Finite Group Databasebeta

← #35374543

Subgroups — #35374543

Order 21248.

638 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917469C211yesmaximal, characteristic
2#24916554C211yesmaximal, characteristic, complemented
3#24916484C211yesmaximal, characteristic
4#24906321C211yesmaximal, characteristic, complemented
5#24906319C211yesmaximal, characteristic, complemented
6#24906302C211yesmaximal, characteristic, complemented
7#24906296C211yesmaximal, characteristic, complemented
8#24906277C211yesmaximal, characteristic, complemented
9C8×#13050577C211yesmaximal, characteristic, complemented
10C8×#13050578C211yesmaximal, characteristic, complemented
11#24906242C211yesmaximal, characteristic, complemented
12#24905904C211yesmaximal, characteristic
13#24905882C211yesmaximal, characteristic, complemented
14#24905759C211yesmaximal, characteristic, complemented
15C83×SD16⋊C8C211yesmaximal, characteristic, complemented
16#19128255C2211yescharacteristic
17C4×D4×D83C2211yescharacteristic
18#19128033C411yescharacteristic
19#19128030C411yescharacteristic
20#19127990C411yescharacteristic
21#19127989C411yescharacteristic
22#19127905C411yescharacteristic
23#19127904C411yescharacteristic
24#19127853C411yescharacteristic
25#19127850C411yescharacteristic
26#19127832C2211yescharacteristic, complemented
27#19127831—21no—
28#19127830—21no—
29#19127828C2211yescharacteristic, complemented
30C8×Dic166C2211yescharacteristic, complemented
31C4×#13050579C2211yescharacteristic
32C4×#13050577C2211yescharacteristic
33C4×#13050578C2211yescharacteristic
34C8×D332C2211yescharacteristic
35D4×C83⋊C8C2211yescharacteristic
36#19127766C2211yescharacteristic
37C8×C22⋊D83—21no—
38(C22⋊D83)⋊C8—21no—
39#19127761—21no—
40#19127760—21no—
41#19127757C2211yescharacteristic
42#19127753C2211yescharacteristic
43#19127752C2211yescharacteristic
44#19127752C2211yescharacteristic
45C2×(C2×D83)⋊C8—21no—
46C22⋊C8×D83—21no—
47C4×#13050573C2211yescharacteristic
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#19127423C2211yescharacteristic, complemented
59C8×C83⋊C8C2211yescharacteristic, complemented
60C8×Q8×C83C2211yescharacteristic, complemented
61C4×SD16×C83C2211yescharacteristic
62C8×D4×C83C2211yescharacteristic
63#335×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×#13050573—21no—
73C22×C83⋊C8—21no—
74C2×C8×D83—21no—
75C2×C8×D83—21no—
76#14581791C2×C411yescharacteristic
77#14581790C2×C411yescharacteristic
78#14581782—21no—
79#14581772C2×C411yescharacteristic
80#14581770—21no—
81#14581768C2×C411yescharacteristic
82C4×Dic166—21no—
83C4×Dic166C2311yescharacteristic
84C4×D332C2311yescharacteristic
85D4×Dic83C2×C411yescharacteristic
86#14581754C2×C411yescharacteristic
87C4×C22⋊D83—21no—
88(C22⋊D83)⋊C4—21no—
89#14581749—21no—
90C4⋊C4×D83C2×C411yescharacteristic
91C22⋊C4×D83—21no—
92C42×D83C2311yescharacteristic
93#14581724C2×C411yescharacteristic
