Finite Group Databasebeta

← #35414692

Subgroups — #35414692

Order 21248.

1162 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127663C211yesmaximal, characteristic
2#24909845C211yesmaximal, characteristic
3C2×#19127558C211yesmaximal, characteristic
4#24907665C212yesmaximal, complemented
5#24907661C212yesmaximal, complemented
6#24907655C211yesmaximal, characteristic, complemented
7#24907653C211yesmaximal, characteristic, complemented
8#24907651C211yesmaximal, characteristic, complemented
9#24907495C212yesmaximal, complemented
10#24907493C212yesmaximal, complemented
11C83×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D83C2211yescharacteristic
13C22×#13050573C2212yes—
14C2×#14581750C2211yescharacteristic
15C2×#14581750C2211yescharacteristic
16C22×C4⋊Dic83C2211yescharacteristic
17C22×(C2×D83)⋊C4—21no—
18C2×#14581726C2212yescomplemented
19C2×#14581725C2212yescomplemented
20C2×#14581723C2212yescomplemented
21#19127663C418yescomplemented
22C2×C22⋊(C83⋊C8)C2211yescharacteristic
23C2×(C2×D83)⋊C8C2212yes—
24#19127563—21no—
25#19127561C2211yescharacteristic
26#19127558—21no—
27#19127558—21no—
28#19127558C2212yes—
29#19127558C2212yes—
30#19127558C2211yescharacteristic
31#19127558C2211yescharacteristic
32#19127556—21no—
33C22⋊C4×Dic83—21no—
34#19127548—21no—
35C2×C83⋊#95—21no—
36#19127464C2212yescomplemented
37#19127463C2212yescomplemented
38#19127463C2212yescomplemented
39C22×C4⋊C4×C83C2211yescharacteristic
40C2×#106×C83C2212yescomplemented
41C2×C22⋊C8×C83C2211yescharacteristic
42#380×C83C2211yescharacteristic
43#340×C83C2212yescomplemented
44C24×D83D411yescharacteristic
45C23×Dic83—21no—
46C23×Dic83D411yescharacteristic
47C22×C4×D83—42no—
48C22×C4×D83—42no—
49C22×C4×D83C2×C414yes—
50C22×C4×D83D411yescharacteristic
51C22×C4×D83D411yescharacteristic
52C22×C4×D83Q814yes—
53C22×C4×D83C2311yescharacteristic
54C22×C4×D83C2311yescharacteristic
55C2×#13050573—42no—
56C2×#13050573—42no—
57C2×#13050573C2×C418yescomplemented
58C22×C83⋊C8C2312yes—
59#14581750—42no—
60#14581750—42no—
61C2×C4⋊Dic83—21no—
62C2×C4⋊Dic83—42no—
63C2×C4⋊Dic83C2312yes—
64C2×C4⋊Dic83C2311yescharacteristic
65C2×C4⋊Dic83C2311yescharacteristic
66C2×C22⋊Dic83—21no—
67C2×(C2×D83)⋊C4—21no—
68C2×(C2×D83)⋊C4—21no—
69C2×(C2×D83)⋊C4—21no—
70C2×(C2×D83)⋊C4—21no—
71C2×(C2×D83)⋊C4—22no—
72C2×(C2×D83)⋊C4—22no—
73C2×(C2×D83)⋊C4—44no—
74C2×C4×Dic83—21no—
75C2×C4×Dic83—21no—
76#14581726—42no—
77#14581725—42no—
78#14581723—42no—
79C22⋊(C83⋊C8)C2×C414yes—
80(C2×D83)⋊C8C2×C418yescomplemented
81C83⋊#95—22no—
82C83⋊#95—22no—
83C83⋊#95—21no—
84C83⋊#95—21no—
85C83⋊#95—21no—
86C23×C4×C83—21no—
87C23×C4×C83C2311yescharacteristic
88C22×C8×C83C2312yes—
