Finite Group Databasebeta

← #25102530

Subgroups — #25102530

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×#14583079C211yesmaximal, characteristic
3C4×#13063358C212yesmaximal, complemented
4#19137307C211yesmaximal, characteristic, complemented
5#19134981C211yesmaximal, characteristic, complemented
6#19134963C212yesmaximal, complemented
7#19134824C212yesmaximal, complemented
8#19134807C211yesmaximal, characteristic, complemented
9#19134551C211yesmaximal, characteristic
10#19134408C212yesmaximal, complemented
11C4×#100×C43C211yesmaximal, characteristic, complemented
12C2×#13063370C2211yescharacteristic
13C2×#13063358C2212yes—
14C2×#13063312C2211yescharacteristic
15#14583380C2212yes—
16#14583378—22no—
17#14583374C2211yescharacteristic
18#14583372—22no—
19#14583371—22no—
20C4×#10621898C2212yes—
21C4×#10621900—22no—
22C2×C4×Dic86C2211yescharacteristic, complemented
23C2×C4×C22⋊D43—22no—
24C4×D4×D43—22no—
25#14583079C418yescomplemented
26#14583007—22no—
27#14583001—22no—
28C4×Dic172—22no—
29C4×#10621873—22no—
30#14582941C2212yescomplemented
31C2×#13063251C2212yes—
32C2×#13063251C2212yes—
33#14582909C2212yescomplemented
34#14582901C2212yes—
35#14582804C2211yescharacteristic, complemented
36#14582802C2211yescharacteristic
37#14582787—22no—
38#14582772—22no—
39#14582768C2211yescharacteristic, complemented
40#14582745C2211yescharacteristic, complemented
41#14582660C2212yescomplemented
42#14582660C2212yescomplemented
43C2×C4×D4×C43C2211yescharacteristic
44C2×#100×C43C2211yescharacteristic
45C2×#100×C43C2211yescharacteristic
46C4×M16×C43C2212yescomplemented
47C43×#343C2212yescomplemented
48#13063370—22no—
49#13063370—24no—
50#13063370C2×C414yes—
51C2×#10621898C2312yes—
52C2×#10621900—22no—
53C22×Dic86C2311yescharacteristic
54C22×C22⋊D43—22no—
55C2×D4×D43—22no—
56C22×C4×D43—22no—
57#13063358—24no—
58#13063358C2×C418yescomplemented
59C2×Dic172—22no—
60C2×#10621873—22no—
61#13063312—22no—
62#13063312C2×C414yescomplemented
63C4×Dic86—22no—
64C4×Dic86—22no—
65C4×Dic86D412yes—
66C4×Dic86D412yes—
67#13063290—22no—
68C4×D172D412yes—
69C4×D172D412yes—
70D4×Dic43—22no—
71D4×Dic43—22no—
72#13063287—22no—
73C4×C22⋊D43—24no—
74C4×C22⋊D43—24no—
75C4×C22⋊D43—22no—
76C4×C22⋊D43—24no—
77C4×C22⋊D43—22no—
78(C22⋊D43)⋊C4—24no—
79(C22⋊D43)⋊C4—24no—
80#13063284C2312yes—
81#13063283—22no—
82#13063282—22no—
83#13063281—24no—
84C2×C4⋊Dic43C2311yescharacteristic
85C2×Dic43⋊C4—22no—
86C2×Dic43⋊C4—21no—
87C2×Dic43⋊C4—22no—
88C4⋊C4×D43—22no—
89C2×C22⋊Dic43—22no—
90C2×C22⋊Dic43—22no—
91C2×C22⋊Dic43—22no—
92C2×(C2×D43)⋊C4—22no—
93C22⋊C4×D43—24no—
94C2×C4×Dic43—22no—
95C2×C4×Dic43—21no—
96C2×C4×Dic43C2311yescharacteristic
97C2×C4×Dic43C2311yescharacteristic
98C42×D43—22no—
99#13063259—22no—
100#13063257—22no—
101#13063251—24no—
102#13063251C2×C418yescomplemented
103#13063250—22no—
104#13063250—22no—
105#13063250—22no—
106#13063247—22no—
107C43⋊#95—21no—
108C43⋊#95—22no—
109C43⋊#95—21no—
