Finite Group Databasebeta

← #35412256

Subgroups — #35412256

Order 21248.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917251C211yesmaximal, characteristic
2#24916403C211yesmaximal, characteristic, complemented
3#24908407C211yesmaximal, characteristic, complemented
4#24908238C211yesmaximal, characteristic, complemented
5#24908062C211yesmaximal, characteristic, complemented
6#24908062C211yesmaximal, characteristic, complemented
7#24907738C211yesmaximal, characteristic
8#24907738C211yesmaximal, characteristic
9#24907483C211yesmaximal, characteristic
10#24907348C211yesmaximal, characteristic, complemented
11#24907343C211yesmaximal, characteristic, complemented
12#24907335C211yesmaximal, characteristic, complemented
13#24907305C211yesmaximal, characteristic
14#24907298C211yesmaximal, characteristic
15C83×#2875C211yesmaximal, characteristic
16#19128258C2211yescharacteristic
17#19128008C411yescharacteristic, complemented
18#19128008C411yescharacteristic, complemented
19#19128001C411yescharacteristic, complemented
20#19128001C411yescharacteristic, complemented
21#19127996C411yescharacteristic, complemented
22#19127991C411yescharacteristic, complemented
23#19127986C411yescharacteristic, complemented
24#19127981C411yescharacteristic, complemented
25#19127829—21no—
26#19127829—21no—
27#19127829—21no—
28#19127829—21no—
29C2×#14581727C2211yescharacteristic
30C2×#14581727C2211yescharacteristic
31#19127737C2211yescharacteristic
32#19127737C2211yescharacteristic
33#19127727C2211yescharacteristic
34#19127727C2211yescharacteristic
35#19127725C2211yescharacteristic
36C2×C4⋊(C83⋊C8)C2211yescharacteristic
37C2×#14581719C2211yescharacteristic
38C2×#14581719C2211yescharacteristic
39#19127697C2211yescharacteristic
40#19127697C2211yescharacteristic
41#19127656C2211yescharacteristic
42#19127656C2211yescharacteristic
43C2×#14581702C2211yescharacteristic
44C2×C8×Dic83C2211yescharacteristic
45C4×C4⋊Dic83C2211yescharacteristic
46#19127465C2211yescharacteristic
47#19127465C2211yescharacteristic
48#19127462C2211yescharacteristic
49#19127462C2211yescharacteristic
50#19127462C2211yescharacteristic
51#19127462C2211yescharacteristic
52#19127460C2211yescharacteristic
53#19127460C2211yescharacteristic
54#19127455C2211yescharacteristic
55#19127455C2211yescharacteristic
56C2×#131×C83C2211yescharacteristic
57C2×#131×C83C2211yescharacteristic
58C4×#41×C83C2211yescharacteristic
59#405×C83—21no—
60#405×C83—21no—
61#405×C83C2211yescharacteristic
62#405×C83C2211yescharacteristic
63C4×M16×C83—21no—
64C4×M16×C83C2211yescharacteristic
65C2×#97×C83—21no—
66C2×#97×C83C2211yescharacteristic
67C2×#13050574C2311yescharacteristic
68C2×#13050574C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70C22×C83⋊C8C2311yescharacteristic
71#14581792C2×C411yescharacteristic
72#14581792C2×C411yescharacteristic
73#14581784C2×C411yescharacteristic
74#14581784C2×C411yescharacteristic
75#14581771—21no—
76#14581771—21no—
77C4×Dic166—21no—
78C4×Dic166—21no—
79#14581751C2311yescharacteristic
80#14581751C2311yescharacteristic
81C2×C4⋊Dic83—21no—
82C2×C4×Dic83C2311yescharacteristic
83C2×C4×Dic83C2311yescharacteristic
84#14581727—22no—
