Finite Group Databasebeta

← #35446265

Subgroups — #35446265

Order 21248.

1132 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127928C212yesmaximal, complemented
2C2×#19127898C211yesmaximal, characteristic, complemented
3C22×C4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
4C4×#14581771C211yesmaximal, characteristic, complemented
5#24908837C212yesmaximal, complemented
6#24908780C212yesmaximal, complemented
7#24908391C211yesmaximal, characteristic, complemented
8#24908224C211yesmaximal, characteristic, complemented
9#24907892C212yesmaximal, complemented
10#24907465C211yesmaximal, characteristic, complemented
11C83×#2856C211yesmaximal, characteristic
12C22×#13050578C2212yes—
13C23×C83⋊C8C2212yes—
14C2×#14581771C2211yescharacteristic
15C2×C4×Dic166C2211yescharacteristic
16#19127928—28no—
17#19127898—24no—
18C2×#14581723C2212yes—
19C2×C4⋊(C83⋊C8)—24no—
20C2×C4⋊(C83⋊C8)C2212yes—
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×#14581715C2212yes—
24C2×#14581713C2212yes—
25C2×#14581713C2212yes—
26C2×#14581713C2212yes—
27#19127607C2211yescharacteristic
28#19127599C2211yescharacteristic
29#19127585—21no—
30#19127574—21no—
31#19127561C2211yescharacteristic
32C4×Dic83⋊C4—21no—
33C4×C4⋊Dic83C2211yescharacteristic
34C4×C22⋊Dic83—21no—
35#19127463C2212yes—
36#19127463C2212yes—
37C2×#127×C83C2211yescharacteristic
38C2×#121×C83—21no—
39C2×#121×C83—21no—
40C2×#121×C83C2212yes—
41C22×C42×C83C2211yescharacteristic
42#390×C83—21no—
43#390×C83—21no—
44#390×C83C2212yes—
45#384×C83C2211yescharacteristic
46C22×Dic166C2311yescharacteristic
47C2×#13050578—24no—
48C2×#13050578—24no—
49C2×#13050578—42no—
50C22×C83⋊C8—24no—
51C22×C83⋊C8—24no—
52C22×C83⋊C8—42no—
53C22×C83⋊C8C2312yes—
54C22×C83⋊C8C2312yes—
55#14581771—22no—
56#14581771D414yescomplemented
57C4×Dic166—22no—
58C4×Dic166—22no—
59C2×C4⋊Dic83C2311yescharacteristic
60C2×C4⋊Dic83C2311yescharacteristic
61C2×C4⋊Dic83C2311yescharacteristic
62C2×Dic83⋊C4—21no—
63C2×Dic83⋊C4—21no—
64C2×Dic83⋊C4—21no—
65C2×C22⋊Dic83—21no—
66C2×C22⋊Dic83—21no—
67C2×C4×Dic83—21no—
68C2×C4×Dic83—21no—
69#14581723—24no—
70C4⋊(C83⋊C8)—24no—
71C4⋊(C83⋊C8)—22no—
72C4⋊(C83⋊C8)D414yescomplemented
73#14581715—24no—
74#14581713—24no—
75#14581713—24no—
76#14581713—42no—
77C83⋊#95—21no—
78C83⋊#95—21no—
79C83⋊#95—21no—
80C83⋊#95—21no—
81C83⋊#95—21no—
82C22×D4×C83—22no—
83C22×D4×C83—21no—
84C22×D4×C83—21no—
85C22×D4×C83C2312yes—
86C23×C4×C83D411yescharacteristic
87C23×C4×C83D411yescharacteristic
88C23×C4×C83C2311yescharacteristic
89#127×C83—22no—
90#127×C83—22no—
91#121×C83—24no—
92#121×C83—24no—
93#121×C83—44no—
94#121×C83—44no—
95C2×C4⋊C4×C83—21no—
96C2×C4⋊C4×C83—21no—
97C2×C4⋊C4×C83C2312yes—
98C2×C22⋊C4×C83—22no—
99C2×C22⋊C4×C83—22no—
100C2×C22⋊C4×C83—22no—
101C2×C42×C83—22no—
102C2×C42×C83—22no—
103C2×C42×C83D411yescharacteristic
104C2×C42×C83D411yescharacteristic
105C2×C42×C83C2311yescharacteristic
106C2×C42×C83C2311yescharacteristic
107C2×Dic166—21no—
108C2×Dic166—21no—
109C2×Dic166—21no—
110C2×Dic166—21no—
111C2×Dic166—21no—
112C2×Dic166C2×D412yes—
113C22×Dic83—21no—
114C22×Dic83—21no—
115C22×Dic83—21no—