110C22×D4×C43C2311yescharacteristic
111C23×C4×C43D412yes—
112C2×M16×C43C2312yes—
113C2×M16×C43C2312yes—
114C4×D4×C43—22no—
115C4×D4×C43—22no—
116C2×C4⋊C4×C43C2311yescharacteristic
117C2×C22⋊C4×C43D412yes—
118C2×C42×C43C2311yescharacteristic
119#100×C43—22no—
120#100×C43C2×C414yescomplemented
121#97×C43—22no—
122C4×C8×C43—22no—
123C24×C43C2×D412yes—
124C23×D43—22no—
125C2×D4×C43—22no—
126C2×D4×C43—22no—
127C2×D4×C43C22×C412yes—
128C2×D4×C43#4112yes—
129C22×C4×C43—22no—
130C22×C4×C43—22no—
131C22×C4×C43—82no—
132C22×C4×C43C2×D412yes—
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2411yescharacteristic
136C2×C22⋊D43—24no—
137C2×C22⋊D43—28no—
138C2×C22⋊D43—28no—
139C2×C22⋊D43—22no—
140C22×Dic43—22no—
141C22×Dic43—22no—
142C22×Dic43—22no—
143C22×Dic43—21no—
144C22×Dic43—21no—
145C22×Dic43—21no—
146C22×Dic43—21no—
147#10621900—28no—
148#10621900—28no—
149D4×D43—28no—
150D4×D43—28no—
151#10621898—22no—
152#10621898—22no—
153#10621898C22×C414yes—
154#10621898#4114yes—
155C2×D172C2×D412yes—
156C2×C4×D43—22no—
157C2×C4×D43—22no—
158C2×C4×D43—24no—
159C2×C4×D43—22no—
160C2×C4×D43—84no—
161C2×Dic86—22no—
162C2×Dic86—22no—
163C2×Dic86—21no—
164C2×Dic86C22×C412yescomplemented
165C2×Dic86C2×D412yes—
166M16×C43—22no—
167M16×C43—22no—
168M16×C43—22no—
169M16×C43C22×C414yescomplemented
170C2×C8×C43—22no—
171C2×C8×C43—22no—
172C4⋊C4×C43—22no—
173C22⋊C4×C43—24no—
174C42×C43C2×D412yes—
175C42×C43C2×D412yes—
176C22⋊Dic43—24no—
177C22⋊Dic43—28no—
178C22⋊Dic43—22no—
179C22⋊Dic43—82no—
180(C2×D43)⋊C4—24no—
181(C2×D43)⋊C4—22no—
182(C2×D43)⋊C4—22no—
183C4⋊Dic43—22no—
184C4⋊Dic43C2×D412yes—
185C4⋊Dic43C2×D412yes—
186Dic43⋊C4—24no—
187Dic43⋊C4—22no—
188Dic43⋊C4—22no—
189Dic43⋊C4—41no—
190Dic43⋊C4—41no—
191Dic43⋊C4—82no—
192C4×Dic43—24no—
193C4×Dic43—22no—
194C4×Dic43—22no—
195C4×Dic43—22no—
196C4×Dic43—21no—
197C4×Dic43—22no—
198C4×Dic43—42no—
199C4×Dic43C22×C412yescomplemented
200Dic172—28no—
201#10621873—28no—
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—
212D4×C43—24no—
213D4×C43—24no—
214C2×C4×C43—22no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43—44no—
218C2×C4×C43—42no—
219C2×C4×C43—82no—
220C2×C4×C43#12011yescharacteristic
221C2×C4×C43#12012yes—
222C2×C4×C43#12011yescharacteristic
223C2×C4×C43C23×C411yescharacteristic
224C2×C4×C43C22×D412yes—
225C2×C4×C43C2×#4112yes—
226C2×C4×C43C2×#4111yescharacteristic
227C22⋊D43—216no—
228C22⋊D43—24no—
229C22⋊D43—24no—
230C2×Dic43—28no—
231C2×Dic43—22no—
232C2×Dic43—22no—
233C2×Dic43—22no—
234C2×Dic43—22no—
235C2×Dic43—22no—
236C2×Dic43—41no—
237C2×Dic43—41no—
238C2×Dic43—41no—
239C2×Dic43—41no—
240C2×Dic43—42no—
241C2×Dic43—41no—
242C2×Dic43—41no—
243C2×Dic43—41no—
244C2×Dic43—41no—
245C2×Dic43—82no—
246C2×Dic43—82no—
247D172—22no—
248D172C4×D414yes—
249C4×D43—24no—
250C4×D43—24no—
251C4×D43—44no—
252C4×D43—82no—
253C4×D43—84no—
254Dic86—22no—
255Dic86—24no—
256Dic86—42no—
257Dic86C4×D414yes—
258C8×C43—24no—
259C8×C43—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×C43C2×C4×D412yes—
