Finite Group Databasebeta

← #35466931

Subgroups — #35466931

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×#19127983C211yesmaximal, characteristic, complemented
2C22×#14581723C211yesmaximal, characteristic
3#24910466C211yesmaximal, characteristic
4#24909007C211yesmaximal, characteristic, complemented
5#24908964C211yesmaximal, characteristic, complemented
6#24908841C211yesmaximal, characteristic, complemented
7#24908793C211yesmaximal, characteristic, complemented
8#24908224C211yesmaximal, characteristic, complemented
9#24908038C211yesmaximal, characteristic, complemented
10#24907884C211yesmaximal, characteristic
11#24907770C211yesmaximal, characteristic
12#24907730C211yesmaximal, characteristic, complemented
13#24907655C211yesmaximal, characteristic
14#24907643C211yesmaximal, characteristic
15C83×#2440C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581771C2211yescharacteristic
18C2×C4×Dic166C2211yescharacteristic
19C22×C4⋊Dic83C2211yescharacteristic
20#19127983—24no—
21C2×#14581723—24no—
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×C4⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581715C2211yescharacteristic
31C2×#14581715C2211yescharacteristic
32C2×#14581715C2211yescharacteristic
33C2×#14581715C2211yescharacteristic
34C2×C22⋊(C83⋊C8)C2211yescharacteristic
35#19127607C2211yescharacteristic
36#19127599C2211yescharacteristic
37#19127598—21no—
38#19127588—21no—
39#19127573—21no—
40C4×C4⋊Dic83C2211yescharacteristic
41C22⋊C4×Dic83—21no—
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×Dic166C2311yescharacteristic
62C23×Dic83—21no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8—22no—
65C22×C83⋊C8—22no—
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70#14581771—22no—
71#14581771Q814yescomplemented
72C4×Dic166—22no—
73C4×Dic166—41no—
74C2×C4⋊Dic83—22no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—41no—
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×Dic83⋊C4—21no—
83C2×C22⋊Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86#14581723—22no—
87#14581723—22no—
88#14581723—22no—
89#14581723—22no—
90#14581723—41no—
91#14581723D414yes—
92C4⋊(C83⋊C8)—22no—
93C4⋊(C83⋊C8)—22no—
94#14581715—22no—
95#14581715—22no—
96#14581715—22no—
97#14581715—41no—
98C22⋊(C83⋊C8)—22no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C22×Q8×C83—21no—
104C23×C4×C83—21no—
105C23×C4×C83C2311yescharacteristic
106C4⋊Q8×C83—22no—
107C4⋊Q8×C83—41no—
108C22⋊Q8×C83—44no—
109C2×C4⋊C4×C83—22no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83—21no—
115C2×C4⋊C4×C83—21no—
116C2×C4⋊C4×C83—41no—
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C22⋊C4×C83—21no—
122C2×C22⋊C4×C83D411yescharacteristic
123C2×C22⋊C4×C83D411yescharacteristic
124C2×C42×C83C2311yescharacteristic
125C2×C42×C83C2311yescharacteristic
126#95×C83—21no—
127#95×C83—21no—
128#95×C83—21no—
129#95×C83D411yescharacteristic
130#95×C83D411yescharacteristic
131C2×Dic166—21no—
132C2×Dic166—21no—
133C2×Dic166—21no—
134C2×Dic166—21no—
135C2×Dic166—21no—
136C2×Dic166C2×Q812yes—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—21no—
140C22×Dic83—21no—
141C22×Dic83—42no—
142C22×Dic83—42no—
143C22×Dic83—81no—
144C2×C83⋊C8—28no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8C2×D412yes—
156C2×C83⋊C8#4112yes—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—42no—
162C4⋊Dic83C2×D412yes—
163C4⋊Dic83C2×Q812yes—
164Dic83⋊C4—24no—
165Dic83⋊C4—42no—
166Dic83⋊C4—81no—
167C22⋊Dic83—24no—
168C4×Dic83—42no—
169C4×Dic83—81no—
170C24×C83C2×D411yescharacteristic
171C2×Q8×C83—42no—
172C2×Q8×C83—42no—
173C2×Q8×C83—81no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—42no—
183C22×C4×C83—42no—
184C22×C4×C83—81no—
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2×Q812yes—
191C22×C4×C83#4112yes—
192C22×C4×C83C2411yescharacteristic
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—42no—
201C4⋊C4×C83—42no—
202C4⋊C4×C83—42no—
203C4⋊C4×C83—42no—
204C4⋊C4×C83—42no—
205C4⋊C4×C83—81no—
206C4⋊C4×C83C2×D412yes—
207C22⋊C4×C83—22no—
208C22⋊C4×C83—42no—
209C22⋊C4×C83SD1614yes—
