Finite Group Databasebeta

← #35567247

Subgroups — #35567247

Order 21248.

2510 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128252C211yesmaximal, characteristic
2#24910618C214yesmaximal, complemented
3#24910503C212yesmaximal, complemented
4#24910500C212yesmaximal, complemented
5#24910408C212yesmaximal, complemented
6#24910387C214yesmaximal, complemented
7#24910364C212yesmaximal, complemented
8#24910275C214yesmaximal, complemented
9#24910212C211yesmaximal, characteristic, complemented
10#24910188C211yesmaximal, characteristic, complemented
11#24910186C211yesmaximal, characteristic, complemented
12#24910054C214yesmaximal, complemented
13#24909869C211yesmaximal, characteristic, complemented
14#24909855C211yesmaximal, characteristic, complemented
15C83×#2331C211yesmaximal, characteristic, complemented
16C22×#13050591C2212yes—
17#19128252C4116yescomplemented
18C2×#14581789C2211yescharacteristic
19C2×#14581781C2212yes—
20C2×#14581767C2212yes—
21C2×#14581767C2212yes—
22C2×C4×D332C2212yes—
23C2×#14581754C2212yes—
24C2×#14581754C2212yes—
25C2×#14581754C2212yes—
26C2×#14581750C2212yes—
27C2×#14581749C2214yes—
28C2×C4⋊C4×D83C2214yes—
29C2×C42×D83C2211yescharacteristic
30#19127608C2214yes—
31#19127603C2214yes—
32#19127603C2212yes—
33#19127602C2214yes—
34#19127602C2212yes—
35#19127600C2212yes—
36#19127593C2214yes—
37#19127592C2214yes—
38#19127591C2214yes—
39#19127590C2214yes—
40#19127587C2214yes—
41#19127579C2214yes—
42#19127579C2214yes—
43#19127579C2214yes—
44#19127579C2214yes—
45#19127579C2214yes—
46#19127579C2214yes—
47#19127575C2214yes—
48#19127572C2214yes—
49#19127570C2212yes—
50#19127568C2212yes—
51#19127567C2211yescharacteristic
52#19127565C2212yes—
53#19127563C2212yes—
54#19127560C2212yes—
55#19127560C2214yes—
56#19127560C2212yes—
57#19127560C2212yes—
58#19127560C2214yes—
59C4⋊C4×Dic83C2212yes—
60C4×Dic83⋊C4C2211yescharacteristic
61C4×Dic83⋊C4C2211yescharacteristic
62#19127552C2212yes—
63#19127552C2212yes—
64C4×(C2×D83)⋊C4C2211yescharacteristic
65C4×(C2×D83)⋊C4C2211yescharacteristic
66C4×(C2×D83)⋊C4C2214yes—
67#19127548C2214yes—
68#19127547C2211yescharacteristic
69C2×C4⋊Q8×C83C2211yescharacteristic
70#391×C83C2214yes—
71#389×C83C2212yes—
72#387×C83C2214yes—
73#382×C83C2211yescharacteristic
74C4×C4⋊C4×C83C2212yes—
75#376×C83C2211yescharacteristic
76C2×#13050591—48no—
77C2×#13050591C2×C4116yescomplemented
78C22×D332C2312yes—
79C22×D332C2312yes—
80C22×D332C2312yes—
81C22×D332C2314yes—
