Finite Group Databasebeta

← #35422099

Subgroups — #35422099

Order 21248.

866 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D83C211yesmaximal, characteristic
2#24910331C211yesmaximal, characteristic
3#24910328C211yesmaximal, characteristic
4#24908200C211yesmaximal, characteristic, complemented
5#24908198C211yesmaximal, characteristic, complemented
6#24908057C211yesmaximal, characteristic, complemented
7#24908056C211yesmaximal, characteristic, complemented
8#24908054C211yesmaximal, characteristic, complemented
9#24907548C211yesmaximal, characteristic
10#24907548C211yesmaximal, characteristic
11#24907487C211yesmaximal, characteristic
12#24907361C211yesmaximal, characteristic, complemented
13#24907359C211yesmaximal, characteristic, complemented
14#24907238C211yesmaximal, characteristic
15C83×#1921C211yesmaximal, characteristic, complemented
16C22×C8×D83C2211yescharacteristic
17C2×#14581750—21no—
18C2×#14581749—21no—
19C2×C4⋊C4×D83C2211yescharacteristic
20C2×C4⋊C4×D83C2211yescharacteristic
21C2×#14581726C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24#107×D83C418yescomplemented
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×(C2×D83)⋊C8C2211yescharacteristic
28C2×#14581703C2211yescharacteristic
29C2×#14581702C2211yescharacteristic
30#19127600C2211yescharacteristic
31#19127599C2211yescharacteristic
32#19127597C2211yescharacteristic
33#19127588—21no—
34#19127585—21no—
35#19127583—21no—
36#19127582—21no—
37#19127563—21no—
38#19127563—21no—
39#19127558C2211yescharacteristic
40#19127558C2211yescharacteristic
41C4⋊C4×Dic83C2211yescharacteristic
42C4⋊C4×Dic83C2211yescharacteristic
43C4×C4⋊Dic83C2211yescharacteristic
44C4×(C2×D83)⋊C4C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127455C2211yescharacteristic
53C2×#107×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55C4×C4⋊C4×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60C22×C4×D83—21no—
61C22×C4×D83—21no—
62C22×C4×D83C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C2×C8×D83—22no—
66C2×C8×D83—22no—
67C2×C8×D83C2×C414yes—
68#14581750—44no—
69#14581749—44no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C4⋊C4×D83—22no—
80C4⋊C4×D83—22no—
81C4⋊C4×D83C2×C414yes—
82C4⋊C4×D83C2×C414yes—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4—21no—
85C2×(C2×D83)⋊C4D411yescharacteristic
86C2×(C2×D83)⋊C4D411yescharacteristic
87C2×C4×Dic83—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83—21no—
90C2×C4×Dic83C2311yescharacteristic
91#14581726C2×C414yes—
92#14581725C2×C414yes—
93#14581725C2×C414yes—
94#14581723—22no—
95#14581723—22no—
96(C2×D83)⋊C8—22no—
97#14581703—22no—
98#14581702—22no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95D411yescharacteristic
108C83⋊#95D411yescharacteristic
109C22×C8×C83C2311yescharacteristic
110C22×C8×C83C2311yescharacteristic
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C42×C83—21no—
117C2×C42×C83C2311yescharacteristic
118#107×C83C2×C414yes—
119#95×C83—21no—
120#95×C83—21no—
121C23×D83C2×D411yescharacteristic
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83C2×D411yescharacteristic
127C22×Dic83C2×D411yescharacteristic
128C22×Dic83C2×D411yescharacteristic
129C2×C4×D83—21no—
130C2×C4×D83—21no—
131C2×C4×D83—21no—
132C2×C4×D83—21no—
133C2×C4×D83—24no—
134C2×C4×D83—24no—
135C2×C4×D83—42no—
136C2×C4×D83—42no—
137C2×C4×D83—42no—
138C2×C4×D83—42no—
139C2×C4×D83C22×C412yes—
140C2×C4×D83#4112yes—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8C22×C412yes—
145C2×C83⋊C8#4112yes—
146C2×C83⋊C8#4112yes—
147C8×D83—28no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—42no—
155C4⋊Dic83C22×C412yes—
156C4⋊Dic83C22×C412yes—
157Dic83⋊C4—24no—
158Dic83⋊C4—24no—
159Dic83⋊C4—42no—
160Dic83⋊C4—42no—
161(C2×D83)⋊C4—22no—
162(C2×D83)⋊C4—22no—
163(C2×D83)⋊C4—42no—
164(C2×D83)⋊C4—42no—
165C4×Dic83—21no—
166C4×Dic83—21no—
167C4×Dic83—42no—
168C4×Dic83—42no—
169C4×Dic83—42no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
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×C83—21no—
