Finite Group Databasebeta

← #35375600

Subgroups — #35375600

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#24917541C211yesmaximal, characteristic
2#24916624C211yesmaximal, characteristic
3#24916551C211yesmaximal, characteristic, complemented
4#24906359C211yesmaximal, characteristic, complemented
5#24906351C211yesmaximal, characteristic, complemented
6#24906336C211yesmaximal, characteristic, complemented
7#24906331C211yesmaximal, characteristic, complemented
8#24906317C211yesmaximal, characteristic, complemented
9C8×#13050580C211yesmaximal, characteristic, complemented
10SD16×C83⋊C8C211yesmaximal, characteristic, complemented
11#24906247C211yesmaximal, characteristic, complemented
12#24905920C211yesmaximal, characteristic
13Q8⋊C8×D83C211yesmaximal, characteristic, complemented
14#24905748C211yesmaximal, characteristic, complemented
15C83×#1588C211yesmaximal, characteristic, complemented
16#19128233C2211yescharacteristic
17C4×Q8×D83C2211yescharacteristic
18#19128033C411yescharacteristic
19#19128013C411yescharacteristic
20#19128011C411yescharacteristic
21#19128010C411yescharacteristic
22#19127922C411yescharacteristic
23#19127921C411yescharacteristic
24#19127858C411yescharacteristic
25#19127838C411yescharacteristic
26#19127831—21no—
27#19127831C2211yescharacteristic
28#19127828—21no—
29#19127828C2211yescharacteristic
30#19127828C2211yescharacteristic
31Q8×C83⋊C8C2211yescharacteristic
32C8×Dic166C2211yescharacteristic
33C4×Dic332C2211yescharacteristic
34C4×#13050580C2211yescharacteristic
35C4×#13050578C2211yescharacteristic
36#19127770—21no—
37D4×C83⋊C8C2211yescharacteristic
38(C22⋊D83)⋊C8—21no—
39#19127759—21no—
40#19127757—21no—
41#19127757C2211yescharacteristic
42#19127752—21no—
43#19127752C2211yescharacteristic
44#19127752C2211yescharacteristic
45#19127729—21no—
46C4⋊C8×D83C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127453C2211yescharacteristic
49#19127452C2211yescharacteristic
50#19127451C2211yescharacteristic
51#19127433C2211yescharacteristic
52#19127433C2211yescharacteristic
53Q8⋊(C83⋊C8)C2211yescharacteristic
54#19127431C2211yescharacteristic
55#19127431C2211yescharacteristic
56D4⋊(C83⋊C8)C2211yescharacteristic
57#19127422C2211yescharacteristic
58C8×C83⋊C8C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60C8×Q8×C83C2211yescharacteristic
61C4×SD16×C83C2211yescharacteristic
62#436×C83C2211yescharacteristic
63#335×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic
67C2×Q8×D83C2×C411yescharacteristic
68C2×Dic332C2×C411yescharacteristic
69C2×#13050580C2×C411yescharacteristic
70C2×#13050578C2×C411yescharacteristic
71C22×C83⋊C8—21no—
72C2×C8×D83—21no—
73C2×C8×D83—21no—
74#14581785—21no—
75#14581783C2×C411yescharacteristic
76#14581779C2×C411yescharacteristic
77#14581774—21no—
78#14581772C2×C411yescharacteristic
79#14581763C2×C411yescharacteristic
80Q8×Dic83C2×C411yescharacteristic
81C4×Dic166—21no—
82C4×Dic166C2311yescharacteristic
83C4×Dic166C2311yescharacteristic
84#14581757—21no—
85#14581757C2×C411yescharacteristic
86C4×C22⋊D83—21no—
87#14581751—21no—
88C4⋊C4×D83—21no—
89C4⋊C4×D83C2×C411yescharacteristic
90C42×D83—21no—
91C42×D83C2311yescharacteristic
92#14581726C2×C411yescharacteristic
93#14581725C2×C411yescharacteristic
94#14581723C2×C411yescharacteristic
95C4⋊(C83⋊C8)—21no—
96C4⋊(C83⋊C8)C2311yescharacteristic
97C4⋊(C83⋊C8)C2311yescharacteristic
98Dic83⋊C8—21no—
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#14581717C2×C411yescharacteristic
109#14581716C2×C411yescharacteristic
110#14581715C2×C411yescharacteristic
111#14581715C2×C411yescharacteristic
112#14581713C2×C411yescharacteristic
113C22⋊(C83⋊C8)—21no—
114(C2×D83)⋊C8—21no—
115(C2×D83)⋊C8C2×C411yescharacteristic