82C22×C4×D83C2311yescharacteristic
83C22×C4×D83C2314yes—
84C22×C4×D83C2311yescharacteristic
85C22×C4×D83C2314yes—
86C22×C4×D83C2314yes—
87C22×C4×D83C2311yescharacteristic
88#14581789C2×C418yes—
89#14581781C2×C4116yescomplemented
90#14581767C2×C4116yescomplemented
91#14581767C2×C4116yescomplemented
92C4×D332—44no—
93#14581754—44no—
94#14581754C2×C4116yescomplemented
95#14581754C2×C4116yescomplemented
96#14581750—44no—
97#14581749—48no—
98C2×C4⋊Dic83C2312yes—
99C2×C4⋊Dic83C2314yes—
100C2×Dic83⋊C4C2314yes—
101C2×Dic83⋊C4C2312yes—
102C2×Dic83⋊C4C2312yes—
103C2×Dic83⋊C4C2314yes—
104C4⋊C4×D83—48no—
105C2×(C2×D83)⋊C4C2314yes—
106C2×(C2×D83)⋊C4C2312yes—
107C2×(C2×D83)⋊C4C2312yes—
108C2×(C2×D83)⋊C4C2314yes—
109C2×(C2×D83)⋊C4C2314yes—
110C2×(C2×D83)⋊C4C2314yes—
111C2×(C2×D83)⋊C4C2314yes—
112C2×(C2×D83)⋊C4C2314yes—
113C2×(C2×D83)⋊C4C2314yes—
114C2×(C2×D83)⋊C4C2314yes—
115C2×(C2×D83)⋊C4C2314yes—
116C2×C4×Dic83C2311yescharacteristic
117C2×C4×Dic83C2311yescharacteristic
118C2×C4×Dic83C2314yes—
119C2×C4×Dic83C2314yes—
120C2×C4×Dic83C2311yescharacteristic
121C2×C4×Dic83C2311yescharacteristic
122C2×C4×Dic83C2311yescharacteristic
123C42×D83C2×C418yescomplemented
124C83⋊#95C2312yes—
125C83⋊#95C2312yes—
126C83⋊#95C2312yes—
127C83⋊#95C2312yes—
128C83⋊#95C2314yes—
129C83⋊#95C2312yes—
130C83⋊#95C2314yes—
131C83⋊#95C2312yes—
132C83⋊#95C2312yes—
133C83⋊#95C2312yes—
134C22×Q8×C83C2312yes—
135C4⋊Q8×C83C2×C418yescomplemented
136C2×C4⋊C4×C83C2312yes—
137C2×C4⋊C4×C83C2314yes—
138C2×C4⋊C4×C83C2312yes—
139C2×C4⋊C4×C83C2312yes—
140C2×C42×C83C2311yescharacteristic
141C2×C42×C83C2311yescharacteristic
142C2×C42×C83C2314yes—
143C2×C42×C83C2311yescharacteristic
144#95×C83C2312yes—
145#95×C83C2312yes—
146#95×C83C2312yes—
147#95×C83C2314yes—
148#95×C83C2314yes—
149#95×C83C2312yes—
150C23×D83C2411yescharacteristic
151C23×D83C2414yes—
152#13050591—432no—
153C2×D332—24no—
154C2×D332—24no—
155C2×D332—44no—
156C2×D332—44no—
157C2×D332—44no—
158C2×D332—44no—
159C2×D332C22×C418yes—
160C2×D332C22×C4116yes—
161C2×D332C22×C418yes—
162C2×D332C22×C418yes—
163C22×Dic83C2412yes—
164C22×Dic83C2414yes—
165C22×Dic83C2412yes—
166C22×Dic83C2411yescharacteristic
167C22×Dic83C2411yescharacteristic
168C22×Dic83C2411yescharacteristic
169C2×C4×D83—22no—
170C2×C4×D83—22no—
171C2×C4×D83—24no—
172C2×C4×D83—28no—
173C2×C4×D83—28no—
174C2×C4×D83—42no—
175C2×C4×D83—44no—
176C2×C4×D83—44no—
177C2×C4×D83—44no—
178C2×C4×D83—44no—
179C2×C4×D83—44no—
180C2×C4×D83—44no—
181C2×C4×D83C22×C414yes—
