Finite Group Databasebeta

← #35407115

Subgroups — #35407115

Order 21248.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127702C211yesmaximal, characteristic
2C2×#19127702C211yesmaximal, characteristic
3C2×C4×#13050574C211yesmaximal, characteristic, complemented
4C2×#19127599C211yesmaximal, characteristic, complemented
5#24908163C212yesmaximal, complemented
6#24908156C212yesmaximal, complemented
7#24908149C212yesmaximal, complemented
8#24908142C212yesmaximal, complemented
9#24907184C211yesmaximal, characteristic, complemented
10#24907184C211yesmaximal, characteristic, complemented
11C83×#2297C211yesmaximal, characteristic
12C22×#13050574C2211yescharacteristic
13C22×#13050574C2211yescharacteristic
14C22×C4⋊Dic83C2211yescharacteristic
15C22×C4⋊Dic83C2211yescharacteristic
16C22×Dic83⋊C4—21no—
17#19127702—24no—
18#19127702C418yescomplemented
19C2×#14581716C2212yes—
20C2×#14581716C2212yes—
21C2×#14581713C2212yes—
22C2×#14581713C2212yes—
23C4×#13050574—24no—
24C2×#14581703C2212yes—
25C2×C4×C83⋊C8C2212yes—
26#19127599—21no—
27#19127599—21no—
28#19127599C2212yes—
29#19127599C2212yes—
30C2×C83⋊#95—21no—
31#19127463C2212yes—
32#19127463C2212yes—
33#19127463C2212yes—
34#19127463C2212yes—
35C22×#41×C83C2211yescharacteristic
36C2×#117×C83C2211yescharacteristic
37C22×C42×C83C2211yescharacteristic
38#391×C83C2211yescharacteristic
39#391×C83C2211yescharacteristic
40#390×C83C2211yescharacteristic
41#390×C83C2211yescharacteristic
42#386×C83—21no—
43#386×C83—21no—
44C4×C22⋊C4×C83—21no—
45C4×C22⋊C4×C83—21no—
46C23×Dic83—21no—
47C23×Dic83—21no—
48C2×#13050574—22no—
49C2×#13050574—22no—
50C2×#13050574—22no—
51C2×#13050574—22no—
52C2×#13050574C2×C414yes—
53C2×#13050574D414yes—
54C22×C83⋊C8C2312yes—
55C22×C83⋊C8C2312yes—
56C2×C4⋊Dic83—21no—
57C2×C4⋊Dic83—22no—
58C2×C4⋊Dic83—21no—
59C2×C4⋊Dic83—22no—
60C2×C4⋊Dic83—22no—
61C2×C4⋊Dic83C2×C414yes—
62C2×C4⋊Dic83C2312yes—
63C2×C4⋊Dic83C2312yes—
64C2×Dic83⋊C4—22no—
65C2×Dic83⋊C4—22no—
66C2×Dic83⋊C4—44no—
67#14581716—24no—
68#14581716C2×C418yescomplemented
69#14581713—24no—
70#14581713C2×C418yescomplemented
71#14581703—24no—
72C4×C83⋊C8—24no—
73C83⋊#95—22no—
74C83⋊#95—22no—
75C2×#41×C83—22no—
76C2×#41×C83—22no—
77C2×#41×C83—22no—
78C2×#41×C83—22no—
79C2×#41×C83—22no—
80C2×#41×C83C2×C414yes—
81C22×Q8×C83C2311yescharacteristic
82C22×D4×C83—21no—
83C22×D4×C83C2311yescharacteristic
84C23×C4×C83—21no—
85C23×C4×C83D411yescharacteristic
86C23×C4×C83D411yescharacteristic
87C23×C4×C83C2311yescharacteristic
88#117×C83—22no—
89#117×C83—22no—
90C2×C4⋊C4×C83C2311yescharacteristic
91C2×C4⋊C4×C83C2311yescharacteristic
92C2×C22⋊C4×C83—21no—
93C2×C22⋊C4×C83—21no—
94C2×C22⋊C4×C83—21no—
95C2×C22⋊C4×C83—21no—
96C2×C22⋊C4×C83—21no—
97C2×C42×C83—22no—
98C2×C42×C83—22no—
99C2×C42×C83—21no—
100C2×C42×C83D411yescharacteristic
101C2×C42×C83D411yescharacteristic
102C2×C42×C83C2311yescharacteristic
103C2×C42×C83C2311yescharacteristic
104#95×C83—21no—
105#95×C83—21no—
106#95×C83—21no—
107#95×C83—21no—
108C22×Dic83—24no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—42no—
112C22×Dic83—42no—
113C22×Dic83—42no—
114C22×Dic83—42no—
