Finite Group Databasebeta

← #35375433

Subgroups — #35375433

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#24917508C211yesmaximal, characteristic
2#24916464C211yesmaximal, characteristic, complemented
3D4×#13050573C211yesmaximal, characteristic, complemented
4#24906374C211yesmaximal, characteristic, complemented
5#24906355C211yesmaximal, characteristic, complemented
6#24906340C211yesmaximal, characteristic, complemented
7#24906328C211yesmaximal, characteristic, complemented
8#24906299C211yesmaximal, characteristic, complemented
9#24906292C211yesmaximal, characteristic, complemented
10#24906272C211yesmaximal, characteristic, complemented
11#24906253C211yesmaximal, characteristic, complemented
12#24905925C211yesmaximal, characteristic, complemented
13#24905852C211yesmaximal, characteristic, complemented
14#24905757C211yesmaximal, characteristic, complemented
15C8×D8×C83C211yesmaximal, characteristic, complemented
16C22×#13050573—21no—
17#19128233C2211yescharacteristic
18C4×D4×D83C2211yescharacteristic
19#19128038C411yescharacteristic
20#19128018C411yescharacteristic
21#19128004C411yescharacteristic
22#19128003C411yescharacteristic
23#19127915C411yescharacteristic
24#19127914C411yescharacteristic
25#19127853C411yescharacteristic
26#19127833C411yescharacteristic
27#19127831C2211yescharacteristic, complemented
28#19127825C2211yescharacteristic, complemented
29C4×#13050579C2211yescharacteristic
30C4×#13050578C2211yescharacteristic
31C4×#13050575C2211yescharacteristic
32#19127780C2211yescharacteristic
33#19127779C2211yescharacteristic, complemented
34#19127778C2211yescharacteristic
35#19127776—21no—
36#19127772—21no—
37#19127772—21no—
38#19127769—21no—
39D4×C83⋊C8C2211yescharacteristic
40#19127759—21no—
41#19127753C2211yescharacteristic
42#19127752C2211yescharacteristic
43#19127752C2211yescharacteristic
44#19127723C2211yescharacteristic
45#19127663—21no—
46#19127656—21no—
47C4×#13050573C2211yescharacteristic
48#19127453C2211yescharacteristic, complemented
49#19127452C2211yescharacteristic, complemented
50#19127451C2211yescharacteristic, complemented
51#19127433C2211yescharacteristic, complemented
52#19127431C2211yescharacteristic, complemented
53#19127430C2211yescharacteristic
54D4⋊(C83⋊C8)C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic, complemented
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic, complemented
58#19127423C2211yescharacteristic, complemented
59#19127422C2211yescharacteristic, complemented
60C4×D8×C83C2211yescharacteristic
61C8×D4×C83C2211yescharacteristic
62C8×D4×C83C2211yescharacteristic, complemented
63#335×C83C2211yescharacteristic, complemented
64D4⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic, complemented
66C82×C83C2211yescharacteristic, complemented
67C2×D4×D83C2×C411yescharacteristic
68C22×C4×D83—21no—
69C2×#13050579C2×C411yescharacteristic
70C2×#13050578C2×C411yescharacteristic
71C2×#13050575C2×C411yescharacteristic
72C2×#13050574—21no—
73C2×#13050573—21no—
74C2×#13050573—21no—
75C2×#13050573—21no—
76C2×#13050573—44no—
77C2×#13050573D411yescharacteristic
78C2×#13050573D411yescharacteristic
79C22×C83⋊C8—21no—
80#14581785—21no—
81#14581783C2×C411yescharacteristic
82#14581779C2×C411yescharacteristic
83#14581774—21no—
84#14581772C2×C411yescharacteristic
85#14581763C2×C411yescharacteristic
86C4×Dic166C2311yescharacteristic
87C4×D332C2311yescharacteristic
88D4×Dic83C2×C411yescharacteristic
89#14581754C2×C411yescharacteristic
90C4×C22⋊D83—21no—
91C4×C22⋊D83—21no—
92(C22⋊D83)⋊C4—21no—
93#14581751—21no—
94C4⋊C4×D83C2×C411yescharacteristic
95C22⋊C4×D83—21no—
96C42×D83C2311yescharacteristic
