Finite Group Databasebeta

← #35375596

Subgroups — #35375596

Order 21248.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917547C211yesmaximal, characteristic
2#24916622C211yesmaximal, characteristic
3#24916549C211yesmaximal, characteristic, complemented
4#24906360C211yesmaximal, characteristic, complemented
5#24906353C211yesmaximal, characteristic, complemented
6#24906343C211yesmaximal, characteristic, complemented
7#24906337C211yesmaximal, characteristic, complemented
8#24906323C211yesmaximal, characteristic, complemented
9C8×#13050580C211yesmaximal, characteristic, complemented
10#24906262C211yesmaximal, characteristic, complemented
11#24906245C211yesmaximal, characteristic, complemented
12#24905922C211yesmaximal, characteristic
13Q8⋊C8×D83C211yesmaximal, characteristic, complemented
14#24905748C211yesmaximal, characteristic, complemented
15C83×#1587C211yesmaximal, characteristic, complemented
16#19128238C2211yescharacteristic
17C4×Q8×D83C2211yescharacteristic
18#19128030C411yescharacteristic
19#19128016C411yescharacteristic
20#19128007C411yescharacteristic
21#19128006C411yescharacteristic
22#19127926C411yescharacteristic
23#19127925C411yescharacteristic
24#19127855C411yescharacteristic
25#19127841C411yescharacteristic
26#19127832C2211yescharacteristic
27#19127832C2211yescharacteristic
28#19127830—21no—
29#19127829—21no—
30C8×Dic166C2211yescharacteristic
31#19127825C2211yescharacteristic
32C4×#13050580C2211yescharacteristic
33C4×#13050579C2211yescharacteristic
34C4×#13050577C2211yescharacteristic
35#19127780C2211yescharacteristic
36#19127771—21no—
37C8×D332—21no—
38#19127766C2211yescharacteristic
39#19127760—21no—
40#19127757C2211yescharacteristic
41#19127756—21no—
42#19127753—21no—
43#19127752C2211yescharacteristic
44#19127752C2211yescharacteristic
45#19127726—21no—
46C4⋊C8×D83C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic
49#19127453C2211yescharacteristic
50#19127451C2211yescharacteristic
51#19127433C2211yescharacteristic
52Q8⋊(C83⋊C8)C2211yescharacteristic
53Q8⋊(C83⋊C8)C2211yescharacteristic
54#19127431C2211yescharacteristic
55#19127430C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127422C2211yescharacteristic
58C8×C83⋊C8C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60#446×C83C2211yescharacteristic
61C8×Q8×C83C2211yescharacteristic
62C4×Q16×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65Q8⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic
67C2×Q8×D83C2×C411yescharacteristic
68C2×#13050580C2×C411yescharacteristic
69C2×#13050579C2×C411yescharacteristic
70C2×#13050577C2×C411yescharacteristic
71C2×#13050573—21no—
72C2×C8×D83—21no—
73C2×C8×D83—21no—
74C2×C8×D83—21no—
75#14581787—21no—
76#14581783C2×C411yescharacteristic
77#14581781C2×C411yescharacteristic
78#14581777—21no—
79#14581768C2×C411yescharacteristic
80#14581767C2×C411yescharacteristic
81Q8×Dic83C2×C411yescharacteristic
82C4×Dic166—21no—
83C4×Dic166C2311yescharacteristic
84#14581757—21no—
85#14581757C2×C411yescharacteristic
86C4×D332—21no—
87C4×D332C2311yescharacteristic
88#14581749—21no—
89C4⋊C4×D83—21no—
90C4⋊C4×D83C2×C411yescharacteristic
91C42×D83—21no—
92C42×D83C2311yescharacteristic
93#14581725C2×C411yescharacteristic
94#14581724C2×C411yescharacteristic
95#14581723C2×C411yescharacteristic
96C4⋊(C83⋊C8)—21no—
97C4⋊(C83⋊C8)—21no—
98C4⋊(C83⋊C8)C2311yescharacteristic
99C4⋊(C83⋊C8)C2311yescharacteristic
100Dic83⋊C8—21no—
