Finite Group Databasebeta

← #35423840

Subgroups — #35423840

Order 21248.

866 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
1#24910412C211yesmaximal, characteristic
2#24910384C211yesmaximal, characteristic, complemented
3#24908162C211yesmaximal, characteristic
4#24908148C211yesmaximal, characteristic
5#24908057C211yesmaximal, characteristic, complemented
6#24908036C211yesmaximal, characteristic, complemented
7#24907820C211yesmaximal, characteristic
8#24907820C211yesmaximal, characteristic
9#24907597C211yesmaximal, characteristic, complemented
10#24907473C211yesmaximal, characteristic
11#24907468C211yesmaximal, characteristic, complemented
12#24907335C211yesmaximal, characteristic, complemented
13#24907295C211yesmaximal, characteristic, complemented
14#103×Dic83C211yesmaximal, characteristic, complemented
15C2×#440×C83C211yesmaximal, characteristic
16C2×Q8×Dic83C2211yescharacteristic
17C2×Q8×Dic83C2211yescharacteristic
18C2×#14581751—21no—
19C2×C4⋊(C83⋊C8)C2211yescharacteristic
20C2×#14581716C2211yescharacteristic
21C2×#14581716C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581702C2211yescharacteristic
25C2×C8×Dic83C2211yescharacteristic
26#19127604C2211yescharacteristic
27#19127604C2211yescharacteristic
28#19127600—21no—
29#19127600—21no—
30#19127600C2211yescharacteristic
31#19127600C2211yescharacteristic
32#19127594—21no—
33#19127594—21no—
34#19127571C2211yescharacteristic
35#19127567—21no—
36#19127567C2211yescharacteristic
37#19127561—21no—
38C4⋊C4×Dic83C2211yescharacteristic
39C4⋊C4×Dic83C2211yescharacteristic
40C4×C4⋊Dic83—21no—
41C4×C4⋊Dic83C2211yescharacteristic
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127464C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
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
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62Q8×Dic83—22no—
63Q8×Dic83—22no—
64Q8×Dic83—41no—
65Q8×Dic83—41no—
66#14581751—44no—
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83D411yescharacteristic
74C2×C4⋊Dic83D411yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C22⋊Dic83—21no—
80C2×C22⋊Dic83—21no—
81C2×C22⋊Dic83—21no—
82C2×C22⋊Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83C2311yescharacteristic
88C2×C4×Dic83C2311yescharacteristic
89C4⋊(C83⋊C8)—22no—
90#14581716—22no—
91#14581716—22no—
92#14581713—41no—
93#14581713—41no—
94#14581702—41no—
95C8×Dic83—41no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95D411yescharacteristic
103C83⋊#95D411yescharacteristic
104C22×Q8×C83C2311yescharacteristic
105C22×D4×C83C2311yescharacteristic
106C23×C4×C83—21no—
107C2×SD16×C83—22no—
108C2×SD16×C83—22no—
109C2×SD16×C83C2×C414yes—
110C22×C8×C83C2311yescharacteristic
111C22×C8×C83C2311yescharacteristic
112C4×Q8×C83—22no—
113C4×Q8×C83C2×C414yes—
114C4×D4×C83—22no—
115C4×D4×C83C2×C414yes—
116C2×C4⋊C4×C83—21no—
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C22⋊C4×C83—21no—
120C2×C42×C83—21no—
121C2×C42×C83C2311yescharacteristic
122#107×C83C2×C414yes—
123#103×C83C2×C414yes—
124#102×C83C2×C414yes—
125#97×C83C2×C414yes—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83C2×D411yescharacteristic
131C22×Dic83C2×D411yescharacteristic
132C22×Dic83C2×D411yescharacteristic
133C22×Dic83C2×D411yescharacteristic
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C4⋊Dic83—22no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—41no—
150C4⋊Dic83—42no—
151C4⋊Dic83—42no—
152C4⋊Dic83—42no—
153C4⋊Dic83—42no—
154C4⋊Dic83—42no—
155C4⋊Dic83—81no—
156C22⋊Dic83—42no—
157C22⋊Dic83—42no—
158C22⋊Dic83—81no—
159C22⋊Dic83—81no—
160C4×Dic83—21no—
161C4×Dic83—21no—
162C4×Dic83—21no—
163C4×Dic83—21no—
164C4×Dic83—42no—
165C4×Dic83—42no—
166C4×Dic83—42no—
167C4×Dic83—42no—
168C24×C83—21no—
169C2×Q8×C83—21no—
170C2×Q8×C83—21no—
171C2×Q8×C83—21no—
172C2×Q8×C83C22×C412yes—
173C2×Q8×C83#4112yes—
174C2×Q8×C83#4112yes—
175C2×D4×C83—21no—
176C2×D4×C83—21no—
177C2×D4×C83—21no—
178C2×D4×C83—21no—
179C2×D4×C83—21no—
180C2×D4×C83C22×C412yes—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—24no—
