Finite Group Databasebeta

← #35559222

Subgroups — #35559222

Order 21248.

584 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#24917046C211yesmaximal, characteristic
2#427×D83C211yesmaximal, characteristic, complemented
3#24909670C211yesmaximal, characteristic, complemented
4#24909669C211yesmaximal, characteristic, complemented
5#24909668C211yesmaximal, characteristic, complemented
6#24909657C211yesmaximal, characteristic, complemented
7#24909656C211yesmaximal, characteristic, complemented
8#24909655C211yesmaximal, characteristic, complemented
9#24909654C211yesmaximal, characteristic, complemented
10#24909250C211yesmaximal, characteristic, complemented
11#24909248C211yesmaximal, characteristic, complemented
12#24909174C211yesmaximal, characteristic, complemented
13#24909174C211yesmaximal, characteristic, complemented
14#24909072C211yesmaximal, characteristic, complemented
15C83×#2237C211yesmaximal, characteristic, complemented
16C2×#14581815C2211yescharacteristic
17C22×C8×D83C2211yescharacteristic
18#19128214—21no—
19#117×D83—21no—
20C2×#14581731C2211yescharacteristic
21C2×#14581731C2211yescharacteristic
22#19127939C2211yescharacteristic
23#19127938C2211yescharacteristic
24#19127937C2211yescharacteristic
25#19127933—21no—
26#19127932—21no—
27#19127739C2211yescharacteristic
28#19127738C2211yescharacteristic
29#19127738C2211yescharacteristic
30#19127738C2211yescharacteristic
31#19127737—21no—
32#19127736—21no—
33#107×D83C2211yescharacteristic
34#107×D83C2211yescharacteristic
35#106×D83—21no—
36#19127696—21no—
37#19127529C2211yescharacteristic, complemented
38#19127528C2211yescharacteristic, complemented
39#19127528C2211yescharacteristic, complemented
40#19127527C2211yescharacteristic, complemented
41#19127526C2211yescharacteristic, complemented
42#19127526C2211yescharacteristic, complemented
43#19127505C2211yescharacteristic, complemented
44#19127505C2211yescharacteristic, complemented
45#19127503C2211yescharacteristic, complemented
46#19127503C2211yescharacteristic, complemented
47#19127501C2211yescharacteristic, complemented
48#19127501C2211yescharacteristic, complemented
49C22⋊(C83⋊C16)C2211yescharacteristic
50(C2×D83)⋊C16C2211yescharacteristic
51(C2×D83)⋊C16C2211yescharacteristic
52#467×C83C2211yescharacteristic
53#427×C83C2211yescharacteristic
54#367×C83C2211yescharacteristic, complemented
55#366×C83C2211yescharacteristic, complemented
56#358×C83C2211yescharacteristic, complemented
57#358×C83C2211yescharacteristic, complemented
58C22⋊C16×C83C2211yescharacteristic
59C2×#13050591—21no—
60C22×C4×D83D411yescharacteristic
61#14581815—22no—
62#14581815—22no—
63C2×#13050577—21no—
64C2×D664C2311yescharacteristic
65C22×C83⋊C8D411yescharacteristic
66C2×C8×D83—22no—
67C2×C8×D83—22no—
68C2×C8×D83D411yescharacteristic
69C2×C8×D83D411yescharacteristic
70C2×C8×D83C2311yescharacteristic
71C2×C8×D83C2311yescharacteristic
72#14581777—41no—
73#14581768—21no—
74#14581767—21no—
75#14581766—21no—
76#14581762—41no—
77#14581754—41no—
78(C22⋊D83)⋊C4—41no—
79#14581751—21no—
80#14581750—21no—
81#14581750—21no—
82#14581750—21no—
83#14581749—41no—
84#14581748—41no—
85C4⋊C4×D83—21no—
86C4⋊C4×D83—21no—
87C22⋊C4×D83—41no—
88C42×D83—41no—
89#14581731—22no—
90#14581731—22no—
91C2×C83⋊C16C2311yescharacteristic
92C2×C83⋊C16C2311yescharacteristic
93#14581726C2311yescharacteristic
94#14581726C2311yescharacteristic
95#14581726C2311yescharacteristic
96#14581725—21no—
97#14581725—21no—
98#14581724—21no—
99#14581723—21no—
100#14581723—21no—
101#14581723—21no—
102#14581716—21no—
103#14581714—21no—
104#14581712—21no—
