Finite Group Databasebeta

← #25251078

Subgroups — #25251078

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583341C211yesmaximal, characteristic, complemented
2C22×#13063264C211yesmaximal, characteristic, complemented
3#19144178C211yesmaximal, characteristic
4#467×D43C211yesmaximal, characteristic, complemented
5#19136711C211yesmaximal, characteristic, complemented
6#19136709C211yesmaximal, characteristic, complemented
7#19136702C211yesmaximal, characteristic, complemented
8#19136700C211yesmaximal, characteristic, complemented
9#19136693C211yesmaximal, characteristic, complemented
10#19136691C211yesmaximal, characteristic, complemented
11#19136684C211yesmaximal, characteristic, complemented
12#19136380C211yesmaximal, characteristic, complemented
13#19136313C211yesmaximal, characteristic, complemented
14#19136306C211yesmaximal, characteristic, complemented
15C43×#2216C211yesmaximal, characteristic, complemented
16C2×#13063348C2211yescharacteristic
17C2×Q16×D43C2211yescharacteristic
18C22×C8×D43C2211yescharacteristic
19#14583412—21no—
20C22⋊Q8×D43—21no—
21#14583341—24no—
22C2×#13063271C2211yescharacteristic, complemented
23C2×#13063270C2211yescharacteristic, complemented
24C2×#13063268C2211yescharacteristic, complemented
25C2×#13063264—24no—
26C2×#13063264C2211yescharacteristic
27C2×#13063264C2211yescharacteristic
28C2×#13063264C2211yescharacteristic
29C2×#13063264C2211yescharacteristic
30C22×C43⋊C16C2211yescharacteristic
31#14583141C2211yescharacteristic
32#14583140C2211yescharacteristic
33#14583139—21no—
34#14583138—21no—
35#14583137C2211yescharacteristic
36#14583137C2211yescharacteristic
37#14583136C2211yescharacteristic
38#14583135C2211yescharacteristic
39#14583131—21no—
40#14583130—21no—
41#14582936C2211yescharacteristic
42#107×D43C2211yescharacteristic
43#14582894—21no—
44#103×D43—21no—
45#14582727C2211yescharacteristic, complemented
46#14582726C2211yescharacteristic, complemented
47#14582724C2211yescharacteristic, complemented
48#14582704C2211yescharacteristic, complemented
49#14582703C2211yescharacteristic, complemented
50#14582703C2211yescharacteristic, complemented
51#14582702C2211yescharacteristic, complemented
52#14582701C2211yescharacteristic, complemented
53#14582699C2211yescharacteristic, complemented
54C2×Q32×C43C2211yescharacteristic, complemented
55C22×C16×C43C2211yescharacteristic
56C43×#467C2211yescharacteristic
57C43×#467C2211yescharacteristic
58C43×#366C2211yescharacteristic, complemented
59C43×#358C2211yescharacteristic, complemented
60C43×#358C2211yescharacteristic, complemented
61C2×#10621902—21no—
62C2×Q8×D43—21no—
63C22×C4×D43D411yescharacteristic
64#13063348—22no—
65#13063348—22no—
66C2×Dic172C2311yescharacteristic
67C2×#10621885—21no—
68Q16×D43—22no—
69Q16×D43—22no—
70C2×#10621884—21no—
71C2×D344C2311yescharacteristic
72C22×C43⋊C8D411yescharacteristic
73C2×C8×D43—22no—
74C2×C8×D43—22no—
75C2×C8×D43D411yescharacteristic
76C2×C8×D43D411yescharacteristic
77C2×C8×D43C2311yescharacteristic
78C2×C8×D43C2311yescharacteristic
79#13063310—41no—
80#13063305—21no—
81#13063304—21no—
82#13063303—41no—
83#13063302—41no—
84#13063301—21no—
85#13063301—21no—
