Finite Group Databasebeta

← #35406108

Subgroups — #35406108

Order 21248.

1986 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128212C211yesmaximal, characteristic
2C2×#19127703C212yesmaximal, complemented
3C2×#19127561C211yesmaximal, characteristic, complemented
4#24907901C212yesmaximal, complemented
5#24907896C212yesmaximal, complemented
6#24907891C212yesmaximal, complemented
7#24907886C212yesmaximal, complemented
8#24907183C212yesmaximal, complemented
9C23×M16×C83C211yesmaximal, characteristic, complemented
10C22×#13050590C2211yescharacteristic
11#19128212—24no—
12#19128212C418yescomplemented
13C2×#14581775—21no—
14C2×#14581771C2211yescharacteristic
15C2×#14581771C2211yescharacteristic
16C2×#14581766C2211yescharacteristic
17C2×#14581766C2211yescharacteristic
18C2×#14581761—21no—
19C2×C4×C22⋊D83—21no—
20C2×C4×C22⋊D83—21no—
21C2×#14581751C2211yescharacteristic
22C22×C4⋊Dic83C2212yes—
23C22×C22⋊Dic83—21no—
24#19127703—28no—
25C2×#14581717C2214yescomplemented
26C2×#14581714C2214yescomplemented
27#19127561—21no—
28#19127561—21no—
29#19127561C2212yes—
30#19127561C2212yes—
31C2×C83⋊#95—21no—
32#19127464C2214yescomplemented
33#19127464C2214yescomplemented
34C24×C4×C83C2211yescharacteristic
35C22×M16×C83—24no—
36C22×M16×C83—28no—
37C22×M16×C83C2212yes—
38C22×M16×C83C2212yes—
39C23×C8×C83C2212yes—
40C2×#13050590—22no—
41C2×#13050590—24no—
42C2×#13050590—24no—
43C2×#13050590C2×C414yes—
44C22×Dic166C2311yescharacteristic
45C22×D332C2311yescharacteristic
46C22×C22⋊D83—21no—
47C22×C22⋊D83—21no—
48C22×C22⋊D83—21no—
49C23×Dic83—22no—
50C22×C4×D83—21no—
51#14581775—28no—
52#14581771—22no—
53#14581771—22no—
54#14581771C2×C414yescomplemented
55#14581771C2×C414yescomplemented
56#14581766—22no—
57#14581766—22no—
58#14581766C2×C414yescomplemented
59#14581766C2×C414yescomplemented
60#14581761—28no—
61C4×C22⋊D83—28no—
62C4×C22⋊D83—28no—
63#14581751—22no—
64#14581751C2×C414yes—
65C2×C4⋊Dic83—24no—
66C2×C4⋊Dic83—22no—
67C2×C4⋊Dic83—24no—
68C2×C4⋊Dic83C2314yes—
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×Dic83⋊C4—21no—
72C2×Dic83⋊C4—21no—
73C2×C22⋊Dic83—21no—
74C2×C22⋊Dic83—21no—
75C2×C22⋊Dic83—21no—
76C2×C22⋊Dic83—22no—
77C2×C22⋊Dic83—22no—
78C2×C22⋊Dic83—44no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×C4×Dic83—21no—
82#14581717—28no—
83#14581714—28no—
84C83⋊#95—22no—
85C83⋊#95—22no—
86C25×C83D411yescharacteristic
87C23×C4×C83—24no—
88C23×C4×C83—24no—
89C23×C4×C83C2×C414yes—
90C23×C4×C83D411yescharacteristic
91C23×C4×C83D411yescharacteristic
92C23×C4×C83D411yescharacteristic
93C23×C4×C83D414yes—
94C23×C4×C83C2311yescharacteristic
95C23×C4×C83C2311yescharacteristic
96C23×C4×C83C2311yescharacteristic
97C2×M16×C83—232no—
98C2×M16×C83—24no—
99C2×M16×C83—24no—
