Finite Group Databasebeta

← #35422151

Subgroups — #35422151

Order 21248.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910356C211yesmaximal, characteristic
2#24910332C211yesmaximal, characteristic
3#24908202C211yesmaximal, characteristic, complemented
4#24908197C211yesmaximal, characteristic, complemented
5#24908057C211yesmaximal, characteristic, complemented
6#24908050C211yesmaximal, characteristic, complemented
7#24908043C211yesmaximal, characteristic, complemented
8#24908036C211yesmaximal, characteristic, complemented
9#24907558C211yesmaximal, characteristic
10#24907553C211yesmaximal, characteristic
11#24907496C211yesmaximal, characteristic
12#24907363C211yesmaximal, characteristic, complemented
13#106×Dic83C211yesmaximal, characteristic, complemented
14#24907244C211yesmaximal, characteristic
15C2×#428×C83C211yesmaximal, characteristic
16C2×#14581751—21no—
17C2×#14581726C2211yescharacteristic
18C2×#14581724C2211yescharacteristic
19C2×C22⋊(C83⋊C8)C2211yescharacteristic
20C2×#14581702C2211yescharacteristic
21C2×C8×Dic83C2211yescharacteristic
22#19127600C2211yescharacteristic
23#19127600C2211yescharacteristic
24#19127600C2211yescharacteristic
25#19127600C2211yescharacteristic
26#19127594—21no—
27#19127594—21no—
28#19127587—21no—
29#19127587—21no—
30#19127569—21no—
31#19127564—21no—
32#19127561C2211yescharacteristic
33#19127561C2211yescharacteristic
34C4⋊C4×Dic83C2211yescharacteristic
35C4⋊C4×Dic83C2211yescharacteristic
36C4⋊C4×Dic83C2211yescharacteristic
37C4⋊C4×Dic83C2211yescharacteristic
38C4×C4⋊Dic83—21no—
39C4×C22⋊Dic83C2211yescharacteristic
40#19127464C2211yescharacteristic
41#19127464C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
52C22×M16×C83C2211yescharacteristic
53C2×#117×C83C2211yescharacteristic
54C22×C4⋊C4×C83C2211yescharacteristic
55#428×C83C418yescomplemented
56C2×#107×C83C2211yescharacteristic
57C2×#107×C83C2211yescharacteristic
58C2×#106×C83C2211yescharacteristic
59C2×#106×C83C2211yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62#14581751—44no—
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C22⋊Dic83—21no—
72C2×C22⋊Dic83—21no—
73C2×C22⋊Dic83D411yescharacteristic
74C2×C22⋊Dic83D411yescharacteristic
75C2×C4×Dic83—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83—21no—
78C2×C4×Dic83—21no—
79C2×C4×Dic83C2311yescharacteristic
80C2×C4×Dic83C2311yescharacteristic
81#14581726—22no—
82#14581724—22no—
83C22⋊(C83⋊C8)—22no—
84#14581702—22no—
85C8×Dic83—22no—
86C83⋊#95—21no—
87C83⋊#95—21no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95D411yescharacteristic
97C83⋊#95D411yescharacteristic
98C23×C4×C83—21no—
99C23×C4×C83C2311yescharacteristic
100C2×M16×C83—22no—
101C2×M16×C83—22no—
102C2×M16×C83C2×C414yes—
103C22×C8×C83C2311yescharacteristic
104C22×C8×C83C2311yescharacteristic
105#117×C83—22no—
106#117×C83C2×C414yes—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—22no—
109C2×C4⋊C4×C83C2×C414yes—
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C22⋊C4×C83—21no—
115C2×C42×C83—21no—
116#107×C83C2×C414yes—
117#107×C83C2×C414yes—
118#106×C83C2×C414yes—
119#106×C83C2×C414yes—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83C2×D411yescharacteristic
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C22×Dic83C2×D411yescharacteristic
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C4⋊Dic83—21no—
135C4⋊Dic83—21no—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—42no—
143C4⋊Dic83—42no—
144C4⋊Dic83—42no—
145C4⋊Dic83—42no—
146C22⋊Dic83—22no—
147C22⋊Dic83—41no—
148C22⋊Dic83—42no—
149C22⋊Dic83—81no—
150C4×Dic83—21no—
151C4×Dic83—21no—
152C4×Dic83—21no—
153C4×Dic83—21no—
154C4×Dic83—42no—
155C4×Dic83—42no—
156C4×Dic83—81no—
157C4×Dic83—81no—
158C24×C83C2×D411yescharacteristic
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—24no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—42no—
169C22×C4×C83—42no—
170C22×C4×C83C22×C412yes—
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83#4112yes—
173C22×C4×C83C2411yescharacteristic
174M16×C83—28no—