94#14581723C2×C411yescharacteristic
95#14581723C2×C411yescharacteristic
96C4⋊(C83⋊C8)—21no—
97C4⋊(C83⋊C8)C2311yescharacteristic
98C4⋊(C83⋊C8)C2311yescharacteristic
99Dic83⋊C8—21no—
100Dic83⋊C8—21no—
101Dic83⋊C8C2×C411yescharacteristic
102Dic83⋊C8C2×C411yescharacteristic
103Dic83⋊C8D411yescharacteristic
104Dic83⋊C8D411yescharacteristic
105#14581717C2×C411yescharacteristic
106#14581716C2×C411yescharacteristic
107#14581715C2×C411yescharacteristic
108#14581714C2×C411yescharacteristic
109#14581713C2×C411yescharacteristic
110#14581712C2×C411yescharacteristic
111C22⋊(C83⋊C8)—21no—
112(C2×D83)⋊C8—21no—
113(C2×D83)⋊C8—21no—
114(C2×D83)⋊C8—21no—
115(C2×D83)⋊C8—21no—
116(C2×D83)⋊C8C2×C411yescharacteristic
117(C2×D83)⋊C8C2×C411yescharacteristic
118(C2×D83)⋊C8D411yescharacteristic
119(C2×D83)⋊C8D411yescharacteristic
120#14581703—21no—
121#14581702—21no—
122#14581702—21no—
123#14581702—21no—
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C8×Dic83—21no—
128C8×Dic83—21no—
129C2×SD16×C83C2×C411yescharacteristic
130C22×C8×C83—21no—
131C4×Q8×C83C2311yescharacteristic
132C4×D4×C83C2311yescharacteristic
133#106×C83C2×C411yescharacteristic
134C4⋊C8×C83—21no—
135C4⋊C8×C83C2311yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137#103×C83C2×C411yescharacteristic
138#102×C83C2×C411yescharacteristic
139C22⋊C8×C83—21no—
140C4×C8×C83—21no—
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C23×D83—21no—
145C2×Dic166—21no—
146C2×Dic166C22×C411yescharacteristic
147C2×D332C22×C411yescharacteristic
148C2×C22⋊D83—21no—
149D4×D83—22no—
150D4×D83—22no—
151C22×Dic83—21no—
152C2×C4×D83—21no—
153C2×C4×D83—21no—
154C2×C4×D83—21no—
155C2×C4×D83—81no—
156C2×C4×D83—81no—
157C2×C4×D83C22×C411yescharacteristic
158C2×C4×D83C2×D411yescharacteristic
159C2×C4×D83#4111yescharacteristic
160#13050579—22no—
161#13050578—22no—
162#13050577—22no—
163#13050573—42no—
164C2×C83⋊C8—21no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—81no—
168C2×C83⋊C8C22×C411yescharacteristic
169C2×C83⋊C8C22×C411yescharacteristic
170C2×C83⋊C8C2×D411yescharacteristic
171C2×C83⋊C8C2×D411yescharacteristic
172C8×D83—81no—
173C8×D83—81no—
174C4⋊Dic83—21no—
175C4⋊Dic83C22×C411yescharacteristic
176C4⋊Dic83C22×C411yescharacteristic
177Dic83⋊C4—21no—
178Dic83⋊C4—21no—
179Dic83⋊C4—21no—
180Dic83⋊C4—21no—
181Dic83⋊C4—21no—
182Dic83⋊C4—21no—
183C22⋊Dic83—21no—
184(C2×D83)⋊C4—21no—
185(C2×D83)⋊C4—21no—
186(C2×D83)⋊C4—21no—
187(C2×D83)⋊C4—21no—
188C4×Dic83—21no—
189C4×Dic83C22×C411yescharacteristic
190C4×Dic83C2×D411yescharacteristic
191C4×Dic83#4111yescharacteristic
192C2×Q8×C83C22×C411yescharacteristic
193C2×D4×C83C22×C411yescharacteristic
194C22×C4×C83—21no—
195SD16×C83—22no—
196C2×C8×C83—21no—
197C2×C8×C83—21no—
198C2×C8×C83—21no—
199C2×C8×C83—81no—
200C2×C8×C83C22×C411yescharacteristic
201C2×C8×C83C22×C411yescharacteristic
202C2×C8×C83C2×D411yescharacteristic
