Finite Group Databasebeta

← #35404992

Subgroups — #35404992

Order 21248.

898 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#97×D83C211yesmaximal, characteristic
2#24909867C211yesmaximal, characteristic
3#24909859C211yesmaximal, characteristic
4#24907630C211yesmaximal, characteristic, complemented
5#24907628C211yesmaximal, characteristic, complemented
6#24907595C211yesmaximal, characteristic, complemented
7#24907595C211yesmaximal, characteristic, complemented
8#24907491C211yesmaximal, characteristic
9#24907489C211yesmaximal, characteristic
10#24907487C211yesmaximal, characteristic
11#24907485C211yesmaximal, characteristic
12#24907361C211yesmaximal, characteristic, complemented
13#24907361C211yesmaximal, characteristic, complemented
14#24907360C211yesmaximal, characteristic, complemented
15C83×#1776C211yesmaximal, characteristic, complemented
16C22×C8×D83C2211yescharacteristic
17C22×C8×D83C2211yescharacteristic
18C2×#14581749—21no—
19C2×C42×D83C2211yescharacteristic
20C2×#14581726C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581724C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581703C2211yescharacteristic
27C2×#14581702C2211yescharacteristic
28C2×#14581702C2211yescharacteristic
29#97×D83C418yescomplemented
30#19127570C2211yescharacteristic
31#19127570C2211yescharacteristic
32#19127568—21no—
33#19127566C2211yescharacteristic
34#19127566C2211yescharacteristic
35#19127565—21no—
36#19127563—21no—
37#19127560—21no—
38#19127558C2211yescharacteristic
39#19127558C2211yescharacteristic
40#19127558C2211yescharacteristic
41#19127558C2211yescharacteristic
42C4×Dic83⋊C4—21no—
43C4×C4⋊Dic83C2211yescharacteristic
44C4×C4⋊Dic83C2211yescharacteristic
45C4×(C2×D83)⋊C4—21no—
46#19127547—21no—
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127463C2211yescharacteristic
53C2×#107×C83C2211yescharacteristic
54C2×#106×C83C2211yescharacteristic
55C2×#97×C83C2211yescharacteristic
56C4×C4⋊C4×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60C22×C4×D83D411yescharacteristic
61C22×C4×D83D411yescharacteristic
62C22×C4×D83C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C2×C8×D83—22no—
66C2×C8×D83—22no—
67C2×C8×D83—22no—
68C2×C8×D83—22no—
69C2×C8×D83C2×C414yes—
70C2×C8×D83C2×C414yes—
71#14581749—44no—
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83D411yescharacteristic
87C2×C4×Dic83D411yescharacteristic
88C2×C4×Dic83C2311yescharacteristic
89C42×D83C2×C414yes—
90C42×D83Q814yes—
91#14581726—22no—
92#14581725—22no—
93#14581725—22no—
94#14581724—22no—
95#14581723—22no—
96#14581723—22no—
97#14581703C2×C414yes—
98#14581702C2×C414yes—
99#14581702C2×C414yes—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C22×C8×C83C2311yescharacteristic
111C22×C8×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C42×C83—21no—
117C2×C42×C83—21no—
118C2×C42×C83C2311yescharacteristic
119#107×C83—22no—
120#106×C83—22no—
121#97×C83C2×C414yes—
122#95×C83—21no—
123#95×C83—21no—
124C23×D83C2×D411yescharacteristic
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83C2×D411yescharacteristic
130C22×Dic83C2×D411yescharacteristic
131C22×Dic83C2×D411yescharacteristic
132C2×C4×D83—22no—
133C2×C4×D83—22no—
134C2×C4×D83—21no—
135C2×C4×D83—21no—
136C2×C4×D83—21no—
137C2×C4×D83—21no—
138C2×C4×D83—41no—
139C2×C4×D83—41no—
140C2×C4×D83C4⋊C414yes—
141C2×C4×D83C4⋊C414yes—
142C2×C4×D83C22×C412yes—
143C2×C4×D83C2×Q812yes—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8C22×C412yes—
147C2×C83⋊C8C22×C412yes—
148C2×C83⋊C8#4112yes—
149C2×C83⋊C8#4112yes—
150C8×D83—28no—
151C8×D83—28no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160Dic83⋊C4—42no—
161Dic83⋊C4—42no—
162Dic83⋊C4—42no—
163Dic83⋊C4—42no—
164(C2×D83)⋊C4—42no—
165(C2×D83)⋊C4—42no—
166(C2×D83)⋊C4—42no—
167(C2×D83)⋊C4—42no—
168C4×Dic83—42no—
169C4×Dic83—42no—
170C4×Dic83C4⋊C414yes—
171C4×Dic83C4⋊C414yes—
172C4×Dic83C22×C412yes—
173C4×Dic83C2×Q812yes—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2411yescharacteristic
181C2×C8×C83—21no—
182C2×C8×C83—21no—
183C2×C8×C83C22×C412yes—
184C2×C8×C83C22×C412yes—
185C2×C8×C83#4112yes—
