Finite Group Databasebeta

← #35567996

Subgroups — #35567996

Order 21248.

2390 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128306C211yesmaximal, characteristic
2#24910616C214yesmaximal, complemented
3#24910559C212yesmaximal, complemented
4#24910527C212yesmaximal, complemented
5#24910476C214yesmaximal, complemented
6#24910438C212yesmaximal, complemented
7#24910363C212yesmaximal, complemented
8#24910252C214yesmaximal, complemented
9#24910152C212yesmaximal, complemented
10#24910136C214yesmaximal, complemented
11#24909969C212yesmaximal, complemented
12#24909947C211yesmaximal, characteristic, complemented
13C83×#2314C211yesmaximal, characteristic, complemented
14#19128306C4116yescomplemented
15C2×#14581789C2211yescharacteristic
16C2×#14581784C2211yescharacteristic
17C2×#14581777C2214yes—
18C2×#14581766C2212yes—
19C2×#14581765C2214yes—
20C2×C4×D332C2214yes—
21C2×C4×D332C2212yes—
22C2×#14581754C2212yes—
23C2×#14581754C2212yes—
24C2×(C22⋊D83)⋊C4C2214yes—
25C2×#14581751C2211yescharacteristic
26C2×C4⋊C4×D83C2214yes—
27C22×C4×Dic83C2211yescharacteristic
28#19127603C2214yes—
29#19127602C2212yes—
30#19127602C2212yes—
31#19127601C2214yes—
32#19127600C2212yes—
33#19127600C2212yes—
34#19127598C2214yes—
35#19127597C2212yes—
36#19127597C2212yes—
37#19127593C2214yes—
38#19127593C2214yes—
39#19127593C2214yes—
40#19127590C2214yes—
41#19127588C2214yes—
42#19127585C2214yes—
43#19127579C2214yes—
44#19127579C2214yes—
45#19127578C2214yes—
46#19127577C2214yes—
47#19127576C2212yes—
48#19127576C2212yes—
49#19127575C2214yes—
50#19127572C2214yes—
51#19127572C2214yes—
52#19127563C2214yes—
53#19127558C2214yes—
54C4⋊C4×Dic83C2212yes—
55C4⋊C4×Dic83C2212yes—
56C4×Dic83⋊C4C2214yes—
57C4×C4⋊Dic83C2212yes—
58C4×C4⋊Dic83C2212yes—
59#19127548C2214yes—
60#19127548C2214yes—
61C2×#117×C83C2211yescharacteristic
62#388×C83C2214yes—
63#387×C83C2214yes—
64#384×C83C2211yescharacteristic
65#382×C83C2211yescharacteristic
66#380×C83C2212yes—
67C4×C4⋊C4×C83C2212yes—
68C22×D332C2312yes—
69C22×D332C2314yes—
70C22×C22⋊D83C2314yes—
71C23×Dic83C2311yescharacteristic
72C23×Dic83C2311yescharacteristic
73C22×C4×D83C2314yes—
74C22×C4×D83C2314yes—
75C22×C4×D83C2314yes—
76#14581789C2×C418yescomplemented
77#14581784C2×C418yescomplemented
78#14581777C2×C4132yescomplemented
79#14581766C2×C4116yescomplemented
80#14581765C2×C4132yescomplemented
81C4×D332—48no—
82C4×D332C2×C4116yescomplemented
83#14581754—44no—
84#14581754—44no—
85(C22⋊D83)⋊C4—48no—
86#14581751—42no—
87C2×C4⋊Dic83C2312yes—
88C2×C4⋊Dic83C2312yes—
89C2×C4⋊Dic83C2312yes—
90C2×C4⋊Dic83C2312yes—
91C2×C4⋊Dic83C2311yescharacteristic
92C2×C4⋊Dic83C2311yescharacteristic
93C2×C4⋊Dic83C2312yes—
