Finite Group Databasebeta

← #35564556

Subgroups — #35564556

Order 21248.

2342 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128169C211yesmaximal, characteristic
2#24910433C214yesmaximal, complemented
3#24910425C214yesmaximal, complemented
4#24910402C214yesmaximal, complemented
5#24910398C214yesmaximal, complemented
6#24910073C211yesmaximal, characteristic, complemented
7#24910072C211yesmaximal, characteristic, complemented
8#24910071C211yesmaximal, characteristic, complemented
9#24910052C212yesmaximal
10#24909868C211yesmaximal, characteristic, complemented
11#24909859C212yesmaximal, complemented
12#24909858C212yesmaximal, complemented
13#382×D83C211yesmaximal, characteristic, complemented
14#24909817C212yesmaximal, complemented
15C83×#2314C211yesmaximal, characteristic, complemented
16C22×#13050590C2211yescharacteristic
17#19128169C4116yescomplemented
18C2×#14581757C2212yes—
19C2×#14581754C2212yes—
20C2×(C22⋊D83)⋊C4C2212yes—
21C2×(C22⋊D83)⋊C4C2212yes—
22C2×#14581750C2212yes—
23C2×#14581749C2211yescharacteristic
24C2×#14581749C2211yescharacteristic
25C2×C4⋊C4×D83C2212yes—
26C22×C4×Dic83C2211yescharacteristic
27C2×C42×D83C2211yescharacteristic
28C2×C42×D83C2211yescharacteristic
29#19127605C2214yes—
30#19127601C2214yes—
31#19127594C2214yes—
32#19127593C2214yes—
33#19127591C2214yes—
34#19127588C2214yes—
35#19127587C2214yes—
36#19127582C2214yes—
37#19127579C2214yes—
38#19127577C2214yes—
39#19127571C2211yescharacteristic
40#19127568C2214yes—
41#19127568C2214yes—
42#19127567C2211yescharacteristic
43#19127566C2212yes—
44#19127566C2212yes—
45#19127565C2214yes—
46#19127563C2214yes—
47#19127562C2214yes—
48#19127561C2212yes—
49#19127558C2212yes—
50#19127558C2212yes—
51#19127558C2214yes—
52C4⋊C4×Dic83C2212yes—
53C4⋊C4×Dic83C2212yes—
54C4⋊C4×Dic83C2214yes—
55C4×Dic83⋊C4C2214yes—
56C4×C4⋊Dic83C2212yes—
57#19127552C2214yes—
58C22⋊C4×Dic83C2214yes—
59C4×(C2×D83)⋊C4C2214yes—
60C4×(C2×D83)⋊C4C2214yes—
61#19127548C2214yes—
62#19127547C2214yes—
63#19127546C2214yes—
64#95×D83C2214yes—
65C2×#117×C83C2211yescharacteristic
66#388×C83C2214yes—
67#387×C83C2214yes—
68#384×C83C2211yescharacteristic
69#382×C83C2211yescharacteristic
70#380×C83C2212yes—
71C4×C4⋊C4×C83C2212yes—
72C2×#13050590—24no—
73C2×#13050590—24no—
74C2×#13050590C2×C418yes—
75C22×Dic166C2311yescharacteristic
76C22×D332C2311yescharacteristic
77C22×C22⋊D83C2311yescharacteristic
78C22×C22⋊D83C2311yescharacteristic
79C23×Dic83C2312yes—
80C22×C4×D83C2311yescharacteristic
81C22×C4×D83C2314yes—
82C22×C4×D83C2312yes—
83C22×C4×D83C2312yes—
84C22×C4×D83C2311yescharacteristic
85#14581757C2×C4116yescomplemented
86#14581754C2×C4116yes—
87(C22⋊D83)⋊C4C2×C4116yes—
88(C22⋊D83)⋊C4C2×C4116yescomplemented
89#14581750—28no—
90#14581749C2×C418yes—
91#14581749C2×C418yescomplemented
92C2×C4⋊Dic83C2312yes—
93C2×C4⋊Dic83C2312yes—
94C2×C4⋊Dic83C2314yes—
95C2×Dic83⋊C4C2314yes—
96C2×Dic83⋊C4C2312yes—
97C2×Dic83⋊C4C2312yes—
98C4⋊C4×D83—28no—
