Finite Group Databasebeta

← #35557745

Subgroups — #35557745

Order 21248.

696 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128138C211yesmaximal, characteristic
2#24916338C211yesmaximal, characteristic, complemented
3#24916310C211yesmaximal, characteristic, complemented
4#24909340C211yesmaximal, characteristic, complemented
5#24909336C211yesmaximal, characteristic, complemented
6#24909330C211yesmaximal, characteristic, complemented
7SD32×Dic83C211yesmaximal, characteristic, complemented
8C4×#14581735C211yesmaximal, characteristic, complemented
9#24909309C211yesmaximal, characteristic, complemented
10C4×#14581733C211yesmaximal, characteristic, complemented
11#24909248C211yesmaximal, characteristic, complemented
12#24909169C211yesmaximal, characteristic, complemented
13#357×D83C211yesmaximal, characteristic, complemented
14#24909061C211yesmaximal, characteristic, complemented
15C83×#2175C211yesmaximal, characteristic, complemented
16C2×#14581815C2211yescharacteristic
17C2×D8×D83C2211yescharacteristic
18#19128183—21no—
19#19128173—21no—
20#19128138C418yescomplemented
21C2×#14581735C2211yescharacteristic
22C2×D1328C2211yescharacteristic
23C2×#14581733C2211yescharacteristic
24C2×#14581731C2211yescharacteristic
25C2×C16×D83C2211yescharacteristic
26#19127811—21no—
27#19127810—21no—
28#19127804—21no—
29#19127799—21no—
30Q16×Dic83C2211yescharacteristic, complemented
31C4×D664C2211yescharacteristic
32C4×D664C2211yescharacteristic, complemented
33D8×Dic83C2211yescharacteristic
34#19127783C2211yescharacteristic
35#19127783C2211yescharacteristic, complemented
36#19127738C2211yescharacteristic
37#107×D83C2211yescharacteristic
38#19127704—21no—
39#19127701—21no—
40#19127655C2211yescharacteristic
41#19127529C2211yescharacteristic, complemented
42#19127528C2211yescharacteristic, complemented
43#19127526C2211yescharacteristic, complemented
44#19127505C2211yescharacteristic, complemented
45#19127503C2211yescharacteristic
46#19127503C2211yescharacteristic, complemented
47#19127502C2211yescharacteristic
48#19127501C2211yescharacteristic
49#19127501C2211yescharacteristic, complemented
50#19127476C2211yescharacteristic, complemented
51C4×C83⋊C16C2211yescharacteristic, complemented
52C16×Dic83C2211yescharacteristic, complemented
53C2×SD32×C83C2211yescharacteristic
54C4×Q16×C83C2211yescharacteristic, complemented
55C4×D8×C83C2211yescharacteristic
56#366×C83C2211yescharacteristic, complemented
57#358×C83C2211yescharacteristic, complemented
58#357×C83C2211yescharacteristic
59#346×C83C2211yescharacteristic, complemented
60C2×#13050591—21no—
61C2×D4×D83—21no—
62#14581815—22no—
63#14581815C2×C414yescomplemented
64C2×#13050577—21no—
65C2×D664C2311yescharacteristic
