Finite Group Databasebeta

← #35375376

Subgroups — #35375376

Order 21248.

740 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917461C211yesmaximal, characteristic
2#24916208C211yesmaximal, characteristic, complemented
3D4×#13050573C211yesmaximal, characteristic, complemented
4#24906376C211yesmaximal, characteristic, complemented
5#24906375C211yesmaximal, characteristic, complemented
6#24906329C211yesmaximal, characteristic, complemented
7#24906324C211yesmaximal, characteristic, complemented
8#24906324C211yesmaximal, characteristic, complemented
9#24906287C211yesmaximal, characteristic, complemented
10#24906285C211yesmaximal, characteristic, complemented
11#24906232C211yesmaximal, characteristic, complemented
12#24905912C211yesmaximal, characteristic, complemented
13D4⋊C8×D83C211yesmaximal, characteristic, complemented
14#24905749C211yesmaximal, characteristic, complemented
15C83×#1586C211yesmaximal, characteristic, complemented
16C2×#14581800—21no—
17C22×#13050573—21no—
18#19128248C2211yescharacteristic
19C4×D4×D83C2211yescharacteristic
20#19128016C411yescharacteristic
21#19128013C411yescharacteristic
22#19127980C411yescharacteristic
23#19127979C411yescharacteristic
24#19127895C411yescharacteristic
25#19127894C411yescharacteristic
26#19127836C411yescharacteristic
27#19127833C411yescharacteristic
28C4×D664C2211yescharacteristic
29C4×#13050575C2211yescharacteristic
30C4×#13050575C2211yescharacteristic
31#19127780C2211yescharacteristic
32#19127778C2211yescharacteristic
33#19127778C2211yescharacteristic, complemented
34#19127772—21no—
35#19127770—21no—
36#19127769—21no—
37#19127769—21no—
38C8×D332C2211yescharacteristic, complemented
39D4×C83⋊C8C2211yescharacteristic
40D4×C83⋊C8C2211yescharacteristic, complemented
41#19127752C2211yescharacteristic
42#19127723C2211yescharacteristic
43C4⋊C8×D83C2211yescharacteristic
44#19127666—21no—
45#19127663—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127452C2211yescharacteristic, complemented
49#19127451C2211yescharacteristic, complemented
50#19127451C2211yescharacteristic, complemented
51#19127430C2211yescharacteristic
52#19127430C2211yescharacteristic, complemented
53D4⋊(C83⋊C8)C2211yescharacteristic
54D4⋊(C83⋊C8)C2211yescharacteristic, complemented
55#19127428C2211yescharacteristic
56#19127428C2211yescharacteristic, complemented
57#19127423C2211yescharacteristic, complemented
58#19127422C2211yescharacteristic, complemented
59#19127422C2211yescharacteristic, complemented
60C4×D8×C83C2211yescharacteristic
61#436×C83C2211yescharacteristic, complemented
62C8×D4×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic, complemented
64D4⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic, complemented
66#322×C83C2211yescharacteristic, complemented
67C2×#13050590—21no—
68C2×D4×D83C2×C411yescharacteristic
69C22×C4×D83—21no—
70C2×D664C2×C411yescharacteristic
71C2×#13050575C2×C411yescharacteristic
72C2×#13050575C2×C411yescharacteristic
73#14581800—44no—
74C2×#13050573—21no—
75C2×#13050573—21no—
76C2×#13050573—21no—
77C2×#13050573—21no—
78C2×#13050573—44no—
79C2×#13050573D411yescharacteristic
80C2×#13050573D411yescharacteristic
81C22×C83⋊C8—21no—
82C22×C83⋊C8—21no—
83C2×C8×D83—21no—
84C2×C8×D83—21no—
