Finite Group Databasebeta

← #35565506

Subgroups — #35565506

Order 21248.

2198 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128177C211yesmaximal, characteristic
2#24910430C213yesmaximal, complemented
3#24910418C213yesmaximal, complemented
4Q8×C4⋊Dic83C211yesmaximal, characteristic, complemented
5#24910355C213yesmaximal
6#24910347C213yesmaximal
7Q8×(C2×D83)⋊C4C211yesmaximal, characteristic
8Q8×(C2×D83)⋊C4C211yesmaximal, characteristic
9#24910110C213yesmaximal, complemented
10#24910073C213yesmaximal
11#24909956C213yesmaximal
12#24909934C213yesmaximal, complemented
13#24909828C211yesmaximal, characteristic, complemented
14C4×Q8×Dic83C211yesmaximal, characteristic, complemented
15C83×#2351C211yesmaximal, characteristic, complemented
16C22×Q8×D83C2211yescharacteristic
17#19128177C4116yescomplemented
18C2×Q8×Dic83C2211yescharacteristic
19C2×Q8×Dic83C2211yescharacteristic
20C2×Q8×Dic83C2211yescharacteristic
21C2×C4×Dic166C2213yes—
22C2×#14581757C2213yes—
23C2×#14581750C2213yes—
24C2×#14581749C2213yes—
25#19127607C2213yes—
26#19127607C2213yes—
27#19127606C2213yes—
28#19127606C2213yes—
29#19127600C2213yes—
30#19127600C2213yes—
31#19127593C2216yes—
32#19127593C2211yescharacteristic
33#19127593C2213yes—
34#19127591C2213yes—
35#19127591C2213yes—
36#19127588C2216yes—
37#19127588C2213yes—
38#19127588C2216yes—
39#19127588C2213yes—
40#19127582C2213yes—
41#19127582C2213yes—
42#19127571C2213yes—
43#19127568C2216yes—
44#19127566C2216yes—
45#19127558C2213yes—
46#19127558C2213yes—
47C4⋊C4×Dic83C2213yes—
48C4⋊C4×Dic83C2213yes—
49C4⋊C4×Dic83C2213yes—
50C4⋊C4×Dic83C2216yes—
51C4⋊C4×Dic83C2213yes—
52C4×C4⋊Dic83C2213yes—
53C4×C4⋊Dic83C2213yes—
54C4×C4⋊Dic83C2213yes—
55C4×(C2×D83)⋊C4C2213yes—
56C4×(C2×D83)⋊C4C2213yes—
57#19127547C2213yes—
58#19127546C2216yes—
59C42×Dic83C2213yes—
60C2×C4×Q8×C83C2211yescharacteristic
61C2×C4×Q8×C83C2211yescharacteristic
62C2×C4×Q8×C83C2211yescharacteristic
63#389×C83C2213yes—
64#389×C83C2213yes—
65#382×C83C2213yes—
66C4×C4⋊C4×C83C2213yes—
67C22×Dic166C2313yes—
68C2×Q8×D83—24no—
69C2×Q8×D83—24no—
70C2×Q8×D83C2×C418yes—
71C22×C4×D83C2313yes—
72Q8×Dic83—24no—
73Q8×Dic83—24no—
74Q8×Dic83C2×C418yes—
75C4×Dic166C2×C4124yes—
76#14581757C2×C4124yes—
77#14581750C2×C4124yes—
78#14581749C2×C4124yes—
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2313yes—
81C2×C4⋊Dic83C2313yes—
82C2×C4⋊Dic83C2313yes—
83C2×C4⋊Dic83C2316yes—
84C2×C4⋊Dic83C2313yes—
85C2×Dic83⋊C4C2313yes—
86C2×Dic83⋊C4C2313yes—
87C2×(C2×D83)⋊C4C2311yescharacteristic
88C2×(C2×D83)⋊C4C2311yescharacteristic