99C2×C22⋊Dic83C2314yes—
100C2×(C2×D83)⋊C4C2314yes—
101C2×(C2×D83)⋊C4C2314yes—
102C2×(C2×D83)⋊C4C2314yes—
103C2×(C2×D83)⋊C4C2312yes—
104C2×(C2×D83)⋊C4C2312yes—
105C2×C4×Dic83C2×C418yescomplemented
106C2×C4×Dic83C2314yes—
107C2×C4×Dic83C2314yes—
108C2×C4×Dic83C2314yes—
109C2×C4×Dic83C2314yes—
110C2×C4×Dic83C2311yescharacteristic
111C2×C4×Dic83C2312yes—
112C2×C4×Dic83C2311yescharacteristic
113C2×C4×Dic83C2312yes—
114C2×C4×Dic83C2312yes—
115C42×D83C2×C418yes—
116C42×D83C2×C418yescomplemented
117C83⋊#95C2314yes—
118C83⋊#95C2314yes—
119C83⋊#95C2314yes—
120C83⋊#95C2314yes—
121C83⋊#95C2314yes—
122C83⋊#95C2314yes—
123C83⋊#95C2314yes—
124C83⋊#95C2314yes—
125C83⋊#95C2314yes—
126C83⋊#95C2314yes—
127C83⋊#95C2314yes—
128C23×C4×C83C2311yescharacteristic
129#117×C83C2×C418yescomplemented
130C2×C4⋊C4×C83C2312yes—
131C2×C4⋊C4×C83C2312yes—
132C2×C4⋊C4×C83C2314yes—
133C2×C4⋊C4×C83C2312yes—
134C2×C22⋊C4×C83C2314yes—
135C2×C22⋊C4×C83C2312yes—
136C2×C42×C83C2314yes—
137C2×C42×C83C2311yescharacteristic
138C2×C42×C83C2311yescharacteristic
139#95×C83C2314yes—
140#95×C83C2314yes—
141#95×C83C2314yes—
142C23×D83C2411yescharacteristic
143C23×D83C2411yescharacteristic
144#13050590—232no—
145C2×Dic166—22no—
146C2×Dic166—22no—
147C2×Dic166C22×C414yes—
148C2×D332—22no—
149C2×D332—22no—
150C2×D332C22×C414yes—
151C2×C22⋊D83—22no—
152C2×C22⋊D83—22no—
153C2×C22⋊D83—22no—
154C2×C22⋊D83—22no—
155C2×C22⋊D83C22×C414yes—
156C2×C22⋊D83C22×C414yes—
157C22×Dic83—42no—
158C22×Dic83—42no—
159C22×Dic83C22×C418yes—
160C22×Dic83C2414yes—
161C22×Dic83C2414yes—
162C22×Dic83C2412yes—
163C22×Dic83C2411yescharacteristic
164C22×Dic83C2412yes—
165C22×Dic83C2411yescharacteristic
166C2×C4×D83—22no—
167C2×C4×D83—22no—
168C2×C4×D83—28no—
169C2×C4×D83—42no—
170C2×C4×D83—42no—
171C2×C4×D83—42no—
172C2×C4×D83—42no—
173C2×C4×D83—44no—
174C2×C4×D83—44no—
175C2×C4×D83C22×C414yes—
176C2×C4×D83C22×C418yes—
177C2×C4×D83C22×C418yes—
178C2×C4×D83C22×C414yes—
179C2×C4×D83#4114yes—
180C2×C4×D83#4114yes—
181C4⋊Dic83—24no—
182C4⋊Dic83—24no—
183C4⋊Dic83—44no—
184Dic83⋊C4—28no—
185Dic83⋊C4C22×C418yes—
186Dic83⋊C4C22×C418yes—
187C22⋊Dic83—44no—
188(C2×D83)⋊C4—28no—
189(C2×D83)⋊C4—44no—
190(C2×D83)⋊C4—44no—
191(C2×D83)⋊C4C22×C418yes—
192(C2×D83)⋊C4C22×C418yes—
193C4×Dic83—28no—
194C4×Dic83—44no—
195C4×Dic83—44no—
196C4×Dic83—44no—
197C4×Dic83C22×C414yes—
198C4×Dic83C22×C418yes—
199C4×Dic83C22×C418yes—
200C4×Dic83C22×C414yes—
201C4×Dic83C22×C418yes—
202C24×C83C2411yescharacteristic
203C22×C4×C83—22no—
204C22×C4×C83—22no—
205C22×C4×C83C22×C414yes—
206C22×C4×C83C2414yes—
207C22×C4×C83C2414yes—
208C22×C4×C83C2412yes—
209C22×C4×C83C2412yes—
210C22×C4×C83C2411yescharacteristic
211C22×C4×C83C2411yescharacteristic
212C4⋊C4×C83—24no—
213C4⋊C4×C83—24no—
214C4⋊C4×C83—44no—
215C4⋊C4×C83C22×C418yes—
216C22⋊C4×C83—44no—
217C22⋊C4×C83C22×C418yes—
218C42×C83—44no—
219C42×C83C22×C414yes—
220C42×C83C22×C414yes—
221C23×C83—21no—
222C23×C83—21no—
223C23×C83—22no—