86#13063300—21no—
87#13063299—21no—
88#13063295—41no—
89#13063287—41no—
90(C22⋊D43)⋊C4—41no—
91#13063283—21no—
92C4⋊C4×D43—21no—
93C4⋊C4×D43—41no—
94C22⋊C4×D43—41no—
95#13063271—22no—
96#13063270—22no—
97#13063268—22no—
98#13063264—22no—
99#13063264—22no—
100#13063264—22no—
101#13063264D414yescomplemented
102C2×C43⋊C16—22no—
103C2×C43⋊C16C2311yescharacteristic
104C2×C43⋊C16C2311yescharacteristic
105#13063259C2311yescharacteristic
106#13063259C2311yescharacteristic
107#13063258—21no—
108#13063256—21no—
109#13063250—21no—
110#13063249—21no—
111#13063249—21no—
112#13063248—21no—
113#13063247—21no—
114#13063245—21no—
115C2×Q16×C43C2311yescharacteristic
116C2×Q16×C43C2311yescharacteristic
117C22×C8×C43C2311yescharacteristic
118C22⋊Q8×C43—21no—
119C22⋊Q8×C43—21no—
120Q32×C43—22no—
121C2×C16×C43—22no—
122C2×C16×C43C2311yescharacteristic
123C2×C16×C43C2311yescharacteristic
124#107×C43C2311yescharacteristic
125#107×C43C2311yescharacteristic
126#103×C43—21no—
127#103×C43—21no—
128C23×D43D811yescharacteristic
129C2×Q8×C43—21no—
130C2×Q8×C43—21no—
131C22×C4×C43C2×D411yescharacteristic
132C2×C22⋊D43—41no—
133C22×Dic43D811yescharacteristic
134#10621902—42no—
135#10621902—42no—
136Q8×D43—42no—
137Q8×D43—42no—
138C2×D172—21no—
139C2×D172—41no—
140C2×D172—41no—
141C2×C4×D43—22no—
142C2×C4×D43—22no—
143C2×C4×D43—41no—
144C2×C4×D43—41no—
145C2×C4×D43—41no—
146C2×C4×D43D811yescharacteristic
147C2×C4×D43D811yescharacteristic
148C2×C4×D43C2×D411yescharacteristic
149C2×C4×D43C2×D411yescharacteristic
150C2×Dic86—21no—
151C2×Dic86—41no—
152Q16×C43—21no—
153Q16×C43—21no—
154Q16×C43—21no—
155Q16×C43—21no—
156C2×C8×C43—21no—
157C2×C8×C43—21no—
158C2×C8×C43C2×D411yescharacteristic
159C2×C8×C43C2411yescharacteristic
160C4⋊C4×C43—21no—
161C4⋊C4×C43—21no—
162C4⋊C4×C43—41no—
163C4⋊C4×C43—41no—
164C22⋊C4×C43—41no—
165C22⋊C4×C43—41no—
166C22⋊Dic43—41no—
167#10621885—42no—
168#10621884—42no—
169(C2×D43)⋊C4—41no—
170(C2×D43)⋊C4—41no—
171(C2×D43)⋊C4—41no—
172(C2×D43)⋊C4—41no—
173(C2×D43)⋊C4—41no—
174(C2×D43)⋊C4—41no—
175C4⋊Dic43—21no—
176C4⋊Dic43—21no—
177C4⋊Dic43—41no—
178Dic43⋊C4—41no—
179Dic43⋊C4—41no—
180Dic43⋊C4—41no—
181Dic43⋊C4—41no—
182C4×Dic43—41no—
183C2×C43⋊C8—22no—
184C2×C43⋊C8C2×D411yescharacteristic
185C2×C43⋊C8C2×D411yescharacteristic
186Dic172—21no—
187Dic172—21no—
188D344—21no—
189D344—21no—
190C8×D43—22no—
191C8×D43—21no—
192C8×D43—21no—
193C8×D43—22no—
194C8×D43C2×D412yes—
195C8×D43#4112yes—
196C16×C43—21no—
197C16×C43C2×D412yes—
198C43⋊C16—21no—
199C43⋊C16C2×D412yes—
200C23×C43C2×D811yescharacteristic
201C22×D43—22no—
202C22×D43—22no—
203C22×D43—41no—
204C22×D43—41no—
205C22×D43C2×D811yescharacteristic
206C22×D43C2×D811yescharacteristic
207Q8×C43—41no—
208Q8×C43—41no—
209Q8×C43—41no—
210Q8×C43—41no—
211C2×C4×C43—21no—
212C2×C4×C43—21no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43C2×D811yescharacteristic
218C2×C4×C43C22×D411yescharacteristic
219C22⋊D43—82no—
220C2×Dic43—22no—
221C2×Dic43—41no—
222C2×Dic43—41no—
223C2×Dic43—41no—
224C2×Dic43C2×D811yescharacteristic
225C2×Dic43C2×D811yescharacteristic
226D172—41no—
227D172—41no—
228D172—82no—
229D172—82no—
230C4×D43—22no—
