Finite Group Databasebeta

← #35422136

Subgroups — #35422136

Order 21248.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910348C211yesmaximal, characteristic
2#24910340C211yesmaximal, 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#24907549C211yesmaximal, characteristic
11#24907496C211yesmaximal, characteristic
12#24907363C211yesmaximal, characteristic, complemented
13#106×Dic83C211yesmaximal, characteristic, complemented
14#24907244C211yesmaximal, characteristic
15C2×#428×C83C211yesmaximal, characteristic
16C2×#14581726C2211yescharacteristic
17C2×#14581724C2211yescharacteristic
18C2×C22⋊(C83⋊C8)C2211yescharacteristic
19C2×#14581702C2211yescharacteristic
20C2×C8×Dic83C2211yescharacteristic
21#19127600C2211yescharacteristic
22#19127600C2211yescharacteristic
23#19127600C2211yescharacteristic
24#19127600C2211yescharacteristic
25#19127594—21no—
26#19127594—21no—
27#19127587—21no—
28#19127587—21no—
29#19127567—21no—
30#19127561—21no—
31#19127561C2211yescharacteristic
32#19127561C2211yescharacteristic
33C4⋊C4×Dic83C2211yescharacteristic
34C4⋊C4×Dic83C2211yescharacteristic
35C4⋊C4×Dic83C2211yescharacteristic
36C4⋊C4×Dic83C2211yescharacteristic
37C4×C22⋊Dic83—21no—
38C4×C22⋊Dic83C2211yescharacteristic
39#19127547—21no—
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
62C2×C4⋊Dic83—21no—
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C22⋊Dic83—21no—
71C2×C22⋊Dic83—21no—
72C2×C22⋊Dic83D411yescharacteristic
73C2×C22⋊Dic83D411yescharacteristic
74C2×C4×Dic83—21no—
75C2×C4×Dic83—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83—21no—
78C2×C4×Dic83C2311yescharacteristic
79C2×C4×Dic83C2311yescharacteristic
80#14581726—22no—
81#14581724—22no—
82C22⋊(C83⋊C8)—22no—
83#14581702—22no—
84C8×Dic83—22no—
85C83⋊#95—21no—
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⋊#95D411yescharacteristic
96C83⋊#95D411yescharacteristic
97C23×C4×C83—21no—
98C23×C4×C83C2311yescharacteristic
99C2×M16×C83—22no—
100C2×M16×C83—22no—
101C2×M16×C83C2×C414yes—
102C22×C8×C83C2311yescharacteristic
103C22×C8×C83C2311yescharacteristic
104#117×C83—22no—
105#117×C83C2×C414yes—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—22no—
108C2×C4⋊C4×C83C2×C414yes—
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C22⋊C4×C83—21no—
114C2×C42×C83—21no—
115#107×C83C2×C414yes—
116#107×C83C2×C414yes—
117#106×C83C2×C414yes—
118#106×C83C2×C414yes—
119C22×Dic83—21no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83C2×D411yescharacteristic
124C22×Dic83C2×D411yescharacteristic
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C2×C83⋊C8—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C4⋊Dic83—21no—
134C4⋊Dic83—21no—
135C4⋊Dic83—21no—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—81no—
142C4⋊Dic83—81no—
143C4⋊Dic83—81no—
144C4⋊Dic83—81no—
145C22⋊Dic83—22no—
146C22⋊Dic83—41no—
147C22⋊Dic83—42no—
148C22⋊Dic83—81no—
149C4×Dic83—21no—
150C4×Dic83—21no—
151C4×Dic83—21no—
152C4×Dic83—21no—
153C4×Dic83—42no—
154C4×Dic83—42no—
155C4×Dic83—81no—
156C4×Dic83—81no—
157C24×C83C2×D411yescharacteristic
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—24no—
167C22×C4×C83—81no—
168C22×C4×C83—81no—
169C22×C4×C83C22×C412yes—
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83#4112yes—
172C22×C4×C83C2411yescharacteristic
173M16×C83—28no—
174C2×C8×C83—21no—
175C2×C8×C83—21no—
176C2×C8×C83C22×C412yes—
177C2×C8×C83C22×C412yes—
178C2×C8×C83#4112yes—
179C2×C8×C83#4112yes—
180C4⋊C4×C83—24no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83C22×C412yes—
186C4⋊C4×C83C22×C412yes—
187C4⋊C4×C83C22×C412yes—