224C23×C83—22no—
225C23×C83C23×C412yes—
226C23×C83C2511yescharacteristic
227C22×D83—22no—
228C22×D83—22no—
229C22×D83—21no—
230C22×D83—21no—
231C22×D83—22no—
232C22×D83—22no—
233C22×D83C23×C412yes—
234C22×D83C23×C412yes—
235C22×D83C2×#4112yes—
236C22×D83C2×#4112yes—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—22no—
240C2×C4×C83—22no—
241C2×C4×C83—22no—
242C2×C4×C83—22no—
243C2×C4×C83—28no—
244C2×C4×C83—24no—
245C2×C4×C83—24no—
246C2×C4×C83—24no—
247C2×C4×C83—24no—
248C2×C4×C83—24no—
249C2×C4×C83—24no—
250C2×C4×C83C23×C414yes—
251C2×C4×C83C23×C414yes—
252C2×C4×C83C23×C412yes—
253C2×C4×C83C23×C412yes—
254C2×C4×C83C2×#4112yes—
255C2×C4×C83C2×#4112yes—
256C22⋊D83—28no—
257C22⋊D83—28no—
258C2×Dic83—21no—
259C2×Dic83—21no—
260C2×Dic83—22no—
261C2×Dic83—22no—
262C2×Dic83—22no—
263C2×Dic83—22no—
264C2×Dic83—24no—
265C2×Dic83—24no—
266C2×Dic83—24no—
267C2×Dic83—24no—
268C2×Dic83—24no—
269C2×Dic83—24no—
270C2×Dic83—48no—
271C2×Dic83C23×C414yes—
272C2×Dic83C23×C414yes—
273C2×Dic83C23×C412yes—
274C2×Dic83C23×C412yes—
275C2×Dic83C2×#4112yes—
276C2×Dic83C2×#4112yes—
277D332—28no—
278C4×D83—28no—
279C4×D83—48no—
280C4×D83—48no—
281C4×D83—88no—
282C4×D83#117116yes—
283Dic166—28no—
284C22×C83—22no—
285C22×C83—24no—
286C22×C83—42no—
287C22×C83—41no—
288C22×C83—41no—
289C22×C83C24×C411yescharacteristic
290C22×C83C22×#4111yescharacteristic
291C22×C83C22×#4111yescharacteristic
292C22×C83#58512yes—
293C22×C83#58512yes—
294C2×D83—24no—
295C2×D83—24no—
296C2×D83—22no—
297C2×D83—22no—
298C2×D83—22no—
299C2×D83—24no—
300C2×D83—42no—
301C2×D83—41no—
302C2×D83—41no—
303C2×D83C2×#11714yes—
304C4×C83—22no—
305C4×C83—24no—
306C4×C83—24no—
307C4×C83—44no—
308C4×C83—44no—
309C4×C83C2×#11714yes—
310Dic83—22no—
311Dic83—24no—
312Dic83—24no—
313Dic83—44no—
314Dic83—44no—
315Dic83C2×#11714yes—
316#34189—831nomaximal, Sylow
317C2×C83—42no—
318C2×C83C22×#11711yescharacteristic
319C2×C83#342812yes—
320C2×C83#351012yes—
321C2×C83#351211yescharacteristic
322C2×C83#351211yescharacteristic
323D83—24no—
324D83—42no—
325C2×C4×#41—831no—
326#2357—834no—
327#2356—834no—
328#2353—834no—
329#2353—834no—
330#2314—831no—
331#2314—831no—
332#2314—831no—
333#2314D8311yescharacteristic, complemented
334#2311—832no—
335#2285—831no—
336#2283—832no—
337#2283—832no—
338C2×#382—831no—
339#2274—832no—
340C83#3418911yescharacteristic, complemented, Sylow
341C22×#41—831no—
342C4×#41—8316no—
343C2×C4×Q8—832no—
344C2×C4×D4—832no—
345C2×C4×D4—832no—
346C2×C4×D4—832no—
347C2×#117—832no—
348C2×#117—831no—
349C2×#117—831no—
350C2×#117C2×D8311yescharacteristic
351C22×C4⋊C4—832no—
352C22×C42—831no—
353C22×C42—831no—
354C22×C42—831no—
355#391—834no—
356#390—834no—
357#388—834no—
358#388—834no—
359#388—834no—
360#388C2×D8314yes—
361#387—834no—
362#387—834no—
363#387—834no—
364#387C2×D8314yes—
365#386—834no—
366#386—834no—
367#384—831no—
368#384C2×D8311yescharacteristic
369#383—834no—
370#383—834no—
371#382—832no—
372#382—832no—
