Finite Group Databasebeta

← #35415008

Subgroups — #35415008

Order 21248.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127759C211yesmaximal, characteristic
2#24910087C211yesmaximal, characteristic
3#24910070C211yesmaximal, characteristic
4#24908232C211yesmaximal, characteristic, complemented
5#24908230C211yesmaximal, characteristic, complemented
6#24908223C211yesmaximal, characteristic, complemented
7#24908220C211yesmaximal, characteristic, complemented
8#24907664C211yesmaximal, characteristic
9#24907660C211yesmaximal, characteristic
10#24907651C211yesmaximal, characteristic
11#24907628C211yesmaximal, characteristic, complemented
12#24907624C211yesmaximal, characteristic, complemented
13#24907595C211yesmaximal, characteristic, complemented
14#24907589C211yesmaximal, characteristic, complemented
15C83×#1853C211yesmaximal, characteristic
16C2×#14581751C2211yescharacteristic
17C2×#14581751C2211yescharacteristic
18#19127759C418yescomplemented
19C2×#14581726C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581724C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×Dic83⋊C8C2211yescharacteristic
26C2×Dic83⋊C8C2211yescharacteristic
27C2×C22⋊(C83⋊C8)C2211yescharacteristic
28C2×#14581702C2211yescharacteristic
29C2×C8×Dic83C2211yescharacteristic
30#19127597—21no—
31#19127597—21no—
32#19127595—21no—
33#19127589—21no—
34#19127586—21no—
35#19127584—21no—
36#19127574—21no—
37#19127574—21no—
38#19127570C2211yescharacteristic
39#19127570C2211yescharacteristic
40#19127570C2211yescharacteristic
41#19127566C2211yescharacteristic
42#19127566C2211yescharacteristic
43#19127566C2211yescharacteristic
44#19127561C2211yescharacteristic
45C4⋊C4×Dic83C2211yescharacteristic
46C4⋊C4×Dic83C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127463C2211yescharacteristic
53C22×C4⋊C4×C83C2211yescharacteristic
54C2×#106×C83C2211yescharacteristic
55C2×#106×C83C2211yescharacteristic
56C2×C22⋊C8×C83C2211yescharacteristic
57#380×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62#14581751—41no—
63#14581751—41no—
64#14581751C2×C414yes—
65#14581751Q814yes—
66C2×C4⋊Dic83D411yescharacteristic
67C2×C4⋊Dic83D411yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×Dic83⋊C4—21no—
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×C22⋊Dic83—21no—
79C2×C22⋊Dic83—21no—
80C2×C22⋊Dic83D411yescharacteristic
81C2×C22⋊Dic83D411yescharacteristic
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83C2311yescharacteristic
86C2×C4×Dic83C2311yescharacteristic
87#14581726—41no—
88#14581725—41no—
89#14581725—41no—
90#14581725—41no—
91#14581724—41no—
92#14581723—41no—
93Dic83⋊C8C2×C414yescomplemented
94Dic83⋊C8C2×C414yescomplemented
95C22⋊(C83⋊C8)C2×C414yes—
96#14581702C2×C414yescomplemented
97C8×Dic83C2×C414yescomplemented
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C23×C4×C83—21no—
108C23×C4×C83C2311yescharacteristic
109C22×C8×C83C2311yescharacteristic
110C22×C8×C83C2311yescharacteristic
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—41no—
113C2×C4⋊C4×C83—41no—
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C22⋊C4×C83—21no—
119#106×C83—41no—
120#106×C83—41no—
121C22⋊C8×C83C2×C414yes—
122#95×C83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83C2×D411yescharacteristic
128C22×Dic83C2×D411yescharacteristic
129C22×Dic83C2×D411yescharacteristic
130C22×Dic83C2×D411yescharacteristic
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8C22×C412yes—
136C2×C83⋊C8C22×C412yes—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83C4⋊C414yes—
146C4⋊Dic83C4⋊C414yes—
147Dic83⋊C4—81no—
148Dic83⋊C4—81no—
149Dic83⋊C4—81no—
150Dic83⋊C4—81no—
151Dic83⋊C4—81no—
152Dic83⋊C4—81no—
153C22⋊Dic83—42no—
154C22⋊Dic83—81no—
155C22⋊Dic83C4⋊C414yes—
156C22⋊Dic83C4⋊C414yes—
157C4×Dic83—42no—
158C4×Dic83—81no—
159C4×Dic83—81no—
160C4×Dic83C22×C412yes—
161C4×Dic83C22×C412yes—
162C4×Dic83C2×Q812yes—
163C4×Dic83C2×Q812yes—
164C24×C83C2×D411yescharacteristic
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—42no—
174C22×C4×C83—81no—
175C22×C4×C83—81no—