175C2×C8×C83—21no—
176C2×C8×C83—21no—
177C2×C8×C83C22×C412yes—
178C2×C8×C83C22×C412yes—
179C2×C8×C83#4112yes—
180C2×C8×C83#4112yes—
181C4⋊C4×C83—24no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83C22×C412yes—
187C4⋊C4×C83C22×C412yes—
188C4⋊C4×C83C22×C412yes—
189C4⋊C4×C83C22×C412yes—
190C22⋊C4×C83—24no—
191C42×C83—24no—
192C23×C83—21no—
193C23×C83—21no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83—21no—
197C23×C83—21no—
198C23×C83C4×D412yes—
199C23×C83C22×D411yescharacteristic
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—22no—
203C2×C4×C83—22no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—22no—
207C2×C4×C83—22no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—48no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83—41no—
219C2×C4×C83C4×D412yes—
220C2×C4×C83C23×C411yescharacteristic
221C2×C4×C83C2×#4111yescharacteristic
222C2×C4×C83C2×#4111yescharacteristic
223C2×C4×C83C2×#4111yescharacteristic
224C2×C4×C832−1+411yescharacteristic
225C2×C4×C832−1+411yescharacteristic
226C2×Dic83—21no—
227C2×Dic83—21no—
228C2×Dic83—21no—
229C2×Dic83—21no—
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83AGL(1,ℤ/8)12yes—
250C8×C83—22no—
251C8×C83#11714yescomplemented
252C83⋊C8—41no—
253C83⋊C8—41no—
254C22×C83—22no—
255C22×C83—22no—
256C22×C83—22no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83—41no—
261C22×C83C2×C4×D411yescharacteristic
262C22×C83#54711yescharacteristic
263C22×C83#54711yescharacteristic
264C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C83#57511yescharacteristic
266C22×C83#57711yescharacteristic
267C22×C83#57811yescharacteristic
268C4×C83—22no—
269C4×C83—42no—
270C4×C83—42no—
271C4×C83—42no—
272C4×C83—42no—
273C4×C83C2×#11711yescharacteristic
274C4×C83#51911yescharacteristic
275C4×C83#55211yescharacteristic
276C4×C83#55211yescharacteristic
277Dic83—22no—
278Dic83—41no—
279Dic83—41no—
280Dic83—41no—
281Dic83—81no—
282Dic83—81no—
283Dic83—81no—
284Dic83—81no—
285#22622—831nomaximal, Sylow
286C2×C83—42no—
287C2×C83#234411yescharacteristic
288C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
289C2×C83#294711yescharacteristic
290C2×C83#334511yescharacteristic
291C2×C83#334911yescharacteristic
292C2×C83#335111yescharacteristic
293C2×C83#335311yescharacteristic
294C2×#428D8311yescharacteristic
295#2347—831no—
296#2344—831no—
297#1943—831no—
298#1942—831no—
299#1921—831no—
300#1920—831no—
301#1919—831no—
302#1918—831no—
303#1825—831no—
304#1824—831no—
305#1809—831no—
306#1776—831no—
307C4×#106—831no—
308#1753—831no—
309C83#2262211yescharacteristic, complemented, Sylow
310C22×M16C2×D8311yescharacteristic
311C2×#117—1661no—
312C2×#117C2×D8311yescharacteristic
313C22×C4⋊C4C2×D8311yescharacteristic
314#428Dic8318yescomplemented
315C2×#107—831no—
316C2×#107C2×D8311yescharacteristic
317C2×#107C2×D8311yescharacteristic
318C2×#106—831no—
319C2×#106C2×D8311yescharacteristic
320C2×#106C2×D8311yescharacteristic
321C2×C22⋊C8—831no—
322C2×#97—831no—
323C2×C4×C8—831no—
324#389—831no—
325#389—831no—
326#389—831no—
327#389—831no—
328#388—1661no—
329#388—1661no—
330#387—1661no—
331#387—1661no—
332#383—1661no—
333#381—1661no—
334#380—831no—
335#380—831no—
336C4×C4⋊C4—831no—
337C4×C4⋊C4—831no—
338C4×C4⋊C4—831no—
339C4×C4⋊C4—831no—
340C4×C4⋊C4—1661no—
341C4×C22⋊C4—831no—
342#340—831no—
343#340—831no—
344#340—831no—
345#340—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#336—831no—
353#336—831no—
354C23×C4—21no—
355C23×C4C22×D8311yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C2×Dic8314yes—
359C22×C8—831no—
360C22×C8—831no—
361C22×C8C22×D8311yescharacteristic
362C22×C8C22×D8311yescharacteristic
363#117—22no—
364#117—3324no—
365#117C2×Dic8314yes—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4C2×Dic8314yes—
377C2×C4⋊C4C22×D8311yescharacteristic
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C4⋊C4C22×D8311yescharacteristic
380C2×C4⋊C4C22×D8311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—831no—
383C2×C22⋊C4—831no—
384C2×C22⋊C4—1661no—