373#382—831no—
374#382—834no—
375#382C2×D8311yescharacteristic
376#381—834no—
377#381—834no—
378#380—832no—
379#380—832no—
380#380—832no—
381#380—834no—
382#380C2×D8312yes—
383C4×C4⋊C4—832no—
384C4×C4⋊C4—832no—
385C4×C4⋊C4—834no—
386C4×C4⋊C4—832no—
387C4×C4⋊C4—834no—
388C4×C4⋊C4—834no—
389C4×C4⋊C4C2×D8312yes—
390C4×C22⋊C4—834no—
391C4×C22⋊C4—834no—
392C4×C22⋊C4—834no—
393C4×C22⋊C4—834no—
394#376—834no—
395#376—834no—
396C2×#95—834no—
397C2×#41—838no—
398C2×#41—1664no—
399C2×#41—1664no—
400C22×Q8—831no—
401C22×D4—831no—
402C22×D4—831no—
403C22×D4—831no—
404C23×C4—832no—
405C23×C4—832no—
406C23×C4—832no—
407C23×C4—834no—
408C23×C4—831no—
409C23×C4—831no—
410C23×C4C22×D8311yescharacteristic
411C4×Q8—8316no—
412C4×D4—8316no—
413C4×D4—8316no—
414C4×D4—8316no—
415#117—838no—
416#117—838no—
417#117—1668no—
418#117C4×D8318yescomplemented
419C2×C4⋊C4—834no—
420C2×C4⋊C4—832no—
421C2×C4⋊C4—832no—
422C2×C4⋊C4—832no—
423C2×C4⋊C4—834no—
424C2×C4⋊C4—832no—
425C2×C4⋊C4—1668no—
426C2×C4⋊C4C22×D8312yes—
427C2×C4⋊C4C22×D8312yes—
428C2×C4⋊C4C22×D8314yes—
429C2×C4⋊C4C22×D8312yes—
430C2×C22⋊C4—834no—
431C2×C22⋊C4—834no—
432C2×C22⋊C4—834no—
433C2×C22⋊C4—832no—
434C2×C22⋊C4—832no—
435C2×C22⋊C4—834no—
436C2×C22⋊C4C22×D8314yes—
437C2×C22⋊C4C22×D8312yes—
438C2×C42—834no—
439C2×C42—834no—
440C2×C42—834no—
441C2×C42—838no—
442C2×C42—831no—
443C2×C42—832no—
444C2×C42—832no—
445C2×C42—832no—
446C2×C42—831no—
447C2×C42—838no—
448C2×C42—838no—
449C2×C42—834no—
450C2×C42C22×D8314yes—
451C2×C42C22×D8311yescharacteristic
452C2×C42C22×D8311yescharacteristic
453#95—834no—
454#95—834no—
455#95—834no—
456#95—834no—
457#95—834no—
458#95—834no—
459#95—834no—
460#95—834no—
461#95—834no—
462#95—834no—
463#95—834no—
464#95C22×D8314yes—
465#95C22×D8314yes—
466#95C22×D8314yes—
467C24—831no—
468C24—831no—
469C24C23×D8311yescharacteristic
470#41—16632no—
471C2×Q8—834no—
472C2×Q8—1662no—
473C2×Q8—1662no—
474C2×D4—834no—
475C2×D4—834no—
476C2×D4—834no—
477C2×D4—1662no—
478C2×D4—1662no—
479C2×D4—1662no—
480C2×D4—1662no—
481C2×D4—1662no—
482C2×D4—1662no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—3322no—
486C22×C4—3322no—
487C22×C4—3322no—
488C22×C4—3322no—
489C22×C4—3324no—
490C22×C4—3324no—
491C22×C4—3322no—
492C22×C4—3322no—
493C22×C4—838no—
494C22×C4—838no—
495C22×C4—838no—
496C22×C4—831no—
497C22×C4—834no—
498C22×C4—834no—
499C22×C4—834no—
500C22×C4—832no—
501C22×C4—831no—
502C22×C4—832no—
503C22×C4—834no—
504C22×C4—834no—
505C22×C4—834no—
506C22×C4—1668no—
507C22×C4—1662no—
508C22×C4—1662no—
509C22×C4C2×C4×D8314yes—
510C22×C4C23×D8314yes—
511C22×C4C23×D8314yes—
512C22×C4C23×D8311yescharacteristic
513C22×C4C23×D8312yes—
514C22×C4C23×D8312yes—
515C22×C4C23×D8311yescharacteristic
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—44no—
519C4⋊C4—3324no—
520C4⋊C4—838no—
521C4⋊C4—838no—
522C4⋊C4—1668no—
523C4⋊C4—1664no—
524C4⋊C4—1664no—
525C4⋊C4C2×C4×D8318yes—
526C22⋊C4—44no—
527C22⋊C4—3324no—
528C22⋊C4—3324no—
