Finite Group Databasebeta

← #35374480

Subgroups — #35374480

Order 21248.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917547C211yesmaximal, characteristic
2#24916611C211yesmaximal, characteristic
3#24916542C211yesmaximal, characteristic, complemented
4#24906338C211yesmaximal, characteristic, complemented
5#24906337C211yesmaximal, characteristic, complemented
6#24906314C211yesmaximal, characteristic, complemented
7#24906305C211yesmaximal, characteristic, complemented
8#24906277C211yesmaximal, characteristic, complemented
9C8×#13050580C211yesmaximal, characteristic, complemented
10C8×#13050577C211yesmaximal, characteristic, complemented
11#24906250C211yesmaximal, characteristic, complemented
12#24905890C211yesmaximal, characteristic, complemented
13#24905870C211yesmaximal, characteristic
14#24905764C211yesmaximal, characteristic, complemented
15C83×#1579C211yesmaximal, 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
27Q8×C83⋊C8C2211yescharacteristic
28C8×Dic166—21no—
29C8×Dic166C2211yescharacteristic
30#19127825—21no—
31#19127825C2211yescharacteristic
32C4×#13050580C2211yescharacteristic
33C4×#13050579C2211yescharacteristic
34C4×#13050577C2211yescharacteristic
35#19127780—21no—
36#19127778C2211yescharacteristic
37#19127771—21no—
38C8×D332C2211yescharacteristic
39#19127760—21no—
40#19127753—21no—
41#19127753C2211yescharacteristic
42#19127752—21no—
43#19127752C2211yescharacteristic
44#19127752C2211yescharacteristic
45#19127752C2211yescharacteristic
46#19127726—21no—
47#19127723C2211yescharacteristic
48#19127454C2211yescharacteristic
49#19127453C2211yescharacteristic
50#19127451C2211yescharacteristic
51#19127433C2211yescharacteristic
52Q8⋊(C83⋊C8)C2211yescharacteristic
53Q8⋊(C83⋊C8)C2211yescharacteristic
54#19127431C2211yescharacteristic
55#19127430C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic
58#19127422C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60C8×Q8×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×#13050573—21no—
73C2×C8×D83—21no—
74#14581787—21no—
75#14581783C2×C411yescharacteristic
76#14581781C2×C411yescharacteristic
77#14581777—21no—
78#14581768C2×C411yescharacteristic
79#14581767C2×C411yescharacteristic
80Q8×Dic83C2×C411yescharacteristic
81C4×Dic166—21no—
82C4×Dic166C2311yescharacteristic
83#14581757—21no—
84#14581757C2×C411yescharacteristic
85C4×D332—21no—
86C4×D332C2311yescharacteristic
87#14581749—21no—
88C4⋊C4×D83—21no—
89C4⋊C4×D83C2×C411yescharacteristic
90C42×D83—21no—
91C42×D83C2311yescharacteristic
92#14581725C2×C411yescharacteristic
93#14581724C2×C411yescharacteristic
94#14581723C2×C411yescharacteristic
95C4⋊(C83⋊C8)—21no—
96C4⋊(C83⋊C8)—21no—
97C4⋊(C83⋊C8)C2311yescharacteristic
98C4⋊(C83⋊C8)C2311yescharacteristic
99Dic83⋊C8—21no—
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)⋊C8—21no—
117(C2×D83)⋊C8C2×C411yescharacteristic
118(C2×D83)⋊C8C2×C411yescharacteristic
119(C2×D83)⋊C8D411yescharacteristic
120(C2×D83)⋊C8D411yescharacteristic
121#14581703—21no—
