Finite Group Databasebeta

← #35408335

Subgroups — #35408335

Order 21248.

1566 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
1C22×#14581823C211yesmaximal, characteristic
2C2×#19127741C211yesmaximal, characteristic, complemented
3C2×#19127700C211yesmaximal, characteristic
4C2×#19127710C211yesmaximal, characteristic, complemented
5C2×C4×#13050574C211yesmaximal, characteristic, complemented
6#24909953C211yesmaximal, characteristic
7#24908347C212yesmaximal, complemented
8#24908333C212yesmaximal, complemented
9#24908323C212yesmaximal, complemented
10#24908320C212yesmaximal, complemented
11C2×#391×C83C211yesmaximal, characteristic, complemented
12C22×#13050590C2211yescharacteristic
13C2×#14581823—24no—
14C2×#14581823—24no—
15C2×#14581823C418yescomplemented
16C22×#13050580C2212yes—
17C22×#13050577C2212yes—
18C22×#13050574C2211yescharacteristic
19C22×#13050574C2211yescharacteristic
20C2×#14581751C2211yescharacteristic
21#19127741—24no—
22C2×#14581725C2212yes—
23C2×#14581723C2212yes—
24#19127710—24no—
25#19127700—24no—
26C2×#14581716C2212yes—
27C2×#14581716C2212yes—
28C2×#14581715C2212yes—
29C2×#14581712C2212yes—
30C4×#13050574—24no—
31C2×#14581703C2212yes—
32C2×C4×C83⋊C8C2212yes—
33#19127607C2211yescharacteristic
34#19127607C2211yescharacteristic
35#19127603C2211yescharacteristic
36#19127603C2211yescharacteristic
37#19127580—22no—
38C4×(C2×D83)⋊C4—22no—
39C23×Q8×C83C2211yescharacteristic
40C22×C4⋊C4×C83C2211yescharacteristic
41C22×C42×C83C2211yescharacteristic
42#391×C83—22no—
43#391×C83C2212yes—
44#391×C83C2212yes—
45C2×#95×C83—22no—
46C2×#13050590—22no—
47C2×#13050590—22no—
48C2×#13050590—22no—
49C2×#13050590—22no—
50C2×#13050590—22no—
51C2×#13050590C2×C414yes—
52C22×Dic166C2311yescharacteristic
53C22×D332C2311yescharacteristic
54C22×C22⋊D83—21no—
55C22×C4×D83—21no—
56#14581823—232no—
57C2×#13050580—24no—
58C2×#13050580—24no—
59C2×#13050580C2×C418yescomplemented
60C2×#13050577—24no—
61C2×#13050577—24no—
62C2×#13050577C2×C418yescomplemented
63C2×#13050574—22no—
64C2×#13050574—22no—
65C2×#13050574C2×C414yes—
66C2×#13050574D414yes—
67C2×#13050574D414yes—
68C2×#13050574D414yes—
69C22×C83⋊C8C2312yes—
70C22×C83⋊C8C2312yes—
71#14581751—22no—
72#14581751—22no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C22⋊Dic83—21no—
76C2×(C2×D83)⋊C4—22no—
77C2×(C2×D83)⋊C4—22no—
78C2×C4×Dic83—21no—
79#14581725—24no—
80#14581723—24no—
81#14581716—24no—
82#14581716—24no—
83#14581715—24no—
84#14581712—24no—
85#14581703—24no—
86C4×C83⋊C8—24no—
87C83⋊#95—22no—
88C83⋊#95—22no—
89C22×Q8×C83—21no—
90C22×Q8×C83—22no—
91C22×Q8×C83—22no—
92C22×Q8×C83—22no—
93C22×Q8×C83—22no—
94C22×Q8×C83—22no—
95C22×Q8×C83C2×C414yes—
96C22×Q8×C83C2312yes—
97C23×C4×C83—21no—
98C23×C4×C83—21no—
99C23×C4×C83D412yes—
100C23×C4×C83C2311yescharacteristic
101C2×C4⋊C4×C83—22no—
102C2×C4⋊C4×C83—22no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83C2312yes—
105C2×C42×C83—22no—
106C2×C42×C83—22no—
107C2×C42×C83D412yes—
108C2×C42×C83C2311yescharacteristic
109C2×C42×C83C2311yescharacteristic
110#95×C83—24no—
111#95×C83—24no—
112C23×D83—21no—
113#13050590—28no—
114#13050590—28no—
115#13050590—44no—
116#13050590—44no—
117C2×Dic166—21no—
118C2×Dic166—21no—
119C2×Dic166—21no—
120C2×Dic166—21no—
121C2×Dic166—21no—
