Finite Group Databasebeta

← #35466963

Subgroups — #35466963

Order 21248.

882 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127997C211yesmaximal, characteristic, complemented
2C22×#14581726C211yesmaximal, characteristic
3#24911165C211yesmaximal, characteristic
4#24910466C211yesmaximal, characteristic
5#24909000C211yesmaximal, characteristic, complemented
6#24908956C211yesmaximal, characteristic, complemented
7#24908857C211yesmaximal, characteristic, complemented
8#24908792C211yesmaximal, characteristic, complemented
9#24908222C211yesmaximal, characteristic, complemented
10#24908056C211yesmaximal, characteristic, complemented
11#24907886C211yesmaximal, characteristic
12#24907768C211yesmaximal, characteristic
13#24907720C211yesmaximal, characteristic, complemented
14#24907653C211yesmaximal, characteristic
15C83×#1850C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581771C2211yescharacteristic
18C2×#14581769—21no—
19C2×C4×Dic166C2211yescharacteristic
20C22×C4⋊Dic83C2211yescharacteristic
21C22×C4⋊Dic83C2211yescharacteristic
22#19127997—24no—
23C2×#14581726—24no—
24C2×#14581726C2211yescharacteristic
25C2×#14581726C2211yescharacteristic
26C2×#14581726C2211yescharacteristic
27C2×#14581726C2211yescharacteristic
28C2×#14581726C2211yescharacteristic
29C2×#14581726C2211yescharacteristic
30C2×#14581717C2211yescharacteristic
31C2×#14581717C2211yescharacteristic
32C2×#14581717C2211yescharacteristic
33C2×#14581717C2211yescharacteristic
34#19127633—21no—
35#19127622—21no—
36#19127607C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127599C2211yescharacteristic
40#19127598—21no—
41#19127588—21no—
42#19127576—21no—
43#19127573—21no—
44#19127571C2211yescharacteristic
45#19127561C2211yescharacteristic
46C4×C4⋊Dic83C2211yescharacteristic
47C22⋊C4×Dic83—21no—
48#19127464C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127464C2211yescharacteristic
51#19127464C2211yescharacteristic
52#19127464C2211yescharacteristic
53#19127464C2211yescharacteristic
54C23×C8×C83C2211yescharacteristic
55C2×C4⋊C8×C83C2211yescharacteristic
56C2×C4⋊C8×C83C2211yescharacteristic
57C2×C22⋊C8×C83C2211yescharacteristic
58C4×C22⋊C4×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60#336×C83C2211yescharacteristic
61C22×Dic166—21no—
62C22×Dic166C2311yescharacteristic
63C23×Dic83—21no—
64C23×Dic83—21no—
65#14581792—22no—
66#14581792—41no—
67#14581771—22no—
68#14581771Q814yescomplemented
69#14581769—44no—
70C4×Dic166—22no—
71C4×Dic166—41no—
72C2×C4⋊Dic83—22no—
73C2×C4⋊Dic83—22no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—21no—
78C2×C4⋊Dic83—21no—
79C2×C4⋊Dic83—41no—
80C2×C4⋊Dic83—41no—
81C2×C4⋊Dic83C2311yescharacteristic
82C2×C4⋊Dic83C2311yescharacteristic
83C2×C4⋊Dic83C2311yescharacteristic
84C2×C4⋊Dic83C2311yescharacteristic
85C2×C4⋊Dic83C2311yescharacteristic
