Finite Group Databasebeta

← #35375436

Subgroups — #35375436

Order 21248.

670 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#24917506C211yesmaximal, characteristic
2M16×D332C211yesmaximal, characteristic, complemented
3#24906375C211yesmaximal, characteristic, complemented
4#24906353C211yesmaximal, characteristic, complemented
5#24906341C211yesmaximal, characteristic, complemented
6#24906325C211yesmaximal, characteristic, complemented
7#24906297C211yesmaximal, characteristic, complemented
8#24906292C211yesmaximal, characteristic, complemented
9#24906273C211yesmaximal, characteristic, complemented
10C8×#13050575C211yesmaximal, characteristic, complemented
11#24906251C211yesmaximal, characteristic, complemented
12#24905928C211yesmaximal, characteristic, complemented
13#24905853C211yesmaximal, characteristic, complemented
14#24905760C211yesmaximal, characteristic, complemented
15C83×#2969C211yesmaximal, characteristic, complemented
16C2×M16×D83—21no—
17C2×C4×D332C2211yescharacteristic
18#19128034C411yescharacteristic
19#19128017C411yescharacteristic
20#19127998C411yescharacteristic
21#19127998C411yescharacteristic
22#19127913C411yescharacteristic
23#19127913C411yescharacteristic
24#19127851C411yescharacteristic
25#19127834C411yescharacteristic
26C4×#13050577C2211yescharacteristic
27C4×#13050577C2211yescharacteristic
28C4×#13050575C2211yescharacteristic
29C4×#13050575C2211yescharacteristic
30#19127780C2211yescharacteristic
31#19127780C2211yescharacteristic
32#19127774—21no—
33#19127774—21no—
34C8×D332C2211yescharacteristic
35C8×D332C2211yescharacteristic
36#19127754C2211yescharacteristic
37#19127754C2211yescharacteristic
38#19127725C2211yescharacteristic
39#19127664—21no—
40#19127453C2211yescharacteristic, complemented
41#19127453C2211yescharacteristic, complemented
42#19127451C2211yescharacteristic, complemented
43#19127451C2211yescharacteristic, complemented
44Q8⋊(C83⋊C8)C2211yescharacteristic, complemented
45Q8⋊(C83⋊C8)C2211yescharacteristic, complemented
46D4⋊(C83⋊C8)C2211yescharacteristic, complemented
47D4⋊(C83⋊C8)C2211yescharacteristic, complemented
48#19127428C2211yescharacteristic
49#19127428C2211yescharacteristic
50#19127428C2211yescharacteristic
51#19127428C2211yescharacteristic
52#19127423C2211yescharacteristic, complemented
53#19127423C2211yescharacteristic, complemented
54#19127422C2211yescharacteristic, complemented
55C8×C83⋊C8C2211yescharacteristic, complemented
56#529×C83C2211yescharacteristic
57#446×C83C2211yescharacteristic, complemented
58C8×Q8×C83C2211yescharacteristic, complemented
59#436×C83—21no—
60#436×C83C2211yescharacteristic, complemented
61#435×C83—21no—
62C8×D4×C83C2211yescharacteristic, complemented
63#433×C83—21no—
64#431×C83C2211yescharacteristic
65#405×C83—21no—
66C4×M16×C83C2211yescharacteristic
67C22×D332C2×C411yescharacteristic
68C22×C4×D83—21no—
69C2×#13050577C2×C411yescharacteristic
70C2×#13050577C2×C411yescharacteristic
71C2×#13050575C2×C411yescharacteristic
72C2×#13050575C2×C411yescharacteristic
73C2×#13050574—21no—
74C2×#13050573—21no—