182C2×C8×C83C22×C412yes—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—42no—
190C4⋊C4×C83C22×C412yes—
191C4⋊C4×C83C22×C412yes—
192C42×C83—21no—
193C42×C83—21no—
194C42×C83—42no—
195C23×C83C22×D411yescharacteristic
196C22×D83—21no—
197C22×D83—21no—
198C22×D83—21no—
199C22×D83—21no—
200C22×D83C4×D412yes—
201C22×D83AGL(1,ℤ/8)12yes—
202C22×D83AGL(1,ℤ/8)12yes—
203C2×C4×C83—22no—
204C2×C4×C83—22no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
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×C83—41no—
221C2×C4×C83C23×C411yescharacteristic
222C2×C4×C83C2×#4111yescharacteristic
223C2×C4×C83C2×#4111yescharacteristic
224C2×C4×C83C2×#4111yescharacteristic
225C2×C4×C832−1+411yescharacteristic
226C2×C4×C832−1+411yescharacteristic
227C2×Dic83—21no—
228C2×Dic83—21no—
229C2×Dic83—22no—
230C2×Dic83—22no—
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×Dic83C4×D412yes—
248C4×D83—22no—
249C4×D83—22no—
250C4×D83—22no—
251C4×D83—22no—
252C4×D83—48no—
253C4×D83—48no—
254C8×C83—22no—
255C8×C83—41no—
256C83⋊C8—41no—
257C83⋊C8#11714yescomplemented
258C22×C83C2×C4×D411yescharacteristic
259C22×C83#54711yescharacteristic
260C22×C83#54711yescharacteristic
261C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C83#57811yescharacteristic
264C22×C83#57811yescharacteristic
265C2×D83—22no—
266C2×D83—22no—
267C2×D83—22no—
268C2×D83—22no—
269C2×D83—22no—
270C2×D83—22no—
271C2×D83#44214yes—
272C4×C83—41no—
273C4×C83—41no—
274C4×C83—42no—
275C4×C83—42no—
276C4×C83—81no—
277C4×C83—81no—
278C4×C83C2×#11711yescharacteristic
279C4×C83#51911yescharacteristic
280C4×C83#55211yescharacteristic
281C4×C83#55211yescharacteristic
282Dic83—22no—
283Dic83—41no—
284Dic83—41no—
285Dic83—42no—
286Dic83—42no—
287Dic83—81no—
288Dic83—81no—
289#22617—831nomaximal, Sylow
290C2×C83#234411yescharacteristic
291C2×C83C2×#44211yescharacteristic
292C2×C83#295011yescharacteristic
293C2×C83#334311yescharacteristic
294C2×C83#334311yescharacteristic
295C2×C83#334311yescharacteristic
296C2×C83#334311yescharacteristic
297D83—24no—
298C22×#107—831no—
299#2344—831no—
300#2344—831no—
301#1943—831no—
302#1943—831no—
303#1921—831no—
304#1921—831no—
305#1921—831no—
306#1921D8311yescharacteristic, complemented
307#1823—831no—
308#1823—831no—
309#1807—831no—
310#1776—831no—
311#1776—831no—
312#1752—831no—
313C83#2261711yescharacteristic, complemented, Sylow
314C23×C8—831no—
315C2×#117—1661no—
316C2×#117—1661no—
317C22×C4⋊C4—831no—
318C22×C4⋊C4—831no—
319C2×#107—831no—
320C2×#107—831no—
321C2×#107—831no—
322C2×#107—838no—
323C2×#107C2×D8311yescharacteristic
324C2×#106—831no—
325C2×#106—831no—
326C2×C22⋊C8—831no—
327C2×#97—831no—
328C2×#97—831no—
329#389—831no—
330#389—831no—
331#389—831no—
332#389C2×D8311yescharacteristic
333#387—1661no—
334#387—1661no—
335#387—1661no—
336#387—1661no—
337#381—1661no—
338#381—1661no—
339#380—831no—
340#380—831no—
341C4×C4⋊C4—831no—
342C4×C4⋊C4—831no—
343C4×C4⋊C4—831no—
344C4×C4⋊C4C2×D8311yescharacteristic
345C4×C22⋊C4—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#340—831no—
353#340C2×D8311yescharacteristic
354#340C2×D8311yescharacteristic
355#340C2×D8311yescharacteristic
356#336—831no—
357#336C2×D8311yescharacteristic
358C23×C4—831no—
359C23×C4—1661no—
360C23×C4—1661no—
361C22×C8—831no—
362C22×C8—834no—
363C22×C8—831no—
364C22×C8—1662no—
365C22×C8—1662no—
366C22×C8C22×D8311yescharacteristic
367C22×C8C22×D8311yescharacteristic
368#117—3324no—
369#117—3324no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—834no—
374C2×C4⋊C4—834no—
375C2×C4⋊C4—831no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—1662no—
378C2×C4⋊C4—1662no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4—1661no—
384C2×C4⋊C4C22×D8311yescharacteristic
385C2×C4⋊C4C22×D8311yescharacteristic
386C2×C4⋊C4C22×D8311yescharacteristic
387C2×C4⋊C4C22×D8311yescharacteristic
388C2×C22⋊C4—831no—
389C2×C22⋊C4—831no—
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1661no—
392C2×C42—21no—
393C2×C42—831no—
394C2×C42—1661no—
395C2×C42—1661no—
396C2×C42—1661no—