115C22×Dic83—42no—
116#13050574—28no—
117#13050574—28no—
118C2×C83⋊C8—22no—
119C2×C83⋊C8—22no—
120C2×C83⋊C8—22no—
121C2×C83⋊C8—22no—
122C2×C83⋊C8C22×C414yes—
123C2×C83⋊C8C2×D414yes—
124C4⋊Dic83—24no—
125C4⋊Dic83—22no—
126C4⋊Dic83—22no—
127C4⋊Dic83—22no—
128C4⋊Dic83—42no—
129C4⋊Dic83C22×C414yes—
130Dic83⋊C4—44no—
131Dic83⋊C4—44no—
132C24×C83—21no—
133C24×C83C2×D411yescharacteristic
134#41×C83—28no—
135#41×C83—28no—
136#41×C83—44no—
137#41×C83—44no—
138C2×Q8×C83—21no—
139C2×Q8×C83—21no—
140C2×Q8×C83—21no—
141C2×Q8×C83—21no—
142C2×Q8×C83—21no—
143C2×Q8×C83C22×C412yes—
144C2×D4×C83—24no—
145C2×D4×C83—21no—
146C2×D4×C83—21no—
147C2×D4×C83—21no—
148C2×D4×C83—21no—
149C2×D4×C83—21no—
150C2×D4×C83—42no—
151C2×D4×C83—42no—
152C2×D4×C83C22×C412yes—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—21no—
156C22×C4×C83—24no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—41no—
162C22×C4×C83—41no—
163C22×C4×C83—42no—
164C22×C4×C83—42no—
165C22×C4×C83C22×C412yes—
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×C83#4112yes—
174C22×C4×C83#4112yes—
175C22×C4×C83#4112yes—
176C22×C4×C83C2411yescharacteristic
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C22⋊C4×C83—42no—
182C22⋊C4×C83—42no—
183C22⋊C4×C83—42no—
184C22⋊C4×C83—42no—
185C22⋊C4×C83—42no—
186C42×C83—22no—
187C42×C83—22no—
188C42×C83—21no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—21no—
192C42×C83—42no—
193C23×C83—21no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83—21no—
197C23×C83—22no—
198C23×C83—41no—
199C23×C83—41no—
200C23×C83—41no—
201C23×C83—41no—
202C23×C83—41no—
203C23×C83—41no—
204C23×C83C4×D412yes—
205C23×C83#12112yes—
206C23×C83#12412yes—
207C23×C83C22×D411yescharacteristic
208Q8×C83—22no—
209Q8×C83—22no—
210Q8×C83—41no—
211Q8×C83—41no—
212D4×C83—22no—
213D4×C83—22no—
214D4×C83—48no—
215D4×C83—41no—
216D4×C83—41no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—22no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—44no—
229C2×C4×C83—44no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—48no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83#11714yes—
240C2×C4×C83C4×D414yes—
241C2×C4×C83C4×D412yes—
242C2×C4×C83#12112yes—
243C2×C4×C83#12314yes—
244C2×C4×C83#12412yes—
245C2×C4×C83#12614yes—
246C2×C4×C83AGL(1,ℤ/8)12yes—
247C2×C4×C83#13712yes—
248C2×C4×C83C23×C411yescharacteristic
249C2×C4×C83C22×D411yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83C2×#4111yescharacteristic
254C2×Dic83—24no—
255C2×Dic83—48no—
256C2×Dic83—42no—
257C2×Dic83—42no—
258C2×Dic83—42no—
259C2×Dic83—42no—
260C2×Dic83—42no—
261C2×Dic83—84no—
262C83⋊C8—24no—
263C83⋊C8—24no—
264C22×C83—22no—
265C22×C83—22no—
266C22×C83—22no—
267C22×C83—22no—
268C22×C83—22no—
269C22×C83—22no—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—42no—
273C22×C83—81no—
274C22×C83—81no—
275C22×C83—81no—
276C22×C83—81no—
277C22×C83#38814yes—
278C22×C83C2×C4×D411yescharacteristic
279C22×C83C2×#12111yescharacteristic
280C22×C83C2×#12411yescharacteristic
281C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C83C2×#13711yescharacteristic
283C22×C83#57511yescharacteristic