182C2×C4×D83C22×C414yes—
183C2×C4×D83C22×C4116yes—
184C2×C4×D83C22×C414yes—
185C4⋊Dic83—24no—
186C4⋊Dic83—44no—
187Dic83⋊C4—24no—
188Dic83⋊C4—28no—
189Dic83⋊C4—24no—
190Dic83⋊C4—44no—
191(C2×D83)⋊C4—44no—
192(C2×D83)⋊C4—42no—
193(C2×D83)⋊C4—44no—
194(C2×D83)⋊C4—42no—
195(C2×D83)⋊C4—44no—
196(C2×D83)⋊C4—44no—
197(C2×D83)⋊C4—44no—
198(C2×D83)⋊C4—44no—
199(C2×D83)⋊C4—44no—
200(C2×D83)⋊C4C22×C4116yes—
201(C2×D83)⋊C4C22×C4116yes—
202C4×Dic83—22no—
203C4×Dic83—22no—
204C4×Dic83—44no—
205C4×Dic83—44no—
206C4×Dic83C22×C414yes—
207C4×Dic83C22×C414yes—
208C4×Dic83C22×C414yes—
209C2×Q8×C83—44no—
210C2×Q8×C83C22×C418yes—
211C22×C4×C83C2412yes—
212C22×C4×C83C2414yes—
213C22×C4×C83C2412yes—
214C22×C4×C83C2411yescharacteristic
215C22×C4×C83C2411yescharacteristic
216C22×C4×C83C2414yes—
217C22×C4×C83C2411yescharacteristic
218C4⋊C4×C83—24no—
219C4⋊C4×C83—44no—
220C4⋊C4×C83C22×C418yes—
221C4⋊C4×C83C22×C418yes—
222C42×C83—22no—
223C42×C83—22no—
224C42×C83—44no—
225C42×C83C22×C414yes—
226C23×C83C2511yescharacteristic
227C22×D83—21no—
228C22×D83—24no—
229C22×D83—24no—
230C22×D83—24no—
231C22×D83—22no—
232C22×D83—21no—
233C22×D83—24no—
234C22×D83—22no—
235C22×D83—24no—
236C22×D83—24no—
237C22×D83C23×C418yes—
238C22×D83C23×C412yes—
239Q8×C83—48no—
240C2×C4×C83—24no—
241C2×C4×C83—24no—
242C2×C4×C83—22no—
243C2×C4×C83—22no—
244C2×C4×C83—22no—
245C2×C4×C83—22no—
246C2×C4×C83—24no—
247C2×C4×C83—24no—
248C2×C4×C83—22no—
249C2×C4×C83—22no—
250C2×C4×C83—22no—
251C2×C4×C83—24no—
252C2×C4×C83—22no—
253C2×C4×C83C23×C412yes—
254C2×C4×C83C23×C412yes—
255C2×C4×C83C23×C418yes—
256C2×C4×C83C23×C412yes—
257C2×C4×C83H(5,𝔽2)12yes—
258C2×C4×C832−1+412yes—
259C2×Dic83—21no—
260C2×Dic83—21no—
261C2×Dic83—22no—
262C2×Dic83—22no—
263C2×Dic83—22no—
264C2×Dic83—24no—
265C2×Dic83—24no—
266C2×Dic83—22no—
267C2×Dic83—22no—
268C2×Dic83—22no—
269C2×Dic83—24no—
270C2×Dic83—22no—
271C2×Dic83C23×C412yes—
272C2×Dic83C23×C412yes—
273C2×Dic83C23×C412yes—
274C2×Dic83H(5,𝔽2)14yes—
275D332—216no—
276D332—48no—
277D332—48no—
278D332—416no—
279C4×D83—28no—
280C4×D83—232no—
281C4×D83—28no—
282C4×D83—44no—
283C4×D83—88no—
284C4×D83—88no—
285C22×C83C24×C411yescharacteristic
286C22×C83C2×H(5,𝔽2)12yes—
287C22×C83C2×H(5,𝔽2)11yescharacteristic
288C22×C83C2×2−1+411yescharacteristic
289C22×C83#58512yes—
290C2×D83—22no—
291C2×D83—28no—
292C2×D83—22no—
293C2×D83—28no—
294C2×D83—22no—
295C2×D83—28no—
296C2×D83—44no—
297C2×D83—42no—
298C2×D83—44no—
299C2×D83—44no—
300C2×D83—44no—