105C2×Q16×C83C2311yescharacteristic
106C22×C8×C83C2311yescharacteristic
107C22⋊Q8×C83—21no—
108#117×C83—21no—
109C2×C16×C83C2311yescharacteristic
110C2×C16×C83C2311yescharacteristic
111#107×C83C2311yescharacteristic
112#107×C83C2311yescharacteristic
113#107×C83C2311yescharacteristic
114#106×C83—21no—
115#103×C83—21no—
116C23×D83D811yescharacteristic
117#13050591—42no—
118#13050591—42no—
119C2×D332—21no—
120C2×D332—41no—
121C2×D332—41no—
122C2×C22⋊D83—41no—
123C22×Dic83D811yescharacteristic
124C2×C4×D83—22no—
125C2×C4×D83—22no—
126C2×C4×D83—41no—
127C2×C4×D83—41no—
128C2×C4×D83—41no—
129C2×C4×D83D811yescharacteristic
130C2×C4×D83D811yescharacteristic
131C2×C4×D83C2×D411yescharacteristic
132C2×C4×D83C2×D411yescharacteristic
133#13050577—42no—
134D664—21no—
135D664—21no—
136C2×C83⋊C8—22no—
137C2×C83⋊C8C2×D411yescharacteristic
138C2×C83⋊C8C2×D411yescharacteristic
139C8×D83—22no—
140C8×D83—21no—
141C8×D83—21no—
142C8×D83—22no—
143C8×D83#4112yes—
144C8×D83#4112yes—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148Dic83⋊C4—41no—
149Dic83⋊C4—41no—
150Dic83⋊C4—41no—
151C22⋊Dic83—41no—
152(C2×D83)⋊C4—41no—
153(C2×D83)⋊C4—41no—
154(C2×D83)⋊C4—41no—
155(C2×D83)⋊C4—41no—
156(C2×D83)⋊C4—41no—
157(C2×D83)⋊C4—41no—
158C4×Dic83—41no—
159C4×Dic83—41no—
160C4×Dic83—41no—
161C4×Dic83—41no—
162C83⋊C16—21no—
163C83⋊C16—21no—
164C2×Q8×C83—21no—
165C22×C4×C83C2×D411yescharacteristic
166Q16×C83—21no—
167Q16×C83—21no—
168C2×C8×C83—21no—
169C2×C8×C83—21no—
170C2×C8×C83C2×D411yescharacteristic
171C2×C8×C83C2411yescharacteristic
172C4⋊C4×C83—21no—
173C4⋊C4×C83—21no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—41no—
176C22⋊C4×C83—41no—
177C22⋊C4×C83—41no—
178C42×C83—41no—
179C16×C83—21no—
180C16×C83—21no—
181C23×C83C2×D811yescharacteristic
182C22×D83—22no—
183C22×D83—41no—
184C22×D83—41no—
185C22×D83—41no—
186C22×D83C2×D811yescharacteristic
187C22×D83C2×D811yescharacteristic
188Q8×C83—41no—
189Q8×C83—41no—
190C2×C4×C83—21no—
191C2×C4×C83—21no—
192C2×C4×C83—41no—
193C2×C4×C83—41no—
194C2×C4×C83—41no—
195C2×C4×C83—41no—
196C2×C4×C83C2×D811yescharacteristic
197C2×C4×C83C22×D411yescharacteristic
198C22⋊D83—82no—
199C2×Dic83—22no—
200C2×Dic83—41no—
201C2×Dic83—41no—
202C2×Dic83—41no—
203C2×Dic83C2×D811yescharacteristic
204C2×Dic83C2×D811yescharacteristic
205D332—41no—
206D332—41no—
207D332—82no—
208D332—82no—
209C4×D83—22no—
210C4×D83—21no—
211C4×D83—21no—
212C4×D83—22no—
213C4×D83—82no—
214C4×D83—82no—
215C4×D83—82no—
216C4×D83#12312yes—
217C4×D83#13712yes—
218C8×C83—21no—
219C8×C83C2×#4111yescharacteristic
220C8×C83C2×#4111yescharacteristic
221C83⋊C8—21no—
222C83⋊C8#12312yes—
223C22×C83—21no—
224C22×C83—21no—
225C22×C83—21no—
226C22×C83C22×D811yescharacteristic
227C2×D83—21no—
228C2×D83—21no—
229C2×D83—21no—
230C2×D83—21no—
231C2×D83—21no—
232C2×D83—44no—
233C2×D83—81no—
234C2×D83—81no—
235C2×D83—81no—
236C2×D83#48012yes—
237C4×C83—21no—
238C4×C83—81no—
239C4×C83—81no—
240C4×C83—81no—
241C4×C83—81no—
242C4×C83C2×#12311yescharacteristic
243C4×C83C2×#13711yescharacteristic
244Dic83—21no—
245Dic83—81no—
246Dic83—81no—
247Dic83—81no—
248Dic83#48012yes—
249#33647—831nomaximal, Sylow
250C2×C83—41no—
251C2×C83C2×#48011yescharacteristic
252C2×C83#341411yescharacteristic
253C2×C83#341411yescharacteristic
254D83—22no—
255D83—41no—
256D83—161no—
257#3050—831no—
258C2×#427—831no—
259#2237—831no—
260#2237—831no—
261#2237—831no—
262#2237D8311yescharacteristic, complemented
263#2234—831no—
264#2234—831no—
265#2234—831no—
266#2234—831no—
267#2159—831no—
268#2159—831no—
269#2139—831no—
270#2139—831no—
271#2112—831no—
272C83#3364711yescharacteristic, complemented, Sylow