231C4×D43—21no—
232C4×D43—21no—
233C4×D43—22no—
234C4×D43—82no—
235C4×D43—82no—
236C4×D43—82no—
237C4×D43#12112yes—
238C4×D43#13512yes—
239Dic86—41no—
240Dic86—41no—
241Dic86—82no—
242C8×C43—21no—
243C8×C43C22×D411yescharacteristic
244C8×C43C2×#4111yescharacteristic
245C43⋊C8—21no—
246C43⋊C8#12112yes—
247#33606—431nomaximal, Sylow
248C22×C43—21no—
249C22×C43—21no—
250C22×C43#51012yescomplemented
251C22×C43C22×D811yescharacteristic
252C2×D43—21no—
253C2×D43—21no—
254C2×D43—21no—
255C2×D43—21no—
256C2×D43—21no—
257C2×D43—44no—
258C2×D43—81no—
259C2×D43—81no—
260C2×D43—81no—
261C2×D43#46612yes—
262C4×C43—21no—
263C4×C43—81no—
264C4×C43—81no—
265C4×C43—81no—
266C4×C43—81no—
267C4×C43C2×#12111yescharacteristic
268C4×C43C2×#13511yescharacteristic
269Dic43—21no—
270Dic43—81no—
271Dic43—81no—
272Dic43—81no—
273Dic43#46612yes—
274C2×#510—431no—
275C22×M32—431no—
276#3050—431no—
277C2×#467—431no—
278#2220—431no—
279#2220—431no—
280#2219—431no—
281#2219—431no—
282#2217—431no—
283#2217—431no—
284#2216—431no—
285#2216D4311yescharacteristic, complemented
286#2159—431no—
287#2141—431no—
288#2139—431no—
289C2×C43—22no—
290C2×C43C2×#46611yescharacteristic
291C2×C43C2×#51011yescharacteristic
292C2×C43#341411yescharacteristic
293D43—22no—
294D43—41no—
295D43—161no—
296C2×#135—431no—
297C22×Q16—431no—
298C23×C8—431no—
299#525—861no—
300C2×C22⋊Q8—861no—
301#510—864no—
302C2×Q32—431no—
303C2×Q32C2×D4311yescharacteristic, complemented
304C2×SD32—431no—
305C2×SD32—431no—
306C2×M32—431no—
307C2×M32—431no—
308C2×M32—431no—
309C2×M32—431no—
310C2×M32—864no—
311C22×C16—431no—
312C22×C16C2×D4311yescharacteristic
313#467—431no—
314#467—431no—
315#467—431no—
316#467—431no—
317#467—861no—
318#467—861no—
319#467C2×D4311yescharacteristic
320#467C2×D4311yescharacteristic
321#466—431no—
322#466—431no—
323#465—861no—
324#465—861no—
325#427—431no—
326C2×#107—431no—
327#416—861no—
328C2×#103—861no—
329#367—431no—
330#367—431no—
331#366—431no—
332#366C2×D4311yescharacteristic, complemented
333#358—431no—
334#358—431no—
335#358—431no—
336#358—431no—
337#358C2×D4311yescharacteristic, complemented
338#358C2×D4311yescharacteristic, complemented
339#357—431no—
340#357—431no—
341C43#3360611yescharacteristic, complemented, Sylow
342C2×#41—861no—
343C22×Q8—861no—
344C23×C4—431no—
345#135—862no—
346#135—862no—
347C2×Q16—431no—
348C2×Q16—862no—
349C2×Q16—862no—
350C2×Q16—861no—
351C2×Q16C22×D4311yescharacteristic
352C2×Q16C22×D4311yescharacteristic
353C2×SD16—861no—
354C2×D8—431no—
355C22×C8—431no—
356C22×C8—431no—
357C22×C8—431no—
358C22×C8—431no—
359C22×C8—431no—
360C22×C8—862no—
361C22×C8—862no—
362C22×C8C22×D4311yescharacteristic
363#123—1721no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—861no—
367C22⋊Q8—861no—
368C22⋊Q8—861no—
369C22⋊Q8—861no—
370C22⋊Q8—1721no—
371C22⋊Q8—1721no—
372#121—861no—
373#121—861no—
374#120—1721no—
375C4×D4—1721no—
376C4×D4—1721no—
377#117—861no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4—1721no—
380C2×C22⋊C4—1721no—
381Q32—22no—
382Q32—862no—
383SD32—862no—
384SD32—862no—
385M32—434no—
386M32—862no—
387M32—862no—
388M32—862no—
389C2×C16—22no—
390C2×C16—431no—
391C2×C16—431no—