176C22×C4×C83C22×C412yes—
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×Q812yes—
179C22×C4×C83C2411yescharacteristic
180C2×C8×C83—21no—
181C2×C8×C83—21no—
182C2×C8×C83—21no—
183C2×C8×C83—21no—
184C2×C8×C83C22×C412yes—
185C2×C8×C83C22×C412yes—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—42no—
195C22⋊C4×C83—81no—
196C23×C83—21no—
197C23×C83—21no—
198C23×C83—21no—
199C23×C83—21no—
200C23×C83—21no—
201C23×C83—21no—
202C23×C83C2×C4⋊C412yes—
203C23×C83C22×D411yescharacteristic
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—22no—
207C2×C4×C83—22no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—41no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—84no—
223C2×C4×C83C2×C4⋊C412yes—
224C2×C4×C83C23×C411yescharacteristic
225C2×C4×C83C22×Q811yescharacteristic
226C2×C4×C83H(5,𝔽2)11yescharacteristic
227C2×C4×C83H(5,𝔽2)11yescharacteristic
228C2×C4×C832−1+411yescharacteristic
229C2×C4×C832−1+411yescharacteristic
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83C2×C4⋊C412yes—
251C2×Dic83C2×C4⋊C412yes—
252C2×Dic83C2×C4⋊C412yes—
253C2×Dic83C2×C4⋊C412yes—
254C8×C83—22no—
255C8×C83—22no—
256C83⋊C8—22no—
257C83⋊C8—22no—
258C22×C83—22no—
259C22×C83—22no—
260C22×C83—22no—
261C22×C83—41no—
262C22×C83—41no—
263C22×C83—41no—
264C22×C83—41no—
265C22×C83C22×C4⋊C411yescharacteristic
266C22×C83#53511yescharacteristic
267C22×C83#53611yescharacteristic
268C22×C83#57611yescharacteristic
269C22×C83#57711yescharacteristic
270C22×C83#57711yescharacteristic
271C22×C83#57811yescharacteristic
272C4×C83—22no—
273C4×C83—81no—
274C4×C83—81no—
275C4×C83—81no—
276C4×C83—81no—
277C4×C83#52011yescharacteristic
278C4×C83#52011yescharacteristic
279C4×C83#54411yescharacteristic
280C4×C83#54411yescharacteristic
281Dic83—22no—
282Dic83—22no—
283Dic83—22no—
284Dic83—22no—
285Dic83—81no—
286Dic83—81no—
287Dic83—81no—
288Dic83—81no—
289#21992—831nomaximal, Sylow
290C2×C83—42no—
291C2×C83#231711yescharacteristic
292C2×C83#291211yescharacteristic
293C2×C83#291311yescharacteristic
294C2×C83#322411yescharacteristic
295C2×C83#322511yescharacteristic
296C2×C83#324111yescharacteristic
297C2×C83#324211yescharacteristic
298C2×#433—831no—
299#2317—831no—
300#2314—831no—
301#1949—831no—
302#1949—831no—
303#1948—831no—
304#1947—831no—
305#1854—831no—
306#1854—831no—
307#1853D8311yescharacteristic
308#1852—831no—
309#1847—831no—
310#1846—831no—
311#1836—831no—
312#1834—831no—
313C83#2199211yescharacteristic, complemented, Sylow
314C2×#117—831no—
315C2×#117—831no—
316C22×C4⋊C4C2×D8311yescharacteristic
317#433—838no—
318C2×#107—831no—
319C2×#107—831no—
320C2×#107—831no—
321C2×#107—831no—
322C2×#106—831no—
323C2×#106—831no—
324C2×#106C2×D8311yescharacteristic
325C2×#106C2×D8311yescharacteristic
326C2×C4⋊C8—831no—
327C2×C4⋊C8—831no—
328C2×C22⋊C8—831no—
329C2×C22⋊C8C2×D8311yescharacteristic
330C2×#97—831no—
331C2×C4×C8—831no—
332#389—1661no—
333#389—1661no—
334#388—1661no—
335#388—1661no—
336#387—1661no—
337#387—1661no—
338#385—1661no—
339#385—1661no—
340#384—831no—
341#384—831no—
342#384—831no—
343#382—831no—
344#382—831no—
345#382—831no—
346#380—831no—
347#380C2×D8311yescharacteristic
348C4×C4⋊C4—831no—
349C4×C4⋊C4—831no—
350#340—831no—
351#340—831no—
352#340—831no—
353#340—831no—
354#340—831no—
355#340—831no—
356#340C2×D8311yescharacteristic
357#340C2×D8311yescharacteristic
358C23×C4—21no—
359C23×C4C22×D8311yescharacteristic
360C22×C8—831no—
361C22×C8—831no—
362C22×C8C22×D8311yescharacteristic
363C22×C8C22×D8311yescharacteristic
364#117—3321no—
365#117—3321no—
366#117—834no—
367#117—834no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4—41no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—831no—
375C2×C4⋊C4—831no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—1661no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4C22×D8311yescharacteristic
384C2×C4⋊C4C22×D8311yescharacteristic
385C2×C4⋊C4C22×D8311yescharacteristic
386C2×C4⋊C4C22×D8311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—831no—
389C2×C22⋊C4—831no—
