Finite Group Databasebeta

← #35407110

Subgroups — #35407110

Order 21248.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127700C211yesmaximal, characteristic
2C2×#19127700C211yesmaximal, characteristic
3C2×C4×#13050574C211yesmaximal, characteristic, complemented
4#24909953C211yesmaximal, characteristic
5#24908159C212yesmaximal, complemented
6#24908152C212yesmaximal, complemented
7#24908145C212yesmaximal, complemented
8#24908138C212yesmaximal, complemented
9#24907182C211yesmaximal, characteristic, complemented
10#24907182C211yesmaximal, characteristic, complemented
11C2×#389×C83C211yesmaximal, characteristic, complemented
12C22×#13050590C2211yescharacteristic
13C22×#13050574C2211yescharacteristic
14C22×#13050574C2211yescharacteristic
15C2×#14581751C2211yescharacteristic
16#19127700—24no—
17#19127700C418yescomplemented
18C2×#14581715C2212yes—
19C2×#14581715C2212yes—
20C2×#14581712C2212yes—
21C2×#14581712C2212yes—
22C4×#13050574—24no—
23C2×#14581703C2212yes—
24C2×C4×C83⋊C8C2212yes—
25#19127607C2211yescharacteristic
26#19127607C2211yescharacteristic
27#19127603C2211yescharacteristic
28#19127603C2211yescharacteristic
29#19127580—21no—
30#19127580—21no—
31C4×(C2×D83)⋊C4—21no—
32C4×(C2×D83)⋊C4—21no—
33#19127463C2212yes—
34#19127463C2212yes—
35#19127463C2212yes—
36#19127463C2212yes—
37C22×C4⋊C4×C83—21no—
38C22×C4⋊C4×C83C2211yescharacteristic
39C22×C4⋊C4×C83C2211yescharacteristic
40C22×C42×C83C2211yescharacteristic
41#389×C83—21no—
42#389×C83—21no—
43#389×C83C2212yes—
44#389×C83C2212yes—
45C2×#95×C83—21no—
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—
56C2×#13050574—22no—
57C2×#13050574—22no—
58C2×#13050574—22no—
59C2×#13050574—22no—
60C2×#13050574C2×C414yes—
61C2×#13050574D414yes—
62C22×C83⋊C8C2312yes—
63C22×C83⋊C8C2312yes—
64#14581751—22no—
65#14581751—22no—
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C22⋊Dic83—21no—
69C2×(C2×D83)⋊C4—21no—
70C2×(C2×D83)⋊C4—21no—
71C2×(C2×D83)⋊C4—21no—
72C2×(C2×D83)⋊C4—21no—
73C2×C4×Dic83—21no—
74#14581715—24no—
75#14581715C2×C418yescomplemented
76#14581712—24no—
77#14581712C2×C418yescomplemented
78#14581703—24no—
79C4×C83⋊C8—24no—
80C83⋊#95—21no—
81C83⋊#95—21no—
82C83⋊#95—21no—
83C83⋊#95—21no—
84C23×C4×C83—21no—
85C23×C4×C83—21no—
86C23×C4×C83D411yescharacteristic
87C23×C4×C83D411yescharacteristic
88C23×C4×C83C2311yescharacteristic
89C2×C4⋊C4×C83—21no—
90C2×C4⋊C4×C83—22no—
91C2×C4⋊C4×C83—22no—
92C2×C4⋊C4×C83—22no—
93C2×C4⋊C4×C83—22no—
94C2×C4⋊C4×C83—22no—
95C2×C4⋊C4×C83—21no—
96C2×C4⋊C4×C83—44no—
97C2×C4⋊C4×C83C2×C414yes—
98C2×C4⋊C4×C83C2312yes—
99C2×C4⋊C4×C83C2312yes—
100C2×C42×C83—22no—
101C2×C42×C83—22no—
102C2×C42×C83D411yescharacteristic
103C2×C42×C83D411yescharacteristic
104C2×C42×C83C2311yescharacteristic
105C2×C42×C83C2311yescharacteristic
106#95×C83—22no—
107#95×C83—22no—
108C23×D83—21no—
109#13050590—28no—
110#13050590—28no—
111#13050590—44no—
112#13050590—44no—
113C2×Dic166—21no—
114C2×Dic166—21no—
115C2×Dic166—21no—
116C2×Dic166—21no—
117C2×Dic166—21no—
118C2×Dic166C22×C412yes—
119C2×D332—21no—
120C2×D332—21no—
121C2×D332—21no—
122C2×D332—21no—
123C2×D332—21no—
124C2×D332C22×C412yes—
125C2×C22⋊D83—24no—