116C22×Dic83—21no—
117#13050578—48no—
118C2×C83⋊C8—22no—
119C2×C83⋊C8—22no—
120C2×C83⋊C8—22no—
121C2×C83⋊C8—22no—
122C2×C83⋊C8—22no—
123C2×C83⋊C8—48no—
124C2×C83⋊C8C2×D414yes—
125C4⋊Dic83—21no—
126C4⋊Dic83—21no—
127C4⋊Dic83—21no—
128C4⋊Dic83—21no—
129C4⋊Dic83—21no—
130C4⋊Dic83C2×D412yes—
131Dic83⋊C4—24no—
132Dic83⋊C4—42no—
133Dic83⋊C4—42no—
134C22⋊Dic83—24no—
135C22⋊Dic83—42no—
136C4×Dic83—42no—
137C4×Dic83—42no—
138C24×C83—22no—
139C24×C83C2×D411yescharacteristic
140C2×D4×C83—22no—
141C2×D4×C83—22no—
142C2×D4×C83—22no—
143C2×D4×C83—22no—
144C2×D4×C83—22no—
145C2×D4×C83—22no—
146C2×D4×C83—42no—
147C2×D4×C83—42no—
148C2×D4×C83—44no—
149C2×D4×C83—44no—
150C2×D4×C83—44no—
151C2×D4×C83—42no—
152C2×D4×C83—42no—
153C2×D4×C83—42no—
154C2×D4×C83—42no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—21no—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—41no—
165C22×C4×C83SD1614yescomplemented
166C22×C4×C83C2×D412yes—
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83#4112yes—
173C22×C4×C83C2411yescharacteristic
174C4⋊C4×C83—22no—
175C4⋊C4×C83—22no—
176C4⋊C4×C83—42no—
177C4⋊C4×C83—42no—
178C22⋊C4×C83—44no—
179C22⋊C4×C83—44no—
180C22⋊C4×C83—44no—
181C42×C83—22no—
182C42×C83—22no—
183C42×C83—21no—
184C42×C83—21no—
185C42×C83C2×D412yes—
186C42×C83#4112yes—
187C23×C83—21no—
188C23×C83—21no—
189C23×C83—21no—
190C23×C83—21no—
191C23×C83—42no—
192C23×C83—42no—
193C23×C83—42no—
194C23×C83—42no—
195C23×C83—42no—
196C23×C83—42no—
197C23×C83—42no—
198C23×C83#12112yes—
199C23×C83C2×SD1612yes—
200C23×C83AGL(1,ℤ/8)12yes—
201C23×C83C22×D411yescharacteristic
202D4×C83—42no—
203D4×C83—42no—
204D4×C83—42no—
205D4×C83—42no—
206D4×C83—84no—
207D4×C83—88no—
208D4×C83—84no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—28no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—44no—
224C2×C4×C83—42no—
225C2×C4×C83—42no—
226C2×C4×C83—42no—
227C2×C4×C83#12112yes—
228C2×C4×C83#12112yes—
229C2×C4×C83#12112yes—
230C2×C4×C83C2×SD1612yes—
231C2×C4×C83C2×SD1612yes—
232C2×C4×C83#13512yes—
233C2×C4×C83C22×D411yescharacteristic
234C2×C4×C83C22×D411yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C2×C4×C83H(5,𝔽2)11yescharacteristic
238C2×C4×C832−1+411yescharacteristic
239C2×Dic83—22no—
240C2×Dic83—22no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251Dic166—22no—
252Dic166—22no—
253Dic166—41no—
254Dic166—41no—
255C83⋊C8—24no—
256C83⋊C8—42no—
257C22×C83—22no—
258C22×C83—22no—
259C22×C83—22no—
260C22×C83—22no—
261C22×C83—22no—
262C22×C83—22no—
263C22×C83—82no—
264C22×C83—82no—
265C22×C83—82no—
266C22×C83—82no—
267C22×C83—82no—
268C22×C83—82no—
269C22×C83—82no—
270C22×C83#46014yescomplemented
271C22×C83C2×#12111yescharacteristic
272C22×C83C2×#12111yescharacteristic
273C22×C83#53711yescharacteristic
274C22×C83C22×SD1611yescharacteristic
275C22×C83C2×#13511yescharacteristic
276C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C83#57511yescharacteristic
278C4×C83—22no—
279C4×C83—22no—
280C4×C83—22no—
281C4×C83—41no—
282C4×C83—82no—
283C4×C83#46514yescomplemented
284C4×C83D4211yescharacteristic
285C4×C83#54711yescharacteristic