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×SD1612yes—
220C23×C83#13712yes—
221C23×C83C22×D411yescharacteristic
222Q8×C83—84no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—22no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
233C2×C4×C83—21no—
234C2×C4×C83—22no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
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×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—84no—
252C2×C4×C83C22⋊Q812yes—
253C2×C4×C83C2×SD1612yes—
254C2×C4×C83C2×SD1612yes—
255C2×C4×C83C22×D411yescharacteristic
256C2×C4×C83C22×Q811yescharacteristic
257C2×C4×C83C2×#4111yescharacteristic
258C2×C4×C83C2×#4111yescharacteristic
259C2×C4×C83H(5,𝔽2)11yescharacteristic
260C2×C4×C832−1+411yescharacteristic
261C2×Dic83—22no—
262C2×Dic83—22no—
263C2×Dic83—22no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—84no—
274Dic166—22no—
275Dic166—22no—
276Dic166—41no—
277Dic166—41no—
278C83⋊C8—22no—
279C83⋊C8—22no—
280C83⋊C8—41no—
281C83⋊C8—41no—
282C22×C83—22no—
283C22×C83—22no—
284C22×C83—22no—
285C22×C83—22no—
286C22×C83—22no—
287C22×C83—22no—
288C22×C83#47514yescomplemented
289C22×C83C2×C22⋊Q811yescharacteristic
290C22×C83D4211yescharacteristic
291C22×C83#54711yescharacteristic
292C22×C83C22×SD1611yescharacteristic
293C22×C83C2×#13711yescharacteristic
294C22×C83#57711yescharacteristic
295C22×C83#57711yescharacteristic
296C4×C83—22no—
297C4×C83—22no—
298C4×C83—22no—
299C4×C83—41no—
300C4×C83—41no—
301C4×C83—42no—
302C4×C83—81no—
303C4×C83—81no—
304C4×C83—81no—
305C4×C83#52511yescharacteristic
306C4×C83#53911yescharacteristic
307C4×C83D4×Q811yescharacteristic
308C4×C83#55411yescharacteristic
309Dic83—42no—
310Dic83—42no—
311Dic83—42no—
312Dic83—81no—
313#26340—831nomaximal, Sylow
314C2×C83—24no—
315C2×C83#244911yescharacteristic
316C2×C83C2×#47511yescharacteristic
317C2×C83#308411yescharacteristic
318C2×C83#326611yescharacteristic
319C2×C83D4×SD1611yescharacteristic
320C2×C83#333511yescharacteristic
321C2×C83#334411yescharacteristic
322C2×#475—831no—
323C22×#106—831no—
324#2440D8311yescharacteristic
325#2360—831no—
326#2096—831no—
327#2088—831no—
328#2066—831no—
329#2058—831no—
330#1948—831no—
331#1919—831no—
332#1892—831no—
333#1872—831no—
334#1865—831no—
335#1853—831no—
336#1850—831no—
337C83#2634011yescharacteristic, complemented, Sylow
338C23×C8—831no—
339C2×C4⋊Q8C2×D8311yescharacteristic
340C2×C22⋊Q8—21no—
341C2×C22⋊Q8—831no—
342C2×C4×Q8—831no—
343C22×C4⋊C4—831no—
344C22×C4⋊C4C2×D8311yescharacteristic
345#475—1664no—
346C2×#106—831no—
347C2×#106—831no—
348C2×#106—831no—
349C2×#106—831no—
350C2×#106—831no—
351C2×#106—831no—
352C2×#106—1664no—
353C2×C4⋊C8—831no—
354C2×C4⋊C8—831no—
355C2×#103—831no—
356C2×#103—831no—
357C2×#103—831no—
358C2×#103—831no—
359C2×C22⋊C8—831no—
360#398—21no—
361#395—21no—
362#391—831no—
363#389—831no—
364#389—1661no—
365#389C2×D8311yescharacteristic
366#389C2×D8311yescharacteristic
367#387—1661no—
368#385—21no—
369#385—1661no—
370#384C2×D8311yescharacteristic
371#380C2×D8311yescharacteristic
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#336—831no—
382#336—831no—
383C22×Q8—21no—
384C22×Q8—831no—
385C23×C4—21no—
386C23×C4—1661no—
387C23×C4C22×D8311yescharacteristic
388C22×C8—831no—
389C22×C8—831no—
390C22×C8—831no—
391C22×C8—831no—
392C22×C8—1662no—
393C22×C8—1662no—
394C22×C8—1662no—
395C4⋊Q8—22no—
396C4⋊Q8—41no—
397C22⋊Q8—44no—
398C22⋊Q8—834no—
399C22⋊Q8—1662no—
400C4×Q8—3321no—
401C4×Q8—1662no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—41no—
410C2×C4⋊C4—3321no—
411C2×C4⋊C4—831no—
412C2×C4⋊C4—831no—
413C2×C4⋊C4—831no—
414C2×C4⋊C4—1661no—
415C2×C4⋊C4—1661no—
416C2×C4⋊C4—1662no—
417C2×C4⋊C4—1661no—
418C2×C4⋊C4—1661no—
419C2×C4⋊C4C22×D8311yescharacteristic
420C2×C4⋊C4C22×D8311yescharacteristic
421C2×C4⋊C4C22×D8311yescharacteristic
422C2×C4⋊C4C22×D8311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1661no—
425C2×C22⋊C4C22⋊D8311yescharacteristic
426C2×C22⋊C4C22⋊D8311yescharacteristic
427C2×C42—1661no—
428C2×C42—1661no—
429C2×C42C22×D8311yescharacteristic