273C2×#135—831no—
274C23×C8—831no—
275#525—1661no—
276C2×#117—1661no—
277C2×M32—831no—
278C2×M32—831no—
279#467—831no—
280#467C2×D8311yescharacteristic
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#367C2×D8311yescharacteristic, complemented
300#366—831no—
301#366—831no—
302#366—831no—
303#366C2×D8311yescharacteristic, complemented
304#358—831no—
305#358—831no—
306#358C2×D8311yescharacteristic, complemented
307#358C2×D8311yescharacteristic, complemented
308#357—831no—
309#357—831no—
310#357—831no—
311#357—831no—
312C22⋊C16—831no—
313C22⋊C16—831no—
314C22⋊C16—831no—
315C22⋊C16C2×D8311yescharacteristic
316C2×#41—1661no—
317C23×C4—831no—
318#135—1662no—
319#135—1662no—
320C2×Q16C22×D8311yescharacteristic
321C2×SD16—1661no—
322C2×D8—831no—
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—21no—
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—
350M32—1662no—
351M32—1662no—
352C2×C16—831no—
353C2×C16—831no—
354C2×C16C22×D8311yescharacteristic
355C2×C16C22×D8311yescharacteristic
356#107—831no—
357#107—831no—
358#107—831no—
359#107—1661no—
360#107—1661no—
361#107C22×D8311yescharacteristic
362#107C22×D8311yescharacteristic
363#107C22×D8311yescharacteristic
364#106—21no—
365#106—1661no—
366#106—1661no—
367#106—1661no—
368#106—1661no—
369#103—21no—
370#103—1661no—
371#102—1661no—
372#102—1661no—
373C24—831no—
374#41—3322no—
375#41—3322no—
376C2×Q8—21no—
377C2×D4—3321no—
378C2×D4—3321no—
379C2×D4—3321no—
380C2×D4—1661no—
381C22×C4—3321no—
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—3322no—
395D8—1661no—
396D8—1661no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—831no—
400C2×C8—832no—
401C2×C8—832no—
402C2×C8—831no—
403C2×C8—1662no—
404C2×C8—1662no—
405C2×C8—1662no—
406C2×C8—1661no—
407C2×C8—1661no—
408C2×C8D4×D8311yescharacteristic
409C2×C8C23×D8311yescharacteristic
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—41no—
414C4⋊C4—3321no—
415C4⋊C4—3321no—
416C4⋊C4—3321no—
417C4⋊C4—1661no—
418C4⋊C4—1661no—
419C4⋊C4—1661no—
420C22⋊C4—41no—
421C22⋊C4—41no—
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—3321no—
439C23—3321no—
440C23—3321no—
441C23—831no—
442C23—831no—
443C23—1662no—
444C23D8×D8311yescharacteristic
445Q8—41no—
446Q8—41no—
447D4—3321no—
448D4—3321no—
449D4—6642no—
450D4—6642no—
451D4—6642no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—3321no—
459C2×C4—3321no—
460C2×C4—3321no—
461C2×C4—831no—
462C2×C4—832no—
463C2×C4—832no—
464C2×C4—831no—
465C2×C4—6642no—
466C2×C4—6642no—
467C2×C4—6642no—
468C2×C4—1662no—
469C2×C4—1662no—
470C2×C4—1662no—
471C2×C4—1661no—
472C2×C4—1661no—
473C2×C4D8×D8311yescharacteristic
474C2×C4C2×D4×D8311yescharacteristic
475C8—21no—
476C8—832no—
477C8—1661no—
478C8#41×D8311yescharacteristic
479C8#41×D8311yescharacteristic
480C22—21no—
481C22—21no—
482C22—21no—
483C22—3324no—
484C22—832no—
485C22—6641no—
486C22—6641no—
487C22—6641no—
488C22—1661no—
489C22—1661no—
490C22—1661no—
491C22—1661no—
492C22—1661no—
493C22C2×D8×D8311yescharacteristic
494C4—21no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—832no—
500C4—6641no—
501C4—6641no—
502C4—6641no—
503C4—1661no—
504C4#123×D8311yescharacteristic
505C4#137×D8311yescharacteristic
506C2—41no—
507C2—13281no—
508C2—3321no—
509C2—1662no—
510C2#480×D8311yescharacteristic
511C2#2492059611yescharacteristic
512C2#2492059811yescharacteristic
subgroupG/H
centerC22C2×D8×D83
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#2237C2
socleC22×C83C22×D8
prime radical#24917046C2
hypercenter#2237D83
hypocenterC83#33647
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2237