392C2×C16—862no—
393C2×C16C22×D4311yescharacteristic
394C2×C16C22×D4311yescharacteristic
395#107—431no—
396#107—431no—
397#107—861no—
398#107C22×D4311yescharacteristic
399#107C22×D4311yescharacteristic
400#106—861no—
401#103—21no—
402#103—21no—
403#103—861no—
404#103—861no—
405#103—861no—
406#103—861no—
407#102—861no—
408#102—861no—
409C24—431no—
410#41—1722no—
411#41—1722no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—861no—
415C2×Q8—1721no—
416C2×Q8—1722no—
417C2×Q8—1722no—
418C2×D4—861no—
419C2×D4—1721no—
420C2×D4—1721no—
421C2×D4—1721no—
422C22×C4—431no—
423C22×C4—431no—
424C22×C4—431no—
425C22×C4—431no—
426C22×C4—431no—
427C22×C4—862no—
428C22×C4—862no—
429C22×C4—1721no—
430C22×C4—1721no—
431C22×C4—1721no—
432C22×C4D4×D4311yescharacteristic
433Q16—21no—
434Q16—21no—
435Q16—21no—
436Q16—21no—
437Q16—861no—
438Q16—861no—
439Q16—1722no—
440SD16—1722no—
441D8—861no—
442D8—861no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—431no—
446C2×C8—432no—
447C2×C8—432no—
448C2×C8—431no—
449C2×C8—862no—
450C2×C8—862no—
451C2×C8—862no—
452C2×C8—861no—
453C2×C8—861no—
454C2×C8D4×D4311yescharacteristic
455C2×C8C23×D4311yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—861no—
461C4⋊C4—861no—
462C4⋊C4—1721no—
463C4⋊C4—1721no—
464C4⋊C4—1721no—
465C4⋊C4—1721no—
466C4⋊C4—1721no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—1721no—
470C22⋊C4—1721no—
471C22⋊C4—1721no—
472C22⋊C4—1721no—
473C22⋊C4—1721no—
474C22⋊C4—1721no—
475C22⋊C4—1721no—
476C42—1721no—
477C16—21no—
478C16—432no—
479C16—861no—
480C16D4×D4312yes—
481C23—431no—
482C23—431no—
483C23—862no—
484C23—862no—
485C23—1721no—
486C23—1721no—
487C23D8×D4311yescharacteristic
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492Q8—3442no—
493Q8—1721no—
494Q8—1721no—
495D4—3442no—
496D4—3442no—
497D4—3442no—
498D4—1721no—
499D4—1721no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—431no—
507C2×C4—432no—
508C2×C4—432no—
509C2×C4—431no—
510C2×C4—862no—
511C2×C4—862no—
512C2×C4—862no—
513C2×C4—861no—
514C2×C4—861no—
515C2×C4—3442no—
516C2×C4—3442no—
517C2×C4—3442no—
518C2×C4—1721no—
519C2×C4—1721no—
520C2×C4—1721no—
521C2×C4D8×D4311yescharacteristic
522C2×C4C2×D4×D4311yescharacteristic
523C8—21no—
524C8—432no—
525C8—861no—
526C8C2×D4×D4311yescharacteristic
527C8#41×D4311yescharacteristic
528C22—21no—
529C22—21no—
530C22—432no—
531C22—861no—
532C22—861no—
533C22—861no—
534C22—861no—
535C22—861no—
536C22—3441no—
537C22—3441no—
538C22—3441no—
539C22—1724no—
540C22#1458334112yescomplemented
541C22C2×D8×D4311yescharacteristic
542C4—21no—
543C4—81no—
544C4—81no—
545C4—81no—
546C4—81no—
547C4—432no—
548C4—861no—
549C4—3441no—
550C4—3441no—
551C4—3441no—
552C4#121×D4311yescharacteristic
553C4#135×D4311yescharacteristic
554C2—22no—
555C2—862no—
556C2—1721no—
557C2—6881no—
558C2#466×D4311yescharacteristic
559C2C2×#1458334111yescharacteristic
560C2#1914772811yescharacteristic
subgroupG/H
centerC22C2×D8×D43
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingC43×#2216C2
socleC22×C43C22×D8
prime radical#19144178C2
hypercenter#2216D43
hypocenterC43#33606
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2216

