Finite Group Databasebeta

← #35420057

Subgroups — #35420057

Order 21248.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128356C211yesmaximal, characteristic
2C2×#19127703C212yesmaximal, complemented
3D4×C4⋊Dic83C211yesmaximal, characteristic, complemented
4#24908385C211yesmaximal, characteristic, complemented
5#24908265C211yesmaximal, characteristic, complemented
6#24907896C212yesmaximal, complemented
7#24907768C212yesmaximal, complemented
8#24907464C211yesmaximal, characteristic, complemented
9#24907404C212yesmaximal, complemented
10C4×#14581714C211yesmaximal, characteristic, complemented
11C2×#436×C83C211yesmaximal, characteristic, complemented
12C22×#13050590C2212yes—
13#19128356—24no—
14#19128356C418yescomplemented
15C2×#14581792C2211yescharacteristic
16C2×#14581774—22no—
17C2×#14581771C2212yes—
18C2×#14581763—21no—
19C2×C4×D332C2211yescharacteristic
20C2×D4×Dic83—21no—
21C22×C4⋊Dic83C2212yes—
22C2×C4⋊C4×D83—21no—
23#19127703—28no—
24C2×#14581717C2212yes—
25C2×#14581717C2212yes—
26C2×#14581717C2212yes—
27C2×#14581717C2212yes—
28C2×#14581714C2212yes—
29C2×#14581714C2211yescharacteristic
30C2×#14581714C2211yescharacteristic
31#19127600—21no—
32#19127576—22no—
33#19127571C2211yescharacteristic
34#19127561C2212yes—
35C4×C4⋊Dic83C2211yescharacteristic
36#19127464C2211yescharacteristic
37#19127464C2211yescharacteristic
38#19127464C2212yes—
39C22×M16×C83C2212yes—
40C2×C4×D4×C83C2211yescharacteristic
41#436×C83—24no—
42C2×C4⋊C8×C83C2211yescharacteristic
43C2×C22⋊C8×C83C2212yes—
44C2×C4×C8×C83C2211yescharacteristic
45C2×#13050590—24no—
46C2×#13050590—24no—
47C2×#13050590—24no—
48C2×#13050590—42no—
49C2×#13050590—42no—
50C2×#13050590C2×C418yescomplemented
51C22×Dic166C2312yes—
52C22×D332C2311yescharacteristic
53C22×C22⋊D83—22no—
54C23×Dic83—22no—
55C22×C4×D83—21no—
56C22×C4×D83—22no—
57#14581792—22no—
58#14581792C2×C414yescomplemented
59#14581774—216no—
60#14581771—24no—
61#14581771C2×C418yescomplemented
62#14581763—28no—
63C4×D332—22no—
64C4×D332C2×C414yescomplemented
65D4×Dic83—44no—
66C2×C4⋊Dic83—24no—
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83—22no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—42no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2312yes—
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×Dic83⋊C4—22no—
78C4⋊C4×D83—28no—
79C2×C22⋊Dic83—22no—
80C2×C22⋊Dic83—22no—
81C2×(C2×D83)⋊C4—21no—
82C2×C4×Dic83—21no—
83#14581717—24no—
84#14581717—24no—
85#14581717—42no—
86#14581717—42no—
87#14581714—24no—
88#14581714—22no—
89#14581714D414yescomplemented
90C83⋊#95—21no—
91C83⋊#95—22no—
92C22×D4×C83D411yescharacteristic
93C23×C4×C83C2312yes—
94C2×M16×C83—24no—
95C2×M16×C83—24no—
96C2×M16×C83—42no—
97C22×C8×C83C2312yes—
98C22×C8×C83C2311yescharacteristic
99C22×C8×C83C2311yescharacteristic
100C4×D4×C83C2×C414yes—
101C4×D4×C83D414yes—
102C2×C4⋊C4×C83D411yescharacteristic
103C2×C22⋊C4×C83D412yes—
104C2×C42×C83C2311yescharacteristic
105C4⋊C8×C83—22no—
106C22⋊C8×C83—24no—
107C4×C8×C83—22no—
108C23×D83—21no—
109#13050590—216no—
110#13050590—48no—
111#13050590—48no—
112#13050590—48no—
113C2×Dic166—22no—
114C2×Dic166—22no—
115C2×Dic166—22no—
116C2×Dic166—22no—
117C2×Dic166—22no—
118C2×Dic166C22×C414yes—
119C2×D332—21no—
120C2×D332—21no—
121C2×D332—21no—
122C2×D332C22×C412yes—
123C2×D332C2×D412yes—
124C2×D332#4112yes—
125C2×C22⋊D83—28no—
126C2×C22⋊D83—44no—
127C2×C22⋊D83—44no—
