Finite Group Databasebeta

← #35375444

Subgroups — #35375444

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#24917512C211yesmaximal, characteristic
2#24916467C211yesmaximal, characteristic, complemented
3M16×D332C211yesmaximal, characteristic, complemented
4#24906378C211yesmaximal, characteristic, complemented
5#24906355C211yesmaximal, characteristic, complemented
6#24906343C211yesmaximal, characteristic, complemented
7#24906329C211yesmaximal, characteristic, complemented
8#24906301C211yesmaximal, characteristic, complemented
9#24906294C211yesmaximal, characteristic, complemented
10#24906277C211yesmaximal, characteristic, complemented
11C8×D664C211yesmaximal, characteristic, complemented
12#24906252C211yesmaximal, characteristic, complemented
13#24905929C211yesmaximal, characteristic, complemented
14#24905855C211yesmaximal, characteristic, complemented
15C83×#1453C211yesmaximal, characteristic, complemented
16C2×M16×D83—21no—
17#19128235C2211yescharacteristic
18C2×C4×D332C2211yescharacteristic
19#19128037C411yescharacteristic
20#19128020C411yescharacteristic
21#19128002C411yescharacteristic
22#19128002C411yescharacteristic
23#19127917C411yescharacteristic
24#19127917C411yescharacteristic
25#19127854C411yescharacteristic
26#19127837C411yescharacteristic
27#19127831C2211yescharacteristic, complemented
28C8×Dic166C2211yescharacteristic, complemented
29C4×#13050579C2211yescharacteristic
30C4×#13050579C2211yescharacteristic
31C4×D664C2211yescharacteristic
32C4×D664C2211yescharacteristic
33#19127780C2211yescharacteristic
34#19127780C2211yescharacteristic
35#19127780C2211yescharacteristic, complemented
36#19127777—21no—
37#19127774—21no—
38#19127774—21no—
39#19127770—21no—
40C8×D332C2211yescharacteristic
41C8×D332C2211yescharacteristic
42C8×D332C2211yescharacteristic, complemented
43#19127763—21no—
44#19127754C2211yescharacteristic
45#19127725C2211yescharacteristic
46#19127664—21no—
47#19127658—21no—
48#19127454C2211yescharacteristic, complemented
49#19127454C2211yescharacteristic, complemented
50#19127452C2211yescharacteristic, complemented
51#19127452C2211yescharacteristic, complemented
52#19127433C2211yescharacteristic, complemented
53#19127433C2211yescharacteristic, complemented
54#19127430C2211yescharacteristic
55#19127430C2211yescharacteristic
56#19127430C2211yescharacteristic
57#19127430C2211yescharacteristic
58#19127430C2211yescharacteristic, complemented
59#19127430C2211yescharacteristic, complemented
60#431×C83C2211yescharacteristic
61#431×C83C2211yescharacteristic
62C4×M16×C83C2211yescharacteristic
63#322×C83C2211yescharacteristic, complemented
64#322×C83C2211yescharacteristic, complemented
65#322×C83C2211yescharacteristic, complemented
66C82×C83C2211yescharacteristic, complemented
67C22×D332C2×C411yescharacteristic
68C22×C4×D83—21no—
69C2×#13050579C2×C411yescharacteristic
70C2×#13050579C2×C411yescharacteristic
71C2×D664C2×C411yescharacteristic
72C2×D664C2×C411yescharacteristic
73C2×#13050574—21no—
74C2×#13050573—21no—
75C2×#13050573—21no—
76C2×#13050573—21no—
77M16×D83—44no—
78C2×C8×D83—21no—
79C2×C8×D83—21no—
80C2×C8×D83—21no—
81#14581786—21no—
82#14581784C2×C411yescharacteristic
83#14581780C2×C411yescharacteristic
84#14581775—21no—
85#14581771C2×C411yescharacteristic
86#14581766C2×C411yescharacteristic
87C4×Dic166C2311yescharacteristic
88C4×D332—21no—
89C4×D332—21no—
90C4×D332C2×C411yescharacteristic
91C4×D332C2×C411yescharacteristic
92C4×D332C2311yescharacteristic
93C4×D332C2311yescharacteristic
94C4×D332C2311yescharacteristic
95C4×C22⋊D83—21no—
96#14581749—21no—
97C2×C4⋊Dic83C2×C411yescharacteristic
98C2×(C2×D83)⋊C4—21no—
99#14581726C2×C411yescharacteristic
100#14581726C2×C411yescharacteristic
