Finite Group Databasebeta

← #35406170

Subgroups — #35406170

Order 21248.

1954 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128212C211yesmaximal, characteristic
2C2×#19127719C212yesmaximal, complemented
3C2×#19127665C211yesmaximal, characteristic, complemented
4#24907911C214yesmaximal, complemented
5#24907906C214yesmaximal, complemented
6#24907173C212yesmaximal, complemented
7C22×#117×C83C211yesmaximal, characteristic, complemented
8C22×#13050590C2211yescharacteristic
9C22×#13050574C2212yes—
10#19128212—22no—
11#19128212—21no—
12#19128212—21no—
13#19128212C414yescomplemented
14#19128212C2212yes—
15#19128212C2212yes—
16C2×#14581775—21no—
17C2×#14581771C412yescomplemented
18C2×#14581766C412yescomplemented
19C2×#14581761—21no—
20C2×C4×C22⋊D83—22no—
21C2×#14581751C2211yescharacteristic
22#19127719—28no—
23C2×#14581720C2218yescomplemented
24#19127665—22no—
25#19127665C2212yescomplemented
26#19127665C2212yescomplemented
27C2×C22⋊(C83⋊C8)—22no—
28#19127461C2218yescomplemented
29C24×C4×C83C2211yescharacteristic
30C2×#117×C83—21no—
31C2×#117×C83—21no—
32C2×#117×C83—28no—
33C2×#117×C83C2212yes—
34C2×#117×C83C2212yescomplemented
35C22×C4⋊C4×C83—21no—
36C22×C22⋊C4×C83—21no—
37C22×C42×C83C2212yescomplemented
38C2×#13050590—21no—
39C2×#13050590—21no—
40C2×#13050590—24no—
41C2×#13050590—24no—
42C2×#13050590C2×C412yescomplemented
43C2×#13050590C2312yes—
44C22×Dic166C2×C411yescharacteristic, complemented
45C22×D332C2×C411yescharacteristic, complemented
46C22×C22⋊D83—22no—
47C22×C22⋊D83—21no—
48C22×C4×D83—21no—
49C2×#13050574—22no—
50C2×#13050574—24no—
51C2×#13050574—24no—
52C2×#13050574C2314yes—
53C22×C83⋊C8—22no—
54#14581775—24no—
55#14581775—22no—
56#14581775—22no—
57#14581771—22no—
58#14581771—22no—
59#14581771C2×C414yes—
60#14581771C2×C414yescomplemented
61#14581766—22no—
62#14581766—22no—
63#14581766C2×C414yes—
64#14581766C2×C414yescomplemented
65#14581761—24no—
66#14581761—22no—
67#14581761—22no—
68C4×C22⋊D83—24no—
69C4×C22⋊D83—24no—
70C4×C22⋊D83—24no—
71C4×C22⋊D83—24no—
72#14581751—21no—
73#14581751—21no—
74#14581751C2×C412yescomplemented
75#14581751C2312yes—
76C2×C4⋊Dic83C2×C412yescomplemented
77C2×Dic83⋊C4—22no—
78C2×C22⋊Dic83—22no—
79C2×C22⋊Dic83—21no—
80C2×(C2×D83)⋊C4—22no—
81C2×C4×Dic83—21no—
82#14581720—216no—
83C22⋊(C83⋊C8)—24no—
84C22⋊(C83⋊C8)—24no—
85C25×C83D411yescharacteristic
86C23×C4×C83—24no—
87C23×C4×C83—24no—
88C23×C4×C83—22no—
89C23×C4×C83C2×C412yes—
90C23×C4×C83C2×C412yes—
91C23×C4×C83D412yes—
92C23×C4×C83D412yes—
93C23×C4×C83D411yescharacteristic
94C23×C4×C83D411yescharacteristic
95C23×C4×C83D411yescharacteristic
96C23×C4×C83C2311yescharacteristic
97C23×C4×C83C2311yescharacteristic
98C23×C4×C83C2311yescharacteristic
99#117×C83—28no—
100#117×C83—28no—
101#117×C83—24no—
102#117×C83—28no—
103#117×C83D418yescomplemented
104C2×C4⋊C4×C83—28no—
105C2×C4⋊C4×C83—22no—
106C2×C4⋊C4×C83—22no—
107C2×C22⋊C4×C83—28no—
108C2×C22⋊C4×C83—22no—
109C2×C22⋊C4×C83—22no—
110C2×C42×C83—22no—