94C2×Dic83⋊C4C2314yes—
95C2×Dic83⋊C4C2314yes—
96C2×Dic83⋊C4C2314yes—
97C4⋊C4×D83—48no—
98C2×C22⋊Dic83C2312yes—
99C2×(C2×D83)⋊C4C2314yes—
100C2×(C2×D83)⋊C4C2314yes—
101C2×(C2×D83)⋊C4C2314yes—
102C2×(C2×D83)⋊C4C2314yes—
103C2×(C2×D83)⋊C4C2314yes—
104C2×(C2×D83)⋊C4C2314yes—
105C2×(C2×D83)⋊C4C2314yes—
106C2×(C2×D83)⋊C4C2314yes—
107C2×C4×Dic83—42no—
108C2×C4×Dic83C2312yes—
109C2×C4×Dic83C2312yes—
110C2×C4×Dic83C2314yes—
111C2×C4×Dic83C2314yes—
112C2×C4×Dic83C2312yes—
113C83⋊#95C2312yes—
114C83⋊#95C2312yes—
115C83⋊#95C2312yes—
116C83⋊#95C2312yes—
117C83⋊#95C2314yes—
118C83⋊#95C2314yes—
119C83⋊#95C2314yes—
120C83⋊#95C2314yes—
121C23×C4×C83C2311yescharacteristic
122#117×C83C2×C418yescomplemented
123C2×C4⋊C4×C83C2312yes—
124C2×C4⋊C4×C83C2312yes—
125C2×C4⋊C4×C83C2314yes—
126C2×C4⋊C4×C83C2312yes—
127C2×C22⋊C4×C83C2314yes—
128C2×C22⋊C4×C83C2312yes—
129C2×C42×C83C2314yes—
130C2×C42×C83C2311yescharacteristic
131C2×C42×C83C2311yescharacteristic
132#95×C83C2314yes—
133#95×C83C2314yes—
134#95×C83C2314yes—
135C23×D83C2414yes—
136C2×D332—24no—
137C2×D332—24no—
138C2×D332—44no—
139C2×D332—44no—
140C2×D332C22×C418yes—
141C2×D332C22×C4116yes—
142C2×C22⋊D83—44no—
143C2×C22⋊D83—44no—
144C2×C22⋊D83C22×C4116yes—
145C22×Dic83—24no—
146C22×Dic83—42no—
147C22×Dic83—41no—
148C22×Dic83—41no—
149C22×Dic83C2412yes—
150C22×Dic83C2412yes—
151C22×Dic83C2412yes—
152C22×Dic83C2414yes—
153C22×Dic83C2412yes—
154C2×C4×D83—28no—
155C2×C4×D83—44no—
156C2×C4×D83—44no—
157C2×C4×D83—44no—
158C2×C4×D83—44no—
159C2×C4×D83—44no—
160C2×C4×D83—44no—
161C2×C4×D83—44no—
162C2×C4×D83C22×C4116yes—
163C4⋊Dic83—24no—
164C4⋊Dic83—24no—
165C4⋊Dic83—42no—
166C4⋊Dic83—42no—
167C4⋊Dic83—41no—
168C4⋊Dic83—41no—
169C4⋊Dic83C22×C418yes—
170Dic83⋊C4—28no—
171Dic83⋊C4—44no—
172Dic83⋊C4C22×C4116yes—
173C22⋊Dic83—42no—
174(C2×D83)⋊C4—28no—
175(C2×D83)⋊C4—44no—
176(C2×D83)⋊C4—44no—
177(C2×D83)⋊C4—44no—
178(C2×D83)⋊C4—44no—
179(C2×D83)⋊C4—44no—
180(C2×D83)⋊C4C22×C4116yes—
181(C2×D83)⋊C4C22×C4116yes—
182C4×Dic83—24no—
183C4×Dic83—24no—
184C4×Dic83—42no—
185C4×Dic83—44no—
186C4×Dic83—44no—
187C24×C83C2411yescharacteristic
188C22×C4×C83—22no—
189C22×C4×C83—22no—
190C22×C4×C83C22×C414yes—
191C22×C4×C83C2414yes—
192C22×C4×C83C2414yes—
193C22×C4×C83C2412yes—
194C22×C4×C83C2412yes—
195C22×C4×C83C2411yescharacteristic
196C22×C4×C83C2411yescharacteristic
197C4⋊C4×C83—24no—
198C4⋊C4×C83—24no—
199C4⋊C4×C83—44no—
200C4⋊C4×C83C22×C418yes—
201C22⋊C4×C83—44no—
202C22⋊C4×C83C22×C418yes—
203C42×C83—44no—
204C42×C83C22×C414yes—
205C42×C83C22×C414yes—
206C23×C83—22no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—22no—
210C23×C83C23×C412yes—
211C23×C83C2511yescharacteristic
212C22×D83—24no—
