Finite Group Databasebeta

← #25102526

Subgroups — #25102526

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
1C2×#14583077C211yesmaximal, characteristic
2C4×#13063355C212yesmaximal, complemented
3#19137307C211yesmaximal, characteristic, complemented
4#19134983C211yesmaximal, characteristic, complemented
5#19134964C212yesmaximal, complemented
6#19134821C212yesmaximal, complemented
7#19134805C211yesmaximal, characteristic, complemented
8#19134549C211yesmaximal, characteristic
9#19134405C212yesmaximal, complemented
10C4×#13063242C211yesmaximal, characteristic, complemented
11C4×H(5,𝔽2)×C43C211yesmaximal, characteristic
12C2×#13063355C2212yes—
13C2×#13063312C2211yescharacteristic
14C2×C4×Dic86C2211yescharacteristic, complemented
15#14583077C418yescomplemented
16#14583004—22no—
17#14582999—22no—
18C4×#10621885—22no—
19C4×#10621883—22no—
20#14582942C2212yescomplemented
21C2×#13063252C2212yes—
22C2×#13063252C2212yes—
23#14582907C2212yescomplemented
24#14582900C2212yes—
25C2×#13063242C2211yescharacteristic
26C2×#13063242C2211yescharacteristic
27C4×#10621877C2212yescomplemented
28#14582804C2211yescharacteristic, complemented
29#14582802C2211yescharacteristic
30#14582787—22no—
31#14582772—22no—
32#14582768C2211yescharacteristic, complemented
33#14582745C2211yescharacteristic, complemented
34#14582668C2212yescomplemented
35#14582658C2212yescomplemented
36#14582658C2212yescomplemented
37C2×H(5,𝔽2)×C43C2211yescharacteristic
38C43×#520—22no—
39C43×#520C2212yes—
40C43×#518—22no—
41C43×#518—22no—
42C43×#518C2211yescharacteristic
43C4×#41×C43—22no—
44C4×#41×C43C2212yes—
45C2×C4×D4×C43—22no—
46C2×C4×D4×C43—22no—
47C2×C4×D4×C43C2211yescharacteristic
48C22×Dic86C2311yescharacteristic
49#13063355—24no—
50#13063355C2×C418yescomplemented
51C2×#10621885—22no—
52C2×#10621883—22no—
53C2×#10621877C2312yes—
54C2×#10621877C2312yes—
55#13063312—22no—
56#13063312C2×C414yescomplemented
57C4×Dic86—22no—
58C4×Dic86—22no—
59C2×C4⋊Dic43C2311yescharacteristic
60C2×Dic43⋊C4—21no—
61C2×Dic43⋊C4—22no—
62C2×C22⋊Dic43—22no—
63C2×C22⋊Dic43—22no—
64C2×C4×Dic43—21no—
65C2×C4×Dic43C2311yescharacteristic
66C2×C4×Dic43C2311yescharacteristic
67#13063258—22no—
68#13063256—22no—
69#13063252—24no—
70#13063252C2×C418yescomplemented
71#13063249—22no—
72#13063248—22no—
73#13063248—22no—
74#13063246—22no—
75#13063242—22no—
76#13063242C2×C414yescomplemented
77#13063236—22no—
78C4×C43⋊C8—22no—
79C43⋊#95—21no—
80C43⋊#95—22no—
81C43⋊#95—21no—
82H(5,𝔽2)×C43—24no—
83H(5,𝔽2)×C43—22no—
84H(5,𝔽2)×C43C2×C414yes—
85C2×#41×C43—22no—
86C2×#41×C43C2312yes—
87C22×D4×C43—22no—
88C22×D4×C43—22no—
89C22×D4×C43C2311yescharacteristic
90C23×C4×C43—22no—
91C23×C4×C43D412yes—
92C4×Q8×C43—22no—
93C4×Q8×C43D412yes—
94C4×Q8×C43D412yes—
95C4×D4×C43—24no—
96C4×D4×C43—24no—
97C4×D4×C43—24no—
98C4×D4×C43—24no—
99C4×D4×C43—22no—
100C4×D4×C43—22no—
101C4×D4×C43—24no—
102C4×D4×C43—22no—
103C4×D4×C43—22no—
104C4×D4×C43—22no—
105C4×D4×C43—22no—
106C4×D4×C43—22no—
107C4×D4×C43D412yes—
108C4×D4×C43D412yes—
109#117×C43—24no—
110#117×C43—22no—
111#117×C43—22no—
112#117×C43C2312yes—
