Finite Group Databasebeta

← #35422144

Subgroups — #35422144

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
1C2×#19127741C211yesmaximal, characteristic
2#24910358C211yesmaximal, characteristic
3#24910332C211yesmaximal, characteristic
4#24908200C211yesmaximal, characteristic, complemented
5#24908195C211yesmaximal, characteristic, complemented
6#24908053C211yesmaximal, characteristic, complemented
7#24908051C211yesmaximal, characteristic, complemented
8#24908044C211yesmaximal, characteristic, complemented
9#24908032C211yesmaximal, characteristic, complemented
10#24907559C211yesmaximal, characteristic
11#24907494C211yesmaximal, characteristic
12#24907359C211yesmaximal, characteristic, complemented
13#24907351C211yesmaximal, characteristic, complemented
14#24907249C211yesmaximal, characteristic
15C83×#1824C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C2×#14581751—21no—
18C2×#14581751C2211yescharacteristic
19#19127741C418yescomplemented
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581702C2211yescharacteristic
27C2×C8×Dic83C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127600C2211yescharacteristic
30#19127597C2211yescharacteristic
31#19127597C2211yescharacteristic
32#19127589—21no—
33#19127589—21no—
34#19127587—21no—
35#19127587—21no—
36#19127568—21no—
37#19127564—21no—
38#19127561C2211yescharacteristic
39C4⋊C4×Dic83C2211yescharacteristic
40C4⋊C4×Dic83C2211yescharacteristic
41C4⋊C4×Dic83C2211yescharacteristic
42C4⋊C4×Dic83C2211yescharacteristic
43#19127552—21no—
44C4×C22⋊Dic83C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127455C2211yescharacteristic
52#19127455C2211yescharacteristic
53C22×C4⋊C4×C83C2211yescharacteristic
54C2×C22⋊C8×C83C2211yescharacteristic
55#380×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60C2×#13050574—22no—
61C2×#13050574—22no—
62C2×#13050574C2×C414yes—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65#14581751—22no—
66#14581751—44no—
67#14581751C2×C414yes—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×C22⋊Dic83—21no—
77C2×C22⋊Dic83—21no—
78C2×C22⋊Dic83D411yescharacteristic
79C2×C22⋊Dic83D411yescharacteristic
80C2×C4×Dic83—21no—
81C2×C4×Dic83—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83C2311yescharacteristic
86C2×C4×Dic83C2311yescharacteristic
87#14581725—22no—
88#14581725C2×C414yes—
89#14581725C2×C414yes—
90#14581723—22no—
91#14581723C2×C414yes—
92#14581723C2×C414yes—
93#14581702—22no—
94C8×Dic83—22no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95D411yescharacteristic
105C83⋊#95D411yescharacteristic
106C23×C4×C83—21no—
107C23×C4×C83C2311yescharacteristic
108C22×C8×C83C2311yescharacteristic
109C22×C8×C83C2311yescharacteristic
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—22no—
112C2×C4⋊C4×C83C2×C414yes—
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C22⋊C4×C83—21no—
118C22⋊C8×C83—22no—
119#95×C83—21no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83C2×D411yescharacteristic
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C22×Dic83C2×D411yescharacteristic
128#13050574—28no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8C22×C412yes—
132C2×C83⋊C8C22×C412yes—
133C2×C83⋊C8#4112yes—
134C2×C83⋊C8#4112yes—
135C4⋊Dic83—21no—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—42no—
142C4⋊Dic83—42no—
143C4⋊Dic83C22×C412yes—
144C4⋊Dic83C22×C412yes—
145Dic83⋊C4—42no—
146Dic83⋊C4—42no—
147C22⋊Dic83—24no—
148C22⋊Dic83—22no—
149C22⋊Dic83—41no—
150C22⋊Dic83—42no—
151C4×Dic83—24no—
152C4×Dic83—21no—
153C4×Dic83—21no—
154C4×Dic83—21no—
155C4×Dic83—21no—
156C4×Dic83—42no—
157C4×Dic83—42no—
158C4×Dic83—81no—
159C4×Dic83—81no—
160C24×C83C2×D411yescharacteristic
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—24no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—42no—
171C22×C4×C83—42no—
172C22×C4×C83C22×C412yes—
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83#4112yes—
175C22×C4×C83C2411yescharacteristic
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83—21no—
179C2×C8×C83—21no—
180C2×C8×C83—21no—
181C2×C8×C83—21no—
182C4⋊C4×C83—24no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83C22×C412yes—
190C4⋊C4×C83C22×C412yes—
191C22⋊C4×C83—81no—
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—48no—
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—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—22no—
229C2×Dic83—22no—
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—21no—
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—41no—
251C8×C83—41no—
252C83⋊C8—22no—
253C83⋊C8#11714yescomplemented
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—81no—
272C4×C83—81no—
273C4×C83C2×#11711yescharacteristic
274C4×C83#51911yescharacteristic
275C4×C83#55211yescharacteristic