66C2×D664C2311yescharacteristic
67C2×#13050575—21no—
68D8×D83—22no—
69D8×D83C2×C414yescomplemented
70C2×#13050573D411yescharacteristic
71C2×C8×D83C2311yescharacteristic
72Q8×Dic83—21no—
73C4×D332—21no—
74C4×D332—21no—
75C4×D332—41no—
76D4×Dic83—21no—
77#14581754—21no—
78#14581754—21no—
79#14581754—41no—
80C4×C22⋊D83—41no—
81(C22⋊D83)⋊C4—41no—
82#14581750—21no—
83#14581749—41no—
84#14581749D411yescharacteristic
85C4⋊C4×D83—21no—
86C4⋊C4×D83—41no—
87C2×(C2×D83)⋊C4—41no—
88C22⋊C4×D83—41no—
89#14581735C2×C414yescomplemented
90D1328C2×C414yescomplemented
91#14581733C2×C414yescomplemented
92#14581731C2×C414yescomplemented
93C2×C83⋊C16C2311yescharacteristic
94C2×C83⋊C16C2311yescharacteristic
95C16×D83—22no—
96#14581726C2311yescharacteristic
97#14581726C2311yescharacteristic
98#14581725—21no—
99#14581723—21no—
100#14581716—21no—
101#14581714—21no—
102#14581714—21no—
103#14581713—21no—
104#14581712—21no—
105#14581712—21no—
106#14581703D411yescharacteristic
107#14581702D411yescharacteristic
108C8×Dic83C2311yescharacteristic
109C2×Q16×C83C2311yescharacteristic
110C2×D8×C83C2311yescharacteristic
111C4×Q8×C83—21no—
112C4×D4×C83—21no—
113SD32×C83C2×C414yescomplemented
114C2×C16×C83C2311yescharacteristic
115C2×C16×C83C2311yescharacteristic
116#107×C83C2311yescharacteristic
117#107×C83C2311yescharacteristic
118#103×C83—21no—
119#102×C83—21no—
120C4×C8×C83C2311yescharacteristic
121C23×D83—41no—
122#13050591—24no—
123#13050591—42no—
124C2×D332—21no—
125C2×D332—21no—
126C2×D332—41no—
127C2×C22⋊D83—41no—
128D4×D83—24no—
129D4×D83—42no—
130C22×Dic83—41no—
131C2×C4×D83—41no—
132C2×C4×D83—41no—
133C2×C4×D83—41no—
134C2×C4×D83—41no—
135C2×C4×D83—41no—
136C2×C4×D83—41no—
137C2×C4×D83C2×D411yescharacteristic
138#13050577—24no—
139D664—21no—
140D664—21no—
141D664C22×C412yes—
142D664C22×C412yes—
143#13050575—24no—
144#13050573—22no—
145C2×C83⋊C8C2×D411yescharacteristic
146C2×C83⋊C8C2×D411yescharacteristic
147C8×D83C22×C412yes—
148C8×D83#4112yes—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—41no—
152Dic83⋊C4—41no—
153Dic83⋊C4D811yescharacteristic
154Dic83⋊C4D811yescharacteristic
155C22⋊Dic83—41no—
156(C2×D83)⋊C4—41no—
157(C2×D83)⋊C4—41no—
158(C2×D83)⋊C4—41no—
159(C2×D83)⋊C4—41no—
160(C2×D83)⋊C4—41no—
161(C2×D83)⋊C4D811yescharacteristic
162(C2×D83)⋊C4D811yescharacteristic
163C4×Dic83—41no—
164C4×Dic83—41no—
165C4×Dic83C2×D411yescharacteristic
166C83⋊C16—21no—
167C83⋊C16C22×C412yes—
168C2×Q8×C83—21no—
169C2×D4×C83—21no—
170C22×C4×C83—41no—
171Q16×C83—21no—
172Q16×C83C22×C412yes—
173D8×C83—21no—
174D8×C83C22×C412yes—
175C2×C8×C83C2×D411yescharacteristic
176C2×C8×C83C2411yescharacteristic
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—41no—
180C22⋊C4×C83—41no—
181C42×C83—41no—
