Finite Group Databasebeta

← #35559149

Subgroups — #35559149

Order 21248.

514 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917988C211yesmaximal, characteristic, complemented
2#24916375C211yesmaximal, characteristic
3#24909640C211yesmaximal, characteristic, complemented
4#24909639C211yesmaximal, characteristic, complemented
5#24909638C211yesmaximal, characteristic, complemented
6#24909637C211yesmaximal, characteristic, complemented
7#24909633C211yesmaximal, characteristic, complemented
8#24909632C211yesmaximal, characteristic, complemented
9#24909631C211yesmaximal, characteristic, complemented
10#24909234C211yesmaximal, characteristic, complemented
11#365×D83C211yesmaximal, characteristic, complemented
12#24909227C211yesmaximal, characteristic, complemented
13#24909197C211yesmaximal, characteristic
14#360×D83C211yesmaximal, characteristic
15C83×#2230C211yesmaximal, characteristic, complemented
16#135×D83C2211yescharacteristic
17#19128466—21no—
18#19128328—21no—
19M32×D83C2211yescharacteristic
20M32×D83C2211yescharacteristic
21#19128108—21no—
22#19128107—21no—
23#19128103C2211yescharacteristic, complemented
24#19128101C2211yescharacteristic, complemented
25#19128098C2211yescharacteristic, complemented
26#19127821—21no—
27#19127820—21no—
28#19127819C2211yescharacteristic
29#19127818C2211yescharacteristic
30#19127817C2211yescharacteristic
31#108×D83C2211yescharacteristic
32#19127740—21no—
33#19127736C2211yescharacteristic
34#106×D83C2211yescharacteristic
35#19127716—21no—
36#19127655C2211yescharacteristic
37#19127525C2211yescharacteristic, complemented
38#19127525C2211yescharacteristic, complemented
39#19127524C2211yescharacteristic, complemented
40#19127524C2211yescharacteristic, complemented
41#19127523C2211yescharacteristic, complemented
42#19127523C2211yescharacteristic, complemented
43#19127522C2211yescharacteristic, complemented
44#19127521C2211yescharacteristic, complemented
45#19127521C2211yescharacteristic, complemented
46#19127512C2211yescharacteristic
47#19127512C2211yescharacteristic
48#19127511C2211yescharacteristic
49#19127511C2211yescharacteristic
50#19127510C2211yescharacteristic
51#19127510C2211yescharacteristic
52#498×C83C2211yescharacteristic, complemented
53#443×C83C2211yescharacteristic
54#365×C83C2211yescharacteristic, complemented
55#365×C83C2211yescharacteristic, complemented
56#364×C83C2211yescharacteristic, complemented
57#360×C83C2211yescharacteristic
58#360×C83C2211yescharacteristic
59#41×D83—21no—
60#14581815Q811yescharacteristic, complemented
61#14581814C2311yescharacteristic
62#14581813—21no—
63#14581812—21no—
64#14581811Q811yescharacteristic, complemented
65Q16×D83Q811yescharacteristic, complemented
66SD16×D83—21no—
67D8×D83Q811yescharacteristic, complemented
68#14581800—21no—
69#14581800—41no—
70#14581799—21no—
71#14581798—21no—
72#14581798—41no—
73C2×#13050573—41no—
74C2×#13050573D411yescharacteristic
75M16×D83—21no—
76M16×D83—41no—
77C2×C8×D83C2311yescharacteristic
78#14581791—21no—
79#14581790—41no—
80#14581784—21no—
81#14581783—21no—
82#14581782—41no—
83#14581770—41no—
84#14581768—41no—
85#14581750—21no—
86#14581749D411yescharacteristic
87C4⋊C4×D83—21no—
88#14581732C2311yescharacteristic
89#14581732C2311yescharacteristic
90#14581731—21no—
91#14581731—21no—
92C16×D83—21no—
93C16×D83—21no—
94#14581728C2311yescharacteristic
95#14581727—21no—
96#14581726—21no—
97#14581725—21no—
98#14581725—21no—
99#14581724C2311yescharacteristic
100#14581724C2311yescharacteristic
101#14581723—21no—
102#14581723—21no—
103#14581720—21no—
104#14581719—21no—
105#14581718—21no—
106#14581703D411yescharacteristic
107#14581702D411yescharacteristic
108C8×Dic83C2311yescharacteristic
109#135×C83C2311yescharacteristic
110#131×C83—21no—
111C4⋊Q8×C83—21no—
112M32×C83C2311yescharacteristic
113M32×C83C2311yescharacteristic
114#108×C83C2311yescharacteristic
115#106×C83—21no—
116#106×C83C2311yescharacteristic
117#106×C83C2311yescharacteristic
118#104×C83—21no—
119C4×C8×C83C2311yescharacteristic