89C2×(C2×D83)⋊C4C2313yes—
90C2×(C2×D83)⋊C4C2313yes—
91C2×(C2×D83)⋊C4C2313yes—
92C2×(C2×D83)⋊C4C2311yescharacteristic
93C2×C4×Dic83C2313yes—
94C2×C4×Dic83C2313yes—
95C2×C4×Dic83C2313yes—
96C2×C4×Dic83C2311yescharacteristic
97C2×C4×Dic83C2313yes—
98C2×C4×Dic83C2316yes—
99C2×C4×Dic83C2313yes—
100C2×C4×Dic83C2313yes—
101C2×C4×Dic83C2316yes—
102C83⋊#95C2313yes—
103C83⋊#95C2316yes—
104C83⋊#95C2313yes—
105C83⋊#95C2313yes—
106C83⋊#95C2313yes—
107C83⋊#95C2313yes—
108C83⋊#95C2313yes—
109C83⋊#95C2316yes—
110C83⋊#95C2316yes—
111C83⋊#95C2313yes—
112C83⋊#95C2313yes—
113C83⋊#95C2311yescharacteristic
114C83⋊#95C2313yes—
115C22×Q8×C83C2311yescharacteristic
116C4×Q8×C83—24no—
117C4×Q8×C83—24no—
118C4×Q8×C83C2×C418yes—
119C2×C4⋊C4×C83C2313yes—
120C2×C4⋊C4×C83C2313yes—
121C2×C4⋊C4×C83C2313yes—
122C2×C4⋊C4×C83C2313yes—
123C2×C4⋊C4×C83C2313yes—
124C2×C42×C83C2313yes—
125C2×C42×C83C2313yes—
126C2×C42×C83C2313yes—
127#95×C83C2316yes—
128#95×C83C2311yescharacteristic
129#95×C83C2313yes—
130C23×D83C2411yescharacteristic
131C2×Dic166—26no—
132C2×Dic166—26no—
133C2×Dic166C22×C4112yes—
134Q8×D83—232no—
135C22×Dic83C2411yescharacteristic
136C22×Dic83C2413yes—
137C22×Dic83C2413yes—
138C22×Dic83C2413yes—
139C22×Dic83C2411yescharacteristic
140C22×Dic83C2413yes—
141C22×Dic83C2411yescharacteristic
142C2×C4×D83—26no—
143C2×C4×D83—26no—
144C2×C4×D83C22×C4112yes—
145C4⋊Dic83—26no—
146C4⋊Dic83—26no—
147C4⋊Dic83—22no—
148C4⋊Dic83—43no—
149C4⋊Dic83—46no—
150C4⋊Dic83C22×C4112yes—
151Dic83⋊C4C22×C4112yes—
152Dic83⋊C4C22×C4112yes—
153(C2×D83)⋊C4—22no—
154(C2×D83)⋊C4—22no—
155(C2×D83)⋊C4—43no—
156(C2×D83)⋊C4—43no—
157(C2×D83)⋊C4C22×C414yes—
158(C2×D83)⋊C4C22×C4112yes—
159C4×Dic83—26no—
160C4×Dic83—26no—
161C4×Dic83—22no—
162C4×Dic83—26no—
163C4×Dic83—46no—
164C4×Dic83—43no—
165C4×Dic83—43no—
166C4×Dic83C22×C4112yes—
167C4×Dic83C22×C4124yes—
168C2×Q8×C83C22×C414yes—
169C2×Q8×C83#4114yes—
170C2×Q8×C83#4114yes—
171C22×C4×C83C2413yes—
172C22×C4×C83C2413yes—
173C22×C4×C83C2411yescharacteristic
174C22×C4×C83C2411yescharacteristic
175C22×C4×C83C2413yes—
176C22×C4×C83C2411yescharacteristic
177C22×C4×C83C2413yes—
178C4⋊C4×C83—26no—
179C4⋊C4×C83—26no—
180C4⋊C4×C83—26no—
181C4⋊C4×C83—43no—
182C4⋊C4×C83C22×C4112yes—
183C42×C83—26no—
184C42×C83—26no—
185C42×C83C22×C4112yes—
186C23×C83C2511yescharacteristic
187C22×D83—21no—
188C22×D83—21no—
189C22×D83—21no—
190C22×D83—21no—
191C22×D83—21no—
192C22×D83—21no—
193C22×D83C23×C412yes—
194Q8×C83#117116yes—
195C2×C4×C83—21no—
196C2×C4×C83—23no—
197C2×C4×C83—21no—
198C2×C4×C83—23no—