75C2×#13050573—21no—
76M16×D83—44no—
77C2×C8×D83—21no—
78C2×C8×D83—21no—
79C4×D332—21no—
80C4×D332—21no—
81C4×D332C2×C411yescharacteristic
82C4×D332C2×C411yescharacteristic
83C4×D332C2311yescharacteristic
84C4×D332C2311yescharacteristic
85C2×C4⋊Dic83C2×C411yescharacteristic
86C2×(C2×D83)⋊C4—21no—
87#14581725C2×C411yescharacteristic
88#14581725C2×C411yescharacteristic
89#14581723C2×C411yescharacteristic
90#14581723C2×C411yescharacteristic
91C4⋊(C83⋊C8)—21no—
92C4⋊(C83⋊C8)—21no—
93C4⋊(C83⋊C8)C2×C411yescharacteristic
94C4⋊(C83⋊C8)C2×C411yescharacteristic
95C4⋊(C83⋊C8)C2311yescharacteristic
96C4⋊(C83⋊C8)C2311yescharacteristic
97#14581716C2×C411yescharacteristic
98#14581716C2×C411yescharacteristic
99#14581713C2×C411yescharacteristic
100#14581713C2×C411yescharacteristic
101#14581712C2×C411yescharacteristic
102#14581712C2×C411yescharacteristic
103#14581712C2×C411yescharacteristic
104#14581712C2×C411yescharacteristic
105C22⋊(C83⋊C8)—21no—
106C22⋊(C83⋊C8)C2×C411yescharacteristic
107C22⋊(C83⋊C8)C2×C411yescharacteristic
108(C2×D83)⋊C8—21no—
109(C2×D83)⋊C8—21no—
110(C2×D83)⋊C8—21no—
111(C2×D83)⋊C8—21no—
112C4×C83⋊C8C2311yescharacteristic
113C4×C83⋊C8C2311yescharacteristic
114C4×C83⋊C8C2311yescharacteristic
115C4×C83⋊C8C2311yescharacteristic
116C2×M16×C83—21no—
117C2×M16×C83D411yescharacteristic
118C2×M16×C83D411yescharacteristic
119C22×C8×C83—21no—
120#126×C83—21no—
121#125×C83C2×C411yescharacteristic
122#124×C83C2×C411yescharacteristic
123#123×C83—21no—
124C22⋊Q8×C83C2×C411yescharacteristic
125#121×C83C2×C411yescharacteristic
126C4×Q8×C83C2311yescharacteristic
127C4×D4×C83—21no—
128C4×D4×C83C2311yescharacteristic
129#117×C83—21no—
130C2×C42×C83C2311yescharacteristic
131C4⋊C8×C83—21no—
132C4⋊C8×C83—21no—
133C4⋊C8×C83C2311yescharacteristic
134C4⋊C8×C83C2311yescharacteristic
135C22⋊C8×C83—21no—
136C22⋊C8×C83—21no—
137C22⋊C8×C83—21no—
138#97×C83—21no—
139#97×C83D411yescharacteristic
140#97×C83D411yescharacteristic
141C4×C8×C83—21no—
142C4×C8×C83—21no—
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C23×D83—21no—
146C2×D332—22no—
147C2×D332—21no—
148C2×D332—41no—
149C2×D332C22×C411yescharacteristic
150C2×D332C22×C411yescharacteristic
151C22×Dic83—21no—
152C2×C4×D83—21no—
153C2×C4×D83—21no—
154C2×C4×D83—21no—
155C2×C4×D83—21no—
156C2×C4×D83—42no—
157C2×C4×D83—42no—
158#13050577—22no—
159#13050577—41no—
160#13050575—22no—
161#13050575—41no—
162#13050574—42no—
163#13050573—42no—
164#13050573—42no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8C22×C411yescharacteristic
170C2×C83⋊C8C22×C411yescharacteristic
171C2×C83⋊C8C22×C411yescharacteristic
172C2×C83⋊C8C22×C411yescharacteristic
173C8×D83—42no—
174C8×D83—42no—
175C4⋊Dic83—21no—
176C4⋊Dic83C22×C411yescharacteristic
177C4⋊Dic83C22×C411yescharacteristic
178(C2×D83)⋊C4—21no—
179(C2×D83)⋊C4—21no—
180(C2×D83)⋊C4—21no—
181(C2×D83)⋊C4—21no—
182C2×Q8×C83C22×C411yescharacteristic
183C2×D4×C83—21no—
184C2×D4×C83C22×C411yescharacteristic
185C22×C4×C83—21no—
186C22×C4×C83C22×C411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83#4111yescharacteristic