120#13050591—21no—
121#13050590—21no—
122#13050590—41no—
123#13050589—21no—
124#13050589—41no—
125C2×Dic166—41no—
126Q8×D83—21no—
127D4×D83—21no—
128D4×D83—41no—
129C2×C4×D83—41no—
130C2×C4×D83—41no—
131C2×C4×D83C2×D411yescharacteristic
132Dic332C2×Q811yescharacteristic
133#13050580—21no—
134#13050579—21no—
135#13050578—21no—
136#13050577—21no—
137D664C2×Q811yescharacteristic
138#13050575—21no—
139#13050574—21no—
140#13050574—41no—
141#13050573—41no—
142#13050573—41no—
143#13050573—41no—
144#13050573SD1611yescharacteristic
145#13050573SD1611yescharacteristic
146#13050573SD1611yescharacteristic
147#13050573SD1611yescharacteristic
148C2×C83⋊C8—41no—
149C2×C83⋊C8C2×D411yescharacteristic
150C2×C83⋊C8C2×D411yescharacteristic
151C8×D83—41no—
152C8×D83—41no—
153C8×D83—41no—
154C8×D83C2×Q811yescharacteristic
155C8×D83C2×Q811yescharacteristic
156C8×D83#4111yescharacteristic
157C8×D83#4111yescharacteristic
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—41no—
161Dic83⋊C4—21no—
162Dic83⋊C4—41no—
163Dic83⋊C4—41no—
164Dic83⋊C4—41no—
165Dic83⋊C4—41no—
166Dic83⋊C4—41no—
167(C2×D83)⋊C4—21no—
168(C2×D83)⋊C4—41no—
169C4×Dic83—41no—
170C4×Dic83C2×D411yescharacteristic
171C83⋊C16—21no—
172C83⋊C16—21no—
173#41×C83—21no—
174C2×Q8×C83—41no—
175Q16×C83C2×Q811yescharacteristic
176SD16×C83—21no—
177D8×C83C2×Q811yescharacteristic
178M16×C83—21no—
179M16×C83—41no—
180C2×C8×C83—41no—
181C2×C8×C83C2×D411yescharacteristic
182C2×C8×C83C2411yescharacteristic
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—41no—
186C42×C83C2×D411yescharacteristic
187C16×C83—21no—
188C16×C83—21no—
189C22×D83—41no—
190C22×D83AGL(1,ℤ/8)11yescharacteristic
191Q8×C83—21no—
192Q8×C83—161no—
193D4×C83—21no—
194D4×C83—41no—
195C2×C4×C83—21no—
196C2×C4×C83—41no—
197C2×C4×C83—41no—
198C2×C4×C83—41no—
199C2×C4×C83C22×D411yescharacteristic
200C22⋊D83—41no—
201C22⋊D83—41no—
202C22⋊D83—41no—
203C2×Dic83—21no—
204C2×Dic83—41no—
205C2×Dic83—41no—
206C2×Dic83—41no—
207C2×Dic83AGL(1,ℤ/8)11yescharacteristic
208D332—21no—
209D332—41no—
210C4×D83—41no—
211C4×D83—41no—
212C4×D83—41no—
213C4×D83—161no—
214C4×D83C22⋊Q811yescharacteristic
215C4×D83C22⋊Q811yescharacteristic
216C4×D83C2×SD1611yescharacteristic
217C4×D83C2×SD1611yescharacteristic
218Dic166—21no—
219Dic166—41no—
220Dic166—161no—
221C8×C83—41no—
222C8×C83C2×SD1611yescharacteristic
223C8×C83C2×SD1611yescharacteristic
224C8×C83C22×Q811yescharacteristic
225C8×C83C2×#4111yescharacteristic
226C83⋊C8—41no—
227C83⋊C8C22⋊Q811yescharacteristic
228C83⋊C8C22⋊Q811yescharacteristic
229C83⋊C8C2×SD1611yescharacteristic
230C83⋊C8C2×SD1611yescharacteristic
231C22×C83—41no—
232C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
233C2×D83—41no—
234C2×D83—41no—
235C2×D83—161no—
236C2×D83#47611yescharacteristic
237C2×D83#47611yescharacteristic
238C4×C83—41no—
239C4×C83—41no—
240C4×C83—81no—
241C4×C83—81no—
242C4×C83C2×C22⋊Q811yescharacteristic
243C4×C83C22×SD1611yescharacteristic
244Dic83—41no—
245Dic83—41no—
246Dic83—81no—
247Dic83—81no—
248Dic83#47611yescharacteristic
249Dic83#47611yescharacteristic
250#33638—831nomaximal, Sylow
251C2×C83—21no—
252C2×C83—81no—
253C2×C83C2×#47611yescharacteristic
254D83—21no—
255D83—21no—
256D83—81no—
257#3163—831no—
258#2957—831no—
259#2231—831no—
260#2231—831no—
261#2231—831no—
262#2231—831no—
263#2230—831no—
264#2230—831no—
265#2230—831no—
266#2230D8311yescharacteristic, complemented
267#2155—831no—
268C2×#365—831no—
269#2153—831no—
270#2145—831no—
271C2×#360—831no—
272C83#3363811yescharacteristic, complemented, Sylow
273C2×#135—831no—
274#568—1661no—
275#544—1661no—
276C2×M32—831no—
277C2×M32—831no—
278#501—1661no—
279#501—1661no—
280#499—831no—
281#499—831no—
282#498—831no—
283#498C2×D8311yescharacteristic, complemented
284#444—1661no—
285#444—1661no—
286#443—831no—
287#443—831no—
288#443—831no—
289#443C2×D8311yescharacteristic
290C2×#108—831no—
291#428—1661no—
292#427—831no—
293C2×#106—831no—
294#421—1661no—
295#405—831no—
296#365—831no—
297#365—831no—
298#365—831no—
299#365—831no—
300#365—831no—