286C4×C83#54811yescharacteristic
287C4×C83#55311yescharacteristic
288Dic83—42no—
289Dic83—42no—
290Dic83—81no—
291Dic83—81no—
292#24753—831nomaximal, Sylow
293C2×C83—24no—
294C2×C83—162no—
295C2×C83#241611yescharacteristic
296C2×C83C2×#46011yescharacteristic
297C2×C83#303511yescharacteristic
298C2×C83C2×#46511yescharacteristic
299C2×C83#305511yescharacteristic
300C2×C83#324611yescharacteristic
301C2×C83#324711yescharacteristic
302C2×#465—832no—
303C2×#460—831no—
304C22×C4⋊C8—831no—
305#2856D8311yescharacteristic
306C4×C22⋊Q8—831no—
307#2065—832no—
308#2055—832no—
309#1982—831no—
310#1948—831no—
311#1893—832no—
312#1803—831no—
313C83#2475311yescharacteristic, complemented, Sylow
314C22×SD16—832no—
315C23×C8—832no—
316C2×#127C2×D8311yescharacteristic
317C2×C22⋊Q8—831no—
318C2×#121—21no—
319C2×#121—21no—
320C2×#121C2×D8312yes—
321C2×C4×Q8—831no—
322C22×C42C2×D8311yescharacteristic
323#465—1668no—
324#460—1664no—
325C2×#106—832no—
326C2×C4⋊C8—831no—
327C2×C4⋊C8—832no—
328C2×C4⋊C8—831no—
329C2×C4⋊C8—1664no—
330C2×#103—832no—
331C2×#102—832no—
332C2×#102—832no—
333C2×#102—832no—
334#391—831no—
335#390—21no—
336#390—21no—
337#390C2×D8312yes—
338#389—831no—
339#387—1661no—
340#385—1661no—
341#384C2×D8311yescharacteristic
342#380—831no—
343C4×C4⋊C4—831no—
344C4×C4⋊C4—1661no—
345C4×C22⋊C4—1661no—
346#340—832no—
347#340—832no—
348C22×Q8—831no—
349C22×D4—22no—
350C22×D4—21no—
351C22×D4—21no—
352C22×D4C22×D8312yes—
353C23×C4C22⋊D8311yescharacteristic
354C23×C4C22⋊D8311yescharacteristic
355C23×C4C22×D8311yescharacteristic
356C2×SD16—3322no—
357C2×SD16—1664no—
358C2×SD16—1664no—
359C22×C8—3322no—
360C22×C8—832no—
361C22×C8—832no—
362C22×C8—1664no—
363C22×C8—1664no—
364#127—22no—
365#127—22no—
366C22⋊Q8—834no—
367C22⋊Q8—1662no—
368#121—24no—
369#121—24no—
370#121—44no—
371#121—44no—
372C4×Q8—1662no—
373C4×Q8—1662no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—831no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4C22×D8312yes—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—1661no—
387C2×C22⋊C4—1661no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—1661no—
391C2×C42—1661no—
392C2×C42C22⋊D8311yescharacteristic
393C2×C42C22⋊D8311yescharacteristic
394C2×C42C22×D8311yescharacteristic
395C2×C42C22×D8311yescharacteristic
396#106—1664no—
397C4⋊C8—834no—
398C4⋊C8—1664no—
399C4⋊C8—1662no—
400#103—1664no—
401#102—3322no—
402#102—1664no—
403#102—1664no—
404#95—1661no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409C24—22no—
410C24C2×C22⋊D8311yescharacteristic
411C2×Q8—832no—
412C2×Q8—1661no—
413C2×Q8—1661no—
414C2×Q8—1661no—
415C2×Q8—1661no—
416C2×Q8—1661no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—1661no—
443C22×C4—1661no—
444C22×C4—1661no—
445C22×C4—1661no—
446C22×C4#1305057814yescomplemented
447C22×C4D4×D8312yes—
448C22×C4C2×C22⋊D8311yescharacteristic
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4#1305058912yes—
454C22×C4C23×D8311yescharacteristic
455SD16—3328no—
456C2×C8—3328no—
457C2×C8—834no—
458C2×C8—1662no—
459C2×C8—1662no—
460C2×C8—1662no—
461C2×C8—1662no—
462C2×C8—1662no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—3322no—
