Finite Group Databasebeta

← #35559237

Subgroups — #35559237

Order 21248.

580 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#467×D83C211yesmaximal, characteristic
2#427×D83C211yesmaximal, characteristic
3#24909672C211yesmaximal, characteristic, complemented
4#24909671C211yesmaximal, characteristic, complemented
5#24909668C211yesmaximal, characteristic, complemented
6#24909667C211yesmaximal, characteristic, complemented
7#24909664C211yesmaximal, characteristic, complemented
8#24909662C211yesmaximal, characteristic, complemented
9#24909659C211yesmaximal, characteristic, complemented
10#24909250C211yesmaximal, characteristic, complemented
11#24909248C211yesmaximal, characteristic, complemented
12#24909181C211yesmaximal, characteristic, complemented
13#24909181C211yesmaximal, characteristic, complemented
14#24909072C211yesmaximal, characteristic
15C83×#2237C211yesmaximal, characteristic, complemented
16C2×Q16×D83C2211yescharacteristic
17C22×C8×D83C2211yescharacteristic
18C22⋊Q8×D83—21no—
19#117×D83—21no—
20C2×#14581731C2211yescharacteristic
21C2×#14581731C2211yescharacteristic
22#19127943C2211yescharacteristic
23#19127942C2211yescharacteristic
24#19127941—21no—
25#19127940—21no—
26#19127939C2211yescharacteristic
27#19127739C2211yescharacteristic
28#19127738C2211yescharacteristic
29#19127738C2211yescharacteristic
30#19127737—21no—
31#19127736—21no—
32#107×D83C2211yescharacteristic
33#107×D83C2211yescharacteristic
34#107×D83C2211yescharacteristic
35#106×D83—21no—
36#103×D83—21no—
37#19127529C2211yescharacteristic, complemented
38#19127528C2211yescharacteristic, complemented
39#19127528C2211yescharacteristic, complemented
40#19127527C2211yescharacteristic, complemented
41#19127526C2211yescharacteristic, complemented
42#19127526C2211yescharacteristic, complemented
43#19127506C2211yescharacteristic, complemented
44#19127506C2211yescharacteristic, complemented
45#19127505C2211yescharacteristic, complemented
46#19127505C2211yescharacteristic, complemented
47#19127504C2211yescharacteristic, complemented
48#19127504C2211yescharacteristic, 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×Q8×D83—21no—
60C22×C4×D83D411yescharacteristic
61C2×Dic332C2311yescharacteristic
62C2×#13050580—21no—
63Q16×D83—22no—
64Q16×D83—22no—
65C22×C83⋊C8D411yescharacteristic
66C2×C8×D83—22no—
67C2×C8×D83—22no—
68C2×C8×D83D411yescharacteristic
69C2×C8×D83D411yescharacteristic
70C2×C8×D83C2311yescharacteristic
71C2×C8×D83C2311yescharacteristic
72#14581772—21no—
73#14581771—21no—
74#14581770—41no—
75#14581769—41no—
76#14581768—21no—
77#14581751—21no—
78#14581750—21no—
79#14581750—21no—
80#14581749—41no—
81#14581748—41no—
82C4⋊C4×D83—21no—
83C4⋊C4×D83—21no—
84C4⋊C4×D83—21no—
85C4⋊C4×D83—41no—
86C22⋊C4×D83—41no—
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#14581725—21no—