529C22⋊C4—3324no—
530C22⋊C4—838no—
531C22⋊C4—838no—
532C22⋊C4—1668no—
533C22⋊C4C2×C4×D8318yes—
534C42—44no—
535C42—3324no—
536C42—3324no—
537C42—3324no—
538C42—834no—
539C42—834no—
540C42—838no—
541C42—838no—
542C42—838no—
543C42—1668no—
544C42C2×C4×D8314yes—
545C42C2×C4×D8314yes—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—832no—
551C23—832no—
552C23—832no—
553C23—832no—
554C23—1661no—
555C23—1661no—
556C23—1662no—
557C23—1662no—
558C23—1662no—
559C23—1662no—
560C23C22×C4×D8312yes—
561C23C24×D8311yescharacteristic
562Q8—1668no—
563D4—1668no—
564D4—1668no—
565D4—1668no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—3328no—
580C2×C4—3328no—
581C2×C4—3328no—
582C2×C4—832no—
583C2×C4—832no—
584C2×C4—832no—
585C2×C4—8316no—
586C2×C4—834no—
587C2×C4—834no—
588C2×C4—832no—
589C2×C4—6648no—
590C2×C4—1664no—
591C2×C4—1664no—
592C2×C4—1664no—
593C2×C4—1664no—
594C2×C4—1664no—
595C2×C4—1664no—
596C2×C4—1661no—
597C2×C4—1661no—
598C2×C4—1662no—
599C2×C4—1662no—
600C2×C4—1662no—
601C2×C4—1662no—
602C2×C4—1668no—
603C2×C4C22×C4×D8312yes—
604C2×C4C22×C4×D8314yes—
605C2×C4C22×C4×D8314yes—
606C2×C4C22×C4×D8312yes—
607C2×C4#41×D8312yes—
608C2×C4#41×D8312yes—
609C22—22no—
610C22—24no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—3322no—
615C22—3321no—
616C22—3321no—
617C22—834no—
618C22—1662no—
619C22—1664no—
620C22—1664no—
621C22—1662no—
622C22—1662no—
623C22—1664no—
624C22C23×C4×D8311yescharacteristic
625C22C2×#41×D8311yescharacteristic
626C22C2×#41×D8311yescharacteristic
627C22#1912853612yes—
628C22#1912853812yes—
629C4—22no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—44no—
634C4—3324no—
635C4—3324no—
636C4—834no—
637C4—1662no—
638C4—1664no—
639C4—1664no—
640C4#117×D8314yes—
641C2—42no—
642C2—3322no—
643C2—1664no—
644C2C2×#117×D8311yescharacteristic
645C2#2492066312yes—
646C2#2492135512yes—
647C2#2492138211yescharacteristic
648C2#2492138611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2314C2
socle, cosocleC23×C83C25
prime radicalC22×#13050590C22
hypercenter#2314D83
hypocenterC83#34189
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #24910433C2semidirect
3. #24910425C2semidirect
4. #24910402C2semidirect
5. #24910398C2semidirect
6. #24910073C2semidirect
7. #24910072C2semidirect
8. #24910071C2semidirect
10. #24909868C2semidirect
11. #24909859C2semidirect
12. #24909858C2semidirect
13. #382×D83C2semidirect
14. #24909817C2semidirect
15. C83×#2314C2semidirect
17. #19128169C4semidirect
85. #14581757C2×C4semidirect
88. (C22⋊D83)⋊C4C2×C4semidirect
91. #14581749C2×C4semidirect
105. C2×C4×Dic83C2×C4semidirect
116. C42×D83C2×C4semidirect
129. #117×C83C2×C4semidirect
333. #2314D83semidirect
340. C83#34189semidirect
418. #117C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281692
#249104332
#249104252
#249104022
#249103982
#249100732
#249100722
#249100712
#249100522
#249098682
#249098592
#249098582
#382×D832
#249098172
C83×#23142
#3418983