301C2×D83—42no—
302C4×C83—22no—
303C4×C83—28no—
304C4×C83—22no—
305C4×C83—42no—
306C4×C83—44no—
307C4×C83—42no—
308C4×C83#52014yes—
309Dic83—22no—
310Dic83—22no—
311Dic83—42no—
312Dic83—44no—
313Dic83—42no—
314Dic83#51814yes—
315#34367—831nomaximal, Sylow
316C2×C83C2×#51811yescharacteristic
317C2×C83C2×#52011yescharacteristic
318C2×C83#342811yescharacteristic
319C2×C83#354612yes—
320C2×C83#354712yes—
321D83—42no—
322D83—48no—
323C2×#532—831no—
324#2377—834no—
325#2363—832no—
326#2363—832no—
327#2354—832no—
328#2352—834no—
329D4×C4⋊C4—832no—
330#2337—834no—
331#2331—831no—
332#2331D8311yescharacteristic, complemented
333#2328—831no—
334#2328—831no—
335#2311—834no—
336#2285—831no—
337#2283—831no—
338C83#3436711yescharacteristic, complemented, Sylow
339C22×#41—832no—
340#532—8316no—
341C2×C4⋊Q8C2×D8311yescharacteristic
342C2×#127—831no—
343C2×#124—832no—
344C2×#121—832no—
345C2×#121—832no—
346C2×C4×D4—832no—
347C2×C4×D4—832no—
348C2×C4×D4—832no—
349C2×C4×D4—832no—
350C2×#117—832no—
351C2×#117—834no—
352C22×C4⋊C4—834no—
353C22×C42—831no—
354#391—834no—
355#391C2×D8314yes—
356#390—834no—
357#390—832no—
358#390—834no—
359#390—832no—
360#389—832no—
361#389C2×D8312yes—
362#388—834no—
363#388—834no—
364#388—834no—
365#388—834no—
366#387—834no—
367#387C2×D8314yes—
368#386—834no—
369#386—834no—
370#386—834no—
371#386—834no—
372#386—834no—
373#386—834no—
374#385—834no—
375#385—834no—
376#384—832no—
377#383—832no—
378#382—831no—
379#382—832no—
380#382C2×D8311yescharacteristic
381#381—832no—
382#380—832no—
383#380—832no—
384#380—834no—
385#380—834no—
386#380—832no—
387C4×C4⋊C4—832no—
388C4×C4⋊C4—831no—
389C4×C4⋊C4—831no—
390C4×C4⋊C4—832no—
391C4×C4⋊C4—832no—
392C4×C4⋊C4C2×D8312yes—
393C4×C22⋊C4—834no—
394C4×C22⋊C4—831no—
395C4×C22⋊C4—831no—
396C4×C22⋊C4—834no—
397#376—831no—
398#376C2×D8311yescharacteristic
399C2×#41—3328no—
400C2×#41—8316no—
401C22×Q8C22×D8312yes—
402C22×D4—832no—
403C22×D4—834no—
404C22×D4—832no—
405C22×D4—832no—
406C23×C4—834no—
407C23×C4—834no—
408C23×C4—831no—
409C23×C4—834no—
410C23×C4—831no—
411C23×C4—831no—
412C4⋊Q8C4×D8318yescomplemented
413#127—838no—
414#124—8316no—
415#121—8316no—
416#121—8316no—
417C4×D4—3324no—
418C4×D4—3324no—
419C4×D4—8316no—
420C4×D4—8316no—
421#117—3328no—
422#117—3324no—
423C2×C4⋊C4—3328no—
424C2×C4⋊C4—834no—
425C2×C4⋊C4—834no—
426C2×C4⋊C4—832no—
427C2×C4⋊C4—834no—
428C2×C4⋊C4—832no—
429C2×C4⋊C4—832no—
430C2×C4⋊C4C22×D8312yes—
431C2×C4⋊C4C22×D8314yes—
432C2×C4⋊C4C22×D8312yes—