182C42×C83C2×D411yescharacteristic
183C16×C83—21no—
184C16×C83C22×C412yes—
185C23×C83—41no—
186C22×D83—41no—
187C22×D83—41no—
188C22×D83—44no—
189C22×D83—161no—
190C22×D83—161no—
191C22×D83C2×D811yescharacteristic
192Q8×C83—22no—
193Q8×C83—41no—
194D4×C83—22no—
195D4×C83—41no—
196C2×C4×C83—41no—
197C2×C4×C83—41no—
198C2×C4×C83—41no—
199C2×C4×C83—161no—
200C2×C4×C83C2×D811yescharacteristic
201C2×C4×C83C2×D811yescharacteristic
202C2×C4×C83C22×D411yescharacteristic
203C22⋊D83—44no—
204C2×Dic83—41no—
205C2×Dic83—41no—
206C2×Dic83—161no—
207C2×Dic83C2×D811yescharacteristic
208C2×Dic83C2×D811yescharacteristic
209C2×Dic83C2×D811yescharacteristic
210D332—22no—
211D332—22no—
212D332—44no—
213D332—41no—
214D332—41no—
215C4×D83—44no—
216C4×D83—161no—
217C4×D83—161no—
218C4×D83—161no—
219C4×D83—161no—
220C4×D83—161no—
221C4×D83C4×D412yes—
222C4×D83#13512yes—
223C8×C83—21no—
224C8×C83C23×C411yescharacteristic
225C8×C83C2×#4111yescharacteristic
226C83⋊C8—21no—
227C83⋊C8C4×D412yes—
228C22×C83—42no—
229C22×C83—81no—
230C22×C83—81no—
231C22×C83C22×D811yescharacteristic
232C2×D83—21no—
233C2×D83—21no—
234C2×D83—42no—
235C2×D83—42no—
236C2×D83—81no—
237C2×D83—81no—
238C2×D83—81no—
239C2×D83—81no—
240C2×D83—164no—
241C2×D83C4×D812yes—
242C4×C83—21no—
243C4×C83—21no—
244C4×C83—42no—
245C4×C83—81no—
246C4×C83—81no—
247C4×C83C2×C4×D411yescharacteristic
248C4×C83C2×#13511yescharacteristic
249Dic83—21no—
250Dic83—21no—
251Dic83—81no—
252Dic83—81no—
253Dic83C4×D812yes—
254#33454—831nomaximal, Sylow
255C2×C83—82no—
256C2×C83C2×C4×D811yescharacteristic
257C2×C83#341511yescharacteristic
258C2×C83#341611yescharacteristic
259D83—22no—
260D83—82no—
261D83—82no—
262C2×#509—831no—
263#2951—831no—
264#2948—831no—
265#2177—831no—
266#2177—831no—
267#2175—831no—
268#2175D8311yescharacteristic, complemented
269C4×SD32—831no—
270C4×SD32—831no—
271C4×D16—831no—
272C4×D16—831no—
273#2159—831no—
274#2138—831no—
275C2×#357—831no—
276#2108—831no—
277C83#3345411yescharacteristic, complemented, Sylow
278C2×#135—831no—
279C22×D8—831no—
280#520—1661no—
281#518—1661no—
282#509—838no—
283C2×SD32—831no—
284C2×SD32C2×D8311yescharacteristic
285C2×D16—831no—
286C2×D16—831no—
287C2×M32—831no—
288C22×C16—831no—
289#442—1661no—
290#442—1661no—
291#440—1661no—
292#440—1661no—
293C4×Q16—831no—
294C4×Q16C2×D8311yescharacteristic, complemented
295C4×D8—831no—
296C4×D8—831no—
297C4×D8—831no—
298C4×D8—831no—
299C4×D8—831no—
300C4×D8C2×D8311yescharacteristic
301#427—831no—
302C2×#107—831no—
303#418—1661no—
304#417—1661no—
305#405—831no—
306#367—831no—
307#367—831no—
308#366—831no—
309#366C2×D8311yescharacteristic, complemented
310#358—831no—
311#358C2×D8311yescharacteristic, complemented
312#357—831no—
313#357—831no—
