Finite Group Databasebeta

← #35559201

Subgroups — #35559201

Order 21248.

590 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#24917040C211yesmaximal, characteristic
2#427×D83C211yesmaximal, characteristic, complemented
3#24909670C211yesmaximal, characteristic, complemented
4#24909669C211yesmaximal, characteristic, complemented
5#24909664C211yesmaximal, characteristic, complemented
6#24909659C211yesmaximal, characteristic, complemented
7#24909653C211yesmaximal, characteristic, complemented
8#24909652C211yesmaximal, characteristic, complemented
9#24909647C211yesmaximal, characteristic, complemented
10#24909642C211yesmaximal, characteristic, complemented
11#24909247C211yesmaximal, characteristic, complemented
12#24909245C211yesmaximal, characteristic, complemented
13#24909170C211yesmaximal, characteristic, complemented
14C2×(C2×D83)⋊C16C211yesmaximal, characteristic, complemented
15C83×#2138C211yesmaximal, characteristic, complemented
16C22×C8×D83C2211yescharacteristic
17#19128208—21no—
18#117×D83—21no—
19C22×C83⋊C16C2211yescharacteristic
20#19127939C2211yescharacteristic
21#19127939C2211yescharacteristic
22#19127934C2211yescharacteristic
23#19127934C2211yescharacteristic
24#19127927—21no—
25#19127927—21no—
26#19127739C2211yescharacteristic
27#19127739C2211yescharacteristic
28#19127738C2211yescharacteristic
29#19127738C2211yescharacteristic
30#19127737—21no—
31#19127736—21no—
32#107×D83C2211yescharacteristic
33#107×D83C2211yescharacteristic
34#106×D83—21no—
35#19127697—21no—
36#19127529C2211yescharacteristic, complemented
37#19127529C2211yescharacteristic, complemented
38#19127528C2211yescharacteristic, complemented
39#19127528C2211yescharacteristic, complemented
40#19127527C2211yescharacteristic, complemented
41#19127527C2211yescharacteristic, complemented
42#19127526C2211yescharacteristic, complemented
43#19127526C2211yescharacteristic, complemented
44#19127505C2211yescharacteristic, complemented
45#19127505C2211yescharacteristic, complemented
46#19127502C2211yescharacteristic, complemented
47#19127502C2211yescharacteristic, complemented
48(C2×D83)⋊C16C2211yescharacteristic
49(C2×D83)⋊C16C2211yescharacteristic
50(C2×D83)⋊C16C2211yescharacteristic
51(C2×D83)⋊C16C2211yescharacteristic
52C2×#135×C83C2211yescharacteristic
53C22×C16×C83C2211yescharacteristic
54#427×C83C2211yescharacteristic
55#358×C83C2211yescharacteristic, complemented
56#358×C83C2211yescharacteristic, complemented
57#357×C83C2211yescharacteristic, complemented
58#357×C83C2211yescharacteristic, complemented
59C22×C4×D83D411yescharacteristic
60C22×C83⋊C8D411yescharacteristic
61C2×C8×D83—22no—
62C2×C8×D83—22no—
63C2×C8×D83D411yescharacteristic
64C2×C8×D83D411yescharacteristic
65C2×C8×D83C2311yescharacteristic
66C2×C8×D83C2311yescharacteristic
67#14581773—41no—
68#14581768—21no—
69#14581768—21no—
70#14581763—21no—
71#14581763—21no—
72#14581760—41no—
73C4×C22⋊D83—41no—
74C4×C22⋊D83—41no—
75#14581751—21no—
76#14581751—21no—
77#14581750—21no—
78#14581750—21no—
79#14581749—41no—
80#14581748—41no—
81C4⋊C4×D83—21no—
82C4⋊C4×D83—21no—
83C22⋊C4×D83—41no—
84C42×D83—41no—
85C2×C83⋊C16—22no—
86C2×C83⋊C16C2311yescharacteristic
87C2×C83⋊C16C2311yescharacteristic
88#14581726C2311yescharacteristic
89#14581726C2311yescharacteristic