433C2×C4⋊C4C22×D8312yes—
434C2×C22⋊C4—834no—
435C2×C22⋊C4—834no—
436C2×C22⋊C4—834no—
437C2×C22⋊C4—834no—
438C2×C22⋊C4—834no—
439C2×C22⋊C4—832no—
440C2×C22⋊C4—832no—
441C2×C22⋊C4—834no—
442C2×C22⋊C4—834no—
443C2×C22⋊C4—834no—
444C2×C22⋊C4—834no—
445C2×C42—834no—
446C2×C42—834no—
447C2×C42—831no—
448C2×C42—831no—
449C2×C42—831no—
450C2×C42—838no—
451C2×C42—831no—
452C2×C42—831no—
453C2×C42C22×D8311yescharacteristic
454C2×C42C22×D8311yescharacteristic
455C2×C42C22×D8314yes—
456C2×C42C22×D8311yescharacteristic
457#95—832no—
458#95—832no—
459#95—832no—
460#95—832no—
461#95—834no—
462#95—832no—
463#95—832no—
464#95—834no—
465#95—832no—
466#95—832no—
467#95C22×D8312yes—
468#95C22×D8312yes—
469#95C22×D8314yes—
470#95C22×D8312yes—
471#95C22×D8312yes—
472#95C22×D8314yes—
473C24—834no—
474C24—831no—
475#41—33232no—
476C2×Q8—44no—
477C2×Q8C2×C4×D8318yes—
478C2×D4—3324no—
479C2×D4—3324no—
480C2×D4—3324no—
481C2×D4—3324no—
482C2×D4—838no—
483C2×D4—8316no—
484C2×D4—838no—
485C2×D4—838no—
486C2×D4—1664no—
487C2×D4—1664no—
488C22×C4—3322no—
489C22×C4—3324no—
490C22×C4—3324no—
491C22×C4—3324no—
492C22×C4—3324no—
493C22×C4—3324no—
494C22×C4—3324no—
495C22×C4—834no—
496C22×C4—831no—
497C22×C4—831no—
498C22×C4—831no—
499C22×C4—834no—
500C22×C4—834no—
501C22×C4—8316no—
502C22×C4—832no—
503C22×C4—832no—
504C22×C4—834no—
505C22×C4—1668no—
506C22×C4—1668no—
507C22×C4—1664no—
508C22×C4—1662no—
509C22×C4—1662no—
510C22×C4C23×D8312yes—
511C22×C4C23×D8312yes—
512C22×C4C23×D8314yes—
513C22×C4C23×D8311yescharacteristic
514C22×C4C23×D8311yescharacteristic
515C22×C4C23×D8314yes—
516C22×C4C23×D8311yescharacteristic
517C4⋊C4—24no—
518C4⋊C4—44no—
519C4⋊C4—3324no—
520C4⋊C4—3324no—
521C4⋊C4—1664no—
522C4⋊C4—1668no—
523C4⋊C4—1664no—
524C4⋊C4—1664no—
525C4⋊C4C2×C4×D8318yes—
526C4⋊C4C2×C4×D8318yes—
527C22⋊C4—3324no—
528C22⋊C4—3324no—
529C22⋊C4—3324no—
530C22⋊C4—3324no—
531C22⋊C4—3324no—
532C22⋊C4—3324no—
533C22⋊C4—3322no—
534C22⋊C4—3324no—
535C22⋊C4—3322no—
536C22⋊C4—8316no—
537C22⋊C4—8316no—
538C42—22no—
539C42—22no—
540C42—44no—
541C42—3324no—
542C42—3324no—
543C42—834no—
544C42—834no—
545C42—834no—
546C42—1662no—
547C42—1662no—
548C42C2×C4×D8314yes—
549C23—838no—
550C23—832no—
551C23—1664no—
552C23—1664no—
553C23—1662no—
554C23—1661no—
555C23—1664no—
556C23—1661no—
557C23—1664no—
558C23—1662no—
559C23—1664no—
560C23—1664no—
561C23C24×D8311yescharacteristic
562Q8—48no—
563D4—3328no—
564D4—33216no—
565D4—3328no—
