Finite Group Databasebeta

← #35409311

Subgroups — #35409311

Order 21248.

826 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×#19127762C211yesmaximal, characteristic
2#24910072C211yesmaximal, characteristic
3#24908231C211yesmaximal, characteristic, complemented
4#24908231C211yesmaximal, characteristic, complemented
5#24908226C211yesmaximal, characteristic, complemented
6#24908221C211yesmaximal, characteristic, complemented
7#24907662C211yesmaximal, characteristic
8#24907662C211yesmaximal, characteristic
9#24907656C211yesmaximal, characteristic
10#24907651C211yesmaximal, characteristic
11#24907361C211yesmaximal, characteristic, complemented
12#24907361C211yesmaximal, characteristic, complemented
13C4×#14581725C211yesmaximal, characteristic, complemented
14C4×#14581723C211yesmaximal, characteristic, complemented
15C83×#2274C211yesmaximal, characteristic
16#19127762C418yescomplemented
17C2×#14581725C2211yescharacteristic
18C2×#14581725C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×C4⋊(C83⋊C8)C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×C22⋊(C83⋊C8)C2211yescharacteristic
28C2×C22⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581703C2211yescharacteristic
30C2×C4×C83⋊C8C2211yescharacteristic
31#19127594—21no—
32#19127594—21no—
33#19127587—21no—
34#19127587—21no—
35#19127571C2211yescharacteristic
36#19127567C2211yescharacteristic
37#19127561C2211yescharacteristic
38#19127561C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49C2×#117×C83C2211yescharacteristic
50C2×#117×C83C2211yescharacteristic
51C22×C4⋊C4×C83C2211yescharacteristic
52C4×C4⋊C4×C83—21no—
53C4×C4⋊C4×C83—21no—
54C4×C4⋊C4×C83C2211yescharacteristic
55C4×C4⋊C4×C83C2211yescharacteristic
56C4×C4⋊C4×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83—21no—
59C4×C22⋊C4×C83—21no—
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C22⋊Dic83—21no—
71C2×C22⋊Dic83—21no—
72C2×C4×Dic83—21no—
73C2×C4×Dic83—21no—
74#14581725—22no—
75#14581725—22no—
76#14581725—22no—
77#14581725—22no—
78#14581723—22no—
79#14581723—22no—
80#14581723—22no—
81#14581723—22no—
82C4⋊(C83⋊C8)C2×C414yescomplemented
83C4⋊(C83⋊C8)C2×C414yescomplemented
84C22⋊(C83⋊C8)C2×C414yes—
85C22⋊(C83⋊C8)C2×C414yes—
86#14581703C2×C414yescomplemented
87C4×C83⋊C8C2×C414yescomplemented
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C23×C4×C83—21no—
95C23×C4×C83C2311yescharacteristic
96#117×C83—22no—
97#117×C83—22no—
98#117×C83C2×C414yes—
99#117×C83Q814yes—
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83—22no—
102C2×C4⋊C4×C83—22no—
103C2×C4⋊C4×C83D411yescharacteristic
104C2×C4⋊C4×C83D411yescharacteristic
105C2×C4⋊C4×C83C2311yescharacteristic
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C22⋊C4×C83—21no—
110C2×C22⋊C4×C83D411yescharacteristic
111C2×C22⋊C4×C83D411yescharacteristic
112C2×C42×C83—21no—
113C2×C42×C83—21no—
114C2×C42×C83—21no—
115C2×C42×C83—21no—
116C2×C42×C83—21no—
117C2×C42×C83C2311yescharacteristic
118C2×C42×C83C2311yescharacteristic
119#95×C83—21no—
120#95×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C2×C83⋊C8—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8C22×C412yes—
136C2×C83⋊C8C22×C412yes—
137C2×C83⋊C8C22×C412yes—
138C2×C83⋊C8C22×C412yes—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—81no—
148C4⋊Dic83—81no—
149C22⋊Dic83—81no—
150C22⋊Dic83—81no—
151C4×Dic83—81no—
152C4×Dic83—81no—
153C24×C83C2×D411yescharacteristic
154C22×C4×C83—21no—
155C22×C4×C83—21no—
156C22×C4×C83—21no—
157C22×C4×C83—21no—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—24no—
163C22×C4×C83—81no—
164C22×C4×C83—81no—
165C22×C4×C83C22×C412yes—
166C22×C4×C83C2×D411yescharacteristic
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×Q812yes—
172C22×C4×C83C2411yescharacteristic
173C4⋊C4×C83—24no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83C4⋊C414yes—
