Finite Group Databasebeta

← #25102524

Subgroups — #25102524

Order 11008.

1422 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13063370C211yesmaximal, characteristic
2C2×#14583076C211yesmaximal, characteristic
3C4×#13063356C212yesmaximal, complemented
4#19134984C211yesmaximal, characteristic, complemented
5#19134967C212yesmaximal, complemented
6#19134820C212yesmaximal, complemented
7#19134804C211yesmaximal, characteristic, complemented
8#19134548C211yesmaximal, characteristic
9#19134406C212yesmaximal, complemented
10C4×#13063242C211yesmaximal, characteristic, complemented
11C43×#2326C211yesmaximal, characteristic, complemented
12C2×#13063370C2211yescharacteristic
13C2×#13063356C2212yes—
14#14583380C2212yes—
15#14583378—22no—
16#14583374C2211yescharacteristic
17#14583372—22no—
18#14583371—22no—
19C4×#10621898C2212yes—
20C4×#10621900—22no—
21C2×C4×C22⋊D43—22no—
22C4×D4×D43—22no—
23#14583076C418yescomplemented
24#14583004—22no—
25#14582997—22no—
26C4×#10621885—22no—
27C4×#10621884—22no—
28#14582939C2212yescomplemented
29C2×#13063253C2212yes—
30C2×#13063253C2212yes—
31#14582908C2212yescomplemented
32#14582898C2212yes—
33C2×#13063242C2211yescharacteristic
34C2×#13063242C2211yescharacteristic
35C4×#10621877C2212yescomplemented
36#14582668C2212yescomplemented
37#14582659C2212yescomplemented
38#14582659C2212yescomplemented
39C2×#124×C43C2211yescharacteristic
40C2×C4×Q8×C43C2211yescharacteristic, complemented
41C2×C4×D4×C43C2211yescharacteristic
42C43×#391C2211yescharacteristic, complemented
43C43×#390C2211yescharacteristic
44C43×#388—22no—
45C43×#385—22no—
46C43×#384C2211yescharacteristic, complemented
47C43×#376C2211yescharacteristic, complemented
48#13063370—22no—
49#13063370—24no—
50#13063370C2×C414yes—
51C2×#10621898C2312yes—
52C2×#10621900—22no—
53C22×C22⋊D43—22no—
54C2×D4×D43—22no—
55C22×C4×D43—22no—
56#13063356—24no—
57#13063356C2×C418yescomplemented
58C2×#10621885—22no—
59C2×#10621884—22no—
60C2×#10621877C2312yes—
61C2×#10621877C2312yes—
62C4×Dic86D412yes—
63C4×Dic86D412yes—
64#13063290—22no—
65C4×D172D412yes—
66C4×D172D412yes—
67D4×Dic43—22no—
68D4×Dic43—22no—
69#13063287—22no—
70C4×C22⋊D43—24no—
71C4×C22⋊D43—24no—
72C4×C22⋊D43—22no—
73C4×C22⋊D43—24no—
74C4×C22⋊D43—22no—
75(C22⋊D43)⋊C4—24no—
76(C22⋊D43)⋊C4—24no—
77#13063284C2312yes—
78#13063283—22no—
79#13063282—22no—
80#13063281—24no—
81C2×Dic43⋊C4—22no—
82C4⋊C4×D43—22no—
83C2×C22⋊Dic43—22no—
84C2×(C2×D43)⋊C4—22no—
85C22⋊C4×D43—24no—
86C2×C4×Dic43—22no—
87C42×D43—22no—
88#13063258—22no—
89#13063256—22no—
90#13063253—24no—
91#13063253C2×C418yescomplemented
92#13063249—22no—
93#13063249—22no—
94#13063248—22no—
95#13063245—22no—
96#13063242—22no—
97#13063242C2×C414yescomplemented
98#13063236—22no—
99C4×C43⋊C8—22no—
100C22×Q8×C43C2311yescharacteristic
101C22×D4×C43C2311yescharacteristic
102C23×C4×C43D412yes—
103#124×C43—22no—
104#124×C43C2×C414yescomplemented
105C4×Q8×C43—22no—
106C4×Q8×C43—22no—
107C4×D4×C43—22no—
108C4×D4×C43—22no—
109C2×C4⋊C4×C43—21no—
110C2×C4⋊C4×C43—22no—
111C2×C4⋊C4×C43C2311yescharacteristic