111C2×C42×C83—28no—
112C2×C42×C83C2314yes—
113C23×D83—21no—
114#13050590—28no—
115#13050590—24no—
116#13050590—24no—
117#13050590—42no—
118#13050590—44no—
119#13050590—42no—
120C2×Dic166—24no—
121C2×Dic166—21no—
122C2×Dic166C22×C412yes—
123C2×D332—24no—
124C2×D332—21no—
125C2×D332C22×C412yes—
126C2×C22⋊D83—24no—
127C2×C22⋊D83—24no—
128C2×C22⋊D83—22no—
129C2×C22⋊D83—22no—
130C2×C22⋊D83—48no—
131C2×C22⋊D83—42no—
132C2×C22⋊D83—42no—
133C22×Dic83—21no—
134C22×Dic83—22no—
135C2×C4×D83—22no—
136C2×C4×D83—22no—
137C2×C4×D83—44no—
138#13050574—24no—
139#13050574—24no—
140#13050574—44no—
141C2×C83⋊C8—24no—
142C2×C83⋊C8—44no—
143C4⋊Dic83—22no—
144C4⋊Dic83C22×C414yes—
145Dic83⋊C4—24no—
146Dic83⋊C4—24no—
147C22⋊Dic83—24no—
148C22⋊Dic83—24no—
149C22⋊Dic83—22no—
150C22⋊Dic83—22no—
151(C2×D83)⋊C4—24no—
152(C2×D83)⋊C4—24no—
153C4×Dic83—22no—
154C4×Dic83—22no—
155C24×C83—22no—
156C24×C83—22no—
157C24×C83—44no—
158C24×C83C22⋊C412yes—
159C24×C83C22⋊C412yes—
160C24×C83C2×D411yescharacteristic
161C24×C83C2×D411yescharacteristic
162C24×C83C2×D411yescharacteristic
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—28no—
166C22×C4×C83—28no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—28no—
170C22×C4×C83—28no—
171C22×C4×C83—22no—
172C22×C4×C83—22no—
173C22×C4×C83—22no—
174C22×C4×C83—22no—
175C22×C4×C83—22no—
176C22×C4×C83—22no—
177C22×C4×C83—28no—
178C22×C4×C83—24no—
179C22×C4×C83—48no—
180C22×C4×C83C22⋊C412yes—
181C22×C4×C83C22⋊C412yes—
182C22×C4×C83C22⋊C412yes—
183C22×C4×C83C22⋊C412yes—
184C22×C4×C83C22⋊C412yes—
185C22×C4×C83C22⋊C412yes—
186C22×C4×C83C22×C412yes—
187C22×C4×C83C22×C412yes—
188C22×C4×C83C22×C411yescharacteristic
189C22×C4×C83C22×C411yescharacteristic
190C22×C4×C83C2×D411yescharacteristic
191C22×C4×C83C2×D411yescharacteristic
192C22×C4×C83C2×D411yescharacteristic
193C22×C4×C83C2×D414yes—
194C22×C4×C83C2×D412yes—
195C22×C4×C83C2×D412yes—
196C22×C4×C83C2×D411yescharacteristic
197C22×C4×C83C2×D411yescharacteristic
198C22×C4×C83#4114yes—
199C22×C4×C83C2411yescharacteristic
200C4⋊C4×C83—28no—
201C4⋊C4×C83—28no—
202C22⋊C4×C83—28no—
203C22⋊C4×C83—28no—
204C42×C83—28no—
205C42×C83—24no—
206C42×C83C2×D418yescomplemented
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—24no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—24no—
213C23×C83—24no—
214C23×C83—24no—
215C23×C83—24no—
216C23×C83—42no—
217C23×C83—42no—
218C23×C83—42no—
219C23×C83—42no—
220C23×C83—48no—
221C23×C83—48no—
222C23×C83C2×C22⋊C412yes—
223C23×C83C2×C22⋊C411yescharacteristic
224C23×C83C2×C22⋊C411yescharacteristic
225C23×C83C2×C22⋊C412yes—
226C23×C83#12012yes—
227C23×C83#12312yes—
228C23×C83C22×D411yescharacteristic
229C22×D83—22no—
230C22×D83—44no—
231C22×D83—42no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—24no—
237C2×C4×C83—24no—
238C2×C4×C83—24no—
239C2×C4×C83—24no—
240C2×C4×C83—24no—
241C2×C4×C83—24no—
242C2×C4×C83—24no—
243C2×C4×C83—24no—
244C2×C4×C83—24no—
245C2×C4×C83—24no—
246C2×C4×C83—28no—