99#14581724—21no—
100#14581723—21no—
101#14581723—21no—
102#14581717—21no—
103#14581716—21no—
104#14581715—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
117C2×Dic166—21no—
118C2×Dic166—41no—
119Q8×D83—42no—
120Q8×D83—42no—
121C22×Dic83D811yescharacteristic
122C2×C4×D83—22no—
123C2×C4×D83—22no—
124C2×C4×D83—41no—
125C2×C4×D83—41no—
126C2×C4×D83—41no—
127C2×C4×D83—41no—
128C2×C4×D83D811yescharacteristic
129C2×C4×D83D811yescharacteristic
130C2×C4×D83C2×D411yescharacteristic
131C2×C4×D83C2×D411yescharacteristic
132Dic332—21no—
133Dic332—21no—
134#13050580—42no—
135C2×C83⋊C8—22no—
136C2×C83⋊C8C2×D411yescharacteristic
137C2×C83⋊C8C2×D411yescharacteristic
138C8×D83—22no—
139C8×D83—21no—
140C8×D83—21no—
141C8×D83—22no—
142C8×D83#4112yes—
143C8×D83#4112yes—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—41no—
148Dic83⋊C4—41no—
149Dic83⋊C4—41no—
150Dic83⋊C4—41no—
151Dic83⋊C4—41no—
152Dic83⋊C4—41no—
153C22⋊Dic83—41no—
154C22⋊Dic83—41no—
155(C2×D83)⋊C4—41no—
156(C2×D83)⋊C4—41no—
157(C2×D83)⋊C4—41no—
158(C2×D83)⋊C4—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—22no—
184C22×D83—41no—
185C22×D83C2×D811yescharacteristic
186C22×D83C2×D811yescharacteristic
187Q8×C83—41no—
188Q8×C83—41no—
189C2×C4×C83—21no—
190C2×C4×C83—21no—
191C2×C4×C83—41no—
192C2×C4×C83—41no—
193C2×C4×C83—41no—
194C2×C4×C83—41no—
195C2×C4×C83C2×D811yescharacteristic
196C2×C4×C83C22×D411yescharacteristic
197C2×Dic83—22no—
198C2×Dic83—41no—
199C2×Dic83—41no—
200C2×Dic83—41no—
201C2×Dic83—41no—
202C2×Dic83C2×D811yescharacteristic
203C2×Dic83C2×D811yescharacteristic
204C4×D83—22no—
205C4×D83—21no—
206C4×D83—21no—
207C4×D83—22no—
208C4×D83—82no—
209C4×D83—82no—
210C4×D83—82no—
211C4×D83—82no—
212C4×D83#12312yes—
213C4×D83#13712yes—
214Dic166—41no—
215Dic166—41no—
216Dic166—82no—
217C8×C83—21no—
218C8×C83C2×#4111yescharacteristic
219C8×C83C2×#4111yescharacteristic
220C83⋊C8—21no—
221C83⋊C8#12312yes—
222C22×C83—21no—
223C22×C83—21no—
224C22×C83—21no—
225C22×C83C22×D811yescharacteristic
226C2×D83—21no—
227C2×D83—21no—
228C2×D83—21no—
229C2×D83—21no—
230C2×D83—44no—
231C2×D83#48012yes—
232C2×D83#51012yes—
233C4×C83—21no—
234C4×C83—81no—
235C4×C83—81no—
236C4×C83—81no—
237C4×C83—81no—
238C4×C83C2×#12311yescharacteristic
239C4×C83C2×#13711yescharacteristic
240Dic83—21no—
241Dic83—81no—
242Dic83—81no—
243Dic83—81no—
244Dic83—81no—
245Dic83#48012yes—
246#33649—831nomaximal, Sylow
247C2×C83—41no—
248C2×C83C2×#48011yescharacteristic
249C2×C83C2×#51011yescharacteristic
250C2×C83#341411yescharacteristic
251D83—22no—
252D83—22no—
253C2×#467—831no—
254C2×#427—831no—
255#2237—831no—
256#2237—831no—
257#2237—831no—
258#2237D8311yescharacteristic, complemented
259#2236—831no—
260#2236—831no—
261#2235—831no—
262#2235—831no—
263#2159—831no—
264#2159—831no—
265#2141—831no—
266#2141—831no—
267#2112—831no—