90#14581726C2311yescharacteristic
91#14581726C2311yescharacteristic
92#14581725—21no—
93#14581725—21no—
94#14581724—21no—
95#14581724—21no—
96#14581723—21no—
97#14581723—21no—
98#14581716—21no—
99#14581716—21no—
100#14581713—21no—
101#14581713—21no—
102C2×#41×C83—21no—
103#135×C83—22no—
104#135×C83—22no—
105C2×Q16×C83C2311yescharacteristic
106C2×SD16×C83—21no—
107C2×D8×C83C2311yescharacteristic
108C22×C8×C83C2311yescharacteristic
109#117×C83—21no—
110C2×C16×C83—22no—
111C2×C16×C83C2311yescharacteristic
112C2×C16×C83C2311yescharacteristic
113#107×C83C2311yescharacteristic
114#107×C83C2311yescharacteristic
115#106×C83—21no—
116C23×D83D811yescharacteristic
117C2×C22⋊D83—41no—
118C2×C22⋊D83—41no—
119C22×Dic83D811yescharacteristic
120C2×C4×D83—22no—
121C2×C4×D83—22no—
122C2×C4×D83—41no—
123C2×C4×D83—41no—
124C2×C4×D83D811yescharacteristic
125C2×C4×D83D811yescharacteristic
126C2×C4×D83C2×D411yescharacteristic
127C2×C4×D83C2×D411yescharacteristic
128C2×C83⋊C8—22no—
129C2×C83⋊C8C2×D411yescharacteristic
130C2×C83⋊C8C2×D411yescharacteristic
131C8×D83—22no—
132C8×D83—22no—
133C8×D83—21no—
134C8×D83—21no—
135C8×D83—21no—
136C8×D83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141Dic83⋊C4—41no—
142Dic83⋊C4—41no—
143Dic83⋊C4—41no—
144Dic83⋊C4—41no—
145C22⋊Dic83—41no—
146C22⋊Dic83—41no—
147C22⋊Dic83—41no—
148C22⋊Dic83—41no—
149(C2×D83)⋊C4—41no—
150(C2×D83)⋊C4—41no—
151(C2×D83)⋊C4—41no—
152(C2×D83)⋊C4—41no—
153C4×Dic83—41no—
154C4×Dic83—41no—
155C4×Dic83—41no—
156C4×Dic83—41no—
157C83⋊C16—21no—
158C83⋊C16—21no—
159#41×C83—42no—
160#41×C83—42no—
161C2×Q8×C83—21no—
162C2×D4×C83—21no—
163C2×D4×C83—41no—
164C22×C4×C83—41no—
165C22×C4×C83C2×D411yescharacteristic
166Q16×C83—21no—
167Q16×C83—21no—
168SD16×C83—42no—
169D8×C83—21no—
170D8×C83—21no—
171C2×C8×C83C2×D411yescharacteristic
172C2×C8×C83#4112yes—
173C2×C8×C83#4112yes—
174C2×C8×C83C2411yescharacteristic
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C22⋊C4×C83—41no—
178C42×C83—41no—
179C16×C83—21no—
180C16×C83—21no—
181C23×C83—41no—
182C23×C83C2×D811yescharacteristic
183C22×D83—22no—
184C22×D83—22no—
185C22×D83—41no—
186C22×D83C2×D811yescharacteristic
187C22×D83C2×D811yescharacteristic
188Q8×C83—41no—
189Q8×C83—41no—
190D4×C83—41no—
191D4×C83—41no—
192D4×C83—82no—
193C2×C4×C83—41no—
194C2×C4×C83—41no—
195C2×C4×C83—41no—
196C2×C4×C83—82no—
197C2×C4×C83#12312yes—
198C2×C4×C83C2×D811yescharacteristic
199C2×C4×C83#13712yes—
200C2×C4×C83C22×D411yescharacteristic
201C22⋊D83—82no—
202C22⋊D83—82no—
203C2×Dic83—22no—
204C2×Dic83—41no—
205C2×Dic83—41no—
206C2×Dic83—41no—
207C2×Dic83—41no—
208C2×Dic83C2×D811yescharacteristic
209C2×Dic83C2×D811yescharacteristic
210C4×D83—21no—
211C4×D83—21no—
212C4×D83—21no—
213C4×D83—21no—
214C4×D83—22no—
215C4×D83—22no—
216C4×D83—82no—
217C4×D83—82no—
218C8×C83#12312yes—
219C8×C83C2×#4111yescharacteristic
220C8×C83C2×#4111yescharacteristic
221C83⋊C8—21no—
222C83⋊C8—21no—
223C22×C83—21no—
224C22×C83—21no—
225C22×C83—81no—
226C22×C83—81no—
227C22×C83—81no—
228C22×C83#48012yes—
229C22×C83C22×D811yescharacteristic
230C2×D83—21no—
231C2×D83—21no—
232C2×D83—21no—
233C2×D83—21no—
234C2×D83—21no—
235C2×D83—44no—
236C2×D83#50812yes—
237C4×C83—81no—
238C4×C83—81no—
239C4×C83—81no—
240C4×C83#48012yes—
241C4×C83C2×#12311yescharacteristic
