Finite Group Databasebeta

← #25086754

Subgroups — #25086754

Order 11008.

632 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#10621884C211yesmaximal, characteristic, complemented
2#19147266C211yesmaximal, characteristic
3#19146262C211yesmaximal, characteristic
4#19134218C211yesmaximal, characteristic, complemented
5#19134115C211yesmaximal, characteristic, complemented
6#19134013C211yesmaximal, characteristic, complemented
7#19133979C211yesmaximal, characteristic, complemented
8#19133943C211yesmaximal, characteristic, complemented
9#19133816C211yesmaximal, characteristic, complemented
10#19133758C211yesmaximal, characteristic, complemented
11#19133720C211yesmaximal, characteristic
12#19133673C211yesmaximal, characteristic, complemented
13#19133438C211yesmaximal, characteristic, complemented
14#19133414C211yesmaximal, characteristic, complemented
15Q8×Q16×C43C211yesmaximal, characteristic, complemented
16C2×#13063357—21no—
17#14583571C2211yescharacteristic
18Q8×D172C2211yescharacteristic
19#14583295—21no—
20#14583295C2211yescharacteristic
21#14583273C2211yescharacteristic
22#14583272C2211yescharacteristic
23#14583249C2211yescharacteristic
24#14583247—21no—
25#14583247C2211yescharacteristic
26#14583186—21no—
27#14583186—21no—
28#14583186C2211yescharacteristic
29#14583186C2211yescharacteristic
30#14583184—21no—
31#14583184C2211yescharacteristic
32#14583146—21no—
33#14583109C2211yescharacteristic
34#14583099—21no—
35#14583099C2211yescharacteristic
36#14583099C2211yescharacteristic
37#14583064—21no—
38#14583039—21no—
39Q8×C43⋊C8C2211yescharacteristic
40Q8×C43⋊C8C2211yescharacteristic
41C4×#10621884—21no—
42C4×#10621884—21no—
43C4×#10621884C2211yescharacteristic
44#14582978C2211yescharacteristic
45#14582650C2211yescharacteristic
46#14582649C2211yescharacteristic
47#14582648C2211yescharacteristic
48#14582648C2211yescharacteristic
49#14582646C2211yescharacteristic
50#14582642C2211yescharacteristic
51Q8⋊(C43⋊C8)C2211yescharacteristic
52Q8⋊(C43⋊C8)C2211yescharacteristic
53Q8⋊(C43⋊C8)C2211yescharacteristic
54#14582626C2211yescharacteristic
55#14582620C2211yescharacteristic
56Q82×C43C2211yescharacteristic
57Q82×C43C2211yescharacteristic
58C43×#500—21no—
59C43×#500C2211yescharacteristic
60C43×#477—21no—
61C43×#477—21no—
62C43×#477C2211yescharacteristic
63C43×#477C2211yescharacteristic
64C8×Q8×C43C2211yescharacteristic
65C4×Q16×C43—21no—
66C4×Q16×C43C2211yescharacteristic
67C2×#10621902—21no—
68C2×Q8×D43—21no—
69#13063357—44no—
70C2×#10621885—21no—
71C2×#10621873—21no—
72C2×#10621873—21no—
73C2×#10621884—21no—
74C2×#10621884—21no—
75C2×#10621884—41no—
76C2×#10621884D411yescharacteristic
77C2×#10621872—21no—
78#13063325—21no—
79#13063325C2311yescharacteristic
80#13063322—21no—
81#13063322C2311yescharacteristic
82#13063305—21no—
83#13063305—41no—
84#13063300—21no—
85#13063300—41no—
86C4×D172—21no—
87C4×D172—21no—
88C4×D172C2311yescharacteristic
89#13063256—21no—
90#13063256—21no—
91#13063256—21no—
92#13063256—21no—
93#13063256—21no—
94#13063256D411yescharacteristic
95C4⋊(C43⋊C8)—21no—
96C4⋊(C43⋊C8)—21no—
97C4⋊(C43⋊C8)C2311yescharacteristic
98C4⋊(C43⋊C8)C2311yescharacteristic
99C4⋊(C43⋊C8)C2311yescharacteristic
100#13063249—21no—
101#13063249—41no—
102#13063249D411yescharacteristic
103#13063248—21no—
104#13063245—21no—
105#13063245—21no—