203C2×C8×C83C2×D411yescharacteristic
204C4⋊C4×C83—21no—
205C4⋊C4×C83C22×C411yescharacteristic
206C4⋊C4×C83C22×C411yescharacteristic
207C22⋊C4×C83—21no—
208C42×C83—21no—
209C42×C83C2411yescharacteristic
210C23×C83—21no—
211C22×D83—21no—
212C22×D83—24no—
213C22×D83—81no—
214C22×D83—81no—
215C22×D83C4×D411yescharacteristic
216Q8×C83—21no—
217Q8×C83—21no—
218D4×C83—21no—
219D4×C83#13112yes—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83C23×C411yescharacteristic
226C2×C4×C83C22×D411yescharacteristic
227C2×C4×C83C2×#4111yescharacteristic
228C22⋊D83—24no—
229C2×Dic83—21no—
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—81no—
233C2×Dic83C4×D411yescharacteristic
234C2×Dic83C4×D411yescharacteristic
235C2×Dic83C4×D411yescharacteristic
236D332—21no—
237D332#13112yes—
238C4×D83—21no—
239C4×D83—21no—
240C4×D83—21no—
241C4×D83—21no—
242C4×D83—41no—
243C4×D83—41no—
244C4×D83—41no—
245C4×D83—81no—
246C4×D83—81no—
247Dic166—21no—
248Dic166—21no—
249Dic166—81no—
250C8×C83—21no—
251C8×C83—21no—
252C8×C83—21no—
253C8×C83—21no—
254C8×C83—41no—
255C8×C83—41no—
256C8×C83—41no—
257C83⋊C8—22no—
258C83⋊C8—21no—
259C83⋊C8—21no—
260C83⋊C8—21no—
261C83⋊C8—21no—
262C83⋊C8—41no—
263C83⋊C8—41no—
264C22×C83—22no—
265C22×C83—41no—
266C22×C83—41no—
267C22×C83C2×C4×D411yescharacteristic
268C2×D83—21no—
269C2×D83—22no—
270C2×D83—21no—
271C2×D83—21no—
272C2×D83—41no—
273C2×D83—41no—
274C2×D83—84no—
275C4×C83—21no—
276C4×C83—41no—
277C4×C83—41no—
278C4×C83—41no—
279C4×C83C2×#13111yescharacteristic
280C4×C83C2×#13111yescharacteristic
281C4×C83#57711yescharacteristic
282C4×C83#57711yescharacteristic
283Dic83—21no—
284Dic83—21no—
285Dic83—21no—
286Dic83—41no—
287Dic83—41no—
288Dic83—41no—
289#17924—831nomaximal, Sylow
290C2×C83—42no—
291C2×C83#292911yescharacteristic
292C2×C83#294911yescharacteristic
293C2×C83#295611yescharacteristic
294D83—41no—
295D83—42no—
296#3101—831no—
297#2985—831no—
298#2972—831no—
299#1585—831no—
300#1585—831no—
301#1582—831no—
302#1582—831no—
303SD16⋊C8—831no—
304SD16⋊C8D8311yescharacteristic, complemented
305C8×SD16—831no—
306C8×SD16—831no—
307#1572—831no—
308#1490—831no—
309#1485—831no—
310#1453—831no—
311C83#1792411yescharacteristic, complemented, Sylow
312#533—831no—
313C2×C4×D4—831no—
314#483—831no—
315#483—831no—
316#476—831no—
317#476—831no—
318#461—831no—
319#461—831no—
320#450—831no—
321#450—831no—
322#446—831no—
323#446—831no—
324#446—1661no—
325#446—1661no—
326C8×Q8—831no—
327C8×Q8C2×D8311yescharacteristic, complemented
328C4×SD16—831no—
329C4×SD16—831no—
330C4×SD16—831no—
331C4×SD16C2×D8311yescharacteristic
332C8×D4—831no—
333C8×D4—831no—
334C8×D4—831no—
335C8×D4—1661no—
336C8×D4—1661no—
337C8×D4C2×D8311yescharacteristic
338#433—1661no—
339#433—1661no—
340#432—831no—
341#431—831no—
342#431—831no—
343#431—831no—
344C2×C22⋊C8—1661no—
345C2×C22⋊C8—1661no—
346C4×M16—831no—
347#335—831no—
348#335C2×D8311yescharacteristic, complemented