185C22×C4×C83—42no—
186C22×C4×C83—42no—
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2411yescharacteristic
190SD16×C83—28no—
191C2×C8×C83—21no—
192C2×C8×C83—21no—
193C2×C8×C83—21no—
194C2×C8×C83—21no—
195C2×C8×C83C22×C412yes—
196C2×C8×C83C22×C412yes—
197C4⋊C4×C83—24no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83C22×C412yes—
201C4⋊C4×C83C22×C412yes—
202C22⋊C4×C83—24no—
203C42×C83—24no—
204C42×C83C22×C412yes—
205C42×C83#4112yes—
206C23×C83—22no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83—41no—
210C23×C83—41no—
211C23×C83—41no—
212C23×C83—41no—
213C23×C83C22×D411yescharacteristic
214Q8×C83—22no—
215Q8×C83—22no—
216Q8×C83—22no—
217Q8×C83#11714yescomplemented
218D4×C83—22no—
219D4×C83—22no—
220D4×C83—41no—
221D4×C83—41no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—22no—
230C2×C4×C83—22no—
231C2×C4×C83—48no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83C4×D412yes—
237C2×C4×C83C4×D412yes—
238C2×C4×C83C23×C411yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C2×C4×C83H(5,𝔽2)11yescharacteristic
243C2×C4×C832−1+411yescharacteristic
244C2×Dic83—21no—
245C2×Dic83—21no—
246C2×Dic83—21no—
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—41no—
266C2×Dic83—41no—
267C2×Dic83AGL(1,ℤ/8)12yes—
268C8×C83—22no—
269C8×C83—22no—
270C83⋊C8—41no—
271C83⋊C8—41no—
272C22×C83—42no—
273C22×C83—42no—
274C22×C83—42no—
275C22×C83—81no—
276C22×C83—81no—
277C22×C83—81no—
278C22×C83—81no—
279C22×C83C2×C4×D411yescharacteristic
280C22×C83#54611yescharacteristic
281C22×C83#54811yescharacteristic
282C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C83#57511yescharacteristic
284C22×C83#57711yescharacteristic
285C22×C83#57811yescharacteristic
286C4×C83—24no—
287C4×C83—22no—
288C4×C83—22no—
289C4×C83—42no—
290C4×C83—42no—
291C4×C83C2×#11711yescharacteristic
292C4×C83#52011yescharacteristic
293C4×C83#55211yescharacteristic
294C4×C83#55311yescharacteristic
295Dic83—22no—
296Dic83—41no—
297Dic83—41no—
298Dic83—41no—
299Dic83—81no—
300Dic83—81no—
301Dic83—81no—
302Dic83—81no—
303#22781—831nomaximal, Sylow
304C2×C83—82no—
305C2×C83#235211yescharacteristic
306C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
307C2×C83#294711yescharacteristic
308C2×C83#330811yescharacteristic
309C2×C83#332111yescharacteristic
310C2×C83#336811yescharacteristic
311C2×C83#337711yescharacteristic
312C2×#440D8311yescharacteristic
313#2354—831no—
314#2351—831no—
315#1936—831no—
316#1934—831no—
317#1921—831no—
318#1918—831no—
319#1880—831no—
320#1880—831no—
321#1836—831no—
322#1804—831no—
323#1803—831no—
324#1771—831no—
325#1763—831no—
326C4×#103—831no—
327C83#2278111yescharacteristic, complemented, Sylow
328C22×SD16C2×D8311yescharacteristic
329C2×C4×Q8—831no—
330C2×C4×Q8—831no—
331C2×C4×Q8C2×D8311yescharacteristic
332C2×C4×D4C2×D8311yescharacteristic
333C2×#117—1661no—
334#440Dic8318yescomplemented
335C2×#107C2×D8311yescharacteristic
336C2×C4⋊C8—831no—
337C2×#103—831no—
338C2×#103—831no—
339C2×#103C2×D8311yescharacteristic
340C2×#102—831no—
341C2×#102—831no—
342C2×#102C2×D8311yescharacteristic
343C2×#97—831no—
344C2×#97C2×D8311yescharacteristic
345C2×C4×C8—831no—
346#390—831no—
347#390—831no—
348#389—831no—
349#389—831no—
350#389—1661no—
351#389—1661no—
352#388—1661no—
353#388—1661no—
354#384—831no—
355#382—831no—
356#382—1661no—
357#380—1661no—
358C4×C4⋊C4—831no—
359C4×C4⋊C4—831no—
360C4×C4⋊C4—831no—
361C4×C4⋊C4—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
373C22×D4C22×D8311yescharacteristic
374C23×C4—21no—
375C2×SD16—22no—
376C2×SD16—22no—
377C2×SD16C2×Dic8314yes—
378C22×C8—831no—
379C22×C8—831no—
380C22×C8C22×D8311yescharacteristic
381C22×C8C22×D8311yescharacteristic
382C4×Q8—22no—
383C4×Q8—3321no—
384C4×Q8—3321no—
385C4×Q8—1662no—
386C4×Q8—1662no—
387C4×Q8C2×Dic8314yes—
388C4×D4—22no—
389C4×D4C2×Dic8314yes—
390#117—3324no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1661no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4C22×D8311yescharacteristic