199C2×C4×C83—21no—
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—23no—
203C2×C4×C83—23no—
204C2×C4×C83—23no—
205C2×C4×C83—23no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—23no—
209C2×C4×C83—23no—
210C2×C4×C83C23×C412yes—
211C2×C4×C83C23×C416yes—
212C2×C4×C83C23×C416yes—
213C2×C4×C83C2×#4116yes—
214C2×C4×C83C2×#4116yes—
215C2×C4×C832−1+412yes—
216C2×Dic83—21no—
217C2×Dic83—23no—
218C2×Dic83—23no—
219C2×Dic83—23no—
220C2×Dic83—21no—
221C2×Dic83—21no—
222C2×Dic83—21no—
223C2×Dic83—23no—
224C2×Dic83—21no—
225C2×Dic83—23no—
226C2×Dic83—23no—
227C2×Dic83—21no—
228C2×Dic83—21no—
229C2×Dic83—23no—
230C2×Dic83—23no—
231C2×Dic83—23no—
232C2×Dic83—23no—
233C2×Dic83C23×C412yes—
234C2×Dic83C23×C416yes—
235C2×Dic83C23×C416yes—
236C2×Dic832−1+412yes—
237C4×D83—224no—
238Dic166—224no—
239C22×C83C24×C411yescharacteristic
240C22×C83C22×#4111yescharacteristic
241C22×C83C22×#4111yescharacteristic
242C22×C83C2×2−1+411yescharacteristic
243C22×C83C2×2−1+411yescharacteristic
244C22×C83#58511yescharacteristic
245C22×C83#58511yescharacteristic
246C2×D83—22no—
247C2×D83—22no—
248C2×D83—22no—
249C2×D83—41no—
250C2×D83—41no—
251C2×D83—41no—
252C2×D83—41no—
253C4×C83—22no—
254C4×C83—26no—
255C4×C83—22no—
256C4×C83—43no—
257C4×C83—41no—
258C4×C83—43no—
259C4×C83C2×#117112yes—
260Dic83—26no—
261Dic83—26no—
262Dic83—22no—
263Dic83—22no—
264Dic83—41no—
265Dic83—43no—
266Dic83—43no—
267#34262—831nomaximal, Sylow
268C2×C83C22×#11711yescharacteristic
269C2×C83C4×2−1+411yescharacteristic
270C2×C83C4×2−1+411yescharacteristic
271C2×C83#351611yescharacteristic
272C2×C83#351811yescharacteristic
273C2×C83#352311yescharacteristic
274C2×C83#352411yescharacteristic
275D83—42no—
276C2×#519—831no—
277#2356—833no—
278#2355—833no—
279#2351D8311yescharacteristic, complemented
280Q8×C4⋊C4—831no—
281#2347—833no—
282#2346—833no—
283Q8×C22⋊C4—831no—
284Q8×C22⋊C4—831no—
285#2320—833no—
286#2314—833no—
287#2298—833no—
288#2295—833no—
289#2276—831no—
290C42×Q8—831no—
291C83#3426211yescharacteristic, complemented, Sylow
292C23×Q8—831no—
293#519—8316no—
294C2×C4×Q8—831no—
295C2×C4×Q8—831no—
296C2×C4×Q8—831no—
297C2×C4×Q8—833no—
298C2×C4×Q8—833no—
299C2×C4×Q8C2×D8311yescharacteristic
300C2×C4×Q8C2×D8311yescharacteristic
301C2×C4×Q8C2×D8311yescharacteristic
302C2×#117—833no—
303C2×#117—833no—
304#391—833no—
305#391—833no—
306#391—833no—
307#391—833no—
308#389—833no—
309#389—833no—
310#389C2×D8313yes—
311#389C2×D8313yes—
312#388—833no—
313#388—833no—
314#388—833no—
315#388—836no—
316#388—831no—
317#387—833no—