89C2×C4⋊C4×C83—21no—
90C2×C4⋊C4×C83—42no—
91C2×C4⋊C4×C83C2312yes—
92C2×C4⋊C4×C83C2311yescharacteristic
93C2×C4⋊C4×C83C2311yescharacteristic
94C2×C22⋊C4×C83—21no—
95#106×C83—42no—
96C22⋊C8×C83C2×C414yes—
97#95×C83—21no—
98C23×D83—22no—
99C23×D83—22no—
100C23×D83—41no—
101C23×D83—41no—
102C23×D83—41no—
103C23×D83—41no—
104C23×D83C4⋊C414yes—
105C23×D83C2×D411yescharacteristic
106C23×D83C2×D411yescharacteristic
107C22×Dic83—22no—
108C22×Dic83—21no—
109C22×Dic83—21no—
110C22×Dic83—41no—
111C22×Dic83—41no—
112C22×Dic83—42no—
113C22×Dic83—42no—
114C22×Dic83—42no—
115C22×Dic83C4⋊C414yes—
116C22×Dic83C2×D411yescharacteristic
117C22×Dic83C2×D411yescharacteristic
118C2×C4×D83—22no—
119C2×C4×D83—22no—
120C2×C4×D83—22no—
121C2×C4×D83—22no—
122C2×C4×D83—48no—
123C2×C4×D83—48no—
124C2×C4×D83—41no—
125C2×C4×D83—41no—
126C2×C4×D83—41no—
127C2×C4×D83—41no—
128C2×C4×D83C4⋊C414yes—
129C2×C4×D83C4⋊C414yes—
130C2×C4×D83C22×C412yes—
131C2×C4×D83C22×C412yes—
132C2×C4×D83C2×Q812yes—
133C2×C4×D83C2×Q812yes—
134#13050573—48no—
135C2×C83⋊C8—22no—
136C2×C83⋊C8—22no—
137C2×C83⋊C8C22×C414yes—
138C4⋊Dic83—22no—
139C4⋊Dic83—22no—
140C4⋊Dic83—22no—
141C4⋊Dic83—22no—
142C4⋊Dic83—42no—
143C22⋊Dic83—42no—
144(C2×D83)⋊C4—42no—
145(C2×D83)⋊C4—42no—
146(C2×D83)⋊C4—44no—
147(C2×D83)⋊C4—44no—
148(C2×D83)⋊C4—42no—
149(C2×D83)⋊C4—42no—
150C4×Dic83—42no—
151C4×Dic83—42no—
152C24×C83C2×D411yescharacteristic
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—21no—
156C22×C4×C83—21no—
157C22×C4×C83—22no—
158C22×C4×C83—42no—
159C22×C4×C83—42no—
160C22×C4×C83—42no—
161C22×C4×C83C22×C412yes—
162C22×C4×C83C2×D411yescharacteristic
163C22×C4×C83C2×Q812yes—
164C22×C4×C83C2411yescharacteristic
165C2×C8×C83—22no—
166C2×C8×C83—22no—
167C2×C8×C83C22×C414yes—
168C4⋊C4×C83—22no—
169C4⋊C4×C83—22no—
170C4⋊C4×C83—22no—
171C4⋊C4×C83—22no—
172C4⋊C4×C83—42no—
173C22⋊C4×C83—42no—
174C23×C83—21no—
175C23×C83—21no—
176C23×C83—21no—
177C23×C83—21no—
178C23×C83C2×C4⋊C412yes—
179C23×C83AGL(1,ℤ/8)12yes—
180C23×C83#13712yes—
181C23×C83C22×D411yescharacteristic
182C22×D83—21no—
183C22×D83—21no—
184C22×D83—21no—
185C22×D83—21no—
186C22×D83—21no—
187C22×D83—21no—
188C22×D83—21no—
189C22×D83—21no—
190C22×D83—21no—
191C22×D83—21no—
192C22×D83—21no—
193C22×D83—21no—
194C22×D83—28no—
195C22×D83—44no—
196C22×D83—44no—
197C22×D83—44no—
198C22×D83—44no—
199C22×D83—44no—
200C22×D83—44no—
201C22×D83C2×C4⋊C412yes—
202C22×D83C2×C4⋊C412yes—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—24no—
206C2×C4×C83—24no—
207C2×C4×C83—42no—
208C2×C4×C83—41no—
209C2×C4×C83—41no—
210C2×C4×C83—42no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—42no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—84no—