405C2×C4⋊C4C22×D8311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1661no—
408C2×C22⋊C4—1661no—
409C2×C22⋊C4—1661no—
410C2×C22⋊C4—1661no—
411C2×C42—21no—
412C2×C42—831no—
413C2×C42—831no—
414C2×C42—1661no—
415C2×C42—1661no—
416C2×C42—1661no—
417C2×C42—1661no—
418C2×C42C22×D8311yescharacteristic
419#107C2×Dic8314yes—
420C4⋊C8—1662no—
421#103—1662no—
422#103—1662no—
423#103C2×Dic8314yes—
424#102—3321no—
425#102—3321no—
426#102C2×Dic8314yes—
427#97—3321no—
428#97C2×Dic8314yes—
429C4×C8—3321no—
430#95—831no—
431#95—831no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95—1661no—
438C24—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C22×Dic8312yes—
443C2×Q8#1305059012yes—
444C2×Q8#1305059012yes—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C22×Dic8312yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—831no—
458C22×C4—831no—
459C22×C4—831no—
460C22×C4—831no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4D4×D8311yescharacteristic
466C22×C4D4×D8311yescharacteristic
467C22×C4C23×D8311yescharacteristic
468SD16—28no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8C22×Dic8312yes—
480C2×C8C22×Dic8312yes—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—3322no—
485C4⋊C4—3322no—
486C4⋊C4—3322no—
487C4⋊C4—3321no—
488C4⋊C4—3322no—
489C4⋊C4—3322no—
490C4⋊C4—6641no—
491C4⋊C4—1662no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4C22×Dic8312yes—
501C4⋊C4C22×Dic8312yes—
502C22⋊C4—24no—
503C22⋊C4—3322no—
504C22⋊C4—3322no—
505C22⋊C4—6641no—
506C22⋊C4—6641no—
507C42—24no—
508C42—3322no—
509C42—3322no—
510C42—3322no—
511C42—3322no—
512C42—1661no—
513C42—1661no—
514C42—1661no—
515C42—1661no—
516C42C22×Dic8312yes—
517C42#1305058912yes—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×D4×D8311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8#1458175114yescomplemented
530D4—22no—
531D4—22no—
532D4—41no—
533D4—41no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—48no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—832no—
561C2×C4—1661no—
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×C4—1661no—
571C2×C4—1661no—
572C2×C4D4×Dic8312yes—
573C2×C4D4×Dic8312yes—
574C2×C4C23×Dic8311yescharacteristic
575C2×C4C2×#1305058911yescharacteristic
576C2×C4C2×#1305059011yescharacteristic
577C2×C4C2×#1305059011yescharacteristic
578C2×C4#1458183811yescharacteristic
579C2×C4#1458183911yescharacteristic
580C8—22no—
581C8—22no—
582C8—3321no—
583C8—3321no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22C2×D4×Dic8311yescharacteristic
592C22#1912834711yescharacteristic
593C22#1912836011yescharacteristic
594C22AGL(1,ℤ/8)×D8311yescharacteristic
595C22#1912849711yescharacteristic
596C22#1912851011yescharacteristic
597C22#1912851711yescharacteristic
598C4—22no—
599C4—24no—
600C4—22no—
601C4—42no—
602C4—42no—
603C4—3321no—
604C4—3321no—
605C4—3321no—
606C4—6641no—
607C4—6641no—
608C4—6641no—
609C4—6641no—
610C4—1662no—
611C4C2×#1458175111yescharacteristic
612C4#1912818411yescharacteristic
613C4#1912839111yescharacteristic
614C4#1912839711yescharacteristic
615C2—82no—
616C2#2491039211yescharacteristic
617C2AGL(1,ℤ/8)×Dic8311yescharacteristic
618C2#2491630711yescharacteristic
619C2#2491945911yescharacteristic
620C2#2491965411yescharacteristic
621C2#2492019311yescharacteristic
622C2#2492030611yescharacteristic
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#22781
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
2. #24910384C2semidirect
5. #24908057C2semidirect
6. #24908036C2semidirect
9. #24907597C2semidirect
11. #24907468C2semidirect
12. #24907335C2semidirect
13. #24907295C2semidirect
14. #103×Dic83C2semidirect
55. #440×C83C4semidirect
217. Q8×C83#117semidirect
327. C83#22781semidirect
334. #440Dic83semidirect
529. Q8#14581751semidirect

Maximal subgroups

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

subgroupindex
#249104122
#249103842
#249081622
#249081482
#249080572
#249080362
#249078202
#249075972
#249074732
#249074682
#249073352
#249072952
#103×Dic832
C2×#440×C832
#2278183