Finite Group Databasebeta

← #35375299

Subgroups — #35375299

Order 21248.

680 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917469C211yesmaximal, characteristic
2#24916405C211yesmaximal, characteristic, complemented
3#24916199C211yesmaximal, characteristic
4#24906354C211yesmaximal, characteristic, complemented
5#24906351C211yesmaximal, characteristic, complemented
6#24906309C211yesmaximal, characteristic, complemented
7#24906302C211yesmaximal, characteristic, complemented
8#24906301C211yesmaximal, characteristic, complemented
9C8×#13050577C211yesmaximal, characteristic, complemented
10#24906262C211yesmaximal, characteristic, complemented
11#24906232C211yesmaximal, characteristic, complemented
12#24905916C211yesmaximal, characteristic, complemented
13D4⋊C8×D83C211yesmaximal, characteristic
14#24905742C211yesmaximal, characteristic, complemented
15C83×#1583C211yesmaximal, characteristic, complemented
16C2×M16×D83—21no—
17#19128255C2211yescharacteristic
18C4×D4×D83C2211yescharacteristic
19#19128033C411yescharacteristic
20#19128030C411yescharacteristic
21#19127990C411yescharacteristic
22#19127989C411yescharacteristic
23#19127905C411yescharacteristic
24#19127904C411yescharacteristic
25#19127853C411yescharacteristic
26#19127850C411yescharacteristic
27#19127832C2211yescharacteristic, complemented
28#19127831—21no—
29#19127831C2211yescharacteristic, complemented
30#19127830—21no—
31#19127825C2211yescharacteristic, complemented
32C4×#13050579C2211yescharacteristic
33C4×#13050577C2211yescharacteristic
34C4×#13050578C2211yescharacteristic
35#19127779C2211yescharacteristic
36#19127775—21no—
37#19127770—21no—
38C8×D332C2211yescharacteristic
39#19127766C2211yescharacteristic
40#19127752C2211yescharacteristic
41#19127726—21no—
42#19127723C2211yescharacteristic
43C4⋊C8×D83C2211yescharacteristic
44#19127663—21no—
45#19127655—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127452C2211yescharacteristic, 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
58C8×C83⋊C8C2211yescharacteristic, complemented
59C8×C83⋊C8C2211yescharacteristic, complemented
60C8×Q8×C83C2211yescharacteristic, complemented
61C4×SD16×C83C2211yescharacteristic
62#436×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic, complemented
64Q8⋊C8×C83C2211yescharacteristic, complemented
65D4⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic, complemented
67C2×D4×D83C2×C411yescharacteristic
68C22×C4×D83—21no—
69C2×#13050579C2×C411yescharacteristic
70C2×#13050578C2×C411yescharacteristic
71C2×#13050577C2×C411yescharacteristic
72C2×#13050574—21no—
73C2×#13050573—21no—
74C2×#13050573D411yescharacteristic
75C2×#13050573D411yescharacteristic
76M16×D83—44no—
77C2×C8×D83—21no—
78C2×C8×D83—21no—
79C2×C8×D83—21no—
80C2×C8×D83—21no—
81#14581791C2×C411yescharacteristic
82#14581790C2×C411yescharacteristic
83#14581782—21no—
84#14581772C2×C411yescharacteristic
85#14581770—21no—
86#14581768C2×C411yescharacteristic
87C4×Dic166—21no—
88C4×Dic166C2311yescharacteristic
89C4×D332C2311yescharacteristic
90D4×Dic83C2×C411yescharacteristic
91#14581754C2×C411yescharacteristic
92C4×C22⋊D83—21no—
93(C22⋊D83)⋊C4—21no—
94#14581749—21no—
95C4⋊C4×D83C2×C411yescharacteristic
96C22⋊C4×D83—21no—
97C42×D83C2311yescharacteristic
98#14581724C2×C411yescharacteristic
99#14581723C2×C411yescharacteristic
100#14581723C2×C411yescharacteristic