106#13063245—21no—
107#13063245—21no—
108#13063245—21no—
109#13063245—21no—
110#13063245D411yescharacteristic
111(C2×D43)⋊C8—21no—
112C4×C43⋊C8—21no—
113C4×C43⋊C8—21no—
114C4×C43⋊C8C2311yescharacteristic
115C4×C43⋊C8C2311yescharacteristic
116C2×Q16×C43D411yescharacteristic
117C4⋊Q8×C43—21no—
118C4⋊Q8×C43—21no—
119C4⋊Q8×C43—21no—
120C4⋊Q8×C43—21no—
121C4⋊Q8×C43—41no—
122C4⋊Q8×C43—41no—
123C4⋊Q8×C43C2311yescharacteristic
124C4⋊Q8×C43C2311yescharacteristic
125C4×Q8×C43—21no—
126C4×Q8×C43—21no—
127C4×Q8×C43—21no—
128C4×Q8×C43—21no—
129C4×Q8×C43C2311yescharacteristic
130C4×Q8×C43C2311yescharacteristic
131C4×Q8×C43C2311yescharacteristic
132#107×C43—21no—
133#107×C43—21no—
134#107×C43—21no—
135#107×C43—21no—
136#107×C43D411yescharacteristic
137C4⋊C8×C43—21no—
138C4⋊C8×C43C2311yescharacteristic
139#103×C43—21no—
140#103×C43—21no—
141#103×C43D411yescharacteristic
142#103×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×C43C2×D411yescharacteristic
150C2×Q8×C43C2×D411yescharacteristic
151C2×Q8×C43C2×D411yescharacteristic
152#10621902—42no—
153#10621902—81no—
154Q8×D43—42no—
155Q8×D43—81no—
156C2×D172—21no—
157C2×D172—21no—
158C2×D172—21no—
159C2×D172—41no—
160C2×D172—41no—
161C2×D172C2×D411yescharacteristic
162C2×C4×D43—21no—
163C2×C4×D43—41no—
164C2×C4×D43—41no—
165C2×Dic86—21no—
166C2×Dic86—41no—
167Q16×C43SD1614yescomplemented
168C2×C8×C43—21no—
169C2×C8×C43C2×D411yescharacteristic
170C2×C8×C43C2×D411yescharacteristic
171C4⋊C4×C43—21no—
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×C43—41no—
182C4⋊C4×C43C2×D411yescharacteristic
183C4⋊C4×C43C2×D411yescharacteristic
184C4⋊C4×C43C2×D411yescharacteristic
185C42×C43—21no—
186C42×C43—21no—
187C42×C43—21no—
188C42×C43—41no—
189C42×C43—41no—
190C42×C43C2411yescharacteristic
191#10621885—42no—
192#10621884—22no—
193#10621884—42no—
194#10621884—42no—
195#10621884—82no—
196(C2×D43)⋊C4—21no—
197(C2×D43)⋊C4—41no—
198(C2×D43)⋊C4—41no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—41no—
202C4⋊Dic43C2×D411yescharacteristic
203C2×C43⋊C8—21no—
204C2×C43⋊C8—21no—
205C2×C43⋊C8—21no—
206C2×C43⋊C8—21no—
207C2×C43⋊C8C2×D411yescharacteristic
208C2×C43⋊C8C2×D411yescharacteristic
209#10621873—42no—
210#10621873—81no—
211#10621872—42no—
212C22×D43—21no—
213C22×D43—41no—
214Q8×C43—21no—
215Q8×C43—21no—
216Q8×C43—21no—
217Q8×C43—81no—
218Q8×C43—81no—
219Q8×C43—161no—
220Q8×C43—161no—
221Q8×C43C2×SD1612yes—
222Q8×C43C2×SD1612yes—
223Q8×C43#13712yescomplemented
224C2×C4×C43—21no—
225C2×C4×C43—21no—
226C2×C4×C43—21no—
227C2×C4×C43—21no—
228C2×C4×C43—21no—
229C2×C4×C43—21no—
230C2×C4×C43—41no—
231C2×C4×C43—41no—
232C2×C4×C43C22×D411yescharacteristic
233C2×C4×C43C22×D411yescharacteristic
234C2×C4×C43H(5,𝔽2)11yescharacteristic
235C2×Dic43—21no—
236C2×Dic43—41no—
237D172—21no—
238D172—21no—
239D172—41no—
240D172—41no—
241D172—41no—
242D172—41no—
243D172—82no—
244D172—81no—
245D172—161no—
246C4×D43—41no—
247C4×D43—41no—
248C4×D43—82no—
249C4×D43—82no—
250Dic86—41no—
251Dic86—41no—
252Dic86—82no—
253C8×C43—21no—
254C8×C43—41no—
255C8×C43C2×SD1612yes—
256C43⋊C8—21no—
257C43⋊C8—21no—
258C43⋊C8—41no—
259C43⋊C8—41no—
260C43⋊C8—41no—
261C43⋊C8—41no—
262#19809—431nomaximal, Sylow
263C22×C43D4211yescharacteristic
264C2×D43—41no—
265C2×D43—41no—
266C2×D43—41no—
267C2×D43—81no—
268C2×D43—81no—
269C2×D43—81no—