128C22×Dic83—22no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83—22no—
132C22×Dic83—21no—
133C22×Dic83—44no—
134C22×Dic83—44no—
135C22×Dic83—82no—
136C2×C4×D83—24no—
137C2×C4×D83—28no—
138C2×C4×D83—44no—
139C2×C4×D83—44no—
140C2×C4×D83—42no—
141C2×C4×D83—42no—
142C4⋊Dic83—24no—
143C4⋊Dic83—24no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—22no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—22no—
150C4⋊Dic83—44no—
151C4⋊Dic83—42no—
152C4⋊Dic83—81no—
153C4⋊Dic83C22×C412yes—
154C4⋊Dic83C2×D412yes—
155Dic83⋊C4—28no—
156C22⋊Dic83—28no—
157C22⋊Dic83—44no—
158(C2×D83)⋊C4—24no—
159C4×Dic83—42no—
160C24×C83C2×D412yes—
161C2×D4×C83—22no—
162C2×D4×C83—22no—
163C2×D4×C83C22⋊C414yes—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83C22×C414yes—
169C22×C4×C83C2×D414yes—
170C22×C4×C83C2×D412yes—
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2411yescharacteristic
174M16×C83—48no—
175C2×C8×C83—21no—
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83—22no—
179C2×C8×C83—22no—
180C2×C8×C83—21no—
181C2×C8×C83—22no—
182C2×C8×C83—21no—
183C2×C8×C83C2×D412yes—
184C4⋊C4×C83C22⋊C414yes—
185C22⋊C4×C83C22⋊C418yes—
186C42×C83C22×C412yes—
187C42×C83C2×D412yes—
188C23×C83—22no—
189C23×C83—22no—
190C23×C83—22no—
191C23×C83—22no—
192C23×C83—22no—
193C23×C83—22no—
194C23×C83C2×C22⋊C414yes—
195C23×C83C22×D411yescharacteristic
196C22×D83—22no—
197C22×D83—22no—
198C22×D83—22no—
199C22×D83—41no—
200C22×D83—41no—
201C22×D83—41no—
202C22×D83—41no—
203D4×C83—28no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—24no—
209C2×C4×C83—24no—
210C2×C4×C83—24no—
211C2×C4×C83—24no—
212C2×C4×C83C2×C22⋊C412yes—
213C2×C4×C83C2×C22⋊C414yes—
214C2×C4×C83C2×C22⋊C412yes—
215C2×C4×C83AGL(1,ℤ/8)12yes—
216C2×C4×C83#13712yes—
217C2×C4×C83C23×C411yescharacteristic
218C2×C4×C83C22×D411yescharacteristic
219C2×C4×C83C22×D411yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C2×C4×C83H(5,𝔽2)11yescharacteristic
222C2×C4×C83H(5,𝔽2)11yescharacteristic
223C22⋊D83—416no—
224C2×Dic83—22no—
225C2×Dic83—24no—
226C2×Dic83—22no—
227C2×Dic83—42no—
228C2×Dic83—42no—
229C2×Dic83—41no—
230C2×Dic83—41no—
231C2×Dic83—41no—
232C2×Dic83—41no—
233C2×Dic83—42no—
234C2×Dic83—41no—
235C2×Dic83—41no—
236C2×Dic83—42no—
237C2×Dic83—42no—
238C2×Dic83—41no—
239C2×Dic83—88no—
240D332—22no—
241D332—22no—
242D332—22no—
243D332C4×D414yescomplemented
244C4×D83—416no—
245C4×D83—48no—
246Dic166—24no—
247Dic166—24no—
248Dic166—42no—
249Dic166—42no—
250C8×C83—22no—
251C8×C83—42no—
252C8×C83—41no—
253C22×C83—24no—
254C22×C83—24no—
255C22×C83—24no—
256C22×C83—42no—
257C22×C83—42no—
258C22×C83—42no—
259C22×C83—42no—
260C22×C83C22×C22⋊C411yescharacteristic
261C22×C83D4211yescharacteristic
262C22×C83#54611yescharacteristic
263C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C83C2×#13711yescharacteristic
265C22×C83#57711yescharacteristic
266C22×C83#57811yescharacteristic
267C2×D83—24no—
268C2×D83—42no—
269C2×D83—42no—
270C2×D83—42no—
271C2×D83—42no—
272C2×D83—42no—
273C2×D83—42no—
274C4×C83—24no—
275C4×C83—22no—
276C4×C83#41714yescomplemented
277C4×C83C2×C4×D411yescharacteristic
278C4×C83#51811yescharacteristic
279C4×C83D4211yescharacteristic
280C4×C83#54611yescharacteristic
281Dic83—44no—
282Dic83—42no—
283Dic83—42no—
284Dic83—81no—
285Dic83—82no—
286#22431—831nomaximal, Sylow
287C2×C83—44no—
288C2×C83D4×C22⋊C411yescharacteristic