101C4⋊(C83⋊C8)—21no—
102C4⋊(C83⋊C8)C2311yescharacteristic
103C4⋊(C83⋊C8)C2311yescharacteristic
104Dic83⋊C8—21no—
105Dic83⋊C8—21no—
106Dic83⋊C8—21no—
107Dic83⋊C8—21no—
108Dic83⋊C8C2×C411yescharacteristic
109Dic83⋊C8C2×C411yescharacteristic
110#14581717C2×C411yescharacteristic
111#14581716C2×C411yescharacteristic
112#14581715C2×C411yescharacteristic
113#14581714C2×C411yescharacteristic
114#14581713C2×C411yescharacteristic
115#14581712C2×C411yescharacteristic
116C22⋊(C83⋊C8)—21no—
117(C2×D83)⋊C8—21no—
118(C2×D83)⋊C8—21no—
119(C2×D83)⋊C8C2×C411yescharacteristic
120(C2×D83)⋊C8C2×C411yescharacteristic
121#14581703—21no—
122#14581702—21no—
123#14581702D411yescharacteristic
124#14581702D411yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C4×C83⋊C8C2311yescharacteristic
128C8×Dic83—21no—
129C8×Dic83—21no—
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—
141C4×C8×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C23×D83—21no—
146C2×Dic166—21no—
147C2×Dic166C22×C411yescharacteristic
148C2×D332C22×C411yescharacteristic
149C2×C22⋊D83—21no—
150D4×D83—22no—
151D4×D83—41no—
152C22×Dic83—21no—
153C2×C4×D83—21no—
154C2×C4×D83—21no—
155C2×C4×D83—21no—
156C2×C4×D83—42no—
157C2×C4×D83—42no—
158C2×C4×D83C22×C411yescharacteristic
159C2×C4×D83C2×D411yescharacteristic
160C2×C4×D83#4111yescharacteristic
161#13050579—22no—
162#13050578—41no—
163#13050577—22no—
164#13050574—42no—
165#13050573—22no—
166#13050573—42no—
167#13050573SD1614yescomplemented
168C2×C83⋊C8—21no—
169C2×C83⋊C8—21no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8C22×C411yescharacteristic
172C2×C83⋊C8C22×C411yescharacteristic
173C2×C83⋊C8C2×D411yescharacteristic
174C2×C83⋊C8C2×D411yescharacteristic
175C8×D83—42no—
176C8×D83—42no—
177C8×D83—42no—
178C8×D83—42no—
179C4⋊Dic83—21no—
180C4⋊Dic83C22×C411yescharacteristic
181C4⋊Dic83C22×C411yescharacteristic
182Dic83⋊C4—21no—
183Dic83⋊C4—21no—
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—
191(C2×D83)⋊C4—21no—
192(C2×D83)⋊C4—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—41no—
201M16×C83—42no—
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—42no—
238C2×Dic83C4×D411yescharacteristic
239C2×Dic83C4×D411yescharacteristic
240C2×Dic83C4×D411yescharacteristic
241D332—21no—
242D332#13112yes—
243C4×D83—21no—
244C4×D83—21no—
245C4×D83—41no—
246C4×D83—42no—
247C4×D83—41no—
248C4×D83—41no—
249C4×D83—42no—
250C4×D83C2×SD1612yes—
251C4×D83#13512yes—
252Dic166—21no—
253Dic166—21no—
254Dic166—81no—
255C8×C83—21no—
256C8×C83—21no—
257C8×C83—21no—
258C8×C83—41no—
259C8×C83—41no—
260C8×C83—41no—
261C8×C83C2×SD1612yes—
262C83⋊C8—22no—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C83⋊C8C2×SD1612yes—
269C22×C83—41no—
270C22×C83—41no—
271C22×C83—41no—
272C22×C83C2×C4×D411yescharacteristic
273C2×D83—21no—
274C2×D83—21no—
275C2×D83—22no—
276C2×D83—41no—
277C2×D83—41no—
278C2×D83—84no—
279C2×D83C4×SD1612yes—
280C4×C83—21no—
281C4×C83—41no—
282C4×C83—41no—
283C4×C83—41no—
284C4×C83C2×#13111yescharacteristic
285C4×C83#56811yescharacteristic
286C4×C83C22×SD1611yescharacteristic
287C4×C83C2×#13511yescharacteristic
288Dic83—21no—
289Dic83—21no—
290Dic83—41no—
291Dic83—41no—
292Dic83—41no—
293Dic83C4×SD1612yes—
294#17995—831nomaximal, Sylow
295C2×C83—81no—
296C2×C83#293311yescharacteristic
297C2×C83C2×C4×SD1611yescharacteristic
298C2×C83#295611yescharacteristic
