Finite Group Databasebeta

← #35467400

Subgroups — #35467400

Order 21248.

780 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128008C211yesmaximal, characteristic
2C2×C4×#13050580C211yesmaximal, characteristic
3#24911324C211yesmaximal, characteristic
4#24909007C211yesmaximal, characteristic
5#24908935C211yesmaximal, characteristic
6#24908841C211yesmaximal, characteristic
7#24908801C211yesmaximal, characteristic
8#24908188C211yesmaximal, characteristic
9#24908022C211yesmaximal, characteristic
10#24907804C211yesmaximal, characteristic
11#24907804C211yesmaximal, characteristic
12#24907731C211yesmaximal, characteristic
13#24907731C211yesmaximal, characteristic
14#24907716C211yesmaximal, characteristic
15Q8×C4⋊C4×C83C211yesmaximal, characteristic
16C22×#13050580C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×C4×Dic166C2211yescharacteristic
19#19128008—24no—
20C4×#13050580—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×#14581716C2211yescharacteristic
28C2×#14581716C2211yescharacteristic
29C2×#14581716C2211yescharacteristic
30C2×#14581715C2211yescharacteristic
31C2×#14581715C2211yescharacteristic
32C2×#14581715C2211yescharacteristic
33C2×C4×C83⋊C8C2211yescharacteristic
34#19127642—21no—
35#19127632—21no—
36#19127607C2211yescharacteristic
37#19127607C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127599C2211yescharacteristic
40#19127596—21no—
41#19127583—21no—
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4×Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52C2×C4×Q8×C83C2211yescharacteristic
53#389×C83—21no—
54#389×C83—21no—
55#389×C83C2211yescharacteristic
56#389×C83C2211yescharacteristic
57#384×C83—21no—
58#384×C83C2211yescharacteristic
59C4×C4⋊C4×C83—21no—
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×Dic166C2311yescharacteristic
62C2×#13050580—22no—
63C2×#13050580—22no—
64C2×#13050580Q814yes—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69#14581784—22no—
70#14581784—41no—
71C4×Dic166—22no—
72C4×Dic166—41no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×C4×Dic83—21no—
82#14581725—22no—
83#14581725—22no—
84#14581723—22no—
85#14581723—22no—
86#14581723—41no—
87C4⋊(C83⋊C8)—22no—
88#14581716—22no—
89#14581716—22no—
90#14581716D414yes—
91#14581715—22no—
92#14581715—22no—
93#14581715—41no—
94C4×C83⋊C8—22no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C22×Q8×C83C2311yescharacteristic
102C4×Q8×C83—22no—
103C4×Q8×C83—22no—
104C4×Q8×C83—22no—
105C4×Q8×C83—22no—
106C4×Q8×C83—41no—
107C4×Q8×C83—41no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83—21no—
115C2×C4⋊C4×C83D411yescharacteristic
116C2×C4⋊C4×C83D411yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C42×C83—21no—
121C2×C42×C83—21no—
122C2×C42×C83—21no—
123C2×C42×C83C2311yescharacteristic
124C2×C42×C83C2311yescharacteristic
125C2×C42×C83C2311yescharacteristic
126#95×C83—21no—
127#95×C83—21no—
128#95×C83D411yescharacteristic
129#95×C83D411yescharacteristic
130C2×Dic166—21no—
131C2×Dic166—21no—
132C2×Dic166—21no—
133C2×Dic166—21no—
134C2×Dic166—21no—
135C2×Dic166C2×Q812yes—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—21no—
140#13050580—28no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8C2×D412yes—
152C2×C83⋊C8C2×Q812yes—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83C2×D412yes—
159Dic83⋊C4—42no—
160Dic83⋊C4—42no—
161Dic83⋊C4—42no—
162Dic83⋊C4—42no—
163Dic83⋊C4—42no—
164C4×Dic83—42no—
165C2×Q8×C83—21no—
166C2×Q8×C83—21no—