85C2×C8×D83—21no—
86#14581791C2×C411yescharacteristic
87#14581788C2×C411yescharacteristic
88#14581778—21no—
89#14581767C2×C411yescharacteristic
90#14581765—21no—
91#14581763C2×C411yescharacteristic
92C4×D332C2311yescharacteristic
93C4×D332C2311yescharacteristic
94D4×Dic83C2×C411yescharacteristic
95#14581754C2×C411yescharacteristic
96C4×C22⋊D83—21no—
97C4×C22⋊D83—21no—
98(C22⋊D83)⋊C4—21no—
99C4⋊C4×D83C2×C411yescharacteristic
100C22⋊C4×D83—21no—
101C42×D83C2311yescharacteristic
102#14581726C2×C411yescharacteristic
103#14581725C2×C411yescharacteristic
104#14581725C2×C411yescharacteristic
105C4⋊(C83⋊C8)C2311yescharacteristic
106C4⋊(C83⋊C8)C2311yescharacteristic
107Dic83⋊C8—21no—
108Dic83⋊C8—21no—
109Dic83⋊C8C2×C411yescharacteristic
110Dic83⋊C8C2×C411yescharacteristic
111#14581714C2×C411yescharacteristic
112#14581714C2×C411yescharacteristic
113#14581713C2×C411yescharacteristic
114#14581713C2×C411yescharacteristic
115#14581712C2×C411yescharacteristic
116#14581712C2×C411yescharacteristic
117C22⋊(C83⋊C8)—21no—
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)⋊C8C2×C411yescharacteristic
124(C2×D83)⋊C8C2×C411yescharacteristic
125#14581702D411yescharacteristic
126#14581702D411yescharacteristic
127C4×C83⋊C8C2311yescharacteristic
128C4×C83⋊C8C2311yescharacteristic
129C4×C83⋊C8C2311yescharacteristic
130C8×Dic83—21no—
131C2×D8×C83C2×C411yescharacteristic
132C2×M16×C83—21no—
133C22×C8×C83—21no—
134C4×D4×C83C2311yescharacteristic
135C4×D4×C83C2311yescharacteristic
136#107×C83C2×C411yescharacteristic
137C4⋊C8×C83C2311yescharacteristic
138C4⋊C8×C83C2311yescharacteristic
139#102×C83C2×C411yescharacteristic
140#102×C83C2×C411yescharacteristic
141C22⋊C8×C83—21no—
142C22⋊C8×C83—21no—
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C4×C8×C83C2311yescharacteristic
146C23×D83—21no—
147#13050590—42no—
148#13050590—42no—
149C2×Dic166—21no—
150C2×D332—21no—
151C2×D332C22×C411yescharacteristic
152C2×D332C22×C411yescharacteristic
153C2×C22⋊D83—21no—
154C2×C22⋊D83—21no—
155C2×C22⋊D83—21no—
156C2×C22⋊D83—21no—
157D4×D83—22no—
158D4×D83—41no—
159C22×Dic83—21no—
160C2×C4×D83—21no—
161C2×C4×D83—21no—
162C2×C4×D83—21no—
163C2×C4×D83—21no—
164C2×C4×D83—42no—
165C2×C4×D83—42no—
166C2×C4×D83C22×C411yescharacteristic
167C2×C4×D83C2×D411yescharacteristic
168C2×C4×D83#4111yescharacteristic
169D664—41no—
170#13050575—22no—
171#13050575—41no—
172#13050573—22no—
173#13050573—22no—
174#13050573—42no—
175#13050573—42no—
176#13050573—42no—
177#13050573—42no—
178C2×C83⋊C8—21no—
179C2×C83⋊C8—21no—
180C2×C83⋊C8—21no—
181C2×C83⋊C8—42no—
182C2×C83⋊C8—42no—
183C2×C83⋊C8C22×C411yescharacteristic
184C2×C83⋊C8C22×C411yescharacteristic
185C2×C83⋊C8C2×D411yescharacteristic
186C2×C83⋊C8C2×D411yescharacteristic
187C8×D83—42no—
188C8×D83—42no—
189C8×D83—42no—
190C4⋊Dic83C22×C411yescharacteristic
191C4⋊Dic83C22×C411yescharacteristic
192Dic83⋊C4—21no—
193Dic83⋊C4—21no—
194C22⋊Dic83—21no—
195C22⋊Dic83—21no—
196(C2×D83)⋊C4—21no—
197(C2×D83)⋊C4—21no—
198(C2×D83)⋊C4—21no—
199(C2×D83)⋊C4—21no—
200C4×Dic83C22×C411yescharacteristic
201C4×Dic83C2×D411yescharacteristic
202C4×Dic83#4111yescharacteristic
203C2×D4×C83C22×C411yescharacteristic
204C2×D4×C83C22×C411yescharacteristic
205C22×C4×C83—21no—
206C22×C4×C83—21no—