301#365—831no—
302#365C2×D8311yescharacteristic, complemented
303#365C2×D8311yescharacteristic, complemented
304#364—831no—
305#364—831no—
306#364—831no—
307#364C2×D8311yescharacteristic, complemented
308#360—831no—
309#360—831no—
310#360—831no—
311#360—831no—
312#360—831no—
313#360—831no—
314#360C2×D8311yescharacteristic
315#360C2×D8311yescharacteristic
316C2×#41—1661no—
317#135—831no—
318#135—831no—
319#135—831no—
320#135—1661no—
321#135—1661no—
322#135C22×D8311yescharacteristic
323C2×Q16—831no—
324C2×SD16—1661no—
325C2×D8—831no—
326#131—21no—
327#131—3321no—
328#131—3321no—
329#131—1661no—
330#131—1661no—
331#131—1661no—
332C2×M16—3321no—
333C2×M16—3321no—
334C2×M16—831no—
335C2×M16—1661no—
336C22×C8—831no—
337C4⋊Q8—21no—
338C4⋊Q8—3321no—
339C4⋊Q8—1661no—
340#125—3321no—
341#125—1661no—
342#125—1661no—
343C22⋊Q8—3321no—
344C22⋊Q8—3321no—
345#117—831no—
346#117—1661no—
347C2×C4⋊C4—1661no—
348M32—831no—
349M32—831no—
350M32—1661no—
351M32—1661no—
352M32C22×D8311yescharacteristic
353M32C22×D8311yescharacteristic
354C2×C16—1661no—
355C2×C16—1661no—
356#108—831no—
357#108—1661no—
358#108C22×D8311yescharacteristic
359#107—1661no—
360#107—1661no—
361#107—1661no—
362#106—21no—
363#106—831no—
364#106—831no—
365#106—1661no—
366#106—1661no—
367#106C22×D8311yescharacteristic
368#106C22×D8311yescharacteristic
369#104—21no—
370#104—1661no—
371#104—1661no—
372#104—1661no—
373#97—831no—
374#97—831no—
375C4×C8—831no—
376C4×C8C22×D8311yescharacteristic
377#41—21no—
378#41—3321no—
379#41—3321no—
380#41—1661no—
381#41—1661no—
382#41—1661no—
383C2×Q8—41no—
384C2×Q8—3321no—
385C2×Q8—1661no—
386C2×D4—3321no—
387C2×D4—1661no—
388C22×C4—3321no—
389C22×C4—3321no—
390C22×C4—831no—
391Q16—831no—
392Q16—1661no—
393Q16Q8×D8311yescharacteristic
394SD16—21no—
395SD16—1661no—
396SD16—1661no—
397SD16—1661no—
398D8—831no—
399D8—1661no—
400D8Q8×D8311yescharacteristic
401M16—21no—
402M16—41no—
403M16—3321no—
404M16—3321no—
405M16—3321no—
406M16—3321no—
407M16—831no—
408M16—831no—
409M16—831no—
410M16—831no—
411M16—1661no—
412C2×C8—41no—
413C2×C8—3321no—
414C2×C8—3321no—
415C2×C8—3321no—
416C2×C8—3321no—
417C2×C8—831no—
418C2×C8—831no—
419C2×C8—831no—
420C2×C8—831no—
421C2×C8—831no—
422C2×C8—831no—
423C2×C8D4×D8311yescharacteristic
424C2×C8C23×D8311yescharacteristic
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—3321no—
429C4⋊C4—3321no—
430C4⋊C4—3321no—
431C4⋊C4—3321no—
432C4⋊C4—3321no—
433C4⋊C4—3321no—
434C4⋊C4—1661no—
435C4⋊C4—1661no—
436C4⋊C4—1661no—
437C22⋊C4—3321no—
438C22⋊C4—1661no—
439C42—3321no—
440C42—831no—
441C42D4×D8311yescharacteristic
442C16—21no—
443C16—21no—
444C16—1661no—
445C16—1661no—
446C23—3321no—
447C23—831no—
448Q8—21no—
449Q8—161no—
450Q8—13281no—
451Q8—3321no—
452Q8—1661no—
453D4—21no—
454D4—41no—
455D4—3321no—
456D4—3321no—
457D4—3321no—
458D4—3321no—
459D4—1661no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—13281no—
465C2×C4—3321no—
466C2×C4—3321no—
467C2×C4—3321no—
468C2×C4—3321no—
469C2×C4—3321no—
470C2×C4—3321no—
471C2×C4—831no—
472C2×C4—831no—
473C2×C4—831no—
474C2×C4—831no—
475C2×C4—831no—
476C2×C4—1661no—
477C2×C4C2×D4×D8311yescharacteristic
478C8—41no—
479C8—3321no—
480C8—831no—
481C8—831no—
482C8—831no—
483C8—831no—
484C8SD16×D8311yescharacteristic
485C8SD16×D8311yescharacteristic
486C8C2×Q8×D8311yescharacteristic
487C8#41×D8311yescharacteristic
488C22—41no—
489C22—13281no—
490C22—3321no—
491C22—3321no—
492C22—831no—
493C22—831no—
494C22AGL(1,ℤ/8)×D8311yescharacteristic
495C4—41no—
496C4—41no—
497C4—81no—
498C4—81no—
499C4—3321no—
500C4—3321no—
501C4—831no—
502C4—831no—
503C4—6641no—
504C4—6641no—
505C4C22⋊Q8×D8311yescharacteristic
506C4C2×SD16×D8311yescharacteristic
507C2—21no—
508C2—81no—
509C2—6641no—
510C2—1661no—
511C2—1661no—
512C2#476×D8311yescharacteristic
subgroupG/H
centerC2#476×D83
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#2230C2
socleC2×C83C2×#476
prime radical#135×D83C22
hypercenter#2230D83
hypocenterC83#33638
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2230

