Finite Group Databasebeta

← #35463197

Subgroups — #35463197

Order 21248.

826 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127742C211yesmaximal, characteristic
2#24911161C211yesmaximal, characteristic
3#24911150C211yesmaximal, characteristic
4#24908947C211yesmaximal, characteristic, complemented
5#24908937C211yesmaximal, characteristic, complemented
6#24908793C211yesmaximal, characteristic, complemented
7#24908788C211yesmaximal, characteristic, complemented
8#24908202C211yesmaximal, characteristic, complemented
9#24908191C211yesmaximal, characteristic, complemented
10#24907659C211yesmaximal, characteristic
11#24907649C211yesmaximal, characteristic
12#24907630C211yesmaximal, characteristic, complemented
13#24907497C211yesmaximal, characteristic
14#24907246C211yesmaximal, characteristic
15C2×#500×C83C211yesmaximal, characteristic, complemented
16C22×#13050573C2211yescharacteristic
17C2×#14581768—21no—
18C2×#14581750C2211yescharacteristic
19C2×C4⋊C4×D83C2211yescharacteristic
20#19127742—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581724C2211yescharacteristic
24C2×#14581724C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×(C2×D83)⋊C8C2211yescharacteristic
28C2×#14581702C2211yescharacteristic
29#19127634—21no—
30#19127623—21no—
31#19127620—21no—
32#19127612—21no—
33#19127608—21no—
34#19127600C2211yescharacteristic
35#19127600C2211yescharacteristic
36#19127599C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127582—21no—
39#19127572—21no—
40#19127570C2211yescharacteristic
41#19127566C2211yescharacteristic
42#19127558C2211yescharacteristic
43#19127558C2211yescharacteristic
44C4×(C2×D83)⋊C4C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127456C2211yescharacteristic
52#19127455C2211yescharacteristic
53C2×C4⋊Q8×C83C2211yescharacteristic
54C2×C4⋊Q8×C83C2211yescharacteristic
55#500×C83—24no—
56C2×#107×C83C2211yescharacteristic
57C2×#107×C83C2211yescharacteristic
58C2×#107×C83C2211yescharacteristic
59C2×#107×C83C2211yescharacteristic
60C2×C4×C8×C83C2211yescharacteristic
61C22×C4×D83—21no—
62C22×C4×D83C2311yescharacteristic
63C2×#13050573—22no—
64C2×#13050573—22no—
65C2×#13050573Q814yescomplemented
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68#14581768—44no—
69#14581750—22no—
70#14581750—41no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C4⋊C4×D83—22no—
83C4⋊C4×D83—41no—
84C2×(C2×D83)⋊C4—21no—
85C2×(C2×D83)⋊C4—21no—
86C2×(C2×D83)⋊C4—21no—
87C2×(C2×D83)⋊C4D411yescharacteristic
88C2×(C2×D83)⋊C4D411yescharacteristic
89C2×C4×Dic83—21no—
90C2×C4×Dic83C2311yescharacteristic
91#14581725—22no—
92#14581725—41no—
93#14581724—22no—
94#14581724—22no—
95#14581723—22no—
96#14581723—41no—
97(C2×D83)⋊C8Q814yescomplemented
98#14581702—22no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95D411yescharacteristic
108C83⋊#95D411yescharacteristic
109C22×Q8×C83—21no—
110C22×Q8×C83—21no—
111C22×C8×C83C2311yescharacteristic
112C22×C8×C83C2311yescharacteristic
113C4⋊Q8×C83—22no—
114C4⋊Q8×C83—22no—
115C4⋊Q8×C83—41no—
116C4⋊Q8×C83—41no—
117C2×C4⋊C4×C83—21no—
118C2×C4⋊C4×C83—21no—
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C4⋊C4×C83C2311yescharacteristic