397C2×C42C22×D8311yescharacteristic
398#107—834no—
399#107—834no—
400#107—834no—
401#107C4×D8314yes—
402#106—1662no—
403#106—1662no—
404C22⋊C8—1662no—
405#97—1662no—
406#97—1662no—
407#95—21no—
408#95—21no—
409#95—831no—
410#95—831no—
411#95—1661no—
412#95—1661no—
413#95—1661no—
414#95—1661no—
415#95—1661no—
416#95—1661no—
417#95—1661no—
418#95—1661no—
419C24—831no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—3322no—
425C22×C4—3322no—
426C22×C4—3322no—
427C22×C4—3322no—
428C22×C4—831no—
429C22×C4—832no—
430C22×C4—832no—
431C22×C4—831no—
432C22×C4—831no—
433C22×C4—1664no—
434C22×C4—1664no—
435C22×C4—1661no—
436C22×C4—1661no—
437C22×C4—1661no—
438C22×C4—1661no—
439C22×C4—1661no—
440C22×C4—1661no—
441C22×C4—1661no—
442C22×C4—1661no—
443C22×C4D4×D8311yescharacteristic
444C22×C4D4×D8311yescharacteristic
445C22×C4C23×D8311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—832no—
452C2×C8—832no—
453C2×C8—832no—
454C2×C8—1668no—
455C2×C8—1661no—
456C2×C8—1661no—
457C2×C8—1661no—
458C2×C8C2×C4×D8312yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—3322no—
467C4⋊C4—3322no—
468C4⋊C4—3322no—
469C4⋊C4—832no—
470C4⋊C4—832no—
471C4⋊C4—1664no—
472C4⋊C4—1664no—
473C4⋊C4—1661no—
474C4⋊C4—1661no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4—1661no—
478C4⋊C4—1661no—
479C4⋊C4C2×C4×D8312yes—
480C4⋊C4C2×C4×D8312yes—
481C22⋊C4—3322no—
482C22⋊C4—3322no—
483C22⋊C4—1662no—
484C22⋊C4—1662no—
485C42—21no—
486C42—21no—
487C42—42no—
488C42—3322no—
489C42—3322no—
490C42—3322no—
491C42—1661no—
492C42—1661no—
493C23—832no—
494C23—832no—
495C23—832no—
496C23—1661no—
497C23—1661no—
498C23—1661no—
499C23—1661no—
500C23C2×D4×D8311yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—3328no—
520C2×C4—3328no—
521C2×C4—3321no—
522C2×C4—3321no—
523C2×C4—3321no—
524C2×C4—3321no—
525C2×C4—3321no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—832no—
532C2×C4—1661no—
533C2×C4—1661no—
534C2×C4—1662no—
535C2×C4—1662no—
536C2×C4—1661no—
537C2×C4—1661no—
538C2×C4—1661no—
539C2×C4—1661no—
540C2×C4—1661no—
541C2×C4—1661no—
542C2×C4—1662no—
543C2×C4—1662no—
544C2×C4—1662no—
545C2×C4—1662no—
546C2×C4C22×C4×D8311yescharacteristic
547C2×C4#41×D8311yescharacteristic
548C2×C4#41×D8311yescharacteristic
549C2×C4#41×D8311yescharacteristic
550C2×C4#1458183911yescharacteristic
551C2×C4#1458184011yescharacteristic
552C8—22no—
553C8—41no—
554C8—3321no—
555C8—834no—
556C22—834no—
557C22—1662no—
558C22—1662no—
559C22—1662no—
560C22—1662no—
561C22—1662no—
562C22—1662no—
563C22C4×D4×D8311yescharacteristic
564C22#1912834911yescharacteristic
565C22#1912835311yescharacteristic
566C22AGL(1,ℤ/8)×D8311yescharacteristic
567C22AGL(1,ℤ/8)×D8311yescharacteristic
568C22#1912851411yescharacteristic
569C22#1912851511yescharacteristic
570C4—42no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—3322no—
577C4—3322no—
578C4—3321no—
579C4—3321no—
580C4—6641no—
581C4—6641no—
582C4—1662no—
583C4#117×D8311yescharacteristic
584C4#1912817711yescharacteristic
585C4#1912838311yescharacteristic
586C4#1912838711yescharacteristic
587C2—1664no—
588C2#2491032911yescharacteristic
589C2#442×D8311yescharacteristic
590C2#2491632611yescharacteristic
591C2#2491989611yescharacteristic
592C2#2491990011yescharacteristic
593C2#2491990611yescharacteristic
594C2#2491991011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1921C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
cosocleC22×C4×C83C24
hypercenter#1921D83
hypocenterC83#22617
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
4. #24908200C2semidirect
5. #24908198C2semidirect
6. #24908057C2semidirect
7. #24908056C2semidirect
8. #24908054C2semidirect
12. #24907361C2semidirect
13. #24907359C2semidirect
15. C83×#1921C2semidirect
24. #107×D83C4semidirect
257. C83⋊C8#117semidirect
306. #1921D83semidirect
313. C83#22617semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D832
#249103312
#249103282
#249082002
#249081982
#249080572
#249080562
#249080542
#249075482
#249074872
#249073612
#249073592
#249072382
C83×#19212
#2261783