318#387—833no—
319#387—836no—
320#387—836no—
321#387—833no—
322#387—833no—
323#384—833no—
324#383—836no—
325#382—836no—
326#382C2×D8313yes—
327#380—833no—
328#380—833no—
329C4×C4⋊C4—833no—
330C4×C4⋊C4—833no—
331C4×C4⋊C4—833no—
332C4×C4⋊C4—836no—
333C4×C4⋊C4—833no—
334C4×C4⋊C4—833no—
335C4×C4⋊C4—833no—
336C4×C4⋊C4—833no—
337C4×C4⋊C4C2×D8313yes—
338C4×C22⋊C4—833no—
339C4×C22⋊C4—833no—
340#376—833no—
341#376—836no—
342C43—833no—
343C22×Q8—838no—
344C22×Q8—833no—
345C22×Q8—1664no—
346C22×Q8—1664no—
347C22×Q8C22×D8311yescharacteristic
348C23×C4—833no—
349C4×Q8—24no—
350C4×Q8—24no—
351C4×Q8—838no—
352C4×Q8—8324no—
353C4×Q8—8324no—
354C4×Q8—1664no—
355C4×Q8—1664no—
356C4×Q8C4×D8318yes—
357#117—8324no—
358#117—8324no—
359C2×C4⋊C4—836no—
360C2×C4⋊C4—833no—
361C2×C4⋊C4—833no—
362C2×C4⋊C4—833no—
363C2×C4⋊C4—833no—
364C2×C4⋊C4—833no—
365C2×C4⋊C4—833no—
366C2×C4⋊C4—831no—
367C2×C4⋊C4C22×D8313yes—
368C2×C4⋊C4C22×D8313yes—
369C2×C4⋊C4C22×D8313yes—
370C2×C4⋊C4C22×D8313yes—
371C2×C4⋊C4C22×D8313yes—
372C2×C22⋊C4—833no—
373C2×C22⋊C4—833no—
374C2×C22⋊C4—833no—
375C2×C22⋊C4—831no—
376C2×C22⋊C4—831no—
377C2×C22⋊C4—831no—
378C2×C42—833no—
379C2×C42—833no—
380C2×C42—836no—
381C2×C42—833no—
382C2×C42—836no—
383C2×C42—833no—
384C2×C42—833no—
385C2×C42—831no—
386C2×C42—833no—
387C2×C42C22×D8313yes—
388C2×C42C22×D8313yes—
389C2×C42C22×D8313yes—
390#95—831no—
391#95—833no—
392#95—833no—
393#95—833no—
394#95—833no—
395#95—833no—
396#95—836no—
397#95—833no—
398#95—833no—
399#95—833no—
400#95—836no—
401#95—836no—
402#95—833no—
403#95C22×D8316yes—
404#95C22×D8311yescharacteristic
405#95C22×D8313yes—
406C24—831no—
407C2×Q8—8312no—
408C2×Q8—16632no—
409C2×Q8—1666no—
410C2×Q8—1666no—
411C2×Q8C2×C4×D8314yes—
412C2×Q8#1305058914yes—
413C2×Q8#1305059114yes—
414C22×C4—833no—
415C22×C4—831no—
416C22×C4—833no—
417C22×C4—831no—
418C22×C4—8312no—
419C22×C4—831no—
420C22×C4—833no—
421C22×C4—833no—
422C22×C4—1666no—
423C22×C4—1666no—
424C22×C4C23×D8311yescharacteristic
425C22×C4C23×D8313yes—
426C22×C4C23×D8313yes—
427C22×C4C23×D8313yes—
428C22×C4C23×D8311yescharacteristic
429C22×C4C23×D8311yescharacteristic
430C22×C4C23×D8313yes—
431C4⋊C4—26no—
432C4⋊C4—26no—
433C4⋊C4—26no—
434C4⋊C4—43no—
435C4⋊C4—3326no—
436C4⋊C4—3323no—
437C4⋊C4—8312no—
438C4⋊C4—8312no—
439C4⋊C4—8312no—
440C4⋊C4—1662no—
441C4⋊C4—1666no—
442C4⋊C4—1666no—
443C4⋊C4C2×C4×D83112yes—
444C22⋊C4—3323no—
445C22⋊C4—3323no—
446C22⋊C4—8312no—
447C22⋊C4—834no—
448C22⋊C4—1662no—
449C22⋊C4—1662no—
450C42—26no—
451C42—26no—
452C42—3323no—
453C42—3323no—
454C42—3326no—
