Finite Group Databasebeta

← #35416491

Subgroups — #35416491

Order 21248.

796 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#125×Dic83C211yesmaximal, characteristic, complemented
2#24909928C211yesmaximal, characteristic
3#24908421C211yesmaximal, characteristic, complemented
4#24908233C211yesmaximal, characteristic, complemented
5#24908197C211yesmaximal, characteristic, complemented
6#24908192C211yesmaximal, characteristic, complemented
7#24908148C211yesmaximal, characteristic
8#24908148C211yesmaximal, characteristic
9#24908092C211yesmaximal, characteristic
10#24908085C211yesmaximal, characteristic
11#24908057C211yesmaximal, characteristic, complemented
12#24908050C211yesmaximal, characteristic, complemented
13#24907473C211yesmaximal, characteristic
14C4⋊C8×Dic83C211yesmaximal, characteristic, complemented
15C2×#478×C83C211yesmaximal, characteristic
16C2×#14581751—21no—
17C2×#14581726C2211yescharacteristic
18C2×#14581724C2211yescharacteristic
19C2×C4⋊(C83⋊C8)C2211yescharacteristic
20C2×#14581713C2211yescharacteristic
21C2×#14581713C2211yescharacteristic
22C2×C8×Dic83C2211yescharacteristic
23C2×C8×Dic83C2211yescharacteristic
24#19127604C2211yescharacteristic
25#19127604C2211yescharacteristic
26#19127600C2211yescharacteristic
27#19127600C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127600C2211yescharacteristic
30#19127594—21no—
31#19127594—21no—
32#19127587—21no—
33#19127587—21no—
34#19127567C2211yescharacteristic
35C4⋊C4×Dic83—21no—
36C4⋊C4×Dic83—21no—
37C4⋊C4×Dic83C2211yescharacteristic
38C4⋊C4×Dic83C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40C4×C22⋊Dic83—21no—
41C42×Dic83C2211yescharacteristic
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
52C2×#125×C83C2211yescharacteristic
53C2×C4×D4×C83C2211yescharacteristic
54#478×C83C418yescomplemented
55C2×#107×C83C2211yescharacteristic
56C2×#106×C83C2211yescharacteristic
57C2×C4⋊C8×C83C2211yescharacteristic
58C2×#102×C83C2211yescharacteristic
59C2×#102×C83C2211yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62#14581751—44no—
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C22⋊Dic83—21no—
72C2×C22⋊Dic83—21no—
73C2×C22⋊Dic83—21no—
74C2×C22⋊Dic83—21no—
75C2×C4×Dic83—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83—21no—
78C2×C4×Dic83D411yescharacteristic
79C2×C4×Dic83D411yescharacteristic
80C2×C4×Dic83D411yescharacteristic
81C2×C4×Dic83D411yescharacteristic
82C2×C4×Dic83C2311yescharacteristic
83C2×C4×Dic83C2311yescharacteristic
84#14581726—41no—
85#14581724—41no—
86C4⋊(C83⋊C8)—22no—
87#14581713—41no—
88#14581713—41no—
89C8×Dic83—41no—
90C8×Dic83—41no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C22×D4×C83C2311yescharacteristic
101C23×C4×C83—21no—
102C22×C8×C83C2311yescharacteristic
103C22×C8×C83C2311yescharacteristic
104#125×C83—22no—
105#125×C83C2×C414yes—
106C4×D4×C83—22no—
107C4×D4×C83C2×C414yes—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C22⋊C4×C83—21no—
114C2×C42×C83C2311yescharacteristic
115#107×C83C2×C414yes—
116#106×C83C2×C414yes—
117C4⋊C8×C83C2×C414yes—
118#102×C83C2×C414yes—
119#102×C83C2×C414yes—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83C2×D411yescharacteristic
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C22×Dic83C2×D411yescharacteristic
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C4⋊Dic83—21no—
135C4⋊Dic83—21no—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—42no—
143C4⋊Dic83—42no—
144C4⋊Dic83—42no—
145C4⋊Dic83—42no—
146C22⋊Dic83—42no—
147C22⋊Dic83—42no—
148C22⋊Dic83—42no—
149C22⋊Dic83—42no—
150C4×Dic83—21no—
151C4×Dic83—21no—
152C4×Dic83—21no—
153C4×Dic83—21no—
154C4×Dic83—22no—
155C4×Dic83—22no—
156C4×Dic83—22no—
157C4×Dic83—41no—
158C4×Dic83—42no—
159C4×Dic83—42no—
160C4×Dic83—42no—
161C24×C83—21no—
162C2×D4×C83—21no—
163C2×D4×C83—21no—
164C2×D4×C83—21no—
165C2×D4×C83—21no—
166C2×D4×C83—21no—
167C2×D4×C83C22×C412yes—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—24no—
172C22×C4×C83—42no—
173C22×C4×C83—42no—
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2411yescharacteristic
