Finite Group Databasebeta

← #35376109

Subgroups — #35376109

Order 21248.

668 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916549C211yesmaximal, characteristic, complemented
2#24916495C211yesmaximal, characteristic
3C4×SD16×D83C211yesmaximal, characteristic
4#24906366C211yesmaximal, characteristic, complemented
5#24906309C211yesmaximal, characteristic, complemented
6#24906304C211yesmaximal, characteristic, complemented
7#24906302C211yesmaximal, characteristic, complemented
8#24906297C211yesmaximal, characteristic, complemented
9#24906277C211yesmaximal, characteristic, complemented
10#24906274C211yesmaximal, characteristic, complemented
11#24906225C211yesmaximal, characteristic, complemented
12#24905908C211yesmaximal, characteristic, complemented
13#24905904C211yesmaximal, characteristic
14#24905749C211yesmaximal, characteristic, complemented
15C83×#1592C211yesmaximal, characteristic, complemented
16C2×SD16×D83C411yescharacteristic
17C4×Q8×D83C2211yescharacteristic
18C4×D4×D83C2211yescharacteristic
19#19127832C2211yescharacteristic, complemented
20#19127830—21no—
21#19127829—21no—
22C8×Dic166C2211yescharacteristic, complemented
23#19127825C2211yescharacteristic, complemented
24#19127792C411yescharacteristic
25C4×#13050579C2211yescharacteristic
26SD16×Dic83C411yescharacteristic
27C4×#13050577C2211yescharacteristic
28#19127788C411yescharacteristic
29C4×#13050578C2211yescharacteristic
30#19127786C411yescharacteristic
31#19127779C2211yescharacteristic
32#19127775—21no—
33#19127770—21no—
34C8×D332C2211yescharacteristic
35#19127766C2211yescharacteristic
36#19127757C2211yescharacteristic
37#19127757C2211yescharacteristic
38#19127756—21no—
39#19127753—21no—
40#19127752C2211yescharacteristic
41#106×D83C411yescharacteristic
42#103×D83C411yescharacteristic
43#102×D83C411yescharacteristic
44#19127664—21no—
45C22⋊C8×D83—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic, complemented
49#19127451C2211yescharacteristic, complemented
50#19127451C2211yescharacteristic, complemented
51#19127433C2211yescharacteristic, complemented
52Q8⋊(C83⋊C8)C2211yescharacteristic, complemented
53#19127431C2211yescharacteristic, complemented
54#19127430C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic, complemented
58#19127422C2211yescharacteristic, complemented
59#19127422C2211yescharacteristic, complemented
60#446×C83C2211yescharacteristic, complemented
61C4×SD16×C83C2211yescharacteristic
62#436×C83C2211yescharacteristic
63#335×C83C2211yescharacteristic, complemented
64Q8⋊C8×C83C2211yescharacteristic, complemented
65D4⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic, complemented
67C2×Q8×D83C2×C411yescharacteristic
68C2×D4×D83C2×C411yescharacteristic
69C22×C4×D83—21no—
70C2×#13050579C2×C411yescharacteristic
71C2×#13050578C2×C411yescharacteristic
72C2×#13050577C2×C411yescharacteristic
73SD16×D83—24no—
74C2×#13050574—21no—
75C2×#13050573—21no—
76C2×C8×D83—21no—
77C2×C8×D83—21no—
78C2×C8×D83C2×C411yescharacteristic
79C2×C8×D83C2×C411yescharacteristic
80Q8×Dic83C2×C411yescharacteristic