85#14581727—22no—
86#14581725C2×C411yescharacteristic
87#14581725C2×C411yescharacteristic
88#14581725C2×C411yescharacteristic
89#14581725C2×C411yescharacteristic
90#14581723C2×C411yescharacteristic
91#14581723C2×C411yescharacteristic
92#14581723C2×C411yescharacteristic
93#14581723C2×C411yescharacteristic
94C4⋊(C83⋊C8)—21no—
95C4⋊(C83⋊C8)—21no—
96C4⋊(C83⋊C8)C2×C411yescharacteristic
97C4⋊(C83⋊C8)C2×C411yescharacteristic
98C4⋊(C83⋊C8)C2×C411yescharacteristic
99C4⋊(C83⋊C8)C2×C411yescharacteristic
100Dic83⋊C8—21no—
101Dic83⋊C8—21no—
102Dic83⋊C8—21no—
103Dic83⋊C8—21no—
104#14581719—22no—
105#14581719—22no—
106#14581716C2×C411yescharacteristic
107#14581716C2×C411yescharacteristic
108#14581716C2×C411yescharacteristic
109#14581716C2×C411yescharacteristic
110#14581713C2×C411yescharacteristic
111#14581713C2×C411yescharacteristic
112#14581713C2×C411yescharacteristic
113#14581713C2×C411yescharacteristic
114#14581702—21no—
115#14581702—21no—
116#14581702—22no—
117C8×Dic83—21no—
118C8×Dic83—21no—
119C8×Dic83—22no—
120C83⋊#95—21no—
121C2×#41×C83C2311yescharacteristic
122#131×C83—22no—
123#131×C83—22no—
124#131×C83—22no—
125#131×C83Q814yescomplemented
126C2×M16×C83—21no—
127C2×M16×C83—21no—
128C2×M16×C83C2311yescharacteristic
129C2×M16×C83C2311yescharacteristic
130C22×C8×C83—21no—
131C22×C8×C83—21no—
132C22×C8×C83C2311yescharacteristic
133C22×C8×C83C2311yescharacteristic
134C4×Q8×C83C2×C411yescharacteristic
135C4×Q8×C83C2×C411yescharacteristic
136C4×D4×C83—21no—
137C4×D4×C83—21no—
138C4×D4×C83C2×C411yescharacteristic
139C4×D4×C83C2×C411yescharacteristic
140#117×C83—21no—
141#117×C83C2311yescharacteristic
142C2×C42×C83—21no—
143C2×C42×C83C2311yescharacteristic
144#97×C83—21no—
145#97×C83—21no—
146#97×C83—21no—
147#97×C83D411yescharacteristic
148#97×C83D411yescharacteristic
149#97×C83D411yescharacteristic
150#97×C83D411yescharacteristic
151C4×C8×C83—21no—
152C4×C8×C83—21no—
153C4×C8×C83—21no—
154C2×Dic166—21no—
155C2×Dic166—21no—
156C22×Dic83—21no—
157C22×Dic83—21no—
158#13050574—21no—
159#13050574—21no—
160#13050574—21no—
161#13050574—21no—
162C2×C83⋊C8—21no—
163C2×C83⋊C8—21no—
164C2×C83⋊C8—21no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8C22×C411yescharacteristic
169C2×C83⋊C8C22×C411yescharacteristic
170C2×C83⋊C8C22×C411yescharacteristic
171C2×C83⋊C8C22×C411yescharacteristic
172C4⋊Dic83—21no—
173C4⋊Dic83—21no—
174C4⋊Dic83—42no—
175C4⋊Dic83C22×C411yescharacteristic
176C4⋊Dic83C22×C411yescharacteristic
177C4⋊Dic83C22×C411yescharacteristic
178C4⋊Dic83C22×C411yescharacteristic
179Dic83⋊C4—21no—
180Dic83⋊C4—21no—
181Dic83⋊C4—21no—
182Dic83⋊C4—21no—
183C22⋊Dic83—21no—
184C22⋊Dic83—21no—
185C4×Dic83—21no—
186C4×Dic83—21no—
187C4×Dic83—21no—
188C4×Dic83—21no—
189C4×Dic83—21no—
190C4×Dic83—21no—
191#41×C83—21no—
192#41×C83—21no—
193#41×C83C2×Q812yes—
194#41×C83#4112yes—
195C2×Q8×C83C22×C411yescharacteristic
196C2×D4×C83—21no—
197C2×D4×C83C22×C411yescharacteristic
198C22×C4×C83—21no—
199C22×C4×C83—21no—
200C22×C4×C83—21no—
201C22×C4×C83#4111yescharacteristic
202C22×C4×C83#4111yescharacteristic
203C22×C4×C83C2411yescharacteristic
204M16×C83—24no—
205M16×C83—21no—
206M16×C83—21no—
207M16×C83—24no—