284C22×C83#57511yescharacteristic
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83—42no—
290C4×C83—81no—
291C4×C83—81no—
292C4×C83#38814yes—
293C4×C83C2×#11711yescharacteristic
294C4×C83C2×C4×D411yescharacteristic
295C4×C83C2×#12311yescharacteristic
296C4×C83C2×#12611yescharacteristic
297Dic83—44no—
298Dic83—82no—
299#21093—831nomaximal, Sylow
300C2×C83—24no—
301C2×C83—82no—
302C2×C83C2×#38811yescharacteristic
303C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C83C4×#13711yescharacteristic
305C2×C83#305511yescharacteristic
306C2×C83#305611yescharacteristic
307C2×C83#314111yescharacteristic
308C2×C83#314211yescharacteristic
309C2×#417—831no—
310C2×#417—831no—
311C2×C4×M16—831no—
312#2297D8311yescharacteristic
313C2×#389—831no—
314#1936—832no—
315#1935—832no—
316#1934—832no—
317#1933—832no—
318#1736—831no—
319#1736—831no—
320C83#2109311yescharacteristic, complemented, Sylow
321C22×#41C2×D8311yescharacteristic
322C22×M16—831no—
323C22×M16—831no—
324C2×#117C2×D8311yescharacteristic
325C22×C4⋊C4—831no—
326C22×C4⋊C4—831no—
327C22×C4⋊C4—1661no—
328C22×C42C2×D8311yescharacteristic
329#417—838no—
330#417—1664no—
331C2×#103—832no—
332C2×#103—832no—
333C2×#102—832no—
334C2×#102—832no—
335C4×M16—1664no—
336C2×#97—832no—
337C2×C4×C8—832no—
338#391C2×D8311yescharacteristic
339#391C2×D8311yescharacteristic
340#390C2×D8311yescharacteristic
341#390C2×D8311yescharacteristic
342#389—832no—
343#389—832no—
344#389—1661no—
345#389—1661no—
346#386—21no—
347#386—21no—
348C4×C22⋊C4—21no—
349C4×C22⋊C4—21no—
350C2×#95—1661no—
351#340—832no—
352#340—832no—
353#340—832no—
354#340—832no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41C4×D8314yes—
361C22×Q8C22×D8311yescharacteristic
362C22×D4—21no—
363C22×D4C22×D8311yescharacteristic
364C23×C4—21no—
365C23×C4—1661no—
366C23×C4—1661no—
367C23×C4C22⋊D8311yescharacteristic
368C23×C4C22⋊D8311yescharacteristic
369C23×C4C22×D8311yescharacteristic
370C2×M16—834no—
371C2×M16—834no—
372C2×M16—1662no—
373C2×M16—1662no—
374C2×M16—1662no—
375C2×M16—1662no—
376C22×C8—832no—
377C22×C8—832no—
378#117—22no—
379#117—22no—
380C2×C4⋊C4—3324no—
381C2×C4⋊C4—834no—
382C2×C4⋊C4—832no—
383C2×C4⋊C4—832no—
384C2×C4⋊C4—1661no—
385C2×C4⋊C4—1662no—
386C2×C4⋊C4—1662no—
387C2×C4⋊C4—1662no—
388C2×C4⋊C4—1662no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4—1662no—
391C2×C4⋊C4C22×D8311yescharacteristic
392C2×C4⋊C4C22×D8311yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—21no—
401C2×C42C22⋊D8311yescharacteristic
402C2×C42C22⋊D8311yescharacteristic
403C2×C42C22×D8311yescharacteristic
404C2×C42C22×D8311yescharacteristic
405#103—838no—
406#103—1664no—
407#102—838no—
408#102—1664no—
409#97—1664no—
410C4×C8—1664no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1662no—
416#95—1662no—
417C24—21no—
418C24C2×C22⋊D8311yescharacteristic
419#41—28no—
420#41—28no—
421#41—44no—
422#41—44no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8C2×C4×D8312yes—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C2×C4×D8312yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—3322no—
451C22×C4—3322no—