113C2×C4⋊C4×C43—22no—
114C2×C4⋊C4×C43—22no—
115C2×C4⋊C4×C43C2311yescharacteristic
116C2×C22⋊C4×C43—22no—
117C2×C22⋊C4×C43—24no—
118C2×C22⋊C4×C43—22no—
119C2×C22⋊C4×C43D412yes—
120C2×C42×C43—22no—
121C2×C42×C43—22no—
122C2×C42×C43C2311yescharacteristic
123C24×C43—22no—
124C24×C43C2×D412yes—
125#41×C43—28no—
126#41×C43—28no—
127#41×C43—22no—
128#41×C43—22no—
129#41×C43C22×C414yes—
130#41×C43#4114yes—
131C2×Q8×C43C2×D412yes—
132C2×D4×C43—24no—
133C2×D4×C43—28no—
134C2×D4×C43—28no—
135C2×D4×C43—28no—
136C2×D4×C43—22no—
137C2×D4×C43—22no—
138C2×D4×C43—28no—
139C2×D4×C43—22no—
140C2×D4×C43C22×C412yes—
141C2×D4×C43C2×D412yes—
142C2×D4×C43#4112yes—
143C22×C4×C43—22no—
144C22×C4×C43—22no—
145C22×C4×C43—22no—
146C22×C4×C43—22no—
147C22×C4×C43—24no—
148C22×C4×C43—22no—
149C22×C4×C43—22no—
150C22×C4×C43—22no—
151C22×C4×C43—22no—
152C22×C4×C43—82no—
153C22×C4×C43—84no—
154C22×C4×C43C2×D412yes—
155C22×C4×C43C2×D411yescharacteristic
156C22×C4×C43C2×D411yescharacteristic
157C22×C4×C43C2411yescharacteristic
158C22×Dic43—21no—
159C22×Dic43—21no—
160C22×Dic43—21no—
161C22×Dic43—21no—
162C2×Dic86—22no—
163C2×Dic86—22no—
164C2×Dic86—21no—
165C2×Dic86C22×C412yescomplemented
166C4⋊C4×C43—24no—
167C4⋊C4×C43—22no—
168C4⋊C4×C43—22no—
169C4⋊C4×C43—22no—
170C4⋊C4×C43C2×D412yes—
171C4⋊C4×C43C2×D412yes—
172C22⋊C4×C43—24no—
173C22⋊C4×C43—24no—
174C22⋊C4×C43—22no—
175C22⋊C4×C43—22no—
176C22⋊C4×C43—24no—
177C22⋊C4×C43—22no—
178C42×C43—24no—
179C42×C43—22no—
180C42×C43—22no—
181C42×C43—22no—
182C42×C43C2×D412yes—
183C42×C43C2×D412yes—
184C22⋊Dic43—28no—
185C22⋊Dic43—82no—
186#10621885—28no—
187#10621883—28no—
188C4⋊Dic43—22no—
189Dic43⋊C4—41no—
190Dic43⋊C4—41no—
191Dic43⋊C4—82no—
192C4×Dic43—21no—
193C4×Dic43—22no—
194C4×Dic43—42no—
195C4×Dic43C22×C412yescomplemented
196#10621877—22no—
197#10621877—22no—
198#10621877—22no—
199#10621877C22×C414yescomplemented
200C2×C43⋊C8—22no—
201C2×C43⋊C8—22no—
202C23×C43—28no—
203C23×C43—22no—
204C23×C43—22no—
205C23×C43—22no—
206C23×C43—42no—
207C23×C43—42no—
208C23×C43—82no—
209C23×C43—82no—
210C23×C43C4×D414yes—
211C23×C43C22×D411yescharacteristic
212Q8×C43—22no—
213Q8×C43C4×D414yes—
214D4×C43—216no—
215D4×C43—24no—
216D4×C43—24no—
217D4×C43—24no—
218D4×C43—22no—
219D4×C43—24no—
220D4×C43C4×D414yes—
221C2×C4×C43—28no—
222C2×C4×C43—22no—
223C2×C4×C43—24no—
224C2×C4×C43—24no—
225C2×C4×C43—22no—
226C2×C4×C43—22no—
227C2×C4×C43—22no—
228C2×C4×C43—41no—
229C2×C4×C43—41no—
230C2×C4×C43—44no—
231C2×C4×C43—44no—
232C2×C4×C43—42no—
233C2×C4×C43—82no—
234C2×C4×C43—82no—
235C2×C4×C43—82no—
236C2×C4×C43—84no—
237C2×C4×C43—82no—
238C2×C4×C43#12012yes—
239C2×C4×C43#12011yescharacteristic
240C2×C4×C43#12011yescharacteristic
241C2×C4×C43C23×C411yescharacteristic
242C2×C4×C43C22×D412yes—
243C2×C4×C43C2×#4112yes—
244C2×C4×C43C2×#4111yescharacteristic
245C2×Dic43—22no—
246C2×Dic43—22no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251C2×Dic43—42no—
252C2×Dic43—41no—
253C2×Dic43—41no—
254C2×Dic43—41no—
255C2×Dic43—41no—
256Dic86—24no—
257Dic86—42no—
258C43⋊C8—24no—
259C43⋊C8—42no—
260#21474—431nomaximal, Sylow
