Finite Group Databasebeta

← #35463198

Subgroups — #35463198

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×#19128104C211yesmaximal, characteristic, complemented
2C2×#19127741C211yesmaximal, characteristic
3#24911154C211yesmaximal, characteristic
4#24908944C211yesmaximal, characteristic, complemented
5#24908935C211yesmaximal, characteristic, complemented
6#24908793C211yesmaximal, characteristic, complemented
7#24908791C211yesmaximal, characteristic, complemented
8#24908198C211yesmaximal, characteristic, complemented
9#24908188C211yesmaximal, characteristic, complemented
10#24907660C211yesmaximal, characteristic
11#24907651C211yesmaximal, characteristic
12#24907629C211yesmaximal, characteristic, complemented
13#24907494C211yesmaximal, characteristic
14#24907247C211yesmaximal, characteristic
15C83×#2440C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581751C2211yescharacteristic
19#19128104—24no—
20#19127741—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581725C2211yescharacteristic
26C2×#14581725C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×#14581723C2211yescharacteristic
29C2×#14581723C2211yescharacteristic
30C2×#14581723C2211yescharacteristic
31C2×C22⋊(C83⋊C8)C2211yescharacteristic
32C2×#14581703C2211yescharacteristic
33C2×C4×C83⋊C8C2211yescharacteristic
34#19127624—21no—
35#19127613—21no—
36#19127605—21no—
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127588—21no—
40#19127567C2211yescharacteristic
41#19127561C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4⋊Q8×C83C2211yescharacteristic
51C2×C22⋊Q8×C83—21no—
52C22×C4⋊C4×C83C2211yescharacteristic
53#398×C83—21no—
54#395×C83—21no—
55#389×C83C2211yescharacteristic
56#389×C83C2211yescharacteristic
57#385×C83—21no—
58#384×C83C2211yescharacteristic
59#380×C83C2211yescharacteristic
60C4×C22⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C2×#13050574—22no—
63C2×#13050574—22no—
64C2×#13050574Q814yescomplemented
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69#14581792—22no—
70#14581792—41no—
71#14581751—22no—
72#14581751—41no—
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×C22⋊Dic83—21no—
81C2×C22⋊Dic83—21no—
82C2×C4×Dic83—21no—
83#14581725—22no—
84#14581725—22no—
85#14581725—22no—
86#14581725—22no—
87#14581725—41no—
88#14581725D414yes—
89#14581723—22no—
90#14581723—22no—
91#14581723—22no—
92#14581723—41no—
93C22⋊(C83⋊C8)Q814yescomplemented
94#14581703—22no—
95C4×C83⋊C8—22no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C22×Q8×C83—21no—
102C23×C4×C83—21no—
103C23×C4×C83C2311yescharacteristic
104C4⋊Q8×C83—22no—
105C4⋊Q8×C83—41no—
106C22⋊Q8×C83—44no—
107C2×C4⋊C4×C83—22no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83—41no—
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C22⋊C4×C83—21no—
120C2×C22⋊C4×C83D411yescharacteristic
121C2×C22⋊C4×C83D411yescharacteristic
122C2×C42×C83C2311yescharacteristic
123C2×C42×C83C2311yescharacteristic
124#95×C83—21no—
125#95×C83—21no—
126#95×C83—21no—
127#95×C83D411yescharacteristic
128#95×C83D411yescharacteristic
129C2×Dic166—42no—
130C2×Dic166—42no—
131C2×Dic166—81no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136#13050574—28no—
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⋊C8C2×D412yes—
144C2×C83⋊C8C2×Q812yes—
145C2×C83⋊C8C2×Q812yes—
146C2×C83⋊C8C2×Q812yes—
147C2×C83⋊C8C2×Q812yes—
148C2×C83⋊C8#4112yes—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—42no—
157C4⋊Dic83C2×D412yes—
158Dic83⋊C4—42no—
159Dic83⋊C4—81no—
160C22⋊Dic83—42no—
161C22⋊Dic83—81no—
162C4×Dic83—42no—
163C24×C83C2×D411yescharacteristic
164C2×Q8×C83—42no—
165C2×Q8×C83—42no—
166C2×Q8×C83—81no—
