Finite Group Databasebeta

← #35407484

Subgroups — #35407484

Order 21248.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910114C211yesmaximal, characteristic, complemented
2#24909824C211yesmaximal, characteristic
3#24908106C211yesmaximal, characteristic, complemented
4#24908099C211yesmaximal, characteristic, complemented
5#24908057C211yesmaximal, characteristic, complemented
6#24908036C211yesmaximal, characteristic, complemented
7#24907799C211yesmaximal, characteristic
8#24907799C211yesmaximal, characteristic
9#24907726C211yesmaximal, characteristic
10#24907712C211yesmaximal, characteristic
11#24907360C211yesmaximal, characteristic, complemented
12C4×#14581716C211yesmaximal, characteristic, complemented
13C4×#14581713C211yesmaximal, characteristic
14#24907216C211yesmaximal, characteristic, complemented
15C2×#440×C83C211yesmaximal, characteristic
16C2×D4×Dic83C2211yescharacteristic
17C2×D4×Dic83C2211yescharacteristic
18C2×#14581726C2211yescharacteristic
19C2×#14581724C2211yescharacteristic
20C2×#14581716C2211yescharacteristic
21C2×#14581716C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×C4×C83⋊C8C2211yescharacteristic
25#19127608C2211yescharacteristic
26#19127608C2211yescharacteristic
27#19127600C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127587—21no—
30#19127587—21no—
31#19127569—21no—
32#19127567—21no—
33C4⋊C4×Dic83C2211yescharacteristic
34C4⋊C4×Dic83C2211yescharacteristic
35C4×C4⋊Dic83C2211yescharacteristic
36C4×C4⋊Dic83C2211yescharacteristic
37C22⋊C4×Dic83—21no—
38C22⋊C4×Dic83—21no—
39C4×C22⋊Dic83—21no—
40#19127547C2211yescharacteristic
41C2×C83⋊#95—21no—
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127455C2211yescharacteristic
49#19127455C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
52C22×SD16×C83C2211yescharacteristic
53C2×C4×Q8×C83C2211yescharacteristic
54C2×C4×D4×C83C2211yescharacteristic
55#440×C83C418yescomplemented
56C2×#107×C83C2211yescharacteristic
57C2×#103×C83C2211yescharacteristic
58C2×#102×C83C2211yescharacteristic
59C2×#97×C83C2211yescharacteristic
60C23×Dic83—21no—
61C23×Dic83—21no—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64D4×Dic83—22no—
65D4×Dic83—22no—
66D4×Dic83—22no—
67D4×Dic83—22no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C22⋊Dic83—21no—
75C2×C22⋊Dic83—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83—21no—
78C2×C4×Dic83—21no—
79C2×C4×Dic83—21no—
80C2×C4×Dic83C2311yescharacteristic
81C2×C4×Dic83C2311yescharacteristic
82#14581726—22no—
83#14581724—22no—
84#14581716—22no—
85#14581716—22no—
86#14581713—22no—
87#14581713—22no—
88C4×C83⋊C8—22no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95D411yescharacteristic
100C83⋊#95D411yescharacteristic
101C83⋊#95D411yescharacteristic
102C83⋊#95D411yescharacteristic
103C22×Q8×C83C2311yescharacteristic
104C22×D4×C83C2311yescharacteristic
105C23×C4×C83—21no—
106C2×SD16×C83—22no—
107C2×SD16×C83—22no—
108C2×SD16×C83C2×C414yes—
109C22×C8×C83C2311yescharacteristic
110C22×C8×C83C2311yescharacteristic
111C4×Q8×C83—22no—
112C4×Q8×C83C2×C414yescomplemented
113C4×D4×C83—22no—
114C4×D4×C83C2×C414yes—
115C2×C4⋊C4×C83—21no—
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C22⋊C4×C83—21no—
119C2×C42×C83—21no—
120C2×C42×C83C2311yescharacteristic
121#107×C83C2×C414yescomplemented
122#103×C83C2×C414yescomplemented
123#102×C83C2×C414yes—
124#97×C83C2×C414yescomplemented
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—24no—
128C22×Dic83—24no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83—81no—
132C22×Dic83—81no—
133C22×Dic83—81no—
134C22×Dic83—81no—
135C22×Dic83C2×D411yescharacteristic
136C22×Dic83C2×D411yescharacteristic
137C22×Dic83C2×D411yescharacteristic
138C22×Dic83C2×D411yescharacteristic
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—81no—
152C4⋊Dic83—81no—
153C4⋊Dic83#4112yes—
154C4⋊Dic83#4112yes—
155C22⋊Dic83—24no—
156C22⋊Dic83—24no—
157C4×Dic83—21no—
158C4×Dic83—21no—
159C4×Dic83—21no—
160C4×Dic83—21no—
161C4×Dic83—81no—
162C4×Dic83—81no—
163C4×Dic83—81no—
164C4×Dic83—81no—
165C24×C83—21no—
166C2×Q8×C83—21no—
167C2×Q8×C83—21no—
168C2×Q8×C83—21no—
169C2×Q8×C83—21no—
170C2×Q8×C83—21no—
171C2×Q8×C83C22×C412yes—