86C2×C4⋊Dic83C2311yescharacteristic
87C2×C4⋊Dic83C2311yescharacteristic
88C2×Dic83⋊C4—21no—
89C2×Dic83⋊C4—21no—
90C2×Dic83⋊C4—21no—
91C2×Dic83⋊C4—21no—
92C2×Dic83⋊C4—21no—
93C2×C22⋊Dic83—21no—
94C2×C22⋊Dic83—21no—
95C2×C4×Dic83—21no—
96C2×C4×Dic83—21no—
97#14581726—22no—
98#14581726—22no—
99#14581726—22no—
100#14581726—22no—
101#14581726—41no—
102#14581726D414yes—
103#14581717—22no—
104#14581717—22no—
105#14581717—22no—
106#14581717—41no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95—21no—
111C83⋊#95—21no—
112C83⋊#95—21no—
113C83⋊#95—21no—
114C23×C4×C83C2311yescharacteristic
115C22×C8×C83—22no—
116C22×C8×C83—22no—
117C22×C8×C83—22no—
118C22×C8×C83C2311yescharacteristic
119C22×C8×C83C2311yescharacteristic
120C22×C8×C83C2311yescharacteristic
121C22×C8×C83C2311yescharacteristic
122C2×C22⋊C4×C83D411yescharacteristic
123C2×C22⋊C4×C83D411yescharacteristic
124C2×C42×C83C2311yescharacteristic
125C2×C42×C83C2311yescharacteristic
126C4⋊C8×C83—22no—
127C4⋊C8×C83—22no—
128C22⋊C8×C83—22no—
129#95×C83D411yescharacteristic
130#95×C83D411yescharacteristic
131C2×Dic166—21no—
132C2×Dic166—21no—
133C2×Dic166—21no—
134C2×Dic166—21no—
135C2×Dic166—21no—
136C2×Dic166—42no—
137C2×Dic166—42no—
138C2×Dic166—81no—
139C2×Dic166C2×Q812yes—
140C22×Dic83—21no—
141C22×Dic83—21no—
142C22×Dic83—21no—
143C22×Dic83—21no—
144C22×Dic83—21no—
145C22×Dic83—21no—
146C22×Dic83—21no—
147C22×Dic83—21no—
148C22×Dic83—42no—
149C22×Dic83—42no—
150C22×Dic83—42no—
151C22×Dic83—42no—
152C22×Dic83—81no—
153C22×Dic83—81no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83—21no—
164C4⋊Dic83—21no—
165C4⋊Dic83—42no—
166C4⋊Dic83—42no—
167C4⋊Dic83—42no—
168C4⋊Dic83—42no—
169C4⋊Dic83—81no—
170C4⋊Dic83C2×D412yes—
171C4⋊Dic83C2×D412yes—
172C4⋊Dic83C2×Q812yes—
173Dic83⋊C4—24no—
174Dic83⋊C4—42no—
175Dic83⋊C4—42no—
176Dic83⋊C4—42no—
177Dic83⋊C4—81no—
178C22⋊Dic83—24no—
179C22⋊Dic83—42no—
180C4×Dic83—42no—
181C4×Dic83—81no—
182C24×C83C2×D411yescharacteristic
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83—21no—
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2×D411yescharacteristic
191C22×C4×C83C2×D411yescharacteristic
192C22×C4×C83C2×Q812yes—
193C22×C4×C83#4112yes—
194C22×C4×C83C2411yescharacteristic
195C2×C8×C83—28no—
196C2×C8×C83—21no—
197C2×C8×C83—21no—
198C2×C8×C83—21no—
199C2×C8×C83—21no—
200C2×C8×C83—21no—
201C2×C8×C83—21no—
202C2×C8×C83—21no—
203C2×C8×C83—21no—
204C2×C8×C83—21no—
205C2×C8×C83—21no—
206C2×C8×C83C2×D412yes—
207C2×C8×C83#4112yes—
208C22⋊C4×C83—22no—
209C22⋊C4×C83Q1614yes—
210C42×C83—21no—
211C42×C83—21no—
212C42×C83—21no—
213C42×C83—21no—
214C23×C83—21no—
215C23×C83—21no—
216C23×C83—21no—
217C23×C83—21no—
218C23×C83C22⋊Q812yes—
219C23×C83C2×Q1612yes—
220C23×C83AGL(1,ℤ/8)12yes—
221C23×C83C22×D411yescharacteristic
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—22no—