276C4×C83#55211yescharacteristic
277Dic83—22no—
278Dic83—41no—
279Dic83—41no—
280Dic83—41no—
281Dic83—42no—
282Dic83—42no—
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×#428—831no—
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#1824D8311yescharacteristic
305#1809—831no—
306#1776—831no—
307C4×#106—831no—
308#1753—831no—
309C83#2262211yescharacteristic, complemented, Sylow
310C22×M16—831no—
311C2×#117—831no—
312C2×#117—1661no—
313C22×C4⋊C4C2×D8311yescharacteristic
314#428—838no—
315C2×#107—831no—
316C2×#107—831no—
317C2×#107—831no—
318C2×#106—831no—
319C2×#106—831no—
320C2×#106—831no—
321C2×C22⋊C8C2×D8311yescharacteristic
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#380C2×D8311yescharacteristic
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#340C2×D8311yescharacteristic
349#340C2×D8311yescharacteristic
350#340C2×D8311yescharacteristic
351#340C2×D8311yescharacteristic
352#336—831no—
353#336—831no—
354C23×C4—21no—
355C23×C4C22×D8311yescharacteristic
356C2×M16—834no—
357C2×M16—1662no—
358C2×M16—1662no—
359C22×C8—831no—
360C22×C8—831no—
361C22×C8C22×D8311yescharacteristic
362C22×C8C22×D8311yescharacteristic
363#117—3324no—
364#117—834no—
365#117—1662no—
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⋊C4C4×D8314yes—
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—831no—
387C2×C42—831no—
388C2×C42—1661no—
389C2×C42—1661no—
390C2×C42—1661no—
391C2×C42—1661no—
392C2×C42—1661no—
393#107—834no—
394#107—834no—
395#107—1662no—
396#106—834no—
397#106—834no—
398#106—1662no—
399C22⋊C8—22no—
400#97—1662no—
401C4×C8—1662no—
402#95—21no—
403#95—831no—
404#95—831no—
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—24no—
422C22×C4—21no—
423C22×C4—21no—
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×C4C2×C4×D8312yes—
435C22×C4D4×D8311yescharacteristic
436C22×C4#1305058912yes—
437C22×C4C23×D8311yescharacteristic
438M16—1668no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—832no—
446C2×C8—832no—
447C2×C8—832no—
448C2×C8—832no—
449C2×C8—1661no—
450C2×C8—1661no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—3322no—
459C4⋊C4—3322no—
460C4⋊C4—3322no—
461C4⋊C4—3322no—
462C4⋊C4—832no—
463C4⋊C4—832no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4C2×C4×D8312yes—
471C4⋊C4C2×C4×D8312yes—
472C22⋊C4—81no—
473C22⋊C4—3322no—
474C22⋊C4—3321no—
475C22⋊C4—1662no—
476C22⋊C4—1664no—
477C42—3322no—
478C42—3322no—
479C42—6641no—
480C42—6641no—
481C42—1661no—
482C42—1661no—
483C42—1661no—
484C42—1661no—
485C42—1664no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23(C22⋊D83)⋊C412yes—
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—48no—
503C2×C4—41no—
504C2×C4—41no—
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—832no—
524C2×C4—1661no—
525C2×C4—1661no—
526C2×C4—1661no—
527C2×C4—1662no—
528C2×C4—1662no—
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×C4(C22⋊D83)⋊C412yes—
538C2×C4C22×C4×D8311yescharacteristic
539C2×C4#41×D8311yescharacteristic
540C2×C4#41×D8311yescharacteristic
541C2×C4C2×#1305058911yescharacteristic
542C2×C4#1458183911yescharacteristic
543C2×C4#1458183911yescharacteristic
544C8—41no—
545C8—41no—
546C8—834no—
547C8—1662no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×(C22⋊D83)⋊C411yescharacteristic
556C22#1912834911yescharacteristic
557C22#1912834911yescharacteristic
558C22AGL(1,ℤ/8)×D8311yescharacteristic
559C22#1912849711yescharacteristic
560C22#1912850711yescharacteristic
561C22#1912851411yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—81no—
567C4—3322no—
568C4—3322no—
569C4—3321no—
570C4—3321no—
571C4—3321no—
572C4—6641no—
573C4—6641no—
574C4—1662no—
575C4#117×D8311yescharacteristic
576C4#1912817811yescharacteristic
577C4#1912838811yescharacteristic
578C4#1912838811yescharacteristic
579C2—42no—
580C2#2491032711yescharacteristic
581C2#2491626711yescharacteristic
582C2#2491630011yescharacteristic
583C2#2491992611yescharacteristic
584C2#2491998611yescharacteristic
585C2#2492002711yescharacteristic
586C2#2492004011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1824C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#1824D83
hypocenterC83#22622
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
4. #24908200C2semidirect
5. #24908195C2semidirect
6. #24908053C2semidirect
7. #24908051C2semidirect
8. #24908044C2semidirect
9. #24908032C2semidirect
12. #24907359C2semidirect
13. #24907351C2semidirect
19. #19127741C4semidirect
253. C83⋊C8#117semidirect
309. C83#22622semidirect

Maximal subgroups

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

subgroupindex
C2×#191277412
#249103582
#249103322
#249082002
#249081952
#249080532
#249080512
#249080442
#249080322
#249075592
#249074942
#249073592
#249073512
#249072492
C83×#18242
#2262283