268C83#3364911yescharacteristic, complemented, Sylow
269C22×Q16—831no—
270C23×C8—831no—
271C2×C22⋊Q8—1661no—
272C2×#117—1661no—
273C2×M32—831no—
274C2×M32—831no—
275#467—831no—
276#467—831no—
277#467—831no—
278#467—1661no—
279#467—1661no—
280#467C2×D8311yescharacteristic
281#427—831no—
282#427—831no—
283#427—831no—
284#427—1661no—
285#427—1661no—
286#427C2×D8311yescharacteristic
287C2×#107—831no—
288C2×#107—831no—
289C2×#107—831no—
290C2×#106—1661no—
291C2×#103—1661no—
292#367—831no—
293#367—831no—
294#367—831no—
295#367C2×D8311yescharacteristic, complemented
296#366—831no—
297#366—831no—
298#366—831no—
299#366C2×D8311yescharacteristic, complemented
300#358—831no—
301#358—831no—
302#358—831no—
303#358—831no—
304#358—831no—
305#358—831no—
306#358C2×D8311yescharacteristic, complemented
307#358C2×D8311yescharacteristic, complemented
308C22⋊C16—831no—
309C22⋊C16—831no—
310C22⋊C16—831no—
311C22⋊C16C2×D8311yescharacteristic
312C22×Q8—1661no—
313C23×C4—831no—
314C2×Q16—831no—
315C2×Q16—1662no—
316C2×Q16—1662no—
317C2×Q16—1661no—
318C2×Q16C22×D8311yescharacteristic
319C22×C8—831no—
320C22×C8—831no—
321C22×C8—831no—
322C22×C8—831no—
323C22×C8—831no—
324C22×C8—1662no—
325C22×C8—1662no—
326C22×C8C22×D8311yescharacteristic
327C22⋊Q8—21no—
328C22⋊Q8—3321no—
329C22⋊Q8—3321no—
330C22⋊Q8—1661no—
331C22⋊Q8—1661no—
332C22⋊Q8—1661no—
333#117—21no—
334#117—3321no—
335#117—3321no—
336#117—1661no—
337#117—1661no—
338#117—1661no—
339C2×C4⋊C4—3321no—
340C2×C4⋊C4—1661no—
341C2×C4⋊C4—1661no—
342C2×C4⋊C4—1661no—
343C2×C22⋊C4—3321no—
344C2×C22⋊C4—3321no—
345C2×C42—3321no—
346M32—1662no—
347M32—1662no—
348C2×C16—831no—
349C2×C16—831no—
350C2×C16C22×D8311yescharacteristic
351C2×C16C22×D8311yescharacteristic
352#107—831no—
353#107—831no—
354#107—831no—
355#107—1661no—
356#107—1661no—
357#107—1661no—
358#107C22×D8311yescharacteristic
359#107C22×D8311yescharacteristic
360#107C22×D8311yescharacteristic
361#106—21no—
362#106—1661no—
363#106—1661no—
364#106—1661no—
365#103—21no—
366#103—1661no—
367#103—1661no—
368#103—1661no—
369C24—831no—
370C2×Q8—21no—
371C2×Q8—3321no—
372C2×Q8—3322no—
373C2×Q8—3322no—
374C2×Q8—1661no—
375C22×C4—3321no—
376C22×C4—3321no—
377C22×C4—3321no—
378C22×C4—3321no—
379C22×C4—831no—
380C22×C4—831no—
381C22×C4—831no—
382C22×C4—831no—
383C22×C4—831no—
384C22×C4—1662no—
385C22×C4—1662no—
386C22×C4D4×D8311yescharacteristic
387Q16—21no—
388Q16—21no—
389Q16—3322no—
390Q16—1661no—
391Q16—1661no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—831no—
395C2×C8—832no—
396C2×C8—832no—
397C2×C8—831no—
398C2×C8—1662no—
399C2×C8—1662no—
400C2×C8—1662no—
401C2×C8—1661no—
402C2×C8—1661no—
403C2×C8D4×D8311yescharacteristic
404C2×C8C23×D8311yescharacteristic
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—41no—
409C4⋊C4—3321no—
410C4⋊C4—3321no—
411C4⋊C4—3321no—
412C4⋊C4—3321no—