261C22×C43—21no—
262C22×C43—21no—
263C22×C43—24no—
264C22×C43—24no—
265C22×C43—44no—
266C22×C43—42no—
267C22×C43—42no—
268C22×C43—42no—
269C22×C43—42no—
270C22×C43—82no—
271C22×C43—88no—
272C22×C43C2×C4×D411yescharacteristic
273C22×C43C2×#12011yescharacteristic
274C22×C43#52511yescharacteristic
275C4×C43—21no—
276C4×C43—21no—
277C4×C43—22no—
278C4×C43—24no—
279C4×C43—42no—
280C4×C43—42no—
281C4×C43—42no—
282C4×C43C2×C4×D412yes—
283C4×C43#52512yes—
284Dic43—42no—
285Dic43—42no—
286Dic43—81no—
287Dic43—81no—
288C4×H(5,𝔽2)D4311yescharacteristic
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×#100—431no—
299C2×C43—21no—
300C2×C43—21no—
301C2×C43—44no—
302C2×C43—44no—
303C2×C43C4×#12011yescharacteristic
304C2×C43#301911yescharacteristic
305C2×C43#302211yescharacteristic
306C2×H(5,𝔽2)C2×D4311yescharacteristic
307C2×#137—432no—
308C2×#124—431no—
309#520—22no—
310#520C2×D4312yes—
311#518—22no—
312#518—22no—
313#518C2×D4311yescharacteristic
314C4×#41—22no—
315C4×#41C2×D4312yes—
316C2×C4×Q8—431no—
317C2×C4×D4—22no—
318C2×C4×D4—22no—
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#391—431no—
334#390—431no—
335#388—862no—
336#385—862no—
337#384—431no—
338#376—431no—
339#343—432no—
340#339—432no—
341#339—432no—
342C43#2147411yescharacteristic, complemented, Sylow
343H(5,𝔽2)—24no—
344H(5,𝔽2)—22no—
345H(5,𝔽2)C4×D4314yes—
346C2×#41—22no—
347C2×#41C22×D4312yes—
348C22×Q8—431no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4C22×D4311yescharacteristic
352C23×C4—22no—
353C23×C4C22⋊D4312yes—
354#137—438no—
355#137—864no—
356C2×Q16—862no—
357C2×SD16—862no—
358C2×M16—432no—
359C2×M16—432no—
360#124—434no—
361#124—862no—
362C4×Q8—22no—
363C4×Q8—862no—
364C4×Q8—862no—
365C4×Q8C22⋊D4312yes—
366C4×Q8C22⋊D4312yes—
367C4×D4—24no—
368C4×D4—24no—
369C4×D4—24no—
370C4×D4—24no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—24no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4C22⋊D4312yes—
380C4×D4C22⋊D4312yes—
381#117—24no—
382#117—22no—
383#117—22no—
384#117C22×D4312yes—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—431no—
388C2×C4⋊C4—862no—
389C2×C4⋊C4—861no—
390C2×C4⋊C4C22×D4311yescharacteristic
391C2×C22⋊C4—22no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—862no—
395C2×C22⋊C4—862no—
396C2×C22⋊C4C22⋊D4312yes—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—431no—
400C2×C42—431no—
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—434no—
412#100—862no—
413#97—862no—
414C4×C8—862no—
415#95—861no—
416#95—861no—
417#95—862no—
418C24—22no—
419C24C2×C22⋊D4312yes—
420#41—28no—
421#41—28no—
422#41—22no—
423#41—22no—
424#41C2×C4×D4314yes—
425#41#1062189814yes—
426C2×Q8—432no—
427C2×Q8—861no—
428C2×Q8—862no—
429C2×Q8—862no—
430C2×Q8C2×C22⋊D4312yes—
431C2×D4—24no—
432C2×D4—28no—
433C2×D4—28no—
434C2×D4—28no—
435C2×D4—22no—
436C2×D4—28no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4C2×C4×D4312yes—
440C2×D4#1062189812yes—
441C2×D4C2×C22⋊D4312yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—82no—
452C22×C4—84no—
453C22×C4—861no—
454C22×C4—861no—
455C22×C4—861no—
