Finite Group Databasebeta

← #35421746

Subgroups — #35421746

Order 21248.

894 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916121C211yesmaximal, characteristic, complemented
2C2×#19127695C211yesmaximal, characteristic
3#24915913C211yesmaximal, characteristic
4#24908175C212yesmaximal, complemented
5#24908109C212yesmaximal, complemented
6#24907739C212yesmaximal, complemented
7#24907499C211yesmaximal, characteristic, complemented
8#24907408C211yesmaximal, characteristic
9#24907304C212yesmaximal, complemented
10#24907241C211yesmaximal, characteristic, complemented
11C83×#1826C211yesmaximal, characteristic, complemented
12C22×#13050590C2211yescharacteristic
13C2×#14581800C2212yes—
14C23×C83⋊C8C2211yescharacteristic
15#19127762—22no—
16#19127758—21no—
17#19127754—21no—
18C2×#14581727C2212yescomplemented
19#19127728C2212yescomplemented
20C2×#14581720C2212yes—
21C2×#14581720C2212yes—
22#19127695C411yescharacteristic, complemented
23#19127695C411yescharacteristic, complemented
24#19127695C411yescharacteristic, complemented
25#19127695C411yescharacteristic, complemented
26#19127695C2211yescharacteristic
27#19127695C2211yescharacteristic
28#19127695C2211yescharacteristic
29#19127695C2211yescharacteristic
30C2×#14581715C411yescharacteristic, complemented
31C2×#14581715C411yescharacteristic, complemented
32C2×#14581712C411yescharacteristic, complemented
33C2×#14581712C411yescharacteristic, complemented
34#19127669C2212yes—
35#19127666—22no—
36#19127665C2211yescharacteristic
37#19127663—21no—
38C2×C22⋊(C83⋊C8)C2211yescharacteristic
39C22⋊C8×D83—21no—
40#19127657C2212yescomplemented
41C2×#14581703C2212yescomplemented
42#19127465C2212yescomplemented
43#19127456C2212yescomplemented
44C2×#117×C83C2211yescharacteristic
45#407×C83C2211yescharacteristic
46C4×C22⋊C4×C83C2211yescharacteristic
47#339×C83C2212yescomplemented
48#339×C83C2212yescomplemented
49C2×#13050590—24no—
50C2×#13050590—21no—
51C2×#13050590—21no—
52C2×#13050590—42no—
53C2×#13050590C2×C411yescharacteristic
54C2×#13050590C2×C411yescharacteristic
55C2×#13050590C2311yescharacteristic
56C2×#13050590C2311yescharacteristic
57C22×Dic166C2×C411yescharacteristic
58C22×D332C2×C411yescharacteristic
59C22×C22⋊D83—21no—
60C22×C4×D83—21no—
61#14581800—24no—
62#14581800—42no—
63C2×#13050574C2312yes—
64C2×#13050573—22no—
65C22×C83⋊C8—42no—
66C22×C83⋊C8C2×C411yescharacteristic
67C22×C83⋊C8C2×C411yescharacteristic
68C22×C83⋊C8C2×C411yescharacteristic
69C22×C83⋊C8C2×C411yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71C22×C83⋊C8C2311yescharacteristic
72C22×C83⋊C8C2312yes—
73C2×C8×D83—22no—
74#14581727—42no—
75C4⋊(C83⋊C8)—22no—
76C4⋊(C83⋊C8)—22no—
77#14581720—24no—
78#14581720—42no—
79#14581715C2×C411yescharacteristic
80#14581715C2×C411yescharacteristic
81#14581715C2×C411yescharacteristic
82#14581715C2×C411yescharacteristic
83#14581715C2×C411yescharacteristic
84#14581715C2×C411yescharacteristic
85#14581715C2×C411yescharacteristic
86#14581715C2×C411yescharacteristic
87#14581712C2×C411yescharacteristic
88#14581712C2×C411yescharacteristic
89#14581712C2×C411yescharacteristic
90#14581712C2×C411yescharacteristic
91#14581712C2×C411yescharacteristic
92#14581712C2×C411yescharacteristic
93#14581712C2×C411yescharacteristic
94#14581712C2×C411yescharacteristic
95C22⋊(C83⋊C8)—21no—
96C22⋊(C83⋊C8)—21no—
97C22⋊(C83⋊C8)D411yescharacteristic
98C22⋊(C83⋊C8)D411yescharacteristic
99C22⋊(C83⋊C8)D412yes—
100(C2×D83)⋊C8—22no—
101(C2×D83)⋊C8—22no—
102#14581703—22no—
103#14581703—42no—
104C4×C83⋊C8—22no—
