Finite Group Databasebeta

← #25086694

Subgroups — #25086694

Order 11008.

640 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#10621884C211yesmaximal, characteristic, complemented
2#19146460C211yesmaximal, characteristic
3#19146242C211yesmaximal, characteristic
4#19134229C211yesmaximal, characteristic, complemented
5#19134091C211yesmaximal, characteristic, complemented
6#19134023C211yesmaximal, characteristic, complemented
7#19133972C211yesmaximal, characteristic, complemented
8#19133965C211yesmaximal, characteristic, complemented
9#19133858C211yesmaximal, characteristic, complemented
10#19133857C211yesmaximal, characteristic, complemented
11#19133816C211yesmaximal, characteristic, complemented
12#19133744C211yesmaximal, characteristic
13#19133438C211yesmaximal, characteristic, complemented
14SD16⋊(C43⋊C8)C211yesmaximal, characteristic, complemented
15Q8×SD16×C43C211yesmaximal, characteristic, complemented
16C2×#13063354—21no—
17Q8×D172C2211yescharacteristic
18#14583553C2211yescharacteristic
19#14583295—21no—
20#14583295C2211yescharacteristic
21#14583295C2211yescharacteristic
22#14583279C2211yescharacteristic
23#14583249C2211yescharacteristic
24#14583249C2211yescharacteristic
25#14583227—21no—
26#14583186—21no—
27#14583186C2211yescharacteristic
28#14583184—21no—
29#14583184C2211yescharacteristic
30#14583184C2211yescharacteristic
31#14583158—21no—
32#14583130—21no—
33#14583109C2211yescharacteristic
34#14583104C2211yescharacteristic
35#14583101C2211yescharacteristic
36#14583062—21no—
37#14583056—21no—
38#14583038—21no—
39Q8×C43⋊C8C2211yescharacteristic
40C4×#10621884—21no—
41C4×#10621884C2211yescharacteristic
42#14582978C2211yescharacteristic
43D4×C43⋊C8C2211yescharacteristic
44C2×#13063249—21no—
45#14582652C2211yescharacteristic
46#14582649C2211yescharacteristic
47#14582648C2211yescharacteristic
48#14582646C2211yescharacteristic
49#14582646C2211yescharacteristic
50#14582645C2211yescharacteristic
51Q8⋊(C43⋊C8)C2211yescharacteristic
52Q8⋊(C43⋊C8)C2211yescharacteristic
53D4⋊(C43⋊C8)C2211yescharacteristic
54#14582626C2211yescharacteristic
55#14582620C2211yescharacteristic
56Q82×C43C2211yescharacteristic
57D4×Q8×C43C2211yescharacteristic
58C43×#498—21no—
59C43×#498C2211yescharacteristic
60C43×#476—21no—
61C43×#476C2211yescharacteristic
62C43×#475—21no—
63C43×#475C2211yescharacteristic
64C8×Q8×C43C2211yescharacteristic
65C4×SD16×C43—21no—
66C4×SD16×C43C2211yescharacteristic
67C2×#10621900—21no—
68C2×Q8×D43—21no—
69#13063354—44no—
70C2×Dic172—21no—
71C2×Dic172—21no—
72C2×#10621885—21no—
73C2×#10621883—21no—
74C2×#10621884—21no—
75C2×#10621884D411yescharacteristic
76C2×#10621872—21no—
77C22×C43⋊C8—21no—
78#13063325—21no—
79#13063325C2311yescharacteristic
