Finite Group Databasebeta

← #35411639

Subgroups — #35411639

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×#19128042C211yesmaximal, characteristic
2#24909991C211yesmaximal, characteristic
3#117×Dic83C211yesmaximal, characteristic
4#24908102C211yesmaximal, characteristic, complemented
5#24908095C211yesmaximal, characteristic, complemented
6#24908053C211yesmaximal, characteristic, complemented
7#24908046C211yesmaximal, characteristic, complemented
8#24907773C211yesmaximal, characteristic
9#24907660C211yesmaximal, characteristic
10#24907359C211yesmaximal, characteristic, complemented
11#24907359C211yesmaximal, characteristic, complemented
12#24907292C211yesmaximal, characteristic, complemented
13#24907292C211yesmaximal, characteristic, complemented
14#24907227C211yesmaximal, characteristic
15C83×#1825C211yesmaximal, characteristic
16C2×#14581771C2211yescharacteristic
17C2×#14581757C2211yescharacteristic
18C2×#14581751C2211yescharacteristic
19C22×C4×Dic83C2211yescharacteristic
20#19128042C418yescomplemented
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581717C2211yescharacteristic
26C2×#14581717C2211yescharacteristic
27C2×#14581715C2211yescharacteristic
28C2×#14581715C2211yescharacteristic
29C2×C22⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581702C2211yescharacteristic
31C2×#14581702C2211yescharacteristic
32#19127606C2211yescharacteristic
33#19127597C2211yescharacteristic
34#19127583—21no—
35#19127573—21no—
36C4⋊C4×Dic83C2211yescharacteristic
37C4⋊C4×Dic83C2211yescharacteristic
38C4⋊C4×Dic83C2211yescharacteristic
39C4⋊C4×Dic83C2211yescharacteristic
40C4⋊C4×Dic83C2211yescharacteristic
41#19127552—21no—
42C22⋊C4×Dic83—21no—
43C22⋊C4×Dic83—21no—
44C22⋊C4×Dic83—21no—
45#19127547—21no—
46C42×Dic83—21no—
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127455C2211yescharacteristic
52#19127455C2211yescharacteristic
53C2×#117×C83C2211yescharacteristic
54C2×C22⋊C8×C83C2211yescharacteristic
55#380×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60C22×Dic166C2311yescharacteristic
61C23×Dic83D411yescharacteristic
62C23×Dic83D411yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65#14581771—22no—
66#14581771C2×C414yes—
67#14581757—22no—
68#14581757—22no—
69#14581751—22no—
70#14581751—22no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×C22⋊Dic83—21no—
78C2×C22⋊Dic83—21no—
79C2×C4×Dic83—22no—
80C2×C4×Dic83—22no—
81C2×C4×Dic83—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83D411yescharacteristic
89C2×C4×Dic83D411yescharacteristic
90C2×C4×Dic83C2311yescharacteristic
91C2×C4×Dic83C2311yescharacteristic
92#14581725—22no—
93#14581725C2×C414yescomplemented
94#14581723—22no—
95#14581723C2×C414yescomplemented
96#14581717—22no—
97#14581717—22no—
98#14581715C2×C414yescomplemented
99#14581715C2×C414yescomplemented
100C22⋊(C83⋊C8)C2×C414yes—
101#14581702—22no—
102#14581702—22no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95—21no—
111C23×C4×C83C2311yescharacteristic
112C22×C8×C83C2311yescharacteristic
113C22×C8×C83C2311yescharacteristic
114#117×C83—22no—
115#117×C83C2×C414yes—
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C22⋊C4×C83—21no—
121C2×C22⋊C4×C83—21no—
122C2×C42×C83—21no—
123C22⋊C8×C83—22no—
124#95×C83—21no—
125C2×Dic166—21no—
126C2×Dic166—21no—
127C2×Dic166—21no—
128C2×Dic166—21no—
129C2×Dic166—21no—
130C2×Dic166C22×C412yes—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—22no—
136C22×Dic83—22no—
137C22×Dic83—22no—
138C22×Dic83—41no—
139C22×Dic83—41no—
140C22×Dic83—41no—
141C22×Dic83C2×D411yescharacteristic
142C22×Dic83C2×D411yescharacteristic
143C22×Dic83C2×D411yescharacteristic
144C22×Dic83C2×D411yescharacteristic
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8C22×C412yes—