105C23×C4×C83C2311yescharacteristic
106C2×M16×C83C2312yes—
107#117×C83—21no—
108#117×C83—21no—
109#117×C83C2×C411yescharacteristic
110#117×C83C2×C411yescharacteristic
111#117×C83C2311yescharacteristic
112#117×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2×C411yescharacteristic
114C2×C4⋊C4×C83C2×C411yescharacteristic
115C2×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83—21no—
117C2×C42×C83—21no—
118C2×C42×C83C2312yes—
119C22⋊C8×C83—21no—
120C22⋊C8×C83—21no—
121#95×C83—21no—
122C23×D83—21no—
123#13050590—22no—
124#13050590—22no—
125#13050590—22no—
126#13050590—22no—
127#13050590—22no—
128#13050590—41no—
129#13050590—41no—
130#13050590—44no—
131#13050590—44no—
132#13050590#4114yes—
133C2×Dic166—22no—
134C2×Dic166—21no—
135C2×Dic166—41no—
136C2×Dic166C22×C411yescharacteristic
137C2×Dic166C22×C411yescharacteristic
138C2×D332—22no—
139C2×D332—21no—
140C2×D332—41no—
141C2×D332C22×C411yescharacteristic
142C2×D332C22×C411yescharacteristic
143C2×C22⋊D83—21no—
144C2×C22⋊D83—21no—
145C2×C22⋊D83—21no—
146C2×C22⋊D83—21no—
147C2×C22⋊D83—44no—
148C22×Dic83—21no—
149C2×C4×D83—21no—
150C2×C4×D83—21no—
151C2×C4×D83—21no—
152C2×C4×D83—21no—
153C2×C4×D83—44no—
154#13050574—22no—
155#13050574—22no—
156#13050573—44no—
157C2×C83⋊C8—22no—
158C2×C83⋊C8—22no—
159C2×C83⋊C8—22no—
160C2×C83⋊C8—22no—
161C2×C83⋊C8—22no—
162C2×C83⋊C8—41no—
163C2×C83⋊C8—41no—
164C2×C83⋊C8—41no—
165C2×C83⋊C8—41no—
166C2×C83⋊C8C22×C411yescharacteristic
167C2×C83⋊C8C22×C411yescharacteristic
168C2×C83⋊C8C22×C411yescharacteristic
169C2×C83⋊C8C22×C411yescharacteristic
170C2×C83⋊C8C2×D412yes—
171C2×C83⋊C8C2×D412yes—
172C8×D83—44no—
173C24×C83#4111yescharacteristic
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—22no—
179C22×C4×C83C22×C411yescharacteristic
180C22×C4×C83C22×C411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83#4111yescharacteristic
184C22×C4×C83C2411yescharacteristic
185M16×C83—22no—
186M16×C83—22no—
187C2×C8×C83—22no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83C22×C411yescharacteristic
191C4⋊C4×C83C22×C411yescharacteristic
192C4⋊C4×C83C22×C411yescharacteristic
193C4⋊C4×C83C22×C411yescharacteristic
194C22⋊C4×C83—21no—
195C22⋊C4×C83—21no—
196C22⋊C4×C83—21no—
197C22⋊C4×C83—21no—
198C22⋊C4×C83—42no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83—22no—
204C42×C83—22no—
205C23×C83—22no—
206C23×C83—22no—
207C23×C83—41no—
208C23×C83C4×D411yescharacteristic
209C23×C83C4×D411yescharacteristic
210C23×C83C2×#4111yescharacteristic
211C22×D83—21no—
212C22×D83—21no—
213C22×D83—42no—
214C22×D83—42no—
215C22×D83—81no—
216C2×C4×C83—22no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—42no—
228C2×C4×C83—42no—
229C2×C4×C83—42no—
230C2×C4×C83—42no—
231C2×C4×C83—81no—
232C2×C4×C83C4×D411yescharacteristic
233C2×C4×C83C4×D411yescharacteristic
234C2×C4×C83C23×C411yescharacteristic
235C2×C4×C83C22×D411yescharacteristic
236C2×C4×C83C2×#4112yes—
237C2×C4×C83H(5,𝔽2)12yes—
238C22⋊D83—24no—
239C22⋊D83—42no—
240C22⋊D83—42no—
241C22⋊D83—42no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—42no—
245D332—21no—
246D332—21no—
247D332—21no—
248D332—42no—
249D332#11712yes—
250C4×D83—24no—
251C4×D83—42no—
252C4×D83—42no—
253C4×D83—42no—
254Dic166—21no—
255Dic166—21no—
256Dic166—21no—
257Dic166—42no—
258Dic166#11712yes—
259C8×C83—42no—
260C83⋊C8—21no—
261C83⋊C8—21no—
262C83⋊C8—21no—
263C83⋊C8—21no—
264C83⋊C8—22no—
265C83⋊C8—22no—
266C83⋊C8—42no—
267C22×C83—21no—
268C22×C83—21no—
269C22×C83—21no—
270C22×C83—21no—
271C22×C83—21no—
272C22×C83—21no—
273C22×C83—21no—