280C4×C43#52512yes—
281Dic43—24no—
282Dic43—42no—
283Dic43—42no—
284Dic43—42no—
285Dic43—42no—
286Dic43—81no—
287Dic43—81no—
288C4×H(5,𝔽2)—431no—
289C2×#455—431no—
290C4×#137—432no—
291#2326—431no—
292#1885—431no—
293#1882—432no—
294#1859—432no—
295#1856—431no—
296#1792—431no—
297#1759—432no—
298C4×#100D4311yescharacteristic, complemented
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×#124—431no—
309#520—432no—
310#520—862no—
311#518—431no—
312#518—862no—
313#518—862no—
314C4×#41—432no—
315C4×#41—862no—
316C2×C4×Q8—431no—
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×#100C2×D4311yescharacteristic
331C2×#100C2×D4311yescharacteristic
332C4×M16C2×D4312yescomplemented
333#391—431no—
334#390—431no—
335#388—862no—
336#385—862no—
337#384—431no—
338#376—431no—
339#343C2×D4312yescomplemented
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×Q8—431no—
349C22×D4—862no—
350C22×D4—862no—
351C22×D4C22×D4311yescharacteristic
352C23×C4—862no—
353C23×C4D17212yes—
354#137—438no—
355#137—864no—
356C2×Q16—862no—
357C2×SD16—862no—
358C2×M16C22×D4312yes—
359C2×M16C22×D4312yes—
360#124—434no—
361#124—862no—
362C4×Q8—432no—
363C4×Q8—432no—
364C4×Q8—862no—
365C4×Q8—862no—
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—431no—
386C2×C4⋊C4—862no—
387C2×C4⋊C4—862no—
388C2×C4⋊C4—862no—
389C2×C4⋊C4—861no—
390C2×C4⋊C4C22×D4311yescharacteristic
391C2×C22⋊C4—864no—
392C2×C22⋊C4—862no—
393C2×C22⋊C4—862no—
394C2×C22⋊C4—862no—
395C2×C22⋊C4—862no—
396C2×C22⋊C4D17212yes—
397C2×C42—431no—
398C2×C42—431no—
399C2×C42—862no—
400C2×C42—862no—
401C2×C42—861no—
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—22no—
412#100C4×D4314yescomplemented
413#97—22no—
414C4×C8—22no—
415#95—861no—
416#95—861no—
417#95—862no—
418C24—862no—
419C24C2×D17212yes—
420#41—434no—
421#41—434no—
422#41—868no—
423#41—868no—
424#41—862no—
425#41—862no—
426C2×Q8—432no—
427C2×Q8—432no—
428C2×Q8—861no—
429C2×Q8—862no—
430C2×Q8—862no—
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—22no—
443C22×C4—22no—
444C22×C4—82no—
445C22×C4—864no—
446C22×C4—862no—
447C22×C4—862no—
448C22×C4—862no—
449C22×C4—862no—
450C22×C4—861no—
451C22×C4—861no—
452C22×C4—862no—
453C22×C4—862no—
454C22×C4—861no—
455C22×C4—861no—
456C22×C4—3444no—
457C22×C4C2×D17212yes—
458C22×C4C2×D17211yescharacteristic
459C22×C4C2×D17211yescharacteristic
460C22×C4C23×D4311yescharacteristic
461Q16—868no—
462SD16—868no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16C2×C4×D4314yescomplemented
467C2×C8—22no—
468C2×C8—22no—
469C4⋊C4—22no—
470C4⋊C4—432no—
471C4⋊C4—432no—
472C4⋊C4—864no—
473C4⋊C4—862no—
474C4⋊C4—862no—
475C4⋊C4—862no—
476C4⋊C4—3442no—
477C4⋊C4—1721no—
478C4⋊C4—1721no—
479C22⋊C4—24no—
480C22⋊C4—864no—
481C22⋊C4—864no—
482C22⋊C4—868no—
483C22⋊C4—862no—
484C22⋊C4—862no—
485C22⋊C4—862no—
486C22⋊C4—3442no—
487C42—432no—
488C42—864no—
489C42—862no—
490C42—862no—
491C42—862no—
492C42—861no—
493C42—862no—