100C2×M16×C83—24no—
101C2×M16×C83—28no—
102C2×M16×C83D418yescomplemented
103C22×C8×C83—24no—
104C22×C8×C83—28no—
105C22×C8×C83C2314yes—
106C23×D83—21no—
107#13050590—216no—
108#13050590—44no—
109#13050590—44no—
110C2×Dic166—22no—
111C2×Dic166—22no—
112C2×Dic166—21no—
113C2×Dic166C22×C412yes—
114C2×D332—22no—
115C2×D332—22no—
116C2×D332—21no—
117C2×D332C22×C412yes—
118C2×C22⋊D83—24no—
119C2×C22⋊D83—24no—
120C2×C22⋊D83—24no—
121C2×C22⋊D83—44no—
122C2×C22⋊D83—44no—
123C2×C22⋊D83—42no—
124C2×C22⋊D83—42no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—24no—
129C22×Dic83—44no—
130C22×Dic83—44no—
131C22×Dic83—44no—
132C2×C4×D83—24no—
133C2×C4×D83—44no—
134C4⋊Dic83—21no—
135C4⋊Dic83—21no—
136C4⋊Dic83—24no—
137C4⋊Dic83—24no—
138C4⋊Dic83—44no—
139C4⋊Dic83C22×C412yes—
140C4⋊Dic83C22×C412yes—
141Dic83⋊C4—24no—
142Dic83⋊C4—24no—
143C22⋊Dic83—24no—
144C22⋊Dic83—24no—
145C22⋊Dic83—24no—
146C22⋊Dic83—44no—
147C22⋊Dic83—44no—
148(C2×D83)⋊C4—24no—
149(C2×D83)⋊C4—24no—
150C4×Dic83—24no—
151C24×C83—24no—
152C24×C83—22no—
153C24×C83—22no—
154C24×C83—42no—
155C24×C83C22⋊C414yes—
156C24×C83C2×D411yescharacteristic
157C24×C83C2×D411yescharacteristic
158C24×C83C2×D411yescharacteristic
159C22×C4×C83—24no—
160C22×C4×C83—216no—
161C22×C4×C83—28no—
162C22×C4×C83—24no—
163C22×C4×C83—24no—
164C22×C4×C83—28no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83C22⋊C414yes—
170C22×C4×C83C22⋊C414yes—
171C22×C4×C83C22⋊C414yes—
172C22×C4×C83C22×C412yes—
173C22×C4×C83C22×C412yes—
174C22×C4×C83C22×C412yes—
175C22×C4×C83C2×D412yes—
176C22×C4×C83C2×D412yes—
177C22×C4×C83C2×D412yes—
178C22×C4×C83C2×D414yes—
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83#4114yes—
183C22×C4×C83C2411yescharacteristic
184M16×C83—216no—
185M16×C83—216no—
186C2×C8×C83—216no—
187C2×C8×C83—24no—
188C2×C8×C83—24no—
189C2×C8×C83C2×D418yes—
190C23×C83—22no—
191C23×C83—21no—
192C23×C83—21no—
193C23×C83—22no—
194C23×C83—22no—
195C23×C83—22no—
196C23×C83—21no—
197C23×C83—21no—
198C23×C83—22no—
199C23×C83—28no—
200C23×C83—28no—
201C23×C83—44no—
202C23×C83—48no—
203C23×C83—48no—
204C23×C83C2×C22⋊C412yes—
205C23×C83C2×C22⋊C412yes—
206C23×C83C2×C22⋊C412yes—
207C23×C83#12114yes—
208C23×C83AGL(1,ℤ/8)12yes—
209C23×C83#13712yes—
210C23×C83C22×D411yescharacteristic
211C22×D83—22no—
212C22×D83—42no—
213C22×D83—41no—
214C22×D83—41no—
215C22×D83—42no—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
218C2×C4×C83—28no—
219C2×C4×C83—24no—
220C2×C4×C83—24no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—28no—
226C2×C4×C83—28no—
227C2×C4×C83C2×C22⋊C412yes—
228C2×C4×C83C2×C22⋊C412yes—
229C2×C4×C83C2×C22⋊C412yes—
230C2×C4×C83C4×D418yes—
231C2×C4×C83#12014yes—
232C2×C4×C83#12114yes—
233C2×C4×C83#12314yes—