189M16×C83—22no—
190M16×C83—41no—
191M16×C83—42no—
192C2×C8×C83—21no—
193C2×C8×C83—21no—
194C2×C8×C83—42no—
195C2×C8×C83C2×D411yescharacteristic
196C2×C8×C83C2×D411yescharacteristic
197C2×C8×C83C2×D411yescharacteristic
198C2×C8×C83C2×D411yescharacteristic
199C4⋊C4×C83—21no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—21no—
203C4⋊C4×C83C22×C411yescharacteristic
204C4⋊C4×C83C22×C411yescharacteristic
205C22⋊C4×C83—21no—
206C22⋊C4×C83—21no—
207C22⋊C4×C83—21no—
208C22⋊C4×C83—21no—
209C22⋊C4×C83—21no—
210C42×C83—21no—
211C42×C83C22×C411yescharacteristic
212C42×C83C2×D411yescharacteristic
213C42×C83#4111yescharacteristic
214C42×C83C2411yescharacteristic
215C23×C83—21no—
216C23×C83C4×D411yescharacteristic
217C22×D83—21no—
218C22×D83—21no—
219C22×D83—42no—
220C22×D83—42no—
221C22×D83—81no—
222Q8×C83—21no—
223Q8×C83—21no—
224D4×C83—21no—
225D4×C83—21no—
226D4×C83—81no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83C4×D411yescharacteristic
238C2×C4×C83C4×D411yescharacteristic
239C2×C4×C83C4×D411yescharacteristic
240C2×C4×C83C23×C411yescharacteristic
241C2×C4×C83C22×D411yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—42no—
246D332—21no—
247D332—21no—
248D332—21no—
249D332—42no—
250D332#13112yes—
251C4×D83—42no—
252C4×D83—42no—
253C4×D83—41no—
254C4×D83—41no—
255C4×D83—41no—
256C4×D83—41no—
257C8×C83—21no—
258C8×C83—21no—
259C8×C83—21no—
260C8×C83—41no—
261C8×C83—41no—
262C8×C83AGL(1,ℤ/8)12yes—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—21no—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83—21no—
272C22×C83—21no—
273C22×C83—21no—
274C22×C83—41no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83C2×C4×D411yescharacteristic
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—21no—
288C4×C83—21no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83—41no—
292C4×C83C2×#13111yescharacteristic
293C4×C83#56811yescharacteristic
294C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
295C4×C83#57511yescharacteristic
296Dic83—41no—
297Dic83—41no—
298#18006—831nomaximal, Sylow
299C2×C83—41no—
300C2×C83—81no—
301C2×C83#293311yescharacteristic
302C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
303C2×C83#295711yescharacteristic
304D83—42no—
305D83—81no—
306#3105—831no—
307#2969D8311yescharacteristic, complemented
308D4×M16—831no—
309#1594—831no—
310#1590—831no—
311#1588—831no—
312#1586—831no—
313#1582—831no—
314#1581—831no—
315SD16⋊C8—831no—
316C8×D8—831no—
317#1574—831no—
318#1495—831no—
319#1478—831no—
320#1453—831no—
321C83#1800611yescharacteristic, complemented, Sylow
322C22×M16—1661no—
323#529C2×D8311yescharacteristic
324C2×C4×D4—831no—
325#484—831no—
326#481—831no—
327#478—831no—
328#478—831no—
329#463—831no—
330#463—831no—
331#450—831no—
332#447—831no—
333#446C2×D8311yescharacteristic, complemented
334C8×Q8C2×D8311yescharacteristic, complemented
335C4×SD16—831no—
336C4×SD16—831no—
337C4×D8—831no—
338C4×D8—831no—
339#436—21no—
340#436—831no—
341#436—831no—
342#436C2×D8311yescharacteristic, complemented
343#435—21no—
344#435—1661no—
