Finite Group Databasebeta

← #35561251

Subgroups — #35561251

Order 21248.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24919832C211yesmaximal, characteristic, complemented
2#24919150C211yesmaximal, characteristic
3#24909695C211yesmaximal, characteristic, complemented
4#24909661C211yesmaximal, characteristic, complemented
5#24909655C211yesmaximal, characteristic, complemented
6#24909621C211yesmaximal, characteristic, complemented
7#24909559C211yesmaximal, characteristic, complemented
8#24909554C211yesmaximal, characteristic, complemented
9#24909526C211yesmaximal, characteristic, complemented
10#24909405C211yesmaximal, characteristic, complemented
11#24909394C211yesmaximal, characteristic, complemented
12C4×#14581737C211yesmaximal, characteristic, complemented
13#24909171C211yesmaximal, characteristic, complemented
14C2×#19127503C211yesmaximal, characteristic, complemented
15D4×C16×C83C211yesmaximal, characteristic, complemented
16C2×#14581814C2211yescharacteristic
17C22×D664C2211yescharacteristic
18#19128391—21no—
19#19128347—21no—
20C2×Dic664C2211yescharacteristic, complemented
21C2×#14581737C2211yescharacteristic
22C2×#14581737C2211yescharacteristic
23#19128103C2211yescharacteristic, complemented
24#19128085C2211yescharacteristic
25#19128043—21no—
26#19128032—21no—
27#19128009—21no—
28#19127942C2211yescharacteristic
29#19127942C2211yescharacteristic
30#19127942C2211yescharacteristic, complemented
31#19127937C2211yescharacteristic
32#19127917—21no—
33#19127854—21no—
34#19127848—21no—
35C4×Dic332C2211yescharacteristic, complemented
36C4×D664C2211yescharacteristic
37#19127739C2211yescharacteristic, complemented
38C2×#14581726C2211yescharacteristic
39#19127529C2211yescharacteristic, complemented
40#19127529C2211yescharacteristic, complemented
41#19127527C2211yescharacteristic, complemented
42#19127506C2211yescharacteristic, complemented
43#19127506C2211yescharacteristic, complemented
44#19127506C2211yescharacteristic, complemented
45#19127506C2211yescharacteristic, complemented
46#19127503C2211yescharacteristic
47#19127503C2211yescharacteristic
48#19127503C2211yescharacteristic
49#19127503C2211yescharacteristic
50#19127503C2211yescharacteristic
51#19127503C2211yescharacteristic
52C22×C16×C83C2211yescharacteristic
53C22×C16×C83C2211yescharacteristic, complemented
54C8×D4×C83C2211yescharacteristic
55C4⋊C16×C83C2211yescharacteristic, complemented
56C22⋊C16×C83C2211yescharacteristic
57C22⋊C16×C83C2211yescharacteristic, complemented
58C4×C16×C83C2211yescharacteristic, complemented
59C2×#13050590—21no—
60C22×D332—21no—
61#14581814—22no—
62#14581814—41no—
63C2×Dic332C2311yescharacteristic
64C2×Dic332C2311yescharacteristic
65C2×#13050579—21no—
66C2×#13050579—21no—
67C2×D664—22no—
68C2×D664—21no—
69C2×D664—41no—
70C2×D664C2311yescharacteristic
71C2×D664C2311yescharacteristic
72#14581785—41no—
73#14581784—21no—
74#14581782—41no—
75#14581780—21no—
76#14581777—41no—
77#14581773—41no—
78#14581771—21no—
79#14581771—21no—
80#14581771—21no—
81#14581766—21no—
82#14581765—41no—
83#14581760—41no—
84C4×Dic166—21no—
85C4×D332—21no—
86#14581751—21no—
87#14581748—41no—
88C2×C4⋊Dic83—21no—
89C22⋊C4×D83—41no—
90Dic664—41no—
91#14581737—22no—
92#14581737—41no—
93#14581726—21no—
94#14581726C2311yescharacteristic
95#14581726C2311yescharacteristic