150C2×C83⋊C8C22×C412yes—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83C22×C412yes—
155C4⋊Dic83#4112yes—
156C4⋊Dic83#4112yes—
157Dic83⋊C4—24no—
158Dic83⋊C4—42no—
159Dic83⋊C4—42no—
160C22⋊Dic83—24no—
161C22⋊Dic83—24no—
162C4×Dic83—24no—
163C4×Dic83—21no—
164C4×Dic83—21no—
165C4×Dic83—22no—
166C4×Dic83—22no—
167C4×Dic83—42no—
168C4×Dic83—42no—
169C4×Dic83—42no—
170C4×Dic83—42no—
171C4×Dic83—42no—
172C4×Dic83—42no—
173C4×Dic83#4112yes—
174C4×Dic83#4112yes—
175C24×C83C2×D411yescharacteristic
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83C22×C412yes—
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83#4112yes—
187C22×C4×C83C2411yescharacteristic
188C2×C8×C83—21no—
189C2×C8×C83—21no—
190C2×C8×C83—21no—
191C2×C8×C83—21no—
192C2×C8×C83—21no—
193C2×C8×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83C22×C412yes—
201C4⋊C4×C83C22×C412yes—
202C22⋊C4×C83—24no—
203C22⋊C4×C83—42no—
204C42×C83—24no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83C4×D412yes—
212C23×C83C22×D411yescharacteristic
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—22no—
220C2×C4×C83—22no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83C4×D412yes—
232C2×C4×C83C23×C411yescharacteristic
233C2×C4×C83C2×#4111yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—22no—
245C2×Dic83—22no—
246C2×Dic83—21no—
247C2×Dic83—22no—
248C2×Dic83—22no—
249C2×Dic83—21no—
250C2×Dic83—21no—
251C2×Dic83—44no—
252C2×Dic83—44no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83#12312yes—
262C2×Dic83#12312yes—
263C2×Dic83#13712yes—
264Dic166—22no—
265Dic166—22no—
266Dic166—41no—
267Dic166—41no—
268C8×C83—41no—
269C8×C83—41no—
270C83⋊C8—22no—
271C83⋊C8—22no—
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—42no—
288C4×C83—42no—
289C4×C83—81no—
290C4×C83—81no—
291C4×C83C4×#4111yescharacteristic
292C4×C83C4×#4111yescharacteristic
293C4×C83#52911yescharacteristic
294C4×C83#52911yescharacteristic
295Dic83—22no—
296Dic83—22no—
297Dic83—22no—
298Dic83—41no—
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⋊Q8—831no—
314C4×#117—831no—
315#1928—831no—
316#1927—831no—
317#1921—831no—
318#1920—831no—
319#1873—831no—
320#1854—831no—
321#1825D8311yescharacteristic
322#1776—831no—
323#1776—831no—
324#1763—831no—
325#1763—831no—
326#1749—831no—
327C83#2158311yescharacteristic, complemented, Sylow
328C2×C22⋊Q8—831no—
329C2×C4×Q8—831no—
330C2×#117—831no—
331C2×#117C2×D8311yescharacteristic
332C22×C42—831no—
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⋊C8C2×D8311yescharacteristic
344C2×#97—831no—
345C2×#97—831no—
346#391—831no—
347#389—831no—
348#387—1661no—
349#385—1661no—
350#380C2×D8311yescharacteristic
351C4×C4⋊C4—831no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—831no—
355C4×C4⋊C4—831no—
356C4×C4⋊C4—1661no—
357C4×C22⋊C4—1661no—
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#340C2×D8311yescharacteristic
367#340C2×D8311yescharacteristic
368#340C2×D8311yescharacteristic
369#340C2×D8311yescharacteristic
370#336—831no—
371#336—831no—
372C22×Q8—831no—
373C23×C4—831no—
374C23×C4—831no—
375C23×C4C22×D8311yescharacteristic
376C22×C8—831no—
377C22×C8—831no—
378C22×C8C22×D8311yescharacteristic
379C22×C8C22×D8311yescharacteristic
380C22⋊Q8—834no—
381C22⋊Q8—1662no—
382C4×Q8—1662no—
383C4×Q8—1662no—
384#117—22no—
385#117—1662no—
386#117—1662no—
387#117C4×D8314yes—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4C22×D8311yescharacteristic
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—21no—
403C2×C42—831no—
404C2×C42—831no—
405C2×C42—831no—
406C2×C42—831no—
407C2×C42—1661no—
408C2×C42—1661no—
409C2×C42—1661no—
410C2×C42—1661no—
411C2×C42—1661no—
412C2×C42—1661no—
413C2×C42—1662no—
