Finite Group Databasebeta

← #35467094

Subgroups — #35467094

Order 21248.

802 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
1C2×#19128008C211yesmaximal, characteristic, complemented
2C2×#19127737C211yesmaximal, characteristic
3#24911176C211yesmaximal, characteristic
4#24909006C211yesmaximal, characteristic, complemented
5#24908965C211yesmaximal, characteristic, complemented
6#24908858C211yesmaximal, characteristic, complemented
7#24908801C211yesmaximal, characteristic, complemented
8#24908221C211yesmaximal, characteristic, complemented
9#24908048C211yesmaximal, characteristic, complemented
10#24907887C211yesmaximal, characteristic
11#24907776C211yesmaximal, characteristic
12#24907717C211yesmaximal, characteristic, complemented
13#24907662C211yesmaximal, characteristic
14#24907643C211yesmaximal, characteristic
15C83×#2362C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×#14581751C2211yescharacteristic
19#19128008—24no—
20#19127737—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581716C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581716C2211yescharacteristic
32C2×#14581716C2211yescharacteristic
33C2×C22⋊(C83⋊C8)C2211yescharacteristic
34#19127637—21no—
35#19127624—21no—
36#19127599C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127598—21no—
39#19127584—21no—
40#19127571C2211yescharacteristic
41#19127561C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C22⋊Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52C2×#117×C83C2211yescharacteristic
53#391×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#388×C83—21no—
56#388×C83—21no—
57#387×C83—21no—
58C4×C4⋊C4×C83—21no—
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C22⋊C4×C83C2211yescharacteristic
61C22×C83⋊C8—22no—
62C22×C83⋊C8—22no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68#14581784—22no—
69#14581784—41no—
70#14581751—22no—
71#14581751—41no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×C22⋊Dic83—21no—
83C2×C22⋊Dic83—21no—
84C2×C4×Dic83—21no—
85#14581725—22no—
86#14581725—22no—
87#14581725—22no—
88#14581723—22no—
89#14581723—41no—
90#14581723D414yes—
91C4⋊(C83⋊C8)—22no—
92C4⋊(C83⋊C8)—22no—
93#14581716—22no—
94#14581716—22no—
95#14581716—22no—
96#14581716—41no—
97C22⋊(C83⋊C8)—22no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C22×Q8×C83C2311yescharacteristic
102C23×C4×C83C2311yescharacteristic
103C22⋊Q8×C83—22no—
104C22⋊Q8×C83Q814yes—
105C4×Q8×C83—22no—
106C4×Q8×C83—41no—
107#117×C83—22no—
108#117×C83—41no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83—21no—
117C2×C22⋊C4×C83D411yescharacteristic
118C2×C22⋊C4×C83D411yescharacteristic
119C2×C42×C83—21no—
120C2×C42×C83—21no—
121C2×C42×C83—21no—
122C2×C42×C83C2311yescharacteristic
123C2×C42×C83C2311yescharacteristic
124#95×C83—21no—
125#95×C83—21no—
126#95×C83—21no—
127#95×C83—21no—
128#95×C83D411yescharacteristic
129#95×C83D411yescharacteristic
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C2×C83⋊C8—28no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8C2×D412yes—
146C2×C83⋊C8#4112yes—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—42no—
155C4⋊Dic83—81no—
156C4⋊Dic83C2×D412yes—
157Dic83⋊C4—42no—
158Dic83⋊C4—42no—
159Dic83⋊C4—42no—
160Dic83⋊C4—81no—
161C22⋊Dic83—42no—
162C22⋊Dic83—81no—
163C4×Dic83—42no—
164C24×C83C2×D411yescharacteristic
165C2×Q8×C83—21no—
166C2×Q8×C83—21no—
167C2×Q8×C83—21no—
168C2×Q8×C83—21no—
169C2×Q8×C83—21no—
170C2×Q8×C83C2×Q812yes—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×Q812yes—
185C22×C4×C83#4112yes—
186C22×C4×C83C2411yescharacteristic