229C2×C4×C83—22no—
230C2×C4×C83—22no—
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×C83C22⋊Q812yes—
239C2×C4×C83C2×Q1612yes—
240C2×C4×C83C2×Q1612yes—
241C2×C4×C83C22×D411yescharacteristic
242C2×C4×C83C22×Q811yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83C2×#4111yescharacteristic
245C2×C4×C83H(5,𝔽2)11yescharacteristic
246C2×C4×C832−1+411yescharacteristic
247C2×Dic83—22no—
248C2×Dic83—22no—
249C2×Dic83—22no—
250C2×Dic83—22no—
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×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—
271C2×Dic83—84no—
272C2×Dic83—84no—
273Dic166—22no—
274Dic166—22no—
275Dic166—41no—
276Dic166—41no—
277Dic166—84no—
278C8×C83—22no—
279C8×C83—22no—
280C8×C83—41no—
281C8×C83—41no—
282C22×C83—22no—
283C22×C83—22no—
284C22×C83—22no—
285C22×C83—22no—
286C22×C83—22no—
287C22×C83—22no—
288C22×C83#47714yescomplemented
289C22×C83C2×C22⋊Q811yescharacteristic
290C22×C83D4211yescharacteristic
291C22×C83#54711yescharacteristic
292C22×C83C22×Q1611yescharacteristic
293C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C83#57611yescharacteristic
295C22×C83#57811yescharacteristic
296C4×C83—22no—
297C4×C83—22no—
298C4×C83—22no—
299C4×C83—41no—
300C4×C83—41no—
301C4×C83#52511yescharacteristic
302C4×C83#53911yescharacteristic
303C4×C83D4×Q811yescharacteristic
304C4×C83#55411yescharacteristic
305Dic83—42no—
306Dic83—42no—
307Dic83—42no—
308Dic83—42no—
309Dic83—81no—
310Dic83—81no—
311Dic83—81no—
312Dic83—81no—
313#26342—831nomaximal, Sylow
314C2×C83—24no—
315C2×C83#244911yescharacteristic
316C2×C83C2×#47711yescharacteristic
317C2×C83#308311yescharacteristic
318C2×C83#326411yescharacteristic
319C2×C83D4×Q1611yescharacteristic
320C2×C83#333811yescharacteristic
321C2×C83#334311yescharacteristic
322C2×#477—831no—
323C22×#107—831no—
324#2440—831no—
325#2360—831no—
326#2095—831no—
327#2087—831no—
328#2068—831no—
329#2057—831no—
330#1947—831no—
331#1921—831no—
332#1892—831no—
333#1872—831no—
334#1863—831no—
335#1852—831no—
336#1850D8311yescharacteristic
337C83#2634211yescharacteristic, complemented, Sylow
338C23×C8C2×D8311yescharacteristic
339C2×C4⋊Q8—831no—
340C2×C22⋊Q8—831no—
341C2×C22⋊Q8—1661no—
342C2×C4×Q8—831no—
343C22×C4⋊C4—831no—
344C22×C4⋊C4—831no—
345#477—1664no—
346C2×#107—831no—
347C2×#107—831no—
348C2×#107—831no—
349C2×#107—831no—
350C2×#107—831no—
351C2×#107—831no—
352C2×#107—1664no—
353C2×C4⋊C8C2×D8311yescharacteristic
354C2×C4⋊C8C2×D8311yescharacteristic
355C2×#103—831no—
356C2×#103—831no—
357C2×#103—831no—
358C2×#103—831no—
359C2×C22⋊C8C2×D8311yescharacteristic
360#398—1661no—
361#395—1661no—
362#391—831no—
363#389—831no—
364#389—831no—
365#389—831no—
366#389—1661no—
367#387—1661no—
368#385—1661no—
369#385—1661no—
370#384—831no—
371#380—831no—
372C4×C4⋊C4—831no—
373C4×C22⋊C4—1661no—