122C4×C83⋊C8—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
139C4×C8×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—22no—
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—22no—
159#13050573—81no—
160#13050573—81no—
161C2×C83⋊C8—21no—
162C2×C83⋊C8—21no—
163C2×C83⋊C8—21no—
164C2×C83⋊C8C22×C411yescharacteristic
165C2×C83⋊C8C22×C411yescharacteristic
166C2×C83⋊C8C2×D411yescharacteristic
167C2×C83⋊C8C2×D411yescharacteristic
168C8×D83—81no—
169C4⋊Dic83—21no—
170C4⋊Dic83—21no—
171C4⋊Dic83C22×C411yescharacteristic
172C4⋊Dic83C22×C411yescharacteristic
173Dic83⋊C4—21no—
174Dic83⋊C4—21no—
175Dic83⋊C4—21no—
176Dic83⋊C4—21no—
177Dic83⋊C4—21no—
178(C2×D83)⋊C4—21no—
179(C2×D83)⋊C4—21no—
180(C2×D83)⋊C4—21no—
181(C2×D83)⋊C4—21no—
182(C2×D83)⋊C4—21no—
183C4×Dic83—21no—
184C4×Dic83—21no—
185C4×Dic83—21no—
186C4×Dic83C22×C411yescharacteristic
187C4×Dic83C2×D411yescharacteristic
188C4×Dic83#4111yescharacteristic
189C2×Q8×C83C22×C411yescharacteristic
190C2×Q8×C83C22×C411yescharacteristic
191Q16×C83—22no—
192C2×C8×C83—21no—
193C2×C8×C83—21no—
194C2×C8×C83—21no—
195C2×C8×C83C22×C411yescharacteristic
196C2×C8×C83C22×C411yescharacteristic
197C2×C8×C83C2×D411yescharacteristic
198C2×C8×C83C2×D411yescharacteristic
199C4⋊C4×C83—21no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83C22×C411yescharacteristic
202C4⋊C4×C83C22×C411yescharacteristic
203C42×C83—21no—
204C42×C83—21no—
205C42×C83C2411yescharacteristic
206C22×D83—21no—
207C22×D83C4×D411yescharacteristic
208Q8×C83—21no—
209Q8×C83—21no—
210Q8×C83—21no—
211Q8×C83#13112yes—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83C23×C411yescharacteristic
217C2×C4×C83C22×D411yescharacteristic
218C2×C4×C83C2×#4111yescharacteristic
219C2×Dic83—21no—
220C2×Dic83—21no—
221C2×Dic83—21no—
222C2×Dic83C4×D411yescharacteristic
223C2×Dic83C4×D411yescharacteristic
224C2×Dic83C4×D411yescharacteristic
225D332—21no—
226D332—21no—
227D332—81no—
228C4×D83—24no—
229C4×D83—21no—
230C4×D83—21no—
231C4×D83—21no—
232C4×D83—21no—
233C4×D83—41no—
234C4×D83—41no—
235C4×D83—41no—
236C4×D83—81no—
237Dic166—24no—
238Dic166—21no—
239Dic166#13112yes—
240C8×C83—21no—
241C8×C83—21no—
242C8×C83—21no—
243C8×C83—21no—
244C8×C83—41no—
245C8×C83—41no—
246C8×C83—41no—
247C83⋊C8—21no—
248C83⋊C8—21no—
249C83⋊C8—21no—
250C83⋊C8—21no—
251C83⋊C8—41no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C22×C83C2×C4×D411yescharacteristic
255C2×D83—21no—
256C2×D83—21no—
257C2×D83—21no—
258C2×D83—41no—
259C2×D83—41no—
260C2×D83—41no—
261C4×C83—22no—
262C4×C83—21no—
263C4×C83—41no—
264C4×C83—41no—
265C4×C83—41no—
266C4×C83C2×#13111yescharacteristic
267C4×C83C2×#13111yescharacteristic
268C4×C83#57711yescharacteristic
269C4×C83#57811yescharacteristic
270Dic83—21no—
271Dic83—22no—
272Dic83—21no—
273Dic83—21no—
274Dic83—41no—
275Dic83—41no—
276#17919—831nomaximal, Sylow
277C2×C83#292911yescharacteristic
278C2×C83#294711yescharacteristic
279C2×C83#295711yescharacteristic
280D83—41no—
281D83—81no—
282#3108—831no—
