Finite Group Databasebeta

← #35375440

Subgroups — #35375440

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#24917510C211yesmaximal, characteristic
2#24916467C211yesmaximal, characteristic, complemented
3#24906375C211yesmaximal, characteristic, complemented
4#24906351C211yesmaximal, characteristic, complemented
5#24906339C211yesmaximal, characteristic, complemented
6#24906327C211yesmaximal, characteristic, complemented
7#24906299C211yesmaximal, characteristic, complemented
8#24906290C211yesmaximal, characteristic, complemented
9#24906275C211yesmaximal, characteristic, complemented
10C8×#13050575C211yesmaximal, characteristic, complemented
11#24906254C211yesmaximal, characteristic, complemented
12#24905926C211yesmaximal, characteristic, complemented
13#24905854C211yesmaximal, characteristic, complemented
14#24905760C211yesmaximal, characteristic, complemented
15D4×M16×C83C211yesmaximal, characteristic, complemented
16#19128235C2211yescharacteristic
17#19128036C411yescharacteristic
18#19128019C411yescharacteristic
19#19128001C411yescharacteristic
20#19128001C411yescharacteristic
21#19127916C411yescharacteristic
22#19127916C411yescharacteristic
23#19127852C411yescharacteristic
24#19127835C411yescharacteristic
25#19127831C2211yescharacteristic, complemented
26C8×Dic166C2211yescharacteristic, complemented
27C4×#13050578C2211yescharacteristic
28C4×#13050578C2211yescharacteristic
29C4×#13050575C2211yescharacteristic
30C4×#13050575C2211yescharacteristic
31#19127780C2211yescharacteristic, complemented
32#19127777—21no—
33#19127770—21no—
34C8×D332C2211yescharacteristic, complemented
35#19127763—21no—
36#19127754C2211yescharacteristic
37#19127754C2211yescharacteristic
38#19127754C2211yescharacteristic
39#19127658—21no—
40#19127453C2211yescharacteristic, complemented
41#19127453C2211yescharacteristic, complemented
42#19127451C2211yescharacteristic, complemented
43#19127451C2211yescharacteristic, complemented
44#19127431C2211yescharacteristic, complemented
45#19127431C2211yescharacteristic, complemented
46D4⋊(C83⋊C8)C2211yescharacteristic
47D4⋊(C83⋊C8)C2211yescharacteristic
48D4⋊(C83⋊C8)C2211yescharacteristic
49D4⋊(C83⋊C8)C2211yescharacteristic
50#19127428C2211yescharacteristic, complemented
51#19127428C2211yescharacteristic, complemented
52#19127423C2211yescharacteristic, complemented
53#19127423C2211yescharacteristic, complemented
54#19127422C2211yescharacteristic, complemented
55C8×C83⋊C8C2211yescharacteristic, complemented
56C22×M16×C83—21no—
57C2×C4×D4×C83C2211yescharacteristic
58#436×C83C2211yescharacteristic
59#436×C83C2211yescharacteristic
60#435×C83—21no—
61#435×C83—21no—
62C8×D4×C83C2211yescharacteristic
63C8×D4×C83C2211yescharacteristic
64#423×C83C2211yescharacteristic
65#407×C83—21no—
66C4×M16×C83C2211yescharacteristic
67C2×#13050578C2×C411yescharacteristic
68C2×#13050578C2×C411yescharacteristic
69C2×#13050575C2×C411yescharacteristic
70C2×#13050575C2×C411yescharacteristic
71C2×#13050573—21no—
72C2×C8×D83—21no—
73#14581786—21no—
74#14581784C2×C411yescharacteristic
75#14581780C2×C411yescharacteristic
76#14581775—21no—
77#14581771C2×C411yescharacteristic
78#14581766C2×C411yescharacteristic
79C4×Dic166C2311yescharacteristic
80C4×D332C2311yescharacteristic
81C4×C22⋊D83—21no—
82#14581749—21no—
83#14581725C2×C411yescharacteristic
84#14581725C2×C411yescharacteristic