122C2×C4⋊C4×C83C2311yescharacteristic
123C2×C42×C83C2311yescharacteristic
124#107×C83—22no—
125#107×C83—22no—
126#107×C83—22no—
127#107×C83D414yes—
128C4×C8×C83—22no—
129C23×D83C2×D411yescharacteristic
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83C2×D411yescharacteristic
135C22×Dic83C2×D411yescharacteristic
136C22×Dic83C2×D411yescharacteristic
137C2×C4×D83—21no—
138C2×C4×D83—21no—
139C2×C4×D83—21no—
140C2×C4×D83—21no—
141C2×C4×D83—42no—
142C2×C4×D83—42no—
143C2×C4×D83—81no—
144C2×C4×D83C2×Q812yes—
145C2×C4×D83C2×Q812yes—
146#13050573—28no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8C2×Q812yes—
152C2×C83⋊C8C2×Q812yes—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—42no—
162C4⋊Dic83—81no—
163Dic83⋊C4—42no—
164Dic83⋊C4—42no—
165Dic83⋊C4—42no—
166Dic83⋊C4—42no—
167Dic83⋊C4—81no—
168(C2×D83)⋊C4—22no—
169(C2×D83)⋊C4—22no—
170(C2×D83)⋊C4—42no—
171(C2×D83)⋊C4—42no—
172(C2×D83)⋊C4—81no—
173C4×Dic83—21no—
174C4×Dic83—21no—
175C4×Dic83—42no—
176C2×Q8×C83—42no—
177C2×Q8×C83—42no—
178C2×Q8×C83—42no—
179C2×Q8×C83—42no—
180C2×Q8×C83—81no—
181C2×Q8×C83—81no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2411yescharacteristic
189C2×C8×C83—21no—
190C2×C8×C83—21no—
191C2×C8×C83C2×D412yes—
192C2×C8×C83C2×Q812yes—
193C2×C8×C83C2×Q812yes—
194C2×C8×C83#4112yes—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83—42no—
202C4⋊C4×C83—42no—
203C4⋊C4×C83C2×D412yes—
204C4⋊C4×C83C2×D412yes—
205C42×C83—21no—
206C42×C83—21no—
207C23×C83C22×D411yescharacteristic
208C22×D83—21no—
209C22×D83—21no—
210C22×D83—21no—
211C22×D83—21no—
212C22×D83—21no—
213C22×D83—21no—
214C22×D83C4⋊Q812yes—
215Q8×C83—84no—
216Q8×C83—84no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
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—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83AGL(1,ℤ/8)12yes—
234C2×C4×C83#13712yes—
235C2×C4×C83C22×D411yescharacteristic
236C2×C4×C83C22×Q811yescharacteristic
237C2×C4×C83C22×Q811yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83H(5,𝔽2)11yescharacteristic
240C2×C4×C832−1+411yescharacteristic
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—21no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83C4⋊Q812yes—
262C4×D83—22no—
263C4×D83—22no—
264C4×D83—22no—
265C4×D83—22no—
266C4×D83—84no—
267C8×C83—22no—
268C8×C83C22⋊Q814yescomplemented
269C83⋊C8—22no—
270C83⋊C8—22no—
271C22×C83C2×C4⋊Q811yescharacteristic
272C22×C83D4211yescharacteristic
273C22×C83#54711yescharacteristic
274C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C83C2×#13711yescharacteristic
276C22×C83#57611yescharacteristic
277C22×C83#57811yescharacteristic
278C2×D83—22no—
279C2×D83—22no—
280C2×D83—22no—
281C2×D83—41no—
282C2×D83—41no—
283C2×D83—41no—
284C2×D83—41no—
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—42no—
288C4×C83—42no—
289C4×C83—81no—
290C4×C83—81no—
291C4×C83C2×C22⋊Q811yescharacteristic
292C4×C83#54411yescharacteristic
293C4×C83D4×Q811yescharacteristic
294C4×C83#55411yescharacteristic
295Dic83—22no—
296Dic83—41no—
297Dic83—41no—
298Dic83—81no—
299Dic83—81no—
300Dic83—81no—
301Dic83—81no—
