419C22⋊C8D17211yescharacteristic
420C22⋊C8D17211yescharacteristic
421C4×C8C22×D4311yescharacteristic
422C24—1721no—
423#41—3441no—
424#41—1722no—
425C2×Q8—861no—
426C2×Q8—861no—
427C2×Q8—861no—
428C2×Q8—861no—
429C2×Q8—3441no—
430C2×Q8—3441no—
431C2×Q8—1722no—
432C2×Q8—1722no—
433C2×Q8—1721no—
434C2×Q8—1721no—
435C2×D4—861no—
436C2×D4—861no—
437C2×D4—3441no—
438C2×D4—1722no—
439C2×D4—1721no—
440C2×D4—1721no—
441C2×D4—1721no—
442C2×D4D34411yescharacteristic
443C22×C4—1721no—
444C22×C4—1721no—
445C22×C4—1721no—
446C22×C4—1721no—
447C22×C4C2×D17211yescharacteristic
448C22×C4C2×D17211yescharacteristic
449Q16—432no—
450Q16—861no—
451Q16—861no—
452Q16—861no—
453Q16—861no—
454Q16—861no—
455Q16—861no—
456Q16—861no—
457Q16—3441no—
458Q16—1722no—
459Q16—1722no—
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×D17211yescharacteristic
472C2×C8C2×D17211yescharacteristic
473C2×C8C2×D17211yescharacteristic
474C2×C8C23×D4311yescharacteristic
475C4⋊C4—861no—
476C4⋊C4—861no—
477C4⋊C4—1721no—
478C4⋊C4—1721no—
479C4⋊C4—1721no—
480C4⋊C4—1721no—
481C4⋊C4—1721no—
482C4⋊C4—1721no—
483C4⋊C4D34411yescharacteristic
484C22⋊C4—1721no—
485C22⋊C4—1721no—
486C22⋊C4—1721no—
487C22⋊C4—1721no—
488C22⋊C4—1721no—
489C22⋊C4—1721no—
490C22⋊C4—1721no—
491C22⋊C4D34411yescharacteristic
492C22⋊C4D34411yescharacteristic
493C42—1721no—
494C42C2×D17211yescharacteristic
495C16—21no—
496C16—21no—
497C16—21no—
498C16D4×D4312yes—
499C23—3442no—
500C23—3442no—
501C23—1721no—
502C23—1721no—
503C23—6881no—
504C23C2×D34411yescharacteristic
505C23C2×D34411yescharacteristic
506Q8—862no—
507Q8—3442no—
508Q8—3442no—
509Q8—1721no—
510Q8—1721no—
511Q8—1721no—
512Q8—1721no—
513Q8—1721no—
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—3442no—
529C2×C4—3442no—
530C2×C4—3442no—
531C2×C4—1721no—
532C2×C4—1721no—
533C2×C4—1721no—
534C2×C4—1721no—
535C2×C4—1721no—
536C2×C4—1721no—
537C2×C4—6881no—
538C2×C4C2×D34411yescharacteristic
539C2×C4C2×D34411yescharacteristic
540C2×C4C2×D34411yescharacteristic
541C2×C4C2×D34411yescharacteristic
542C2×C4C22×D17211yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8C2×D4×D4311yescharacteristic
547C8#1306337111yescharacteristic
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#1458334212yescomplemented
561C22C22×D34411yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—3441no—
567C4—3441no—
568C4—3441no—
569C4—3441no—
570C4—3441no—
571C4—1722no—
572C4D4×D17211yescharacteristic
573C4#1458370411yescharacteristic
574C2—22no—
575C2—41no—
576C2—3442no—
577C2—6881no—
578C2D4×D34411yescharacteristic
579C2C2×#1458334211yescharacteristic
580C2#1914774311yescharacteristic
subgroupG/H
centerC22C22×D344
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingC43×#2168C2
socleC22×C43C22×D8
prime radical#19146282C2
hypercenter#2168D43
hypocenterC43#33832
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. C2×#14583342C2semidirect
2. C22×#13063270C2semidirect
3. D4×Dic172C2semidirect
5. #19136866C2semidirect
6. #19136708C2semidirect
7. #19136686C2semidirect
8. #19136658C2semidirect
9. #19136641C2semidirect
10. #19136537C2semidirect
11. #19136531C2semidirect
12. #19136526C2semidirect
13. #19136514C2semidirect
14. #19136461C2semidirect
15. C43×#2168C2semidirect
18. C22×Dic172C22semidirect
23. C2×Dic344C22semidirect
24. C2×Dic344C22semidirect
25. C2×Dic344C22semidirect
34. #14583279C22semidirect
37. #14583140C22semidirect
45. C4×Dic172C22semidirect
47. #14582725C22semidirect
48. #14582704C22semidirect
49. #14582704C22semidirect
50. #14582704C22semidirect
51. #14582704C22semidirect
54. C2×M32×C43C22semidirect
57. C43×C4⋊C16C22semidirect
59. C43×C22⋊C16C22semidirect
60. C43×#346C22semidirect
109. #13063270D4semidirect
263. C22×C43#510semidirect
298. #2168D43semidirect
323. C2×M32C2×D43semidirect
341. C4⋊C16C2×D43semidirect
349. C22⋊C16C2×D43semidirect
350. #346C2×D43semidirect
351. C43#33832semidirect
560. C22#14583342semidirect

Maximal subgroups

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

subgroupindex
C2×#145833422
C22×#130632702
D4×Dic1722
#191462822
#191368662
#191367082
#191366862
#191366582
#191366412
#191365372
#191365312
#191365262
#191365142
#191364612
C43×#21682
#3383243