96#14581726C2311yescharacteristic
97#14581726C2311yescharacteristic
98#14581724—21no—
99#14581724—21no—
100#14581717—21no—
101#14581717—21no—
102#14581717—21no—
103#14581717—21no—
104#14581714—21no—
105#14581714—21no—
106C22×C8×C83C2311yescharacteristic
107C22×C8×C83C2311yescharacteristic
108C4×D4×C83D411yescharacteristic
109C2×C16×C83—41no—
110C2×C16×C83—41no—
111C2×C16×C83C2311yescharacteristic
112C2×C16×C83C2311yescharacteristic
113C2×C16×C83C2311yescharacteristic
114C2×C16×C83C2311yescharacteristic
115C4⋊C8×C83D411yescharacteristic
116C22⋊C8×C83D411yescharacteristic
117C22⋊C8×C83D411yescharacteristic
118C4×C8×C83C2311yescharacteristic
119C23×D83—41no—
120#13050590—42no—
121#13050590—81no—
122C2×Dic166—21no—
123C2×Dic166—21no—
124C2×D332—21no—
125C2×D332—21no—
126C2×D332—42no—
127C2×D332—41no—
128C2×D332—81no—
129C2×C22⋊D83—41no—
130C2×C22⋊D83—41no—
131C22×Dic83—41no—
132C2×C4×D83—41no—
133C2×C4×D83—41no—
134Dic332—21no—
135Dic332—21no—
136Dic332—21no—
137Dic332—21no—
138#13050579—42no—
139#13050579—81no—
140D664—21no—
141D664—21no—
142D664—21no—
143D664—42no—
144D664#4112yes—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—41no—
150Dic83⋊C4—41no—
151Dic83⋊C4—41no—
152Dic83⋊C4—41no—
153Dic83⋊C4—41no—
154Dic83⋊C4—41no—
155Dic83⋊C4—41no—
156C22⋊Dic83—41no—
157C22⋊Dic83—41no—
158C22⋊Dic83—41no—
159C22⋊Dic83—41no—
160(C2×D83)⋊C4—41no—
161(C2×D83)⋊C4—41no—
162(C2×D83)⋊C4—41no—
163(C2×D83)⋊C4—41no—
164C4×Dic83—41no—
165C4×Dic83—41no—
166C2×D4×C83D811yescharacteristic
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C2×C8×C83—21no—
170C2×C8×C83—21no—
171C2×C8×C83—21no—
172C2×C8×C83—21no—
173C2×C8×C83C2×D411yescharacteristic
174C2×C8×C83C2×D411yescharacteristic
175C2×C8×C83C2×D411yescharacteristic
176C2×C8×C83C2411yescharacteristic
177C4⋊C4×C83D811yescharacteristic
178C22⋊C4×C83D811yescharacteristic
179C22⋊C4×C83D811yescharacteristic
180C42×C83C2×D411yescharacteristic
181C16×C83—21no—
182C16×C83—21no—
183C16×C83—21no—
184C16×C83—21no—
185C23×C83C2×D811yescharacteristic
186C23×C83C2×D811yescharacteristic
187C22×D83—41no—
188C22×D83—41no—
189C22×D83—82no—
190C22×D83—82no—
191C22×D83—161no—
192D4×C83—22no—
193C2×C4×C83—21no—
194C2×C4×C83—21no—
195C2×C4×C83—21no—
196C2×C4×C83—21no—
197C2×C4×C83C2×D811yescharacteristic
198C2×C4×C83C2×D811yescharacteristic
199C2×C4×C83C2×D811yescharacteristic
200C2×C4×C83C2×D811yescharacteristic
201C2×C4×C83C22×D411yescharacteristic
202C22⋊D83—82no—
203C22⋊D83—161no—
204C2×Dic83—41no—
205C2×Dic83—41no—
206C2×Dic83—41no—
207C2×Dic83—41no—
208C2×Dic83—41no—
209C2×Dic83—41no—
210C2×Dic83—82no—
211D332—22no—
212D332—41no—
213D332—41no—
214D332—41no—
215D332—82no—
216C4×D83—82no—
217C4×D83—161no—
218Dic166—41no—
219Dic166—41no—
220Dic166—41no—
221Dic166—41no—
222C8×C83—21no—
223C8×C83—21no—
224C8×C83—21no—
225C8×C83C2×#4111yescharacteristic
226C8×C83H(5,𝔽2)11yescharacteristic
227C22×C83—21no—
228C22×C83—21no—
229C22×C83—21no—
230C22×C83—21no—
231C22×C83—21no—
232C22×C83#50812yes—
233C22×C83C22×D811yescharacteristic
234C2×D83—42no—
235C2×D83—81no—
236C2×D83—81no—
237C2×D83—81no—
238C2×D83—81no—
239C2×D83—81no—
240C2×D83—164no—
241C4×C83—21no—
242C4×C83—21no—
243C4×C83—21no—
244C4×C83—21no—
245C4×C83#54611yescharacteristic
246C4×C83#57711yescharacteristic
247Dic83—81no—
248Dic83—81no—
249Dic83—81no—
250Dic83—81no—