217C2×C4×C83C2×C4⋊C412yes—
218C2×C4×C83C23×C411yescharacteristic
219C2×C4×C83C22×Q811yescharacteristic
220C2×C4×C83H(5,𝔽2)12yes—
221C2×C4×C83H(5,𝔽2)12yes—
222C2×Dic83—21no—
223C2×Dic83—21no—
224C2×Dic83—21no—
225C2×Dic83—21no—
226C2×Dic83—44no—
227C2×Dic83—41no—
228C2×Dic83—41no—
229C2×Dic83—42no—
230C2×Dic83—42no—
231C2×Dic83—41no—
232C2×Dic83—41no—
233C2×Dic83—41no—
234C2×Dic83—41no—
235C2×Dic83—42no—
236C2×Dic83—84no—
237C2×Dic83C2×C4⋊C412yes—
238C2×Dic83C2×C4⋊C412yes—
239C4×D83—24no—
240C4×D83—24no—
241C4×D83—24no—
242C4×D83—24no—
243C4×D83—44no—
244C4×D83—44no—
245C8×C83—24no—
246C83⋊C8—24no—
247C22×C83—22no—
248C22×C83—22no—
249C22×C83—22no—
250C22×C83—22no—
251C22×C83—22no—
252C22×C83—22no—
253C22×C83#42814yes—
254C22×C83C22×C4⋊C411yescharacteristic
255C22×C83#53411yescharacteristic
256C22×C83#53411yescharacteristic
257C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C83C2×#13711yescharacteristic
259C22×C83#57511yescharacteristic
260C22×C83#57511yescharacteristic
261C2×D83—22no—
262C2×D83—22no—
263C2×D83—22no—
264C2×D83—22no—
265C2×D83—22no—
266C2×D83—22no—
267C2×D83—416no—
268C2×D83—41no—
269C2×D83—41no—
270C2×D83—41no—
271C2×D83—41no—
272C2×D83—41no—
273C2×D83—41no—
274C2×D83—41no—
275C2×D83—41no—
276C4×C83—22no—
277C4×C83—82no—
278C4×C83—81no—
279C4×C83—81no—
280C4×C83#51812yes—
281C4×C83#54312yes—
282Dic83—22no—
283Dic83—22no—
284Dic83—81no—
285Dic83—81no—
286Dic83—82no—
287#21944—831nomaximal, Sylow
288C2×C83—24no—
289C2×C83#231511yescharacteristic
290C2×C83C2×#42811yescharacteristic
291C2×C83#291111yescharacteristic
292C2×C83#319411yescharacteristic
293C2×C83#319411yescharacteristic
294C2×C83#319611yescharacteristic
295C2×C83#319611yescharacteristic
296D83—42no—
297D83—42no—
298C2×#407—831no—
299#2280—831no—
300C2×#380—831no—
301#1854—832no—
302#1854—832no—
303#1853—831no—
304#1853D8311yescharacteristic, complemented
305#1852—831no—
306#1852—831no—
307#1809—832no—
308#1809—832no—
309C83#2194411yescharacteristic, complemented, Sylow
310C24×C4—831no—
311C22×M16—832no—
312C2×#117—831no—
313C2×#117—831no—
314C22×C4⋊C4—831no—
315C22×C4⋊C4C2×D8311yescharacteristic
316C22×C22⋊C4—1661no—
317C2×#107—832no—
318C2×#107—832no—
319C2×#106—832no—
320C2×#106C2×D8312yescomplemented
321#407—838no—
322C2×C22⋊C8—831no—
323C2×C22⋊C8—832no—
324C2×C22⋊C8C2×D8311yescharacteristic
325#381—1661no—
326#380—831no—
327#380—832no—
328#380—831no—
329#380—831no—
330#380—832no—
331#380—1661no—
332#380—1661no—
333#380C2×D8311yescharacteristic
334#379—1661no—
335C4×C22⋊C4—1661no—
336C4×C22⋊C4—1661no—
337C2×#95—1661no—
338#340—832no—
339#340—832no—
340#340—832no—
341#340C2×D8312yescomplemented
342C25—831no—