172C2×D4×C83—21no—
173C2×D4×C83—21no—
174C2×D4×C83—21no—
175C2×D4×C83C22×C412yes—
176C2×D4×C83#4112yes—
177C2×D4×C83#4112yes—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83—24no—
182C22×C4×C83—81no—
183C22×C4×C83—81no—
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2411yescharacteristic
187SD16×C83—28no—
188C2×C8×C83—21no—
189C2×C8×C83—21no—
190C2×C8×C83—21no—
191C2×C8×C83—21no—
192C2×C8×C83C22×C412yes—
193C2×C8×C83C22×C412yes—
194C4⋊C4×C83—24no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83C22×C412yes—
198C4⋊C4×C83C22×C412yes—
199C22⋊C4×C83—24no—
200C42×C83—24no—
201C42×C83C22×C412yes—
202C42×C83#4112yes—
203C23×C83—22no—
204C23×C83—22no—
205C23×C83—22no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83—41no—
210C23×C83C22×D411yescharacteristic
211Q8×C83—22no—
212Q8×C83—22no—
213Q8×C83—41no—
214Q8×C83—41no—
215D4×C83—22no—
216D4×C83—22no—
217D4×C83—22no—
218D4×C83#11714yes—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—84no—
233C2×C4×C83C4×D412yes—
234C2×C4×C83C4×D412yes—
235C2×C4×C83C23×C411yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—22no—
246C2×Dic83—22no—
247C2×Dic83—21no—
248C2×Dic83—21no—
249C2×Dic83—21no—
250C2×Dic83—21no—
251C2×Dic83—21no—
252C2×Dic83—21no—
253C2×Dic83—21no—
254C2×Dic83—21no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—84no—
266C2×Dic83—84no—
267C8×C83—22no—
268C8×C83—22no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83—24no—
272C22×C83—42no—
273C22×C83—42no—
274C22×C83—42no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—42no—
278C22×C83C2×C4×D411yescharacteristic
279C22×C83#52511yescharacteristic
280C22×C83#52511yescharacteristic
281C22×C83#57611yescharacteristic
282C22×C83#57711yescharacteristic
283C22×C83#57711yescharacteristic
284C22×C83#57811yescharacteristic
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—42no—
288C4×C83—42no—
289C4×C83—42no—
290C4×C83C2×#11711yescharacteristic
291C4×C83C4×#4111yescharacteristic
292C4×C83#52911yescharacteristic
293C4×C83#52911yescharacteristic
294Dic83—41no—
295Dic83—42no—
296Dic83—41no—
297Dic83—41no—
298Dic83—42no—
299Dic83—41no—
300Dic83—81no—
301Dic83—81no—
302#21125—831nomaximal, Sylow
303C2×C83—44no—
304C2×C83#229011yescharacteristic
305C2×C83#294611yescharacteristic
306C2×C83#294711yescharacteristic
307C2×C83#311811yescharacteristic
308C2×C83#312011yescharacteristic
309C2×C83#312311yescharacteristic
310C2×C83#312411yescharacteristic
311C2×#440D8311yescharacteristic
312#2320—831no—
313#2275—831no—
314#1928—831no—
315#1927—831no—
316#1921—831no—
317#1918—831no—
318#1877—831no—
319#1877—831no—
320#1864—831no—
321#1862—831no—
322#1776—831no—
323C4×#103—831no—
324C4×#102—831no—
325#1746—831no—
326C83#2112511yescharacteristic, complemented, Sylow
327C22×SD16C2×D8311yescharacteristic
328C2×C4×Q8C2×D8311yescharacteristic
329C2×C4×D4—831no—
330C2×C4×D4—831no—
331C2×C4×D4C2×D8311yescharacteristic
332#440Dic8318yescomplemented
333C2×#107—831no—
334C2×#107C2×D8311yescharacteristic
335C2×#106—831no—
336C2×#103—831no—
337C2×#103—831no—
338C2×#103C2×D8311yescharacteristic
339C2×#102—831no—
340C2×#102—831no—
341C2×#102C2×D8311yescharacteristic
342C2×#97C2×D8311yescharacteristic
343C2×C4×C8—831no—
344#391—831no—
345#391—831no—
346#389—831no—
347#389—831no—
348#387—1661no—
349#387—1661no—
350#383—1661no—
351#382—1661no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—831no—
355C4×C4⋊C4—831no—
356C4×C22⋊C4—1661no—
357C4×C22⋊C4—1661no—
358C4×C22⋊C4—1661no—
359#376—831no—
360C2×#95—1661no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#336—831no—
368#336—831no—
369#336—831no—
370#336—831no—
371C22×Q8C22×D8311yescharacteristic
372C22×D4C22×D8311yescharacteristic
373C23×C4—21no—
374C23×C4—1661no—
375C23×C4—1661no—
376C2×SD16—22no—
377C2×SD16—22no—
378C2×SD16C2×Dic8314yes—
379C22×C8—831no—
380C22×C8—831no—
381C22×C8C22×D8311yescharacteristic
382C22×C8C22×D8311yescharacteristic
383C4×Q8—22no—
384C4×Q8C2×Dic8314yescomplemented
385C4×D4—22no—
386C4×D4—1662no—
387C4×D4—1662no—
388C4×D4—1662no—
389C4×D4—1662no—
390C4×D4C2×Dic8314yes—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—1661no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1661no—