283#2993—831no—
284#2982—831no—
285#1588—831no—
286#1587—831no—
287#1584—831no—
288#1583—831no—
289#1579D8311yescharacteristic, complemented
290SD16⋊C8—831no—
291C8×Q16—831no—
292C8×SD16—831no—
293#1574—831no—
294#1487—831no—
295#1482—831no—
296#1454—831no—
297C83#1791911yescharacteristic, complemented, Sylow
298#529—831no—
299C2×C4×Q8—831no—
300#483—831no—
301#480—831no—
302#479—831no—
303#479—831no—
304#464—831no—
305#464—831no—
306#451—831no—
307#448—831no—
308#446—831no—
309C8×Q8—831no—
310C8×Q8—831no—
311C8×Q8—831no—
312C8×Q8—1661no—
313C8×Q8—1661no—
314C8×Q8C2×D8311yescharacteristic
315C8×Q8C2×D8311yescharacteristic
316C4×Q16—831no—
317C4×Q16C2×D8311yescharacteristic
318C4×SD16—831no—
319C4×SD16—831no—
320#436—831no—
321#436—1661no—
322#435—1661no—
323C8×D4—831no—
324#433—1661no—
325#431—831no—
326#431—831no—
327#431—831no—
328#431—831no—
329#431—1661no—
330#431—1661no—
331#424—1661no—
332#423—831no—
333#335—831no—
334#335—831no—
335#334—831no—
336#334C2×D8311yescharacteristic
337Q8⋊C8—831no—
338Q8⋊C8—831no—
339Q8⋊C8—831no—
340Q8⋊C8—831no—
341Q8⋊C8C2×D8311yescharacteristic
342Q8⋊C8C2×D8311yescharacteristic
343D4⋊C8—831no—
344D4⋊C8—831no—
345#322—831no—
346#322—831no—
347#322C2×D8311yescharacteristic
348C82—831no—
349C22×Q8—831no—
350C2×Q16—831no—
351C2×Q16C4×D8311yescharacteristic
352C2×SD16—831no—
353C2×SD16—831no—
354C2×M16—1661no—
355C2×M16—1661no—
356C22×C8—1661no—
357#126—1661no—
358#125—831no—
359#124—831no—
360#123—1661no—
361C22⋊Q8—831no—
362#121—831no—
363C4×Q8—831no—
364C4×Q8—831no—
365C4×Q8—831no—
366C4×Q8—1661no—
367C4×Q8—1661no—
368C4×Q8C22×D8311yescharacteristic
369C4×Q8C22×D8311yescharacteristic
370C4×D4—831no—
371C4×D4—1661no—
372#117—1661no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—1661no—
375C2×C42—831no—
376C2×C42—1661no—
377#107—831no—
378#107C4×D8311yescharacteristic
379#106—831no—
380#106—831no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—831no—
384C4⋊C8—831no—
385C4⋊C8—831no—
386C4⋊C8—831no—
387C4⋊C8—831no—
388C4⋊C8—831no—
389C4⋊C8—1661no—
390C4⋊C8—1661no—
391C4⋊C8—1661no—
392C4⋊C8—1661no—
393C4⋊C8—1661no—
394C4⋊C8—1661no—
395C4⋊C8C22×D8311yescharacteristic
396C4⋊C8C22×D8311yescharacteristic
397#103—831no—
398#103—831no—
399#103—831no—
400#103—831no—
401#103C4×D8311yescharacteristic
402#103C4×D8311yescharacteristic
403#102—831no—
404#102—831no—
405C22⋊C8—831no—
406C22⋊C8—831no—
407C22⋊C8—831no—
408C22⋊C8—831no—
409C22⋊C8—1661no—
410C22⋊C8—1661no—
411C22⋊C8—1661no—
412C22⋊C8—1661no—
413#97—1661no—
414C4×C8—21no—
415C4×C8—21no—
416C4×C8—831no—
417C4×C8—831no—
418C4×C8—831no—
419C4×C8—1661no—
420C4×C8—1661no—
421C4×C8—1661no—
422C4×C8—1661no—
423C4×C8C22×D8311yescharacteristic
424C4×C8C22×D8311yescharacteristic
425C4×C8C22×D8311yescharacteristic
426C2×Q8—831no—
427C2×Q8—1661no—
428C2×Q8—1662no—
429C2×Q8—1662no—
430C2×Q8C2×C4×D8311yescharacteristic
431C2×Q8C2×C4×D8311yescharacteristic
432C2×D4—831no—
433C2×D4—1661no—
434C22×C4—831no—
435C22×C4—831no—
436C22×C4—831no—