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—21no—
174C22×C4×C83—21no—
175C22×C4×C83—42no—
176C22×C4×C83—42no—
177C22×C4×C83—81no—
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×Q812yes—
184C22×C4×C83C2×Q812yes—
185C22×C4×C83C2411yescharacteristic
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—42no—
194C4⋊C4×C83—42no—
195C4⋊C4×C83—42no—
196C4⋊C4×C83—42no—
197C4⋊C4×C83—42no—
198C4⋊C4×C83—81no—
199C4⋊C4×C83C2×D412yes—
200C22⋊C4×C83—22no—
201C22⋊C4×C83—22no—
202C22⋊C4×C83—42no—
203C42×C83—21no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—21no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83C4⋊Q812yes—
214C23×C83C22×D411yescharacteristic
215Q8×C83—84no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—84no—
245C2×C4×C83C4⋊Q812yes—
246C2×C4×C83AGL(1,ℤ/8)12yes—
247C2×C4×C83#13712yes—
248C2×C4×C83C22×D411yescharacteristic
249C2×C4×C83C22×Q811yescharacteristic
250C2×C4×C83C22×Q811yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C2×C4×C83H(5,𝔽2)11yescharacteristic
253C2×C4×C832−1+411yescharacteristic
254C2×Dic83—22no—
255C2×Dic83—41no—
256C2×Dic83—41no—
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—
266Dic166—84no—
267C83⋊C8—22no—
268C83⋊C8—22no—
269C83⋊C8—22no—
270C83⋊C8C22⋊Q814yescomplemented
271C22×C83—22no—
272C22×C83—22no—
273C22×C83—22no—
274C22×C83—41no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83C2×C4⋊Q811yescharacteristic
279C22×C83D4211yescharacteristic
280C22×C83#54711yescharacteristic
281C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C83C2×#13711yescharacteristic
283C22×C83#57611yescharacteristic
284C22×C83#57811yescharacteristic
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—22no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83—42no—
291C4×C83—81no—
292C4×C83—81no—
293C4×C83—81no—
294C4×C83C2×C22⋊Q811yescharacteristic
295C4×C83#54411yescharacteristic
296C4×C83D4×Q811yescharacteristic
297C4×C83#55411yescharacteristic
298Dic83—42no—
299Dic83—81no—
300Dic83—81no—
301Dic83—81no—
302#26074—831nomaximal, Sylow
303C2×C83—42no—
304C2×C83#244011yescharacteristic
305C2×C83#316411yescharacteristic
306C2×C83#316511yescharacteristic
307C2×C83#326411yescharacteristic
308C2×C83#327011yescharacteristic
309C2×C83#335311yescharacteristic
310C2×C83#335711yescharacteristic
311C2×#500—831no—
312C2×#428—831no—
313#2440D8311yescharacteristic
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⋊Q8—831no—
329C2×C4⋊Q8C2×D8311yescharacteristic
330C2×C22⋊Q8—21no—
331C2×#117—831no—
332C22×C4⋊C4C2×D8311yescharacteristic
333#500—1664no—
334#428—1664no—
335C2×#107—831no—
336C2×#107—831no—
337C2×#107—831no—
338C2×#107—831no—
339C2×#107—831no—
340C2×#107—831no—
341C2×#106—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×#106—831no—
345C2×C22⋊C8—831no—
346C2×#97—831no—
347C2×C4×C8—831no—
348#398—21no—
349#396—1661no—
350#395—21no—
351#393—1661no—
352#391—1661no—
353#389—831no—
354#389—831no—
355#389C2×D8311yescharacteristic
356#389C2×D8311yescharacteristic
357#387—1661no—
358#385—21no—
359#384C2×D8311yescharacteristic
360#382—831no—
361#380—831no—
362#380C2×D8311yescharacteristic
363C4×C22⋊C4C2×D8311yescharacteristic
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—1661no—
374C23×C4—21no—
375C23×C4C22×D8311yescharacteristic
376C2×M16—834no—
377C2×M16—1662no—
378C2×M16—1662no—
379C22×C8—831no—
380C22×C8—831no—
381C22×C8—831no—
382C22×C8—831no—
383C4⋊Q8—22no—
384C4⋊Q8—41no—
385C4⋊Q8—3321no—
386C4⋊Q8—1662no—
387C22⋊Q8—44no—
388#117—3321no—
389#117—1662no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—831no—
400C2×C4⋊C4—831no—
401C2×C4⋊C4—831no—