112C2×C4⋊C4×C43C2311yescharacteristic
113C2×C22⋊C4×C43—22no—
114C2×C22⋊C4×C43—22no—
115C2×C22⋊C4×C43D412yes—
116C2×C42×C43—21no—
117C2×C42×C43C2311yescharacteristic
118C2×C42×C43C2311yescharacteristic
119C2×C42×C43C2311yescharacteristic
120#95×C43—21no—
121#95×C43—21no—
122#95×C43—22no—
123C24×C43C2×D412yes—
124C23×D43—22no—
125C2×Q8×C43—21no—
126C2×Q8×C43—22no—
127C2×Q8×C43—22no—
128C2×Q8×C43C22×C412yescomplemented
129C2×D4×C43—22no—
130C2×D4×C43—22no—
131C2×D4×C43C22×C412yes—
132C2×D4×C43#4112yes—
133C22×C4×C43—22no—
134C22×C4×C43—22no—
135C22×C4×C43—21no—
136C22×C4×C43—21no—
137C22×C4×C43—21no—
138C22×C4×C43—21no—
139C22×C4×C43—82no—
140C22×C4×C43C2×D412yes—
141C22×C4×C43C2×D411yescharacteristic
142C22×C4×C43C2×D411yescharacteristic
143C22×C4×C43C2411yescharacteristic
144C2×C22⋊D43—24no—
145C2×C22⋊D43—28no—
146C2×C22⋊D43—28no—
147C2×C22⋊D43—22no—
148C22×Dic43—22no—
149C22×Dic43—22no—
150C22×Dic43—22no—
151#10621900—28no—
152#10621900—28no—
153D4×D43—28no—
154D4×D43—28no—
155#10621898—22no—
156#10621898—22no—
157#10621898C22×C414yes—
158#10621898#4114yes—
159C2×D172C2×D412yes—
160C2×C4×D43—22no—
161C2×C4×D43—22no—
162C2×C4×D43—24no—
163C2×C4×D43—22no—
164C2×C4×D43—84no—
165C2×Dic86C2×D412yes—
166C4⋊C4×C43—22no—
167C4⋊C4×C43—22no—
168C4⋊C4×C43—41no—
169C4⋊C4×C43—41no—
170C4⋊C4×C43—82no—
171C22⋊C4×C43—24no—
172C22⋊C4×C43—28no—
173C22⋊C4×C43—82no—
174C42×C43—21no—
175C42×C43—22no—
176C42×C43—42no—
177C42×C43C22×C412yescomplemented
178C42×C43C2×D412yes—
179C42×C43C2×D412yes—
180C22⋊Dic43—24no—
181C22⋊Dic43—22no—
182#10621885—28no—
183#10621884—28no—
184(C2×D43)⋊C4—24no—
185(C2×D43)⋊C4—22no—
186(C2×D43)⋊C4—22no—
187C4⋊Dic43C2×D412yes—
188C4⋊Dic43C2×D412yes—
189Dic43⋊C4—24no—
190Dic43⋊C4—22no—
191Dic43⋊C4—22no—
192C4×Dic43—24no—
193C4×Dic43—22no—
194C4×Dic43—22no—
195C4×Dic43—22no—
196#10621877—22no—
197#10621877—22no—
198#10621877—22no—
199#10621877C22×C414yescomplemented
200C2×C43⋊C8—22no—
201C2×C43⋊C8—22no—
202C23×C43—22no—
203C23×C43—22no—
204C23×C43—42no—
205C23×C43—42no—
206C23×C43C4×D414yes—
207C23×C43C22×D411yescharacteristic
208C22×D43—22no—
209C22×D43—28no—
210C22×D43—82no—
211C22×D43—82no—
212Q8×C43—24no—
213Q8×C43—42no—
214D4×C43—24no—
215D4×C43—24no—
216C2×C4×C43—22no—
217C2×C4×C43—22no—
218C2×C4×C43—22no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43—44no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43—41no—
225C2×C4×C43—41no—
226C2×C4×C43—42no—
227C2×C4×C43—41no—
228C2×C4×C43—42no—
229C2×C4×C43—41no—
230C2×C4×C43—41no—
231C2×C4×C43—41no—
232C2×C4×C43—82no—
233C2×C4×C43#12011yescharacteristic
234C2×C4×C43#12012yes—
235C2×C4×C43#12011yescharacteristic
236C2×C4×C43C23×C411yescharacteristic
237C2×C4×C43C22×D412yes—
238C2×C4×C43C2×#4112yes—
239C2×C4×C43C2×#4111yescharacteristic
240C22⋊D43—216no—
241C22⋊D43—24no—
242C22⋊D43—24no—
243C2×Dic43—28no—
244C2×Dic43—22no—
245C2×Dic43—22no—
246C2×Dic43—22no—
247C2×Dic43—82no—
248C2×Dic43—82no—
249D172—22no—
250D172C4×D414yes—
251C4×D43—24no—
252C4×D43—24no—
253C4×D43—44no—
254C4×D43—82no—
255C4×D43—84no—
256Dic86—22no—
257Dic86C4×D414yes—