208M16×C83C2×Q812yes—
209M16×C83#4112yes—
210C2×C8×C83—24no—
211C2×C8×C83—21no—
212C2×C8×C83—21no—
213C2×C8×C83—21no—
214C2×C8×C83—21no—
215C2×C8×C83—24no—
216C2×C8×C83C2×D411yescharacteristic
217C2×C8×C83C2×D411yescharacteristic
218C2×C8×C83C2×D411yescharacteristic
219C2×C8×C83C2×D411yescharacteristic
220C2×C8×C83C2×Q812yes—
221C2×C8×C83#4112yes—
222C4⋊C4×C83—21no—
223C4⋊C4×C83—21no—
224C4⋊C4×C83C22×C411yescharacteristic
225C4⋊C4×C83C22×C411yescharacteristic
226C22⋊C4×C83—21no—
227C22⋊C4×C83—21no—
228C22⋊C4×C83—21no—
229C42×C83—21no—
230C42×C83—21no—
231C42×C83—21no—
232C42×C83C22×C411yescharacteristic
233C42×C83C22×C411yescharacteristic
234C42×C83C2×D411yescharacteristic
235C42×C83C2×D411yescharacteristic
236C23×C83—21no—
237C23×C83C2×#4111yescharacteristic
238Q8×C83—21no—
239Q8×C83C4×Q812yes—
240D4×C83—22no—
241D4×C83—22no—
242D4×C83—21no—
243D4×C83C4×Q812yes—
244C2×C4×C83—21no—
245C2×C4×C83—22no—
246C2×C4×C83—22no—
247C2×C4×C83—21no—
248C2×C4×C83—21no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—81no—
252C2×C4×C83—81no—
253C2×C4×C83C4×D411yescharacteristic
254C2×C4×C83C4×D411yescharacteristic
255C2×C4×C83C4×D411yescharacteristic
256C2×C4×C83C4×D411yescharacteristic
257C2×C4×C83C23×C411yescharacteristic
258C2×C4×C83C22×D411yescharacteristic
259C2×C4×C83C22×Q811yescharacteristic
260C2×C4×C83C2×#4111yescharacteristic
261C2×C4×C83C2×#4111yescharacteristic
262C2×C4×C83C2×#4111yescharacteristic
263C2×Dic83—22no—
264C2×Dic83—21no—
265C2×Dic83—21no—
266C2×Dic83—21no—
267C2×Dic83—21no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273Dic166—81no—
274Dic166—81no—
275C8×C83—22no—
276C8×C83—21no—
277C8×C83—22no—
278C8×C83—21no—
279C8×C83C22⋊Q812yes—
280C8×C83C22⋊Q812yes—
281C83⋊C8—21no—
282C83⋊C8—21no—
283C83⋊C8—21no—
284C83⋊C8—21no—
285C83⋊C8—41no—
286C83⋊C8—41no—
287C22×C83—21no—
288C22×C83—21no—
289C22×C83—22no—
290C22×C83—41no—
291C22×C83—41no—
292C22×C83C2×C4×D411yescharacteristic
293C22×C83#52911yescharacteristic
294C22×C83#53311yescharacteristic
295C4×C83—21no—
296C4×C83—21no—
297C4×C83—21no—
298C4×C83—21no—
299C4×C83—22no—
300C4×C83—41no—
301C4×C83—41no—
302C4×C83C2×C4×Q811yescharacteristic
303C4×C83C4×#4111yescharacteristic
304C4×C83C2×C22⋊Q811yescharacteristic
305C4×C83#52511yescharacteristic
306Dic83—42no—
307Dic83—41no—
308Dic83—41no—
309Dic83—41no—
310Dic83—41no—
311Dic83—81no—
312#21629—831nomaximal, Sylow
313C2×C83—21no—
314C2×C83—21no—
315C2×C83—42no—
316C2×C83C4×C22⋊Q811yescharacteristic
317C2×C83#295611yescharacteristic
318C2×C83#295711yescharacteristic
319#3076—831no—
320#2961—831no—
321#2875D8311yescharacteristic
322#1986—831no—
323#1950—831no—
324#1922—831no—
325#1922—831no—
326#1866—831no—
327#1866—831no—
328#1806—831no—
329#1773—831no—
330#1772—831no—
331#1771—831no—
332#1765—831no—
333#1764—831no—
334C83#2162911yescharacteristic, complemented, Sylow
335C2×#131C2×D8311yescharacteristic
336C2×#131C2×D8311yescharacteristic
337#533—831no—
338C4×#41C2×D8311yescharacteristic
339#479—831no—
340#479—831no—
341#478—831no—
342#478—831no—
343#477—831no—
344#476—831no—
345#475—831no—
346#474—831no—
347#446—1661no—
348#446—1661no—
349#446—1661no—
350#446—1661no—