270C4×C43—22no—
271C4×C43—22no—
272C4×C43—22no—
273C4×C43—21no—
274C4×C43—41no—
275C4×C43—41no—
276C4×C43—41no—
277C4×C43—81no—
278C4×C43—81no—
279C4×C43C22×SD1611yescharacteristic
280C4×C43C2×#13711yescharacteristic
281C4×C43#57711yescharacteristic
282C4×C43#57811yescharacteristic
283Dic43—41no—
284Dic43—81no—
285Q8×Q16D4311yescharacteristic, complemented
286Q8×SD16—431no—
287#3362—431no—
288#3278—431no—
289#1714—431no—
290#1695—431no—
291#1678—431no—
292#1672—431no—
293#1666—431no—
294#1645—431no—
295#1637—431no—
296#1631—431no—
297#1623—431no—
298#1583—431no—
299#1579—431no—
300C2×C43#327011yescharacteristic
301C2×C43D4×SD1611yescharacteristic
302C2×C43#328811yescharacteristic
303D43—81no—
304D43—161no—
305C2×#137—861no—
306Q82C2×D4311yescharacteristic
307Q82C2×D4311yescharacteristic
308#550—431no—
309D4×Q8—431no—
310#500—21no—
311#500C2×D4311yescharacteristic
312#498—431no—
313#498—861no—
314#492—431no—
315#492—431no—
316#487—431no—
317#486—431no—
318#486—861no—
319#477—21no—
320#477—21no—
321#477C2×D4311yescharacteristic
322#477C2×D4311yescharacteristic
323#476—431no—
324#476—431no—
325#476—861no—
326#476—861no—
327#475—431no—
328#475—861no—
329#468—861no—
330#462—431no—
331#460—431no—
332#460—431no—
333#460—861no—
334#452—861no—
335#448—861no—
336C8×Q8—431no—
337C8×Q8—431no—
338C8×Q8C2×D4311yescharacteristic
339C4×Q16—21no—
340C4×Q16C2×D4311yescharacteristic
341C4×SD16—431no—
342C4×SD16—861no—
343C4×SD16—861no—
344#436—431no—
345#334—431no—
346#334—431no—
347#333—431no—
348#333—431no—
349#332—431no—
350#331—431no—
351Q8⋊C8—431no—
352Q8⋊C8—431no—
353Q8⋊C8—431no—
354D4⋊C8—431no—
355#322—431no—
356C43#1980911yescharacteristic, complemented, Sylow
357C2×#41—861no—
358C22×Q8—861no—
359#137—1724no—
360C2×Q16—861no—
361C2×Q16C22⋊D4311yescharacteristic
362C2×SD16—431no—
363C2×SD16—861no—
364C2×SD16—861no—
365C2×SD16—861no—
366C2×SD16—861no—
367C2×SD16—1721no—
368C2×M16—861no—
369C4⋊Q8—21no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—41no—
374C4⋊Q8—41no—
375C4⋊Q8—431no—
376C4⋊Q8—861no—
377C4⋊Q8C22×D4311yescharacteristic
378C4⋊Q8C22×D4311yescharacteristic
379#127—431no—
380#127—861no—
381C22⋊Q8—861no—
382C22⋊Q8—1721no—
383#121—861no—
384#121—1721no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8C22×D4311yescharacteristic
390C4×Q8C22×D4311yescharacteristic
391C4×Q8C22×D4311yescharacteristic
392C4×D4—431no—
393C4×D4—861no—
394C4×D4—861no—
395#107—21no—
396#107—21no—
397#107—21no—
398#107—21no—
399#107C22⋊D4311yescharacteristic
400#106—431no—
401#106—861no—
402#106—861no—
403#106—861no—
404#106—861no—
405#106—861no—
406C4⋊C8—21no—
407C4⋊C8—431no—
408C4⋊C8—431no—
409C4⋊C8—431no—
410C4⋊C8—861no—
411C4⋊C8—861no—
412C4⋊C8C22×D4311yescharacteristic
413#103—21no—
414#103—21no—
415#103—431no—
416#103—861no—
417#103—861no—
418#103—1721no—
419#103C22⋊D4311yescharacteristic
420#103C22⋊D4311yescharacteristic
421#102—431no—
422#102—861no—
423#102—861no—
424#102—861no—
425#102—861no—
426#102—861no—
427#102—861no—
428C22⋊C8—861no—
429C4×C8—21no—
430C4×C8—431no—
431C4×C8—431no—
432C4×C8—861no—
433C4×C8—861no—
434C4×C8C22×D4311yescharacteristic
435#41—3441no—
436#41—1722no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—41no—
440C2×Q8—41no—
441C2×Q8—861no—
442C2×Q8—3441no—
443C2×Q8—1721no—
444C2×Q8—1722no—
445C2×Q8D4×D4311yescharacteristic
