Finite Group Databasebeta

← #35411640

Subgroups — #35411640

Order 21248.

876 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128039C211yesmaximal, characteristic
2C4×#14581751C211yesmaximal, characteristic
3#24908101C211yesmaximal, characteristic, complemented
4#24908094C211yesmaximal, characteristic, complemented
5#24908052C211yesmaximal, characteristic, complemented
6#24908045C211yesmaximal, characteristic, complemented
7#24907776C211yesmaximal, characteristic
8#24907660C211yesmaximal, characteristic
9#24907555C211yesmaximal, characteristic
10#24907361C211yesmaximal, characteristic, complemented
11#24907361C211yesmaximal, characteristic, complemented
12#24907293C211yesmaximal, characteristic, complemented
13#24907293C211yesmaximal, characteristic, complemented
14#24907225C211yesmaximal, characteristic
15C4×C22⋊Q8×C83C211yesmaximal, characteristic
16C2×#14581751C2211yescharacteristic
17C2×#14581751C2211yescharacteristic
18#19128039C418yescomplemented
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581716C2211yescharacteristic
24C2×#14581716C2211yescharacteristic
25C2×#14581716C2211yescharacteristic
26C2×#14581716C2211yescharacteristic
27C2×C22⋊(C83⋊C8)C2211yescharacteristic
28C2×C22⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581703C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31C4×C4⋊Dic83C2211yescharacteristic
32C4×C4⋊Dic83C2211yescharacteristic
33C4×C4⋊Dic83C2211yescharacteristic
34C4×C4⋊Dic83C2211yescharacteristic
35C4×C22⋊Dic83—21no—
36C4×C22⋊Dic83—21no—
37#19127547—21no—
38C42×Dic83—21no—
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49C2×C22⋊Q8×C83C2211yescharacteristic
50C2×C4×Q8×C83C2211yescharacteristic
51C22×C42×C83C2211yescharacteristic
52#391×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#387×C83—21no—
55#385×C83—21no—
56#380×C83C2211yescharacteristic
57C4×C4⋊C4×C83—21no—
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C22⋊C4×C83—21no—
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64#14581751—22no—
65#14581751—22no—
66#14581751—22no—
67#14581751C2×C414yes—
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C22⋊Dic83—21no—
73C2×C22⋊Dic83—21no—
74C2×C4×Dic83—21no—
75C2×C4×Dic83—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83—21no—
78C2×C4×Dic83—21no—
79C2×C4×Dic83—21no—
80#14581725—22no—
81#14581725C2×C414yescomplemented
82#14581723—22no—
83#14581723C2×C414yescomplemented
84#14581716—22no—
85#14581716—22no—
86#14581716C2×C414yescomplemented
87#14581716C2×C414yescomplemented
88C22⋊(C83⋊C8)—22no—
89C22⋊(C83⋊C8)C2×C414yes—
90#14581703—22no—
91#14581703—22no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C22×Q8×C83C2311yescharacteristic
97C23×C4×C83D411yescharacteristic
98C23×C4×C83D411yescharacteristic
99C23×C4×C83C2311yescharacteristic
100C22⋊Q8×C83—22no—
101C22⋊Q8×C83C2×C414yes—
102C4×Q8×C83—22no—
103C4×Q8×C83—22no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C22⋊C4×C83—21no—
111C2×C22⋊C4×C83—21no—
112C2×C42×C83—22no—
113C2×C42×C83—22no—
114C2×C42×C83—21no—
115C2×C42×C83—21no—
116C2×C42×C83D411yescharacteristic
117C2×C42×C83D411yescharacteristic
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120#95×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124#95×C83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8C22×C412yes—
140C2×C83⋊C8C22×C412yes—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83C22×C412yes—
146C4⋊Dic83C22×C412yes—
147C4⋊Dic83#4112yes—
148C4⋊Dic83#4112yes—
149C22⋊Dic83—24no—
150C22⋊Dic83—24no—
151C4×Dic83—24no—
152C4×Dic83—24no—
153C4×Dic83—42no—
154C4×Dic83—42no—
155C4×Dic83—42no—
156C4×Dic83—42no—
157C24×C83C2×D411yescharacteristic
158C2×Q8×C83—21no—
159C2×Q8×C83—21no—
160C2×Q8×C83—21no—
161C2×Q8×C83—21no—
162C2×Q8×C83—21no—