494C42—1722no—
495C42D4×D4312yes—
496C42D4×D4312yes—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—868no—
502C23—862no—
503C23—3442no—
504C23—3442no—
505C23C4×D17214yes—
506C23C22×D17211yescharacteristic
507Q8—434no—
508Q8—862no—
509Q8—864no—
510Q8—1722no—
511D4—24no—
512D4—24no—
513D4—434no—
514D4—8616no—
515D4—864no—
516D4—864no—
517D4—862no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—82no—
524C2×C4—868no—
525C2×C4—862no—
526C2×C4—864no—
527C2×C4—864no—
528C2×C4—862no—
529C2×C4—862no—
530C2×C4—862no—
531C2×C4—862no—
532C2×C4—3442no—
533C2×C4—3444no—
534C2×C4—3442no—
535C2×C4—3442no—
536C2×C4—1724no—
537C2×C4—1721no—
538C2×C4—1722no—
539C2×C4—1721no—
540C2×C4—1721no—
541C2×C4—1721no—
542C2×C4—1721no—
543C2×C4—1721no—
544C2×C4—1721no—
545C2×C4—1721no—
546C2×C4#1306329511yescharacteristic
547C2×C4#1306329512yes—
548C2×C4#1306329511yescharacteristic
549C2×C4C22×C4×D4311yescharacteristic
550C2×C4C2×D4×D4312yes—
551C2×C4#41×D4312yes—
552C2×C4C2×#1062189811yescharacteristic
553C8—24no—
554C8—42no—
555C22—21no—
556C22—21no—
557C22—24no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22—864no—
563C22—3448no—
564C22—1722no—
565C22—1722no—
566C22C2×C4×D17211yescharacteristic
567C22C2×#1306329511yescharacteristic
568C22#1458341111yescharacteristic
569C4—21no—
570C4—21no—
571C4—22no—
572C4—42no—
573C4—864no—
574C4—3441no—
575C4—3441no—
576C4—1722no—
577C4—1722no—
578C4—1722no—
579C4—1722no—
580C4C4×D4×D4312yes—
581C4#1458340812yes—
582C2—21no—
583C2—21no—
584C2—44no—
585C2—1724no—
586C2C4×#1306329511yescharacteristic
587C2#1914395511yescharacteristic
588C2#1914397611yescharacteristic
subgroupG/H
centerC22C2×#13063295
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC4×#100×C43C2
socleC22×C43C2×#120
prime radicalC2×#13063370C22
cosocleC22×C4×C43C24
hypercenterC4×#100D43
hypocenterC43#21474
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
3. C4×#13063358C2semidirect
4. #19137307C2semidirect
5. #19134981C2semidirect
6. #19134963C2semidirect
7. #19134824C2semidirect
8. #19134807C2semidirect
10. #19134408C2semidirect
11. C4×#100×C43C2semidirect
22. C2×C4×Dic86C22semidirect
25. #14583079C4semidirect
30. #14582941C22semidirect
33. #14582909C22semidirect
35. #14582804C22semidirect
39. #14582768C22semidirect
40. #14582745C22semidirect
41. #14582660C22semidirect
42. #14582660C22semidirect
46. C4×M16×C43C22semidirect
47. C43×#343C22semidirect
58. #13063358C2×C4semidirect
62. #13063312C2×C4semidirect
102. #13063251C2×C4semidirect
120. #100×C43C2×C4semidirect
164. C2×Dic86C22×C4semidirect
169. M16×C43C22×C4semidirect
199. C4×Dic43C22×C4semidirect
298. C4×#100D43semidirect
332. C4×M16C2×D43semidirect
339. #343C2×D43semidirect
342. C43#21474semidirect
412. #100C4×D43semidirect
466. M16C2×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×#145830792
C4×#130633582
#191373072
#191349812
#191349632
#191348242
#191348072
#191345512
#191344082
C4×#100×C432
#2147443