81C4×Dic166—21no—
82C4×Dic166C2311yescharacteristic
83#14581757—21no—
84#14581757C2×C411yescharacteristic
85C4×D332C2311yescharacteristic
86D4×Dic83C2×C411yescharacteristic
87#14581754C2×C411yescharacteristic
88C4×C22⋊D83—21no—
89(C22⋊D83)⋊C4—21no—
90C4⋊C4×D83—21no—
91C4⋊C4×D83C2×C411yescharacteristic
92C4⋊C4×D83C2×C411yescharacteristic
93C22⋊C4×D83—21no—
94C42×D83—21no—
95C42×D83C2311yescharacteristic
96#14581724C2×C411yescharacteristic
97#14581723C2×C411yescharacteristic
98#14581723C2×C411yescharacteristic
99C4⋊(C83⋊C8)—21no—
100C4⋊(C83⋊C8)C2311yescharacteristic
101C4⋊(C83⋊C8)C2311yescharacteristic
102Dic83⋊C8—21no—
103Dic83⋊C8—21no—
104Dic83⋊C8D411yescharacteristic
105Dic83⋊C8D411yescharacteristic
106#14581717C2×C411yescharacteristic
107#14581716C2×C411yescharacteristic
108#14581715C2×C411yescharacteristic
109#14581714C2×C411yescharacteristic
110#14581713C2×C411yescharacteristic
111#14581712C2×C411yescharacteristic
112C22⋊(C83⋊C8)—21no—
113(C2×D83)⋊C8—21no—
114(C2×D83)⋊C8—21no—
115(C2×D83)⋊C8—21no—
116(C2×D83)⋊C8—21no—
117(C2×D83)⋊C8D411yescharacteristic
118(C2×D83)⋊C8D411yescharacteristic
119#14581703—21no—
120#14581702—21no—
121#14581702—21no—
122#14581702—21no—
123C4×C83⋊C8C2311yescharacteristic
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C8×Dic83—21no—
127C8×Dic83—21no—
128C8×Dic83C2×C411yescharacteristic
129C8×Dic83C2×C411yescharacteristic
130C2×SD16×C83C2×C411yescharacteristic
131C2×M16×C83—21no—
132C4×Q8×C83C2311yescharacteristic
133C4×D4×C83C2311yescharacteristic
134#106×C83C2×C411yescharacteristic
135C4⋊C8×C83—21no—
136C4⋊C8×C83C2311yescharacteristic
137C4⋊C8×C83C2311yescharacteristic
138#103×C83C2×C411yescharacteristic
139#102×C83C2×C411yescharacteristic
140C22⋊C8×C83—21no—
141#97×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C23×D83—21no—
146C2×Dic166—21no—
147C2×Dic166C22×C411yescharacteristic
148Q8×D83—22no—
149Q8×D83—41no—
150C2×D332C22×C411yescharacteristic
151C2×C22⋊D83—21no—
152D4×D83—22no—
153D4×D83—41no—
154C22×Dic83—21no—
155C2×C4×D83—21no—
156C2×C4×D83—21no—
157C2×C4×D83—21no—
158C2×C4×D83—21no—
159C2×C4×D83—21no—
160C2×C4×D83—81no—
161C2×C4×D83—81no—
162C2×C4×D83C22×C411yescharacteristic
163C2×C4×D83C2×D411yescharacteristic
164C2×C4×D83#4111yescharacteristic
165#13050579M1614yescomplemented
166#13050578—22no—
167#13050577—22no—
168#13050574—81no—
169#13050573—42no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8—21no—
172C2×C83⋊C8—21no—
173C2×C83⋊C8C22×C411yescharacteristic
174C2×C83⋊C8C22×C411yescharacteristic
175C2×C83⋊C8C2×D411yescharacteristic
176C2×C83⋊C8C2×D411yescharacteristic
177C8×D83—22no—
178C8×D83—41no—
179C8×D83—81no—
180C8×D83—81no—
181C4⋊Dic83—21no—
182C4⋊Dic83C22×C411yescharacteristic
183C4⋊Dic83C22×C411yescharacteristic
184Dic83⋊C4—21no—
185Dic83⋊C4—21no—
186Dic83⋊C4—21no—
187Dic83⋊C4—21no—
188C22⋊Dic83—21no—
189(C2×D83)⋊C4—21no—
190(C2×D83)⋊C4—21no—
191C4×Dic83—21no—
192C4×Dic83—21no—
193C4×Dic83—21no—
194C4×Dic83C22×C411yescharacteristic
195C4×Dic83C2×D411yescharacteristic
196C4×Dic83#4111yescharacteristic
197C2×Q8×C83C22×C411yescharacteristic