413C4⋊C4—3321no—
414C4⋊C4—3321no—
415C4⋊C4—1661no—
416C4⋊C4—1661no—
417C4⋊C4—1661no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—3321no—
421C22⋊C4—3321no—
422C22⋊C4—3321no—
423C22⋊C4—3321no—
424C22⋊C4—3321no—
425C22⋊C4—3321no—
426C42—41no—
427C42—3321no—
428C42—3321no—
429C42—3321no—
430C16—21no—
431C16—21no—
432C16—1661no—
433C16—1661no—
434C23—3321no—
435C23—831no—
436C23—831no—
437C23—1662no—
438C23—1662no—
439C23D8×D8311yescharacteristic
440Q8—41no—
441Q8—41no—
442Q8—3321no—
443Q8—3321no—
444Q8—6642no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—3321no—
452C2×C4—3321no—
453C2×C4—3321no—
454C2×C4—3321no—
455C2×C4—831no—
456C2×C4—832no—
457C2×C4—832no—
458C2×C4—831no—
459C2×C4—6642no—
460C2×C4—6642no—
461C2×C4—6642no—
462C2×C4—6642no—
463C2×C4—1662no—
464C2×C4—1662no—
465C2×C4—1662no—
466C2×C4—1661no—
467C2×C4—1661no—
468C2×C4D8×D8311yescharacteristic
469C2×C4C2×D4×D8311yescharacteristic
470C8—21no—
471C8—832no—
472C8—1661no—
473C8#41×D8311yescharacteristic
474C8#41×D8311yescharacteristic
475C22—21no—
476C22—21no—
477C22—21no—
478C22—3324no—
479C22—832no—
480C22—832no—
481C22—1661no—
482C22—1661no—
483C22—1661no—
484C22—1661no—
485C22C2×D8×D8311yescharacteristic
486C4—21no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—832no—
492C4—6641no—
493C4—6641no—
494C4—6641no—
495C4—6641no—
496C4—1661no—
497C4#123×D8311yescharacteristic
498C4#137×D8311yescharacteristic
499C2—41no—
500C2—1662no—
501C2—1662no—
502C2#480×D8311yescharacteristic
503C2#510×D8311yescharacteristic
504C2#2492059611yescharacteristic
subgroupG/H
centerC22C2×D8×D83
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#2237C2
socleC22×C83C22×D8
prime radicalC23×D83D8
hypercenter#2237D83
hypocenterC83#33649
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
3. #24909672C2semidirect
4. #24909671C2semidirect
5. #24909668C2semidirect
6. #24909667C2semidirect
7. #24909664C2semidirect
8. #24909662C2semidirect
9. #24909659C2semidirect
10. #24909250C2semidirect
11. #24909248C2semidirect
12. #24909181C2semidirect
13. #24909181C2semidirect
15. C83×#2237C2semidirect
37. #19127529C22semidirect
38. #19127528C22semidirect
39. #19127528C22semidirect
40. #19127527C22semidirect
41. #19127526C22semidirect
42. #19127526C22semidirect
43. #19127506C22semidirect
44. #19127506C22semidirect
45. #19127505C22semidirect
46. #19127505C22semidirect
47. #19127504C22semidirect
48. #19127504C22semidirect
54. #367×C83C22semidirect
55. #366×C83C22semidirect
56. #358×C83C22semidirect
57. #358×C83C22semidirect
258. #2237D83semidirect
268. C83#33649semidirect
295. #367C2×D83semidirect
299. #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
#467×D832
#427×D832
#249096722
#249096712
#249096682
#249096672
#249096642
#249096622
#249096592
#249092502
#249092482
#249091812
#249090722
C83×#22372
#3364983