343C23×C4—21no—
344C23×C4—3322no—
345C23×C4—3322no—
346C23×C4—831no—
347C23×C4—834no—
348C23×C4—831no—
349C23×C4—831no—
350C23×C4—834no—
351C23×C4—831no—
352C23×C4—831no—
353C23×C4—1661no—
354C23×C4C22×D8311yescharacteristic
355C2×M16—3322no—
356C2×M16—3322no—
357C2×M16—838no—
358C22×C8—832no—
359C22×C8C22×D8312yes—
360#117—3322no—
361#117—3322no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—3322no—
365C2×C4⋊C4—832no—
366C2×C4⋊C4—831no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—1661no—
369C2×C4⋊C4C22×D8312yes—
370C2×C4⋊C4C22×D8311yescharacteristic
371C2×C4⋊C4C22×D8311yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—3324no—
374C2×C22⋊C4—1662no—
375C2×C22⋊C4—1662no—
376C2×C22⋊C4—1661no—
377C2×C22⋊C4—1661no—
378C2×C22⋊C4—1661no—
379C2×C22⋊C4—1661no—
380C2×C22⋊C4—1661no—
381C2×C42—1661no—
382C2×C42—1661no—
383#107—3322no—
384#107—3322no—
385#106—42no—
386#106—3322no—
387C22⋊C8—834no—
388C22⋊C8—838no—
389C22⋊C8C4×D8314yes—
390#95—21no—
391#95—1662no—
392#95—1662no—
393#95—1661no—
394#95—1661no—
395#95—1661no—
396C24—3321no—
397C24—3321no—
398C24—3321no—
399C24—3321no—
400C24—834no—
401C24—831no—
402C24—831no—
403C24—1662no—
404C24—1662no—
405C24D4×D8311yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—3321no—
415C22×C4—3321no—
416C22×C4—3321no—
417C22×C4—3321no—
418C22×C4—3328no—
419C22×C4—3328no—
420C22×C4—3321no—
421C22×C4—3321no—
422C22×C4—3322no—
423C22×C4—3322no—
424C22×C4—3322no—
425C22×C4—834no—
426C22×C4—834no—
427C22×C4—834no—
428C22×C4—831no—
429C22×C4—831no—
430C22×C4—832no—
431C22×C4—832no—
432C22×C4—832no—
433C22×C4—832no—
434C22×C4—1661no—
435C22×C4—1661no—
436C22×C4—1662no—
437C22×C4—1662no—
438C22×C4—1662no—
439C22×C4—1662no—
440C22×C4—1662no—
441C22×C4C2×C4×D8312yes—
442C22×C4D4×D8311yescharacteristic
443C22×C4Q8×D8312yes—
444C22×C4C23×D8311yescharacteristic
445M16—3328no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—834no—
449C2×C8—1662no—
450C2×C8—1662no—
451C2×C8C2×C4×D8314yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—3322no—
458C4⋊C4—1662no—
459C4⋊C4—1662no—
460C4⋊C4—1662no—
461C4⋊C4—1662no—
462C22⋊C4—42no—
463C22⋊C4—3324no—
464C22⋊C4—3324no—
465C22⋊C4—3322no—
466C22⋊C4—3322no—
467C22⋊C4—3322no—
468C22⋊C4—3322no—
469C22⋊C4—3322no—
470C42—3322no—
471C42—3322no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—3324no—
477C23—3324no—
478C23—3324no—
479C23—3324no—
480C23—3324no—
481C23—3324no—
482C23—832no—
483C23—832no—
484C23—1661no—
485C23—1661no—
486C23—1661no—
487C23—1661no—
488C23—1661no—
489C23—1661no—
490C23—1661no—
491C23—1661no—
492C23—1668no—
493C23—1661no—
494C23—1661no—
495C23—1661no—
496C23—1661no—
497C23C4⋊C4×D8312yes—