101Dic83⋊C8—21no—
102Dic83⋊C8—21no—
103Dic83⋊C8C2×C411yescharacteristic
104Dic83⋊C8C2×C411yescharacteristic
105Dic83⋊C8D411yescharacteristic
106Dic83⋊C8D411yescharacteristic
107#14581717C2×C411yescharacteristic
108#14581716C2×C411yescharacteristic
109#14581716C2×C411yescharacteristic
110#14581715C2×C411yescharacteristic
111#14581714C2×C411yescharacteristic
112#14581712C2×C411yescharacteristic
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#14581703—21no—
122#14581702—21no—
123C4×C83⋊C8C2311yescharacteristic
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C8×Dic83—21no—
127C8×Dic83—21no—
128C8×Dic83—21no—
129C2×Q16×C83C2×C411yescharacteristic
130C4×Q8×C83C2311yescharacteristic
131C4×Q8×C83C2311yescharacteristic
132#107×C83C2×C411yescharacteristic
133C4⋊C8×C83—21no—
134C4⋊C8×C83—21no—
135C4⋊C8×C83C2311yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137#103×C83C2×C411yescharacteristic
138#103×C83C2×C411yescharacteristic
139#97×C83—21no—
140C4×C8×C83—21no—
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C2×Dic166—21no—
145C2×Dic166C22×C411yescharacteristic
146Q8×D83—22no—
147Q8×D83—41no—
148C2×D332—21no—
149C2×D332C22×C411yescharacteristic
150C2×C4×D83—21no—
151C2×C4×D83—21no—
152C2×C4×D83—21no—
153C2×C4×D83C22×C411yescharacteristic
154C2×C4×D83C2×D411yescharacteristic
155C2×C4×D83#4111yescharacteristic
156#13050580—22no—
157#13050579—22no—
158#13050577—41no—
159#13050573—81no—
160C2×C83⋊C8—21no—
161C2×C83⋊C8—21no—
162C2×C83⋊C8—21no—
163C2×C83⋊C8C22×C411yescharacteristic
164C2×C83⋊C8C22×C411yescharacteristic
165C2×C83⋊C8C2×D411yescharacteristic
166C2×C83⋊C8C2×D411yescharacteristic
167C8×D83—24no—
168C8×D83—81no—
169C8×D83—81no—
170C4⋊Dic83—21no—
171C4⋊Dic83—21no—
172C4⋊Dic83C22×C411yescharacteristic
173C4⋊Dic83C22×C411yescharacteristic
174Dic83⋊C4—21no—
175Dic83⋊C4—21no—
176Dic83⋊C4—21no—
177Dic83⋊C4—21no—
178Dic83⋊C4—21no—
179(C2×D83)⋊C4—21no—
180(C2×D83)⋊C4—21no—
181(C2×D83)⋊C4—21no—
182(C2×D83)⋊C4—21no—
183(C2×D83)⋊C4—21no—
184C4×Dic83—21no—
185C4×Dic83—21no—
186C4×Dic83—21no—
187C4×Dic83C22×C411yescharacteristic
188C4×Dic83C2×D411yescharacteristic
189C4×Dic83#4111yescharacteristic
190C2×Q8×C83C22×C411yescharacteristic
191C2×Q8×C83C22×C411yescharacteristic
192Q16×C83—41no—
193C2×C8×C83—21no—
194C2×C8×C83—21no—
195C2×C8×C83—21no—
196C2×C8×C83C22×C411yescharacteristic
197C2×C8×C83C22×C411yescharacteristic
198C2×C8×C83C2×D411yescharacteristic
199C2×C8×C83C2×D411yescharacteristic
200C4⋊C4×C83—21no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83C22×C411yescharacteristic
203C4⋊C4×C83C22×C411yescharacteristic
204C42×C83—21no—
205C42×C83—21no—
206C42×C83C2411yescharacteristic
207C22×D83—21no—
208C22×D83C4×D411yescharacteristic
209Q8×C83—21no—
210Q8×C83—21no—
211Q8×C83—21no—
212Q8×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83C23×C411yescharacteristic
218C2×C4×C83C22×D411yescharacteristic
219C2×C4×C83C2×#4111yescharacteristic
220C2×Dic83—21no—
221C2×Dic83—21no—
222C2×Dic83—21no—
223C2×Dic83C4×D411yescharacteristic
224C2×Dic83C4×D411yescharacteristic
225C2×Dic83C4×D411yescharacteristic
226D332—21no—
227D332—21no—
228D332—42no—
229C4×D83—22no—
230C4×D83—21no—
231C4×D83—21no—
232C4×D83—21no—
233C4×D83—21no—
234C4×D83—42no—
235C4×D83—42no—
236C4×D83—41no—
237C4×D83—41no—
238Dic166—21no—
239Dic166—42no—
240Dic166#13112yes—
241C8×C83—22no—
242C8×C83—21no—
243C8×C83—21no—
244C8×C83—21no—
245C8×C83—21no—