122C2×Dic166C22×C412yes—
123C2×D332—21no—
124C2×D332—21no—
125C2×D332—21no—
126C2×D332—21no—
127C2×D332—21no—
128C2×D332C22×C412yes—
129C2×C22⋊D83—24no—
130C2×C22⋊D83—42no—
131C2×C22⋊D83—42no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C2×C4×D83—24no—
136C2×C4×D83—42no—
137C2×C4×D83—42no—
138#13050580—216no—
139#13050577—216no—
140#13050574—28no—
141#13050574C22⋊C4116yescomplemented
142C2×C83⋊C8—22no—
143C2×C83⋊C8—22no—
144C2×C83⋊C8C22×C414yes—
145C2×C83⋊C8C2×D414yes—
146C2×C83⋊C8C2×D414yes—
147C2×C83⋊C8C2×D414yes—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C22⋊Dic83—42no—
153(C2×D83)⋊C4—44no—
154(C2×D83)⋊C4—44no—
155C4×Dic83—42no—
156C24×C83C2×D411yescharacteristic
157C2×Q8×C83—24no—
158C2×Q8×C83—22no—
159C2×Q8×C83—28no—
160C2×Q8×C83—28no—
161C2×Q8×C83—22no—
162C2×Q8×C83—22no—
163C2×Q8×C83—22no—
164C2×Q8×C83—22no—
165C2×Q8×C83—42no—
166C2×Q8×C83—44no—
167C2×Q8×C83—44no—
168C2×Q8×C83—42no—
169C2×Q8×C83C22×C414yes—
170C22×C4×C83—24no—
171C22×C4×C83—24no—
172C22×C4×C83—22no—
173C22×C4×C83—24no—
174C22×C4×C83—22no—
175C22×C4×C83—42no—
176C22×C4×C83—42no—
177C22×C4×C83—42no—
178C22×C4×C83—44no—
179C22×C4×C83—42no—
180C22×C4×C83C22×C412yes—
181C22×C4×C83C2×D412yes—
182C22×C4×C83C2×D412yes—
183C22×C4×C83C2×D412yes—
184C22×C4×C83C2×D412yes—
185C22×C4×C83C2×D412yes—
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83#4112yes—
188C22×C4×C83#4112yes—
189C22×C4×C83C2411yescharacteristic
190C4⋊C4×C83—22no—
191C4⋊C4×C83—22no—
192C4⋊C4×C83—42no—
193C42×C83—24no—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83—21no—
197C42×C83—21no—
198C23×C83—22no—
199C23×C83—22no—
200C23×C83C4×D412yes—
201C23×C83#12112yes—
202C23×C83#12712yes—
203C23×C83C22×D411yescharacteristic
204C22×D83—22no—
205C22×D83—41no—
206C22×D83—41no—
207C22×D83—42no—
208C22×D83—42no—
209Q8×C83—24no—
210Q8×C83—24no—
211Q8×C83—42no—
212Q8×C83—42no—
213Q8×C83—48no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—24no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—48no—
221C2×C4×C83—42no—
222C2×C4×C83—42no—
223C2×C4×C83—48no—
224C2×C4×C83—42no—
225C2×C4×C83—44no—
226C2×C4×C83—84no—
227C2×C4×C83C2×C22⋊C414yes—
228C2×C4×C83C4×D414yes—
229C2×C4×C83C4×D412yes—
230C2×C4×C83#12112yes—
231C2×C4×C83#12114yes—
232C2×C4×C83#12414yes—
233C2×C4×C83#12712yes—
234C2×C4×C83#13714yescomplemented
235C2×C4×C83C23×C411yescharacteristic
236C2×C4×C83C22×D411yescharacteristic
237C2×C4×C83C22×D411yescharacteristic
238C2×C4×C83C22×D411yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C22⋊D83—48no—
242C2×Dic83—22no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—42no—
248C2×Dic83—42no—
249D332—22no—
250D332—22no—
251D332—41no—
252D332—41no—
253C4×D83—48no—
254Dic166—22no—
255Dic166—22no—
256Dic166—41no—
257Dic166—41no—
258C83⋊C8—24no—
259C83⋊C8C2×C22⋊C418yes—
260C22×C83—22no—
261C22×C83—22no—
262C22×C83—24no—
263C22×C83—24no—
264C22×C83#39014yes—
265C22×C83C2×C4×D411yescharacteristic
266C22×C83C2×#12111yescharacteristic
267C22×C83C2×#12711yescharacteristic
268C22×C83C2×#13712yes—
269C22×C83#57512yes—
270C2×D83—44no—
271C2×D83—42no—
272C2×D83—81no—
273C2×D83—81no—
274C2×D83—82no—
275C4×C83—24no—
276C4×C83—42no—
277C4×C83—44no—
278C4×C83—82no—
279C4×C83#39014yes—
280C4×C83C22×C22⋊C411yescharacteristic