247C2×C4×C83—416no—
248C2×C4×C83—48no—
249C2×C4×C83C2×C22⋊C412yes—
250C2×C4×C83C2×C22⋊C412yes—
251C2×C4×C83C2×C22⋊C411yescharacteristic
252C2×C4×C83C2×C22⋊C411yescharacteristic
253C2×C4×C83C4×D414yes—
254C2×C4×C83C4×D414yes—
255C2×C4×C83#12012yes—
256C2×C4×C83#12114yes—
257C2×C4×C83#12114yes—
258C2×C4×C83#12312yes—
259C2×C4×C83C23×C411yescharacteristic
260C2×C4×C83C22×D412yes—
261C2×C4×C83C22×D411yescharacteristic
262C2×C4×C83C2×#4112yes—
263C22⋊D83—48no—
264C22⋊D83—42no—
265C22⋊D83—42no—
266C22⋊D83—44no—
267C22⋊D83—48no—
268C2×Dic83—24no—
269C2×Dic83—22no—
270C2×Dic83—44no—
271C2×Dic83—42no—
272D332—24no—
273D332—42no—
274C4×D83—44no—
275C4×D83—44no—
276Dic166—24no—
277Dic166—42no—
278C83⋊C8—44no—
279C22×C83—24no—
280C22×C83—24no—
281C22×C83—22no—
282C22×C83—22no—
283C22×C83—22no—
284C22×C83—22no—
285C22×C83—22no—
286C22×C83—22no—
287C22×C83—22no—
288C22×C83—22no—
289C22×C83—42no—
290C22×C83—42no—
291C22×C83—48no—
292C22×C83—48no—
293C22×C83—42no—
294C22×C83—42no—
295C22×C83#38612yes—
296C22×C83#38612yes—
297C22×C83#42114yescomplemented
298C22×C83C22×C22⋊C411yescharacteristic
299C22×C83C2×#12011yescharacteristic
300C22×C83C2×#12311yescharacteristic
301C2×D83—44no—
302C2×D83—42no—
303C2×D83—84no—
304C4×C83—22no—
305C4×C83—22no—
306C4×C83—48no—
307C4×C83#38612yes—
308C4×C83#38612yes—
309C4×C83C2×C4×D412yes—
310C4×C83C2×#12112yes—
311Dic83—44no—
312Dic83—42no—
313#20979—831nomaximal, Sylow
314C2×C83—21no—
315C2×C83—22no—
316C2×C83—21no—
317C2×C83—24no—
318C2×C83—22no—
319C2×C83—42no—
320C2×C83C2×#38611yescharacteristic
321C2×C83C2×#42112yes—
322D83—82no—
323C2×#525—831no—
324C22×#117D8311yescharacteristic, complemented
325C2×#421—832no—
326C2×#407—831no—
327#1897—834no—
328#1896—834no—
329#1733—832no—
330C83#2097911yescharacteristic, complemented, Sylow
331C22×#41—831no—
332C24×C4C2×D8311yescharacteristic
333C22×M16—832no—
334#525—834no—
335#525—832no—
336#525—832no—
337#525—1662no—
338#525—1661no—
339#525—1661no—
340C2×#123—1661no—
341C2×C22⋊Q8—832no—
342C2×#121—832no—
343C2×#120—1661no—
344C2×C4×D4—1662no—
345C2×#117—21no—
346C2×#117—21no—
347C2×#117—28no—
348C2×#117—831no—
349C2×#117C2×D8312yes—
350C2×#117C2×D8312yescomplemented
351C22×C4⋊C4—21no—
352C22×C22⋊C4—21no—
353C22×C42C2×D8312yescomplemented
354#421—1668no—
355C2×#104—838no—
356#407—832no—
357#407—832no—
358#407—1662no—
359C2×C22⋊C8—1662no—
360#339—838no—
361C25C22⋊D8311yescharacteristic
362C2×#41—832no—
363C2×#41—832no—
364C2×#41—1664no—
365C2×#41—1664no—
366C2×#41—1661no—
367C2×#41—1661no—
368C22×Q8—831no—
369C22×D4—831no—
370C22×D4—1662no—
371C22×D4—1661no—
372C23×C4—24no—
373C23×C4—24no—
374C23×C4—22no—
375C23×C4—1661no—
376C23×C4C4×D8312yes—
377C23×C4C4×D8312yes—
378C23×C4D33212yes—
379C23×C4D33212yes—
380C23×C4C22⋊D8311yescharacteristic
381C23×C4C22⋊D8311yescharacteristic
382C23×C4C22⋊D8311yescharacteristic
383C23×C4C22×D8311yescharacteristic
384C23×C4C22×D8311yescharacteristic
385C23×C4C22×D8311yescharacteristic
386C2×M16—834no—