385C2×C22⋊C4—1661no—
386C2×C42—21no—
387C2×C42—831no—
388C2×C42—831no—
389C2×C42—1661no—
390C2×C42—1661no—
391C2×C42—1661no—
392C2×C42—1661no—
393#107—1662no—
394#107C2×Dic8314yes—
395#107C2×Dic8314yes—
396#106—1662no—
397#106C2×Dic8314yes—
398#106C2×Dic8314yes—
399C22⋊C8—1662no—
400#97—1662no—
401C4×C8—1662no—
402#95—831no—
403#95—831no—
404#95—1661no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412#95—1661no—
413#95—1661no—
414C24D4×D8311yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—831no—
427C22×C4—831no—
428C22×C4—831no—
429C22×C4—831no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4—1661no—
433C22×C4—1661no—
434C22×C4C22×Dic8312yes—
435C22×C4D4×D8311yescharacteristic
436C22×C4#1305058912yes—
437C22×C4C23×D8311yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—1661no—
442C2×C8—1661no—
443C2×C8—1661no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8C22×Dic8312yes—
448C2×C8C22×Dic8312yes—
449C2×C8#1305059012yes—
450C2×C8#1305059012yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—3322no—
457C4⋊C4—3322no—
458C4⋊C4—3322no—
459C4⋊C4—3322no—
460C4⋊C4—1661no—
461C4⋊C4—1661no—
462C4⋊C4—1661no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4C22×Dic8312yes—
469C4⋊C4C22×Dic8312yes—
470C4⋊C4C22×Dic8312yes—
471C4⋊C4C22×Dic8312yes—
472C22⋊C4—24no—
473C22⋊C4—3322no—
474C22⋊C4—3321no—
475C22⋊C4—6641no—
476C22⋊C4—1662no—
477C42—24no—
478C42—3322no—
479C42—3322no—
480C42—6641no—
481C42—6641no—
482C42—1661no—
483C42—1661no—
484C42—1661no—
485C42—1661no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23D4×Dic8312yes—
493C23C2×D4×D8311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—3321no—
514C2×C4—3321no—
515C2×C4—3321no—
516C2×C4—3321no—
517C2×C4—3321no—
518C2×C4—3321no—
519C2×C4—3321no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—3321no—
523C2×C4—3321no—
524C2×C4—3321no—
525C2×C4—832no—
526C2×C4—1661no—
527C2×C4—1661no—
528C2×C4—1661no—
529C2×C4—1661no—
530C2×C4—1661no—
531C2×C4—1661no—
532C2×C4—1661no—
533C2×C4—1661no—
534C2×C4—1661no—
535C2×C4—1661no—
536C2×C4—1661no—
537C2×C4D4×Dic8312yes—
538C2×C4C23×Dic8311yescharacteristic
539C2×C4C2×#1305058911yescharacteristic
540C2×C4C2×#1305059011yescharacteristic
541C2×C4C2×#1305059011yescharacteristic
542C2×C4#1458184011yescharacteristic
543C2×C4#1458184011yescharacteristic
544C8—22no—
545C8—3321no—
546C8—3321no—
547C8#1458175114yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×D4×Dic8311yescharacteristic
556C22#1912835211yescharacteristic
557C22#1912835211yescharacteristic
558C22AGL(1,ℤ/8)×D8311yescharacteristic
559C22#1912849711yescharacteristic
560C22#1912851011yescharacteristic
561C22#1912851711yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—3321no—
568C4—3321no—
569C4—3321no—
570C4—6641no—
571C4—6641no—
572C4—6641no—
573C4—6641no—
574C4—1662no—
575C4C2×#1458175111yescharacteristic
576C4#1912817911yescharacteristic
577C4#1912838611yescharacteristic
578C4#1912838611yescharacteristic
579C2—42no—
580C2#2491033211yescharacteristic
581C2AGL(1,ℤ/8)×Dic8311yescharacteristic
582C2#2491630711yescharacteristic
583C2#2491993411yescharacteristic
584C2#2491999411yescharacteristic
585C2#2492001911yescharacteristic
586C2#2492004311yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#428×C83C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenterC2×#428D83
hypocenterC83#22622
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #24908202C2semidirect
4. #24908197C2semidirect
5. #24908057C2semidirect
6. #24908050C2semidirect
7. #24908043C2semidirect
8. #24908036C2semidirect
12. #24907363C2semidirect
13. #106×Dic83C2semidirect
55. #428×C83C4semidirect
251. C8×C83#117semidirect
309. C83#22622semidirect
314. #428Dic83semidirect
547. C8#14581751semidirect

Maximal subgroups

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

subgroupindex
#249103562
#249103322
#249082022
#249081972
#249080572
#249080502
#249080432
#249080362
#249075582
#249075532
#249074962
#249073632
#106×Dic832
#249072442
C2×#428×C832
#2262283