Finite Group Databasebeta

← #25074181

Subgroups — #25074181

Order 11008.

980 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145907C211yesmaximal, characteristic, complemented
2#19145155C211yesmaximal, characteristic, complemented
3#19144059C211yesmaximal, characteristic
4#19133858C212yesmaximal, complemented
5#19133802C212yesmaximal, complemented
6#19133698C212yesmaximal, complemented
7#19133667C212yesmaximal, complemented
8#19133285C211yesmaximal, characteristic, complemented
9#19133050C211yesmaximal, characteristic, complemented
10C2×D4⋊(C43⋊C8)C211yesmaximal, characteristic, complemented
11C43×#3501C211yesmaximal, characteristic, complemented
12C2×#13063351—21no—
13#14583449C2211yescharacteristic
14#14583295C2211yescharacteristic, complemented
15#14583273C2211yescharacteristic, complemented
16#14583249C2211yescharacteristic, complemented
17#14583247C2211yescharacteristic, complemented
18#14583186—21no—
19#14583184C2212yescomplemented
20#14583130—22no—
21#14583130—22no—
22#14583114—22no—
23#14583104C2212yes—
24#14583101—21no—
25#14583099C2212yescomplemented
26#14583094C2212yes—
27C4×#10621884—22no—
28#14582923C2211yescharacteristic
29C2×C4⋊(C43⋊C8)C2211yescharacteristic
30#14582898—21no—
31C2×C4×C43⋊C8C2211yescharacteristic
32#14582648C2212yescomplemented
33#14582645C2212yescomplemented
34Q8⋊(C43⋊C8)C2212yescomplemented
35Q8⋊(C43⋊C8)C2212yescomplemented
36D4⋊(C43⋊C8)C2212yes—
37D4⋊(C43⋊C8)C2212yes—
38C43×#554—21no—
39C43×#554—22no—
40C43×#554—22no—
41C43×#554C2212yes—
42D4×Q8×C43—21no—
43D4×Q8×C43C2212yes—
44C43×#544—21no—
45C43×#543—22no—
46C43×#533C2211yescharacteristic
47C43×#533C2211yescharacteristic
48C2×C22⋊Q8×C43—22no—
49C43×#518—21no—
50C2×C4×D4×C43C2211yescharacteristic
51C2×#10621898—21no—
52#13063351—44no—
53C2×#10621883—22no—
54C2×#10621884—22no—
55C2×#10621884—22no—
56C2×#10621882—22no—
57C2×#10621877—21no—
58C22×C43⋊C8—21no—
59C22×C43⋊C8—21no—
60#13063325C2311yescharacteristic
61#13063322C2311yescharacteristic
62#13063313—21no—
63#13063299—21no—
64#13063299—21no—
65C4×D172—21no—
66C4×D172—21no—
67#13063256—22no—
68#13063256—22no—
69C4⋊(C43⋊C8)—21no—
70C4⋊(C43⋊C8)—21no—
71C4⋊(C43⋊C8)C2312yes—
72C4⋊(C43⋊C8)C2312yes—
73#13063249—22no—
74#13063249—22no—
75#13063248—22no—
76#13063245—22no—
77#13063245—22no—
78#13063245—22no—
79C4×C43⋊C8—21no—
80C4×C43⋊C8C2311yescharacteristic
81C4×C43⋊C8C2311yescharacteristic
82C22×Q8×C43—22no—
83C22×D4×C43D411yescharacteristic
84C23×C4×C43—21no—
85C4⋊Q8×C43—22no—
86C4⋊Q8×C43—22no—
87C4⋊Q8×C43D411yescharacteristic
88C4⋊Q8×C43D411yescharacteristic
89C4⋊Q8×C43C2311yescharacteristic
90C4⋊Q8×C43C2311yescharacteristic
91#125×C43—22no—
92#125×C43—22no—
93#125×C43—21no—
94#125×C43—21no—
95C22⋊Q8×C43—24no—
96C22⋊Q8×C43—24no—
97C22⋊Q8×C43—24no—
98C22⋊Q8×C43—24no—
99C22⋊Q8×C43—22no—
100C22⋊Q8×C43—24no—
101C22⋊Q8×C43—21no—
102C22⋊Q8×C43—21no—
103C22⋊Q8×C43—22no—
104C22⋊Q8×C43—24no—
105C22⋊Q8×C43D412yes—
106C22⋊Q8×C43D412yes—
107C4×Q8×C43—21no—
108C4×Q8×C43—21no—
109C4×Q8×C43D412yes—
110C4×Q8×C43C2312yes—
111C4×D4×C43—22no—
112C4×D4×C43—22no—
113C4×D4×C43—22no—
114C4×D4×C43—22no—
115C4×D4×C43—21no—
116C4×D4×C43—21no—