281C4×C83C2×C4×D411yescharacteristic
282C4×C83C2×#12111yescharacteristic
283C4×C83C2×#12411yescharacteristic
284Dic83—42no—
285Dic83—81no—
286Dic83—81no—
287#21237—831nomaximal, Sylow
288C2×C83—24no—
289C2×C83C2×#39011yescharacteristic
290C2×C83C4×#13712yes—
291C2×C83#305612yes—
292C2×C83#315211yescharacteristic
293C2×C83#315311yescharacteristic
294D83—82no—
295C22×#137—831no—
296C2×#428—831no—
297C2×#417—831no—
298C2×#419—831no—
299C2×C4×M16—831no—
300#2297—831no—
301C2×#391D8311yescharacteristic, complemented
302#1973—832no—
303#1971—832no—
304#1969—832no—
305#1968—832no—
306C83#2123711yescharacteristic, complemented, Sylow
307C22×#41—831no—
308C23×Q8C2×D8311yescharacteristic
309C2×#137—838no—
310C2×#137—1664no—
311C2×#137—1664no—
312C22×Q16—832no—
313C22×SD16—832no—
314C22×M16—831no—
315C22×M16—831no—
316C2×#117—831no—
317C22×C4⋊C4C2×D8311yescharacteristic
318C22×C42C2×D8311yescharacteristic
319#428—1664no—
320C2×#107—832no—
321C2×#106—832no—
322#419—1664no—
323#417—1664no—
324C2×#103—832no—
325C2×#103—832no—
326C2×#103—832no—
327C2×#102—832no—
328C4×M16—1664no—
329C2×#97—832no—
330C2×C4×C8—832no—
331#391—22no—
332#391—831no—
333#391—831no—
334#391C2×D8312yes—
335#391C2×D8312yes—
336#390—831no—
337#390—831no—
338#386—1662no—
339C4×C22⋊C4—1662no—
340C2×#95—22no—
341C2×#41—834no—
342C2×#41—1662no—
343C2×#41—1662no—
344C2×#41—1662no—
345C2×#41—1662no—
346C2×#41—1662no—
347C22×Q8—21no—
348C22×Q8—22no—
349C22×Q8—22no—
350C22×Q8—22no—
351C22×Q8—22no—
352C22×Q8—22no—
353C22×Q8—831no—
354C22×Q8C4×D8314yes—
355C22×Q8C22×D8312yes—
356C22×D4—831no—
357C22×D4—1661no—
358C23×C4—21no—
359C23×C4—21no—
360C23×C4—1661no—
361C23×C4C22⋊D8312yes—
362C23×C4C22×D8311yescharacteristic
363#137—16632no—
364C2×Q16—838no—
365C2×Q16—1664no—
366C2×Q16—1664no—
367C2×SD16—838no—
368C2×SD16—1664no—
369C2×SD16—1664no—
370C2×M16—834no—
371C2×M16—834no—
372C2×M16—834no—
373C2×M16—834no—
374C2×M16—1662no—
375C2×M16—1662no—
376C22×C8—832no—
377C22×C8—832no—
378#117—1662no—
379#117—1662no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—831no—
384C2×C4⋊C4—831no—
385C2×C4⋊C4C22×D8312yes—
386C2×C22⋊C4—1661no—
387C2×C22⋊C4—1662no—
388C2×C22⋊C4—1662no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—1661no—
392C2×C42C22⋊D8312yes—
393C2×C42C22×D8311yescharacteristic
394C2×C42C22×D8311yescharacteristic
395#107—1664no—
396#106—1664no—
397#103—1664no—
398#103—1664no—
399#103—1664no—
400#102—1664no—
401#97—1664no—
402C4×C8—1664no—
403#95—24no—
404#95—24no—
405#95—1662no—
406#95—1662no—
407C24—1661no—
408C24C2×C22⋊D8311yescharacteristic
409#41—3324no—
410#41—3324no—
411#41—1668no—
412#41—1668no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—832no—
426C2×Q8—1661no—
427C2×Q8—1661no—
428C2×Q8—1661no—
429C2×Q8—1661no—
430C2×Q8—1661no—
431C2×Q8C2×C4×D8314yes—
432C2×D4—3322no—
433C2×D4—3322no—
434C2×D4—832no—
435C2×D4—1664no—
436C2×D4—1661no—
437C2×D4—1661no—
438C2×D4—1661no—
439C2×D4—1661no—
440C2×D4—1661no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—3322no—
452C22×C4—3322no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4—1664no—
457C22×C4C2×C4×D8312yes—
458C22×C4D4×D8312yes—
459C22×C4D4×D8312yes—
460C22×C4D4×D8312yes—
461C22×C4C2×C22⋊D8312yes—
462C22×C4C2×C22⋊D8312yes—