430C2×C42C22×D8311yescharacteristic
431#106—3321no—
432#106—834no—
433#106—1662no—
434#106—1662no—
435#106—1662no—
436#106—1662no—
437C4⋊C8—1662no—
438C4⋊C8—1662no—
439#103—3321no—
440#103—1662no—
441#103—1662no—
442#103—1662no—
443C22⋊C8—1662no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1661no—
448#95—1661no—
449#95—1661no—
450#95—1661no—
451#95C22⋊D8311yescharacteristic
452#95C22⋊D8311yescharacteristic
453C24C2×C22⋊D8311yescharacteristic
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—81no—
457C2×Q8—832no—
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—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—3322no—
475C22×C4—3322no—
476C22×C4—6641no—
477C22×C4—1661no—
478C22×C4—1661no—
479C22×C4—1661no—
480C22×C4—1661no—
481C22×C4C2×C22⋊D8311yescharacteristic
482C22×C4C2×C22⋊D8311yescharacteristic
483C22×C4C2×C22⋊D8311yescharacteristic
484C22×C4C2×C22⋊D8311yescharacteristic
485C22×C4C2×C22⋊D8311yescharacteristic
486C22×C4Q8×D8312yes—
487C22×C4#1305059112yes—
488C22×C4C23×D8311yescharacteristic
489C2×C8—832no—
490C2×C8—832no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8—1661no—
496C2×C8—1661no—
497C2×C8—1668no—
498C2×C8—1661no—
499C2×C8—1661no—
500C2×C8—1661no—
501C2×C8—1661no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—3322no—
516C4⋊C4—3322no—
517C4⋊C4—832no—
518C4⋊C4—832no—
519C4⋊C4—6641no—
520C4⋊C4—1661no—
521C4⋊C4—1661no—
522C4⋊C4—1664no—
523C4⋊C4—1661no—
524C4⋊C4—1661no—
525C4⋊C4D4×D8312yes—
526C22⋊C4—22no—
527C22⋊C4—42no—
528C22⋊C4—1664no—
529C22⋊C4#1305057814yes—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—3322no—
535C42—6641no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23#1458176812yes—
541C23C2×#1305057812yes—
542C23#1458182312yes—
543C23C22×C22⋊D8311yescharacteristic
544Q8—84no—
545Q8—3321no—
546Q8—3321no—
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—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—3321no—
584C2×C4—3321no—
585C2×C4—3321no—
586C2×C4—3321no—
587C2×C4—6644no—
588C2×C4—1662no—
589C2×C4—1662no—
590C2×C4—1662no—
591C2×C4#1458176812yes—
592C2×C4C2×#1305057812yes—
593C2×C4C2×#1305057812yes—
594C2×C4C2×D4×D8311yescharacteristic
595C2×C4C2×Q8×D8311yescharacteristic
596C2×C4#41×D8311yescharacteristic
597C2×C4C2×#1305059111yescharacteristic
598C2×C4#1458183711yescharacteristic
599C2×C4#1458183911yescharacteristic
600C8—3321no—
601C8—3321no—
602C8—1662no—
603C8—1662no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22#1912798314yescomplemented
611C22C2×#1458176811yescharacteristic
612C22D4×C22⋊D8311yescharacteristic
613C22#1912835011yescharacteristic
614C22C22×#1305057811yescharacteristic
615C22C2×#1458182311yescharacteristic
616C22#1912850911yescharacteristic
617C22#1912851211yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—42no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—3322no—
628C4—3322no—
629C4—3322no—
630C4—6641no—
631C4#1912821411yescharacteristic
632C4#1912830011yescharacteristic
633C4#1912836411yescharacteristic
634C4#1912840511yescharacteristic
635C2—24no—
636C2#2491126811yescharacteristic
637C2C2×#1912798311yescharacteristic
638C2#2491732111yescharacteristic
639C2#2491897411yescharacteristic
640C2D4×#1305057811yescharacteristic
641C2#2491982011yescharacteristic
642C2#2491991211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2440C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2440D83
hypocenterC83#26340
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×#19127983C2semidirect
4. #24909007C2semidirect
5. #24908964C2semidirect
6. #24908841C2semidirect
7. #24908793C2semidirect
8. #24908224C2semidirect
9. #24908038C2semidirect
12. #24907730C2semidirect
71. #14581771Q8semidirect
288. C22×C83#475semidirect
337. C83#26340semidirect
610. C22#19127983semidirect

Maximal subgroups

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

subgroupindex
C2×#191279832
C22×#145817232
#249104662
#249090072
#249089642
#249088412
#249087932
#249082242
#249080382
#249078842
#249077702
#249077302
#249076552
#249076432
C83×#24402
#2634083