126C2×C22⋊D83—42no—
127C2×C22⋊D83—42no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C2×C4×D83—24no—
132C2×C4×D83—42no—
133C2×C4×D83—42no—
134#13050574—28no—
135#13050574—28no—
136C2×C83⋊C8—22no—
137C2×C83⋊C8—22no—
138C2×C83⋊C8—22no—
139C2×C83⋊C8—22no—
140C2×C83⋊C8C22×C414yes—
141C2×C83⋊C8C2×D414yes—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C22⋊Dic83—42no—
147(C2×D83)⋊C4—42no—
148(C2×D83)⋊C4—42no—
149(C2×D83)⋊C4—42no—
150(C2×D83)⋊C4—42no—
151C4×Dic83—42no—
152C24×C83C2×D411yescharacteristic
153C22×C4×C83—24no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—41no—
161C22×C4×C83—41no—
162C22×C4×C83—42no—
163C22×C4×C83—42no—
164C22×C4×C83—42no—
165C22×C4×C83—42no—
166C22×C4×C83—42no—
167C22×C4×C83C22×C412yes—
168C22×C4×C83C2×D412yes—
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83#4112yes—
175C22×C4×C83#4112yes—
176C22×C4×C83#4112yes—
177C22×C4×C83#4112yes—
178C22×C4×C83C2411yescharacteristic
179C4⋊C4×C83—24no—
180C4⋊C4×C83—22no—
181C4⋊C4×C83—22no—
182C4⋊C4×C83—22no—
183C4⋊C4×C83—44no—
184C4⋊C4×C83—42no—
185C4⋊C4×C83—44no—
186C4⋊C4×C83C22×C414yes—
187C42×C83—22no—
188C42×C83—22no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—21no—
192C42×C83—21no—
193C23×C83—21no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83—21no—
197C23×C83C4×D412yes—
198C23×C83#12112yes—
199C23×C83#12412yes—
200C23×C83C22×D411yescharacteristic
201C22×D83—22no—
202C22×D83—41no—
203C22×D83—41no—
204C22×D83—41no—
205C22×D83—41no—
206C22×D83—41no—
207C22×D83—41no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—24no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—44no—
220C2×C4×C83—44no—
221C2×C4×C83—48no—
222C2×C4×C83—42no—
223C2×C4×C83—42no—
224C2×C4×C83—42no—
225C2×C4×C83—42no—
226C2×C4×C83—42no—
227C2×C4×C83—84no—
228C2×C4×C83#11714yes—
229C2×C4×C83C4×D412yes—
230C2×C4×C83C4×D414yes—
231C2×C4×C83#12112yes—
232C2×C4×C83#12314yes—
233C2×C4×C83#12412yes—
234C2×C4×C83#12614yes—
235C2×C4×C83AGL(1,ℤ/8)12yes—
236C2×C4×C83#13712yes—
237C2×C4×C83C23×C411yescharacteristic
238C2×C4×C83C22×D411yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C22⋊D83—48no—
244C2×Dic83—22no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253D332—22no—
254D332—22no—
255D332—41no—
256D332—41no—
257C4×D83—48no—
258Dic166—22no—
259Dic166—22no—
260Dic166—41no—
261Dic166—41no—
262C83⋊C8—24no—
263C83⋊C8—24no—
264C22×C83—22no—
265C22×C83—22no—
266C22×C83—22no—
267C22×C83—22no—
268C22×C83—22no—
269C22×C83—22no—
270C22×C83#38814yes—
271C22×C83C2×C4×D411yescharacteristic
272C22×C83C2×#12111yescharacteristic
273C22×C83C2×#12411yescharacteristic
274C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C83C2×#13711yescharacteristic
276C22×C83#57511yescharacteristic
277C22×C83#57511yescharacteristic
278C2×D83—42no—
279C2×D83—42no—
280C2×D83—42no—
281C2×D83—81no—
282C2×D83—81no—
283C2×D83—81no—
284C2×D83—81no—
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—44no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83—82no—
291C4×C83#38814yes—
292C4×C83C2×#11711yescharacteristic
293C4×C83C2×C4×D411yescharacteristic
294C4×C83C2×#12311yescharacteristic
295C4×C83C2×#12611yescharacteristic
296Dic83—42no—