80#13063325C2311yescharacteristic
81#13063309—41no—
82#13063305—21no—
83#13063305—21no—
84#13063305—41no—
85#13063299—21no—
86C4×D172—21no—
87C4×D172C2311yescharacteristic
88C2×C4⋊Dic43—21no—
89#13063256—21no—
90#13063256—21no—
91#13063256—21no—
92#13063256—21no—
93#13063256—21no—
94#13063256—41no—
95#13063256D411yescharacteristic
96C4⋊(C43⋊C8)—21no—
97C4⋊(C43⋊C8)C2311yescharacteristic
98C4⋊(C43⋊C8)C2311yescharacteristic
99C4⋊(C43⋊C8)C2311yescharacteristic
100#13063249—21no—
101#13063249—21no—
102#13063249—21no—
103#13063249—21no—
104#13063249—21no—
105#13063249—41no—
106#13063249D411yescharacteristic
107#13063248—21no—
108#13063248—21no—
109#13063245—21no—
110#13063245D411yescharacteristic
111C22⋊(C43⋊C8)—21no—
112(C2×D43)⋊C8—21no—
113C4×C43⋊C8—21no—
114C4×C43⋊C8C2311yescharacteristic
115C4×C43⋊C8C2311yescharacteristic
116C22×Q8×C43—21no—
117C2×SD16×C43D411yescharacteristic
118C4⋊Q8×C43—21no—
119C4⋊Q8×C43—21no—
120C4⋊Q8×C43—21no—
121C4⋊Q8×C43—41no—
122C4⋊Q8×C43C2311yescharacteristic
123C4⋊Q8×C43C2311yescharacteristic
124C22⋊Q8×C43—21no—
125C22⋊Q8×C43—41no—
126C4×Q8×C43—21no—
127C4×Q8×C43—21no—
128C4×Q8×C43C2311yescharacteristic
129C4×Q8×C43C2311yescharacteristic
130C4×D4×C43—21no—
131C4×D4×C43C2311yescharacteristic
132#106×C43—21no—
133#106×C43—21no—
134#106×C43—21no—
135#106×C43—21no—
136#106×C43D411yescharacteristic
137C4⋊C8×C43—21no—
138C4⋊C8×C43C2311yescharacteristic
139#103×C43—21no—
140#103×C43D411yescharacteristic
141#102×C43—21no—
142#102×C43D411yescharacteristic
143C4×C8×C43—21no—
144C4×C8×C43C2311yescharacteristic
145C2×Q8×C43—21no—
146C2×Q8×C43—21no—
147C2×Q8×C43—41no—
148C2×Q8×C43—41no—
149C2×Q8×C43—42no—
150C2×Q8×C43—81no—
151C2×Q8×C43C2×D411yescharacteristic
152C2×Q8×C43C2×D411yescharacteristic
153C2×D4×C43C2×D411yescharacteristic
154C22×C4×C43—21no—
155C22×C4×C43—41no—
156C2×C22⋊D43—41no—
157C22×Dic43—41no—
158Q8×D43—42no—
159Q8×D43—81no—
160#10621900—42no—
161#10621900—81no—
162C2×D172C2×D411yescharacteristic
163C2×C4×D43—21no—
164C2×C4×D43—41no—
165C2×Dic86—21no—
166C2×Dic86—21no—
167C2×Dic86—41no—
168SD16×C43SD1614yescomplemented
169C2×C8×C43—21no—
170C2×C8×C43C2×D411yescharacteristic
171C2×C8×C43C2×D411yescharacteristic
172C4⋊C4×C43—21no—
173C4⋊C4×C43—21no—
174C4⋊C4×C43—21no—
175C4⋊C4×C43—21no—
176C4⋊C4×C43—21no—
177C4⋊C4×C43—21no—
178C4⋊C4×C43—41no—
179C4⋊C4×C43—41no—
180C4⋊C4×C43—41no—
181C4⋊C4×C43C2×D411yescharacteristic
182C4⋊C4×C43C2×D411yescharacteristic
183C4⋊C4×C43C2×D411yescharacteristic
184C22⋊C4×C43—21no—
185C22⋊C4×C43—41no—
186C42×C43—21no—
187C42×C43—21no—
188C42×C43—41no—
189C42×C43C2411yescharacteristic
190#10621885—42no—
191#10621884—22no—
192#10621884—81no—
193#10621883—42no—
194(C2×D43)⋊C4—21no—