177C2×C8×C83—21no—
178C2×C8×C83—21no—
179C2×C8×C83—21no—
180C2×C8×C83—21no—
181C2×C8×C83C22×C412yes—
182C2×C8×C83C22×C412yes—
183C4⋊C4×C83—24no—
184C4⋊C4×C83—24no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83C22×C412yes—
189C4⋊C4×C83C22×C412yes—
190C4⋊C4×C83C22×C412yes—
191C22⋊C4×C83—24no—
192C42×C83C22×C412yes—
193C42×C83#4112yes—
194C23×C83—22no—
195C23×C83—41no—
196C23×C83—41no—
197C23×C83—41no—
198C23×C83—41no—
199C23×C83—41no—
200C23×C83—41no—
201C23×C83C22×D411yescharacteristic
202D4×C83—22no—
203D4×C83—22no—
204D4×C83—41no—
205D4×C83—41no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—22no—
211C2×C4×C83—22no—
212C2×C4×C83—22no—
213C2×C4×C83—48no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83C4×D412yes—
221C2×C4×C83C4×D412yes—
222C2×C4×C83C23×C411yescharacteristic
223C2×C4×C83C2×#4111yescharacteristic
224C2×C4×C83H(5,𝔽2)11yescharacteristic
225C2×C4×C832−1+411yescharacteristic
226C2×C4×C832−1+411yescharacteristic
227C2×C4×C832−1+411yescharacteristic
228C2×Dic83—21no—
229C2×Dic83—21no—
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83#13512yes—
250C2×Dic83#13512yes—
251C2×Dic83AGL(1,ℤ/8)12yes—
252C8×C83—22no—
253C8×C83—22no—
254C83⋊C8—41no—
255C83⋊C8—41no—
256C22×C83—42no—
257C22×C83—42no—
258C22×C83—42no—
259C22×C83—81no—
260C22×C83—81no—
261C22×C83—81no—
262C22×C83—81no—
263C22×C83C2×C4×D411yescharacteristic
264C22×C83#53611yescharacteristic
265C22×C83#53711yescharacteristic
266C22×C83C2×#13511yescharacteristic
267C22×C83C2×#13511yescharacteristic
268C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C83#57511yescharacteristic
270C4×C83—22no—
271C4×C83—22no—
272C4×C83—42no—
273C4×C83—42no—
274C4×C83—42no—
275C4×C83#51911yescharacteristic
276C4×C83#52011yescharacteristic
277C4×C83#54111yescharacteristic
278C4×C83#54211yescharacteristic
279Dic83—22no—
280Dic83—22no—
281Dic83—22no—
282Dic83—41no—
283Dic83—81no—
284Dic83—81no—
285Dic83—81no—
286Dic83—81no—
287#22114—831nomaximal, Sylow
288C2×C83—82no—
289C2×C83#232311yescharacteristic
290C2×C83C4×#13511yescharacteristic
291C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
292C2×C83#322811yescharacteristic
293C2×C83#323211yescharacteristic
294C2×C83#324711yescharacteristic
295C2×C83#325011yescharacteristic
296C2×#478D8311yescharacteristic
297C4×#125—831no—
298#2293—831no—
299#1989—831no—
300#1949—831no—
301#1942—831no—
302#1941—831no—
303#1934—831no—
304#1934—831no—
305#1926—831no—
306#1925—831no—
307#1921—831no—
308#1920—831no—
309#1804—831no—
310C4×C4⋊C8—831no—
311C83#2211411yescharacteristic, complemented, Sylow
312C2×#125C2×D8311yescharacteristic
313C2×C4×D4C2×D8311yescharacteristic
314C2×#117—1661no—
315#478Dic8318yescomplemented
316C2×#107—831no—
317C2×#107C2×D8311yescharacteristic
318C2×#106—831no—
319C2×#106C2×D8311yescharacteristic
320C2×C4⋊C8—831no—
321C2×C4⋊C8C2×D8311yescharacteristic
322C2×#102—831no—
323C2×#102—831no—
324C2×#102C2×D8311yescharacteristic
325C2×#102C2×D8311yescharacteristic
326C2×C4×C8—831no—
327C2×C4×C8—831no—
328#390—831no—
329#390—831no—
330#389—831no—
331#389—831no—
332#389—831no—
333#389—831no—
334#388—1661no—
335#388—1661no—
336#387—1661no—
337#387—1661no—
338#382—831no—
339C4×C4⋊C4—831no—
340C4×C4⋊C4—831no—
341C4×C4⋊C4—831no—
342C4×C4⋊C4—1661no—
343C4×C4⋊C4—1661no—
344C4×C22⋊C4—1661no—
345C43—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#340—831no—
353#340—831no—
354#336—831no—
355#336—831no—
356C22×D4C22×D8311yescharacteristic
357C23×C4—21no—
358C22×C8—831no—
359C22×C8—831no—
360C22×C8C22×D8311yescharacteristic
361C22×C8C22×D8311yescharacteristic
362#125—22no—
363#125C2×Dic8314yes—
364C4×D4—22no—
365C4×D4C2×Dic8314yes—
366#117—3324no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4C22×D8311yescharacteristic
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C4⋊C4C22×D8311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1661no—
382C2×C22⋊C4—1661no—
383C2×C22⋊C4—1661no—
384C2×C22⋊C4—1661no—
385C2×C42—831no—
386C2×C42—831no—
387C2×C42—831no—