456C22×C4—861no—
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—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—862no—
474C4⋊C4—3442no—
475C4⋊C4—1721no—
476C4⋊C4—1721no—
477C4⋊C4C2×C22⋊D4312yes—
478C4⋊C4C2×C22⋊D4312yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—24no—
485C22⋊C4—868no—
486C22⋊C4—3442no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—432no—
492C42—861no—
493C42—862no—
494C42—1722no—
495C42C2×C22⋊D4312yes—
496C42C2×C22⋊D4312yes—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—42no—
502C23—42no—
503C23—82no—
504C23—82no—
505C23C4×C22⋊D4314yes—
506C23C22×C22⋊D4311yescharacteristic
507Q8—22no—
508Q8—864no—
509Q8—1722no—
510Q8C4×C22⋊D4314yes—
511D4—216no—
512D4—24no—
513D4—24no—
514D4—24no—
515D4—24no—
516D4—22no—
517D4C4×C22⋊D4314yes—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—84no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—862no—
536C2×C4—862no—
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#1306329411yescharacteristic
547C2×C4#1306329412yes—
548C2×C4#1306329411yescharacteristic
549C2×C4C22×C4×D4311yescharacteristic
550C2×C4C22×C22⋊D4312yes—
551C2×C4C2×#1062189812yes—
552C2×C4C2×#1062189811yescharacteristic
553C8—864no—
554C8—1722no—
555C22—21no—
556C22—21no—
557C22—24no—
558C22—24no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—82no—
565C22—88no—
566C22C2×C4×C22⋊D4311yescharacteristic
567C22C2×#1306329411yescharacteristic
568C22#1458341011yescharacteristic
569C4—21no—
570C4—21no—
571C4—22no—
572C4—24no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—3441no—
577C4—3441no—
578C4—1722no—
579C4—1722no—
580C4C2×C4×C22⋊D4312yes—
581C4#1458341012yes—
582C2—21no—
583C2—21no—
584C2—44no—
585C2—44no—
586C2C4×#1306329411yescharacteristic
587C2#1914395211yescharacteristic
588C2#1914397411yescharacteristic
subgroupG/H
centerC22C2×#13063294
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC4×H(5,𝔽2)×C43C2
socleC22×C43C2×#120
prime radicalC2×H(5,𝔽2)×C43C22
cosocleC22×C4×C43C24
hypercenterC4×H(5,𝔽2)D43
hypocenterC43#21474
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C4×#13063355C2semidirect
3. #19137307C2semidirect
4. #19134983C2semidirect
5. #19134964C2semidirect
6. #19134821C2semidirect
7. #19134805C2semidirect
9. #19134405C2semidirect
10. C4×#13063242C2semidirect
14. C2×C4×Dic86C22semidirect
15. #14583077C4semidirect
20. #14582942C22semidirect
23. #14582907C22semidirect
27. C4×#10621877C22semidirect
28. #14582804C22semidirect
32. #14582768C22semidirect
33. #14582745C22semidirect
34. #14582668C22semidirect
35. #14582658C22semidirect
36. #14582658C22semidirect
50. #13063355C2×C4semidirect
56. #13063312C2×C4semidirect
70. #13063252C2×C4semidirect
76. #13063242C2×C4semidirect
165. C2×Dic86C22×C4semidirect
195. C4×Dic43C22×C4semidirect
199. #10621877C22×C4semidirect
342. C43#21474semidirect

Maximal subgroups

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

subgroupindex
C2×#145830772
C4×#130633552
#191373072
#191349832
#191349642
#191348212
#191348052
#191345492
#191344052
C4×#130632422
C4×H(5,𝔽2)×C432
#2147443