186C2×C8×C83#4112yes—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C42×C83—42no—
196C42×C83—42no—
197C42×C83C22×C412yes—
198C42×C83C2×Q812yes—
199C23×C83C22×D411yescharacteristic
200C22×D83—21no—
201C22×D83—21no—
202C22×D83—21no—
203C22×D83—21no—
204C22×D83C2×C4⋊C412yes—
205C22×D83AGL(1,ℤ/8)12yes—
206C22×D83#13712yes—
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×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83C2×C4⋊C412yes—
224C2×C4×C83C2×C4⋊C412yes—
225C2×C4×C83C23×C411yescharacteristic
226C2×C4×C83C22×Q811yescharacteristic
227C2×C4×C83C2×#4111yescharacteristic
228C2×C4×C83C2×#4111yescharacteristic
229C2×C4×C83C2×#4111yescharacteristic
230C2×C4×C83C2×#4111yescharacteristic
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×Dic83C2×C4⋊C412yes—
250C2×Dic83C2×C4⋊C412yes—
251C2×Dic83C2×C4⋊C412yes—
252C4×D83—28no—
253C4×D83—22no—
254C4×D83—22no—
255C4×D83—22no—
256C4×D83—22no—
257C4×D83—44no—
258C8×C83—22no—
259C8×C83#11714yescomplemented
260C83⋊C8—22no—
261C83⋊C8#11714yescomplemented
262C22×C83C22×C4⋊C411yescharacteristic
263C22×C83#53211yescharacteristic
264C22×C83#53211yescharacteristic
265C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C83C2×#13711yescharacteristic
267C22×C83#57511yescharacteristic
268C22×C83#57511yescharacteristic
269C2×D83—22no—
270C2×D83—22no—
271C2×D83—22no—
272C2×D83—22no—
273C2×D83—22no—
274C2×D83—22no—
275C2×D83#42814yes—
276C4×C83—22no—
277C4×C83—22no—
278C4×C83—81no—
279C4×C83—81no—
280C4×C83—81no—
281C4×C83—81no—
282C4×C83C2×#11711yescharacteristic
283C4×C83C2×#11711yescharacteristic
284C4×C83#53311yescharacteristic
285C4×C83#53311yescharacteristic
286Dic83—22no—
287Dic83—22no—
288Dic83—22no—
289Dic83—81no—
290Dic83—81no—
291Dic83—81no—
292Dic83—81no—
293#20823—831nomaximal, Sylow
294C2×C83#228511yescharacteristic
295C2×C83C2×#42811yescharacteristic
296C2×C83#291111yescharacteristic
297C2×C83#318011yescharacteristic
298C2×C83#318011yescharacteristic
299C2×C83#318111yescharacteristic
300C2×C83#318111yescharacteristic
301D83—24no—
302C22×#97—831no—
303#2285—831no—
304#2283—831no—
305#1847—831no—
306#1847—831no—
307#1836—831no—
308#1836—831no—
309#1808—831no—
310#1808—831no—
311#1807—831no—
312#1807—831no—
313#1776—831no—
314#1776—831no—
315#1776—831no—
316#1776D8311yescharacteristic, complemented
317C83#2082311yescharacteristic, complemented, Sylow
318C23×C8—831no—
319C23×C8—831no—
320C2×#117—1661no—
321C22×C42—831no—
322C2×#107—831no—
323C2×#107—831no—
324C2×#107—831no—
325C2×#107C2×D8311yescharacteristic
326C2×#106—831no—
327C2×#106—831no—
328C2×#106—831no—
329C2×#106C2×D8311yescharacteristic
330C2×#97—831no—
331C2×#97—831no—
332C2×#97—831no—
333C2×#97—838no—
334C2×#97C2×D8311yescharacteristic
335#384—831no—
336#384—831no—
337#383—1661no—
338#382—831no—
339#382—831no—
340#382—1661no—
341#381—1661no—
342#380—831no—
343#380—831no—
344#380—831no—
345#380—831no—
346#380—1661no—
347C4×C4⋊C4—831no—
348C4×C4⋊C4—831no—
349C4×C4⋊C4—1661no—
350C4×C4⋊C4C2×D8311yescharacteristic
351C4×C4⋊C4C2×D8311yescharacteristic
352C4×C22⋊C4—1661no—
353#376—1661no—
354#340—831no—
355#340—831no—
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340C2×D8311yescharacteristic
361#340C2×D8311yescharacteristic
362C23×C4—831no—
363C23×C4—831no—
364C23×C4—831no—
365C22×C8—831no—
366C22×C8—831no—
367C22×C8—834no—
368C22×C8—834no—
369C22×C8—1662no—
370C22×C8—1662no—
371C22×C8—1662no—
372C22×C8—1662no—
373C22×C8C22×D8311yescharacteristic
374C22×C8C22×D8311yescharacteristic
375#117—3324no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—831no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—831no—
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—1661no—
389C2×C22⋊C4—1661no—
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1661no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—831no—
395C2×C42—831no—
396C2×C42—831no—
397C2×C42—834no—
398C2×C42—834no—
399C2×C42—1661no—
400C2×C42—1661no—
401C2×C42C22×D8311yescharacteristic
402#107—22no—
403#107—1662no—
404#107—1662no—
405#107—1662no—
406#106—22no—
407#106—1662no—
408#106—1662no—
409#106—1662no—