198C2×D4×C83C22×C411yescharacteristic
199C22×C4×C83—21no—
200SD16×C83—22no—
201M16×C83—81no—
202C2×C8×C83—21no—
203C2×C8×C83—21no—
204C2×C8×C83—21no—
205C2×C8×C83C22×C411yescharacteristic
206C2×C8×C83C22×C411yescharacteristic
207C2×C8×C83C2×D411yescharacteristic
208C2×C8×C83C2×D411yescharacteristic
209C4⋊C4×C83—21no—
210C4⋊C4×C83C22×C411yescharacteristic
211C4⋊C4×C83C22×C411yescharacteristic
212C22⋊C4×C83—21no—
213C42×C83—21no—
214C42×C83C2411yescharacteristic
215C23×C83—21no—
216C22×D83—21no—
217C22×D83—42no—
218C22×D83—42no—
219C22×D83—81no—
220C22×D83C4×D411yescharacteristic
221Q8×C83—21no—
222Q8×C83—21no—
223D4×C83—21no—
224D4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83C23×C411yescharacteristic
231C2×C4×C83C22×D411yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C22⋊D83—42no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—81no—
238C2×Dic83C4×D411yescharacteristic
239C2×Dic83C4×D411yescharacteristic
240C2×Dic83C4×D411yescharacteristic
241D332—21no—
242D332C2×M1612yes—
243C4×D83—22no—
244C4×D83—21no—
245C4×D83—21no—
246C4×D83—21no—
247C4×D83—21no—
248C4×D83—42no—
249C4×D83—42no—
250C4×D83—41no—
251C4×D83—41no—
252C4×D83—81no—
253C4×D83—81no—
254Dic166—21no—
255Dic166—42no—
256Dic166C2×M1612yes—
257C8×C83—21no—
258C8×C83—21no—
259C8×C83—21no—
260C8×C83—41no—
261C8×C83—41no—
262C8×C83—41no—
263C8×C83C2×M1612yes—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—21no—
267C83⋊C8—21no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83—41no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83C2×C4×D411yescharacteristic
275C2×D83—21no—
276C2×D83—21no—
277C2×D83—22no—
278C2×D83—41no—
279C2×D83—41no—
280C2×D83—84no—
281C2×D83#44412yes—
282C4×C83—21no—
283C4×C83—41no—
284C4×C83—41no—
285C4×C83—41no—
286C4×C83C22×M1611yescharacteristic
287C4×C83#56811yescharacteristic
288C4×C83#57711yescharacteristic
289C4×C83#57711yescharacteristic
290Dic83—22no—
291Dic83—21no—
292Dic83—21no—
293Dic83—21no—
294Dic83—41no—
295Dic83—41no—
296#18060—831nomaximal, Sylow
297C2×C83—81no—
298C2×C83D4×M1611yescharacteristic
299C2×C83#294911yescharacteristic
300C2×C83C2×#44411yescharacteristic
301D83—22no—
302D83—42no—
303#2984—831no—
304#2974—831no—
305C2×C4×SD16—831no—
306#1592—831no—
307#1592D8311yescharacteristic, complemented
308#1583—831no—
309#1583—831no—
310#1582—831no—
311#1582—831no—
312SD16⋊C8—831no—
313SD16⋊C8—831no—
314#1568—831no—
315#1491—831no—
316#1490—831no—
317#1451—831no—
318C83#1806011yescharacteristic, complemented, Sylow
319C22×SD16—831no—
320C2×C4×Q8—831no—
321C2×C4×D4—831no—
322#446—831no—
323#446—1661no—
324#446—1661no—
325#446C2×D8311yescharacteristic, complemented
326C8×Q8—831no—
327C8×Q8—831no—
328C4×SD16—831no—
329C4×SD16—831no—
330C4×SD16—831no—
331C4×SD16—831no—
332C4×SD16—831no—
333C4×SD16—831no—
334C4×SD16—831no—
335C4×SD16C2×D8311yescharacteristic
336#436—831no—
337#436C2×D8311yescharacteristic
338#435—1661no—
339#435—1661no—
340C8×D4—831no—
341C8×D4—831no—
342#432—831no—
343#432—831no—
344#432—1661no—
345#431—831no—
346#431—1661no—
347C2×#106—831no—