85#14581723C2×C411yescharacteristic
86#14581723C2×C411yescharacteristic
87C4⋊(C83⋊C8)—21no—
88C4⋊(C83⋊C8)C2×C411yescharacteristic
89C4⋊(C83⋊C8)C2×C411yescharacteristic
90C4⋊(C83⋊C8)C2311yescharacteristic
91C4⋊(C83⋊C8)C2311yescharacteristic
92Dic83⋊C8—21no—
93Dic83⋊C8—21no—
94#14581715C2×C411yescharacteristic
95#14581715C2×C411yescharacteristic
96#14581713C2×C411yescharacteristic
97#14581713C2×C411yescharacteristic
98#14581713C2×C411yescharacteristic
99#14581713C2×C411yescharacteristic
100#14581712C2×C411yescharacteristic
101#14581712C2×C411yescharacteristic
102C22⋊(C83⋊C8)—21no—
103C22⋊(C83⋊C8)—21no—
104C22⋊(C83⋊C8)C2×C411yescharacteristic
105C22⋊(C83⋊C8)C2×C411yescharacteristic
106(C2×D83)⋊C8—21no—
107(C2×D83)⋊C8—21no—
108#14581702—21no—
109C4×C83⋊C8—21no—
110C4×C83⋊C8C2311yescharacteristic
111C4×C83⋊C8C2311yescharacteristic
112C4×C83⋊C8C2311yescharacteristic
113C4×C83⋊C8C2311yescharacteristic
114C8×Dic83—21no—
115C22×D4×C83C2×C411yescharacteristic
116C23×C4×C83—21no—
117C2×M16×C83—21no—
118C2×M16×C83—21no—
119C2×M16×C83—21no—
120C2×M16×C83—44no—
121C2×M16×C83D411yescharacteristic
122C2×M16×C83D411yescharacteristic
123C22×C8×C83—21no—
124C22×C8×C83—21no—
125C4×D4×C83—21no—
126C4×D4×C83—21no—
127C4×D4×C83C2×C411yescharacteristic
128C4×D4×C83C2×C411yescharacteristic
129C4×D4×C83C2311yescharacteristic
130C4×D4×C83C2311yescharacteristic
131C2×C4⋊C4×C83C2×C411yescharacteristic
132C2×C22⋊C4×C83—21no—
133C2×C42×C83C2311yescharacteristic
134C4⋊C8×C83—21no—
135C4⋊C8×C83C2311yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137C22⋊C8×C83—21no—
138C22⋊C8×C83—21no—
139C22⋊C8×C83—21no—
140C22⋊C8×C83—21no—
141#97×C83D411yescharacteristic
142#97×C83D411yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C2×Dic166C22×C411yescharacteristic
146C2×D332C22×C411yescharacteristic
147C2×C22⋊D83—21no—
148C2×C4×D83—21no—
149#13050578—22no—
150#13050578—41no—
151#13050575—22no—
152#13050575—41no—
153#13050573—42no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8—21no—
158C2×C83⋊C8C22×C411yescharacteristic
159C2×C83⋊C8C22×C411yescharacteristic
160C2×C83⋊C8C22×C411yescharacteristic
161C2×C83⋊C8C22×C411yescharacteristic
162C8×D83—42no—
163C4⋊Dic83C22×C411yescharacteristic
164C4⋊Dic83C22×C411yescharacteristic
165Dic83⋊C4—21no—
166Dic83⋊C4—21no—
167Dic83⋊C4—21no—
168Dic83⋊C4—21no—
169C22⋊Dic83—21no—
170(C2×D83)⋊C4—21no—
171(C2×D83)⋊C4—21no—
172(C2×D83)⋊C4—21no—
173(C2×D83)⋊C4—21no—
174C4×Dic83—21no—
175C24×C83—21no—
176C2×D4×C83—22no—
177C2×D4×C83—21no—
178C2×D4×C83—41no—
179C2×D4×C83C22×C411yescharacteristic
180C2×D4×C83C22×C411yescharacteristic
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83—42no—
187C22×C4×C83—42no—
188C22×C4×C83C22×C411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83#4111yescharacteristic
191M16×C83—22no—
192M16×C83—41no—
193M16×C83—42no—
194M16×C83—42no—
195M16×C83—42no—
196C2×C8×C83—21no—
197C2×C8×C83—21no—
198C2×C8×C83—42no—
199C2×C8×C83—42no—
200C2×C8×C83C2×D411yescharacteristic
201C2×C8×C83C2×D411yescharacteristic
202C2×C8×C83C2×D411yescharacteristic
203C2×C8×C83C2×D411yescharacteristic
204C4⋊C4×C83—21no—
205C4⋊C4×C83C22×C411yescharacteristic
206C4⋊C4×C83C22×C411yescharacteristic
207C22⋊C4×C83—21no—
208C22⋊C4×C83—21no—
209C22⋊C4×C83—21no—
210C22⋊C4×C83—21no—
211C42×C83C22×C411yescharacteristic
212C42×C83C2×D411yescharacteristic
213C42×C83#4111yescharacteristic