446C2×Q8C2×C22⋊D4311yescharacteristic
447C2×Q8C2×C22⋊D4311yescharacteristic
448C2×D4—431no—
449C2×D4—861no—
450C2×D4—861no—
451C2×D4—861no—
452C2×D4—1721no—
453C2×D4—1721no—
454C22×C4—861no—
455C22×C4—1721no—
456C22×C4—1721no—
457Q16—1722no—
458Q16#1062188414yescomplemented
459SD16—862no—
460SD16—3441no—
461SD16—3442no—
462SD16—1722no—
463SD16—1722no—
464SD16—1722no—
465M16—1722no—
466C2×C8—21no—
467C2×C8—431no—
468C2×C8—431no—
469C2×C8—861no—
470C2×C8—861no—
471C2×C8—861no—
472C2×C8—861no—
473C2×C8C2×C22⋊D4311yescharacteristic
474C2×C8C2×C22⋊D4311yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—431no—
487C4⋊C4—861no—
488C4⋊C4—861no—
489C4⋊C4—1721no—
490C4⋊C4D4×D4311yescharacteristic
491C4⋊C4C2×C22⋊D4311yescharacteristic
492C4⋊C4C2×C22⋊D4311yescharacteristic
493C22⋊C4—861no—
494C22⋊C4—1721no—
495C22⋊C4—1721no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—41no—
500C42—41no—
501C42C23×D4311yescharacteristic
502C23—861no—
503C23—1721no—
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8—81no—
508Q8—81no—
509Q8—161no—
510Q8—161no—
511Q8—3442no—
512Q8—1721no—
513Q8—1721no—
514Q8C2×#1062188412yes—
515Q8C2×#1062188412yes—
516Q8#1306335712yescomplemented
517D4—861no—
518D4—861no—
519D4—3441no—
520D4—3442no—
521D4—1721no—
522D4—1721no—
523D4—1721no—
524D4—1721no—
525D4—6881no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—861no—
535C2×C4—3442no—
536C2×C4—3442no—
537C2×C4—1721no—
538C2×C4—1721no—
539C2×C4—1721no—
540C2×C4C2×D4×D4311yescharacteristic
541C2×C4C22×C22⋊D4311yescharacteristic
542C2×C4#1306337011yescharacteristic
543C8—21no—
544C8—41no—
545C8—861no—
546C8—861no—
547C8—1721no—
548C8—1721no—
549C8—1721no—
550C8—1721no—
551C8C2×#1062188412yes—
552C22—3441no—
553C22—3441no—
554C22—3441no—
555C22—1721no—
556C22—1721no—
557C22—1721no—
558C22D4×C22⋊D4311yescharacteristic
559C4—22no—
560C4—22no—
561C4—21no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—3441no—
569C4—1721no—
570C4C22×#1062188411yescharacteristic
571C4C2×#1306335711yescharacteristic
572C4#1458370711yescharacteristic
573C4#1458371211yescharacteristic
574C2—3441no—
575C2—6881no—
576C2#1914616511yescharacteristic
577C2D4×#1062188411yescharacteristic
578C2#1914635911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D43
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingQ8×Q16×C43C2
socleC22×C43D42
prime radical#14583571C22
hypercenterQ8×Q16D43
hypocenterC43#19809
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C42 ◁ Q8×Q16

Complements

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

NH ≅ G/Ntype
1. Q8×#10621884C2semidirect
4. #19134218C2semidirect
5. #19134115C2semidirect
6. #19134013C2semidirect
7. #19133979C2semidirect
8. #19133943C2semidirect
9. #19133816C2semidirect
10. #19133758C2semidirect
12. #19133673C2semidirect
13. #19133438C2semidirect
14. #19133414C2semidirect
15. Q8×Q16×C43C2semidirect
167. Q16×C43SD16semidirect
223. Q8×C43#137semidirect
285. Q8×Q16D43semidirect
356. C43#19809semidirect
458. Q16#10621884semidirect
516. Q8#13063357semidirect

Maximal subgroups

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

subgroupindex
Q8×#106218842
#191472662
#191462622
#191342182
#191341152
#191340132
#191339792
#191339432
#191338162
#191337582
#191337202
#191336732
#191334382
#191334142
Q8×Q16×C432
#1980943