101#14581724C2×C411yescharacteristic
102#14581724C2×C411yescharacteristic
103C4⋊(C83⋊C8)—21no—
104C4⋊(C83⋊C8)C2311yescharacteristic
105C4⋊(C83⋊C8)C2311yescharacteristic
106C4⋊(C83⋊C8)C2311yescharacteristic
107C4⋊(C83⋊C8)C2311yescharacteristic
108Dic83⋊C8—21no—
109Dic83⋊C8—21no—
110#14581717C2×C411yescharacteristic
111#14581717C2×C411yescharacteristic
112#14581714C2×C411yescharacteristic
113#14581714C2×C411yescharacteristic
114#14581714C2×C411yescharacteristic
115#14581714C2×C411yescharacteristic
116#14581714C2×C411yescharacteristic
117#14581714C2×C411yescharacteristic
118C22⋊(C83⋊C8)—21no—
119(C2×D83)⋊C8—21no—
120(C2×D83)⋊C8—21no—
121(C2×D83)⋊C8—21no—
122(C2×D83)⋊C8—21no—
123(C2×D83)⋊C8—21no—
124(C2×D83)⋊C8—21no—
125#14581702—21no—
126C4×C83⋊C8—21no—
127C8×Dic83—21no—
128C2×M16×C83D411yescharacteristic
129C2×M16×C83D411yescharacteristic
130C2×C42×C83C2311yescharacteristic
131C4⋊C8×C83—21no—
132C4⋊C8×C83C2×C411yescharacteristic
133C4⋊C8×C83C2×C411yescharacteristic
134C22⋊C8×C83—21no—
135C22⋊C8×C83C2×C411yescharacteristic
136C22⋊C8×C83C2×C411yescharacteristic
137#97×C83D411yescharacteristic
138#97×C83D411yescharacteristic
139C4×C8×C83C2311yescharacteristic
140C4×C8×C83C2311yescharacteristic
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C23×D83—21no—
146C2×Dic166C22×C411yescharacteristic
147C2×D332—22no—
148C2×D332—21no—
149C2×D332—41no—
150C2×D332C22×C411yescharacteristic
151C2×D332C22×C411yescharacteristic
152C2×D332C22×C411yescharacteristic
153C2×C22⋊D83—21no—
154C22×Dic83—21no—
155C2×C4×D83—21no—
156C2×C4×D83—21no—
157C2×C4×D83—21no—
158C2×C4×D83—21no—
159C2×C4×D83—21no—
160C2×C4×D83—42no—
161C2×C4×D83—42no—
162#13050579—22no—
163#13050579—41no—
164D664—22no—
165D664—41no—
166#13050574—42no—
167#13050573—42no—
168#13050573—42no—
169#13050573—42no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8—21no—
172C2×C83⋊C8—21no—
173C2×C83⋊C8—21no—
174C8×D83—42no—
175C8×D83—42no—
176C8×D83—42no—
177C4⋊Dic83—21no—
178C4⋊Dic83C22×C411yescharacteristic
179C4⋊Dic83C22×C411yescharacteristic
180C4⋊Dic83C22×C411yescharacteristic
181C4⋊Dic83C22×C411yescharacteristic
182Dic83⋊C4—21no—
183Dic83⋊C4—21no—
184Dic83⋊C4—21no—
185Dic83⋊C4—21no—
186C22⋊Dic83—21no—
187(C2×D83)⋊C4—21no—
188(C2×D83)⋊C4—21no—
189(C2×D83)⋊C4—21no—
190(C2×D83)⋊C4—21no—
191(C2×D83)⋊C4—21no—
192(C2×D83)⋊C4—21no—
193(C2×D83)⋊C4—21no—
194(C2×D83)⋊C4—21no—
195C4×Dic83—21no—
196C22×C4×C83C22×C411yescharacteristic
197C22×C4×C83C2×D411yescharacteristic
198C22×C4×C83#4111yescharacteristic
199M16×C83—22no—
200M16×C83—41no—
201C2×C8×C83—21no—
202C2×C8×C83—21no—
203C2×C8×C83C22×C411yescharacteristic
204C2×C8×C83C22×C411yescharacteristic
205C2×C8×C83C22×C411yescharacteristic
206C2×C8×C83C22×C411yescharacteristic
207C2×C8×C83C2×D411yescharacteristic
208C2×C8×C83C2×D411yescharacteristic
209C2×C8×C83C2×D411yescharacteristic
210C2×C8×C83C2×D411yescharacteristic
211C42×C83C22×C411yescharacteristic
212C42×C83C2×D411yescharacteristic
213C42×C83#4111yescharacteristic
214C42×C83C2411yescharacteristic
215C23×C83C4×D411yescharacteristic
216C22×D83—21no—
217C22×D83—21no—
218C22×D83—21no—
219C22×D83—42no—
220C22×D83—42no—
221C22×D83—81no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—41no—
227C2×C4×C83C4×D411yescharacteristic
228C2×C4×C83C4×D411yescharacteristic
229C2×C4×C83C4×D411yescharacteristic
230C2×C4×C83C23×C411yescharacteristic
231C2×C4×C83C22×D411yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C22⋊D83—81no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—21no—
239C2×Dic83—42no—
240D332—21no—
241D332—21no—
242D332—21no—
243D332—21no—
244D332—21no—
245D332—42no—
246D332#13112yes—
247C4×D83—42no—
248C4×D83—42no—
249C4×D83—41no—