97#14581725C2×C411yescharacteristic
98#14581724C2×C411yescharacteristic
99#14581723C2×C411yescharacteristic
100C4⋊(C83⋊C8)C2311yescharacteristic
101C4⋊(C83⋊C8)C2311yescharacteristic
102Dic83⋊C8—21no—
103Dic83⋊C8—21no—
104Dic83⋊C8—21no—
105Dic83⋊C8—21no—
106Dic83⋊C8C2×C411yescharacteristic
107Dic83⋊C8C2×C411yescharacteristic
108#14581717C2×C411yescharacteristic
109#14581715C2×C411yescharacteristic
110#14581714C2×C411yescharacteristic
111#14581713C2×C411yescharacteristic
112#14581713C2×C411yescharacteristic
113#14581712C2×C411yescharacteristic
114C22⋊(C83⋊C8)—21no—
115C22⋊(C83⋊C8)—21no—
116(C2×D83)⋊C8—21no—
117(C2×D83)⋊C8—21no—
118(C2×D83)⋊C8—21no—
119(C2×D83)⋊C8—21no—
120(C2×D83)⋊C8C2×C411yescharacteristic
121(C2×D83)⋊C8C2×C411yescharacteristic
122#14581702—21no—
123#14581702D411yescharacteristic
124#14581702D411yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C4×C83⋊C8C2311yescharacteristic
128C8×Dic83—21no—
129C8×Dic83—21no—
130C2×D8×C83C2×C411yescharacteristic
131C22×C8×C83—21no—
132C22×C8×C83—21no—
133C4×D4×C83C2311yescharacteristic
134C4×D4×C83C2311yescharacteristic
135#107×C83C2×C411yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137C4⋊C8×C83C2311yescharacteristic
138#102×C83C2×C411yescharacteristic
139#102×C83C2×C411yescharacteristic
140C22⋊C8×C83—21no—
141C22⋊C8×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C23×D83—21no—
146C2×Dic166C22×C411yescharacteristic
147C2×D332C22×C411yescharacteristic
148C2×C22⋊D83—21no—
149C2×C22⋊D83—21no—
150D4×D83—22no—
151D4×D83—41no—
152C22×Dic83—21no—
153C2×C4×D83—21no—
154C2×C4×D83—21no—
155C2×C4×D83—21no—
156C2×C4×D83—42no—
157C2×C4×D83—42no—
158C2×C4×D83C22×C411yescharacteristic
159C2×C4×D83C2×D411yescharacteristic
160C2×C4×D83#4111yescharacteristic
161#13050579—41no—
162#13050578—22no—
163#13050575—41no—
164#13050574—42no—
165#13050573—22no—
166#13050573—41no—
167#13050573—42no—
168#13050573—42no—
169#13050573—42no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8—21no—
172C2×C83⋊C8—21no—
173C2×C83⋊C8—42no—
174C2×C83⋊C8C22×C411yescharacteristic
175C2×C83⋊C8C22×C411yescharacteristic
176C2×C83⋊C8C2×D411yescharacteristic
177C2×C83⋊C8C2×D411yescharacteristic
178C4⋊Dic83—21no—
179C4⋊Dic83—21no—
180C4⋊Dic83C22×C411yescharacteristic
181C4⋊Dic83C22×C411yescharacteristic
182Dic83⋊C4—21no—
183Dic83⋊C4—21no—
184Dic83⋊C4—21no—
185C22⋊Dic83—21no—
186C22⋊Dic83—21no—
187C22⋊Dic83—21no—
188C22⋊Dic83—21no—
189(C2×D83)⋊C4—21no—
190(C2×D83)⋊C4—21no—
191(C2×D83)⋊C4—21no—
192C4×Dic83—21no—
193C4×Dic83C22×C411yescharacteristic
194C4×Dic83C2×D411yescharacteristic
195C4×Dic83#4111yescharacteristic
196C2×D4×C83C22×C411yescharacteristic
197C2×D4×C83C22×C411yescharacteristic
198C22×C4×C83—21no—
199C22×C4×C83—21no—
200D8×C83—22no—
201C2×C8×C83—21no—
202C2×C8×C83—21no—
203C2×C8×C83—21no—
204C2×C8×C83—42no—
205C2×C8×C83—42no—
206C2×C8×C83C22×C411yescharacteristic
207C2×C8×C83C22×C411yescharacteristic
208C2×C8×C83C2×D411yescharacteristic
209C2×C8×C83C2×D411yescharacteristic
210C4⋊C4×C83C22×C411yescharacteristic
211C4⋊C4×C83C22×C411yescharacteristic
212C22⋊C4×C83—21no—
213C22⋊C4×C83—21no—
214C42×C83C2411yescharacteristic
215C23×C83—21no—
216C23×C83—21no—
217C22×D83—21no—
218C22×D83—42no—
219C22×D83—42no—
220C22×D83—81no—
221C22×D83C4×D411yescharacteristic
222D4×C83—21no—
223D4×C83—21no—
224D4×C83—21no—
225D4×C83#13112yes—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83C23×C411yescharacteristic
233C2×C4×C83C22×D411yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C22⋊D83—42no—
236C22⋊D83—81no—
237C2×Dic83—21no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—42no—