351C2×#108—831no—
352C2×#108—831no—
353#427—831no—
354#427—831no—
355#424—831no—
356#424—831no—
357#423—831no—
358C2×C4⋊C8—831no—
359C2×#104—831no—
360C2×#104—831no—
361#416—831no—
362#416—831no—
363#405—21no—
364#405—21no—
365#405—831no—
366#405—831no—
367#405C2×D8311yescharacteristic
368#405C2×D8311yescharacteristic
369C4×M16—21no—
370C4×M16C2×D8311yescharacteristic
371C2×#97—21no—
372C2×#97—831no—
373C2×#97C2×D8311yescharacteristic
374C2×C4×C8—831no—
375C4×C4⋊C4—831no—
376#341—831no—
377#341—831no—
378#339—831no—
379#339—831no—
380#339—831no—
381#339—831no—
382#339—831no—
383#339—831no—
384#336—831no—
385#336—831no—
386C2×#41C22×D8311yescharacteristic
387#131—22no—
388#131—22no—
389#131—22no—
390#131Dic16614yescomplemented
391C2×M16—21no—
392C2×M16—21no—
393C2×M16—831no—
394C2×M16—831no—
395C2×M16C22×D8311yescharacteristic
396C2×M16C22×D8311yescharacteristic
397C22×C8—21no—
398C22×C8—21no—
399C22×C8—831no—
400C22×C8—831no—
401C22×C8C22×D8311yescharacteristic
402C22×C8C22×D8311yescharacteristic
403C4⋊Q8—831no—
404C4⋊Q8—831no—
405#125—831no—
406#125—831no—
407C22⋊Q8—1661no—
408C22⋊Q8—1661no—
409C4×Q8—1661no—
410C4×Q8—1661no—
411C4×Q8C4×D8311yescharacteristic
412C4×Q8C4×D8311yescharacteristic
413C4×D4—21no—
414C4×D4—21no—
415C4×D4C4×D8311yescharacteristic
416C4×D4C4×D8311yescharacteristic
417#117—21no—
418#117—831no—
419#117—831no—
420#117C22×D8311yescharacteristic
421C2×C4⋊C4—1661no—
422C2×C42—21no—
423C2×C42—831no—
424C2×C42—831no—
425C2×C42C22×D8311yescharacteristic
426#108—1662no—
427#108—1662no—
428#107—831no—
429#107—831no—
430#107—831no—
431#107—831no—
432#106—831no—
433#106—831no—
434#106—831no—
435#106—831no—
436C4⋊C8—831no—
437C4⋊C8—831no—
438C4⋊C8—831no—
439C4⋊C8—831no—
440C4⋊C8—1661no—
441C4⋊C8—1661no—
442C4⋊C8—1661no—
443C4⋊C8—1661no—
444C4⋊C8—1661no—
445C4⋊C8—1661no—
446#104—1662no—
447#104—1662no—
448#103—831no—
449#103—831no—
450#103—831no—
451#103—831no—
452#102—831no—
453#102—831no—
454#102—831no—
455#102—831no—
456#97—21no—
457#97—21no—
458#97—21no—
459#97—1661no—
460#97—1661no—
461#97—1662no—
462#97C22⋊D8311yescharacteristic
463#97C22⋊D8311yescharacteristic
464#97C22⋊D8311yescharacteristic
465#97C22⋊D8311yescharacteristic
466C4×C8—21no—
467C4×C8—21no—
468C4×C8—21no—
469C4×C8—1661no—
470C4×C8—1661no—
471C4×C8—1662no—
472#95—1661no—
473#41—21no—
474#41—21no—
475#41C2×Dic16612yes—
476#41#1305059012yes—
477C2×Q8—1661no—
478C2×Q8—1661no—
479C2×Q8C2×C4×D8311yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D8311yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—1661no—
486C22×C4—1661no—
487C22×C4#1305059011yescharacteristic
488C22×C4#1305059011yescharacteristic
489C22×C4C23×D8311yescharacteristic
490M16—24no—
491M16—21no—
492M16—21no—
493M16—24no—
494M16—1661no—
495M16—1661no—
496M16—1661no—
497M16—1661no—
498M16C2×Dic16612yes—
499M16#1305059012yes—
500C2×C8—24no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—24no—
506C2×C8—831no—
507C2×C8—831no—
508C2×C8—831no—
509C2×C8—831no—
510C2×C8—1661no—
511C2×C8—1661no—
512C2×C8—1661no—
513C2×C8—1661no—
514C2×C8—1661no—
515C2×C8—1661no—
516C2×C8C2×C22⋊D8311yescharacteristic
517C2×C8C2×C22⋊D8311yescharacteristic