234C2×C4×C83C23×C411yescharacteristic
235C2×C4×C83C22×D412yes—
236C2×C4×C83C22×D411yescharacteristic
237C2×C4×C83C2×#4112yes—
238C22⋊D83—48no—
239C22⋊D83—48no—
240C22⋊D83—48no—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—22no—
244C2×Dic83—42no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—42no—
248C2×Dic83—44no—
249C2×Dic83—44no—
250C2×Dic83—44no—
251C2×Dic83—88no—
252D332—24no—
253D332—41no—
254D332—41no—
255C4×D83—48no—
256Dic166—24no—
257Dic166—41no—
258Dic166—41no—
259C8×C83—28no—
260C22×C83—22no—
261C22×C83—22no—
262C22×C83—22no—
263C22×C83—22no—
264C22×C83—24no—
265C22×C83—24no—
266C22×C83—24no—
267C22×C83—24no—
268C22×C83—22no—
269C22×C83—22no—
270C22×C83—22no—
271C22×C83—416no—
272C22×C83—42no—
273C22×C83—42no—
274C22×C83#38614yes—
275C22×C83#41714yescomplemented
276C22×C83C22×C22⋊C411yescharacteristic
277C22×C83C2×#12112yes—
278C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C83C2×#13711yescharacteristic
280C22×C83#57512yes—
281C2×D83—42no—
282C2×D83—42no—
283C2×D83—42no—
284C2×D83—82no—
285C2×D83—82no—
286C4×C83—22no—
287C4×C83—22no—
288C4×C83#38614yes—
289C4×C83C2×C4×D412yes—
290C4×C83C2×#12011yescharacteristic
291C4×C83C2×#12311yescharacteristic
292Dic83—42no—
293Dic83—42no—
294Dic83—42no—
295Dic83—84no—
296#20971—831nomaximal, Sylow
297C2×C83—24no—
298C2×C83—24no—
299C2×C83—42no—
300C2×C83C2×#38611yescharacteristic
301C2×C83C2×#41711yescharacteristic
302C2×C83#289611yescharacteristic
303C2×C83#305512yes—
304C2×C83#305612yes—
305D83—82no—
306C23×M16D8311yescharacteristic, complemented
307C2×#525—831no—
308C2×#417—832no—
309C2×#380—831no—
310#1895—832no—
311#1894—832no—
312#1893—832no—
313#1892—832no—
314#1736—832no—
315C83#2097111yescharacteristic, complemented, Sylow
316C22×#41—831no—
317C24×C4C2×D8311yescharacteristic
318C22×M16—24no—
319C22×M16—28no—
320C22×M16C2×D8312yes—
321C22×M16C2×D8312yes—
322C23×C8C2×D8312yes—
323#525—838no—
324#525—1664no—
325C2×#123—1661no—
326C2×C22⋊Q8—831no—
327C2×C22⋊Q8—831no—
328C2×#121—831no—
329C2×#121—831no—
330C2×#120—1661no—
331C2×C4×D4—1661no—
332C2×C4×D4—1661no—
333C2×#117—831no—
334C22×C4⋊C4—832no—
335C22×C22⋊C4—1661no—
336#417—1668no—
337C2×#103—834no—
338C2×#102—834no—
339#380—832no—
340#380—832no—
341#380—1661no—
342#380—1661no—
343C2×#95—1661no—
344#340—834no—
345#340—834no—
346C25D33211yescharacteristic
347C2×#41—834no—
348C2×#41—1664no—
349C2×#41—1664no—
350C2×#41—1662no—
351C22×Q8—831no—
352C22×D4—831no—
353C22×D4—1661no—
354C22×D4—1661no—
355C22×D4—1661no—
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—1662no—
359C23×C4—1661no—
360C23×C4C4×D8314yes—
361C23×C4D33211yescharacteristic
362C23×C4D33211yescharacteristic
363C23×C4D33211yescharacteristic
364C23×C4C22⋊D8314yes—
365C23×C4C22×D8311yescharacteristic