387C2×M16—1664no—
388C2×M16—1664no—
389C2×M16—1662no—
390C22×C8—1662no—
391#123—1664no—
392#123—1662no—
393#123—1662no—
394C22⋊Q8—834no—
395C22⋊Q8—834no—
396C22⋊Q8—1662no—
397C22⋊Q8—1662no—
398#121—834no—
399#121—834no—
400#121—1662no—
401#121—1662no—
402#120—1664no—
403#120—1662no—
404#120—1662no—
405C4×D4—1664no—
406C4×D4—1664no—
407C4×D4—1664no—
408C4×D4—1664no—
409#117—28no—
410#117—28no—
411#117—24no—
412#117—28no—
413#117—832no—
414#117—832no—
415#117—1661no—
416#117—1661no—
417#117C22⋊D8318yescomplemented
418C2×C4⋊C4—28no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—832no—
422C2×C4⋊C4—1662no—
423C2×C22⋊C4—28no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—1662no—
427C2×C22⋊C4—1661no—
428C2×C22⋊C4—1662no—
429C2×C42—22no—
430C2×C42—28no—
431C2×C42—1661no—
432C2×C42C22×D8314yes—
433#104—16616no—
434C22⋊C8—1664no—
435C22⋊C8—1664no—
436C24—22no—
437C24—22no—
438C24—44no—
439C24—1661no—
440C24(C2×D83)⋊C412yes—
441C24(C2×D83)⋊C412yes—
442C24C2×C22⋊D8311yescharacteristic
443C24C2×C22⋊D8311yescharacteristic
444C24C2×C22⋊D8311yescharacteristic
445#41—3322no—
446#41—3324no—
447#41—3322no—
448#41—1664no—
449#41—1664no—
450#41—1668no—
451C2×Q8—832no—
452C2×Q8—1664no—
453C2×Q8—1661no—
454C2×D4—3328no—
455C2×D4—3322no—
456C2×D4—3322no—
457C2×D4—832no—
458C2×D4—1662no—
459C2×D4—1664no—
460C2×D4—1664no—
461C2×D4—1662no—
462C2×D4—1661no—
463C2×D4—1664no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—28no—
469C22×C4—28no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—28no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—24no—
480C22×C4—48no—
481C22×C4—3324no—
482C22×C4—1662no—
483C22×C4—1661no—
484C22×C4—1662no—
485C22×C4—1662no—
486C22×C4(C2×D83)⋊C412yes—
487C22×C4(C2×D83)⋊C412yes—
488C22×C4(C2×D83)⋊C412yes—
489C22×C4(C2×D83)⋊C412yes—
490C22×C4(C2×D83)⋊C412yes—
491C22×C4(C2×D83)⋊C412yes—
492C22×C4C2×C4×D8312yes—
493C22×C4C2×C4×D8312yes—
494C22×C4C2×C4×D8311yescharacteristic
495C22×C4C2×C4×D8311yescharacteristic
496C22×C4C2×C22⋊D8311yescharacteristic
497C22×C4C2×C22⋊D8311yescharacteristic
498C22×C4C2×C22⋊D8311yescharacteristic
499C22×C4C2×C22⋊D8314yes—
500C22×C4C2×D33212yes—
501C22×C4C2×D33212yes—
502C22×C4C2×D33211yescharacteristic
503C22×C4C2×D33211yescharacteristic
504C22×C4#1305059014yes—
505C22×C4C23×D8311yescharacteristic
506M16—3324no—
507M16—1664no—
508M16—1664no—
509C2×C8—3324no—
510C2×C8—1664no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—834no—
514C4⋊C4—1662no—
515C4⋊C4—1664no—
516C4⋊C4—1664no—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—1662no—
520C22⋊C4—1662no—
521C22⋊C4—1664no—
522C22⋊C4—1664no—
523C22⋊C4—1664no—
524C22⋊C4—1664no—
525C42—28no—
526C42—24no—
527C42—1662no—
528C42—1662no—
529C42C2×C22⋊D8318yescomplemented
530C23—21no—
531C23—21no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—21no—
536C23—21no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—48no—
544C23—48no—
545C23—3324no—
546C23—3322no—
547C23—1662no—