179C4⋊C4×C83C4⋊C414yes—
180C4⋊C4×C83#4112yes—
181C4⋊C4×C83#4112yes—
182C4⋊C4×C83#4112yes—
183C4⋊C4×C83#4112yes—
184C22⋊C4×C83—24no—
185C22⋊C4×C83C4⋊C414yes—
186C22⋊C4×C83C4⋊C414yes—
187C42×C83—24no—
188C42×C83—81no—
189C42×C83—81no—
190C42×C83—81no—
191C42×C83—81no—
192C42×C83C22×C412yes—
193C42×C83C22×C412yes—
194C42×C83C2×Q812yes—
195C42×C83C2×Q812yes—
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—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—22no—
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—22no—
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—84no—
231C2×C4×C83C2×C4⋊C412yes—
232C2×C4×C83C2×C4⋊C412yes—
233C2×C4×C83C2×C4⋊C412yes—
234C2×C4×C83C2×C4⋊C412yes—
235C2×C4×C83C2×C4⋊C412yes—
236C2×C4×C83C23×C411yescharacteristic
237C2×C4×C83C22×Q811yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
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×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C83⋊C8—22no—
255C83⋊C8—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#52511yescharacteristic
267C22×C83#52511yescharacteristic
268C22×C83#57611yescharacteristic
269C22×C83#57711yescharacteristic
270C22×C83#57711yescharacteristic
271C22×C83#57811yescharacteristic
272C4×C83—22no—
273C4×C83—22no—
274C4×C83—22no—
275C4×C83—22no—
276C4×C83—22no—
277C4×C83—42no—
278C4×C83—42no—
279C4×C83—42no—
280C4×C83—42no—
281C4×C83C4×#4111yescharacteristic
282C4×C83C4×#4111yescharacteristic
283C4×C83#53311yescharacteristic
284C4×C83#53311yescharacteristic
285Dic83—81no—
286Dic83—81no—
287Dic83—81no—
288Dic83—81no—
289#21367—831nomaximal, Sylow
290C2×C83—42no—
291C2×C83#229611yescharacteristic
292C2×C83#291211yescharacteristic
293C2×C83#291311yescharacteristic
294C2×C83#311811yescharacteristic
295C2×C83#311911yescharacteristic
296C2×C83#312011yescharacteristic
297C2×C83#312111yescharacteristic
298C2×#433—831no—
299#2314—831no—
300#2274D8311yescharacteristic
301#1949—831no—
302#1949—831no—
303#1948—831no—
304#1947—831no—
305#1854—831no—
306#1854—831no—
307#1853—831no—
308#1852—831no—
309#1776—831no—
310#1776—831no—
311C4×#107—831no—
312C4×#106—831no—
313C83#2136711yescharacteristic, complemented, Sylow
314C2×#117C2×D8311yescharacteristic
315C2×#117C2×D8311yescharacteristic
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×#106—831no—
325C2×#106—831no—
326C2×C4⋊C8—831no—
327C2×C4⋊C8—831no—
328C2×C22⋊C8—831no—
329C2×C22⋊C8—831no—
330C2×#97—831no—
331C2×C4×C8—831no—
332#388—1661no—
333#388—1661no—
334#387—1661no—
335#387—1661no—
336#384—831no—
337#382—831no—
338#380—831no—
339#380—831no—
340C4×C4⋊C4—21no—
341C4×C4⋊C4—21no—
342C4×C4⋊C4—831no—
343C4×C4⋊C4—831no—
344C4×C4⋊C4C2×D8311yescharacteristic
345C4×C4⋊C4C2×D8311yescharacteristic
346C4×C4⋊C4C2×D8311yescharacteristic
347C4×C4⋊C4C2×D8311yescharacteristic
348C4×C22⋊C4—21no—
349C4×C22⋊C4—21no—
350#340—831no—
351#340—831no—
352#340—831no—
353#340—831no—
354#340—831no—
355#340—831no—
356#340—831no—
357#340—831no—
358C23×C4—21no—
359C23×C4C22×D8311yescharacteristic
360C22×C8—831no—
361C22×C8—831no—
362C22×C8—831no—
363C22×C8—831no—
364#117—22no—
365#117—22no—
366#117Dic16614yes—
367#117C4×D8314yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—831no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4C22⋊D8311yescharacteristic
378C2×C4⋊C4C22⋊D8311yescharacteristic
379C2×C4⋊C4C22×D8311yescharacteristic
380C2×C4⋊C4C22×D8311yescharacteristic
381C2×C4⋊C4C22×D8311yescharacteristic
382C2×C4⋊C4C22×D8311yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1661no—
385C2×C22⋊C4—1661no—
386C2×C22⋊C4C22⋊D8311yescharacteristic
387C2×C22⋊C4C22⋊D8311yescharacteristic
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—1661no—
394C2×C42—1661no—
395C2×C42C22×D8311yescharacteristic
396C2×C42C22×D8311yescharacteristic
397#107—1662no—
398#107—1662no—