117C4×D4×C43D412yes—
118C4×D4×C43C2312yes—
119#117×C43—21no—
120#117×C43—21no—
121C2×C4⋊C4×C43—24no—
122C2×C4⋊C4×C43—22no—
123C2×C4⋊C4×C43D411yescharacteristic
124C2×C22⋊C4×C43—24no—
125C2×C22⋊C4×C43—21no—
126C2×C42×C43C2311yescharacteristic
127C24×C43—21no—
128C2×Q8×C43—24no—
129C2×Q8×C43—21no—
130C2×Q8×C43—21no—
131C2×Q8×C43—82no—
132C2×Q8×C43—82no—
133C2×Q8×C43C2×D412yes—
134C2×Q8×C43C2×D412yes—
135C2×D4×C43—21no—
136C2×D4×C43—22no—
137C2×D4×C43SD1614yescomplemented
138C2×D4×C43C2×D412yes—
139C22×C4×C43—24no—
140C22×C4×C43—24no—
141C22×C4×C43—22no—
142C22×C4×C43—22no—
143C22×C4×C43—21no—
144C22×C4×C43—81no—
145C22×C4×C43—81no—
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2×D411yescharacteristic
148C22×C4×C43C2×D411yescharacteristic
149C2×C22⋊D43—41no—
150#10621898—42no—
151#10621898—42no—
152C2×D172—21no—
153C2×D172—21no—
154C2×D172—21no—
155C2×C4×D43—41no—
156C2×Dic86—21no—
157C4⋊C4×C43—24no—
158C4⋊C4×C43—24no—
159C4⋊C4×C43—24no—
160C4⋊C4×C43—22no—
161C4⋊C4×C43—22no—
162C4⋊C4×C43—21no—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—21no—
165C4⋊C4×C43—21no—
166C4⋊C4×C43—22no—
167C4⋊C4×C43—21no—
168C4⋊C4×C43—21no—
169C4⋊C4×C43—21no—
170C4⋊C4×C43—22no—
171C4⋊C4×C43C2×D412yes—
172C4⋊C4×C43C2×D412yes—
173C4⋊C4×C43C2×D412yes—
174C22⋊C4×C43—24no—
175C22⋊C4×C43—24no—
176C22⋊C4×C43—22no—
177C22⋊C4×C43—22no—
178C22⋊C4×C43—22no—
179C22⋊C4×C43—22no—
180C42×C43—22no—
181C42×C43—22no—
182C42×C43C2×D411yescharacteristic
183C42×C43C2×D411yescharacteristic
184C42×C43C2×D411yescharacteristic
185C42×C43C2411yescharacteristic
186#10621884—44no—
187#10621884—82no—
188#10621883—44no—
189#10621882—44no—
190(C2×D43)⋊C4—41no—
191(C2×D43)⋊C4—41no—
192C4⋊Dic43—21no—
193C4⋊Dic43—21no—
194#10621877—42no—
195C2×C43⋊C8—21no—
196C2×C43⋊C8—21no—
197C2×C43⋊C8—22no—
198C2×C43⋊C8—22no—
199C2×C43⋊C8—42no—
200C2×C43⋊C8—81no—
201C23×C43—22no—
202C23×C43—24no—
203C23×C43—81no—
204C23×C43—81no—
205C23×C43#12011yescharacteristic
206C22×D43—41no—
207Q8×C43—22no—
208Q8×C43—22no—
209Q8×C43—42no—
210Q8×C43—41no—
211Q8×C43—41no—
212Q8×C43—81no—
213Q8×C43—84no—
214D4×C43—22no—
215D4×C43—24no—
216D4×C43C2×SD1614yes—
217C2×C4×C43—21no—
218C2×C4×C43—21no—
219C2×C4×C43—22no—
220C2×C4×C43—22no—
221C2×C4×C43—22no—
222C2×C4×C43—22no—
223C2×C4×C43—22no—
224C2×C4×C43—41no—
225C2×C4×C43—42no—
226C2×C4×C43—42no—
227C2×C4×C43—82no—
228C2×C4×C43—81no—
229C2×C4×C43—84no—
230C2×C4×C43—84no—
231C2×C4×C43#12011yescharacteristic
232C2×C4×C43#12011yescharacteristic
233C2×C4×C43#12011yescharacteristic
234C2×C4×C43C2×SD1612yes—
235C2×C4×C43#13712yes—
236C2×C4×C43C22×D411yescharacteristic
237C2×C4×C43C22×D411yescharacteristic
238C2×C4×C43C22×D411yescharacteristic
239C22⋊D43—82no—
240C2×Dic43—41no—
241D172—42no—
242D172—41no—
243D172—41no—
244D172—41no—
245D172—41no—
246C4×D43—82no—
247Dic86—41no—
248Dic86—41no—
249C43⋊C8—22no—
250C43⋊C8—41no—
251C43⋊C8—42no—
252C43⋊C8—42no—
253#18664—431nomaximal, Sylow
254C22×C43—21no—
255C22×C43—21no—
256C22×C43—24no—
257C22×C43—42no—
258C22×C43—42no—
259C22×C43—84no—
260C22×C43#44812yes—
261C22×C43C2×#12011yescharacteristic
262C2×D43—81no—
263C2×D43—81no—