452C22×C4—3322no—
453C22×C4—3322no—
454C22×C4—3322no—
455C22×C4—1662no—
456C22×C4—1662no—
457C22×C4—1664no—
458C22×C4C2×C4×D8312yes—
459C22×C4D4×D8312yes—
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4#1305058912yes—
466C22×C4#1305058912yes—
467C22×C4#1305058912yes—
468C22×C4#1305059012yes—
469C22×C4C23×D8311yescharacteristic
470M16—1668no—
471M16—1668no—
472C2×C8—834no—
473C2×C8—834no—
474C2×C8—1662no—
475C2×C8—1662no—
476C2×C8—1662no—
477C2×C8—1662no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—3324no—
483C4⋊C4—3322no—
484C4⋊C4—3324no—
485C4⋊C4—834no—
486C4⋊C4—1662no—
487C4⋊C4—1662no—
488C4⋊C4—1662no—
489C4⋊C4—1664no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C42—22no—
496C42—22no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—42no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C4×C22⋊D8312yes—
514C23#1458176412yes—
515C23#1458177912yes—
516C23C22×C22⋊D8311yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—3328no—
549C2×C4—3322no—
550C2×C4—3322no—
551C2×C4—3322no—
552C2×C4—3322no—
553C2×C4—3322no—
554C2×C4—6644no—
555C2×C4—1664no—
556C2×C4#1458174814yes—
557C2×C4(C22⋊D83)⋊C414yes—
558C2×C4C4×C22⋊D8312yes—
559C2×C4#1458176412yes—
560C2×C4#1458177414yes—
561C2×C4#1458177912yes—
562C2×C4#1458178514yes—
563C2×C4#1458181712yes—
564C2×C4#1458182212yes—
565C2×C4C22×C4×D8311yescharacteristic
566C2×C4C2×D4×D8311yescharacteristic
567C2×C4C2×#1305058911yescharacteristic
568C2×C4C2×#1305058911yescharacteristic
569C2×C4C2×#1305058911yescharacteristic
570C2×C4C2×#1305059011yescharacteristic
571C8—1664no—
572C8—1664no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#1912759514yes—
587C22C2×C4×C22⋊D8311yescharacteristic
588C22C2×#1458176411yescharacteristic
589C22C2×#1458177911yescharacteristic
590C22C2×#1458181711yescharacteristic
591C22C2×#1458182211yescharacteristic
592C22#1912849911yescharacteristic
593C22#1912849911yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—3324no—
602C4—6642no—
603C4#1912759514yes—
604C4C2×#1458174811yescharacteristic
605C4C2×(C22⋊D83)⋊C411yescharacteristic
606C4C2×#1458177411yescharacteristic
607C4C2×#1458178511yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#1912759511yescharacteristic
611C2C4×#1458181711yescharacteristic
612C2C4×#1458182211yescharacteristic
613C2#2491708411yescharacteristic
614C2#2491709511yescharacteristic
615C2#2491785611yescharacteristic
616C2#2491786311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2297C2
socleC23×C83C22×D4
prime radicalC22×#41×C83C22
cosocleC22×C4×C83C24
hypercenter#2297D83
hypocenterC83#21093
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
3. C2×C4×#13050574C2semidirect
4. C2×#19127599C2semidirect
5. #24908163C2semidirect
6. #24908156C2semidirect
7. #24908149C2semidirect
8. #24908142C2semidirect
9. #24907184C2semidirect
10. #24907184C2semidirect
18. #19127702C4semidirect
68. #14581716C2×C4semidirect
70. #14581713C2×C4semidirect
320. C83#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#191277022
C2×C4×#130505742
C2×#191275992
#249081632
#249081562
#249081492
#249081422
#249071842
C83×#22972
#2109383