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×C4⋊C4C22×D8311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1661no—
411C2×C22⋊C4—1661no—
412C2×C22⋊C4C22⋊D8311yescharacteristic
413C2×C22⋊C4C22⋊D8311yescharacteristic
414C2×C42—1661no—
415C2×C42C22×D8311yescharacteristic
416C2×C42C22×D8311yescharacteristic
417#107—3321no—
418#107—834no—
419#107—1662no—
420#107—1662no—
421#107—1662no—
422#107—1662no—
423#106—3321no—
424#106—1662no—
425#106—1662no—
426#106—1662no—
427C22⋊C8—834no—
428#97—1662no—
429C4×C8—1662no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95—1661no—
438#95C22⋊D8311yescharacteristic
439#95C22⋊D8311yescharacteristic
440C24C2×C22⋊D8311yescharacteristic
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—81no—
444C2×Q8—3322no—
445C2×Q8—3322no—
446C2×Q8—6641no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4—1661no—
461C22×C4—1661no—
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C2×C22⋊D8311yescharacteristic
467C22×C4Q8×D8312yes—
468C22×C4Q8×D8312yes—
469C22×C4C23×D8311yescharacteristic
470M16—1668no—
471C2×C8—832no—
472C2×C8—832no—
473C2×C8—832no—
474C2×C8—832no—
475C2×C8—832no—
476C2×C8—832no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—3322no—
497C4⋊C4—3322no—
498C4⋊C4—832no—
499C4⋊C4—6641no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4—1661no—
507C4⋊C4D4×D8312yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—42no—
511C22⋊C4—3322no—
512C22⋊C4—6641no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—3322no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23#1458179012yes—
525C23C22×C22⋊D8311yescharacteristic
526Q8—84no—
527Q8—6644no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—84no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
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—1662no—
569C2×C4#1458179012yes—
570C2×C4#1458181812yes—
571C2×C4#1458182312yes—
572C2×C4C2×D4×D8311yescharacteristic
573C2×C4C2×Q8×D8311yescharacteristic
574C2×C4C2×Q8×D8311yescharacteristic
575C2×C4#41×D8311yescharacteristic
576C2×C4#1458183711yescharacteristic
577C2×C4#1458183911yescharacteristic
578C8—834no—
579C8—1662no—
580C8—1662no—
581C8—1662no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1458179011yescharacteristic
590C22D4×C22⋊D8311yescharacteristic
591C22#1912835011yescharacteristic
592C22C2×#1458181811yescharacteristic
593C22C2×#1458182311yescharacteristic
594C22#1912850411yescharacteristic
595C22#1912851611yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—3322no—
606C4—6641no—
607C4—6641no—
608C4—6641no—
609C4C22⋊Q8×D8311yescharacteristic
610C4#1912833011yescharacteristic
611C4#1912836411yescharacteristic
612C4#1912840511yescharacteristic
613C2—42no—
614C2#2491116211yescharacteristic
615C2#2491799711yescharacteristic
616C2#2491800511yescharacteristic
617C2#2491893911yescharacteristic
618C2#2491903411yescharacteristic
619C2#2492004711yescharacteristic
620C2#2492008911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2440C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2440D83
hypocenterC83#26074
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
1. C2×#19128104C2semidirect
4. #24908944C2semidirect
5. #24908935C2semidirect
6. #24908793C2semidirect
7. #24908791C2semidirect
8. #24908198C2semidirect
9. #24908188C2semidirect
12. #24907629C2semidirect
64. C2×#13050574Q8semidirect
93. C22⋊(C83⋊C8)Q8semidirect
270. C83⋊C8C22⋊Q8semidirect
326. C83#26074semidirect

Maximal subgroups

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

subgroupindex
C2×#191281042
C2×#191277412
#249111542
#249089442
#249089352
#249087932
#249087912
#249081982
#249081882
#249076602
#249076512
#249076292
#249074942
#249072472
C83×#24402
#2607483