251Dic83—81no—
252Dic83—81no—
253#33863—831nomaximal, Sylow
254C2×C83—22no—
255C2×C83—41no—
256C2×C83#329411yescharacteristic
257C2×C83C2×#50811yescharacteristic
258C2×C83#341611yescharacteristic
259D83—82no—
260D83—161no—
261#3336—831no—
262#3280—831no—
263#2242—831no—
264#2235—831no—
265#2234—831no—
266#2228—831no—
267#2217—831no—
268#2216—831no—
269#2211—831no—
270#2189—831no—
271#2187—831no—
272C4×SD32—831no—
273D4×C16D8311yescharacteristic, complemented
274#2138—831no—
275C2×#357—831no—
276C83#3386311yescharacteristic, complemented, Sylow
277C2×#135—831no—
278C22×D8—831no—
279#552—1661no—
280#546—1661no—
281C2×Q32—831no—
282C2×SD32—831no—
283C2×SD32—831no—
284C22×C16C2×D8311yescharacteristic
285C22×C16C2×D8311yescharacteristic, complemented
286#499—831no—
287#495—831no—
288#485—1661no—
289#483—1661no—
290#479—1661no—
291#467—831no—
292#467—831no—
293#467—831no—
294#466—831no—
295#463—1661no—
296#450—1661no—
297#449—1661no—
298C4×Q16—831no—
299C4×D8—831no—
300C8×D4C2×D8311yescharacteristic
301#427—831no—
302C2×#107—831no—
303#367—831no—
304#367—831no—
305#366—831no—
306C4⋊C16C2×D8311yescharacteristic, complemented
307#358—831no—
308#358—831no—
309#358—831no—
310#358—831no—
311#357—831no—
312#357—831no—
313#357—831no—
314#357—831no—
315#357—831no—
316#357—831no—
317C22⋊C16C2×D8311yescharacteristic
318C22⋊C16C2×D8311yescharacteristic, complemented
319C4×C16C2×D8311yescharacteristic, complemented
320C2×#41—1661no—
321C22×D4—1661no—
322#135—3321no—
323#135—1662no—
324C2×Q16—831no—
325C2×Q16—831no—
326C2×SD16—1661no—
327C2×SD16—1661no—
328C2×D8—3321no—
329C2×D8—831no—
330C2×D8—831no—
331C2×D8—1662no—
332C2×D8—1661no—
333C22×C8C22×D8311yescharacteristic
334C22×C8C22×D8311yescharacteristic
335#126—3321no—
336#125—3321no—
337#125—1661no—
338#124—1661no—
339#123—3321no—
340#123—3321no—
341C22⋊Q8—1661no—
342C22⋊Q8—1661no—
343C22⋊Q8—1661no—
344#121—3321no—
345#121—1661no—
346#120—3321no—
347C4×Q8—1661no—
348C4×D4—1661no—
349C4×D4D33211yescharacteristic
350#117—3321no—
351#117—1661no—
352C2×C4⋊C4—1661no—
353C2×C22⋊C4—3321no—
354Q32—3321no—
355SD32—3321no—
356SD32—1662no—
357C2×C16—41no—
358C2×C16—41no—
359C2×C16C22×D8311yescharacteristic
360C2×C16C22×D8311yescharacteristic
361C2×C16C22×D8311yescharacteristic
362C2×C16C22×D8311yescharacteristic
363#107—831no—
364#107—831no—
365#107—831no—
366#107—831no—
367#107—1661no—
368#106—1661no—
369#106—1661no—
370C4⋊C8D33211yescharacteristic
371#103—1661no—
372#103—1661no—
373#103—1661no—
374#103—1661no—
375#102—1661no—
376#102—1661no—
377C22⋊C8D33211yescharacteristic
378C22⋊C8D33211yescharacteristic
379C4×C8C22×D8311yescharacteristic
380C24—3321no—
381#41—3322no—
382#41—6641no—
383C2×Q8—1661no—
384C2×Q8—1661no—
385C2×D4—3322no—
386C2×D4—3321no—
387C2×D4—3321no—
388C2×D4—3321no—
389C2×D4—6641no—
390C2×D4—1661no—
391C2×D4—1661no—
392C2×D4D66411yescharacteristic
393C22×C4—3321no—
394C22×C4—3321no—
395C22×C4—3321no—
396C22×C4C2×D33211yescharacteristic
397C22×C4C2×D33211yescharacteristic
398Q16—1661no—
399Q16—1661no—
400Q16—1661no—
401Q16—1661no—
402SD16—3322no—
403SD16—6641no—
404D8—3322no—
405D8—832no—
406D8—1661no—
407D8—1661no—
408D8—1661no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×D33211yescharacteristic
414C2×C8C2×D33211yescharacteristic
415C2×C8C2×D33211yescharacteristic