402C2×C22⋊C4—1661no—
403C2×C42—21no—
404C2×C42—831no—
405C2×C42—831no—
406C2×C42—1661no—
407C2×C42—1661no—
408C2×C42—1661no—
409C2×C42—1661no—
410C2×C42C22×D8311yescharacteristic
411#107—1662no—
412#107C2×Dic8314yescomplemented
413#106—1662no—
414#103—1662no—
415#103—1662no—
416#103C2×Dic8314yescomplemented
417#102—1662no—
418#102—1662no—
419#102C2×Dic8314yes—
420#97C2×Dic8314yescomplemented
421C4×C8—1662no—
422#95—831no—
423#95—831no—
424#95—831no—
425#95—831no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C22×Dic8312yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C22×Dic8312yes—
447C2×D4#1305059012yes—
448C2×D4#1305059012yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—831no—
456C22×C4—831no—
457C22×C4—831no—
458C22×C4—831no—
459C22×C4—6641no—
460C22×C4—6641no—
461C22×C4—6641no—
462C22×C4—6641no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4—1664no—
468C22×C4—1664no—
469C22×C4D4×D8311yescharacteristic
470C22×C4D4×D8311yescharacteristic
471C22×C4C23×D8311yescharacteristic
472SD16—28no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8C22×Dic8312yes—
484C2×C8C22×Dic8312yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—832no—
489C4⋊C4—832no—
490C4⋊C4—6641no—
491C4⋊C4—6641no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4C22×Dic8312yes—
499C4⋊C4C22×Dic8312yes—
500C22⋊C4—24no—
501C22⋊C4—1664no—
502C22⋊C4—1664no—
503C42—24no—
504C42—6641no—
505C42—6641no—
506C42—6641no—
507C42—6641no—
508C42—1661no—
509C42—1661no—
510C42—1661no—
511C42—1661no—
512C42C22×Dic8312yes—
513C42#1305058912yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D8311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4#1458175114yes—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
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—84no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—6644no—
555C2×C4—6644no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4—1661no—
559C2×C4—1661no—
560C2×C4—1662no—
561C2×C4—1662no—
562C2×C4—1661no—
563C2×C4—1661no—
564C2×C4—1661no—
565C2×C4—1661no—
566C2×C4—1661no—
567C2×C4—1661no—
568C2×C4—1661no—
569C2×C4—1661no—
570C2×C4D4×Dic8312yes—
571C2×C4D4×Dic8312yes—
572C2×C4C23×Dic8311yescharacteristic
573C2×C4#41×D8311yescharacteristic
574C2×C4#41×D8311yescharacteristic
575C2×C4C2×#1305058911yescharacteristic
576C2×C4C2×#1305059011yescharacteristic
577C2×C4C2×#1305059011yescharacteristic
578C8—22no—
579C8—22no—
580C8—3321no—
581C8—3321no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C2×D4×Dic8311yescharacteristic
590C22#1912821011yescharacteristic
591C22#1912821011yescharacteristic
592C22#1912850511yescharacteristic
593C22#1912851011yescharacteristic
594C22#1912851111yescharacteristic
595C22#1912851711yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—3322no—
602C4—3321no—
603C4—3321no—
604C4—3322no—
605C4—3321no—
606C4—3321no—
607C4—6641no—
608C4—6641no—
609C4C2×#1458175111yescharacteristic
610C4#41×Dic8311yescharacteristic
611C4#1912823111yescharacteristic
612C4#1912823111yescharacteristic
613C2—44no—
614C2#2490990811yescharacteristic
615C2#2491629611yescharacteristic
616C2#2491630711yescharacteristic
617C2#2491765111yescharacteristic
618C2#2491768111yescharacteristic
619C2#2491771411yescharacteristic
620C2#2491772911yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#440×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#440D83
hypocenterC83#21125
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #24910114C2semidirect
3. #24908106C2semidirect
4. #24908099C2semidirect
5. #24908057C2semidirect
6. #24908036C2semidirect
11. #24907360C2semidirect
12. C4×#14581716C2semidirect
14. #24907216C2semidirect
55. #440×C83C4semidirect
112. C4×Q8×C83C2×C4semidirect
121. #107×C83C2×C4semidirect
122. #103×C83C2×C4semidirect
124. #97×C83C2×C4semidirect
326. C83#21125semidirect
332. #440Dic83semidirect
384. C4×Q8C2×Dic83semidirect
412. #107C2×Dic83semidirect
416. #103C2×Dic83semidirect
420. #97C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
#249101142
#249098242
#249081062
#249080992
#249080572
#249080362
#249077992
#249077262
#249077122
#249073602
C4×#145817162
C4×#145817132
#249072162
C2×#440×C832
#2112583