302#26074—831nomaximal, Sylow
303C2×C83#244011yescharacteristic
304C2×C83#316411yescharacteristic
305C2×C83#316511yescharacteristic
306C2×C83#326411yescharacteristic
307C2×C83#327011yescharacteristic
308C2×C83#335311yescharacteristic
309C2×C83#335711yescharacteristic
310D83—42no—
311C2×#500D8311yescharacteristic, complemented
312C2×#428—831no—
313#2440—831no—
314#2439—831no—
315#2085—831no—
316#2083—831no—
317#2058—831no—
318#2057—831no—
319#1943—831no—
320#1941—831no—
321#1854—831no—
322#1852—831no—
323#1847—831no—
324#1809—831no—
325#1753—831no—
326C83#2607411yescharacteristic, complemented, Sylow
327C22×M16—831no—
328C2×C4⋊Q8C2×D8311yescharacteristic
329C2×C4⋊Q8C2×D8311yescharacteristic
330C2×C22⋊Q8—1661no—
331C2×#117—831no—
332C22×C4⋊C4—831no—
333#500—24no—
334#428—1664no—
335C2×#107—831no—
336C2×#107—831no—
337C2×#107C2×D8311yescharacteristic
338C2×#107C2×D8311yescharacteristic
339C2×#107C2×D8311yescharacteristic
340C2×#107C2×D8311yescharacteristic
341C2×#106—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×#106—831no—
345C2×C22⋊C8—831no—
346C2×#97—831no—
347C2×C4×C8C2×D8311yescharacteristic
348#398—1661no—
349#396—1661no—
350#395—1661no—
351#393—1661no—
352#391—1661no—
353#389—831no—
354#389—831no—
355#389—831no—
356#389—831no—
357#387—1661no—
358#385—1661no—
359#384—831no—
360#382—831no—
361#380—831no—
362#380—831no—
363C4×C22⋊C4—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×Q8—21no—
373C22×Q8—21no—
374C23×C4—831no—
375C23×C4—1661no—
376C2×M16—834no—
377C2×M16—1662no—
378C2×M16—1662no—
379C22×C8—831no—
380C22×C8—831no—
381C22×C8C22×D8311yescharacteristic
382C22×C8C22×D8311yescharacteristic
383C4⋊Q8—22no—
384C4⋊Q8—22no—
385C4⋊Q8—41no—
386C4⋊Q8—41no—
387C22⋊Q8—3324no—
388#117—3321no—
389#117—1662no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—3321no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1662no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4C22×D8311yescharacteristic
406C2×C4⋊C4C22×D8311yescharacteristic
407C2×C4⋊C4C22×D8311yescharacteristic
408C2×C4⋊C4C22×D8311yescharacteristic
409C2×C22⋊C4—831no—
410C2×C22⋊C4—831no—
411C2×C22⋊C4—1661no—
412C2×C22⋊C4—1661no—
413C2×C22⋊C4—1661no—
414C2×C42—831no—
415C2×C42—1661no—
416C2×C42C22×D8311yescharacteristic
417#107—22no—
418#107—22no—
419#107—22no—
420#107—3321no—
421#107—1662no—
422#107C22⋊D8314yes—
423#106—3321no—
424#106—1662no—
425#106—1662no—
426#106—1662no—
427C22⋊C8—834no—
428#97—1662no—
429C4×C8—22no—
430#95—831no—
431#95—831no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95—1661no—
438#95—1661no—
439#95—1661no—
440C24—831no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—81no—
446C2×Q8—81no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—3322no—
452C22×C4—3322no—
453C22×C4—831no—
454C22×C4—832no—
455C22×C4—832no—
456C22×C4—831no—
457C22×C4—831no—
458C22×C4—6641no—
459C22×C4—1661no—
460C22×C4—1661no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4D4×D8311yescharacteristic
468C22×C4D4×D8311yescharacteristic
469C22×C4C23×D8311yescharacteristic
470M16—1668no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—832no—