399#107—1662no—
400#107—1662no—
401#106—1662no—
402#106—1662no—
403#106—1662no—
404#106—1662no—
405C4⋊C8—834no—
406C4⋊C8—834no—
407C22⋊C8—834no—
408C22⋊C8—834no—
409#97—834no—
410C4×C8—834no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1661no—
416#95—1661no—
417#95—1661no—
418#95—1661no—
419#95—1661no—
420#95—1661no—
421C24C2×C22⋊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—24no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—1661no—
434C22×C4—1661no—
435C22×C4—1661no—
436C22×C4—1661no—
437C22×C4C2×C4×D8312yes—
438C22×C4C2×C22⋊D8311yescharacteristic
439C22×C4C2×C22⋊D8311yescharacteristic
440C22×C4C2×C22⋊D8311yescharacteristic
441C22×C4C2×C22⋊D8311yescharacteristic
442C22×C4C2×C22⋊D8311yescharacteristic
443C22×C4C2×Dic16612yes—
444C22×C4C23×D8311yescharacteristic
445C2×C8—832no—
446C2×C8—832no—
447C2×C8—832no—
448C2×C8—832no—
449C2×C8—1661no—
450C2×C8—1661no—
451C2×C8—1661no—
452C2×C8—1661no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8—1661no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—6641no—
463C4⋊C4—6641no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4Dic83⋊C414yes—
473C4⋊C4Dic83⋊C414yes—
474C4⋊C4#1305059012yes—
475C4⋊C4#1305059012yes—
476C4⋊C4#1305059012yes—
477C4⋊C4#1305059012yes—
478C22⋊C4—24no—
479C22⋊C4—6641no—
480C22⋊C4—6641no—
481C22⋊C4Dic83⋊C414yes—
482C22⋊C4Dic83⋊C414yes—
483C42—24no—
484C42—81no—
485C42—81no—
486C42—81no—
487C42—81no—
488C42—6641no—
489C42—6641no—
490C42C2×C4×D8312yes—
491C42C2×C4×D8312yes—
492C42C2×Dic16612yes—
493C42C2×Dic16612yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×Dic83⋊C412yes—
501C23C22×C22⋊D8311yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—3321no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4C2×Dic83⋊C412yes—
542C2×C4C2×Dic83⋊C412yes—
543C2×C4C2×Dic83⋊C412yes—
544C2×C4C2×Dic83⋊C412yes—
545C2×C4C2×Dic83⋊C412yes—
546C2×C4C22×C4×D8311yescharacteristic
547C2×C4C22×Dic16611yescharacteristic
548C2×C4C2×#1305059011yescharacteristic
549C2×C4C2×#1305059011yescharacteristic
550C2×C4C2×#1305059011yescharacteristic
551C2×C4C2×#1305059011yescharacteristic
552C8—1662no—
553C8—1662no—
554C8—1662no—
555C8—1662no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×Dic83⋊C411yescharacteristic
564C22#1912821211yescharacteristic
565C22#1912821211yescharacteristic
566C22#1912850411yescharacteristic
567C22#1912850911yescharacteristic
568C22#1912851211yescharacteristic
569C22#1912851611yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—6641no—
580C4—6641no—
581C4—6641no—
582C4—6641no—
583C4C4×#1305059011yescharacteristic
584C4C4×#1305059011yescharacteristic
585C4#1912825811yescharacteristic
586C4#1912825811yescharacteristic
587C2—42no—
588C2#2490994611yescharacteristic
589C2#2491605611yescharacteristic
590C2#2491606111yescharacteristic
591C2#2491764711yescharacteristic
592C2#2491766211yescharacteristic
593C2#2491767711yescharacteristic
594C2#2491769211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2274C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#2274D83
hypocenterC83#21367
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2274

Complements

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

NH ≅ G/Ntype
3. #24908231C2semidirect
4. #24908231C2semidirect
5. #24908226C2semidirect
6. #24908221C2semidirect
11. #24907361C2semidirect
12. #24907361C2semidirect
13. C4×#14581725C2semidirect
14. C4×#14581723C2semidirect
16. #19127762C4semidirect
82. C4⋊(C83⋊C8)C2×C4semidirect
83. C4⋊(C83⋊C8)C2×C4semidirect
86. #14581703C2×C4semidirect
87. C4×C83⋊C8C2×C4semidirect
313. C83#21367semidirect

Maximal subgroups

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

subgroupindex
C2×#191277622
#249100722
#249082312
#249082262
#249082212
#249076622
#249076562
#249076512
#249073612
C4×#145817252
C4×#145817232
C83×#22742
#2136783