410#97—834no—
411#97—834no—
412#97—834no—
413#97C4×D8314yes—
414#95—21no—
415#95—21no—
416#95—1661no—
417#95—1661no—
418#95—1661no—
419#95—1661no—
420#95—1661no—
421#95—1661no—
422#95—1661no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426C24—831no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—3321no—
432C22×C4—3321no—
433C22×C4—834no—
434C22×C4—831no—
435C22×C4—831no—
436C22×C4—831no—
437C22×C4—832no—
438C22×C4—832no—
439C22×C4—834no—
440C22×C4—1662no—
441C22×C4—1662no—
442C22×C4—1661no—
443C22×C4—1661no—
444C22×C4—1661no—
445C22×C4—1661no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4D4×D8311yescharacteristic
451C22×C4D4×D8311yescharacteristic
452C22×C4C23×D8311yescharacteristic
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—832no—
456C2×C8—832no—
457C2×C8—832no—
458C2×C8—832no—
459C2×C8—1668no—
460C2×C8—1661no—
461C2×C8—1661no—
462C2×C8—1668no—
463C2×C8C2×C4×D8312yes—
464C2×C8C2×C4×D8312yes—
465C2×C8#1305059012yes—
466C2×C8#1305059012yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—3322no—
476C4⋊C4—3322no—
477C4⋊C4—3322no—
478C4⋊C4—3322no—
479C4⋊C4—1661no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C4⋊C4—1661no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C22⋊C4—3322no—
488C22⋊C4—3322no—
489C22⋊C4—3322no—
490C22⋊C4—3322no—
491C42—42no—
492C42—42no—
493C42—3322no—
494C42—3322no—
495C42—832no—
496C42—832no—
497C42—834no—
498C42—834no—
499C42C2×C4×D8312yes—
500C42Q8×D8312yes—
501C23—832no—
502C23—832no—
503C23—832no—
504C23—1661no—
505C23—1661no—
506C23—1661no—
507C23—1661no—
508C23C2×D4×D8311yescharacteristic
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—3324no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—3321no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—832no—
539C2×C4—832no—
540C2×C4—832no—
541C2×C4—1668no—
542C2×C4—1661no—
543C2×C4—1661no—
544C2×C4—1661no—
545C2×C4—1661no—
546C2×C4—1661no—
547C2×C4—1661no—
548C2×C4—1662no—
549C2×C4—1662no—
550C2×C4—1662no—
551C2×C4—1662no—
552C2×C4C4⋊C4×D8312yes—
553C2×C4C4⋊C4×D8312yes—
554C2×C4C22×C4×D8311yescharacteristic
555C2×C4C2×Q8×D8311yescharacteristic
556C2×C4#41×D8311yescharacteristic
557C2×C4#41×D8311yescharacteristic
558C2×C4C2×#1305059011yescharacteristic
559C2×C4C2×#1305059011yescharacteristic
560C8—22no—
561C8—834no—
562C8—1662no—
563C8#1458174914yescomplemented
564C22—834no—
565C22—1662no—
566C22—1662no—
567C22—1662no—
568C22—1662no—
569C22—1662no—
570C22—1662no—
571C22C2×C4⋊C4×D8311yescharacteristic
572C22#1912824811yescharacteristic
573C22#1912824811yescharacteristic
574C22AGL(1,ℤ/8)×D8311yescharacteristic
575C22#137×D8311yescharacteristic
576C22#1912849711yescharacteristic
577C22#1912850011yescharacteristic
578C4—22no—
579C4—22no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—6641no—
585C4—6641no—
586C4—6641no—
587C4—6641no—
588C4—1662no—
589C4—1662no—
590C4—1662no—
591C4#117×D8311yescharacteristic
592C4C2×#1458174911yescharacteristic
593C4#1912825511yescharacteristic
594C4#1912825511yescharacteristic
595C2—1664no—
596C2#2490986711yescharacteristic
597C2#428×D8311yescharacteristic
598C2#2491605011yescharacteristic
599C2#2491815711yescharacteristic
600C2#2491815911yescharacteristic
601C2#2491816811yescharacteristic
602C2#2491817011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1776C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
cosocleC22×C4×C83C24
hypercenter#1776D83
hypocenterC83#20823
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #24907630C2semidirect
5. #24907628C2semidirect
6. #24907595C2semidirect
7. #24907595C2semidirect
12. #24907361C2semidirect
13. #24907361C2semidirect
14. #24907360C2semidirect
15. C83×#1776C2semidirect
29. #97×D83C4semidirect
259. C8×C83#117semidirect
261. C83⋊C8#117semidirect
316. #1776D83semidirect
317. C83#20823semidirect
563. C8#14581749semidirect

Maximal subgroups

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

subgroupindex
C2×#97×D832
#249098672
#249098592
#249076302
#249076282
#249075952
#249074912
#249074892
#249074872
#249074852
#249073612
#249073602
C83×#17762
#2082383