188C4⋊C4×C83C22×C412yes—
189C22⋊C4×C83—24no—
190C42×C83—24no—
191C23×C83—21no—
192C23×C83—21no—
193C23×C83—21no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83—21no—
197C23×C83C4×D412yes—
198C23×C83C22×D411yescharacteristic
199C2×C4×C83—21no—
200C2×C4×C83—21no—
201C2×C4×C83—22no—
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—41no—
210C2×C4×C83—41no—
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—84no—
218C2×C4×C83C4×D412yes—
219C2×C4×C83C23×C411yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C2×C4×C83C2×#4111yescharacteristic
222C2×C4×C83C2×#4111yescharacteristic
223C2×C4×C832−1+411yescharacteristic
224C2×C4×C832−1+411yescharacteristic
225C2×Dic83—21no—
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—41no—
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#13712yes—
249C8×C83—22no—
250C8×C83#11714yes—
251C83⋊C8—41no—
252C83⋊C8—41no—
253C22×C83—22no—
254C22×C83—22no—
255C22×C83—22no—
256C22×C83—41no—
257C22×C83—41no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83C2×C4×D411yescharacteristic
261C22×C83#54711yescharacteristic
262C22×C83#54711yescharacteristic
263C22×C83C2×#13711yescharacteristic
264C22×C83#57511yescharacteristic
265C22×C83#57611yescharacteristic
266C22×C83#57711yescharacteristic
267C4×C83—22no—
268C4×C83—42no—
269C4×C83—42no—
270C4×C83—42no—
271C4×C83—42no—
272C4×C83C2×#11711yescharacteristic
273C4×C83#51911yescharacteristic
274C4×C83#55211yescharacteristic
275C4×C83#55211yescharacteristic
276Dic83—22no—
277Dic83—41no—
278Dic83—41no—
279Dic83—41no—
280Dic83—81no—
281Dic83—81no—
282Dic83—81no—
283Dic83—81no—
284#22621—831nomaximal, Sylow
285C2×C83—42no—
286C2×C83#234411yescharacteristic
287C2×C83C4×#13711yescharacteristic
288C2×C83#294611yescharacteristic
289C2×C83#334411yescharacteristic
290C2×C83#334711yescharacteristic
291C2×C83#335111yescharacteristic
292C2×C83#335711yescharacteristic
293C2×#428D8311yescharacteristic
294#2346—831no—
295#2345—831no—
296#1943—831no—
297#1942—831no—
298#1921—831no—
299#1920—831no—
300#1919—831no—
301#1918—831no—
302#1825—831no—
303#1823—831no—
304#1809—831no—
305#1776—831no—
306C4×#106—831no—
307#1753—831no—
308C83#2262111yescharacteristic, complemented, Sylow
309C22×M16C2×D8311yescharacteristic
310C2×#117C2×D8311yescharacteristic
311C22×C4⋊C4C2×D8311yescharacteristic
312#428Dic8318yescomplemented
313C2×#107—831no—
314C2×#107C2×D8311yescharacteristic
315C2×#107C2×D8311yescharacteristic
316C2×#106—831no—
317C2×#106C2×D8311yescharacteristic
318C2×#106C2×D8311yescharacteristic
319C2×C22⋊C8—831no—
320C2×#97—831no—
321C2×C4×C8—831no—
322#389—831no—
323#389—831no—
324#389—831no—
325#389—831no—
326#388—1661no—
327#388—1661no—
328#387—1661no—
329#387—1661no—
330#382—1661no—
331#380—831no—
332#380—831no—
333#380—1661no—
334C4×C4⋊C4—831no—
335C4×C4⋊C4—831no—
336C4×C4⋊C4—831no—
337C4×C4⋊C4—831no—
338C4×C22⋊C4—831no—
339C4×C22⋊C4—1661no—
340#376—1661no—
341#340—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#336—831no—
352#336—831no—
353C23×C4—21no—
354C23×C4C22×D8311yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C2×Dic8314yes—
358C22×C8—831no—
359C22×C8—831no—
360C22×C8C22×D8311yescharacteristic
361C22×C8C22×D8311yescharacteristic
362#117—22no—
363#117C2×Dic8314yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—831no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—1661no—
371C2×C4⋊C4—1661no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4C2×Dic8314yes—
375C2×C4⋊C4C22×D8311yescharacteristic
376C2×C4⋊C4C22×D8311yescharacteristic
377C2×C4⋊C4C22×D8311yescharacteristic