Complements

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

NH ≅ G/Ntype
2. #427×D83C2semidirect
3. #24909670C2semidirect
4. #24909669C2semidirect
5. #24909668C2semidirect
6. #24909657C2semidirect
7. #24909656C2semidirect
8. #24909655C2semidirect
9. #24909654C2semidirect
10. #24909250C2semidirect
11. #24909248C2semidirect
12. #24909174C2semidirect
13. #24909174C2semidirect
14. #24909072C2semidirect
15. C83×#2237C2semidirect
37. #19127529C22semidirect
38. #19127528C22semidirect
39. #19127528C22semidirect
40. #19127527C22semidirect
41. #19127526C22semidirect
42. #19127526C22semidirect
43. #19127505C22semidirect
44. #19127505C22semidirect
45. #19127503C22semidirect
46. #19127503C22semidirect
47. #19127501C22semidirect
48. #19127501C22semidirect
54. #367×C83C22semidirect
55. #366×C83C22semidirect
56. #358×C83C22semidirect
57. #358×C83C22semidirect
262. #2237D83semidirect
272. C83#33647semidirect
299. #367C2×D83semidirect
303. #366C2×D83semidirect
306. #358C2×D83semidirect
307. #358C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249170462
#427×D832
#249096702
#249096692
#249096682
#249096572
#249096562
#249096552
#249096542
#249092502
#249092482
#249091742
#249090722
C83×#22372
#3364783