297Dic83—81no—
298Dic83—81no—
299#21093—831nomaximal, Sylow
300C2×C83—24no—
301C2×C83C2×#38811yescharacteristic
302C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
303C2×C83C4×#13711yescharacteristic
304C2×C83#305511yescharacteristic
305C2×C83#305611yescharacteristic
306C2×C83#314111yescharacteristic
307C2×C83#314211yescharacteristic
308D83—82no—
309C2×#417—831no—
310C2×#417—831no—
311C2×C4×M16—831no—
312#2297—831no—
313C2×#389D8311yescharacteristic, complemented
314#1936—832no—
315#1935—832no—
316#1934—832no—
317#1933—832no—
318#1736—831no—
319#1736—831no—
320C83#2109311yescharacteristic, complemented, Sylow
321C22×#41—831no—
322C22×M16—831no—
323C22×M16—831no—
324C2×#117—831no—
325C22×C4⋊C4—21no—
326C22×C4⋊C4C2×D8311yescharacteristic
327C22×C4⋊C4C2×D8311yescharacteristic
328C22×C42C2×D8311yescharacteristic
329#417—838no—
330#417—1664no—
331C2×#103—832no—
332C2×#103—832no—
333C2×#102—832no—
334C2×#102—832no—
335C4×M16—1664no—
336C2×#97—832no—
337C2×C4×C8—832no—
338#391—831no—
339#391—831no—
340#390—831no—
341#390—831no—
342#389—21no—
343#389—21no—
344#389C2×D8312yes—
345#389C2×D8312yes—
346#386—1661no—
347#386—1661no—
348C4×C22⋊C4—1661no—
349C4×C22⋊C4—1661no—
350C2×#95—21no—
351#340—832no—
352#340—832no—
353#340—832no—
354#340—832no—
355C2×#41—834no—
356C2×#41—1662no—
357C2×#41—1662no—
358C2×#41—1662no—
359C2×#41—1662no—
360C2×#41—1662no—
361C22×Q8—831no—
362C22×D4—831no—
363C22×D4—1661no—
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—1661no—
367C23×C4C22⋊D8311yescharacteristic
368C23×C4C22⋊D8311yescharacteristic
369C23×C4C22×D8311yescharacteristic
370C2×M16—834no—
371C2×M16—834no—
372C2×M16—1662no—
373C2×M16—1662no—
374C2×M16—1662no—
375C2×M16—1662no—
376C22×C8—832no—
377C22×C8—832no—
378#117—1662no—
379#117—1662no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—44no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4C4×D8314yes—
391C2×C4⋊C4C22×D8312yes—
392C2×C4⋊C4C22×D8312yes—
393C2×C22⋊C4—1661no—
394C2×C22⋊C4—1661no—
395C2×C22⋊C4—1661no—
396C2×C22⋊C4—1661no—
397C2×C22⋊C4—1661no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—1661no—
401C2×C42C22⋊D8311yescharacteristic
402C2×C42C22⋊D8311yescharacteristic
403C2×C42C22×D8311yescharacteristic
404C2×C42C22×D8311yescharacteristic
405#103—838no—
406#103—1664no—
407#102—838no—
408#102—1664no—
409#97—1664no—
410C4×C8—1664no—
411#95—22no—
412#95—22no—
413#95—1661no—
414#95—1661no—
415#95—1661no—
416#95—1661no—
417C24—1661no—
418C24C2×C22⋊D8311yescharacteristic
419#41—3324no—
420#41—3324no—
421#41—1668no—
422#41—1668no—
423C2×Q8—832no—
424C2×Q8—1661no—
425C2×Q8—1661no—
426C2×Q8—1661no—
427C2×Q8—1661no—
428C2×Q8—1661no—
429C2×D4—3322no—
430C2×D4—3322no—
431C2×D4—832no—
432C2×D4—1664no—
433C2×D4—1661no—
434C2×D4—1661no—
435C2×D4—1661no—
436C2×D4—1661no—
437C2×D4—1661no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—3322no—
453C22×C4—3322no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4—1661no—
457C22×C4—1664no—
458C22×C4C2×C4×D8312yes—
459C22×C4D4×D8312yes—
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4#1305058912yes—
466C22×C4#1305059012yes—
467C22×C4#1305059112yes—