374C4×C22⋊C4C2×D8311yescharacteristic
375#340—831no—
376#340—831no—
377#340—831no—
378#340—831no—
379#340—831no—
380#340—831no—
381#336C2×D8311yescharacteristic
382#336C2×D8311yescharacteristic
383C22×Q8—831no—
384C22×Q8—1661no—
385C23×C4—1661no—
386C23×C4—1661no—
387C23×C4C22×D8311yescharacteristic
388C22×C8—22no—
389C22×C8—22no—
390C22×C8—22no—
391C22×C8C22×D8311yescharacteristic
392C22×C8C22×D8311yescharacteristic
393C22×C8C22×D8311yescharacteristic
394C22×C8C22×D8311yescharacteristic
395C4⋊Q8—3321no—
396C4⋊Q8—1662no—
397C22⋊Q8—3324no—
398C22⋊Q8—834no—
399C22⋊Q8—1662no—
400C4×Q8—3321no—
401C4×Q8—1662no—
402C2×C4⋊C4—3321no—
403C2×C4⋊C4—3321no—
404C2×C4⋊C4—831no—
405C2×C4⋊C4—831no—
406C2×C4⋊C4—831no—
407C2×C4⋊C4—831no—
408C2×C4⋊C4—831no—
409C2×C4⋊C4—831no—
410C2×C4⋊C4—831no—
411C2×C4⋊C4—1661no—
412C2×C4⋊C4—1661no—
413C2×C4⋊C4—1661no—
414C2×C4⋊C4—1661no—
415C2×C4⋊C4—1661no—
416C2×C4⋊C4—1661no—
417C2×C4⋊C4—1661no—
418C2×C4⋊C4—1662no—
419C2×C4⋊C4—1662no—
420C2×C4⋊C4—1661no—
421C2×C4⋊C4—1661no—
422C2×C4⋊C4—1661no—
423C2×C22⋊C4—1661no—
424C2×C22⋊C4—1661no—
425C2×C22⋊C4D33211yescharacteristic
426C2×C22⋊C4D33211yescharacteristic
427C2×C42—1661no—
428C2×C42—1661no—
429C2×C42C22×D8311yescharacteristic
430C2×C42C22×D8311yescharacteristic
431#107—3321no—
432#107—834no—
433#107—1662no—
434#107—1662no—
435#107—1662no—
436#107—1662no—
437C4⋊C8—22no—
438C4⋊C8—22no—
439#103—3321no—
440#103—1662no—
441#103—1662no—
442#103—1662no—
443C22⋊C8—22no—
444#95—1661no—
445#95—1661no—
446#95—1661no—
447#95—1661no—
448#95—1661no—
449#95—1661no—
450#95—1661no—
451#95D33211yescharacteristic
452#95D33211yescharacteristic
453C24C2×D33211yescharacteristic
454C2×Q8—3322no—
455C2×Q8—3322no—
456C2×Q8—832no—
457C2×Q8—6641no—
458C2×Q8—1661no—
459C2×Q8—1661no—
460C2×Q8—1661no—
461C2×Q8—1661no—
462C2×Q8—1661no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—3322no—
468C22×C4—3322no—
469C22×C4—3322no—
470C22×C4—3322no—
471C22×C4—6641no—
472C22×C4—6641no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4—1661no—
476C22×C4—1661no—
477C22×C4—1661no—
478C22×C4—1661no—
479C22×C4—1661no—
480C22×C4—1661no—
481C22×C4C2×D33211yescharacteristic
482C22×C4C2×D33211yescharacteristic
483C22×C4C2×D33211yescharacteristic
484C22×C4C2×D33211yescharacteristic
485C22×C4C2×D33211yescharacteristic
486C22×C4Q8×D8312yes—
487C22×C4#1305058912yes—
488C22×C4C23×D8311yescharacteristic
489C2×C8—28no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8D4×D8312yes—
501C2×C8#1305058912yes—
502C4⋊C4—3322no—
503C4⋊C4—3322no—
504C4⋊C4—3322no—
505C4⋊C4—3322no—
506C4⋊C4—3322no—
507C4⋊C4—3322no—
508C4⋊C4—3322no—
509C4⋊C4—832no—
510C4⋊C4—832no—
511C4⋊C4—832no—
512C4⋊C4—6641no—
513C4⋊C4—6641no—
514C4⋊C4—1661no—
515C4⋊C4—1661no—
516C4⋊C4—1661no—
517C4⋊C4—1661no—
518C4⋊C4—1661no—
519C4⋊C4—1661no—