414C2×C42—1662no—
415C2×C42—1661no—
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—22no—
425C22⋊C8—834no—
426#97—1662no—
427#97—1662no—
428#95—21no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437C24D4×D8311yescharacteristic
438C2×Q8—832no—
439C2×Q8—1661no—
440C2×Q8—1661no—
441C2×Q8—1661no—
442C2×Q8—1661no—
443C2×Q8—1661no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—3321no—
453C22×C4—3321no—
454C22×C4—3321no—
455C22×C4—831no—
456C22×C4—831no—
457C22×C4—831no—
458C22×C4—831no—
459C22×C4—1662no—
460C22×C4—1662no—
461C22×C4—1662no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4C2×C4×D8312yes—
467C22×C4D4×D8311yescharacteristic
468C22×C4#1305058912yes—
469C22×C4C23×D8311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—832no—
477C2×C8—832no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—3322no—
489C4⋊C4—3322no—
490C4⋊C4—832no—
491C4⋊C4—832no—
492C4⋊C4—832no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1664no—
497C4⋊C4C2×C4×D8312yes—
498C4⋊C4C2×C4×D8312yes—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1664no—
502C22⋊C4—1664no—
503C42—24no—
504C42—3322no—
505C42—3322no—
506C42—3322no—
507C42—3322no—
508C42—3322no—
509C42—3322no—
510C42—832no—
511C42—832no—
512C42—1662no—
513C42—1662no—
514C42—1664no—
515C42—1661no—
516C42—1661no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23(C22⋊D83)⋊C412yes—
524C23C2×D4×D8311yescharacteristic
525Q8—3321no—
526Q8—3321no—
527Q8—1662no—
528Q8—1662no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3324no—
556C2×C4—3324no—
557C2×C4—832no—
558C2×C4—832no—
559C2×C4—832no—
560C2×C4—1662no—
561C2×C4—1662no—
562C2×C4—1661no—
563C2×C4—1662no—
564C2×C4—1662no—
565C2×C4—1661no—
566C2×C4—1661no—
567C2×C4—1661no—
568C2×C4—1661no—
569C2×C4—1661no—
570C2×C4—1661no—
571C2×C4—1661no—
572C2×C4—1661no—
573C2×C4(C22⋊D83)⋊C412yes—
574C2×C4C22×C4×D8311yescharacteristic
575C2×C4#41×D8311yescharacteristic
576C2×C4#41×D8311yescharacteristic
577C2×C4#41×D8311yescharacteristic
578C2×C4#41×D8311yescharacteristic
579C2×C4C2×#1305058911yescharacteristic
580C8—41no—
581C8—41no—
582C8—1662no—
583C8—1662no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×(C22⋊D83)⋊C411yescharacteristic
592C22#123×D8311yescharacteristic
593C22#1912820911yescharacteristic
594C22#137×D8311yescharacteristic
595C22#1912849711yescharacteristic
596C22#1912849811yescharacteristic
597C22#1912849811yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—81no—
602C4—81no—
603C4—3322no—
604C4—3322no—
605C4—3322no—
606C4—3321no—
607C4—3322no—
608C4—1662no—
609C4—1662no—
610C4—1662no—
611C4#1912816911yescharacteristic
612C4#1912816911yescharacteristic
613C4#1912822811yescharacteristic
614C4#1912822811yescharacteristic
615C2—42no—
616C2#2490998011yescharacteristic
617C2#2491625911yescharacteristic
618C2#2491628711yescharacteristic
619C2#2491739111yescharacteristic
620C2#2491741011yescharacteristic
621C2#2491756711yescharacteristic
622C2#2491762011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1825C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#1825D83
hypocenterC83#21583
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
4. #24908102C2semidirect
5. #24908095C2semidirect
6. #24908053C2semidirect
7. #24908046C2semidirect
10. #24907359C2semidirect
11. #24907359C2semidirect
12. #24907292C2semidirect
13. #24907292C2semidirect
20. #19128042C4semidirect
93. #14581725C2×C4semidirect
95. #14581723C2×C4semidirect
98. #14581715C2×C4semidirect
99. #14581715C2×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×#191280422
#249099912
#117×Dic832
#249081022
#249080952
#249080532
#249080462
#249077732
#249076602
#249073592
#249072922
#249072272
C83×#18252
#2158383