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4#1305058912yes—
465C22×C4#1305059012yes—
466C22×C4C23×D8311yescharacteristic
467Q16—16616no—
468SD16—16616no—
469M16—8316no—
470M16—1668no—
471C2×C8—834no—
472C2×C8—834no—
473C2×C8—834no—
474C2×C8—834no—
475C2×C8—1662no—
476C2×C8—1662no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C4⋊C4—1661no—
483C4⋊C4—1661no—
484C22⋊C4—3324no—
485C22⋊C4—3322no—
486C22⋊C4—3324no—
487C42—24no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—3322no—
493C23—22no—
494C23—22no—
495C23—3321no—
496C23—3322no—
497C23—3321no—
498C23—3322no—
499C23—1662no—
500C23C4×C22⋊D8312yes—
501C23#1458176412yes—
502C23#1458178812yes—
503C23C22×C22⋊D8311yescharacteristic
504Q8—24no—
505Q8—24no—
506Q8—48no—
507Q8—42no—
508Q8—42no—
509Q8—3321no—
510Q8—3321no—
511Q8—1662no—
512Q8—1662no—
513D4—3328no—
514D4—3321no—
515D4—3321no—
516D4—1662no—
517D4—1662no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—84no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—3328no—
534C2×C4—3322no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3322no—
538C2×C4—1662no—
539C2×C4C22⋊C4×D8314yes—
540C2×C4(C22⋊D83)⋊C414yes—
541C2×C4C4×C22⋊D8312yes—
542C2×C4#1458176412yes—
543C2×C4#1458176514yes—
544C2×C4#1458177814yes—
545C2×C4#1458178812yes—
546C2×C4#1458182314yescomplemented
547C2×C4C22×C4×D8311yescharacteristic
548C2×C4C2×D4×D8311yescharacteristic
549C2×C4C2×D4×D8311yescharacteristic
550C2×C4C2×D4×D8311yescharacteristic
551C2×C4C2×#1305058911yescharacteristic
552C2×C4C2×#1305059011yescharacteristic
553C8—838no—
554C8—1664no—
555C22—24no—
556C22—24no—
557C22—22no—
558C22—22no—
559C22—3324no—
560C22—3322no—
561C22—6642no—
562C22—6641no—
563C22—6641no—
564C22#1912760114yes—
565C22C2×C4×C22⋊D8311yescharacteristic
566C22C2×#1458176411yescharacteristic
567C22C2×#1458178811yescharacteristic
568C22C2×#1458182312yes—
569C22#1912849912yes—
570C4—24no—
571C4—44no—
572C4—42no—
573C4—82no—
574C4—3322no—
575C4—6641no—
576C4—6641no—
577C4#1912760114yes—
578C4C2×C22⋊C4×D8311yescharacteristic
579C4C2×(C22⋊D83)⋊C411yescharacteristic
580C4C2×#1458176511yescharacteristic
581C4C2×#1458177811yescharacteristic
582C2—24no—
583C2—6642no—
584C2C2×#1912760111yescharacteristic
585C2C4×#1458182312yes—
586C2#2491708912yes—
587C2#2491792011yescharacteristic
588C2#2491792711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#391×C83C2
socleC23×C83C22×D4
prime radicalC22×#13050590C22
cosocleC22×C4×C83C24
hypercenterC2×#391D83
hypocenterC83#21237
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C2×#19127741C2semidirect
4. C2×#19127710C2semidirect
5. C2×C4×#13050574C2semidirect
7. #24908347C2semidirect
8. #24908333C2semidirect
9. #24908323C2semidirect
10. #24908320C2semidirect
11. C2×#391×C83C2semidirect
15. C2×#14581823C4semidirect
59. C2×#13050580C2×C4semidirect
62. C2×#13050577C2×C4semidirect
141. #13050574C22⋊C4semidirect
234. C2×C4×C83#137semidirect
301. C2×#391D83semidirect
306. C83#21237semidirect
546. C2×C4#14581823semidirect

Maximal subgroups

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

subgroupindex
C22×#145818232
C2×#191277412
C2×#191277002
C2×#191277102
C2×C4×#130505742
#249099532
#249083472
#249083332
#249083232
#249083202
C2×#391×C832
#2123783