498C23#1458181912yes—
499C23#1458182112yes—
500C23C2×D4×D8311yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—3324no—
516C2×C4—3324no—
517C2×C4—3322no—
518C2×C4—3321no—
519C2×C4—3321no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—3322no—
523C2×C4—3321no—
524C2×C4—3321no—
525C2×C4—3322no—
526C2×C4—3324no—
527C2×C4—832no—
528C2×C4—832no—
529C2×C4—6644no—
530C2×C4—1661no—
531C2×C4—1661no—
532C2×C4—1661no—
533C2×C4—1661no—
534C2×C4—1664no—
535C2×C4—1664no—
536C2×C4—1664no—
537C2×C4—1664no—
538C2×C4C4⋊C4×D8312yes—
539C2×C4C22×C4×D8311yescharacteristic
540C2×C4C2×Q8×D8311yescharacteristic
541C2×C4#1458183712yes—
542C2×C4#1458183812yes—
543C8—24no—
544C8—1664no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—33216no—
552C22—3321no—
553C22—3321no—
554C22—3321no—
555C22—3321no—
556C22—3321no—
557C22—3321no—
558C22—3321no—
559C22—3321no—
560C22—1662no—
561C22—1662no—
562C22—1662no—
563C22—1662no—
564C22—1662no—
565C22—1662no—
566C22#1912774014yes—
567C22C2×C4⋊C4×D8311yescharacteristic
568C22#1912826111yescharacteristic
569C22#1912826211yescharacteristic
570C22C2×#1458181911yescharacteristic
571C22C2×#1458182111yescharacteristic
572C22#1912849711yescharacteristic
573C22#1912850011yescharacteristic
574C4—22no—
575C4—82no—
576C4—81no—
577C4—81no—
578C4—6641no—
579C4—6642no—
580C4—6641no—
581C4—1662no—
582C4—1662no—
583C4#1912817312yes—
584C4#1912832612yes—
585C2—24no—
586C2—3322no—
587C2—3322no—
588C2#2491007411yescharacteristic
589C2C2×#1912774011yescharacteristic
590C2#2491605011yescharacteristic
591C2#2491828011yescharacteristic
592C2#2491828211yescharacteristic
593C2#2491829011yescharacteristic
594C2#2491829211yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1853C2
socleC23×C83C22×D4
prime radicalC24×D83D4
cosocleC22×C4×C83C24
hypercenter#1853D83
hypocenterC83#21944
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #24907665C2semidirect
5. #24907661C2semidirect
6. #24907655C2semidirect
7. #24907653C2semidirect
8. #24907651C2semidirect
9. #24907495C2semidirect
10. #24907493C2semidirect
11. C83×#1853C2semidirect
18. C2×#14581726C22semidirect
19. C2×#14581725C22semidirect
20. C2×#14581723C22semidirect
21. #19127663C4semidirect
36. #19127464C22semidirect
37. #19127463C22semidirect
38. #19127463C22semidirect
40. C2×#106×C83C22semidirect
43. #340×C83C22semidirect
57. C2×#13050573C2×C4semidirect
80. (C2×D83)⋊C8C2×C4semidirect
304. #1853D83semidirect
309. C83#21944semidirect
320. C2×#106C2×D83semidirect
341. #340C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191276632
#249098452
C2×#191275582
#249076652
#249076612
#249076552
#249076532
#249076512
#249074952
#249074932
C83×#18532
#2194483