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—831no—
381C2×C22⋊C4—831no—
382C2×C22⋊C4—1661no—
383C2×C22⋊C4—1661no—
384C2×C42—21no—
385C2×C42—831no—
386C2×C42—831no—
387C2×C42—1661no—
388C2×C42—1661no—
389C2×C42—1661no—
390C2×C42—1661no—
391#107—1662no—
392#107C2×Dic8314yes—
393#107C2×Dic8314yes—
394#106—1662no—
395#106C2×Dic8314yes—
396#106C2×Dic8314yes—
397C22⋊C8—1662no—
398#97—1662no—
399C4×C8—1662no—
400#95—831no—
401#95—831no—
402#95—1661no—
403#95—1661no—
404#95—1661no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412C24D4×D8311yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—831no—
425C22×C4—831no—
426C22×C4—831no—
427C22×C4—831no—
428C22×C4—1661no—
429C22×C4—1661no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4C22×Dic8312yes—
433C22×C4D4×D8311yescharacteristic
434C22×C4#1305058912yes—
435C22×C4C23×D8311yescharacteristic
436M16—28no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—1661no—
440C2×C8—1661no—
441C2×C8—1661no—
442C2×C8—1661no—
443C2×C8—1661no—
444C2×C8—1661no—
445C2×C8C22×Dic8312yes—
446C2×C8C22×Dic8312yes—
447C2×C8#1305059012yes—
448C2×C8#1305059012yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—24no—
454C4⋊C4—6641no—
455C4⋊C4—6641no—
456C4⋊C4—6641no—
457C4⋊C4—6641no—
458C4⋊C4—1661no—
459C4⋊C4—1661no—
460C4⋊C4—1661no—
461C4⋊C4—1661no—
462C4⋊C4—1661no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4C22×Dic8312yes—
467C4⋊C4C22×Dic8312yes—
468C4⋊C4C22×Dic8312yes—
469C4⋊C4C22×Dic8312yes—
470C22⋊C4—24no—
471C22⋊C4—3322no—
472C22⋊C4—3321no—
473C22⋊C4—6641no—
474C22⋊C4—1662no—
475C42—24no—
476C42—3322no—
477C42—3322no—
478C42—6641no—
479C42—6641no—
480C42—1661no—
481C42—1661no—
482C42—1661no—
483C42—1661no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23D4×Dic8312yes—
491C23C2×D4×D8311yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—84no—
511C2×C4—3321no—
512C2×C4—3321no—
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—832no—
524C2×C4—1661no—
525C2×C4—1661no—
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×C4D4×Dic8312yes—
536C2×C4C23×Dic8311yescharacteristic
537C2×C4C2×#1305058911yescharacteristic
538C2×C4C2×#1305059011yescharacteristic
539C2×C4C2×#1305059011yescharacteristic
540C2×C4#1458184011yescharacteristic
541C2×C4#1458184011yescharacteristic
542C8—22no—
543C8—3321no—
544C8—3321no—
545C8#1458175114yes—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×D4×Dic8311yescharacteristic
554C22#1912835211yescharacteristic
555C22#1912835211yescharacteristic
556C22#137×D8311yescharacteristic
557C22#1912850011yescharacteristic
558C22#1912850511yescharacteristic
559C22#1912851111yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—3321no—
566C4—3321no—
567C4—3321no—
568C4—6641no—
569C4—6641no—
570C4—6641no—
571C4—6641no—
572C4—1662no—
573C4C2×#1458175111yescharacteristic
574C4#1912817911yescharacteristic
575C4#1912838611yescharacteristic
576C4#1912838611yescharacteristic
577C2—42no—
578C2#2491033211yescharacteristic
579C2#137×Dic8311yescharacteristic
580C2#2491629611yescharacteristic
581C2#2491991911yescharacteristic
582C2#2491996411yescharacteristic
583C2#2492002411yescharacteristic
584C2#2492008511yescharacteristic
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#22621
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
308. C83#22621semidirect
312. #428Dic83semidirect

Maximal subgroups

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

subgroupindex
#249103482
#249103402
#249082022
#249081972
#249080572
#249080502
#249080432
#249080362
#249075582
#249075492
#249074962
#249073632
#106×Dic832
#249072442
C2×#428×C832
#2262183