468C22×C4#1305059112yes—
469C22×C4C23×D8311yescharacteristic
470M16—1668no—
471M16—1668no—
472C2×C8—834no—
473C2×C8—834no—
474C2×C8—1662no—
475C2×C8—1662no—
476C2×C8—1662no—
477C2×C8—1662no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4C2×C4×D8314yes—
490C22⋊C4—3322no—
491C22⋊C4—3322no—
492C22⋊C4—3322no—
493C22⋊C4—3322no—
494C22⋊C4—3322no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—3322no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—3321no—
507C23—3321no—
508C23—3321no—
509C23—3321no—
510C23—3321no—
511C23—3321no—
512C23—1662no—
513C23C4×C22⋊D8312yes—
514C23#1458176412yes—
515C23#1458178112yes—
516C23C22×C22⋊D8311yescharacteristic
517Q8—3321no—
518Q8—3321no—
519Q8—1662no—
520Q8—1662no—
521D4—3328no—
522D4—3321no—
523D4—3321no—
524D4—1662no—
525D4—1662no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—84no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3328no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—1662no—
556C2×C4#1458175014yes—
557C2×C4C4×C22⋊D8312yes—
558C2×C4#1458175414yes—
559C2×C4#1458176412yes—
560C2×C4#1458177714yes—
561C2×C4#1458178112yes—
562C2×C4#1458178714yes—
563C2×C4#1458181812yes—
564C2×C4#1458182312yes—
565C2×C4C22×C4×D8311yescharacteristic
566C2×C4C2×D4×D8311yescharacteristic
567C2×C4C2×#1305058911yescharacteristic
568C2×C4C2×#1305059011yescharacteristic
569C2×C4C2×#1305059111yescharacteristic
570C2×C4C2×#1305059111yescharacteristic
571C8—1664no—
572C8—1664no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—3322no—
580C22—3322no—
581C22—3322no—
582C22—6641no—
583C22—6641no—
584C22—6641no—
585C22—6641no—
586C22#1912759114yes—
587C22C2×C4×C22⋊D8311yescharacteristic
588C22C2×#1458176411yescharacteristic
589C22C2×#1458178111yescharacteristic
590C22C2×#1458181811yescharacteristic
591C22C2×#1458182311yescharacteristic
592C22#1912849911yescharacteristic
593C22#1912849911yescharacteristic
594C4—22no—
595C4—22no—
596C4—44no—
597C4—41no—
598C4—41no—
599C4—82no—
600C4—3322no—
601C4—6641no—
602C4—6641no—
603C4#1912759114yes—
604C4C2×#1458175011yescharacteristic
605C4C2×#1458175411yescharacteristic
606C4C2×#1458177711yescharacteristic
607C4C2×#1458178711yescharacteristic
608C2—24no—
609C2—6642no—
610C2C2×#1912759111yescharacteristic
611C2C4×#1458181811yescharacteristic
612C2C4×#1458182311yescharacteristic
613C2#2491707811yescharacteristic
614C2#2491708911yescharacteristic
615C2#2491785911yescharacteristic
616C2#2491786611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#389×C83C2
socleC23×C83C22×D4
prime radicalC22×#13050590C22
cosocleC22×C4×C83C24
hypercenterC2×#389D83
hypocenterC83#21093
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. C2×C4×#13050574C2semidirect
5. #24908159C2semidirect
6. #24908152C2semidirect
7. #24908145C2semidirect
8. #24908138C2semidirect
9. #24907182C2semidirect
10. #24907182C2semidirect
11. C2×#389×C83C2semidirect
17. #19127700C4semidirect
75. #14581715C2×C4semidirect
77. #14581712C2×C4semidirect
313. C2×#389D83semidirect
320. C83#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#191277002
C2×C4×#130505742
#249099532
#249081592
#249081522
#249081452
#249081382
#249071822
C2×#389×C832
#2109383