207D8×C83—22no—
208M16×C83—42no—
209C2×C8×C83—21no—
210C2×C8×C83—21no—
211C2×C8×C83—21no—
212C2×C8×C83—42no—
213C2×C8×C83C22×C411yescharacteristic
214C2×C8×C83C22×C411yescharacteristic
215C2×C8×C83C2×D411yescharacteristic
216C2×C8×C83C2×D411yescharacteristic
217C4⋊C4×C83C22×C411yescharacteristic
218C4⋊C4×C83C22×C411yescharacteristic
219C22⋊C4×C83—21no—
220C22⋊C4×C83—21no—
221C42×C83C2411yescharacteristic
222C23×C83—21no—
223C23×C83—21no—
224C22×D83—21no—
225C22×D83—21no—
226C22×D83—42no—
227C22×D83—42no—
228C22×D83—81no—
229C22×D83C4×D411yescharacteristic
230D4×C83—21no—
231D4×C83—21no—
232D4×C83—21no—
233D4×C83#13112yescomplemented
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83C23×C411yescharacteristic
241C2×C4×C83C22×D411yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C22⋊D83—42no—
244C22⋊D83—42no—
245C22⋊D83—42no—
246C22⋊D83—81no—
247C2×Dic83—21no—
248C2×Dic83—21no—
249C2×Dic83—42no—
250C2×Dic83C4×D411yescharacteristic
251C2×Dic83C4×D411yescharacteristic
252C2×Dic83C4×D411yescharacteristic
253D332—21no—
254D332—21no—
255D332—21no—
256D332—21no—
257D332—42no—
258C4×D83—21no—
259C4×D83—21no—
260C4×D83—41no—
261C4×D83—42no—
262C4×D83—41no—
263C4×D83—41no—
264C4×D83—41no—
265C4×D83—41no—
266C4×D83—42no—
267C4×D83AGL(1,ℤ/8)12yes—
268C4×D83AGL(1,ℤ/8)12yes—
269Dic166—42no—
270C8×C83—22no—
271C8×C83—21no—
272C8×C83—21no—
273C8×C83—21no—
274C8×C83—21no—
275C8×C83—41no—
276C8×C83—41no—
277C83⋊C8—21no—
278C83⋊C8—21no—
279C83⋊C8—21no—
280C83⋊C8—21no—
281C83⋊C8—41no—
282C83⋊C8—41no—
283C83⋊C8—41no—
284C22×C83—22no—
285C22×C83—41no—
286C22×C83—41no—
287C22×C83—41no—
288C22×C83—41no—
289C22×C83—41no—
290C22×C83C2×C4×D411yescharacteristic
291C2×D83—21no—
292C2×D83—21no—
293C2×D83—41no—
294C2×D83—41no—
295C2×D83—41no—
296C2×D83—41no—
297C2×D83—41no—
298C2×D83—41no—
299C2×D83—84no—
300C2×D83#44212yes—
301C4×C83—21no—
302C4×C83—41no—
303C4×C83—41no—
304C4×C83C2×#13111yescharacteristic
305C4×C83#56811yescharacteristic
306C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
307C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
308Dic83—21no—
309Dic83—21no—
310Dic83—41no—
311Dic83—41no—
312Dic83#44212yes—
313#18001—831nomaximal, Sylow
314C2×C83—42no—
315C2×C83—81no—
316C2×C83#293311yescharacteristic
317C2×C83C2×#44211yescharacteristic
318C2×C83#295611yescharacteristic
319D83—22no—
320D83—81no—
321D83—81no—
322#3100—831no—
323#2935—831no—
324D4×M16—831no—
325#1594—831no—
326#1594—831no—
327#1586—831no—
328#1586—831no—
329#1586—831no—
330#1586D8311yescharacteristic, complemented
331#1580—831no—
332#1580—831no—
333#1570—831no—
334#1492—831no—
335C2×D4⋊C8—831no—
336#1451—831no—
337C83#1800111yescharacteristic, complemented, Sylow
338C2×#131—1661no—
339C22×M16—1661no—
340#532—831no—
341C2×C4×D4—831no—
342#480—831no—
343#480—831no—
344#474—831no—
345#474—831no—
346#459—831no—
347#459—831no—
348#447—831no—
349#447—831no—
350C4×D8—831no—
351C4×D8—831no—
352C4×D8—831no—
353C4×D8C2×D8311yescharacteristic
354#436—831no—
355#436—831no—
356#436—831no—
357#436C2×D8311yescharacteristic, complemented
358#435—1661no—
359#435—1661no—
360#435—1661no—
361#435—1661no—
362C8×D4—831no—
363C8×D4—831no—
364C8×D4—831no—
365C8×D4C2×D8311yescharacteristic
