Finite Group Databasebeta

← #35467259

Subgroups — #35467259

Order 21248.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128008C211yesmaximal, characteristic, complemented
2C2×#19127741C211yesmaximal, characteristic
3#24911176C211yesmaximal, characteristic
4#24909001C211yesmaximal, characteristic, complemented
5#24908954C211yesmaximal, characteristic, complemented
6#24908844C211yesmaximal, characteristic, complemented
7#24908801C211yesmaximal, characteristic, complemented
8#24908221C211yesmaximal, characteristic, complemented
9#24908041C211yesmaximal, characteristic, complemented
10#24907897C211yesmaximal, characteristic
11#24907776C211yesmaximal, characteristic
12#24907717C211yesmaximal, characteristic, complemented
13#24907651C211yesmaximal, characteristic
14#24907648C211yesmaximal, characteristic
15C83×#2360C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×#14581751C2211yescharacteristic
19#19128008—24no—
20#19127741—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
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
52C22×C4⋊C4×C83C2211yescharacteristic
53#391×C83C2211yescharacteristic
54#389×C83—21no—
55#389×C83C2211yescharacteristic
56#387×C83—21no—
57#385×C83—21no—
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C22⋊C4×C83—21no—
60C4×C22⋊C4×C83C2211yescharacteristic
61C2×#13050574—22no—
62C2×#13050574—22no—
63C2×#13050574—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#14581725—41no—
89#14581723—22no—
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×C83—21no—
103C23×C4×C83C2311yescharacteristic
104C22⋊Q8×C83—22no—
105C22⋊Q8×C83Q814yes—
106C4×Q8×C83—22no—
107C4×Q8×C83—41no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83—22no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—41no—
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83D411yescharacteristic
119C2×C22⋊C4×C83D411yescharacteristic
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—
134#13050574—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×C83—42no—
180C22×C4×C83—42no—
181C22×C4×C83—81no—
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×Q812yes—
188C22×C4×C83#4112yes—
189C22×C4×C83C2411yescharacteristic
190C4⋊C4×C83—24no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—42no—
196C4⋊C4×C83—42no—
197C4⋊C4×C83—81no—
198C4⋊C4×C83C2×D412yes—
199C4⋊C4×C83C2×Q812yes—
200C22⋊C4×C83—24no—
201C22⋊C4×C83—22no—
202C22⋊C4×C83—41no—
203C42×C83—21no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83—42no—
208C42×C83—81no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83—21no—
215C23×C83C22⋊Q812yes—
216C23×C83C22×D411yescharacteristic
217Q8×C83—22no—
218Q8×C83—22no—
219Q8×C83—41no—
220Q8×C83—41no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
229C2×C4×C83—22no—
230C2×C4×C83—22no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—22no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
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×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—84no—
251C2×C4×C83C22⋊Q812yes—
252C2×C4×C83#13712yes—
253C2×C4×C83C22×D411yescharacteristic
254C2×C4×C83C22×Q811yescharacteristic
255C2×C4×C83C2×#4111yescharacteristic
256C2×C4×C83C2×#4111yescharacteristic
257C2×C4×C83H(5,𝔽2)11yescharacteristic
258C2×C4×C832−1+411yescharacteristic
259C2×Dic83—22no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C83⋊C8—22no—
272C83⋊C8—22no—
273C83⋊C8—41no—
274C83⋊C8—41no—
275C22×C83—22no—
276C22×C83—22no—
277C22×C83—22no—
278C22×C83—41no—
279C22×C83—41no—
280C22×C83—41no—
281C22×C83—41no—
282C22×C83C2×C22⋊Q811yescharacteristic
283C22×C83D4211yescharacteristic
284C22×C83#54711yescharacteristic
285C22×C83C2×#13711yescharacteristic
286C22×C83#57511yescharacteristic
287C22×C83#57611yescharacteristic
288C22×C83#57811yescharacteristic
289C4×C83—22no—
290C4×C83—22no—
291C4×C83—42no—
292C4×C83—41no—
293C4×C83—42no—
294C4×C83—41no—
295C4×C83—42no—
296C4×C83—41no—
297C4×C83—81no—
298C4×C83#52511yescharacteristic
299C4×C83#53911yescharacteristic
300C4×C83D4×Q811yescharacteristic
301C4×C83#55411yescharacteristic
302Dic83—42no—
303Dic83—81no—
304Dic83—81no—
305Dic83—81no—
306#26362—831nomaximal, Sylow
307C2×C83—42no—
308C2×C83#244911yescharacteristic
309C2×C83#307811yescharacteristic
310C2×C83#308311yescharacteristic
311C2×C83#326811yescharacteristic
312C2×C83#327611yescharacteristic
313C2×C83#334611yescharacteristic
314C2×C83#335211yescharacteristic
315C2×#479—831no—
316C2×#428—831no—
317#2441—831no—
318#2360D8311yescharacteristic
319#2095—831no—
320#2086—831no—
321#2066—831no—
322#2059—831no—
323#1947—831no—
324#1919—831no—
325#1894—831no—
326#1873—831no—
327#1863—831no—
328#1852—831no—
329#1851—831no—
330C83#2636211yescharacteristic, complemented, Sylow
331C22×M16—831no—
332C2×#125—831no—
333C2×C22⋊Q8C2×D8311yescharacteristic
334C2×C4×Q8C2×D8311yescharacteristic
335C2×#117—831no—
336C22×C4⋊C4C2×D8311yescharacteristic
337#479—1664no—
338#428—1664no—
339C2×#107—831no—
340C2×#107—831no—
341C2×#107—831no—
342C2×#107—831no—