241C2×Dic83C4×D411yescharacteristic
242C2×Dic83C4×D411yescharacteristic
243C2×Dic83C4×D411yescharacteristic
244D332—21no—
245D332—21no—
246C4×D83—21no—
247C4×D83—21no—
248C4×D83—21no—
249C4×D83—21no—
250C4×D83—41no—
251C4×D83—42no—
252C4×D83—42no—
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×C83—41no—
263C8×C83AGL(1,ℤ/8)12yes—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—21no—
267C83⋊C8—21no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83—22no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83—41no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83C2×C4×D411yescharacteristic
278C2×D83—21no—
279C2×D83—21no—
280C2×D83—21no—
281C2×D83—41no—
282C2×D83—41no—
283C2×D83—41no—
284C2×D83—84no—
285C4×C83—21no—
286C4×C83—41no—
287C4×C83—41no—
288C4×C83C2×#13111yescharacteristic
289C4×C83#56811yescharacteristic
290C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
291C4×C83#57511yescharacteristic
292Dic83—21no—
293Dic83—21no—
294Dic83—21no—
295Dic83—41no—
296Dic83—41no—
297Dic83—41no—
298#18006—831nomaximal, Sylow
299C2×C83—42no—
300C2×C83—81no—
301C2×C83#293311yescharacteristic
302C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
303C2×C83#295711yescharacteristic
304D83—41no—
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×D8D8311yescharacteristic, complemented
317#1574—831no—
318#1495—831no—
319#1478—831no—
320#1453—831no—
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×D8C2×D8311yescharacteristic
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×D4C2×D8311yescharacteristic
348C8×D4C2×D8311yescharacteristic, complemented
349#433—1661no—
350#431—831no—
351#431—831no—
352#431—831no—
353#423—831no—
354#407—1661no—
355#405—1661no—
356C4×M16—831no—
357#335—831no—
358#335C2×D8311yescharacteristic, complemented
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⋊C8C2×D8311yescharacteristic
368D4⋊C8C2×D8311yescharacteristic, complemented
369#322—831no—
370#322—831no—
371#322—831no—
372C82C2×D8311yescharacteristic, complemented
373C22×D4—831no—
374C23×C4—1661no—
375C2×SD16—831no—
376C2×SD16—831no—
377C2×D8—831no—
378C2×D8C4×D8311yescharacteristic
379C2×M16—3324no—
380C2×M16—831no—
381C2×M16—831no—
382C2×M16—1661no—
383C2×M16—1661no—
384C2×M16—1661no—
385C2×M16—1661no—
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—831no—
397C4×D4—831no—
398C4×D4—831no—
399C4×D4—1661no—
400C4×D4—1661no—
401C4×D4—1661no—
402C4×D4C22×D8311yescharacteristic
403C4×D4C22×D8311yescharacteristic
404#117—1661no—
405C2×C4⋊C4—831no—
406C2×C22⋊C4—1661no—
407C2×C42—831no—
408#107—831no—
409#107C4×D8311yescharacteristic
410#106—831no—
411#106—831no—
412C4⋊C8—831no—
413C4⋊C8—831no—
414C4⋊C8—831no—
415C4⋊C8—831no—
416C4⋊C8—1661no—
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#102C4×D8311yescharacteristic
429#102C4×D8311yescharacteristic
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—831no—
433C22⋊C8—831no—
434C22⋊C8—1661no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438C22⋊C8—1661no—
439C22⋊C8—1661no—
440#97—831no—
441#97—831no—
442#97—1661no—
443C4×C8—831no—
444C4×C8—831no—
445C4×C8—831no—
446C4×C8—1661no—
447C4×C8—1661no—
448C4×C8C22×D8311yescharacteristic
449C4×C8C22×D8311yescharacteristic
450C4×C8C22×D8311yescharacteristic
451C24—1661no—
452C2×Q8—831no—
453C2×D4—3321no—
454C2×D4—831no—
455C2×D4—1661no—
456C2×D4—1662no—
457C2×D4—1661no—
458C2×D4C2×C4×D8311yescharacteristic
459C2×D4C2×C4×D8311yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—3322no—
463C22×C4—3322no—
464C22×C4—831no—
465C22×C4—831no—
466C22×C4—831no—
467C22×C4—1661no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1661no—