349#334—831no—
350#334—831no—
351Q8⋊C8—831no—
352Q8⋊C8—831no—
353Q8⋊C8—831no—
354Q8⋊C8C2×D8311yescharacteristic, complemented
355D4⋊C8—831no—
356D4⋊C8—831no—
357D4⋊C8—831no—
358D4⋊C8C2×D8311yescharacteristic
359#322—831no—
360#322—831no—
361#322C2×D8311yescharacteristic, complemented
362C82—831no—
363C22×D4—831no—
364C23×C4—1661no—
365C2×SD16—831no—
366C2×SD16—831no—
367C2×SD16—831no—
368C2×SD16C4×D8311yescharacteristic
369C2×M16—1661no—
370C22×C8—21no—
371C22×C8—1661no—
372C22×C8—1661no—
373C22×C8—1661no—
374C4⋊Q8—831no—
375C4⋊Q8—831no—
376#125—1661no—
377C22⋊Q8—831no—
378C22⋊Q8—831no—
379C22⋊Q8—1661no—
380C4×Q8—831no—
381C4×Q8—1661no—
382C4×Q8C22×D8311yescharacteristic
383C4×D4—831no—
384C4×D4—831no—
385C4×D4—831no—
386C4×D4—1661no—
387C4×D4—1661no—
388C4×D4C22×D8311yescharacteristic
389#117—1661no—
390C2×C4⋊C4—831no—
391C2×C22⋊C4—1661no—
392C2×C42—831no—
393#106—831no—
394#106—831no—
395#106—831no—
396#106C4×D8311yescharacteristic
397C4⋊C8—21no—
398C4⋊C8—831no—
399C4⋊C8—831no—
400C4⋊C8—831no—
401C4⋊C8—831no—
402C4⋊C8—831no—
403C4⋊C8—831no—
404C4⋊C8—1661no—
405C4⋊C8—1661no—
406C4⋊C8—1661no—
407C4⋊C8C22×D8311yescharacteristic
408C4⋊C8C22×D8311yescharacteristic
409#103—831no—
410#103—831no—
411#103—831no—
412#103C4×D8311yescharacteristic
413#102—831no—
414#102—831no—
415#102—831no—
416#102C4×D8311yescharacteristic
417C22⋊C8—21no—
418C22⋊C8—831no—
419C22⋊C8—831no—
420C22⋊C8—831no—
421C22⋊C8—831no—
422C22⋊C8—1661no—
423C22⋊C8—1661no—
424C22⋊C8—1661no—
425C22⋊C8—1661no—
426C22⋊C8—1661no—
427#97—1661no—
428#97—1661no—
429#97—1661no—
430#97—1661no—
431C4×C8—21no—
432C4×C8—831no—
433C4×C8—831no—
434C4×C8—831no—
435C4×C8—1661no—
436C4×C8—1661no—
437C4×C8C22×D8311yescharacteristic
438C4×C8C22×D8311yescharacteristic
439C4×C8C22×D8311yescharacteristic
440C24—1661no—
441C2×Q8—831no—
442C2×Q8—1661no—
443C2×Q8C2×C4×D8311yescharacteristic
444C2×D4—831no—
445C2×D4—1661no—
446C2×D4—1662no—
447C2×D4—1662no—
448C2×D4C2×C4×D8311yescharacteristic
449C22×C4—21no—
450C22×C4—831no—
451C22×C4—831no—
452C22×C4—831no—
453C22×C4—6641no—
454C22×C4—6641no—
455C22×C4—1661no—
456C22×C4—1661no—
457C22×C4—1661no—
458C22×C4—1661no—
459SD16—22no—
460SD16—1662no—
461SD16—1662no—
462SD16—1662no—
463M16—3322no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—81no—
468C2×C8—831no—
469C2×C8—831no—
470C2×C8—831no—
471C2×C8—831no—
472C2×C8—6641no—
473C2×C8—6641no—
474C2×C8—6641no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8C2×C4×D8311yescharacteristic
479C2×C8C2×C4×D8311yescharacteristic
480C2×C8D4×D8311yescharacteristic
481C2×C8D4×D8311yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—831no—
484C4⋊C4—831no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4C2×C4×D8311yescharacteristic
493C4⋊C4C2×C4×D8311yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—1661no—
496C22⋊C4—1661no—
497C22⋊C4—1661no—
498C22⋊C4—1661no—