289C2×C83C2×#41711yescharacteristic
290C2×C83#289811yescharacteristic
291C2×C83#327211yescharacteristic
292C2×C83#327611yescharacteristic
293C2×C83#333011yescharacteristic
294C2×C83#333111yescharacteristic
295D83—44no—
296C2×#547—831no—
297C2×#436D8311yescharacteristic, complemented
298C2×#417—832no—
299D4×C4⋊C4—831no—
300#1981—831no—
301#1956—831no—
302#1894—832no—
303#1872—832no—
304#1802—831no—
305#1788—832no—
306C4×#102—831no—
307C83#2243111yescharacteristic, complemented, Sylow
308C22×#41—832no—
309C22×M16C2×D8312yes—
310#547—838no—
311#547—1664no—
312C2×C4⋊Q8—831no—
313C2×#123—1662no—
314C2×C22⋊Q8—832no—
315C2×#121—1661no—
316C2×C4×D4—831no—
317C2×C4×D4—1661no—
318C2×C4×D4C2×D8311yescharacteristic
319C22×C4⋊C4—832no—
320C22×C4⋊C4—1661no—
321#436—24no—
322C2×C4⋊C8C2×D8311yescharacteristic
323#417—1668no—
324C2×#103—832no—
325C2×#103—832no—
326C2×#103—832no—
327C2×#103—832no—
328C2×#102—831no—
329C2×#102—832no—
330C2×#102—831no—
331C2×C22⋊C8C2×D8312yes—
332C2×C4×C8C2×D8311yescharacteristic
333#389—1661no—
334#385—1662no—
335#384—831no—
336#380—832no—
337C4×C4⋊C4—831no—
338#340—832no—
339#340—831no—
340#340—831no—
341C2×#41—3322no—
342C2×#41—3322no—
343C2×#41—838no—
344C2×#41—1664no—
345C2×#41—1664no—
346C2×#41—1664no—
347C22×Q8—832no—
348C22×D4—831no—
349C22×D4—1662no—
350C22×D4D33211yescharacteristic
351C23×C4—1662no—
352C23×C4—1661no—
353C23×C4—1662no—
354C23×C4C22×D8312yes—
355C2×M16—24no—
356C2×M16—24no—
357C2×M16—42no—
358C22×C8C22×D8312yes—
359C22×C8C22×D8311yescharacteristic
360C22×C8C22×D8311yescharacteristic
361C4⋊Q8—834no—
362C4⋊Q8—1662no—
363#123—16616no—
364C22⋊Q8—838no—
365C22⋊Q8—1664no—
366#121—1668no—
367C4×D4—3324no—
368C4×D4—834no—
369C4×D4—1662no—
370C4×D4C4×D8314yes—
371C4×D4C22⋊D8314yes—
372C2×C4⋊C4—3322no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—831no—
375C2×C4⋊C4—832no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—1662no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1664no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1662no—
383C2×C4⋊C4—1661no—
384C2×C4⋊C4—1668no—
385C2×C4⋊C4D33211yescharacteristic
386C2×C22⋊C4—1662no—
387C2×C22⋊C4—1662no—
388C2×C22⋊C4—1661no—
389C2×C22⋊C4D33212yes—
390C2×C42—1661no—
391C2×C42C22×D8311yescharacteristic
392C4⋊C8—22no—
393#103—3322no—
394#103—3322no—
395#103—1664no—
396#103—1664no—
397#102—834no—
398#102—1664no—
399#102—1662no—
400C22⋊C8—24no—
401C4×C8—22no—
402#95—1661no—
403#95—1662no—
404C24—1661no—
405C24C2×D33212yes—
406#41—3328no—
407#41—3328no—
408#41—3328no—
409#41—16616no—
410C2×Q8—834no—
411C2×Q8—1662no—
412C2×Q8—1662no—
413C2×Q8—1662no—
414C2×Q8—1662no—
415C2×Q8—1662no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—3324no—
419C2×D4—3324no—
420C2×D4—832no—
421C2×D4—832no—
422C2×D4—832no—
423C2×D4—1668no—
424C2×D4—1661no—
425C2×D4—1661no—
426C2×D4—1661no—
427C2×D4(C2×D83)⋊C414yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—3324no—
433C22×C4—3324no—
434C22×C4—3322no—
435C22×C4—3322no—
436C22×C4—3324no—
437C22×C4—3324no—
438C22×C4—6642no—
439C22×C4—1661no—
440C22×C4—1662no—
441C22×C4—1661no—
442C22×C4—1661no—
443C22×C4—1662no—
444C22×C4—1664no—
445C22×C4—1668no—
446C22×C4C2×C4×D8314yes—
447C22×C4C2×C22⋊D8314yes—
448C22×C4C2×D33211yescharacteristic
449C22×C4C2×D33212yes—
450C22×C4C2×D33211yescharacteristic
451C22×C4C23×D8311yescharacteristic
452M16—48no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—22no—