348C2×#103—831no—
349C2×#102—831no—
350#407—1661no—
351C2×C22⋊C8—1661no—
352C4×M16—831no—
353C2×C4×C8—831no—
354#335—831no—
355#335C2×D8311yescharacteristic, complemented
356#334—831no—
357#334—831no—
358Q8⋊C8—831no—
359Q8⋊C8—831no—
360Q8⋊C8—831no—
361Q8⋊C8C2×D8311yescharacteristic, complemented
362D4⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8C2×D8311yescharacteristic
366#322—831no—
367#322—831no—
368#322—831no—
369#322C2×D8311yescharacteristic, complemented
370C22×Q8—831no—
371C22×D4—831no—
372C23×C4—1661no—
373C2×SD16—831no—
374C2×SD16—831no—
375C2×SD16—831no—
376C2×SD16—1664no—
377C2×SD16C4×D8311yescharacteristic
378C2×M16—21no—
379C2×M16—1661no—
380C2×M16—1661no—
381C22×C8—831no—
382C22×C8—831no—
383C22×C8—1661no—
384C22×C8—1661no—
385C4×Q8—831no—
386C4×Q8—831no—
387C4×Q8—831no—
388C4×Q8—1661no—
389C4×Q8—1661no—
390C4×Q8C22×D8311yescharacteristic
391C4×D4—831no—
392C4×D4—831no—
393C4×D4—831no—
394C4×D4—1661no—
395C4×D4—1661no—
396C4×D4C22×D8311yescharacteristic
397C2×C4⋊C4—831no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—1661no—
400C2×C22⋊C4—1661no—
401C2×C42—831no—
402C2×C42—1661no—
403#106—831no—
404#106—831no—
405#106—831no—
406#106C4×D8311yescharacteristic
407C4⋊C8—21no—
408C4⋊C8—831no—
409C4⋊C8—831no—
410C4⋊C8—831no—
411C4⋊C8—831no—
412C4⋊C8—1661no—
413C4⋊C8—1661no—
414C4⋊C8—1661no—
415C4⋊C8C22×D8311yescharacteristic
416C4⋊C8C22×D8311yescharacteristic
417#103—831no—
418#103—831no—
419#103—831no—
420#103C4×D8311yescharacteristic
421#102—831no—
422#102—831no—
423#102—831no—
424#102C4×D8311yescharacteristic
425C22⋊C8—21no—
426C22⋊C8—831no—
427C22⋊C8—831no—
428C22⋊C8—1661no—
429C22⋊C8—1661no—
430C22⋊C8—1661no—
431C22⋊C8—1661no—
432C22⋊C8—1661no—
433#97—21no—
434#97—1661no—
435#97—1661no—
436#97—1661no—
437#97—1661no—
438C4×C8—831no—
439C4×C8—831no—
440C4×C8—831no—
441C4×C8—831no—
442C4×C8—831no—
443C4×C8—1661no—
444C4×C8—1661no—
445C4×C8C22×D8311yescharacteristic
446C4×C8C22×D8311yescharacteristic
447C4×C8C22×D8311yescharacteristic
448C24—1661no—
449C2×Q8—3321no—
450C2×Q8—831no—
451C2×Q8—1661no—
452C2×Q8—1662no—
453C2×Q8C2×C4×D8311yescharacteristic
454C2×D4—3321no—
455C2×D4—831no—
456C2×D4—1661no—
457C2×D4—1662no—
458C2×D4C2×C4×D8311yescharacteristic
459C22×C4—21no—
460C22×C4—831no—
461C22×C4—831no—
462C22×C4—831no—
463C22×C4—6641no—
464C22×C4—6641no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4—1661no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1661no—
471SD16—22no—
472SD16—834no—
473SD16—1662no—
474SD16—1662no—
475M16—81no—
476M16—3322no—
477M16—6641no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—3321no—
482C2×C8—831no—
483C2×C8—831no—
484C2×C8—831no—
485C2×C8—831no—
486C2×C8—6641no—
487C2×C8—6641no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1662no—
492C2×C8C2×C4×D8311yescharacteristic
493C2×C8C2×C4×D8311yescharacteristic
494C2×C8D4×D8311yescharacteristic
495C2×C8D4×D8311yescharacteristic
496C4⋊C4—21no—
497C4⋊C4—831no—
498C4⋊C4—831no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4C2×C4×D8311yescharacteristic