343C2×#106—831no—
344C2×#106—831no—
345C2×C4⋊C8—831no—
346C2×C4⋊C8—831no—
347C2×#103—831no—
348C2×#103—831no—
349C2×#103—831no—
350C2×#103—831no—
351C2×C22⋊C8—831no—
352#399—1661no—
353#396—1661no—
354#391C2×D8311yescharacteristic
355#389—21no—
356#389—831no—
357#389—831no—
358#389—1661no—
359#389C2×D8311yescharacteristic
360#387—21no—
361#387—1661no—
362#385—21no—
363#384—831no—
364#380—831no—
365C4×C4⋊C4C2×D8311yescharacteristic
366C4×C22⋊C4—21no—
367C4×C22⋊C4C2×D8311yescharacteristic
368#340—831no—
369#340—831no—
370#340—831no—
371#340—831no—
372#340—831no—
373#340—831no—
374#336—831no—
375#336—831no—
376C22×Q8C22×D8311yescharacteristic
377C23×C4—21no—
378C23×C4C22×D8311yescharacteristic
379C2×M16—1662no—
380C2×M16—1662no—
381C2×M16—1662no—
382C22×C8—831no—
383C22×C8—831no—
384C22×C8—831no—
385C22×C8—831no—
386#125—3321no—
387#125—1662no—
388C22⋊Q8—22no—
389C22⋊Q8Dic16614yes—
390C4×Q8—22no—
391C4×Q8—41no—
392#117—3321no—
393#117—1662no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4—831no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—831no—
403C2×C4⋊C4—831no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4—1661no—
410C2×C4⋊C4C22×D8311yescharacteristic
411C2×C4⋊C4C22×D8311yescharacteristic
412C2×C4⋊C4C22×D8311yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1661no—
415C2×C22⋊C4—1661no—
416C2×C22⋊C4C22⋊D8311yescharacteristic
417C2×C22⋊C4C22⋊D8311yescharacteristic
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—1661no—
421C2×C42C22×D8311yescharacteristic
422C2×C42C22×D8311yescharacteristic
423#107—3321no—
424#107—1662no—
425#107—1662no—
426#107—1662no—
427#106—834no—
428#106—1662no—
429C4⋊C8—1662no—
430C4⋊C8—1662no—
431#103—3321no—
432#103—1662no—
433#103—1662no—
434#103—1662no—
435C22⋊C8—1662no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—1661no—
441#95—1661no—
442#95—1661no—
443#95C22⋊D8311yescharacteristic
444#95C22⋊D8311yescharacteristic
445C24C2×C22⋊D8311yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×Dic16612yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—81no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4C2×C22⋊D8311yescharacteristic
468C22×C4C2×C22⋊D8311yescharacteristic
469C22×C4C2×C22⋊D8311yescharacteristic
470C22×C4C2×C22⋊D8311yescharacteristic
471C22×C4C2×C22⋊D8311yescharacteristic
472C22×C4C2×Dic16612yes—
473C22×C4#1305059012yes—
474C22×C4C23×D8311yescharacteristic
475M16—1668no—
476C2×C8—832no—
477C2×C8—832no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1661no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—3322no—
497C4⋊C4—3322no—
498C4⋊C4—3322no—
499C4⋊C4—3322no—
500C4⋊C4—832no—
501C4⋊C4—6641no—
502C4⋊C4—6641no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4—1661no—
507C4⋊C4—1661no—
508C4⋊C4—1661no—
509C4⋊C4—1661no—
510C4⋊C4D4×D8312yes—
511C4⋊C4C2×Dic16612yes—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—41no—
515C22⋊C4—3322no—
516C22⋊C4—6641no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—42no—
522C42—81no—
523C42—3322no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1458177112yes—
531C23C22×C22⋊D8311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—3321no—
571C2×C4—3321no—
572C2×C4—3321no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—1662no—
578C2×C4#1458177112yes—
579C2×C4#1458182312yes—
580C2×C4C2×D4×D8311yescharacteristic
581C2×C4C22×Dic16611yescharacteristic
582C2×C4#41×D8311yescharacteristic
583C2×C4C2×#1305059011yescharacteristic
584C2×C4#1458183711yescharacteristic
585C2×C4#1458183911yescharacteristic
586C8—3321no—
587C8—3321no—
588C8—1662no—
589C8—1662no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#1458177111yescharacteristic
598C22D4×C22⋊D8311yescharacteristic
599C22#1912835011yescharacteristic
600C22C2×#1458182311yescharacteristic
601C22#1912849911yescharacteristic
602C22#1912850411yescharacteristic
603C22#1912851611yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—81no—
613C4—3322no—
614C4—6641no—
615C4—6641no—
616C4—6641no—
617C4#1912821011yescharacteristic
618C4#1912829311yescharacteristic
619C4D4×Dic16611yescharacteristic
620C4#1912841011yescharacteristic
621C2—42no—
622C2#2491126611yescharacteristic
623C2#2491726811yescharacteristic
624C2#2491731411yescharacteristic
625C2#2491899911yescharacteristic
626C2#2491910911yescharacteristic
627C2#2491994211yescharacteristic
628C2#2492003211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2360C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2360D83
hypocenterC83#26362
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
1. C2×#19128008C2semidirect
4. #24909001C2semidirect
5. #24908954C2semidirect
6. #24908844C2semidirect
7. #24908801C2semidirect
8. #24908221C2semidirect
9. #24908041C2semidirect
12. #24907717C2semidirect
330. C83#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#191280082
C2×#191277412
#249111762
#249090012
#249089542
#249088442
#249088012
#249082212
#249080412
#249078972
#249077762
#249077172
#249076512
#249076482
C83×#23602
#2636283