195(C2×D43)⋊C4—41no—
196(C2×D43)⋊C4—41no—
197C4⋊Dic43—21no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—41no—
202C4⋊Dic43—41no—
203C4⋊Dic43C2×D411yescharacteristic
204C2×C43⋊C8—21no—
205C2×C43⋊C8—21no—
206C2×C43⋊C8—21no—
207C2×C43⋊C8—21no—
208C2×C43⋊C8—81no—
209C2×C43⋊C8C2×D411yescharacteristic
210C2×C43⋊C8C2×D411yescharacteristic
211Dic172—42no—
212Dic172—81no—
213#10621872—42no—
214C23×C43—21no—
215C22×D43—21no—
216Q8×C43—21no—
217Q8×C43—21no—
218Q8×C43—41no—
219Q8×C43—41no—
220Q8×C43—82no—
221Q8×C43—81no—
222Q8×C43—161no—
223Q8×C43C2×SD1612yes—
224Q8×C43#13712yescomplemented
225D4×C43—21no—
226D4×C43C2×SD1612yes—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—21no—
231C2×C4×C43—21no—
232C2×C4×C43—41no—
233C2×C4×C43—41no—
234C2×C4×C43—41no—
235C2×C4×C43—41no—
236C2×C4×C43—82no—
237C2×C4×C43C22×D411yescharacteristic
238C2×C4×C43C22×D411yescharacteristic
239C2×C4×C43H(5,𝔽2)11yescharacteristic
240C22⋊D43—82no—
241C2×Dic43—21no—
242C2×Dic43—41no—
243C2×Dic43—41no—
244C2×Dic43—82no—
245D172—21no—
246D172—21no—
247C4×D43—41no—
248C4×D43—41no—
249C4×D43—82no—
250Dic86—41no—
251Dic86—41no—
252Dic86—41no—
253Dic86—41no—
254Dic86—82no—
255C8×C43—21no—
256C8×C43—41no—
257C8×C43C2×SD1612yes—
258C43⋊C8—21no—
259C43⋊C8—21no—
260C43⋊C8—41no—
261C43⋊C8—41no—
262C43⋊C8—41no—
263C43⋊C8—41no—
264#19805—431nomaximal, Sylow
265C22×C43—22no—
266C22×C43—41no—
267C22×C43—41no—
268C22×C43D4211yescharacteristic
269C2×D43—41no—
270C2×D43—41no—
271C2×D43—41no—
272C4×C43—22no—
273C4×C43—22no—
274C4×C43—21no—
275C4×C43—41no—
276C4×C43—41no—
277C4×C43—41no—
278C4×C43—81no—
279C4×C43—81no—
280C4×C43C22×SD1611yescharacteristic
281C4×C43C2×#13711yescharacteristic
282C4×C43#57711yescharacteristic
283C4×C43#57711yescharacteristic
284Dic43—41no—
285Dic43—81no—
286Dic43—81no—
287Q8×SD16—431no—
288Q8×SD16D4311yescharacteristic, complemented
289#3298—431no—
290#3276—431no—
291#1716—431no—
292#1691—431no—
293#1680—431no—
294#1671—431no—
295#1670—431no—
296#1651—431no—
297#1651—431no—
298#1645—431no—
299#1635—431no—
300#1583—431no—
301SD16⋊C8—431no—
302C2×C43—42no—
303C2×C43#326611yescharacteristic
304C2×C43D4×SD1611yescharacteristic
305C2×C43#328911yescharacteristic
306D43—81no—
307C2×#137—861no—
308Q82C2×D4311yescharacteristic
309D4×Q8—431no—
310D4×Q8C2×D4311yescharacteristic
311#547—431no—
312#498—21no—
313#498—431no—
314#498—431no—
315#498—861no—
316#498C2×D4311yescharacteristic
317#494—431no—
318#487—431no—
319#487—431no—
320#483—861no—
321#476—21no—
322#476—431no—
323#476—861no—
324#476C2×D4311yescharacteristic
325#475—21no—
326#475—431no—
327#475—431no—
328#475—861no—
329#475C2×D4311yescharacteristic
330#470—861no—
331#465—861no—
332#462—431no—
333#461—431no—