274C22×C83—44no—
275C22×C83#44412yes—
276C22×C83C2×C4×D411yescharacteristic
277C22×C83#55112yes—
278C2×D83—22no—
279C2×D83—41no—
280C2×D83—41no—
281C2×D83—41no—
282C2×D83—41no—
283C2×D83—41no—
284C2×D83—84no—
285C4×C83—21no—
286C4×C83—21no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83—82no—
292C4×C83C2×#11711yescharacteristic
293C4×C83#51811yescharacteristic
294C4×C83#54612yes—
295Dic83—22no—
296Dic83—41no—
297#22586—831nomaximal, Sylow
298C2×C83—22no—
299C2×C83—22no—
300C2×C83—41no—
301C2×C83#234211yescharacteristic
302C2×C83C2×#44411yescharacteristic
303C2×C83#295711yescharacteristic
304D83—42no—
305D83—81no—
306#2924—831no—
307C2×#416—831no—
308#2880—831no—
309#1938—832no—
310#1929—832no—
311#1866—832no—
312#1826D8311yescharacteristic, complemented
313#1810—831no—
314#1789—831no—
315#1765—832no—
316#1752—831no—
317C83#2258611yescharacteristic, complemented, Sylow
318C22×#41—831no—
319C2×#131—832no—
320C23×C8—831no—
321C2×#117C2×D8311yescharacteristic
322#433—1662no—
323#432—1661no—
324#431—1661no—
325C2×#108—832no—
326#424—832no—
327C2×#104—832no—
328C2×#104—832no—
329#416—831no—
330#416—831no—
331#416—831no—
332#416—831no—
333#416—831no—
334#416—831no—
335#416—831no—
336#416—831no—
337C2×#103—831no—
338C2×#103—831no—
339C2×#102—831no—
340C2×#102—831no—
341#408—832no—
342#408—1662no—
343#407—831no—
344#407—1661no—
345#407C2×D8311yescharacteristic
346C2×C22⋊C8—831no—
347C2×C22⋊C8—1661no—
348#405—832no—
349C2×#97—832no—
350C4×C22⋊C4C2×D8311yescharacteristic
351#341—832no—
352#339C2×D8312yescomplemented
353#339C2×D8312yescomplemented
354#336—832no—
355C2×#41—3322no—
356C2×#41—831no—
357C2×#41—831no—
358C2×#41—831no—
359C2×#41—831no—
360C2×#41—1664no—
361C2×#41—1661no—
362C2×#41—1661no—
363C22×Q8—831no—
364C22×D4—831no—
365C22×D4—1661no—
366C23×C4—1661no—
367C23×C4C22×D8311yescharacteristic
368#131—3322no—
369#131—1664no—
370C2×M16—832no—
371C2×M16—1662no—
372C2×M16C22×D8312yes—
373C22×C8—3322no—
374C22×C8—831no—
375C22×C8—831no—
376C22×C8—831no—
377C22×C8—831no—
378C22×C8—831no—
379C22×C8—831no—
380C22×C8—832no—
381C22×C8—1662no—
382#117—21no—
383#117—21no—
384#117C4×D8311yescharacteristic
385#117C4×D8311yescharacteristic
386#117C22×D8311yescharacteristic
387#117C22×D8311yescharacteristic
388C2×C4⋊C4C4×D8311yescharacteristic
389C2×C4⋊C4C4×D8311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C42—21no—
393C2×C42C22×D8312yes—
394#108—3322no—
395C4⋊C8—1662no—
396C4⋊C8—1662no—
397#104—3322no—
398#104—1664no—
399#103—831no—
400#103—831no—
401#103—831no—
402#103—831no—
403#103—831no—
404#103—831no—
405#103—831no—
406#103—831no—
407#102—831no—
408#102—831no—
409#102—831no—
410#102—831no—
411#102—831no—
412#102—831no—
413#102—831no—
414#102—831no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—831no—
418C22⋊C8—831no—
419C22⋊C8—832no—
420C22⋊C8—1662no—
421C22⋊C8—1662no—
422C22⋊C8—1661no—
423C22⋊C8—1661no—
424#97—3322no—
425#97—1662no—
426C4×C8—1662no—
427#95—21no—
428C24—1661no—
429C24#1305058911yescharacteristic
430#41—3324no—
431#41—3321no—
432#41—3321no—
433#41—3324no—
434#41—834no—
435#41—1662no—
436#41—1662no—
437#41—1662no—
438#41—1662no—
439#41—1662no—
440C2×Q8—3321no—
441C2×Q8—831no—
442C2×Q8—831no—
443C2×Q8—1662no—
444C2×Q8—1661no—
445C2×D4—3321no—
446C2×D4—3324no—
447C2×D4—831no—
448C2×D4—831no—
449C2×D4—1662no—
450C2×D4—1661no—
451C2×D4—1661no—
452C2×D4—1661no—
453C2×D4—1661no—
454C2×D4—1661no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—3324no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4C2×C4×D8311yescharacteristic