250C4×D83—41no—
251C4×D83—41no—
252C4×D83—41no—
253C4×D83—41no—
254C4×D83—41no—
255Dic166—21no—
256Dic166—21no—
257C8×C83—21no—
258C8×C83—21no—
259C8×C83—21no—
260C8×C83—21no—
261C8×C83—21no—
262C8×C83—21no—
263C8×C83—21no—
264C8×C83—41no—
265C8×C83—41no—
266C8×C83AGL(1,ℤ/8)12yes—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83—21no—
272C22×C83—21no—
273C22×C83—21no—
274C22×C83C2×C4×D411yescharacteristic
275C2×D83—22no—
276C2×D83—41no—
277C2×D83—41no—
278C2×D83—41no—
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×C83C2×#13111yescharacteristic
290C4×C83#56811yescharacteristic
291C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C83#57511yescharacteristic
293Dic83—41no—
294Dic83—41no—
295Dic83—41no—
296Dic83—41no—
297Dic83—41no—
298#18006—831nomaximal, Sylow
299C2×C83—41no—
300C2×C83#293311yescharacteristic
301C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
302C2×C83#295711yescharacteristic
303D83—42no—
304D83—81no—
305D83—81no—
306#3105—831no—
307#2969—831no—
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#1453D8311yescharacteristic, complemented
321C83#1800611yescharacteristic, complemented, Sylow
322C22×M16—1661no—
323#529—831no—
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#446—831no—
334C8×Q8—831no—
335C4×SD16—831no—
336C4×SD16—831no—
337C4×D8—831no—
338C4×D8—831no—
339#436—831no—
340#436—831no—
341#436—831no—
342#436—1661no—
343#435—1661no—
344#435—1661no—
345#435—1661no—
346C8×D4—831no—
347C8×D4—831no—
348C8×D4—831no—
349#433—1661no—
350#431—831no—
351#431C2×D8311yescharacteristic
352#431C2×D8311yescharacteristic
353#423—831no—
354#407—1661no—
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#322C2×D8311yescharacteristic, complemented
370#322C2×D8311yescharacteristic, complemented
371#322C2×D8311yescharacteristic, complemented
372C82C2×D8311yescharacteristic, complemented
373C22×D4—831no—
374C23×C4—1661no—
375C2×SD16—831no—
376C2×SD16—831no—
377C2×D8—831no—
378C2×D8—831no—
379C2×M16—3324no—
380C2×M16—1661no—
381C2×M16—1661no—
382C2×M16—1661no—
383C2×M16—1661no—
384C2×M16D33211yescharacteristic
385C2×M16D33211yescharacteristic
386C22×C8—1661no—
387C22×C8—1661no—
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—831no—
397C4×D4—831no—
398C4×D4—831no—
399C4×D4—831no—
400C4×D4—831no—
401C4×D4—1661no—
402C4×D4—1661no—
403C4×D4—1661no—
404#117—1661no—
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—831no—
414C4⋊C8—831no—
415C4⋊C8—831no—
416C4⋊C8—831no—
417C4⋊C8—1661no—
418C4⋊C8—1661no—
419C4⋊C8—1661no—
420C4⋊C8C4×D8311yescharacteristic
421C4⋊C8C4×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—1661no—
432C22⋊C8—1661no—
433C22⋊C8—1661no—
434C22⋊C8—1661no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438C22⋊C8C4×D8311yescharacteristic
439C22⋊C8C4×D8311yescharacteristic
440#97—1661no—
441#97D33211yescharacteristic
442#97D33211yescharacteristic
443C4×C8—1661no—
444C4×C8—1661no—
445C4×C8C22×D8311yescharacteristic
446C4×C8C22×D8311yescharacteristic
447C4×C8C22×D8311yescharacteristic
448C4×C8C22×D8311yescharacteristic
449C4×C8C22×D8311yescharacteristic
450C4×C8C22×D8311yescharacteristic
451C24—1661no—
452C2×Q8—831no—
453C2×D4—3321no—
454C2×D4—831no—
455C2×D4—831no—
456C2×D4—831no—
457C2×D4—1662no—
458C2×D4—1661no—
459C2×D4—1661no—
460C22×C4—3322no—
461C22×C4—3322no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4—1661no—
468C22×C4C2×C4×D8311yescharacteristic
469C22×C4C2×D33211yescharacteristic
470C22×C4#1305059011yescharacteristic
471SD16—3321no—
472SD16—1662no—
473D8—3321no—
474D8—1662no—
475M16—22no—
476M16—41no—
477M16—3322no—
478M16—3322no—
479M16—3322no—
480M16—3322no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—3322no—