163C2×Q8×C83C22×C412yes—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—22no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—41no—
176C22×C4×C83—41no—
177C22×C4×C83—41no—
178C22×C4×C83C22×C412yes—
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83#4112yes—
185C22×C4×C83C2411yescharacteristic
186C4⋊C4×C83—24no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—42no—
193C4⋊C4×C83—42no—
194C4⋊C4×C83C22×C412yes—
195C22⋊C4×C83—24no—
196C22⋊C4×C83—42no—
197C42×C83—22no—
198C42×C83—22no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—42no—
202C42×C83—42no—
203C42×C83#4112yes—
204C42×C83#4112yes—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83C4×D412yes—
212C23×C83C22×D411yescharacteristic
213Q8×C83—22no—
214Q8×C83—22no—
215Q8×C83—41no—
216Q8×C83—41no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
229C2×C4×C83—22no—
230C2×C4×C83—22no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—44no—
235C2×C4×C83—44no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83C4×D412yes—
247C2×C4×C83#12312yes—
248C2×C4×C83#12312yes—
249C2×C4×C83#13712yes—
250C2×C4×C83C23×C411yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83C2×#4111yescharacteristic
254C2×C4×C83C2×#4111yescharacteristic
255C2×C4×C83C2×#4111yescharacteristic
256C2×Dic83—22no—
257C2×Dic83—22no—
258C2×Dic83—22no—
259C2×Dic83—22no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C83⋊C8—22no—
269C83⋊C8—22no—
270C83⋊C8—41no—
271C83⋊C8—41no—
272C22×C83—22no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83C2×C4×D411yescharacteristic
280C22×C83C2×#12311yescharacteristic
281C22×C83#52511yescharacteristic
282C22×C83C2×#13711yescharacteristic
283C22×C83#57511yescharacteristic
284C22×C83#57511yescharacteristic
285C22×C83#57511yescharacteristic
286C4×C83—22no—
287C4×C83—22no—
288C4×C83—22no—
289C4×C83—22no—
290C4×C83—41no—
291C4×C83—42no—
292C4×C83—42no—
293C4×C83—81no—
294C4×C83—81no—
295C4×C83C4×#4111yescharacteristic
296C4×C83C4×#4111yescharacteristic
297C4×C83#52911yescharacteristic
298C4×C83#52911yescharacteristic
299Dic83—42no—
300Dic83—42no—
301Dic83—42no—
302Dic83—42no—
303#21583—831nomaximal, Sylow
304C2×C83—42no—
305C2×C83C4×#12311yescharacteristic
306C2×C83#294311yescharacteristic
307C2×C83C4×#13711yescharacteristic
308C2×C83#309311yescharacteristic
309C2×C83#309511yescharacteristic
310C2×C83#311111yescharacteristic
311C2×C83#311611yescharacteristic
312C2×#485—831no—
313C4×C22⋊Q8D8311yescharacteristic
314C4×#117—831no—
315#1928—831no—
316#1927—831no—
317#1921—831no—
318#1920—831no—
319#1873—831no—
320#1854—831no—
321#1825—831no—
322#1776—831no—
323#1776—831no—
324#1763—831no—
325#1763—831no—
326#1749—831no—
327C83#2158311yescharacteristic, complemented, Sylow
328C2×C22⋊Q8C2×D8311yescharacteristic
329C2×C4×Q8C2×D8311yescharacteristic
330C2×#117—831no—
331C2×#117—831no—
332C22×C42C2×D8311yescharacteristic
333#485—838no—
334C2×#107—831no—
335C2×#107—831no—
336C2×#106—831no—
337C2×#106—831no—
338C2×#103—831no—
339C2×#103—831no—
340C2×#103—831no—
341C2×#103—831no—
342C2×C22⋊C8—831no—
343C2×C22⋊C8—831no—
344C2×#97—831no—
345C2×#97—831no—
346#391C2×D8311yescharacteristic
347#389C2×D8311yescharacteristic
348#387—21no—
349#385—21no—
350#380C2×D8311yescharacteristic
351C4×C4⋊C4—21no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—831no—
355C4×C4⋊C4—831no—
356C4×C4⋊C4C2×D8311yescharacteristic
357C4×C22⋊C4—21no—
358C4×C22⋊C4—1661no—
359C4×C22⋊C4—1661no—
360#376—1661no—
361C43—1661no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370#336—831no—
371#336—831no—
372C22×Q8C22×D8311yescharacteristic
373C23×C4C22⋊D8311yescharacteristic
374C23×C4C22⋊D8311yescharacteristic
375C23×C4C22×D8311yescharacteristic
376C22×C8—831no—
377C22×C8—831no—
378C22×C8—831no—
379C22×C8—831no—
380C22⋊Q8—22no—
381C22⋊Q8C4×D8314yes—
382C4×Q8—22no—
383C4×Q8—22no—
384#117—834no—