345#435—1661no—
346C8×D4—831no—
347C8×D4—831no—
348C8×D4C2×D8311yescharacteristic, complemented
349#433—21no—
350#431—831no—
351#431—831no—
352#431C2×D8311yescharacteristic
353#423—831no—
354#407—1661no—
355#405—21no—
356C4×M16C2×D8311yescharacteristic
357#335—831no—
358#335—831no—
359#334—831no—
360#334—831no—
361Q8⋊C8—831no—
362Q8⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8—831no—
366D4⋊C8—831no—
367D4⋊C8—831no—
368D4⋊C8—831no—
369#322—831no—
370#322—831no—
371#322—831no—
372C82—831no—
373C22×D4—831no—
374C23×C4—1661no—
375C2×SD16—831no—
376C2×SD16—831no—
377C2×D8—831no—
378C2×D8—831no—
379C2×M16—21no—
380C2×M16—3324no—
381C2×M16—1661no—
382C2×M16—1661no—
383C2×M16—1661no—
384C2×M16C22⋊D8311yescharacteristic
385C2×M16C22⋊D8311yescharacteristic
386C22×C8—21no—
387C22×C8—1661no—
388C22×C8—1661no—
389#126—21no—
390#125C4×D8311yescharacteristic
391#124C4×D8311yescharacteristic
392#123—21no—
393C22⋊Q8C4×D8311yescharacteristic
394#121C4×D8311yescharacteristic
395C4×Q8C22×D8311yescharacteristic
396C4×D4—21no—
397C4×D4—831no—
398C4×D4—831no—
399C4×D4—831no—
400C4×D4—831no—
401C4×D4—1661no—
402C4×D4—1661no—
403C4×D4C22×D8311yescharacteristic
404#117—21no—
405C2×C4⋊C4—831no—
406C2×C22⋊C4—1661no—
407C2×C42C22×D8311yescharacteristic
408#107—831no—
409#107—831no—
410#106—831no—
411#106—831no—
412C4⋊C8—21no—
413C4⋊C8—21no—
414C4⋊C8—831no—
415C4⋊C8—831no—
416C4⋊C8—831no—
417C4⋊C8—831no—
418C4⋊C8—1661no—
419C4⋊C8—1661no—
420C4⋊C8C22×D8311yescharacteristic
421C4⋊C8C22×D8311yescharacteristic
422#103—831no—
423#103—831no—
424#102—831no—
425#102—831no—
426#102—831no—
427#102—831no—
428#102—831no—
429#102—831no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8—831no—
434C22⋊C8—831no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438C22⋊C8—1661no—
439C22⋊C8—1661no—
440#97—21no—
441#97C22⋊D8311yescharacteristic
442#97C22⋊D8311yescharacteristic
443C4×C8—21no—
444C4×C8—21no—
445C4×C8—831no—
446C4×C8—831no—
447C4×C8—831no—
448C4×C8—831no—
449C4×C8C22×D8311yescharacteristic
450C4×C8C22×D8311yescharacteristic
451C24—1661no—
452C2×Q8C2×C4×D8311yescharacteristic
453C2×D4—21no—
454C2×D4—3321no—
455C2×D4—831no—
456C2×D4—831no—
457C2×D4—1662no—
458C2×D4—1661no—
459C2×D4C2×C4×D8311yescharacteristic
460C22×C4—21no—
461C22×C4—3322no—
462C22×C4—3322no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4—1661no—
468C22×C4C2×C4×D8311yescharacteristic
469C22×C4C2×C22⋊D8311yescharacteristic
470C22×C4#1305059011yescharacteristic
471SD16—3321no—
472SD16—1662no—
473D8—3321no—
474D8—1662no—
475M16—22no—
476M16—41no—
477M16—42no—
478M16—3322no—
479M16—3322no—
480M16—3322no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—42no—
484C2×C8—3322no—
485C2×C8—3322no—
486C2×C8—831no—
487C2×C8—831no—
488C2×C8—831no—
489C2×C8—831no—
490C2×C8—1661no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8C2×C22⋊D8311yescharacteristic
495C2×C8C2×C22⋊D8311yescharacteristic
496C2×C8C2×C22⋊D8311yescharacteristic
497C2×C8C2×C22⋊D8311yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—831no—
503C4⋊C4—831no—
504C4⋊C4—1661no—