258C43⋊C8—24no—
259C43⋊C8—42no—
260#21474—431nomaximal, Sylow
261C22×C43—21no—
262C22×C43—21no—
263C22×C43—24no—
264C22×C43—44no—
265C22×C43—42no—
266C22×C43—42no—
267C22×C43—82no—
268C22×C43C2×C4×D411yescharacteristic
269C22×C43C2×#12011yescharacteristic
270C22×C43#52511yescharacteristic
271C2×D43—24no—
272C2×D43—42no—
273C2×D43—42no—
274C2×D43—88no—
275C4×C43—21no—
276C4×C43—21no—
277C4×C43—22no—
278C4×C43—42no—
279C4×C43—42no—
280C4×C43—42no—
281C4×C43—81no—
282C4×C43—81no—
283C4×C43C2×C4×D412yes—
284C4×C43#52512yes—
285Dic43—24no—
286Dic43—42no—
287Dic43—42no—
288C4×H(5,𝔽2)—431no—
289C2×#455—431no—
290C4×#137—432no—
291#2326D4311yescharacteristic, complemented
292#1885—431no—
293#1882—432no—
294#1859—432no—
295#1856—431no—
296#1792—431no—
297#1759—432no—
298C4×#100—431no—
299C2×C43—21no—
300C2×C43—21no—
301C2×C43—44no—
302C2×C43C4×#12011yescharacteristic
303C2×C43#301911yescharacteristic
304C2×C43#302211yescharacteristic
305D43—44no—
306C2×H(5,𝔽2)—431no—
307C2×#137—432no—
308C2×#124C2×D4311yescharacteristic
309#520—432no—
310#520—862no—
311#518—431no—
312#518—862no—
313#518—862no—
314C4×#41—432no—
315C4×#41—862no—
316C2×C4×Q8C2×D4311yescharacteristic, complemented
317C2×C4×D4—862no—
318C2×C4×D4—862no—
319C2×C4×D4C2×D4311yescharacteristic
320#455—438no—
321#441—862no—
322#440—862no—
323C4×Q16—862no—
324C4×SD16—862no—
325#428—432no—
326C2×#104—432no—
327C2×#104—432no—
328#419—432no—
329#417—432no—
330C2×#100—431no—
331C2×#100—431no—
332C4×M16—432no—
333#391C2×D4311yescharacteristic, complemented
334#390C2×D4311yescharacteristic
335#388—22no—
336#385—22no—
337#384C2×D4311yescharacteristic, complemented
338#376C2×D4311yescharacteristic, complemented
339#343—432no—
340#339—432no—
341#339—432no—
342C43#2147411yescharacteristic, complemented, Sylow
343H(5,𝔽2)—434no—
344H(5,𝔽2)—864no—
345H(5,𝔽2)—862no—
346C2×#41—432no—
347C2×#41—862no—
348C22×Q8C22×D4311yescharacteristic
349C22×D4—862no—
350C22×D4—862no—
351C22×D4C22×D4311yescharacteristic
352C23×C4—862no—
353C23×C4C22⋊D4312yes—
354#137—438no—
355#137—864no—
356C2×Q16—862no—
357C2×SD16—862no—
358C2×M16—432no—
359C2×M16—432no—
360#124—22no—
361#124C4×D4314yescomplemented
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—432no—
365C4×Q8—432no—
366C4×Q8—862no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—432no—
370C4×D4—432no—
371C4×D4—864no—
372C4×D4—864no—
373C4×D4—862no—
374C4×D4—862no—
375C4×D4—862no—
376C4×D4—864no—
377C4×D4—864no—
378C4×D4—862no—
379C4×D4—864no—
380C4×D4—862no—
381#117—432no—
382#117—864no—
383#117—862no—
384#117—862no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—862no—
388C2×C4⋊C4—862no—
389C2×C4⋊C4C22×D4311yescharacteristic
390C2×C4⋊C4C22×D4311yescharacteristic
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—864no—
394C2×C22⋊C4—862no—
395C2×C22⋊C4—862no—
396C2×C22⋊C4C22⋊D4312yes—
397C2×C42—21no—
398C2×C42—862no—
399C2×C42—862no—
400C2×C42C22×D4311yescharacteristic
401C2×C42C22×D4311yescharacteristic
402C2×C42C22×D4311yescharacteristic
403#107—862no—
404#106—862no—
405#104—438no—
406#104—864no—