366C23×C4C22×D8311yescharacteristic
367C23×C4C22×D8311yescharacteristic
368C2×M16—232no—
369C2×M16—24no—
370C2×M16—24no—
371C2×M16—24no—
372C2×M16—28no—
373C2×M16C22⋊D8318yescomplemented
374C22×C8—24no—
375C22×C8—28no—
376C22×C8C22×D8314yes—
377#123—1668no—
378C22⋊Q8—834no—
379C22⋊Q8—834no—
380C22⋊Q8—1662no—
381C22⋊Q8—1662no—
382#121—834no—
383#121—834no—
384#121—1662no—
385#121—1662no—
386#120—1668no—
387C4×D4—1668no—
388C4×D4—1668no—
389#117—834no—
390#117—1662no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—834no—
394C2×C4⋊C4—1662no—
395C2×C4⋊C4—1664no—
396C2×C4⋊C4—1664no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C22⋊C4—3324no—
400C2×C22⋊C4—1662no—
401C2×C22⋊C4—1662no—
402C2×C22⋊C4—1661no—
403C2×C22⋊C4—1661no—
404C2×C22⋊C4—1661no—
405C2×C22⋊C4—1661no—
406C2×C22⋊C4—1661no—
407C2×C42—1661no—
408#103—1668no—
409#102—1668no—
410#95—1662no—
411#95—1662no—
412C24—24no—
413C24—22no—
414C24—22no—
415C24—42no—
416C24—1661no—
417C24(C2×D83)⋊C414yes—
418C24C2×D33211yescharacteristic
419C24C2×D33211yescharacteristic
420C24C2×D33211yescharacteristic
421#41—3324no—
422#41—3324no—
423#41—16616no—
424C2×Q8—832no—
425C2×Q8—1662no—
426C2×Q8—1662no—
427C2×Q8—1661no—
428C2×D4—3324no—
429C2×D4—3324no—
430C2×D4—3322no—
431C2×D4—3322no—
432C2×D4—832no—
433C2×D4—1664no—
434C2×D4—1664no—
435C2×D4—1664no—
436C2×D4—1661no—
437C2×D4—1662no—
438C2×D4—1662no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—216no—
449C22×C4—3324no—
450C22×C4—3324no—
451C22×C4—3324no—
452C22×C4—3324no—
453C22×C4—1664no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4—1661no—
457C22×C4—1664no—
458C22×C4(C2×D83)⋊C414yes—
459C22×C4(C2×D83)⋊C414yes—
460C22×C4(C2×D83)⋊C414yes—
461C22×C4C2×C4×D8312yes—
462C22×C4C2×C4×D8312yes—
463C22×C4C2×C4×D8312yes—
464C22×C4C2×C22⋊D8312yes—
465C22×C4C2×C22⋊D8312yes—
466C22×C4C2×C22⋊D8314yes—
467C22×C4C2×C22⋊D8312yes—
468C22×C4C2×D33211yescharacteristic
469C22×C4C2×D33211yescharacteristic
470C22×C4C2×D33211yescharacteristic
471C22×C4#1305059014yes—
472C22×C4C23×D8311yescharacteristic
473M16—216no—
474M16—216no—
475C2×C8—216no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8C2×C22⋊D8318yes—
479C4⋊C4—3324no—
480C4⋊C4—832no—
481C4⋊C4—832no—
482C4⋊C4—1664no—
483C4⋊C4—1664no—
484C4⋊C4—1661no—
485C4⋊C4—1661no—
486C4⋊C4—1664no—
487C4⋊C4—1664no—
488C22⋊C4—3324no—
489C22⋊C4—3324no—
490C22⋊C4—1664no—
491C22⋊C4—1664no—
492C22⋊C4—1664no—
493C22⋊C4—1664no—
494C22⋊C4—1664no—
495C42—1664no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—28no—
502C23—22no—
503C23—28no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—3322no—
511C23—3321no—
512C23—3321no—
513C23—3322no—
514C23—1662no—
515C23C2×(C2×D83)⋊C412yes—
516C23C2×(C2×D83)⋊C412yes—