484C2×C8—3322no—
485C2×C8—3322no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8C2×C4×D8311yescharacteristic
491C2×C8C2×C4×D8311yescharacteristic
492C2×C8C2×C4×D8311yescharacteristic
493C2×C8C2×C4×D8311yescharacteristic
494C2×C8C2×D33211yescharacteristic
495C2×C8C2×D33211yescharacteristic
496C2×C8C2×D33211yescharacteristic
497C2×C8C2×D33211yescharacteristic
498C4⋊C4—831no—
499C4⋊C4—831no—
500C4⋊C4—831no—
501C4⋊C4—831no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4—1661no—
507C22⋊C4—1661no—
508C22⋊C4—1661no—
509C22⋊C4—1661no—
510C22⋊C4—1661no—
511C22⋊C4—1661no—
512C22⋊C4—1661no—
513C22⋊C4—1661no—
514C22⋊C4—1661no—
515C22⋊C4—1661no—
516C42—1661no—
517C42C2×C4×D8311yescharacteristic
518C42C2×D33211yescharacteristic
519C42#1305059011yescharacteristic
520C42C23×D8311yescharacteristic
521C23—3322no—
522C23—3322no—
523C23—6641no—
524C23—1661no—
525C23—1661no—
526C23—1661no—
527C23C4×D33211yescharacteristic
528Q8—1661no—
529Q8—1661no—
530D4—3322no—
531D4—832no—
532D4—6641no—
533D4—1661no—
534D4—1661no—
535D4—1661no—
536D4—1661no—
537D4—1661no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—3322no—
544C2×C4—3322no—
545C2×C4—3322no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4C4×D33211yescharacteristic
558C2×C4C4×D33211yescharacteristic
559C2×C4C4×D33211yescharacteristic
560C2×C4C22×C4×D8311yescharacteristic
561C2×C4C22×D33211yescharacteristic
562C2×C4C2×#1305059011yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—21no—
570C8—41no—
571C8—41no—
572C8—3321no—
573C8—3321no—
574C8—3321no—
575C8—3321no—
576C8#1458182012yes—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—3321no—
581C22—3321no—
582C22—3321no—
583C22—3321no—
584C22—3321no—
585C22—3321no—
586C22—3321no—
587C22—3321no—
588C22—6644no—
589C22—1662no—
590C22C2×C4×D33211yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—3321no—
596C4—3321no—
597C4—3321no—
598C4—3321no—
599C4—3321no—
600C4#131×D8311yescharacteristic
601C4#1912846611yescharacteristic
602C4C2×#1458182011yescharacteristic
603C4#1912850111yescharacteristic
604C2—41no—
605C2—3322no—
606C2—6641no—
607C2—6641no—
608C2#2491619111yescharacteristic
609C2C4×#1458182011yescharacteristic
610C2#2491638111yescharacteristic
subgroupG/H
centerC2×C4C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1453C2
socleC22×C83C2×C4×D4
prime radical#19127837C4
cosocleC42×C83C24
hypercenter#1453D83
hypocenterC83#18006
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1453

Complements

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

NH ≅ G/Ntype
2. #24916467C2semidirect
3. M16×D332C2semidirect
4. #24906378C2semidirect
5. #24906355C2semidirect
6. #24906343C2semidirect
7. #24906329C2semidirect
8. #24906301C2semidirect
9. #24906294C2semidirect
10. #24906277C2semidirect
11. C8×D664C2semidirect
12. #24906252C2semidirect
13. #24905929C2semidirect
14. #24905855C2semidirect
15. C83×#1453C2semidirect
27. #19127831C22semidirect
28. C8×Dic166C22semidirect
35. #19127780C22semidirect
42. C8×D332C22semidirect
48. #19127454C22semidirect
49. #19127454C22semidirect
50. #19127452C22semidirect
51. #19127452C22semidirect
52. #19127433C22semidirect
53. #19127433C22semidirect
58. #19127430C22semidirect
59. #19127430C22semidirect
63. #322×C83C22semidirect
64. #322×C83C22semidirect
65. #322×C83C22semidirect
66. C82×C83C22semidirect
320. #1453D83semidirect
321. C83#18006semidirect
369. #322C2×D83semidirect
370. #322C2×D83semidirect
371. #322C2×D83semidirect
372. C82C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249175122
#249164672
M16×D3322
#249063782
#249063552
#249063432
#249063292
#249063012
#249062942
#249062772
C8×D6642
#249062522
#249059292
#249058552
C83×#14532
#1800683