213C22×D83—24no—
214C22×D83—24no—
215C22×D83—24no—
216C22×D83—24no—
217C22×D83—24no—
218C22×D83C23×C418yes—
219C2×C4×C83—22no—
220C2×C4×C83—22no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—28no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—24no—
229C2×C4×C83—24no—
230C2×C4×C83—24no—
231C2×C4×C83—24no—
232C2×C4×C83C23×C414yes—
233C2×C4×C83C23×C414yes—
234C2×C4×C83C23×C412yes—
235C2×C4×C83C23×C412yes—
236C2×C4×C83H(5,𝔽2)12yes—
237C2×C4×C832−1+412yes—
238C22⋊D83—416no—
239C2×Dic83—22no—
240C2×Dic83—22no—
241C2×Dic83—24no—
242C2×Dic83—24no—
243C2×Dic83—22no—
244C2×Dic83—22no—
245C2×Dic83—22no—
246C2×Dic83—22no—
247C2×Dic83—22no—
248C2×Dic83—22no—
249C2×Dic83—22no—
250C2×Dic83—22no—
251C2×Dic83—22no—
252C2×Dic83—44no—
253C2×Dic83—44no—
254C2×Dic83C23×C418yes—
255C2×Dic83H(5,𝔽2)14yes—
256D332—216no—
257D332—416no—
258C4×D83—416no—
259C4×D83—88no—
260C4×D83—88no—
261C22×C83—22no—
262C22×C83—24no—
263C22×C83—41no—
264C22×C83—41no—
265C22×C83—42no—
266C22×C83C24×C411yescharacteristic
267C22×C83C2×H(5,𝔽2)12yes—
268C22×C83C2×H(5,𝔽2)11yescharacteristic
269C22×C83C2×2−1+411yescharacteristic
270C22×C83#58512yes—
271C2×D83—28no—
272C2×D83—28no—
273C2×D83—28no—
274C2×D83—44no—
275C2×D83—44no—
276C2×D83—44no—
277C2×D83—44no—
278C4×C83—22no—
279C4×C83—24no—
280C4×C83—24no—
281C4×C83—44no—
282C4×C83—44no—
283C4×C83#52014yes—
284Dic83—28no—
285Dic83—24no—
286Dic83—42no—
287Dic83—42no—
288Dic83—42no—
289#34418—831nomaximal, Sylow
290C2×C83—42no—
291C2×C83C2×#52011yescharacteristic
292C2×C83C4×H(5,𝔽2)12yes—
293C2×C83#353711yescharacteristic
294C2×C83#353911yescharacteristic
295C2×C83#355412yes—
296D83—48no—
297C2×#540—831no—
298#2377—834no—
299#2370—832no—
300#2366—832no—
301#2361—834no—
302#2357—832no—
303D4×C4⋊C4—832no—
304#2335—834no—
305#2324—832no—
306#2322—834no—
307#2314D8311yescharacteristic, complemented
308C4×#121—832no—
309#2296—831no—
310C83#3441811yescharacteristic, complemented, Sylow
311#540—8316no—
312C2×#127—831no—
313C2×#125—831no—
314C2×#123—834no—
315C2×#121—832no—
316C2×#121—834no—
317C2×C4×D4—832no—
318C2×C4×D4—832no—
319C2×C4×D4—834no—
320C2×C4×D4—834no—
321C2×C4×D4—832no—
322C2×#117—831no—
323C2×#117C2×D8311yescharacteristic
324C22×C4⋊C4—834no—
325C22×C42—831no—
326#390—832no—
327#390—832no—
328#390—834no—
329#390—834no—
330#389—832no—
331#389—832no—
332#389—832no—
333#389—832no—
334#389—834no—
335#388—834no—
336#388—834no—
337#388—834no—
338#388—834no—
339#388C2×D8314yes—
340#387—834no—
341#387—834no—
342#387C2×D8314yes—
343#386—834no—
344#386—834no—
345#386—834no—
346#386—834no—
347#385—834no—
348#385—834no—
349#385—832no—
350#385—832no—
351#385—834no—
352#384C2×D8311yescharacteristic