Complements

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

NH ≅ G/Ntype
1. C2×#14583341C2semidirect
2. C22×#13063264C2semidirect
4. #467×D43C2semidirect
5. #19136711C2semidirect
6. #19136709C2semidirect
7. #19136702C2semidirect
8. #19136700C2semidirect
9. #19136693C2semidirect
10. #19136691C2semidirect
11. #19136684C2semidirect
12. #19136380C2semidirect
13. #19136313C2semidirect
14. #19136306C2semidirect
15. C43×#2216C2semidirect
22. C2×#13063271C22semidirect
23. C2×#13063270C22semidirect
24. C2×#13063268C22semidirect
45. #14582727C22semidirect
46. #14582726C22semidirect
47. #14582724C22semidirect
48. #14582704C22semidirect
49. #14582703C22semidirect
50. #14582703C22semidirect
51. #14582702C22semidirect
52. #14582701C22semidirect
53. #14582699C22semidirect
54. C2×Q32×C43C22semidirect
58. C43×#366C22semidirect
59. C43×#358C22semidirect
60. C43×#358C22semidirect
101. #13063264D4semidirect
250. C22×C43#510semidirect
285. #2216D43semidirect
303. C2×Q32C2×D43semidirect
332. #366C2×D43semidirect
337. #358C2×D43semidirect
338. #358C2×D43semidirect
341. C43#33606semidirect
540. C22#14583341semidirect

Maximal subgroups

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

subgroupindex
C2×#145833412
C22×#130632642
#191441782
#467×D432
#191367112
#191367092
#191367022
#191367002
#191366932
#191366912
#191366842
#191363802
#191363132
#191363062
C43×#22162
#3360643