366#431—831no—
367#423—831no—
368C2×C4⋊C8—831no—
369#408—1661no—
370#407—1661no—
371C4×M16—831no—
372C2×C4×C8—831no—
373#334—831no—
374#334—831no—
375#334—831no—
376#334C2×D8311yescharacteristic, complemented
377D4⋊C8—831no—
378D4⋊C8—831no—
379D4⋊C8—831no—
380D4⋊C8—831no—
381D4⋊C8—831no—
382D4⋊C8—831no—
383D4⋊C8C2×D8311yescharacteristic
384D4⋊C8C2×D8311yescharacteristic, complemented
385#322—831no—
386#322—831no—
387#322—831no—
388#322C2×D8311yescharacteristic, complemented
389C2×#41—1661no—
390C22×D4—831no—
391C23×C4—1661no—
392C2×D8—831no—
393C2×D8—831no—
394C2×D8—831no—
395C2×D8C4×D8311yescharacteristic
396#131—3324no—
397C2×M16—21no—
398C2×M16—3324no—
399C2×M16—831no—
400C2×M16—831no—
401C2×M16—1661no—
402C2×M16—1661no—
403C2×M16—1661no—
404C2×M16—1661no—
405C22×C8—21no—
406C22×C8—1661no—
407C22×C8—1661no—
408C22×C8—1661no—
409C22×C8—1661no—
410C22×C8—1661no—
411C4⋊Q8—831no—
412#127—831no—
413#124—1661no—
414#121—831no—
415#121—831no—
416#121—1661no—
417C4×D4—831no—
418C4×D4—831no—
419C4×D4—831no—
420C4×D4—831no—
421C4×D4—1661no—
422C4×D4—1661no—
423C4×D4—1661no—
424C4×D4C22×D8311yescharacteristic
425C4×D4C22×D8311yescharacteristic
426C2×C4⋊C4—831no—
427C2×C22⋊C4—1661no—
428C2×C42—831no—
429#107—831no—
430#107—831no—
431#107—831no—
432#107C4×D8311yescharacteristic
433C4⋊C8—831no—
434C4⋊C8—831no—
435C4⋊C8—831no—
436C4⋊C8—831no—
437C4⋊C8—1661no—
438C4⋊C8—1661no—
439C4⋊C8C22×D8311yescharacteristic
440C4⋊C8C22×D8311yescharacteristic
441#102—831no—
442#102—831no—
443#102—831no—
444#102—831no—
445#102—831no—
446#102—831no—
447#102C4×D8311yescharacteristic
448#102C4×D8311yescharacteristic
449C22⋊C8—21no—
450C22⋊C8—21no—
451C22⋊C8—831no—
452C22⋊C8—831no—
453C22⋊C8—1661no—
454C22⋊C8—1661no—
455C22⋊C8—1661no—
456C22⋊C8—1661no—
457C22⋊C8—1661no—
458C22⋊C8—1661no—
459#97—831no—
460#97—831no—
461C4×C8—831no—
462C4×C8—831no—
463C4×C8—831no—
464C4×C8—1661no—
465C4×C8C22×D8311yescharacteristic
466C4×C8C22×D8311yescharacteristic
467C4×C8C22×D8311yescharacteristic
468C24—1661no—
469#41—3322no—
470#41—3322no—
471C2×Q8—1661no—
472C2×D4—3321no—
473C2×D4—831no—
474C2×D4—831no—
475C2×D4—1661no—
476C2×D4—1661no—
477C2×D4—1661no—
478C2×D4—1661no—
479C2×D4—1661no—
480C2×D4—1662no—
481C2×D4C2×C4×D8311yescharacteristic
482C2×D4C2×C4×D8311yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—3322no—
486C22×C4—3322no—
487C22×C4—831no—
488C22×C4—831no—
489C22×C4—831no—
490C22×C4—1661no—
491C22×C4—1661no—
492C22×C4—1661no—
493C22×C4—1661no—
494C22×C4—1661no—
495D8—22no—
496D8—3321no—
497D8—3321no—
498D8—1662no—
499M16—42no—
500M16—3322no—
501M16—3322no—
502M16—3322no—
503M16—3322no—
504M16—1662no—
505M16—1662no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—42no—
510C2×C8—3322no—
511C2×C8—3322no—
512C2×C8—3322no—
513C2×C8—3322no—
514C2×C8—3322no—
515C2×C8—831no—
516C2×C8—831no—
517C2×C8—831no—
518C2×C8—831no—
519C2×C8—1661no—
520C2×C8—1661no—
521C2×C8—1661no—
522C2×C8C2×C4×D8311yescharacteristic
523C2×C8C2×C4×D8311yescharacteristic
524C2×C8D4×D8311yescharacteristic
525C2×C8D4×D8311yescharacteristic
526C4⋊C4—831no—
527C4⋊C4—831no—
528C4⋊C4—1661no—
529C4⋊C4—1661no—
530C4⋊C4C2×C4×D8311yescharacteristic
531C4⋊C4C2×C4×D8311yescharacteristic
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—1661no—