187C4⋊C4×C83—24no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—42no—
193C4⋊C4×C83—42no—
194C4⋊C4×C83C2×D412yes—
195C4⋊C4×C83C2×Q812yes—
196C22⋊C4×C83—24no—
197C22⋊C4×C83—22no—
198C22⋊C4×C83—41no—
199C22⋊C4×C83—42no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83—21no—
204C42×C83—42no—
205C42×C83—42no—
206C42×C83—42no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83C22⋊Q812yes—
214C23×C83C22×D411yescharacteristic
215Q8×C83—22no—
216Q8×C83—22no—
217Q8×C83—41no—
218Q8×C83—41no—
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—22no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—22no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83C22⋊Q812yes—
249C2×C4×C83#13512yes—
250C2×C4×C83C22×D411yescharacteristic
251C2×C4×C83C22×Q811yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83C2×#4111yescharacteristic
254C2×C4×C83H(5,𝔽2)11yescharacteristic
255C2×C4×C832−1+411yescharacteristic
256C2×Dic83—22no—
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×Dic83—41no—
268C83⋊C8—22no—
269C83⋊C8—22no—
270C83⋊C8—41no—
271C83⋊C8—41no—
272C22×C83—22no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83C2×C22⋊Q811yescharacteristic
280C22×C83D4211yescharacteristic
281C22×C83#54711yescharacteristic
282C22×C83C2×#13511yescharacteristic
283C22×C83#57511yescharacteristic
284C22×C83#57711yescharacteristic
285C22×C83#57811yescharacteristic
286C4×C83—22no—
287C4×C83—22no—
288C4×C83—42no—
289C4×C83—41no—
290C4×C83—42no—
291C4×C83—41no—
292C4×C83—41no—
293C4×C83—42no—
294C4×C83—81no—
295C4×C83#52511yescharacteristic
296C4×C83#53911yescharacteristic
297C4×C83D4×Q811yescharacteristic
298C4×C83#55411yescharacteristic
299Dic83—42no—
300Dic83—81no—
301Dic83—81no—
302Dic83—81no—
303#26351—831nomaximal, Sylow
304C2×C83—42no—
305C2×C83#244911yescharacteristic
306C2×C83#307611yescharacteristic
307C2×C83#308211yescharacteristic
308C2×C83#326911yescharacteristic
309C2×C83#328411yescharacteristic
310C2×C83#333611yescharacteristic
311C2×C83#335011yescharacteristic
312C2×#479—831no—
313C2×#427—831no—
314#2441—831no—
315#2362D8311yescharacteristic
316#2096—831no—
317#2088—831no—
318#2068—831no—
319#2059—831no—
320#1947—831no—
321#1920—831no—
322#1892—831no—
323#1873—831no—
324#1863—831no—
325#1854—831no—
326#1850—831no—
327C83#2635111yescharacteristic, complemented, Sylow
328C23×C8—831no—
329C2×#125—831no—
330C2×C22⋊Q8C2×D8311yescharacteristic
331C2×C4×Q8C2×D8311yescharacteristic
332C2×#117—831no—
333C2×#117C2×D8311yescharacteristic
334#479—1664no—
335#427—1664no—
336C2×#107—831no—
337C2×#107—831no—
338C2×#107—831no—
339C2×#106—831no—
340C2×#106—831no—
341C2×#106—831no—
342C2×C4⋊C8—831no—
343C2×C4⋊C8—831no—
344C2×#103—831no—
345C2×#103—831no—
346C2×#103—831no—
347C2×#103—831no—
348C2×C22⋊C8—831no—
349#399—1661no—
350#396—1661no—
351#391C2×D8311yescharacteristic
352#389—831no—
353#389—831no—
354#389—1661no—
355#389C2×D8311yescharacteristic
356#388—21no—
357#388—21no—
358#387—21no—
359#387—1661no—
360#384—831no—
361#380—831no—
362C4×C4⋊C4—21no—
363C4×C4⋊C4C2×D8311yescharacteristic
364C4×C22⋊C4C2×D8311yescharacteristic
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370#340—831no—
371#336—831no—
372#336—831no—
373C22×Q8C22×D8311yescharacteristic
374C23×C4C22×D8311yescharacteristic
375C22×C8—831no—
376C22×C8—831no—
377C22×C8—831no—
378C22×C8—831no—
379C22×C8—1662no—
380C22×C8—1662no—
381C22×C8—1662no—
382#125—3321no—
383#125—1662no—
384C22⋊Q8—22no—
385C22⋊Q8Dic16614yes—
386C4×Q8—22no—
387C4×Q8—41no—
388#117—22no—
389#117—41no—
390#117—3321no—
391#117—1662no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1661no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4C22×D8311yescharacteristic