167C2×Q8×C83—21no—
168C2×Q8×C83C2×D412yes—
169C2×Q8×C83C2×Q812yes—
170C2×Q8×C83#4112yes—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2411yescharacteristic
182C4⋊C4×C83—24no—
183C4⋊C4×C83—22no—
184C4⋊C4×C83—22no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—42no—
191C4⋊C4×C83—42no—
192C4⋊C4×C83—42no—
193C4⋊C4×C83—81no—
194C4⋊C4×C83—81no—
195C4⋊C4×C83—81no—
196C4⋊C4×C83#4112yes—
197C42×C83—24no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83—21no—
204C42×C83—42no—
205C42×C83—42no—
206C23×C83C22×D411yescharacteristic
207Q8×C83—22no—
208Q8×C83—22no—
209Q8×C83—22no—
210Q8×C83C22⋊Q814yes—
211C2×C4×C83—21no—
212C2×C4×C83—22no—
213C2×C4×C83—21no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—22no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83#13512yes—
240C2×C4×C83#13712yes—
241C2×C4×C83C22×D411yescharacteristic
242C2×C4×C83C22×Q811yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83C2×#4111yescharacteristic
245C2×C4×C83H(5,𝔽2)11yescharacteristic
246C2×C4×C832−1+411yescharacteristic
247C2×Dic83—22no—
248C2×Dic83—22no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259Dic166—22no—
260Dic166—22no—
261Dic166—41no—
262Dic166—41no—
263C83⋊C8—22no—
264C83⋊C8—22no—
265C83⋊C8—41no—
266C83⋊C8—41no—
267C22×C83#52511yescharacteristic
268C22×C83D4211yescharacteristic
269C22×C83D4×Q811yescharacteristic
270C22×C83C2×#13511yescharacteristic
271C22×C83C2×#13711yescharacteristic
272C22×C83#57711yescharacteristic
273C22×C83#57811yescharacteristic
274C4×C83—22no—
275C4×C83—24no—
276C4×C83—22no—
277C4×C83—41no—
278C4×C83—42no—
279C4×C83—41no—
280C4×C83—41no—
281C4×C83—41no—
282C4×C83—81no—
283C4×C83—81no—
284C4×C83C2×C22⋊Q811yescharacteristic
285C4×C83#53911yescharacteristic
286C4×C83#54711yescharacteristic
287C4×C83#55411yescharacteristic
288Dic83—42no—
289Dic83—42no—
290Dic83—81no—
291Dic83—81no—
292#26371—831nomaximal, Sylow
293C2×C83#244911yescharacteristic
294C2×C83#310811yescharacteristic
295C2×C83#312011yescharacteristic
296C2×C83#327011yescharacteristic
297C2×C83#328411yescharacteristic
298C2×C83#338311yescharacteristic
299C2×C83#338711yescharacteristic
300C2×#479—831no—
301C2×C4×Q16—831no—
302#2453—831no—
303Q8×C4⋊C4D8311yescharacteristic
304#2096—831no—
305#2083—831no—
306#2066—831no—
307#2059—831no—
308#1941—831no—
309#1916—831no—
310#1878—831no—
311#1878—831no—
312#1865—831no—
313#1865—831no—
314#1863—831no—
315C83#2637111yescharacteristic, complemented, Sylow
316C22×Q16—831no—
317C2×#125—831no—
318C2×C4×Q8—831no—
319C2×C4×Q8C2×D8311yescharacteristic
320C2×C4×Q8C2×D8311yescharacteristic
321C2×C4×Q8C2×D8311yescharacteristic
322#479—1664no—
323C4×Q16—1664no—
324C2×#107—831no—
325C2×#107—831no—
326C2×#106—831no—
327C2×#106—831no—
328C2×#106—831no—
329C2×C4⋊C8—831no—
330C2×#103—831no—
331C2×#103—831no—
332C2×#103—831no—
333C2×#103—831no—
334C2×#103—831no—
335C2×#103—831no—
336C2×C4×C8—831no—
337#400—1661no—
338#398—1661no—
339#391—831no—
340#391—831no—
341#389—21no—
342#389—21no—
343#389—831no—
344#389—831no—
345#389—1661no—
346#389C2×D8311yescharacteristic
347#389C2×D8311yescharacteristic
348#387—1661no—
349#384—21no—
350#384C2×D8311yescharacteristic
351C4×C4⋊C4—21no—
352C4×C4⋊C4C2×D8311yescharacteristic
353#340—831no—
354#340—831no—
355#340—831no—
356#340—831no—
357#336—831no—
358#336—831no—
359#336—831no—
360#336—831no—
361C22×Q8—831no—
362C22×Q8C22×D8311yescharacteristic
363C2×Q16—834no—
364C2×Q16—1662no—
365C2×Q16—1662no—