214C42×C83C2411yescharacteristic
215C23×C83—21no—
216C23×C83—21no—
217C23×C83—42no—
218C23×C83—42no—
219C23×C83—81no—
220C23×C83C4×D411yescharacteristic
221C22×D83—21no—
222D4×C83—21no—
223D4×C83—21no—
224D4×C83—21no—
225D4×C83—42no—
226D4×C83#13112yes—
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—41no—
234C2×C4×C83—42no—
235C2×C4×C83—42no—
236C2×C4×C83—42no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83C4×D411yescharacteristic
242C2×C4×C83C4×D411yescharacteristic
243C2×C4×C83C4×D411yescharacteristic
244C2×C4×C83C23×C411yescharacteristic
245C2×C4×C83C22×D411yescharacteristic
246C2×C4×C83C2×#4111yescharacteristic
247C22⋊D83—81no—
248C2×Dic83—21no—
249C2×Dic83—21no—
250C2×Dic83—21no—
251D332—21no—
252D332—21no—
253C4×D83—41no—
254C4×D83—41no—
255Dic166—21no—
256Dic166—21no—
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—22no—
273C22×C83—21no—
274C22×C83—21no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83—41no—
280C22×C83—84no—
281C22×C83C2×C4×D411yescharacteristic
282C2×D83—41no—
283C2×D83—41no—
284C2×D83—41no—
285C4×C83—21no—
286C4×C83—21no—
287C4×C83—21no—
288C4×C83—21no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83C2×#13111yescharacteristic
292C4×C83#56811yescharacteristic
293C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
294C4×C83#57511yescharacteristic
295Dic83—41no—
296Dic83—41no—
297Dic83—41no—
298#18006—831nomaximal, Sylow
299C2×C83—41no—
300C2×C83—42no—
301C2×C83—81no—
302C2×C83#293311yescharacteristic
303C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C83#295711yescharacteristic
305D83—81no—
306#3105—831no—
307#2969—831no—
308D4×M16D8311yescharacteristic, complemented
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—21no—
323#529—831no—
324C2×C4×D4C2×D8311yescharacteristic
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#446—831no—
334C8×Q8—831no—
335C4×SD16—831no—
336C4×SD16—831no—
337C4×D8—831no—
338C4×D8—831no—
339#436—831no—
340#436—1661no—
341#436C2×D8311yescharacteristic
342#436C2×D8311yescharacteristic
343#435—21no—
344#435—21no—
345#435—1661no—
346C8×D4—831no—
347C8×D4C2×D8311yescharacteristic
348C8×D4C2×D8311yescharacteristic
349#433—1661no—
350#431—831no—
351#431—831no—
352#431—831no—
353#423C2×D8311yescharacteristic
354#407—21no—
355#405—1661no—
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×D4C4×D8311yescharacteristic
374C23×C4—21no—
375C2×SD16—831no—
376C2×SD16—831no—
377C2×D8—831no—
378C2×D8—831no—
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—44no—
383C2×M16—1661no—
384C2×M16C22⋊D8311yescharacteristic
385C2×M16C22⋊D8311yescharacteristic
386C22×C8—21no—
387C22×C8—21no—
388C22×C8—1661no—
389#126—1661no—
390#125—831no—
391#124—831no—
392#123—1661no—
393C22⋊Q8—831no—
394#121—831no—
395C4×Q8—831no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—831no—
399C4×D4—1661no—
400C4×D4C4×D8311yescharacteristic
401C4×D4C4×D8311yescharacteristic
402C4×D4C22×D8311yescharacteristic
403C4×D4C22×D8311yescharacteristic
404#117—1661no—
405C2×C4⋊C4C4×D8311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C42C22×D8311yescharacteristic
408#107—831no—
409#107—831no—
410#106—831no—
411#106—831no—
412C4⋊C8—21no—
413C4⋊C8—831no—
414C4⋊C8—831no—
415C4⋊C8—831no—
416C4⋊C8—831no—
417C4⋊C8—1661no—
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—21no—
434C22⋊C8—831no—