353#382C2×D8311yescharacteristic
354#381—834no—
355#380—834no—
356#380C2×D8312yes—
357C4×C4⋊C4—832no—
358C4×C4⋊C4—832no—
359C4×C4⋊C4—832no—
360C4×C4⋊C4—832no—
361C4×C4⋊C4—834no—
362C4×C4⋊C4C2×D8312yes—
363C4×C22⋊C4—834no—
364C4×C22⋊C4—834no—
365C22×D4—834no—
366C22×D4—834no—
367C22×D4—832no—
368C23×C4—831no—
369C23×C4—834no—
370C23×C4—834no—
371C23×C4—834no—
372C23×C4—831no—
373C23×C4C22×D8311yescharacteristic
374#127—838no—
375#125—838no—
376#123—8332no—
377#121—8316no—
378#121—8332no—
379C4×D4—3328no—
380C4×D4—3328no—
381C4×D4—3324no—
382C4×D4—3324no—
383C4×D4—8316no—
384#117—3322no—
385#117C4×D8318yescomplemented
386C2×C4⋊C4—3328no—
387C2×C4⋊C4—834no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—832no—
391C2×C4⋊C4—832no—
392C2×C4⋊C4—832no—
393C2×C4⋊C4—834no—
394C2×C4⋊C4—832no—
395C2×C4⋊C4—834no—
396C2×C4⋊C4—832no—
397C2×C4⋊C4C22×D8312yes—
398C2×C4⋊C4C22×D8312yes—
399C2×C4⋊C4C22×D8314yes—
400C2×C4⋊C4C22×D8312yes—
401C2×C22⋊C4—832no—
402C2×C22⋊C4—834no—
403C2×C22⋊C4—834no—
404C2×C22⋊C4—834no—
405C2×C22⋊C4—834no—
406C2×C22⋊C4—834no—
407C2×C22⋊C4—834no—
408C2×C22⋊C4—834no—
409C2×C22⋊C4—834no—
410C2×C22⋊C4C22×D8314yes—
411C2×C22⋊C4C22×D8312yes—
412C2×C42—3322no—
413C2×C42—834no—
414C2×C42—832no—
415C2×C42—834no—
416C2×C42—832no—
417C2×C42—832no—
418C2×C42C22×D8314yes—
419C2×C42C22×D8311yescharacteristic
420C2×C42C22×D8311yescharacteristic
421#95—832no—
422#95—834no—
423#95—832no—
424#95—834no—
425#95—832no—
426#95—834no—
427#95—834no—
428#95—832no—
429#95C22×D8314yes—
430#95C22×D8314yes—
431#95C22×D8314yes—
432C24—834no—
433C24C23×D8311yescharacteristic
434C2×D4—3324no—
435C2×D4—3324no—
436C2×D4—3324no—
437C2×D4—3324no—
438C2×D4—8316no—
439C2×D4—8316no—
440C2×D4—838no—
441C2×D4—1664no—
442C2×D4—1664no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—3324no—
446C22×C4—3324no—
447C22×C4—3324no—
448C22×C4—3324no—
449C22×C4—3324no—
450C22×C4—3324no—
451C22×C4—3324no—
452C22×C4—3321no—
453C22×C4—3321no—
454C22×C4—3322no—
455C22×C4—8316no—
456C22×C4—834no—
457C22×C4—832no—
458C22×C4—832no—
459C22×C4—832no—
460C22×C4—832no—
461C22×C4—1668no—
462C22×C4—1664no—
463C22×C4C2×C4×D8314yes—
464C22×C4C23×D8314yes—
465C22×C4C23×D8314yes—
466C22×C4C23×D8312yes—
467C22×C4C23×D8312yes—
468C22×C4C23×D8311yescharacteristic
469C22×C4C23×D8311yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—3324no—
474C4⋊C4—3321no—
475C4⋊C4—3321no—
476C4⋊C4—3322no—
477C4⋊C4—3322no—
478C4⋊C4—838no—
479C4⋊C4—8316no—
480C4⋊C4—1664no—
481C4⋊C4—1668no—
482C4⋊C4—1664no—
483C4⋊C4C2×C4×D8318yes—
484C22⋊C4—44no—
485C22⋊C4—3324no—
486C22⋊C4—3324no—
487C22⋊C4—3324no—
488C22⋊C4—3324no—
489C22⋊C4—3324no—
490C22⋊C4—3322no—