116(C2×D83)⋊C8C2×C411yescharacteristic
117(C2×D83)⋊C8D411yescharacteristic
118(C2×D83)⋊C8D411yescharacteristic
119#14581702—21no—
120#14581702—21no—
121#14581702—21no—
122#14581702—21no—
123C4×C83⋊C8—21no—
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C8×Dic83—21no—
128C8×Dic83—21no—
129C2×SD16×C83C2×C411yescharacteristic
130C2×M16×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
144C2×Dic166—21no—
145C2×Dic166C22×C411yescharacteristic
146C2×Dic166C22×C411yescharacteristic
147Q8×D83—22no—
148Q8×D83—41no—
149C2×C22⋊D83—21no—
150C2×C4×D83—21no—
151C2×C4×D83—21no—
152C2×C4×D83C22×C411yescharacteristic
153C2×C4×D83C2×D411yescharacteristic
154C2×C4×D83#4111yescharacteristic
155Dic332—41no—
156#13050580—22no—
157#13050578—41no—
158C2×C83⋊C8—21no—
159C2×C83⋊C8—21no—
160C2×C83⋊C8—21no—
161C2×C83⋊C8—81no—
162C2×C83⋊C8C22×C411yescharacteristic
163C2×C83⋊C8C22×C411yescharacteristic
164C2×C83⋊C8C2×D411yescharacteristic
165C2×C83⋊C8C2×D411yescharacteristic
166C8×D83—24no—
167C8×D83—81no—
168C4⋊Dic83—21no—
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—
178C22⋊Dic83—21no—
179C22⋊Dic83—21no—
180C22⋊Dic83—21no—
181(C2×D83)⋊C4—21no—
182C4×Dic83—21no—
183C4×Dic83—21no—
184C4×Dic83—21no—
185C4×Dic83—21no—
186C4×Dic83C22×C411yescharacteristic
187C4×Dic83C2×D411yescharacteristic
188C4×Dic83#4111yescharacteristic
189C2×Q8×C83C22×C411yescharacteristic
190C2×D4×C83C22×C411yescharacteristic
191C22×C4×C83—21no—
192SD16×C83—22no—
193M16×C83—81no—
194C2×C8×C83—21no—
195C2×C8×C83—21no—
196C2×C8×C83—21no—
197C2×C8×C83C22×C411yescharacteristic
198C2×C8×C83C22×C411yescharacteristic
199C2×C8×C83C2×D411yescharacteristic
200C2×C8×C83C2×D411yescharacteristic
201C4⋊C4×C83—21no—
202C4⋊C4×C83C22×C411yescharacteristic
203C4⋊C4×C83C22×C411yescharacteristic
204C22⋊C4×C83—21no—
205C42×C83—21no—
206C42×C83C2411yescharacteristic
207C23×C83—21no—
208C22×D83C4×D411yescharacteristic
209Q8×C83—21no—
210Q8×C83#13112yes—
211D4×C83—21no—
212D4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83C23×C411yescharacteristic
219C2×C4×C83C22×D411yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C22⋊D83—42no—
222C2×Dic83—21no—
223C2×Dic83—21no—
224C2×Dic83—21no—
225C2×Dic83—21no—
226C2×Dic83C4×D411yescharacteristic
227C2×Dic83C4×D411yescharacteristic
228C2×Dic83C4×D411yescharacteristic
229C4×D83—22no—
230C4×D83—21no—
231C4×D83—21no—
232C4×D83—21no—
233C4×D83—21no—
234C4×D83—42no—
235C4×D83—42no—
236Dic166—21no—
237Dic166—21no—
238Dic166—21no—
239Dic166—21no—
240Dic166—42no—
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×C83—41no—
256C22×C83—41no—
257C22×C83—41no—
258C22×C83C2×C4×D411yescharacteristic
259C2×D83—21no—
260C2×D83—21no—
261C2×D83#44312yes—
262C4×C83—22no—
263C4×C83—21no—
264C4×C83—41no—
265C4×C83—41no—
266C4×C83C2×#13111yescharacteristic
267C4×C83#56811yescharacteristic
268C4×C83#57711yescharacteristic
269C4×C83#57811yescharacteristic
270Dic83—21no—
271Dic83—21no—
272Dic83—21no—
273Dic83—41no—
274Dic83—41no—
275Dic83—41no—
276Dic83—41no—
277#18017—831nomaximal, Sylow
278C2×C83—81no—
279C2×C83#293311yescharacteristic
280C2×C83#294711yescharacteristic
281C2×C83C2×#44311yescharacteristic
282D83—22no—
283#3108—831no—
284#2994—831no—
285#2984—831no—
286#1591—831no—
287#1590—831no—
288#1588D8311yescharacteristic, complemented
289#1587—831no—
290#1587—831no—
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—831no—
312#446—1661no—
313#446—1661no—
314C8×Q8—831no—
315C8×Q8—831no—
316C8×Q8C2×D8311yescharacteristic
317C4×Q16—831no—
318C4×Q16—831no—
319C4×SD16—831no—
320C4×SD16C2×D8311yescharacteristic
321#436C2×D8311yescharacteristic
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#335C2×D8311yescharacteristic
336#334—831no—
337#334—831no—