471SD16—3321no—
472SD16—1662no—
473D8—22no—
474D8—3321no—
475M16—3321no—
476M16—3322no—
477M16—3322no—
478M16—3322no—
479M16—3322no—
480M16—1662no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—3322no—
487C2×C8—831no—
488C2×C8—831no—
489C2×C8—831no—
490C2×C8—831no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8C2×C4×D8311yescharacteristic
495C2×C8C2×C4×D8311yescharacteristic
496C2×C8D4×D8311yescharacteristic
497C2×C8D4×D8311yescharacteristic
498C4⋊C4—831no—
499C4⋊C4—831no—
500C4⋊C4—1661no—
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—1661no—
510C22⋊C4—1661no—
511C22⋊C4—1661no—
512C22⋊C4—1661no—
513C22⋊C4—1661no—
514C22⋊C4—1661no—
515C22⋊C4—1661no—
516C42—831no—
517C42—831no—
518C42—831no—
519C42—1661no—
520C42C23×D8311yescharacteristic
521C23—21no—
522C23—21no—
523C23—3322no—
524C23—3322no—
525C23—831no—
526C23—6641no—
527C23—1661no—
528Q8—1661no—
529Q8—1661no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—3322no—
534D4—6641no—
535D4—1661no—
536D4—1661no—
537D4#1458179812yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—3321no—
545C2×C4—3322no—
546C2×C4—3322no—
547C2×C4—3322no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—831no—
551C2×C4—831no—
552C2×C4—831no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4—1661no—
559C2×C4—1661no—
560C2×C4C22×C4×D8311yescharacteristic
561C2×C4C2×D4×D8311yescharacteristic
562C2×C4#41×D8311yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—3321no—
570C8—3321no—
571C8—3321no—
572C8—1661no—
573C8—1661no—
574C8—1661no—
575C8—1661no—
576C8#1458181612yes—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—3321no—
584C22—3321no—
585C22—3321no—
586C22—6644no—
587C22—1661no—
588C22—1661no—
589C22—1661no—
590C22C4×D4×D8311yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—3321no—
595C4—3321no—
596C4—3321no—
597C4—1661no—
598C4—1661no—
599C4—1661no—
600C4C2×#1458179811yescharacteristic
601C4#1912846611yescharacteristic
602C4C2×#1458181611yescharacteristic
603C4#1912849811yescharacteristic
604C2—42no—
605C2—81no—
606C2—3321no—
607C2—6641no—
608C2#2491618311yescharacteristic
609C2C4×#1458181611yescharacteristic
610C2#2491637511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC8×D8×C83C2
socleC22×C83C2×C4×D4
prime radical#19127833C4
cosocleC42×C83C24
hypercenterC8×D8D83
hypocenterC83#18006
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D8

Complements

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

NH ≅ G/Ntype
2. #24916464C2semidirect
3. D4×#13050573C2semidirect
4. #24906374C2semidirect
5. #24906355C2semidirect
6. #24906340C2semidirect
7. #24906328C2semidirect
8. #24906299C2semidirect
9. #24906292C2semidirect
10. #24906272C2semidirect
11. #24906253C2semidirect
12. #24905925C2semidirect
13. #24905852C2semidirect
14. #24905757C2semidirect
15. C8×D8×C83C2semidirect
27. #19127831C22semidirect
28. #19127825C22semidirect
33. #19127779C22semidirect
48. #19127453C22semidirect
49. #19127452C22semidirect
50. #19127451C22semidirect
51. #19127433C22semidirect
52. #19127431C22semidirect
55. D4⋊(C83⋊C8)C22semidirect
57. #19127423C22semidirect
58. #19127423C22semidirect
59. #19127422C22semidirect
62. C8×D4×C83C22semidirect
63. #335×C83C22semidirect
65. D4⋊C8×C83C22semidirect
66. C82×C83C22semidirect
316. C8×D8D83semidirect
321. C83#18006semidirect
348. C8×D4C2×D83semidirect
358. #335C2×D83semidirect
368. D4⋊C8C2×D83semidirect
372. C82C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249175082
#249164642
D4×#130505732
#249063742
#249063552
#249063402
#249063282
#249062992
#249062922
#249062722
#249062532
#249059252
#249058522
#249057572
C8×D8×C832
#1800683