299D83—22no—
300D83—42no—
301#3101—831no—
302#2962—831no—
303D4×M16—831no—
304#1590—831no—
305#1590—831no—
306#1583—831no—
307#1583D8311yescharacteristic, complemented
308#1582—831no—
309#1582—831no—
310C8×SD16—831no—
311C8×SD16—831no—
312#1570—831no—
313#1493—831no—
314C2×D4⋊C8—831no—
315C8×M16—831no—
316C83#1799511yescharacteristic, complemented, Sylow
317C22×M16—1661no—
318#533—831no—
319C2×C4×D4—831no—
320#483—831no—
321#483—831no—
322#476—831no—
323#476—831no—
324#461—831no—
325#461—831no—
326#450—831no—
327#450—831no—
328#446—831no—
329#446—831no—
330#446—1661no—
331#446—1661no—
332C8×Q8—831no—
333C8×Q8C2×D8311yescharacteristic, complemented
334C4×SD16—831no—
335C4×SD16—831no—
336C4×SD16—831no—
337C4×SD16C2×D8311yescharacteristic
338#436—831no—
339#436C2×D8311yescharacteristic
340#435—1661no—
341#435—1661no—
342C8×D4—831no—
343C8×D4—831no—
344#431—831no—
345#424—1661no—
346#423—831no—
347C2×C4⋊C8—831no—
348#407—1661no—
349#405—1661no—
350C4×M16—831no—
351C2×C4×C8—831no—
352#334—831no—
353#334—831no—
354#334—831no—
355#334C2×D8311yescharacteristic, complemented
356Q8⋊C8—831no—
357Q8⋊C8—831no—
358Q8⋊C8—831no—
359Q8⋊C8C2×D8311yescharacteristic, complemented
360D4⋊C8—831no—
361D4⋊C8—831no—
362D4⋊C8—831no—
363D4⋊C8C2×D8311yescharacteristic
364#322—831no—
365#322C2×D8311yescharacteristic, complemented
366C82—831no—
367C82—831no—
368C22×D4—831no—
369C23×C4—1661no—
370C2×SD16—831no—
371C2×SD16—831no—
372C2×SD16—831no—
373C2×SD16C4×D8311yescharacteristic
374C2×M16—21no—
375C2×M16—3324no—
376C2×M16—831no—
377C2×M16—831no—
378C2×M16—1661no—
379C2×M16—1661no—
380C22×C8—1661no—
381C22×C8—1661no—
382C22×C8—1661no—
383C22×C8—1661no—
384C4⋊Q8—831no—
385C4⋊Q8—831no—
386#125—1661no—
387C22⋊Q8—831no—
388C22⋊Q8—831no—
389C22⋊Q8—1661no—
390C4×Q8—831no—
391C4×Q8—1661no—
392C4×Q8C22×D8311yescharacteristic
393C4×D4—831no—
394C4×D4—831no—
395C4×D4—831no—
396C4×D4—1661no—
397C4×D4—1661no—
398C4×D4C22×D8311yescharacteristic
399#117—1661no—
400C2×C4⋊C4—831no—
401C2×C22⋊C4—1661no—
402C2×C42—831no—
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⋊C8—1661no—
416C4⋊C8—1661no—
417C4⋊C8C22×D8311yescharacteristic
418C4⋊C8C22×D8311yescharacteristic
419#103—831no—
420#103—831no—
421#103—831no—
422#103C4×D8311yescharacteristic
423#102—831no—
424#102—831no—
425#102—831no—
426#102C4×D8311yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—831no—
429C22⋊C8—831no—
430C22⋊C8—1661no—
431C22⋊C8—1661no—
432C22⋊C8—1661no—
433#97—831no—
434#97—831no—
435#97—1661no—
436#97—1661no—
437C4×C8—21no—
438C4×C8—831no—
439C4×C8—831no—
440C4×C8—831no—
441C4×C8—1661no—
442C4×C8—1661no—
443C4×C8C22×D8311yescharacteristic
444C4×C8C22×D8311yescharacteristic
445C4×C8C22×D8311yescharacteristic
446C24—1661no—
447C2×Q8—831no—
448C2×Q8—1661no—
449C2×Q8C2×C4×D8311yescharacteristic
450C2×D4—3321no—
451C2×D4—831no—
452C2×D4—1661no—
453C2×D4—1662no—
454C2×D4C2×C4×D8311yescharacteristic
455C22×C4—21no—
456C22×C4—3322no—
457C22×C4—3322no—
458C22×C4—831no—
459C22×C4—831no—
460C22×C4—831no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465SD16—41no—
466SD16—3321no—
467SD16—1662no—
468SD16—1662no—
469M16—42no—
470M16—3322no—
471M16—3322no—
472M16—834no—
473M16—1662no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—3322no—
478C2×C8—3322no—
479C2×C8—3322no—
480C2×C8—3322no—
481C2×C8—831no—
482C2×C8—831no—