388C2×C42—831no—
389C2×C42—831no—
390C2×C42—831no—
391C2×C42—1661no—
392C2×C42—1661no—
393C2×C42—1661no—
394C2×C42C22×D8311yescharacteristic
395#107—3321no—
396#107C2×Dic8314yes—
397#106—3321no—
398#106C2×Dic8314yes—
399C4⋊C8—1662no—
400C4⋊C8C2×Dic8314yes—
401#102—3321no—
402#102—3321no—
403#102C2×Dic8314yes—
404#102C2×Dic8314yes—
405C4×C8—3321no—
406C4×C8—3321no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412#95—1661no—
413#95—1661no—
414#95—1661no—
415#95—1661no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C22×Dic8312yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—831no—
430C22×C4—831no—
431C22×C4—831no—
432C22×C4—831no—
433C22×C4—1661no—
434C22×C4—1661no—
435C22×C4—1661no—
436C22×C4—1661no—
437C22×C4D4×D8311yescharacteristic
438C22×C4D4×D8311yescharacteristic
439C22×C4C23×D8311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8—1661no—
448C2×C8—1661no—
449C2×C8—1661no—
450C2×C8C22×Dic8312yes—
451C2×C8C22×Dic8312yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—3322no—
458C4⋊C4—3322no—
459C4⋊C4—3322no—
460C4⋊C4—3322no—
461C4⋊C4—1661no—
462C4⋊C4—1661no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4C22×Dic8312yes—
470C4⋊C4C22×Dic8312yes—
471C4⋊C4C22×Dic8312yes—
472C22⋊C4—24no—
473C22⋊C4—3322no—
474C22⋊C4—3322no—
475C22⋊C4—3322no—
476C22⋊C4—3322no—
477C42—3322no—
478C42—3321no—
479C42—3322no—
480C42—3322no—
481C42—1662no—
482C42—1662no—
483C42—1662no—
484C42—1661no—
485C42—1661no—
486C42—1661no—
487C42—1661no—
488C42C22×Dic8312yes—
489C42#1305058912yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D8311yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—48no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—3321no—
517C2×C4—3321no—
518C2×C4—3321no—
519C2×C4—3321no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—3321no—
523C2×C4—3321no—
524C2×C4—3321no—
525C2×C4—3321no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—832no—
529C2×C4—832no—
530C2×C4—832no—
531C2×C4—1661no—
532C2×C4—1661no—
533C2×C4—1661no—
534C2×C4—1661no—
535C2×C4—1661no—
536C2×C4—1661no—
537C2×C4—1661no—
538C2×C4—1661no—
539C2×C4—1661no—
540C2×C4D4×Dic8312yes—
541C2×C4D4×Dic8312yes—
542C2×C4C23×Dic8311yescharacteristic
543C2×C4C2×#1305058911yescharacteristic
544C2×C4#1458183811yescharacteristic
545C2×C4#1458183911yescharacteristic
546C2×C4#1458184011yescharacteristic
547C2×C4#1458184011yescharacteristic
548C8—22no—
549C8—22no—
550C8—3321no—
551C8—3321no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×D4×Dic8311yescharacteristic
560C22#1912827411yescharacteristic
561C22#1912828111yescharacteristic
562C22#135×D8311yescharacteristic
563C22#135×D8311yescharacteristic
564C22AGL(1,ℤ/8)×D8311yescharacteristic
565C22#1912849711yescharacteristic
566C4—22no—
567C4—22no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—3321no—
572C4—6641no—
573C4—6641no—
574C4—6641no—
575C4—6641no—
576C4—1662no—
577C4—1662no—
578C4—1662no—
579C4#1912817911yescharacteristic
580C4#1912818411yescharacteristic
581C4#1912831111yescharacteristic
582C4#1912832411yescharacteristic
583C2—82no—
584C2#2491014711yescharacteristic
585C2#135×Dic8311yescharacteristic
586C2AGL(1,ℤ/8)×Dic8311yescharacteristic
587C2#2491859811yescharacteristic
588C2#2491862911yescharacteristic
589C2#2491878811yescharacteristic
590C2#2491881411yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#478×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#478D83
hypocenterC83#22114
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #125×Dic83C2semidirect
3. #24908421C2semidirect
4. #24908233C2semidirect
5. #24908197C2semidirect
6. #24908192C2semidirect
11. #24908057C2semidirect
12. #24908050C2semidirect
14. C4⋊C8×Dic83C2semidirect
54. #478×C83C4semidirect
311. C83#22114semidirect
315. #478Dic83semidirect

Maximal subgroups

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

subgroupindex
#125×Dic832
#249099282
#249084212
#249082332
#249081972
#249081922
#249081482
#249080922
#249080852
#249080572
#249080502
#249074732
C4⋊C8×Dic832
C2×#478×C832
#2211483