406C2×C4⋊C4C22×D8311yescharacteristic
407C2×C4⋊C4C22×D8311yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1661no—
411C2×C22⋊C4—1661no—
412C2×C22⋊C4C22⋊D8311yescharacteristic
413C2×C22⋊C4C22⋊D8311yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42—1661no—
418C2×C42C22×D8311yescharacteristic
419C2×C42C22×D8311yescharacteristic
420#107—1662no—
421#107—1662no—
422#107—1662no—
423#106—3321no—
424#106—834no—
425#106—1662no—
426C4⋊C8—1662no—
427C4⋊C8—1662no—
428#103—3321no—
429#103—1662no—
430#103—1662no—
431#103—1662no—
432C22⋊C8—1662no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—1661no—
438#95—1661no—
439#95—1661no—
440#95C22⋊D8311yescharacteristic
441#95C22⋊D8311yescharacteristic
442C24C2×C22⋊D8311yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×Dic16612yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4—1661no—
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C2×Dic16612yes—
467C22×C4#1305059012yes—
468C22×C4C23×D8311yescharacteristic
469C2×C8—832no—
470C2×C8—832no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1668no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—3322no—
490C4⋊C4—3322no—
491C4⋊C4—3322no—
492C4⋊C4—3322no—
493C4⋊C4—832no—
494C4⋊C4—6641no—
495C4⋊C4—6641no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4D4×D8312yes—
504C4⋊C4C2×Dic16612yes—
505C22⋊C4—24no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C22⋊C4—42no—
509C22⋊C4—3322no—
510C22⋊C4—6641no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—3322no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23#1458177112yes—
526C23C22×C22⋊D8311yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—3321no—
571C2×C4—1662no—
572C2×C4#1458177112yes—
573C2×C4#1458181212yes—
574C2×C4C2×D4×D8311yescharacteristic
575C2×C4C22×Dic16611yescharacteristic
576C2×C4#41×D8311yescharacteristic
577C2×C4C2×#1305059011yescharacteristic
578C2×C4#1458183711yescharacteristic
579C2×C4#1458183911yescharacteristic
580C8—3321no—
581C8—3321no—
582C8—1662no—
583C8—1662no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×#1458177111yescharacteristic
592C22D4×C22⋊D8311yescharacteristic
593C22#1912835011yescharacteristic
594C22C2×#1458181211yescharacteristic
595C22#1912849911yescharacteristic
596C22#1912850911yescharacteristic
597C22#1912851611yescharacteristic
598C4—22no—
599C4—22no—
600C4—42no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—3322no—
608C4—6641no—
609C4—6641no—
610C4—6641no—
611C4#1912821011yescharacteristic
612C4#1912829311yescharacteristic
613C4D4×Dic16611yescharacteristic
614C4#1912841011yescharacteristic
615C2—42no—
616C2#2491126611yescharacteristic
617C2#2491725111yescharacteristic
618C2#2491730511yescharacteristic
619C2#2491901911yescharacteristic
620C2#2491918911yescharacteristic
621C2#2491982511yescharacteristic
622C2#2492000811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2362C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2362D83
hypocenterC83#26351
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
1. C2×#19128008C2semidirect
4. #24909006C2semidirect
5. #24908965C2semidirect
6. #24908858C2semidirect
7. #24908801C2semidirect
8. #24908221C2semidirect
9. #24908048C2semidirect
12. #24907717C2semidirect
327. C83#26351semidirect

Maximal subgroups

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

subgroupindex
C2×#191280082
C2×#191277372
#249111762
#249090062
#249089652
#249088582
#249088012
#249082212
#249080482
#249078872
#249077762
#249077172
#249076622
#249076432
C83×#23622
#2635183