416C2×C8C23×D8311yescharacteristic
417C4⋊C4—3321no—
418C4⋊C4—3321no—
419C4⋊C4—3321no—
420C4⋊C4—3321no—
421C4⋊C4—3321no—
422C4⋊C4—3321no—
423C4⋊C4—3321no—
424C4⋊C4—1661no—
425C4⋊C4—1661no—
426C4⋊C4—1661no—
427C4⋊C4—1661no—
428C4⋊C4D66411yescharacteristic
429C22⋊C4—3321no—
430C22⋊C4—3321no—
431C22⋊C4—3321no—
432C22⋊C4—3321no—
433C22⋊C4—3321no—
434C22⋊C4—3321no—
435C22⋊C4—3321no—
436C22⋊C4—3321no—
437C22⋊C4D66411yescharacteristic
438C22⋊C4D66411yescharacteristic
439C42—3321no—
440C42—3321no—
441C42C2×D33211yescharacteristic
442C16—21no—
443C16—21no—
444C16—21no—
445C16—21no—
446C23—13281no—
447C23—3321no—
448C23—3321no—
449C23—6642no—
450C23—6642no—
451C23C2×D66411yescharacteristic
452C23C2×D66411yescharacteristic
453Q8—3321no—
454Q8—3321no—
455Q8—3321no—
456Q8—3321no—
457D4—22no—
458D4—13281no—
459D4—3321no—
460D4—3321no—
461D4—3321no—
462D4—6642no—
463D4—6642no—
464D4—1662no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—13281no—
470C2×C4—3321no—
471C2×C4—3321no—
472C2×C4—3321no—
473C2×C4—3321no—
474C2×C4—3321no—
475C2×C4—3321no—
476C2×C4—6642no—
477C2×C4—6642no—
478C2×C4C2×D66411yescharacteristic
479C2×C4C2×D66411yescharacteristic
480C2×C4C2×D66411yescharacteristic
481C2×C4C2×D66411yescharacteristic
482C2×C4C22×D33211yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D8311yescharacteristic
487C8#1458183711yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—13284no—
494C22—3322no—
495C22—6641no—
496C22—6641no—
497C22—6641no—
498C22—6641no—
499C22—6641no—
500C22#1912813312yes—
501C22C22×D66411yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—6641no—
507C4—6641no—
508C4—6641no—
509C4—6641no—
510C4—6641no—
511C4—6641no—
512C4#1912834611yescharacteristic
513C4#1912851311yescharacteristic
514C2—22no—
515C2—41no—
516C2—13281no—
517C2—6642no—
518C2#2491927711yescharacteristic
519C2C2×#1912813311yescharacteristic
520C2#2492061111yescharacteristic
subgroupG/H
centerC22C22×D664
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingD4×C16×C83C2
socleC22×C83C22×D8
prime radical#24919150C2
hypercenterD4×C16D83
hypocenterC83#33863
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #24919832C2semidirect
3. #24909695C2semidirect
4. #24909661C2semidirect
5. #24909655C2semidirect
6. #24909621C2semidirect
7. #24909559C2semidirect
8. #24909554C2semidirect
9. #24909526C2semidirect
10. #24909405C2semidirect
11. #24909394C2semidirect
12. C4×#14581737C2semidirect
13. #24909171C2semidirect
14. C2×#19127503C2semidirect
15. D4×C16×C83C2semidirect
20. C2×Dic664C22semidirect
23. #19128103C22semidirect
30. #19127942C22semidirect
35. C4×Dic332C22semidirect
37. #19127739C22semidirect
39. #19127529C22semidirect
40. #19127529C22semidirect
41. #19127527C22semidirect
42. #19127506C22semidirect
43. #19127506C22semidirect
44. #19127506C22semidirect
45. #19127506C22semidirect
53. C22×C16×C83C22semidirect
55. C4⋊C16×C83C22semidirect
57. C22⋊C16×C83C22semidirect
58. C4×C16×C83C22semidirect
273. D4×C16D83semidirect
276. C83#33863semidirect
285. C22×C16C2×D83semidirect
306. C4⋊C16C2×D83semidirect
318. C22⋊C16C2×D83semidirect
319. C4×C16C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249198322
#249191502
#249096952
#249096612
#249096552
#249096212
#249095592
#249095542
#249095262
#249094052
#249093942
C4×#145817372
#249091712
C2×#191275032
D4×C16×C832
#3386383