366C22×C8—831no—
367C22×C8—831no—
368C22×C8—831no—
369C22×C8—831no—
370#125—3321no—
371#125—1662no—
372C4×Q8—22no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×Q8—41no—
377C4×Q8—41no—
378C4×Q8—3321no—
379C4×Q8—1662no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4C22⋊D8311yescharacteristic
396C2×C4⋊C4C22⋊D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—1661no—
404C2×C42C22×D8311yescharacteristic
405C2×C42C22×D8311yescharacteristic
406C2×C42C22×D8311yescharacteristic
407#107—1662no—
408#107—1662no—
409#106—3321no—
410#106—1662no—
411#106—1662no—
412C4⋊C8—1662no—
413#103—3321no—
414#103—834no—
415#103—1662no—
416#103—1662no—
417#103—1662no—
418#103—1662no—
419C4×C8—1662no—
420#95—21no—
421#95—21no—
422#95—1661no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95C22⋊D8311yescharacteristic
429#95C22⋊D8311yescharacteristic
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—832no—
434C2×Q8—1661no—
435C2×Q8—1661no—
436C2×Q8—1661no—
437C2×Q8—1661no—
438C2×Q8—1661no—
439C2×Q8D4×D8312yes—
440C2×Q8Q8×D8312yes—
441C2×Q8#1305059012yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4C2×C22⋊D8311yescharacteristic
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4C2×C22⋊D8311yescharacteristic
456C22×C4C23×D8311yescharacteristic
457Q16—1668no—
458C2×C8—832no—
459C2×C8—832no—
460C2×C8—1661no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—3322no—
485C4⋊C4—3322no—
486C4⋊C4—3322no—
487C4⋊C4—3322no—
488C4⋊C4—3322no—
489C4⋊C4—832no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4#1305059012yes—
496C42—24no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—42no—
504C42—42no—
505C42—3322no—
506C23C22×C22⋊D8311yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8—3321no—
511Q8—3321no—
512Q8—1662no—
513Q8—1662no—
514Q8#1458177014yes—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—1662no—
554C2×C4—1662no—
555C2×C4#1458181212yes—
556C2×C4#1458182312yes—
557C2×C4C2×D4×D8311yescharacteristic
558C2×C4C2×Q8×D8311yescharacteristic
559C2×C4C2×#1305059011yescharacteristic
560C2×C4C2×#1305059011yescharacteristic
561C2×C4#1458183711yescharacteristic
562C2×C4#1458184011yescharacteristic
563C8—3321no—
564C8—3321no—
565C8—1662no—
566C8—1662no—
567C22#1912821211yescharacteristic
568C22D4×C22⋊D8311yescharacteristic
569C22Q8×C22⋊D8311yescharacteristic
570C22C2×#1458181211yescharacteristic
571C22C2×#1458182311yescharacteristic
572C22#1912851211yescharacteristic
573C22#1912851611yescharacteristic
574C4—22no—
575C4—24no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—3322no—
585C4—3322no—
586C4—6641no—
587C4—6641no—
588C4C2×#1458177011yescharacteristic
589C4#1912829811yescharacteristic
590C4#1912835211yescharacteristic
591C4#1912840611yescharacteristic
592C2#2491127211yescharacteristic
593C2#2491754311yescharacteristic
594C2#2491767711yescharacteristic
595C2#2491903411yescharacteristic
596C2#2491919311yescharacteristic
597C2#2492035011yescharacteristic
598C2#2492039111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingQ8×C4⋊C4×C83C2
socle, prime radicalC23×C83C22×D4
hypercenterQ8×C4⋊C4D83
hypocenterC83#26371
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
315. C83#26371semidirect

Maximal subgroups

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

subgroupindex
C2×#191280082
C2×C4×#130505802
#249113242
#249090072
#249089352
#249088412
#249088012
#249081882
#249080222
#249078042
#249077312
#249077162
Q8×C4⋊C4×C832
#2637183