246C8×C83—41no—
247C8×C83—41no—
248C83⋊C8—22no—
249C83⋊C8—21no—
250C83⋊C8—21no—
251C83⋊C8—21no—
252C83⋊C8—21no—
253C83⋊C8—41no—
254C83⋊C8—41no—
255C22×C83C2×C4×D411yescharacteristic
256C2×D83—21no—
257C2×D83—21no—
258C2×D83—41no—
259C2×D83—41no—
260C2×D83—41no—
261C2×D83#44312yes—
262C4×C83—21no—
263C4×C83—41no—
264C4×C83—41no—
265C4×C83—41no—
266C4×C83—41no—
267C4×C83C2×#13111yescharacteristic
268C4×C83#56811yescharacteristic
269C4×C83#57711yescharacteristic
270C4×C83#57811yescharacteristic
271Dic83—22no—
272Dic83—21no—
273Dic83—21no—
274Dic83—21no—
275Dic83—41no—
276Dic83—41no—
277#18017—831nomaximal, Sylow
278C2×C83#293311yescharacteristic
279C2×C83#294711yescharacteristic
280C2×C83C2×#44311yescharacteristic
281D83—22no—
282D83—81no—
283#3108—831no—
284#2994—831no—
285#2984—831no—
286#1591—831no—
287#1590—831no—
288#1588—831no—
289#1587—831no—
290#1587D8311yescharacteristic, complemented
291#1585—831no—
292C8×Q16—831no—
293C8×SD16—831no—
294#1573—831no—
295#1494—831no—
296C2×Q8⋊C8—831no—
297#1450—831no—
298C83#1801711yescharacteristic, complemented, Sylow
299#529—831no—
300C2×C4×Q8—831no—
301#483—831no—
302#480—831no—
303#479—831no—
304#479—831no—
305#464—831no—
306#464—831no—
307#451—831no—
308#448—831no—
309#446—831no—
310#446—831no—
311#446—1661no—
312#446—1661no—
313#446C2×D8311yescharacteristic
314C8×Q8—831no—
315C8×Q8—831no—
316C8×Q8C2×D8311yescharacteristic
317C4×Q16—831no—
318C4×Q16C2×D8311yescharacteristic
319C4×SD16—831no—
320C4×SD16—831no—
321#436—831no—
322#435—1661no—
323C8×D4—831no—
324C8×D4—1661no—
325#433—1661no—
326#432—831no—
327#432—1661no—
328#431—831no—
329#431—831no—
330#431—1661no—
331#424—1661no—
332C2×C4⋊C8—831no—
333C2×C4×C8—831no—
334#335—831no—
335#335—831no—
336#334—831no—
337#334C2×D8311yescharacteristic
338Q8⋊C8—831no—
339Q8⋊C8—831no—
340Q8⋊C8—831no—
341Q8⋊C8—831no—
342Q8⋊C8C2×D8311yescharacteristic
343Q8⋊C8C2×D8311yescharacteristic
344D4⋊C8—831no—
345D4⋊C8—831no—
346#322—831no—
347#322C2×D8311yescharacteristic
348C82—831no—
349C82—831no—
350C22×Q8—831no—
351C2×Q16—831no—
352C2×Q16C4×D8311yescharacteristic
353C2×SD16—831no—
354C2×SD16—831no—
355C2×M16—1661no—
356C22×C8—1661no—
357C22×C8—1661no—
358C22×C8—1661no—
359#126—1661no—
360#125—831no—
361#124—831no—
362#123—1661no—
363C22⋊Q8—831no—
364#121—831no—
365C4×Q8—831no—
366C4×Q8—831no—
367C4×Q8—831no—
368C4×Q8—1661no—
369C4×Q8—1661no—
370C4×Q8C22×D8311yescharacteristic
371C4×Q8C22×D8311yescharacteristic
372C4×D4—831no—
373C4×D4—1661no—
374#117—1661no—
375C2×C4⋊C4—831no—
376C2×C4⋊C4—1661no—
377C2×C42—831no—
378C2×C42—1661no—
379#107—831no—
380#107C4×D8311yescharacteristic
381#106—831no—
382#106—831no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—831no—
386C4⋊C8—831no—
387C4⋊C8—831no—
388C4⋊C8—831no—
389C4⋊C8—831no—
390C4⋊C8—831no—
391C4⋊C8—1661no—
392C4⋊C8—1661no—
393C4⋊C8—1661no—
394C4⋊C8—1661no—
395C4⋊C8—1661no—
396C4⋊C8C22×D8311yescharacteristic
397C4⋊C8C22×D8311yescharacteristic
398#103—831no—
399#103—831no—
400#103—831no—
401#103—831no—
402#103C4×D8311yescharacteristic
403#103C4×D8311yescharacteristic
404#102—831no—
405#102—831no—
406C22⋊C8—831no—
407C22⋊C8—831no—
408C22⋊C8—831no—
409C22⋊C8—831no—
410C22⋊C8—1661no—
411C22⋊C8—1661no—
412C22⋊C8—1661no—
413#97—21no—
414#97—1661no—
415#97—1661no—
416#97—1661no—
417C4×C8—21no—
418C4×C8—831no—
419C4×C8—831no—
420C4×C8—831no—
421C4×C8—1661no—
422C4×C8—1661no—
423C4×C8—1661no—