566D4—16616no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—3324no—
581C2×C4—832no—
582C2×C4—834no—
583C2×C4—832no—
584C2×C4—832no—
585C2×C4—6648no—
586C2×C4—6648no—
587C2×C4—1668no—
588C2×C4—16632no—
589C2×C4—1662no—
590C2×C4—1664no—
591C2×C4—1662no—
592C2×C4—1664no—
593C2×C4—1662no—
594C2×C4—1662no—
595C2×C4—1662no—
596C2×C4—1664no—
597C2×C4—1662no—
598C2×C4—1661no—
599C2×C4—1661no—
600C2×C4—1662no—
601C2×C4—1668no—
602C2×C4C22×C4×D8312yes—
603C2×C4C22×C4×D8312yes—
604C2×C4C22×C4×D8318yes—
605C2×C4C22×C4×D8312yes—
606C2×C4#1458183812yes—
607C2×C4#1458184012yes—
608C22—3324no—
609C22—3324no—
610C22—3322no—
611C22—3324no—
612C22—3324no—
613C22—3322no—
614C22—1662no—
615C22—1668no—
616C22—1662no—
617C22—1668no—
618C22—1662no—
619C22—1668no—
620C22C23×C4×D8311yescharacteristic
621C22H(5,𝔽2)×D8312yes—
622C22C2×#1458183811yescharacteristic
623C22C2×#1458184011yescharacteristic
624C22#1912853812yes—
625C4—22no—
626C4—28no—
627C4—22no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—3322no—
632C4—3324no—
633C4—3322no—
634C4—834no—
635C4—1662no—
636C4—1662no—
637C4#1912818314yes—
638C2—3328no—
639C2—3322no—
640C2#518×D8311yescharacteristic
641C2C2×#1912818311yescharacteristic
642C2#2492066511yescharacteristic
643C2#2492183212yes—
644C2#2492184912yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2331C2
socle, cosocleC23×C83C25
prime radicalC2×#19128252C2
hypercenter#2331D83
hypocenterC83#34367
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2331

Complements

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

NH ≅ G/Ntype
2. #24910618C2semidirect
3. #24910503C2semidirect
4. #24910500C2semidirect
5. #24910408C2semidirect
6. #24910387C2semidirect
7. #24910364C2semidirect
8. #24910275C2semidirect
9. #24910212C2semidirect
10. #24910188C2semidirect
11. #24910186C2semidirect
12. #24910054C2semidirect
13. #24909869C2semidirect
14. #24909855C2semidirect
15. C83×#2331C2semidirect
17. #19128252C4semidirect
77. C2×#13050591C2×C4semidirect
89. #14581781C2×C4semidirect
90. #14581767C2×C4semidirect
91. #14581767C2×C4semidirect
94. #14581754C2×C4semidirect
95. #14581754C2×C4semidirect
123. C42×D83C2×C4semidirect
135. C4⋊Q8×C83C2×C4semidirect
332. #2331D83semidirect
338. C83#34367semidirect
412. C4⋊Q8C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191282522
#249106182
#249105032
#249105002
#249104082
#249103872
#249103642
#249102752
#249102122
#249101882
#249101862
#249100542
#249098692
#249098552
C83×#23312
#3436783