518C2×C8C2×C22⋊D8311yescharacteristic
519C2×C8C2×C22⋊D8311yescharacteristic
520C2×C8C2×Dic16612yes—
521C2×C8#1305059012yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—3322no—
525C4⋊C4—831no—
526C4⋊C4—831no—
527C4⋊C4—831no—
528C4⋊C4—831no—
529C4⋊C4—1661no—
530C4⋊C4—1661no—
531C4⋊C4—1661no—
532C4⋊C4—1661no—
533C4⋊C4—1661no—
534C4⋊C4—1661no—
535C4⋊C4C2×C4×D8311yescharacteristic
536C4⋊C4C2×C4×D8311yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—1661no—
541C22⋊C4—1661no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—1661no—
546C42—1661no—
547C42—1661no—
548C42—1661no—
549C42—1661no—
550C42—1661no—
551C42C2×C4×D8311yescharacteristic
552C42C2×C4×D8311yescharacteristic
553C42C2×C22⋊D8311yescharacteristic
554C42C2×C22⋊D8311yescharacteristic
555C23—21no—
556C23C2×#1305059011yescharacteristic
557Q8—21no—
558Q8—6641no—
559Q8—6641no—
560Q8C4×Dic16612yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4C4×Dic16612yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4—1662no—
580C2×C4—1661no—
581C2×C4—1661no—
582C2×C4—1661no—
583C2×C4—1661no—
584C2×C4C4×C22⋊D8311yescharacteristic
585C2×C4C4×C22⋊D8311yescharacteristic
586C2×C4C4×C22⋊D8311yescharacteristic
587C2×C4C4×C22⋊D8311yescharacteristic
588C2×C4C22×C4×D8311yescharacteristic
589C2×C4C22×C22⋊D8311yescharacteristic
590C2×C4C22×Dic16611yescharacteristic
591C2×C4C2×#1305059011yescharacteristic
592C2×C4C2×#1305059011yescharacteristic
593C2×C4C2×#1305059011yescharacteristic
594C8—22no—
595C8—21no—
596C8—22no—
597C8—21no—
598C8—3321no—
599C8—3321no—
600C8—1661no—
601C8—1661no—
602C8—1661no—
603C8—1661no—
604C8#1458177112yes—
605C8#1458177112yes—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C2×C4×C22⋊D8311yescharacteristic
612C22#1912823511yescharacteristic
613C22#1912825811yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—41no—
620C4—41no—
621C4—3322no—
622C4—3321no—
623C4—3321no—
624C4—3321no—
625C4—3321no—
626C4—6641no—
627C4C2×C4×Dic16611yescharacteristic
628C4C4×#1305059011yescharacteristic
629C4C2×#1458177111yescharacteristic
630C4#1912821211yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2C4×#1458177111yescharacteristic
635C2#2491637011yescharacteristic
636C2#2491637811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2875C2
socleC22×C83C2×C4×D4
prime radicalC2×#41×C83C23
cosocleC22×C4×C83C24
hypercenter#2875D83
hypocenterC83#21629
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #24916403C2semidirect
3. #24908407C2semidirect
4. #24908238C2semidirect
5. #24908062C2semidirect
6. #24908062C2semidirect
10. #24907348C2semidirect
11. #24907343C2semidirect
12. #24907335C2semidirect
17. #19128008C4semidirect
18. #19128008C4semidirect
19. #19128001C4semidirect
20. #19128001C4semidirect
21. #19127996C4semidirect
22. #19127991C4semidirect
23. #19127986C4semidirect
24. #19127981C4semidirect
125. #131×C83Q8semidirect
334. C83#21629semidirect
390. #131Dic166semidirect

Maximal subgroups

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

subgroupindex
#249172512
#249164032
#249084072
#249082382
#249080622
#249077382
#249074832
#249073482
#249073432
#249073352
#249073052
#249072982
C83×#28752
#2162983