424C4×C8C22×D8311yescharacteristic
425C4×C8C22×D8311yescharacteristic
426C4×C8C22×D8311yescharacteristic
427C2×Q8—3321no—
428C2×Q8—831no—
429C2×Q8—1661no—
430C2×Q8—1662no—
431C2×Q8C2×C4×D8311yescharacteristic
432C2×Q8C2×C4×D8311yescharacteristic
433C2×D4—831no—
434C2×D4—1661no—
435C22×C4—831no—
436C22×C4—831no—
437C22×C4—831no—
438C22×C4—1661no—
439C22×C4—1661no—
440C22×C4—1661no—
441Q16—41no—
442Q16—1662no—
443SD16—3321no—
444SD16—1662no—
445M16—6641no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—831no—
450C2×C8—831no—
451C2×C8—831no—
452C2×C8—831no—
453C2×C8—6641no—
454C2×C8—6641no—
455C2×C8—1664no—
456C2×C8—1661no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8C2×C4×D8311yescharacteristic
460C2×C8C2×C4×D8311yescharacteristic
461C2×C8D4×D8311yescharacteristic
462C2×C8D4×D8311yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—831no—
466C4⋊C4—831no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C4⋊C4C2×C4×D8311yescharacteristic
475C4⋊C4C2×C4×D8311yescharacteristic
476C22⋊C4—1661no—
477C22⋊C4—1661no—
478C22⋊C4—1661no—
479C22⋊C4—1661no—
480C22⋊C4—1661no—
481C42—21no—
482C42—21no—
483C42—831no—
484C42—831no—
485C42—831no—
486C42—1661no—
487C42—1661no—
488C42—1661no—
489C42C23×D8311yescharacteristic
490C23—831no—
491C23—1661no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—3322no—
497Q8—832no—
498Q8—1661no—
499D4—3322no—
500D4—1661no—
501D4—1661no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—3322no—
507C2×C4—3322no—
508C2×C4—3321no—
509C2×C4—3321no—
510C2×C4—831no—
511C2×C4—831no—
512C2×C4—831no—
513C2×C4—1662no—
514C2×C4—1661no—
515C2×C4—1661no—
516C2×C4—1661no—
517C2×C4—1661no—
518C2×C4—1661no—
519C2×C4—1661no—
520C2×C4—1661no—
521C2×C4C22×C4×D8311yescharacteristic
522C2×C4C2×D4×D8311yescharacteristic
523C2×C4#41×D8311yescharacteristic
524C8—22no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—3321no—
532C8—3321no—
533C8—1662no—
534C8—1661no—
535C8—1661no—
536C8—1661no—
537C8—1661no—
538C22—3321no—
539C22—3321no—
540C22—3321no—
541C22—832no—
542C22—1661no—
543C22—1661no—
544C22C4×D4×D8311yescharacteristic
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—3321no—
551C4—3321no—
552C4—1661no—
553C4—1661no—
554C4—1662no—
555C4—1661no—
556C4#131×D8311yescharacteristic
557C4#1912846711yescharacteristic
558C4#1912851011yescharacteristic
559C4#1912851711yescharacteristic
560C2—6641no—
561C2—1662no—
562C2#2491618611yescharacteristic
563C2#2491630511yescharacteristic
564C2#443×D8311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1587C2
socleC22×C83C2×C4×D4
prime radical#14581767C2×C4
cosocleC42×C83C24
hypercenter#1587D83
hypocenterC83#18017
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1587

Complements

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

NH ≅ G/Ntype
3. #24916549C2semidirect
4. #24906360C2semidirect
5. #24906353C2semidirect
6. #24906343C2semidirect
7. #24906337C2semidirect
8. #24906323C2semidirect
9. C8×#13050580C2semidirect
10. #24906262C2semidirect
11. #24906245C2semidirect
13. Q8⋊C8×D83C2semidirect
14. #24905748C2semidirect
15. C83×#1587C2semidirect
290. #1587D83semidirect
298. C83#18017semidirect

Maximal subgroups

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

subgroupindex
#249175472
#249166222
#249165492
#249063602
#249063532
#249063432
#249063372
#249063232
C8×#130505802
#249062622
#249062452
#249059222
Q8⋊C8×D832
#249057482
C83×#15872
#1801783