314#357—831no—
315#357—831no—
316#357—831no—
317#357C2×D8311yescharacteristic
318#346—831no—
319#346C2×D8311yescharacteristic, complemented
320C4×C16—831no—
321C4×C16—831no—
322C2×#41—1661no—
323C22×D4—1661no—
324#135—834no—
325#135—1662no—
326C2×Q16C22×D8311yescharacteristic
327C2×SD16—1661no—
328C2×D8—834no—
329C2×D8—831no—
330C2×D8—831no—
331C2×D8—1662no—
332C2×D8—1661no—
333C2×D8C22×D8311yescharacteristic
334C2×M16—831no—
335C22×C8—831no—
336C4×Q8—21no—
337C4×Q8—1661no—
338C4×D4—21no—
339C4×D4—3321no—
340C4×D4—3321no—
341C4×D4—3321no—
342C4×D4—3321no—
343C4×D4—1661no—
344C4×D4—1661no—
345C4×D4—1661no—
346C4×D4—1661no—
347C4×D4—1661no—
348#117—3321no—
349#117—831no—
350#117—1661no—
351C2×C4⋊C4—3321no—
352C2×C4⋊C4—1661no—
353C2×C22⋊C4—3321no—
354C2×C22⋊C4—3321no—
355SD32—834no—
356SD32C4×D8314yescomplemented
357D16—834no—
358D16—834no—
359M32—834no—
360C2×C16—831no—
361C2×C16—831no—
362C2×C16—1662no—
363C2×C16C22×D8311yescharacteristic
364C2×C16C22×D8311yescharacteristic
365#107—831no—
366#107—831no—
367#107—1661no—
368#107C22×D8311yescharacteristic
369#107C22×D8311yescharacteristic
370#106—1661no—
371#103—21no—
372#103—1661no—
373#102—21no—
374#102—1661no—
375#102—1661no—
376#102—1661no—
377#102—1661no—
378#102—1661no—
379#97—831no—
380#97—831no—
381C4×C8—831no—
382C4×C8C22×D8311yescharacteristic
383C24—3321no—
384#41—3322no—
385#41—1664no—
386C2×Q8—21no—
387C2×D4—21no—
388C2×D4—3321no—
389C2×D4—3322no—
390C2×D4—3321no—
391C2×D4—1664no—
392C2×D4—1661no—
393C2×D4—1661no—
394C22×C4—41no—
395C22×C4—3321no—
396C22×C4—3321no—
397C22×C4—3321no—
398C22×C4—3321no—
399C22×C4—3321no—
400C22×C4—3321no—
401C22×C4—3321no—
402C22×C4—831no—
403Q16—21no—
404Q16C2×C4×D8312yes—
405SD16—1664no—
406D8—21no—
407D8—832no—
408D8—832no—
409D8—1664no—
410D8—1661no—
411D8—1661no—
412D8C2×C4×D8312yes—
413M16—1662no—
414C2×C8—831no—
415C2×C8—832no—
416C2×C8—832no—
417C2×C8—831no—
418C2×C8D4×D8311yescharacteristic
419C2×C8C23×D8311yescharacteristic
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—3321no—
424C4⋊C4—3321no—
425C4⋊C4—831no—
426C4⋊C4—831no—
427C4⋊C4—1661no—
428C4⋊C4—1661no—
429C22⋊C4—41no—
430C22⋊C4—3321no—
431C22⋊C4—3321no—
432C22⋊C4—3321no—
433C22⋊C4—3321no—
434C22⋊C4—3321no—
435C22⋊C4—3321no—
436C22⋊C4—831no—
437C22⋊C4—831no—
438C42—41no—
439C42—3321no—
440C42—3321no—
441C42—831no—
442C42D4×D8311yescharacteristic
443C16—21no—
444C16—832no—
445C16—1661no—
446C16C2×C4×D8312yes—
447C23—41no—
448C23—13281no—
449C23—13281no—
450C23—3324no—
451C23—3321no—
452C23—3321no—
453C23—831no—
454Q8—22no—
455Q8—41no—
456D4—22no—
457D4—41no—
458D4—3324no—
459D4—3324no—
460D4—3321no—
461D4—3321no—
462D4—1662no—
463D4—1662no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—161no—