474C2×C8—832no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8C2×C22⋊D8312yes—
480C2×C8Q8×D8312yes—
481C2×C8Q8×D8312yes—
482C2×C8#1305059112yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—3322no—
492C4⋊C4—3322no—
493C4⋊C4—3322no—
494C4⋊C4—3322no—
495C4⋊C4—3322no—
496C4⋊C4—6641no—
497C4⋊C4—6641no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4C2×C22⋊D8312yes—
507C4⋊C4C2×C22⋊D8312yes—
508C22⋊C4—3322no—
509C22⋊C4—3322no—
510C22⋊C4—6641no—
511C22⋊C4—1662no—
512C22⋊C4—1662no—
513C42—21no—
514C42—21no—
515C42—3322no—
516C42—1661no—
517C42—1661no—
518C23—832no—
519C23—1661no—
520C23—1661no—
521C23—1661no—
522C23—1661no—
523C23—1661no—
524C23—1661no—
525C23C2×D4×D8311yescharacteristic
526Q8—84no—
527Q8—84no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
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—832no—
557C2×C4—6644no—
558C2×C4—1661no—
559C2×C4—1661no—
560C2×C4—1661no—
561C2×C4—1661no—
562C2×C4—1661no—
563C2×C4—1661no—
564C2×C4—1661no—
565C2×C4—1661no—
566C2×C4—1662no—
567C2×C4—1662no—
568C2×C4—1662no—
569C2×C4—1662no—
570C2×C4#1458181612yes—
571C2×C4#1458182412yes—
572C2×C4C22×C22⋊D8311yescharacteristic
573C2×C4C2×Q8×D8311yescharacteristic
574C2×C4C2×Q8×D8311yescharacteristic
575C2×C4C2×#1305059111yescharacteristic
576C2×C4#1458183711yescharacteristic
577C2×C4#1458184011yescharacteristic
578C8—22no—
579C8—1662no—
580C8—1662no—
581C8#1458176814yescomplemented
582C22—3321no—
583C22—3321no—
584C22—3321no—
585C22—3321no—
586C22—1662no—
587C22—1662no—
588C22—1662no—
589C22C4⋊Q8×D8311yescharacteristic
590C22D4×C22⋊D8311yescharacteristic
591C22#1912835411yescharacteristic
592C22C2×#1458181611yescharacteristic
593C22C2×#1458182411yescharacteristic
594C22#1912850211yescharacteristic
595C22#1912851411yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—3321no—
603C4—3321no—
604C4—6641no—
605C4—6641no—
606C4—6641no—
607C4—6641no—
608C4—1662no—
609C4C2×#1458176811yescharacteristic
610C4#1912832811yescharacteristic
611C4Q8×C22⋊D8311yescharacteristic
612C4#1912841211yescharacteristic
613C2—3322no—
614C2#2491116111yescharacteristic
615C2#2491799611yescharacteristic
616C2#2491800611yescharacteristic
617C2#2491894311yescharacteristic
618C2#2491903311yescharacteristic
619C2#2492004611yescharacteristic
620C2#2492008811yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC2×#500×C83C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
hypercenterC2×#500D83
hypocenterC83#26074
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
4. #24908947C2semidirect
5. #24908937C2semidirect
6. #24908793C2semidirect
7. #24908788C2semidirect
8. #24908202C2semidirect
9. #24908191C2semidirect
12. #24907630C2semidirect
15. C2×#500×C83C2semidirect
65. C2×#13050573Q8semidirect
97. (C2×D83)⋊C8Q8semidirect
268. C8×C83C22⋊Q8semidirect
311. C2×#500D83semidirect
326. C83#26074semidirect
581. C8#14581768semidirect

Maximal subgroups

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

subgroupindex
C2×#191277422
#249111612
#249111502
#249089472
#249089372
#249087932
#249087882
#249082022
#249081912
#249076592
#249076492
#249076302
#249074972
#249072462
C2×#500×C832
#2607483