467C22×C4C2×C4×D8311yescharacteristic
468C22×C4D4×D8311yescharacteristic
469C22×C4D4×D8311yescharacteristic
470C22×C4#1305058911yescharacteristic
471C22×C4C23×D8311yescharacteristic
472M16—22no—
473M16—22no—
474M16—3324no—
475M16—1662no—
476M16—1662no—
477C2×C8—22no—
478C2×C8—3324no—
479C2×C8—3321no—
480C2×C8—3321no—
481C2×C8—3321no—
482C2×C8—3321no—
483C2×C8—831no—
484C2×C8—831no—
485C2×C8—831no—
486C2×C8—831no—
487C2×C8—832no—
488C2×C8—832no—
489C2×C8—1662no—
490C2×C8—1662no—
491C2×C8—1662no—
492C2×C8—1662no—
493C2×C8—1662no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C2×C4×D8311yescharacteristic
497C4⋊C4C2×C4×D8311yescharacteristic
498C4⋊C4C2×C4×D8311yescharacteristic
499C4⋊C4C2×C4×D8311yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—42no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—22no—
510C42—22no—
511C23—22no—
512C23—22no—
513C23—41no—
514C23—3322no—
515C23—3322no—
516C23—6641no—
517C23—1661no—
518C23—1661no—
519C23(C22⋊D83)⋊C411yescharacteristic
520C23(C22⋊D83)⋊C411yescharacteristic
521C23C2×#1305058911yescharacteristic
522Q8—3322no—
523Q8—832no—
524Q8—1661no—
525Q8—1661no—
526Q8—1661no—
527D4—3322no—
528D4—3322no—
529D4—3322no—
530D4—3322no—
531D4—832no—
532D4—1664no—
533D4—1661no—
534D4—1661no—
535D4—1661no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—81no—
552C2×C4—3322no—
553C2×C4—3322no—
554C2×C4—3322no—
555C2×C4—3322no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4—1664no—
559C2×C4(C22⋊D83)⋊C411yescharacteristic
560C2×C4(C22⋊D83)⋊C411yescharacteristic
561C2×C4C22×C4×D8311yescharacteristic
562C2×C4C2×D4×D8311yescharacteristic
563C2×C4#41×D8312yes—
564C2×C4#1458183812yes—
565C8—42no—
566C8—3322no—
567C8—1662no—
568C8—1662no—
569C8—1661no—
570C8—1661no—
571C8—1661no—
572C8—1661no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—44no—
581C22—3321no—
582C22—3321no—
583C22—3321no—
584C22—3321no—
585C22—3321no—
586C22—6644no—
587C22—1662no—
588C22#1912782012yes—
589C22C2×(C22⋊D83)⋊C411yescharacteristic
590C22#1912837412yes—
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—82no—
598C4—3321no—
599C4—1662no—
600C4#117×D8311yescharacteristic
601C4#1912817511yescharacteristic
602C4#1912834312yes—
603C2—22no—
604C2—22no—
605C2—41no—
606C2—3322no—
607C2—6641no—
608C2#2491030511yescharacteristic
609C2C2×#1912782011yescharacteristic
610C2#2491637611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1826C2
socleC22×C83C2×C4×D4
prime radicalC22×#13050590C22
cosocleC22×C4×C83C24
hypercenter#1826D83
hypocenterC83#22586
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
1. #24916121C2semidirect
4. #24908175C2semidirect
5. #24908109C2semidirect
6. #24907739C2semidirect
7. #24907499C2semidirect
9. #24907304C2semidirect
10. #24907241C2semidirect
11. C83×#1826C2semidirect
18. C2×#14581727C22semidirect
19. #19127728C22semidirect
22. #19127695C4semidirect
23. #19127695C4semidirect
24. #19127695C4semidirect
25. #19127695C4semidirect
30. C2×#14581715C4semidirect
31. C2×#14581715C4semidirect
32. C2×#14581712C4semidirect
33. C2×#14581712C4semidirect
40. #19127657C22semidirect
41. C2×#14581703C22semidirect
42. #19127465C22semidirect
43. #19127456C22semidirect
47. #339×C83C22semidirect
48. #339×C83C22semidirect
312. #1826D83semidirect
317. C83#22586semidirect
352. #339C2×D83semidirect
353. #339C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249161212
C2×#191276952
#249159132
#249081752
#249081092
#249077392
#249074992
#249074082
#249073042
#249072412
C83×#18262
#2258683