468C2×C4—13281no—
469C2×C4—13281no—
470C2×C4—13281no—
471C2×C4—13281no—
472C2×C4—13281no—
473C2×C4—13281no—
474C2×C4—3324no—
475C2×C4—3321no—
476C2×C4—3321no—
477C2×C4—831no—
478C2×C4—832no—
479C2×C4—832no—
480C2×C4—831no—
481C2×C4—831no—
482C2×C4D8×D8311yescharacteristic
483C2×C4D8×D8311yescharacteristic
484C2×C4C2×D4×D8311yescharacteristic
485C8—21no—
486C8—832no—
487C8—1661no—
488C8C22×C4×D8311yescharacteristic
489C8#41×D8311yescharacteristic
490C22—42no—
491C22—81no—
492C22—81no—
493C22—13284no—
494C22—3322no—
495C22—3322no—
496C22—832no—
497C22—6641no—
498C22—6641no—
499C22—6641no—
500C22—6641no—
501C22—1661no—
502C22—1661no—
503C22C2×D8×D8311yescharacteristic
504C4—21no—
505C4—21no—
506C4—42no—
507C4—81no—
508C4—81no—
509C4—832no—
510C4—6641no—
511C4—6641no—
512C4—1661no—
513C4—1661no—
514C4C4×D4×D8311yescharacteristic
515C4#135×D8311yescharacteristic
516C2—82no—
517C2—6642no—
518C2—6642no—
519C2—1662no—
520C2C4×D8×D8311yescharacteristic
521C2#2492060311yescharacteristic
522C2#2492060811yescharacteristic
subgroupG/H
centerC22C2×D8×D83
commutatorC8×C83C23×C4
FrattiniC2×C8C23×D83
FittingC83×#2175C2
socleC22×C83C22×D8
prime radicalC2×#19128138C2
cosocleC2×C8×C83C24
hypercenter#2175D83
hypocenterC83#33454
LCSG ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. #24916338C2semidirect
3. #24916310C2semidirect
4. #24909340C2semidirect
5. #24909336C2semidirect
6. #24909330C2semidirect
7. SD32×Dic83C2semidirect
8. C4×#14581735C2semidirect
9. #24909309C2semidirect
10. C4×#14581733C2semidirect
11. #24909248C2semidirect
12. #24909169C2semidirect
13. #357×D83C2semidirect
14. #24909061C2semidirect
15. C83×#2175C2semidirect
20. #19128138C4semidirect
30. Q16×Dic83C22semidirect
32. C4×D664C22semidirect
35. #19127783C22semidirect
41. #19127529C22semidirect
42. #19127528C22semidirect
43. #19127526C22semidirect
44. #19127505C22semidirect
46. #19127503C22semidirect
49. #19127501C22semidirect
50. #19127476C22semidirect
51. C4×C83⋊C16C22semidirect
52. C16×Dic83C22semidirect
54. C4×Q16×C83C22semidirect
56. #366×C83C22semidirect
57. #358×C83C22semidirect
59. #346×C83C22semidirect
63. #14581815C2×C4semidirect
69. D8×D83C2×C4semidirect
89. #14581735C2×C4semidirect
90. D1328C2×C4semidirect
91. #14581733C2×C4semidirect
92. #14581731C2×C4semidirect
113. SD32×C83C2×C4semidirect
268. #2175D83semidirect
277. C83#33454semidirect
294. C4×Q16C2×D83semidirect
309. #366C2×D83semidirect
311. #358C2×D83semidirect
319. #346C2×D83semidirect
356. SD32C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281382
#249163382
#249163102
#249093402
#249093362
#249093302
SD32×Dic832
C4×#145817352
#249093092
C4×#145817332
#249092482
#249091692
#357×D832
#249090612
C83×#21752
#3345483