242C4×C83C2×#13711yescharacteristic
243Dic83—21no—
244Dic83—21no—
245Dic83—81no—
246Dic83—81no—
247Dic83—81no—
248Dic83—81no—
249#33645—831nomaximal, Sylow
250C2×C83—22no—
251C2×C83—161no—
252C2×C83C2×#48011yescharacteristic
253C2×C83C2×#50811yescharacteristic
254C2×C83#341411yescharacteristic
255D83—22no—
256D83—41no—
257#3050—831no—
258C2×#427—831no—
259#2237—831no—
260#2237—831no—
261#2236—831no—
262#2235—831no—
263#2234—831no—
264#2234—831no—
265#2233—831no—
266#2232—831no—
267#2158—831no—
268#2158—831no—
269#2138—831no—
270#2138D8311yescharacteristic, complemented
271C2×C22⋊C16—831no—
272C83#3364511yescharacteristic, complemented, Sylow
273C2×#135C2×D8311yescharacteristic
274C23×C8—831no—
275#525—1661no—
276C2×#117—1661no—
277C22×C16—831no—
278C22×C16C2×D8311yescharacteristic
279#467—831no—
280#467—831no—
281#466—831no—
282#466—831no—
283#465—1661no—
284#465—1661no—
285#427—831no—
286#427—831no—
287#427—831no—
288#427—831no—
289#427—1661no—
290#427—1661no—
291#427C2×D8311yescharacteristic
292C2×#107—831no—
293C2×#107—831no—
294C2×#106—1661no—
295#416—1661no—
296#367—831no—
297#367—831no—
298#367—831no—
299#367—831no—
300#366—831no—
301#366—831no—
302#366—831no—
303#366—831no—
304#358—831no—
305#358—831no—
306#358C2×D8311yescharacteristic, complemented
307#358C2×D8311yescharacteristic, complemented
308#357—831no—
309#357—831no—
310#357C2×D8311yescharacteristic, complemented
311#357C2×D8311yescharacteristic, complemented
312C22⋊C16—831no—
313C22⋊C16—831no—
314C22⋊C16—831no—
315C22⋊C16—831no—
316C2×#41—21no—
317C23×C4—831no—
318#135—22no—
319#135—22no—
320C2×Q16C22×D8311yescharacteristic
321C2×SD16—21no—
322C2×D8C22×D8311yescharacteristic
323C22×C8—831no—
324C22×C8—831no—
325C22×C8—831no—
326C22×C8—831no—
327C22×C8—831no—
328C22×C8—1662no—
329C22×C8—1662no—
330C22×C8C22×D8311yescharacteristic
331#123—3321no—
332C22⋊Q8—1661no—
333C22⋊Q8—1661no—
334#121—1661no—
335#121—1661no—
336#120—3321no—
337C4×D4—3321no—
338C4×D4—3321no—
339#117—21no—
340#117—3321no—
341#117—3321no—
342#117—1661no—
343#117—1661no—
344#117—1661no—
345#117—1661no—
346C2×C4⋊C4—1661no—
347C2×C4⋊C4—1661no—
348C2×C22⋊C4—3321no—
349C2×C42—3321no—
350C2×C16—22no—
351C2×C16—831no—
352C2×C16—831no—
353C2×C16—1662no—
354C2×C16C22×D8311yescharacteristic
355C2×C16C22×D8311yescharacteristic
356#107—831no—
357#107—831no—
358#107—831no—
359#107—831no—
360#107—1661no—
361#107—1661no—
362#107C22×D8311yescharacteristic
363#107C22×D8311yescharacteristic
364#106—21no—
365#106—1661no—
366#106—1661no—
367#106—1661no—
368#106—1661no—
369#103—1661no—
370#103—1661no—
371#102—1661no—
372#102—1661no—
373C24—831no—
374#41—42no—
375#41—42no—
376C2×Q8—21no—
377C2×D4—21no—
378C2×D4—41no—
379C2×D4—3321no—
380C2×D4—3321no—
381C22×C4—41no—
382C22×C4—3321no—
383C22×C4—3321no—
384C22×C4—831no—
385C22×C4—831no—
386C22×C4—831no—
387C22×C4—831no—
388C22×C4—831no—
389C22×C4—1662no—
390C22×C4—1662no—
391C22×C4D4×D8311yescharacteristic
392Q16—21no—
393Q16—21no—
394SD16—42no—
395D8—21no—
396D8—21no—
397C2×C8—831no—
398C2×C8—831no—
399C2×C8—1662no—
400C2×C8—1662no—
401C2×C8—1662no—
402C2×C8—1661no—
403C2×C8—1661no—
404C2×C8—1661no—
405C2×C8—1661no—
406C2×C8D4×D8311yescharacteristic
407C2×C8#1305058912yes—
408C2×C8#1305059012yes—