338Q8⋊C8—831no—
339Q8⋊C8—831no—
340Q8⋊C8—831no—
341Q8⋊C8—831no—
342Q8⋊C8—831no—
343Q8⋊C8C2×D8311yescharacteristic
344D4⋊C8—831no—
345D4⋊C8C2×D8311yescharacteristic
346#322—831no—
347#322C2×D8311yescharacteristic
348C82—831no—
349C82—831no—
350C22×Q8—831no—
351C2×Q16—831no—
352C2×Q16—831no—
353C2×SD16—831no—
354C2×SD16C4×D8311yescharacteristic
355C2×M16—21no—
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—831no—
369C4×Q8—1661no—
370C4×Q8—1661no—
371C4×Q8C22×D8311yescharacteristic
372C4×D4—1661no—
373C4×D4C22×D8311yescharacteristic
374#117—1661no—
375C2×C4⋊C4—831no—
376C2×C4⋊C4—1661no—
377C2×C42—831no—
378C2×C42—1661no—
379#107—831no—
380#107—831no—
381#106—831no—
382#106C4×D8311yescharacteristic
383C4⋊C8—21no—
384C4⋊C8—831no—
385C4⋊C8—831no—
386C4⋊C8—831no—
387C4⋊C8—831no—
388C4⋊C8—831no—
389C4⋊C8—831no—
390C4⋊C8—1661no—
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#103—831no—
403#103C4×D8311yescharacteristic
404#102—831no—
405#102C4×D8311yescharacteristic
406C22⋊C8—21no—
407C22⋊C8—831no—
408C22⋊C8—831no—
409C22⋊C8—831no—
410C22⋊C8—831no—
411C22⋊C8—1661no—
412C22⋊C8—1661no—
413#97—1661no—
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—831no—
430C2×Q8—1661no—
431C2×Q8—1662no—
432C2×Q8C2×C4×D8311yescharacteristic
433C2×D4—1661no—
434C2×D4C2×C4×D8311yescharacteristic
435C22×C4—21no—
436C22×C4—831no—
437C22×C4—831no—
438C22×C4—831no—
439C22×C4—1661no—
440C22×C4—1661no—
441Q16—3321no—
442Q16—1662no—
443SD16—22no—
444SD16—3321no—
445M16—81no—
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—831no—
465C4⋊C4—831no—
466C4⋊C4—1661no—
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—21no—
477C22⋊C4—1661no—
478C22⋊C4—1661no—
479C22⋊C4—1661no—
480C22⋊C4—1661no—
481C42—21no—
482C42—831no—
483C42—831no—
484C42—831no—
485C42—1661no—
486C42—1661no—
487C42—1661no—
488C42—1661no—
489C42C23×D8311yescharacteristic
490C23—21no—
491C23—831no—
492Q8—21no—
493Q8—3322no—
494Q8—1661no—
495Q8—1661no—
496Q8—1661no—
497Q8—1661no—
498Q8#1458180012yes—
499D4—21no—
500D4—21no—
501D4—3322no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—3322no—
508C2×C4—3322no—
509C2×C4—831no—
510C2×C4—831no—
511C2×C4—831no—
512C2×C4—1662no—
513C2×C4—1661no—
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—41no—
539C22—41no—
540C22—41no—
541C22—832no—
542C22—1661no—
543C22—1661no—
544C22C4×D4×D8311yescharacteristic
545C4—21no—
546C4—22no—
547C4—41no—
548C4—41no—
549C4—3321no—
550C4—3321no—
551C4—3321no—
552C4—3321no—
553C4—1661no—
554C4—1661no—
555C4—1661no—
556C4C2×#1458180011yescharacteristic
557C4#1912846611yescharacteristic
558C4#1912851111yescharacteristic
559C4#1912851711yescharacteristic
560C2—81no—
561C2—1662no—
562C2#2491618911yescharacteristic
563C2#2491630311yescharacteristic
564C2#443×D8311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1588C2
socleC22×C83C2×C4×D4
prime radical#14581763C2×C4
cosocleC42×C83C24
hypercenter#1588D83
hypocenterC83#18017
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1588

Complements

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

NH ≅ G/Ntype
3. #24916551C2semidirect
4. #24906359C2semidirect
5. #24906351C2semidirect
6. #24906336C2semidirect
7. #24906331C2semidirect
8. #24906317C2semidirect
9. C8×#13050580C2semidirect
10. SD16×C83⋊C8C2semidirect
11. #24906247C2semidirect
13. Q8⋊C8×D83C2semidirect
14. #24905748C2semidirect
15. C83×#1588C2semidirect
288. #1588D83semidirect
298. C83#18017semidirect

Maximal subgroups

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

subgroupindex
#249175412
#249166242
#249165512
#249063592
#249063512
#249063362
#249063312
#249063172
C8×#130505802
SD16×C83⋊C82
#249062472
#249059202
Q8⋊C8×D832
#249057482
C83×#15882
#1801783