548C23C2×(C2×D83)⋊C412yes—
549C23C2×(C2×D83)⋊C411yescharacteristic
550C23C2×(C2×D83)⋊C411yescharacteristic
551C23C2×(C2×D83)⋊C412yes—
552C23#1458176112yes—
553C23#1458177512yes—
554C23C22×C22⋊D8311yescharacteristic
555Q8—3322no—
556Q8—1664no—
557D4—3328no—
558D4—3324no—
559D4—3322no—
560D4—3322no—
561D4—3328no—
562D4—3322no—
563D4—1664no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—416no—
580C2×C4—48no—
581C2×C4—3324no—
582C2×C4—3322no—
583C2×C4—3324no—
584C2×C4—3324no—
585C2×C4—1664no—
586C2×C4—1662no—
587C2×C4C2×(C2×D83)⋊C412yes—
588C2×C4C2×(C2×D83)⋊C412yes—
589C2×C4C2×(C2×D83)⋊C411yescharacteristic
590C2×C4C2×(C2×D83)⋊C411yescharacteristic
591C2×C4C4×C22⋊D8314yes—
592C2×C4C4×C22⋊D8314yes—
593C2×C4#1458176112yes—
594C2×C4#1458176614yes—
595C2×C4#1458176614yes—
596C2×C4#1458177512yes—
597C2×C4C22×C4×D8311yescharacteristic
598C2×C4C22×C22⋊D8312yes—
599C2×C4C22×D33211yescharacteristic
600C2×C4C2×#1305059012yes—
601C8—3324no—
602C22—24no—
603C22—24no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—48no—
615C22—48no—
616C22—42no—
617C22—42no—
618C22—3324no—
619C22—3322no—
620C22—6644no—
621C22#1912758012yes—
622C22#1912758012yes—
623C22#1912771914yescomplemented
624C22C22×(C2×D83)⋊C411yescharacteristic
625C22C2×#1458176111yescharacteristic
626C22C2×#1458177511yescharacteristic
627C4—22no—
628C4—22no—
629C4—48no—
630C4—3324no—
631C4—3322no—
632C4#1912758012yes—
633C4#1912758012yes—
634C4C2×C4×C22⋊D8312yes—
635C4C2×#1458176612yes—
636C2—21no—
637C2—21no—
638C2—24no—
639C2—22no—
640C2—22no—
641C2—42no—
642C2—6642no—
643C2C2×#1912758011yescharacteristic
644C2C2×#1912771912yes—
subgroupG/H
centerC2×C4C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC22×#117×C83C2
socleC22×C83C22×C22⋊C4
prime radicalC2×#19128212C2
cosocleC22×C4×C83C24
hypercenterC22×#117D83
hypocenterC83#20979
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#19127719C2semidirect
3. C2×#19127665C2semidirect
4. #24907911C2semidirect
5. #24907906C2semidirect
6. #24907173C2semidirect
7. C22×#117×C83C2semidirect
13. #19128212C4semidirect
17. C2×#14581771C4semidirect
18. C2×#14581766C4semidirect
23. C2×#14581720C22semidirect
25. #19127665C22semidirect
26. #19127665C22semidirect
28. #19127461C22semidirect
34. C2×#117×C83C22semidirect
37. C22×C42×C83C22semidirect
42. C2×#13050590C2×C4semidirect
44. C22×Dic166C2×C4semidirect
45. C22×D332C2×C4semidirect
60. #14581771C2×C4semidirect
64. #14581766C2×C4semidirect
74. #14581751C2×C4semidirect
76. C2×C4⋊Dic83C2×C4semidirect
103. #117×C83D4semidirect
206. C42×C83C2×D4semidirect
297. C22×C83#421semidirect
324. C22×#117D83semidirect
330. C83#20979semidirect
350. C2×#117C2×D83semidirect
353. C22×C42C2×D83semidirect
417. #117C22⋊D83semidirect
529. C42C2×C22⋊D83semidirect
623. C22#19127719semidirect

Maximal subgroups

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

subgroupindex
C2×#191282122
C2×#191277192
C2×#191276652
#249079112
#249079062
#249071732
C22×#117×C832
#2097983