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1661no—
392C2×C42—831no—
393C2×C42—831no—
394C2×C42—1661no—
395C2×C42—1661no—
396C2×C42—1661no—
397#107—3321no—
398#107—3321no—
399#107—3321no—
400#107—3321no—
401#106—41no—
402#106—41no—
403#106—3321no—
404#106—3321no—
405C4⋊C8—834no—
406C4⋊C8—834no—
407C22⋊C8—834no—
408C22⋊C8C4×D8314yes—
409#97—834no—
410C4×C8—834no—
411#95—21no—
412#95—1661no—
413#95—1661no—
414#95—1661no—
415#95—1661no—
416#95—1661no—
417#95—1661no—
418#95—1661no—
419#95—1661no—
420#95—1661no—
421C24D4×D8311yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—831no—
434C22×C4—831no—
435C22×C4—831no—
436C22×C4—831no—
437C22×C4—1661no—
438C22×C4—1661no—
439C22×C4—1661no—
440C22×C4—1661no—
441C22×C4C2×C4×D8312yes—
442C22×C4D4×D8311yescharacteristic
443C22×C4Q8×D8312yes—
444C22×C4C23×D8311yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—832no—
450C2×C8—832no—
451C2×C8—1661no—
452C2×C8—1661no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8C2×C4×D8312yes—
456C2×C8C2×C4×D8312yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—834no—
467C4⋊C4—834no—
468C4⋊C4—6641no—
469C4⋊C4—6641no—
470C4⋊C4—6641no—
471C4⋊C4—6641no—
472C4⋊C4—6641no—
473C4⋊C4—6641no—
474C4⋊C4—1661no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4—1661no—
478C4⋊C4—1661no—
479C4⋊C4—1661no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C22⋊C4—81no—
483C22⋊C4—3322no—
484C22⋊C4—834no—
485C22⋊C4—834no—
486C22⋊C4—6641no—
487C42—3322no—
488C42—832no—
489C42—832no—
490C42—832no—
491C42—832no—
492C42—6641no—
493C42—6641no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C4⋊C4×D8312yes—
501C23C2×D4×D8311yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—84no—
521C2×C4—3321no—
522C2×C4—3321no—
523C2×C4—3321no—
524C2×C4—3321no—
525C2×C4—3321no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—832no—
534C2×C4—832no—
535C2×C4—832no—
536C2×C4—832no—
537C2×C4—1661no—
538C2×C4—1661no—
539C2×C4—1661no—
540C2×C4—1661no—
541C2×C4—1661no—
542C2×C4—1661no—
543C2×C4—1661no—
544C2×C4—1661no—
545C2×C4C4⋊C4×D8312yes—
546C2×C4C22×C4×D8311yescharacteristic
547C2×C4C2×Q8×D8311yescharacteristic
548C2×C4#1458183711yescharacteristic
549C2×C4#1458183711yescharacteristic
550C2×C4#1458183911yescharacteristic
551C2×C4#1458183911yescharacteristic
552C8—22no—
553C8—22no—
554C8—1662no—
555C8—1662no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C2×C4⋊C4×D8311yescharacteristic
564C22#1912826611yescharacteristic
565C22#1912827311yescharacteristic
566C22#1912850511yescharacteristic
567C22#1912850711yescharacteristic
568C22#1912850711yescharacteristic
569C22#1912851711yescharacteristic
570C4—22no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—6641no—
576C4—6641no—
577C4—6641no—
578C4—6641no—
579C4—1662no—
580C4—1662no—
581C4—1662no—
582C4—1662no—
583C4#1912818011yescharacteristic
584C4#1912818011yescharacteristic
585C4#1912833011yescharacteristic
586C4#1912833011yescharacteristic
587C2—42no—
588C2#2491008211yescharacteristic
589C2#2491605411yescharacteristic
590C2#2491606211yescharacteristic
591C2#2491856111yescharacteristic
592C2#2491857011yescharacteristic
593C2#2491872011yescharacteristic
594C2#2491872711yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1853C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#1853D83
hypocenterC83#21992
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #24908232C2semidirect
5. #24908230C2semidirect
6. #24908223C2semidirect
7. #24908220C2semidirect
11. #24907628C2semidirect
12. #24907624C2semidirect
13. #24907595C2semidirect
14. #24907589C2semidirect
18. #19127759C4semidirect
93. Dic83⋊C8C2×C4semidirect
94. Dic83⋊C8C2×C4semidirect
96. #14581702C2×C4semidirect
97. C8×Dic83C2×C4semidirect
313. C83#21992semidirect

Maximal subgroups

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

subgroupindex
C2×#191277592
#249100872
#249100702
#249082322
#249082302
#249082232
#249082202
#249076642
#249076602
#249076512
#249076282
#249076242
#249075952
#249075892
C83×#18532
#2199283