Complements

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

NH ≅ G/Ntype
1. #24917988C2semidirect
3. #24909640C2semidirect
4. #24909639C2semidirect
5. #24909638C2semidirect
6. #24909637C2semidirect
7. #24909633C2semidirect
8. #24909632C2semidirect
9. #24909631C2semidirect
10. #24909234C2semidirect
11. #365×D83C2semidirect
12. #24909227C2semidirect
15. C83×#2230C2semidirect
23. #19128103C22semidirect
24. #19128101C22semidirect
25. #19128098C22semidirect
37. #19127525C22semidirect
38. #19127525C22semidirect
39. #19127524C22semidirect
40. #19127524C22semidirect
41. #19127523C22semidirect
42. #19127523C22semidirect
43. #19127522C22semidirect
44. #19127521C22semidirect
45. #19127521C22semidirect
52. #498×C83C22semidirect
54. #365×C83C22semidirect
55. #365×C83C22semidirect
56. #364×C83C22semidirect
60. #14581815Q8semidirect
64. #14581811Q8semidirect
65. Q16×D83Q8semidirect
67. D8×D83Q8semidirect
266. #2230D83semidirect
272. C83#33638semidirect
283. #498C2×D83semidirect
302. #365C2×D83semidirect
303. #365C2×D83semidirect
307. #364C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249179882
#249163752
#249096402
#249096392
#249096382
#249096372
#249096332
#249096322
#249096312
#249092342
#365×D832
#249092272
#249091972
#360×D832
C83×#22302
#3363883