407#103—862no—
408#103—862no—
409#103—862no—
410#102—862no—
411#100—434no—
412#100—862no—
413#97—862no—
414C4×C8—862no—
415#95—21no—
416#95—21no—
417#95—22no—
418C24—862no—
419C24C2×C22⋊D4312yes—
420#41—434no—
421#41—434no—
422#41—868no—
423#41—868no—
424#41—862no—
425#41—862no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—432no—
430C2×Q8C2×C4×D4312yescomplemented
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—432no—
434C2×D4—868no—
435C2×D4—868no—
436C2×D4—864no—
437C2×D4—868no—
438C2×D4—868no—
439C2×D4—862no—
440C2×D4C2×C4×D4312yes—
441C2×D4#1062189812yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—82no—
449C22×C4—864no—
450C22×C4—862no—
451C22×C4—862no—
452C22×C4—862no—
453C22×C4—862no—
454C22×C4—862no—
455C22×C4—862no—
456C22×C4—3444no—
457C22×C4C2×C22⋊D4312yes—
458C22×C4C2×C22⋊D4311yescharacteristic
459C22×C4C2×C22⋊D4311yescharacteristic
460C22×C4C23×D4311yescharacteristic
461Q16—868no—
462SD16—868no—
463M16—434no—
464M16—862no—
465M16—862no—
466M16—862no—
467C2×C8—862no—
468C2×C8—862no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—82no—
474C4⋊C4—432no—
475C4⋊C4—432no—
476C4⋊C4—864no—
477C4⋊C4—862no—
478C4⋊C4—862no—
479C22⋊C4—28no—
480C22⋊C4—24no—
481C22⋊C4—82no—
482C22⋊C4—864no—
483C22⋊C4—864no—
484C22⋊C4—862no—
485C22⋊C4—862no—
486C22⋊C4—862no—
487C42—21no—
488C42—22no—
489C42—42no—
490C42—864no—
491C42—862no—
492C42—862no—
493C42—862no—
494C42C2×C4×D4312yescomplemented
495C42D4×D4312yes—
496C42D4×D4312yes—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—868no—
502C23—862no—
503C23—3442no—
504C23—3442no—
505C23C4×C22⋊D4314yes—
506C23C22×C22⋊D4311yescharacteristic
507Q8—24no—
508Q8—42no—
509Q8—434no—
510Q8—862no—
511D4—24no—
512D4—24no—
513D4—434no—
514D4—8616no—
515D4—864no—
516D4—864no—
517D4—862no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—82no—
535C2×C4—868no—
536C2×C4—862no—
537C2×C4—864no—
538C2×C4—864no—
539C2×C4—862no—
540C2×C4—862no—
541C2×C4—3442no—
542C2×C4—3444no—
543C2×C4—3442no—
544C2×C4—3442no—
545C2×C4—1724no—
546C2×C4#1306329312yes—
547C2×C4#1306329311yescharacteristic
548C2×C4#1306329311yescharacteristic
549C2×C4C22×C4×D4311yescharacteristic
550C2×C4C2×D4×D4312yes—
551C2×C4#41×D4312yes—
552C2×C4C2×#1062189811yescharacteristic
553C8—864no—
554C8—1722no—
555C22—21no—
556C22—21no—
557C22—24no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22—864no—
563C22—3448no—
564C22—1722no—
565C22—1722no—
566C22C2×C4×C22⋊D4311yescharacteristic
567C22C2×#1306329311yescharacteristic
568C22#1458340611yescharacteristic
569C4—21no—
570C4—21no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—864no—
578C4—1722no—
579C4—1722no—
580C4C4×D4×D4312yes—
581C4#1458340812yes—
582C2—21no—
583C2—21no—
584C2—44no—
585C2—1724no—
586C2C4×#1306329311yescharacteristic
587C2#1914395111yescharacteristic
588C2#1914397311yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC43×#2326C2
socleC22×C43C2×#120
prime radicalC2×#13063370C22
cosocleC22×C4×C43C24
hypercenter#2326D43
hypocenterC43#21474
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2326