468C4⋊C4—3322no—
469C4⋊C4—832no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C4⋊C4—1664no—
475C4⋊C4—1661no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—3322no—
480C22⋊C4—1664no—
481C42—22no—
482C42—22no—
483C42—21no—
484C42—21no—
485C42—3322no—
486C42—3322no—
487C42D4×D8312yes—
488C42#1305058912yes—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23#1458176312yes—
501C23C2×#1305057812yes—
502C23#1458181812yes—
503C23C22×C22⋊D8311yescharacteristic
504Q8—3321no—
505Q8—3321no—
506Q8—1662no—
507Q8—1662no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—84no—
513D4—88no—
514D4—84no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—3321no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—1662no—
544C2×C4—1662no—
545C2×C4#1458176312yes—
546C2×C4#1458176412yes—
547C2×C4#1458176412yes—
548C2×C4C2×#1305057812yes—
549C2×C4C2×#1305057812yes—
550C2×C4#1458181212yes—
551C2×C4C2×D4×D8311yescharacteristic
552C2×C4C2×D4×D8311yescharacteristic
553C2×C4C2×#1305058911yescharacteristic
554C2×C4C2×#1305058911yescharacteristic
555C2×C4#1458183811yescharacteristic
556C2×C4#1458183911yescharacteristic
557C8—3322no—
558C8—1664no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22#1912789814yescomplemented
573C22C2×#1458176311yescharacteristic
574C22C2×#1458176411yescharacteristic
575C22#1912827911yescharacteristic
576C22C22×#1305057811yescharacteristic
577C22C2×#1458181211yescharacteristic
578C22C2×#1458181811yescharacteristic
579C22#1912849911yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—41no—
584C4—82no—
585C4—3322no—
586C4—3322no—
587C4—6641no—
588C4—6641no—
589C4#1912792814yescomplemented
590C4#1912833511yescharacteristic
591C4#1912835111yescharacteristic
592C4#1912835711yescharacteristic
593C4#1912840311yescharacteristic
594C2—24no—
595C2—162no—
596C2#2491090411yescharacteristic
597C2C2×#1912789811yescharacteristic
598C2#2491691211yescharacteristic
599C2C2×#1912792811yescharacteristic
600C2#2491707811yescharacteristic
601C2#2491877911yescharacteristic
602C2#2491878311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
Fitting, prime radicalC83×#2856C2
socleC23×C83C22×D4
hypercenter#2856D83
hypocenterC83#24753
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2856

Complements

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

NH ≅ G/Ntype
1. C2×#19127928C2semidirect
2. C2×#19127898C2semidirect
3. C22×C4⋊(C83⋊C8)C2semidirect
4. C4×#14581771C2semidirect
5. #24908837C2semidirect
6. #24908780C2semidirect
7. #24908391C2semidirect
8. #24908224C2semidirect
9. #24907892C2semidirect
10. #24907465C2semidirect
56. #14581771D4semidirect
72. C4⋊(C83⋊C8)D4semidirect
165. C22×C4×C83SD16semidirect
270. C22×C83#460semidirect
283. C4×C83#465semidirect
313. C83#24753semidirect
446. C22×C4#13050578semidirect
572. C22#19127898semidirect
589. C4#19127928semidirect

Maximal subgroups

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

subgroupindex
C2×#191279282
C2×#191278982
C22×C4⋊(C83⋊C8)2
C4×#145817712
#249088372
#249087802
#249083912
#249082242
#249078922
#249074652
C83×#28562
#2475383