334#461—431no—
335#452—861no—
336#451—861no—
337#448—861no—
338C8×Q8—431no—
339C8×Q8C2×D4311yescharacteristic
340C4×SD16—21no—
341C4×SD16—431no—
342C4×SD16—861no—
343C4×SD16C2×D4311yescharacteristic
344#436—431no—
345C8×D4—431no—
346C2×#103—861no—
347#335—431no—
348#334—431no—
349#333—431no—
350#332—431no—
351#332—431no—
352#332—431no—
353Q8⋊C8—431no—
354Q8⋊C8—431no—
355D4⋊C8—431no—
356D4⋊C8—431no—
357#322—431no—
358C43#1980511yescharacteristic, complemented, Sylow
359C2×#41—861no—
360C22×Q8—21no—
361C22×Q8—861no—
362#137—1724no—
363C2×Q16—861no—
364C2×Q16—861no—
365C2×Q16—861no—
366C2×SD16—431no—
367C2×SD16—861no—
368C2×SD16—861no—
369C2×SD16C22⋊D4311yescharacteristic
370C2×M16—861no—
371C22×C8—861no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—41no—
376C4⋊Q8—431no—
377C4⋊Q8—431no—
378C4⋊Q8—861no—
379C4⋊Q8C22×D4311yescharacteristic
380C4⋊Q8C22×D4311yescharacteristic
381#123—1721no—
382C22⋊Q8—21no—
383C22⋊Q8—41no—
384C22⋊Q8—861no—
385C22⋊Q8—861no—
386C22⋊Q8—1721no—
387#121—861no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8C22×D4311yescharacteristic
391C4×Q8C22×D4311yescharacteristic
392C4×D4—21no—
393C4×D4—431no—
394C4×D4—861no—
395C4×D4C22×D4311yescharacteristic
396C2×C4⋊C4—861no—
397#106—21no—
398#106—21no—
399#106—21no—
400#106—21no—
401#106—431no—
402#106—861no—
403#106—861no—
404#106—861no—
405#106—861no—
406#106—861no—
407#106—1721no—
408#106C22⋊D4311yescharacteristic
409C4⋊C8—21no—
410C4⋊C8—431no—
411C4⋊C8—431no—
412C4⋊C8—431no—
413C4⋊C8—861no—
414C4⋊C8C22×D4311yescharacteristic
415#103—21no—
416#103—431no—
417#103—861no—
418#103—861no—
419#103—861no—
420#103—861no—
421#103—861no—
422#103—861no—
423#103—861no—
424#103—1721no—
425#103C22⋊D4311yescharacteristic
426#102—21no—
427#102—431no—
428#102—861no—
429#102C22⋊D4311yescharacteristic
430C22⋊C8—861no—
431C22⋊C8—861no—
432C4×C8—21no—
433C4×C8—431no—
434C4×C8—431no—
435C4×C8—861no—
436C4×C8C22×D4311yescharacteristic
437#41—3441no—
438#41—1722no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—41no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×Q8—81no—
445C2×Q8—861no—
446C2×Q8—861no—
447C2×Q8—3441no—
448C2×Q8—1721no—
449C2×Q8—1722no—
450C2×Q8D4×D4311yescharacteristic
451C2×Q8C2×C22⋊D4311yescharacteristic
452C2×D4—431no—
453C2×D4—1721no—
454C2×D4C2×C22⋊D4311yescharacteristic
455C22×C4—21no—
456C22×C4—41no—
457C22×C4—861no—
458C22×C4—1721no—
459C22×C4—1721no—
460Q16—3441no—
461Q16—1722no—
462Q16—1722no—
463SD16—862no—
464SD16—3441no—
465SD16—1722no—
466SD16#1062188414yescomplemented
467M16—1722no—
468C2×C8—21no—
469C2×C8—431no—
470C2×C8—431no—
471C2×C8—861no—
472C2×C8—861no—
473C2×C8—861no—
474C2×C8—861no—
475C2×C8—3441no—
476C2×C8C2×C22⋊D4311yescharacteristic