409C2×C8C23×D8311yescharacteristic
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—3321no—
413C4⋊C4—3321no—
414C4⋊C4—3321no—
415C4⋊C4—3321no—
416C4⋊C4—1661no—
417C4⋊C4—1661no—
418C4⋊C4—1661no—
419C4⋊C4—1661no—
420C22⋊C4—41no—
421C22⋊C4—3321no—
422C22⋊C4—3321no—
423C22⋊C4—3321no—
424C22⋊C4—3321no—
425C22⋊C4—3321no—
426C22⋊C4—3321no—
427C22⋊C4—3321no—
428C22⋊C4—3321no—
429C42—41no—
430C42—3321no—
431C42—3321no—
432C42—3321no—
433C42—3321no—
434C16—21no—
435C16—21no—
436C16—1661no—
437C16—1661no—
438C23—41no—
439C23—3321no—
440C23—831no—
441C23—831no—
442C23—1662no—
443C23—1662no—
444C23D8×D8311yescharacteristic
445Q8—41no—
446Q8—41no—
447D4—41no—
448D4—41no—
449D4—82no—
450D4—6642no—
451D4—6642no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—82no—
456C2×C4—3321no—
457C2×C4—3321no—
458C2×C4—3321no—
459C2×C4—3321no—
460C2×C4—831no—
461C2×C4—831no—
462C2×C4—6642no—
463C2×C4—6642no—
464C2×C4—1662no—
465C2×C4—1661no—
466C2×C4—1661no—
467C2×C4—1661no—
468C2×C4—1661no—
469C2×C4—1662no—
470C2×C4—1662no—
471C2×C4#1458177412yes—
472C2×C4D8×D8311yescharacteristic
473C2×C4#1458182112yes—
474C2×C4C2×D4×D8311yescharacteristic
475C8—1661no—
476C8—1661no—
477C8#1458177412yes—
478C8C2×#1305058911yescharacteristic
479C8C2×#1305059011yescharacteristic
480C22—21no—
481C22—21no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—3324no—
486C22—832no—
487C22—1661no—
488C22—1661no—
489C22—1661no—
490C22—1661no—
491C22—1661no—
492C22#1912801312yes—
493C22C2×D8×D8311yescharacteristic
494C4—81no—
495C4—81no—
496C4—81no—
497C4—6641no—
498C4—6641no—
499C4—6641no—
500C4—6641no—
501C4—1661no—
502C4—1661no—
503C4#1912801312yes—
504C4C2×#1458177411yescharacteristic
505C4C2×#1458182111yescharacteristic
506C2—22no—
507C2—161no—
508C2—3321no—
509C2—1662no—
510C2C2×#1912801311yescharacteristic
511C2#508×D8311yescharacteristic
512C2#2492059611yescharacteristic
subgroupG/H
centerC2×C4D8×D83
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#2138C2
socleC22×C83C22×D8
prime radical#24917040C2
hypercenter#2138D83
hypocenterC83#33645
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. #427×D83C2semidirect
3. #24909670C2semidirect
4. #24909669C2semidirect
5. #24909664C2semidirect
6. #24909659C2semidirect
7. #24909653C2semidirect
8. #24909652C2semidirect
9. #24909647C2semidirect
10. #24909642C2semidirect
11. #24909247C2semidirect
12. #24909245C2semidirect
13. #24909170C2semidirect
14. C2×(C2×D83)⋊C16C2semidirect
15. C83×#2138C2semidirect
36. #19127529C22semidirect
37. #19127529C22semidirect
38. #19127528C22semidirect
39. #19127528C22semidirect
40. #19127527C22semidirect
41. #19127527C22semidirect
42. #19127526C22semidirect
43. #19127526C22semidirect
44. #19127505C22semidirect
45. #19127505C22semidirect
46. #19127502C22semidirect
47. #19127502C22semidirect
55. #358×C83C22semidirect
56. #358×C83C22semidirect
57. #357×C83C22semidirect
58. #357×C83C22semidirect
270. #2138D83semidirect
272. C83#33645semidirect
306. #358C2×D83semidirect
307. #358C2×D83semidirect
310. #357C2×D83semidirect
311. #357C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249170402
#427×D832
#249096702
#249096692
#249096642
#249096592
#249096532
#249096522
#249096472
#249096422
#249092472
#249092452
#249091702
C2×(C2×D83)⋊C162
C83×#21382
#3364583