264C2×D43—81no—
265C4×C43—21no—
266C4×C43—21no—
267C4×C43—21no—
268C4×C43—42no—
269C4×C43—42no—
270C4×C43—42no—
271C4×C43—42no—
272C4×C43—42no—
273C4×C43#44812yes—
274C4×C43C22×SD1611yescharacteristic
275C4×C43C2×#13711yescharacteristic
276C4×C43#57611yescharacteristic
277C4×C43#57711yescharacteristic
278Dic43—81no—
279#3501D4311yescharacteristic, complemented
280#3246—431no—
281#3167—431no—
282#3036—431no—
283#1651—432no—
284#1643—432no—
285#1627—432no—
286#1622—432no—
287#1550—431no—
288#1493—431no—
289C2×D4⋊C8—431no—
290C2×C43—22no—
291C2×C43—44no—
292C2×C43C2×#44811yescharacteristic
293C2×C43#301211yescharacteristic
294C2×C43#301811yescharacteristic
295D43—161no—
296C2×AGL(1,ℤ/8)—861no—
297#554—21no—
298#554—22no—
299#554—22no—
300#554C2×D4312yes—
301D4×Q8—21no—
302D4×Q8C2×D4312yes—
303#544—21no—
304#543—22no—
305#533C2×D4311yescharacteristic
306#533C2×D4311yescharacteristic
307#532—431no—
308C2×C22⋊Q8—22no—
309#518—21no—
310C2×C4×D4C2×D4311yescharacteristic
311#498—431no—
312#492—431no—
313#487—431no—
314#486—431no—
315#476—861no—
316#475—432no—
317#465—862no—
318#465—862no—
319#463—862no—
320#461—432no—
321#461—861no—
322#460—432no—
323#459—432no—
324C4×SD16—862no—
325#423—431no—
326C2×C4⋊C8—431no—
327#417—861no—
328C2×C4×C8—431no—
329#333—432no—
330#332—432no—
331Q8⋊C8—432no—
332Q8⋊C8—432no—
333D4⋊C8—432no—
334D4⋊C8—432no—
335C43#1866411yescharacteristic, complemented, Sylow
336C2×#41—861no—
337C22×Q8—22no—
338C22×D4C22⋊D4311yescharacteristic
339C23×C4—21no—
340AGL(1,ℤ/8)—1724no—
341C2×SD16—862no—
342C2×SD16—862no—
343C2×SD16—862no—
344C2×D8—862no—
345C2×M16—861no—
346C22×C8—861no—
347C22×C8—861no—
348C4⋊Q8—22no—
349C4⋊Q8—22no—
350C4⋊Q8—431no—
351C4⋊Q8C22⋊D4311yescharacteristic
352C4⋊Q8C22⋊D4311yescharacteristic
353C4⋊Q8C22×D4311yescharacteristic
354C4⋊Q8C22×D4311yescharacteristic
355#127—431no—
356#125—22no—
357#125—22no—
358#125—21no—
359#125—21no—
360#124—861no—
361C22⋊Q8—24no—
362C22⋊Q8—24no—
363C22⋊Q8—24no—
364C22⋊Q8—24no—
365C22⋊Q8—22no—
366C22⋊Q8—24no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—22no—
370C22⋊Q8—24no—
371C22⋊Q8C22⋊D4312yes—
372C22⋊Q8C22⋊D4312yes—
373#121—861no—
374#121—861no—
375C4×Q8—21no—
376C4×Q8—21no—
377C4×Q8C22⋊D4312yes—
378C4×Q8C22×D4312yes—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—861no—
386C4×D4—861no—
387C4×D4C22⋊D4312yes—
388C4×D4C22×D4312yes—
389#117—21no—
390#117—21no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4C22⋊D4311yescharacteristic
394C2×C22⋊C4—24no—
395C2×C22⋊C4—21no—
396C2×C42C22×D4311yescharacteristic
397#106—862no—
398#106—862no—
399C4⋊C8—432no—
400C4⋊C8—432no—
401C4⋊C8—861no—
402C4⋊C8—861no—
403#103—862no—
404#103—862no—
405#103—862no—
406#102—862no—
407#102—862no—
408#102—862no—
409C4×C8—431no—
410C4×C8—431no—
411C4×C8—861no—
412C24—21no—
413#41—1722no—
414#41—1722no—
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8—861no—
421C2×Q8C2×C22⋊D4312yes—
422C2×Q8C2×C22⋊D4312yes—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—861no—
426C2×D4—861no—
427C2×D4—861no—
428C2×D4—1721no—
429C2×D4#1062188414yescomplemented
430C2×D4C2×C22⋊D4312yes—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—1721no—