477C2×C8C2×C22⋊D4311yescharacteristic
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—431no—
488C4⋊C4—861no—
489C4⋊C4—861no—
490C4⋊C4—861no—
491C4⋊C4—861no—
492C4⋊C4—1721no—
493C4⋊C4—1721no—
494C4⋊C4D4×D4311yescharacteristic
495C4⋊C4C2×C22⋊D4311yescharacteristic
496C4⋊C4C2×C22⋊D4311yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—41no—
499C22⋊C4—861no—
500C22⋊C4—1721no—
501C22⋊C4—1721no—
502C42—21no—
503C42—21no—
504C42—41no—
505C42C23×D4311yescharacteristic
506C23—21no—
507C23—861no—
508Q8—21no—
509Q8—21no—
510Q8—41no—
511Q8—41no—
512Q8—82no—
513Q8—81no—
514Q8—161no—
515Q8—3442no—
516Q8—1721no—
517Q8—1721no—
518Q8—1721no—
519Q8—1721no—
520Q8C2×#1062188412yes—
521Q8#1306335412yescomplemented
522D4—21no—
523D4—861no—
524D4—861no—
525D4—3442no—
526D4C2×#1062188412yes—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—861no—
538C2×C4—3442no—
539C2×C4—3442no—
540C2×C4—1721no—
541C2×C4—1721no—
542C2×C4—1721no—
543C2×C4—1721no—
544C2×C4C2×D4×D4311yescharacteristic
545C2×C4C22×C22⋊D4311yescharacteristic
546C2×C4#1306337011yescharacteristic
547C8—21no—
548C8—41no—
549C8—861no—
550C8—861no—
551C8—1721no—
552C8—1721no—
553C8—1721no—
554C8—1721no—
555C8C2×#1062188412yes—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—1721no—
560C22—1721no—
561C22—1721no—
562C22D4×C22⋊D4311yescharacteristic
563C4—22no—
564C4—22no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—3441no—
572C4—3441no—
573C4—1721no—
574C4C22×#1062188411yescharacteristic
575C4C2×#1306335411yescharacteristic
576C4#1458370511yescharacteristic
577C4#1458370711yescharacteristic
578C2—42no—
579C2—3441no—
580C2#1914610511yescharacteristic
581C2D4×#1062188411yescharacteristic
582C2#1914636811yescharacteristic
subgroupG/H
centerC22D4×C22⋊D43
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingQ8×SD16×C43C2
socleC22×C43D42
prime radical#14583553C22
hypercenterQ8×SD16D43
hypocenterC43#19805
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×#10621884C2semidirect
4. #19134229C2semidirect
5. #19134091C2semidirect
6. #19134023C2semidirect
7. #19133972C2semidirect
8. #19133965C2semidirect
9. #19133858C2semidirect
10. #19133857C2semidirect
11. #19133816C2semidirect
13. #19133438C2semidirect
14. SD16⋊(C43⋊C8)C2semidirect
15. Q8×SD16×C43C2semidirect
168. SD16×C43SD16semidirect
224. Q8×C43#137semidirect
288. Q8×SD16D43semidirect
358. C43#19805semidirect
466. SD16#10621884semidirect
521. Q8#13063354semidirect

Maximal subgroups

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

subgroupindex
Q8×#106218842
#191464602
#191462422
#191342292
#191340912
#191340232
#191339722
#191339652
#191338582
#191338572
#191338162
#191337442
#191334382
SD16⋊(C43⋊C8)2
Q8×SD16×C432
#1980543