Complements

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

NH ≅ G/Ntype
3. C4×#13063356C2semidirect
4. #19134984C2semidirect
5. #19134967C2semidirect
6. #19134820C2semidirect
7. #19134804C2semidirect
9. #19134406C2semidirect
10. C4×#13063242C2semidirect
11. C43×#2326C2semidirect
23. #14583076C4semidirect
28. #14582939C22semidirect
31. #14582908C22semidirect
35. C4×#10621877C22semidirect
36. #14582668C22semidirect
37. #14582659C22semidirect
38. #14582659C22semidirect
40. C2×C4×Q8×C43C22semidirect
42. C43×#391C22semidirect
46. C43×#384C22semidirect
47. C43×#376C22semidirect
57. #13063356C2×C4semidirect
91. #13063253C2×C4semidirect
97. #13063242C2×C4semidirect
104. #124×C43C2×C4semidirect
128. C2×Q8×C43C22×C4semidirect
177. C42×C43C22×C4semidirect
199. #10621877C22×C4semidirect
291. #2326D43semidirect
316. C2×C4×Q8C2×D43semidirect
333. #391C2×D43semidirect
337. #384C2×D43semidirect
338. #376C2×D43semidirect
342. C43#21474semidirect
361. #124C4×D43semidirect
430. C2×Q8C2×C4×D43semidirect
494. C42C2×C4×D43semidirect

Maximal subgroups

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

subgroupindex
C4×#130633702
C2×#145830762
C4×#130633562
#191349842
#191349672
#191348202
#191348042
#191345482
#191344062
C4×#130632422
C43×#23262
#2147443