491C22⋊C4—8316no—
492C22⋊C4—8316no—
493C22⋊C4—1668no—
494C22⋊C4C2×C4×D8318yes—
495C42—44no—
496C42—3322no—
497C42—3324no—
498C42—3324no—
499C42—1664no—
500C42—1664no—
501C42C2×C4×D8314yes—
502C42C2×C4×D8314yes—
503C23—22no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—838no—
508C23—1664no—
509C23—1664no—
510C23—1664no—
511C23—1664no—
512C23—1664no—
513C23—1664no—
514C23C22×C4×D8312yes—
515C23C24×D8311yescharacteristic
516D4—33216no—
517D4—33216no—
518D4—16616no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—33216no—
533C2×C4—3324no—
534C2×C4—3324no—
535C2×C4—838no—
536C2×C4—834no—
537C2×C4—6648no—
538C2×C4—6648no—
539C2×C4—1662no—
540C2×C4—1662no—
541C2×C4—1662no—
542C2×C4—1662no—
543C2×C4—1662no—
544C2×C4—1662no—
545C2×C4—1662no—
546C2×C4—1662no—
547C2×C4—1662no—
548C2×C4—1662no—
549C2×C4—1664no—
550C2×C4—1662no—
551C2×C4—1664no—
552C2×C4C22×C4×D8314yes—
553C2×C4C22×C4×D8314yes—
554C2×C4C22×C4×D8312yes—
555C2×C4C22×C4×D8312yes—
556C2×C4#1458183812yes—
557C2×C4#1458183912yes—
558C22—22no—
559C22—24no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—3324no—
564C22—3324no—
565C22—3324no—
566C22—3324no—
567C22—1668no—
568C22—1668no—
569C22—1668no—
570C22C23×C4×D8311yescharacteristic
571C22H(5,𝔽2)×D8312yes—
572C22C2×#1458183811yescharacteristic
573C22C2×#1458183911yescharacteristic
574C22#1912853812yes—
575C4—22no—
576C4—24no—
577C4—24no—
578C4—44no—
579C4—44no—
580C4—3322no—
581C4—3322no—
582C4—3322no—
583C4—1668no—
584C4—1664no—
585C4#1912818114yes—
586C2—42no—
587C2—3328no—
588C2C2×#1912818111yescharacteristic
589C2#2492066912yes—
590C2#2492170111yescharacteristic
591C2#2492172511yescharacteristic
592C2#2492193212yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2314C2
socle, cosocleC23×C83C25
prime radicalC2×#19128306C2
hypercenter#2314D83
hypocenterC83#34418
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #24910616C2semidirect
3. #24910559C2semidirect
4. #24910527C2semidirect
5. #24910476C2semidirect
6. #24910438C2semidirect
7. #24910363C2semidirect
8. #24910252C2semidirect
9. #24910152C2semidirect
10. #24910136C2semidirect
11. #24909969C2semidirect
12. #24909947C2semidirect
13. C83×#2314C2semidirect
14. #19128306C4semidirect
76. #14581789C2×C4semidirect
77. #14581784C2×C4semidirect
78. #14581777C2×C4semidirect
79. #14581766C2×C4semidirect
80. #14581765C2×C4semidirect
82. C4×D332C2×C4semidirect
122. #117×C83C2×C4semidirect
307. #2314D83semidirect
310. C83#34418semidirect
385. #117C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191283062
#249106162
#249105592
#249105272
#249104762
#249104382
#249103632
#249102522
#249101522
#249101362
#249099692
#249099472
C83×#23142
#3441883