385#117—1662no—
386#117—1662no—
387#117—1662no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4C22×D8311yescharacteristic
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1661no—
401C2×C22⋊C4—1661no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—1661no—
407C2×C42—1661no—
408C2×C42—1661no—
409C2×C42—1661no—
410C2×C42—1661no—
411C2×C42—1661no—
412C2×C42C22⋊D8311yescharacteristic
413C2×C42C22⋊D8311yescharacteristic
414C2×C42C22×D8311yescharacteristic
415C2×C42C22×D8311yescharacteristic
416#107—834no—
417#107—1662no—
418#106—834no—
419#106—1662no—
420#103—834no—
421#103—834no—
422#103—1662no—
423#103—1662no—
424C22⋊C8—834no—
425C22⋊C8—1662no—
426#97—1662no—
427#97—1662no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437C24C2×C22⋊D8311yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×C4×D8312yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4—1661no—
461C22×C4—1661no—
462C22×C4C2×C4×D8312yes—
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C2×C22⋊D8311yescharacteristic
467C22×C4C2×C22⋊D8311yescharacteristic
468C22×C4#1305059012yes—
469C22×C4C23×D8311yescharacteristic
470C2×C8—832no—
471C2×C8—832no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—832no—
491C4⋊C4—832no—
492C4⋊C4—832no—
493C4⋊C4—832no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4C2×C4×D8312yes—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1664no—
502C22⋊C4—1664no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—42no—
509C42—3322no—
510C42—3322no—
511C42—3322no—
512C42—3322no—
513C42—1664no—
514C42—1664no—
515C42#1305058912yes—
516C42#1305058912yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23C4×C22⋊D8312yes—
524C23C22×C22⋊D8311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—1662no—
567C2×C4—1662no—
568C2×C4—1662no—
569C2×C4—1662no—
570C2×C4C4×C22⋊D8312yes—
571C2×C4#1458177312yes—
572C2×C4#1458177312yes—
573C2×C4#1458182312yes—
574C2×C4C22×C4×D8311yescharacteristic
575C2×C4#41×D8311yescharacteristic
576C2×C4#41×D8311yescharacteristic
577C2×C4C2×#1305058911yescharacteristic
578C2×C4C2×#1305058911yescharacteristic
579C2×C4C2×#1305059011yescharacteristic
580C8—3321no—
581C8—3321no—
582C8—1662no—
583C8—1662no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×C4×C22⋊D8311yescharacteristic
592C22C2×#1458177311yescharacteristic
593C22#1912820811yescharacteristic
594C22C2×#1458182311yescharacteristic
595C22#1912849911yescharacteristic
596C22#1912849911yescharacteristic
597C22#1912849911yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—3322no—
608C4—3322no—
609C4—3322no—
610C4—3322no—
611C4C4×#1305058911yescharacteristic
612C4C4×#1305058911yescharacteristic
613C4#1912823111yescharacteristic
614C4#1912823111yescharacteristic
615C2—42no—
616C2C4×#1458177311yescharacteristic
617C2#2491626111yescharacteristic
618C2C4×#1458182311yescharacteristic
619C2#2491739211yescharacteristic
620C2#2491740711yescharacteristic
621C2#2491756611yescharacteristic
622C2#2491761911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC4×C22⋊Q8×C83C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenterC4×C22⋊Q8D83
hypocenterC83#21583
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #24908101C2semidirect
4. #24908094C2semidirect
5. #24908052C2semidirect
6. #24908045C2semidirect
10. #24907361C2semidirect
11. #24907361C2semidirect
12. #24907293C2semidirect
13. #24907293C2semidirect
18. #19128039C4semidirect
81. #14581725C2×C4semidirect
83. #14581723C2×C4semidirect
86. #14581716C2×C4semidirect
87. #14581716C2×C4semidirect
327. C83#21583semidirect

Maximal subgroups

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

subgroupindex
C2×#191280392
C4×#145817512
#249081012
#249080942
#249080522
#249080452
#249077762
#249076602
#249075552
#249073612
#249072932
#249072252
C4×C22⋊Q8×C832
#2158383