439C22×C4C2×C22⋊D4311yescharacteristic
440C22×C4C2×C22⋊D4311yescharacteristic
441C22×C4C2×C22⋊D4311yescharacteristic
442SD16—3442no—
443SD16—1724no—
444SD16—1724no—
445D8—1724no—
446M16—1722no—
447C2×C8—861no—
448C2×C8—861no—
449C2×C8—862no—
450C2×C8—862no—
451C2×C8—3441no—
452C2×C8—1722no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—861no—
468C4⋊C4—861no—
469C4⋊C4C2×C22⋊D4312yes—
470C4⋊C4C2×C22⋊D4312yes—
471C4⋊C4C2×C22⋊D4312yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—1721no—
479C22⋊C4—1721no—
480C42—22no—
481C42—22no—
482C42C2×C22⋊D4311yescharacteristic
483C42C2×C22⋊D4311yescharacteristic
484C42C2×C22⋊D4311yescharacteristic
485C42C23×D4311yescharacteristic
486C23—22no—
487C23—24no—
488C23—81no—
489C23—81no—
490C23—1721no—
491C23#1306329411yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—42no—
495Q8—41no—
496Q8—41no—
497Q8—81no—
498Q8—84no—
499Q8—1721no—
500Q8—1721no—
501D4—22no—
502D4—24no—
503D4—3442no—
504D4—1722no—
505D4—1721no—
506D4—1721no—
507D4—1721no—
508D4—1721no—
509D4C2×#1062188414yes—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—3442no—
525C2×C4—1721no—
526C2×C4#1306329411yescharacteristic
527C2×C4#1306329411yescharacteristic
528C2×C4#1306329411yescharacteristic
529C2×C4C2×#1062188412yes—
530C2×C4#1306335612yes—
531C2×C4C22×C22⋊D4311yescharacteristic
532C2×C4C22×C22⋊D4311yescharacteristic
533C2×C4C22×C22⋊D4311yescharacteristic
534C8—862no—
535C8—1721no—
536C8—1722no—
537C8—1722no—
538C22—21no—
539C22—21no—
540C22—24no—
541C22—42no—
542C22—42no—
543C22—84no—
544C22—3441no—
545C22—3441no—
546C22—3441no—
547C22#1458303812yes—
548C22C2×#1306329411yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—3441no—
558C4#1458303812yes—
559C4C22×#1062188411yescharacteristic
560C4C2×#1306335611yescharacteristic
561C4#1458370211yescharacteristic
562C4#1458371011yescharacteristic
563C2—22no—
564C2—44no—
565C2—6881no—
566C2C2×#1458303811yescharacteristic
567C2#1914389911yescharacteristic
568C2#1914394411yescharacteristic
subgroupG/H
centerC22C2×#13063294
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3501C2
socleC22×C43C2×#120
prime radical#19144059C2
hypercenter#3501D43
hypocenterC43#18664
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #19145907C2semidirect
2. #19145155C2semidirect
4. #19133858C2semidirect
5. #19133802C2semidirect
6. #19133698C2semidirect
7. #19133667C2semidirect
8. #19133285C2semidirect
9. #19133050C2semidirect
10. C2×D4⋊(C43⋊C8)C2semidirect
11. C43×#3501C2semidirect
14. #14583295C22semidirect
15. #14583273C22semidirect
16. #14583249C22semidirect
17. #14583247C22semidirect
19. #14583184C22semidirect
25. #14583099C22semidirect
32. #14582648C22semidirect
33. #14582645C22semidirect
34. Q8⋊(C43⋊C8)C22semidirect
35. Q8⋊(C43⋊C8)C22semidirect
137. C2×D4×C43SD16semidirect
279. #3501D43semidirect
335. C43#18664semidirect
429. C2×D4#10621884semidirect

Maximal subgroups

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

subgroupindex
#191459072
#191451552
#191440592
#191338582
#191338022
#191336982
#191336672
#191332852
#191330502
C2×D4⋊(C43⋊C8)2
C43×#35012
#1866443