499C22⋊C4—1661no—
500C42—21no—
501C42—831no—
502C42—831no—
503C42—831no—
504C42—1661no—
505C42C23×D8311yescharacteristic
506C23—21no—
507C23—831no—
508C23—6641no—
509C23—6641no—
510C23—1664no—
511C23—1661no—
512Q8—21no—
513Q8—21no—
514Q8—6641no—
515Q8—1661no—
516Q8—1661no—
517D4—21no—
518D4—832no—
519D4—1664no—
520D4—1661no—
521D4#1458179812yes—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—3321no—
530C2×C4—831no—
531C2×C4—831no—
532C2×C4—831no—
533C2×C4—6641no—
534C2×C4—6641no—
535C2×C4—6641no—
536C2×C4—1661no—
537C2×C4—1661no—
538C2×C4—1661no—
539C2×C4—1661no—
540C2×C4—1661no—
541C2×C4—1661no—
542C2×C4—1661no—
543C2×C4C22×C4×D8311yescharacteristic
544C2×C4C2×D4×D8311yescharacteristic
545C2×C4#41×D8311yescharacteristic
546C8—21no—
547C8—21no—
548C8—21no—
549C8—21no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—3321no—
554C8—3321no—
555C8—1662no—
556C8—1661no—
557C8—1661no—
558C8—1661no—
559C8—1661no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—3321no—
564C22—3321no—
565C22—6644no—
566C22—1661no—
567C22—1661no—
568C22—1661no—
569C22—1662no—
570C22C4×D4×D8311yescharacteristic
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—3321no—
576C4—3321no—
577C4—3321no—
578C4—1661no—
579C4—1661no—
580C4—1661no—
581C4#131×D8311yescharacteristic
582C4C2×#1458179811yescharacteristic
583C4#1912851011yescharacteristic
584C4#1912851111yescharacteristic
585C2—42no—
586C2—3322no—
587C2—3321no—
588C2#2491615211yescharacteristic
589C2#2491631911yescharacteristic
590C2#2491637111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×SD16⋊C8C2
socleC22×C83C2×C4×D4
prime radicalC2×D4×D83C2×C4
cosocleC42×C83C24
hypercenterSD16⋊C8D83
hypocenterC83#17924
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #24916554C2semidirect
4. #24906321C2semidirect
5. #24906319C2semidirect
6. #24906302C2semidirect
7. #24906296C2semidirect
8. #24906277C2semidirect
9. C8×#13050577C2semidirect
10. C8×#13050578C2semidirect
11. #24906242C2semidirect
13. #24905882C2semidirect
14. #24905759C2semidirect
15. C83×SD16⋊C8C2semidirect
26. #19127832C22semidirect
29. #19127828C22semidirect
30. C8×Dic166C22semidirect
48. #19127454C22semidirect
49. #19127451C22semidirect
50. #19127451C22semidirect
51. #19127433C22semidirect
52. Q8⋊(C83⋊C8)C22semidirect
53. #19127431C22semidirect
57. #19127423C22semidirect
58. #19127423C22semidirect
59. C8×C83⋊C8C22semidirect
60. C8×Q8×C83C22semidirect
63. #335×C83C22semidirect
64. Q8⋊C8×C83C22semidirect
66. #322×C83C22semidirect
304. SD16⋊C8D83semidirect
311. C83#17924semidirect
327. C8×Q8C2×D83semidirect
348. #335C2×D83semidirect
354. Q8⋊C8C2×D83semidirect
361. #322C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249174692
#249165542
#249164842
#249063212
#249063192
#249063022
#249062962
#249062772
C8×#130505772
C8×#130505782
#249062422
#249059042
#249058822
#249057592
C83×SD16⋊C82
#1792483