505C4⋊C4C2×C4×D8311yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—1661no—
508C22⋊C4—1661no—
509C22⋊C4—1661no—
510C42—21no—
511C42—831no—
512C42—831no—
513C42—831no—
514C42—1661no—
515C42—1661no—
516C42—1661no—
517C42C23×D8311yescharacteristic
518C23—21no—
519C23—3322no—
520C23—3322no—
521C23—831no—
522C23—6641no—
523C23—1661no—
524Q8—21no—
525Q8—21no—
526Q8—3322no—
527Q8—832no—
528Q8—1661no—
529D4—21no—
530D4—21no—
531D4—3322no—
532D4—832no—
533D4—1661no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—3322no—
540C2×C4—3322no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—831no—
544C2×C4—831no—
545C2×C4—831no—
546C2×C4—6641no—
547C2×C4—6641no—
548C2×C4—6641no—
549C2×C4—1662no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4C22×C4×D8311yescharacteristic
558C2×C4C2×D4×D8311yescharacteristic
559C2×C4#41×D8311yescharacteristic
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—3321no—
567C8—3321no—
568C8—3321no—
569C8—1661no—
570C8—1661no—
571C8—1661no—
572C8—1661no—
573C8M16×D8312yes—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—3321no—
578C22—3321no—
579C22—832no—
580C22—6644no—
581C22—1662no—
582C22—1661no—
583C22—1661no—
584C22C4×D4×D8311yescharacteristic
585C4—21no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—3321no—
590C4—3321no—
591C4—1661no—
592C4—1661no—
593C4—1662no—
594C4—1661no—
595C4C2×M16×D8311yescharacteristic
596C4#1912846611yescharacteristic
597C4#1912851011yescharacteristic
598C4#1912851111yescharacteristic
599C2—81no—
600C2—3322no—
601C2—1662no—
602C2#2491619911yescharacteristic
603C2#2491631911yescharacteristic
604C2#444×D8311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1592C2
socleC22×C83C2×C4×D4
prime radicalC2×D4×D83C2×C4
cosocleC42×C83C24
hypercenter#1592D83
hypocenterC83#18060
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1592

Complements

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

NH ≅ G/Ntype
1. #24916549C2semidirect
4. #24906366C2semidirect
5. #24906309C2semidirect
6. #24906304C2semidirect
7. #24906302C2semidirect
8. #24906297C2semidirect
9. #24906277C2semidirect
10. #24906274C2semidirect
11. #24906225C2semidirect
12. #24905908C2semidirect
14. #24905749C2semidirect
15. C83×#1592C2semidirect
19. #19127832C22semidirect
22. C8×Dic166C22semidirect
23. #19127825C22semidirect
48. #19127454C22semidirect
49. #19127451C22semidirect
50. #19127451C22semidirect
51. #19127433C22semidirect
52. Q8⋊(C83⋊C8)C22semidirect
53. #19127431C22semidirect
57. #19127423C22semidirect
58. #19127422C22semidirect
59. #19127422C22semidirect
60. #446×C83C22semidirect
63. #335×C83C22semidirect
64. Q8⋊C8×C83C22semidirect
66. #322×C83C22semidirect
165. #13050579M16semidirect
307. #1592D83semidirect
318. C83#18060semidirect
325. #446C2×D83semidirect
355. #335C2×D83semidirect
361. Q8⋊C8C2×D83semidirect
369. #322C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249165492
#249164952
C4×SD16×D832
#249063662
#249063092
#249063042
#249063022
#249062972
#249062772
#249062742
#249062252
#249059082
#249059042
#249057492
C83×#15922
#1806083