505C4⋊C4C2×C4×D8311yescharacteristic
506C4⋊C4C2×C4×D8311yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—1661no—
513C22⋊C4—1661no—
514C22⋊C4—1661no—
515C22⋊C4—1661no—
516C42—21no—
517C42C2×C4×D8311yescharacteristic
518C42C2×C22⋊D8311yescharacteristic
519C42#1305059011yescharacteristic
520C42C23×D8311yescharacteristic
521C23—21no—
522C23—3322no—
523C23—3322no—
524C23—6641no—
525C23—1661no—
526C23—1661no—
527C23C4×C22⋊D8311yescharacteristic
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—21no—
532D4—81no—
533D4—3322no—
534D4—832no—
535D4—1661no—
536D4—1661no—
537D4—1661no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—3322no—
549C2×C4—3322no—
550C2×C4—3322no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4C4×C22⋊D8311yescharacteristic
558C2×C4C4×C22⋊D8311yescharacteristic
559C2×C4C4×C22⋊D8311yescharacteristic
560C2×C4C22×C4×D8311yescharacteristic
561C2×C4C22×C22⋊D8311yescharacteristic
562C2×C4C2×#1305059011yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—3321no—
569C8—3321no—
570C8—3321no—
571C8—3321no—
572C8—1661no—
573C8—1661no—
574C8—1661no—
575C8—1661no—
576C8#1458181712yes—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—3321no—
584C22—3321no—
585C22—3321no—
586C22—3321no—
587C22—3321no—
588C22—6644no—
589C22—1662no—
590C22C2×C4×C22⋊D8311yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—3321no—
599C4—3321no—
600C4#131×D8311yescharacteristic
601C4#1912846611yescharacteristic
602C4C2×#1458181711yescharacteristic
603C4#1912849911yescharacteristic
604C2—41no—
605C2—81no—
606C2—3322no—
607C2—6641no—
608C2#2491618511yescharacteristic
609C2C4×#1458181711yescharacteristic
610C2#2491637811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#2969C2
socleC22×C83C2×C4×D4
prime radical#19127834C4
cosocleC42×C83C24
hypercenter#2969D83
hypocenterC83#18006
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. M16×D332C2semidirect
3. #24906375C2semidirect
4. #24906353C2semidirect
5. #24906341C2semidirect
6. #24906325C2semidirect
7. #24906297C2semidirect
8. #24906292C2semidirect
9. #24906273C2semidirect
10. C8×#13050575C2semidirect
11. #24906251C2semidirect
12. #24905928C2semidirect
13. #24905853C2semidirect
14. #24905760C2semidirect
15. C83×#2969C2semidirect
40. #19127453C22semidirect
41. #19127453C22semidirect
42. #19127451C22semidirect
43. #19127451C22semidirect
44. Q8⋊(C83⋊C8)C22semidirect
45. Q8⋊(C83⋊C8)C22semidirect
46. D4⋊(C83⋊C8)C22semidirect
47. D4⋊(C83⋊C8)C22semidirect
52. #19127423C22semidirect
53. #19127423C22semidirect
54. #19127422C22semidirect
55. C8×C83⋊C8C22semidirect
57. #446×C83C22semidirect
58. C8×Q8×C83C22semidirect
60. #436×C83C22semidirect
62. C8×D4×C83C22semidirect
307. #2969D83semidirect
321. C83#18006semidirect
333. #446C2×D83semidirect
334. C8×Q8C2×D83semidirect
342. #436C2×D83semidirect
348. C8×D4C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249175062
M16×D3322
#249063752
#249063532
#249063412
#249063252
#249062972
#249062922
#249062732
C8×#130505752
#249062512
#249059282
#249058532
#249057602
C83×#29692
#1800683