520C4⋊C4—1661no—
521C4⋊C4—1661no—
522C4⋊C4—1664no—
523C4⋊C4—1661no—
524C4⋊C4—1661no—
525C4⋊C4—1661no—
526C22⋊C4—22no—
527C22⋊C4—3322no—
528C22⋊C4—1664no—
529C22⋊C4Dic33214yes—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—3322no—
535C42—6641no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#1458177212yes—
541C23C2×Dic33212yes—
542C23#1458182012yes—
543C23C22×D33211yescharacteristic
544Q8—3321no—
545Q8—3321no—
546Q8—6644no—
547Q8—1662no—
548Q8—1662no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—3321no—
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—3321no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3321no—
584C2×C4—3321no—
585C2×C4—6644no—
586C2×C4—6644no—
587C2×C4—1662no—
588C2×C4—1662no—
589C2×C4—1662no—
590C2×C4—1662no—
591C2×C4#1458177212yes—
592C2×C4C2×Dic33212yes—
593C2×C4C2×Dic33212yes—
594C2×C4C2×D4×D8311yescharacteristic
595C2×C4C2×Q8×D8311yescharacteristic
596C2×C4C2×#1305058911yescharacteristic
597C2×C4C2×#1305058911yescharacteristic
598C2×C4#1458183811yescharacteristic
599C2×C4#1458183911yescharacteristic
600C8—22no—
601C8—22no—
602C8—41no—
603C8—41no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22#1912799714yescomplemented
611C22C2×#1458177211yescharacteristic
612C22D4×D33211yescharacteristic
613C22#1912835511yescharacteristic
614C22C22×Dic33211yescharacteristic
615C22C2×#1458182011yescharacteristic
616C22#1912850611yescharacteristic
617C22#1912851811yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—41no—
623C4—3322no—
624C4—3322no—
625C4—3322no—
626C4—3322no—
627C4—6641no—
628C4—6641no—
629C4—6641no—
630C4—6641no—
631C4#1912820711yescharacteristic
632C4#1912829511yescharacteristic
633C4#1912836411yescharacteristic
634C4#1912840911yescharacteristic
635C2—24no—
636C2#2491127011yescharacteristic
637C2C2×#1912799711yescharacteristic
638C2#2491731611yescharacteristic
639C2#2491894811yescharacteristic
640C2D4×Dic33211yescharacteristic
641C2#2491985611yescharacteristic
642C2#2491990511yescharacteristic
subgroupG/H
centerC23C22×D332
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1850C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1850D83
hypocenterC83#26342
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. C2×#19127997C2semidirect
5. #24909000C2semidirect
6. #24908956C2semidirect
7. #24908857C2semidirect
8. #24908792C2semidirect
9. #24908222C2semidirect
10. #24908056C2semidirect
13. #24907720C2semidirect
68. #14581771Q8semidirect
288. C22×C83#477semidirect
337. C83#26342semidirect
610. C22#19127997semidirect

Maximal subgroups

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

subgroupindex
C2×#191279972
C22×#145817262
#249111652
#249104662
#249090002
#249089562
#249088572
#249087922
#249082222
#249080562
#249078862
#249077682
#249077202
#249076532
C83×#18502
#2634283