535C22⋊C4—1661no—
536C22⋊C4—1661no—
537C22⋊C4—1661no—
538C22⋊C4—1661no—
539C22⋊C4—1661no—
540C42—831no—
541C42—831no—
542C42—831no—
543C42C23×D8311yescharacteristic
544C23—21no—
545C23—21no—
546C23—3322no—
547C23—3322no—
548C23—831no—
549C23—6641no—
550C23—1661no—
551C23—1661no—
552Q8—3322no—
553D4—21no—
554D4—21no—
555D4—21no—
556D4—3322no—
557D4—3322no—
558D4—3322no—
559D4—3322no—
560D4—6641no—
561D4—1661no—
562D4—1661no—
563D4—1661no—
564D4—1661no—
565D4#1458180012yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—3321no—
573C2×C4—3322no—
574C2×C4—3322no—
575C2×C4—3322no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—831no—
581C2×C4—832no—
582C2×C4—832no—
583C2×C4—831no—
584C2×C4—831no—
585C2×C4—1661no—
586C2×C4—1661no—
587C2×C4—1661no—
588C2×C4—1661no—
589C2×C4C22×C4×D8311yescharacteristic
590C2×C4C2×D4×D8311yescharacteristic
591C2×C4#41×D8311yescharacteristic
592C8—22no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—21no—
597C8—41no—
598C8—41no—
599C8—3321no—
600C8—3321no—
601C8—3321no—
602C8—1661no—
603C8—1661no—
604C8—1661no—
605C8—1661no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—3321no—
613C22—3321no—
614C22—3321no—
615C22—3321no—
616C22—3321no—
617C22—3321no—
618C22—832no—
619C22—6644no—
620C22—1661no—
621C22—1661no—
622C22C4×D4×D8311yescharacteristic
623C4—21no—
624C4—41no—
625C4—41no—
626C4—3321no—
627C4—3321no—
628C4—832no—
629C4—1661no—
630C4—1661no—
631C4C2×#1458180011yescharacteristic
632C4#1912846611yescharacteristic
633C4AGL(1,ℤ/8)×D8311yescharacteristic
634C4AGL(1,ℤ/8)×D8311yescharacteristic
635C2—42no—
636C2—81no—
637C2—6641no—
638C2—6641no—
639C2—1662no—
640C2#2491618911yescharacteristic
641C2#442×D8311yescharacteristic
642C2#2491636811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1586C2
socleC22×C83C2×C4×D4
prime radical#24917461C2
cosocleC42×C83C24
hypercenter#1586D83
hypocenterC83#18001
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. #24916208C2semidirect
3. D4×#13050573C2semidirect
4. #24906376C2semidirect
5. #24906375C2semidirect
6. #24906329C2semidirect
7. #24906324C2semidirect
8. #24906324C2semidirect
9. #24906287C2semidirect
10. #24906285C2semidirect
11. #24906232C2semidirect
12. #24905912C2semidirect
13. D4⋊C8×D83C2semidirect
14. #24905749C2semidirect
15. C83×#1586C2semidirect
33. #19127778C22semidirect
38. C8×D332C22semidirect
40. D4×C83⋊C8C22semidirect
48. #19127452C22semidirect
49. #19127451C22semidirect
50. #19127451C22semidirect
52. #19127430C22semidirect
54. D4⋊(C83⋊C8)C22semidirect
56. #19127428C22semidirect
57. #19127423C22semidirect
58. #19127422C22semidirect
59. #19127422C22semidirect
61. #436×C83C22semidirect
63. #334×C83C22semidirect
65. D4⋊C8×C83C22semidirect
66. #322×C83C22semidirect
233. D4×C83#131semidirect
330. #1586D83semidirect
337. C83#18001semidirect
357. #436C2×D83semidirect
376. #334C2×D83semidirect
384. D4⋊C8C2×D83semidirect
388. #322C2×D83semidirect
565. D4#14581800semidirect

Maximal subgroups

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

subgroupindex
#249174612
#249162082
D4×#130505732
#249063762
#249063752
#249063292
#249063242
#249062872
#249062852
#249062322
#249059122
D4⋊C8×D832
#249057492
C83×#15862
#1800183