435C22⋊C8—831no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438C22⋊C8—1661no—
439C22⋊C8—1661no—
440#97—1661no—
441#97C22⋊D8311yescharacteristic
442#97C22⋊D8311yescharacteristic
443C4×C8—831no—
444C4×C8—831no—
445C4×C8—831no—
446C4×C8—831no—
447C4×C8—1661no—
448C4×C8—1661no—
449C4×C8C22×D8311yescharacteristic
450C4×C8C22×D8311yescharacteristic
451C24—21no—
452C2×Q8—831no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4—831no—
457C2×D4—1661no—
458C2×D4C2×C4×D8311yescharacteristic
459C2×D4C2×C4×D8311yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
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—42no—
479M16—42no—
480M16—3322no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—42no—
484C2×C8—42no—
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—831no—
500C4⋊C4—831no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
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—1661no—
512C22⋊C4—1661no—
513C22⋊C4—1661no—
514C22⋊C4—1661no—
515C22⋊C4—1661no—
516C42—1661no—
517C42C2×C4×D8311yescharacteristic
518C42C2×C22⋊D8311yescharacteristic
519C42#1305059011yescharacteristic
520C42C23×D8311yescharacteristic
521C23—21no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—81no—
526C23—1661no—
527C23C4×C22⋊D8311yescharacteristic
528Q8—1661no—
529Q8—1661no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—42no—
534D4—6641no—
535D4—1661no—
536D4—1661no—
537D4#1458180012yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—1661no—
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#1458181812yes—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—84no—
587C22—3321no—
588C22—3321no—
589C22—3321no—
590C22C2×C4×C22⋊D8311yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—3321no—
598C4—3321no—
599C4—3321no—
600C4C2×#1458180011yescharacteristic
601C4#1912846711yescharacteristic
602C4C2×#1458181811yescharacteristic
603C4#1912849911yescharacteristic
604C2—41no—
605C2—42no—
606C2—81no—
607C2—6641no—
608C2#2491618811yescharacteristic
609C2C4×#1458181811yescharacteristic
610C2#2491637811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingD4×M16×C83C2
socleC22×C83C2×C4×D4
prime radical#19127835C4
cosocleC42×C83C24
hypercenterD4×M16D83
hypocenterC83#18006
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
2. #24916467C2semidirect
3. #24906375C2semidirect
4. #24906351C2semidirect
5. #24906339C2semidirect
6. #24906327C2semidirect
7. #24906299C2semidirect
8. #24906290C2semidirect
9. #24906275C2semidirect
10. C8×#13050575C2semidirect
11. #24906254C2semidirect
12. #24905926C2semidirect
13. #24905854C2semidirect
14. #24905760C2semidirect
15. D4×M16×C83C2semidirect
25. #19127831C22semidirect
26. C8×Dic166C22semidirect
31. #19127780C22semidirect
34. C8×D332C22semidirect
40. #19127453C22semidirect
41. #19127453C22semidirect
42. #19127451C22semidirect
43. #19127451C22semidirect
44. #19127431C22semidirect
45. #19127431C22semidirect
50. #19127428C22semidirect
51. #19127428C22semidirect
52. #19127423C22semidirect
53. #19127423C22semidirect
54. #19127422C22semidirect
55. C8×C83⋊C8C22semidirect
308. D4×M16D83semidirect
321. C83#18006semidirect

Maximal subgroups

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

subgroupindex
#249175102
#249164672
#249063752
#249063512
#249063392
#249063272
#249062992
#249062902
#249062752
C8×#130505752
#249062542
#249059262
#249058542
#249057602
D4×M16×C832
#1800683