455C42—8312no—
456C42—8324no—
457C42—1666no—
458C42—1666no—
459C42—1666no—
460C42—1662no—
461C42C2×C4×D83112yes—
462C23—832no—
463C23—1661no—
464C23—1661no—
465C23—1661no—
466C23—1661no—
467C23—1661no—
468C23—1661no—
469C23C24×D8311yescharacteristic
470Q8—16624no—
471Q8#14581750116yes—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—23no—
480C2×C4—23no—
481C2×C4—23no—
482C2×C4—21no—
483C2×C4—23no—
484C2×C4—23no—
485C2×C4—21no—
486C2×C4—23no—
487C2×C4—836no—
488C2×C4—832no—
489C2×C4—836no—
490C2×C4—832no—
491C2×C4—1661no—
492C2×C4—1663no—
493C2×C4—1663no—
494C2×C4—1661no—
495C2×C4—1661no—
496C2×C4—1663no—
497C2×C4—1663no—
498C2×C4—1663no—
499C2×C4—1663no—
500C2×C4—1663no—
501C2×C4—1661no—
502C2×C4—1663no—
503C2×C4—1663no—
504C2×C4—1661no—
505C2×C4—1661no—
506C2×C4—1661no—
507C2×C4—1663no—
508C2×C4—16624no—
509C2×C4C22×C4×D8316yes—
510C2×C4C22×C4×D8312yes—
511C2×C4C22×C4×D8316yes—
512C2×C4C2×#1305058916yes—
513C2×C4C2×#1305059116yes—
514C2×C4#1458183912yes—
515C22—3321no—
516C22—3321no—
517C22—3321no—
518C22—3321no—
519C22—1662no—
520C22—1662no—
521C22—1662no—
522C22C23×C4×D8311yescharacteristic
523C22C22×#1305058911yescharacteristic
524C22C22×#1305059111yescharacteristic
525C222−1+4×D8311yescharacteristic
526C22C2×#1458183911yescharacteristic
527C22#1912853711yescharacteristic
528C22#1912853811yescharacteristic
529C4—22no—
530C4—26no—
531C4—22no—
532C4—43no—
533C4—41no—
534C4—43no—
535C4—3321no—
536C4—3323no—
537C4—3323no—
538C4—1666no—
539C4—1666no—
540C4—1662no—
541C4—1662no—
542C4C2×#14581750112yes—
543C2—3322no—
544C2C22×#1458175011yescharacteristic
545C2#2492067211yescharacteristic
546C2C4×#1458183911yescharacteristic
547C2#2492144511yescharacteristic
548C2#2492147111yescharacteristic
549C2#2492156011yescharacteristic
550C2#2492157511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2351C2
socle, cosocleC23×C83C25
prime radicalC23×D83C24
hypercenter#2351D83
hypocenterC83#34262
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. #24910430C2semidirect
3. #24910418C2semidirect
4. Q8×C4⋊Dic83C2semidirect
9. #24910110C2semidirect
12. #24909934C2semidirect
13. #24909828C2semidirect
14. C4×Q8×Dic83C2semidirect
15. C83×#2351C2semidirect
17. #19128177C4semidirect
279. #2351D83semidirect
291. C83#34262semidirect

Maximal subgroups

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

subgroupindex
C2×#191281772
#249104302
#249104182
Q8×C4⋊Dic832
#249103552
#249103472
Q8×(C2×D83)⋊C42
#249101102
#249100732
#249099562
#249099342
#249098282
C4×Q8×Dic832
C83×#23512
#3426283