517C23C2×(C2×D83)⋊C412yes—
518C23#1458176614yes—
519C23#1458182012yes—
520C23#1458182512yes—
521C23C22×D33211yescharacteristic
522Q8—3321no—
523Q8—3321no—
524Q8—1664no—
525D4—3328no—
526D4—3328no—
527D4—3328no—
528D4—3321no—
529D4—3321no—
530D4—1664no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—3322no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3322no—
546C2×C4—3328no—
547C2×C4—3324no—
548C2×C4—3324no—
549C2×C4—3324no—
550C2×C4—6648no—
551C2×C4—1662no—
552C2×C4—1662no—
553C2×C4—1662no—
554C2×C4C2×(C2×D83)⋊C412yes—
555C2×C4C2×(C2×D83)⋊C412yes—
556C2×C4C2×(C2×D83)⋊C412yes—
557C2×C4C4×C22⋊D8318yes—
558C2×C4#1458176114yes—
559C2×C4#1458176614yes—
560C2×C4#1458177514yes—
561C2×C4C22×C4×D8311yescharacteristic
562C2×C4C22×C22⋊D8312yes—
563C2×C4C22×C22⋊D8311yescharacteristic
564C2×C4C2×#1305059012yes—
565C8—28no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—416no—
578C22—42no—
579C22—42no—
580C22—3322no—
581C22—3322no—
582C22—3322no—
583C22—6642no—
584C22—6642no—
585C22#1912758014yes—
586C22#1912770314yescomplemented
587C22C22×(C2×D83)⋊C411yescharacteristic
588C22C2×#1458176612yes—
589C22C2×#1458182011yescharacteristic
590C22C2×#1458182511yescharacteristic
591C22#1912850112yes—
592C4—22no—
593C4—22no—
594C4—3322no—
595C4—3322no—
596C4—3322no—
597C4—6644no—
598C4#1912758014yes—
599C4C2×C4×C22⋊D8312yes—
600C4C2×#1458176111yescharacteristic
601C4C2×#1458177511yescharacteristic
602C2—24no—
603C2—24no—
604C2—42no—
605C2—6642no—
606C2C2×#1912758011yescharacteristic
607C2C2×#1912770311yescharacteristic
608C2#2491598311yescharacteristic
609C2#2491708212yes—
610C2#2491709312yes—
subgroupG/H
centerC23C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC23×M16×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128212C2
cosocleC22×C4×C83C24
hypercenterC23×M16D83
hypocenterC83#20971
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
2. C2×#19127703C2semidirect
3. C2×#19127561C2semidirect
4. #24907901C2semidirect
5. #24907896C2semidirect
6. #24907891C2semidirect
7. #24907886C2semidirect
8. #24907183C2semidirect
9. C23×M16×C83C2semidirect
12. #19128212C4semidirect
25. C2×#14581717C22semidirect
26. C2×#14581714C22semidirect
32. #19127464C22semidirect
33. #19127464C22semidirect
54. #14581771C2×C4semidirect
55. #14581771C2×C4semidirect
58. #14581766C2×C4semidirect
59. #14581766C2×C4semidirect
102. C2×M16×C83D4semidirect
275. C22×C83#417semidirect
306. C23×M16D83semidirect
315. C83#20971semidirect
373. C2×M16C22⋊D83semidirect
586. C22#19127703semidirect

Maximal subgroups

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

subgroupindex
C2×#191282122
C2×#191277032
C2×#191275612
#249079012
#249078962
#249078912
#249078862
#249071832
C23×M16×C832
#2097183