483C2×C8—831no—
484C2×C8—831no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8C2×C4×D8311yescharacteristic
489C2×C8C2×C4×D8311yescharacteristic
490C2×C8D4×D8311yescharacteristic
491C2×C8D4×D8311yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—831no—
494C4⋊C4—831no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4C2×C4×D8311yescharacteristic
503C4⋊C4C2×C4×D8311yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1661no—
506C22⋊C4—1661no—
507C22⋊C4—1661no—
508C22⋊C4—1661no—
509C22⋊C4—1661no—
510C42—21no—
511C42—831no—
512C42—831no—
513C42—831no—
514C42—1661no—
515C42C23×D8311yescharacteristic
516C23—21no—
517C23—3322no—
518C23—3322no—
519C23—831no—
520C23—6641no—
521C23—1661no—
522Q8—21no—
523Q8—21no—
524Q8—6641no—
525Q8—1661no—
526Q8—1661no—
527D4—21no—
528D4—21no—
529D4—3322no—
530D4—832no—
531D4—1661no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—3321no—
538C2×C4—3322no—
539C2×C4—3322no—
540C2×C4—3322no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—831no—
544C2×C4—832no—
545C2×C4—832no—
546C2×C4—831no—
547C2×C4—831no—
548C2×C4—1661no—
549C2×C4—1661no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4C22×C4×D8311yescharacteristic
554C2×C4C2×D4×D8311yescharacteristic
555C2×C4#41×D8311yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—3321no—
563C8—3321no—
564C8—832no—
565C8—1662no—
566C8—1661no—
567C8—1661no—
568C8—1661no—
569C8SD16×D8312yes—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—3321no—
574C22—3321no—
575C22—832no—
576C22—6644no—
577C22—1662no—
578C22—1661no—
579C22—1661no—
580C22C4×D4×D8311yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—3321no—
586C4—3321no—
587C4—3321no—
588C4—832no—
589C4—1661no—
590C4—1661no—
591C4#131×D8311yescharacteristic
592C4#1912846711yescharacteristic
593C4C2×SD16×D8311yescharacteristic
594C4#135×D8311yescharacteristic
595C2—81no—
596C2—3322no—
597C2—1662no—
598C2#2491618611yescharacteristic
599C2C4×SD16×D8311yescharacteristic
600C2#2491637111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1583C2
socleC22×C83C2×C4×D4
prime radicalC2×D4×D83C2×C4
cosocleC42×C83C24
hypercenter#1583D83
hypocenterC83#17995
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
2. #24916405C2semidirect
4. #24906354C2semidirect
5. #24906351C2semidirect
6. #24906309C2semidirect
7. #24906302C2semidirect
8. #24906301C2semidirect
9. C8×#13050577C2semidirect
10. #24906262C2semidirect
11. #24906232C2semidirect
12. #24905916C2semidirect
14. #24905742C2semidirect
15. C83×#1583C2semidirect
27. #19127832C22semidirect
29. #19127831C22semidirect
31. #19127825C22semidirect
48. #19127452C22semidirect
49. #19127451C22semidirect
50. #19127451C22semidirect
51. #19127433C22semidirect
52. Q8⋊(C83⋊C8)C22semidirect
53. #19127431C22semidirect
57. #19127423C22semidirect
58. C8×C83⋊C8C22semidirect
59. C8×C83⋊C8C22semidirect
60. C8×Q8×C83C22semidirect
63. #334×C83C22semidirect
64. Q8⋊C8×C83C22semidirect
66. #322×C83C22semidirect
167. #13050573SD16semidirect
307. #1583D83semidirect
316. C83#17995semidirect
333. C8×Q8C2×D83semidirect
355. #334C2×D83semidirect
359. Q8⋊C8C2×D83semidirect
365. #322C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249174692
#249164052
#249161992
#249063542
#249063512
#249063092
#249063022
#249063012
C8×#130505772
#249062622
#249062322
#249059162
D4⋊C8×D832
#249057422
C83×#15832
#1799583