437C22×C4—1661no—
438C22×C4—1661no—
439C22×C4—1661no—
440Q16—22no—
441Q16—1662no—
442SD16—1662no—
443SD16—1662no—
444M16—6641no—
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—1661no—
455C2×C8—1661no—
456C2×C8—1661no—
457C2×C8C2×C4×D8311yescharacteristic
458C2×C8C2×C4×D8311yescharacteristic
459C2×C8D4×D8311yescharacteristic
460C2×C8D4×D8311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—831no—
464C4⋊C4—831no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4C2×C4×D8311yescharacteristic
473C4⋊C4C2×C4×D8311yescharacteristic
474C22⋊C4—1661no—
475C22⋊C4—1661no—
476C22⋊C4—1661no—
477C22⋊C4—1661no—
478C22⋊C4—1661no—
479C42—21no—
480C42—21no—
481C42—831no—
482C42—831no—
483C42—831no—
484C42—1661no—
485C42—1661no—
486C42—1661no—
487C42C23×D8311yescharacteristic
488C23—831no—
489C23—1661no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—832no—
494Q8—1664no—
495Q8—1661no—
496Q8#1458179812yes—
497D4—6641no—
498D4—1661no—
499D4—1661no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—3321no—
505C2×C4—3321no—
506C2×C4—3321no—
507C2×C4—831no—
508C2×C4—831no—
509C2×C4—831no—
510C2×C4—6641no—
511C2×C4—1664no—
512C2×C4—1661no—
513C2×C4—1661no—
514C2×C4—1661no—
515C2×C4—1661no—
516C2×C4—1661no—
517C2×C4—1661no—
518C2×C4—1661no—
519C2×C4C22×C4×D8311yescharacteristic
520C2×C4C2×D4×D8311yescharacteristic
521C2×C4#41×D8311yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—3321no—
530C8—3321no—
531C8—3321no—
532C8—1661no—
533C8—1661no—
534C8—1661no—
535C8—1661no—
536C22—3321no—
537C22—3321no—
538C22—3321no—
539C22—1661no—
540C22—1661no—
541C22—1661no—
542C22C4×D4×D8311yescharacteristic
543C4—21no—
544C4—22no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—3321no—
549C4—3321no—
550C4—1661no—
551C4—1661no—
552C4—1661no—
553C4—1662no—
554C4#131×D8311yescharacteristic
555C4C2×#1458179811yescharacteristic
556C4#1912851011yescharacteristic
557C4#1912851711yescharacteristic
558C2—3321no—
559C2—6641no—
560C2#2491615211yescharacteristic
561C2#2491630511yescharacteristic
562C2#2491637711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1579C2
socleC22×C83C2×C4×D4
prime radical#14581767C2×C4
cosocleC42×C83C24
hypercenter#1579D83
hypocenterC83#17919
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1579

Complements

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

NH ≅ G/Ntype
3. #24916542C2semidirect
4. #24906338C2semidirect
5. #24906337C2semidirect
6. #24906314C2semidirect
7. #24906305C2semidirect
8. #24906277C2semidirect
9. C8×#13050580C2semidirect
10. C8×#13050577C2semidirect
11. #24906250C2semidirect
12. #24905890C2semidirect
14. #24905764C2semidirect
15. C83×#1579C2semidirect
289. #1579D83semidirect
297. C83#17919semidirect

Maximal subgroups

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

subgroupindex
#249175472
#249166112
#249165422
#249063382
#249063372
#249063142
#249063052
#249062772
C8×#130505802
C8×#130505772
#249062502
#249058902
#249058702
#249057642
C83×#15792
#1791983