456C2×C8—21no—
457C2×C8—22no—
458C2×C8—21no—
459C2×C8—22no—
460C2×C8—21no—
461C2×C8D4×D8312yes—
462C4⋊C4—3324no—
463C4⋊C4—3322no—
464C4⋊C4—832no—
465C4⋊C4—832no—
466C4⋊C4—6641no—
467C4⋊C4—1662no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1662no—
471C4⋊C4—1661no—
472C4⋊C4—1661no—
473C4⋊C4—1664no—
474C4⋊C4—1668no—
475C4⋊C4—1664no—
476C4⋊C4(C2×D83)⋊C414yes—
477C22⋊C4—3324no—
478C22⋊C4—1668no—
479C22⋊C4—1664no—
480C22⋊C4(C2×D83)⋊C418yes—
481C42—3322no—
482C42C2×C4×D8312yes—
483C42C2×C22⋊D8312yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—3321no—
491C23—3321no—
492C23—3321no—
493C23—3321no—
494C23—1662no—
495C23—1662no—
496C23—1662no—
497C23C2×(C2×D83)⋊C414yes—
498C23C22×D33211yescharacteristic
499Q8—3322no—
500Q8—3322no—
501Q8—1664no—
502Q8—1664no—
503D4—28no—
504D4—33216no—
505D4—834no—
506D4—1662no—
507D4—1662no—
508D4—1662no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—33216no—
518C2×C4—3322no—
519C2×C4—3322no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—3321no—
523C2×C4—3321no—
524C2×C4—3328no—
525C2×C4—3322no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—3322no—
529C2×C4—3322no—
530C2×C4—3321no—
531C2×C4—6648no—
532C2×C4—1662no—
533C2×C4—1662no—
534C2×C4—1664no—
535C2×C4C2×(C2×D83)⋊C412yes—
536C2×C4C2×(C2×D83)⋊C414yes—
537C2×C4C2×(C2×D83)⋊C412yes—
538C2×C4#1458182012yes—
539C2×C4#1458182512yes—
540C2×C4C22×C4×D8311yescharacteristic
541C2×C4C2×D4×D8311yescharacteristic
542C2×C4C22×C22⋊D8311yescharacteristic
543C2×C4#41×D8311yescharacteristic
544C2×C4#1458183711yescharacteristic
545C2×C4#1458183811yescharacteristic
546C8—22no—
547C8—42no—
548C8—41no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—3322no—
557C22—3322no—
558C22—3322no—
559C22—3322no—
560C22—3322no—
561C22—3322no—
562C22—1664no—
563C22C22×(C2×D83)⋊C411yescharacteristic
564C22D4×D33211yescharacteristic
565C22#1912834611yescharacteristic
566C22C2×#1458182011yescharacteristic
567C22C2×#1458182511yescharacteristic
568C22#1912851311yescharacteristic
569C22#1912851811yescharacteristic
570C4—22no—
571C4—24no—
572C4—3322no—
573C4—3322no—
574C4—3324no—
575C4—6642no—
576C4—6641no—
577C4#1912770314yescomplemented
578C4C4×D4×D8311yescharacteristic
579C4#1912817311yescharacteristic
580C4D4×C22⋊D8311yescharacteristic
581C4#1912833911yescharacteristic
582C2—44no—
583C2—3324no—
584C2D4×(C2×D83)⋊C411yescharacteristic
585C2C2×#1912770311yescharacteristic
586C2#2491599711yescharacteristic
587C2#2491906511yescharacteristic
588C2#2491911111yescharacteristic
589C2#2491976911yescharacteristic
590C2#2491978011yescharacteristic
subgroupG/H
centerC23C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#436×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128356C2
cosocleC22×C4×C83C24
hypercenterC2×#436D83
hypocenterC83#22431
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
2. C2×#19127703C2semidirect
3. D4×C4⋊Dic83C2semidirect
4. #24908385C2semidirect
5. #24908265C2semidirect
6. #24907896C2semidirect
7. #24907768C2semidirect
8. #24907464C2semidirect
9. #24907404C2semidirect
10. C4×#14581714C2semidirect
11. C2×#436×C83C2semidirect
14. #19128356C4semidirect
50. C2×#13050590C2×C4semidirect
58. #14581792C2×C4semidirect
61. #14581771C2×C4semidirect
64. C4×D332C2×C4semidirect
89. #14581714D4semidirect
243. D332C4×D4semidirect
276. C4×C83#417semidirect
297. C2×#436D83semidirect
307. C83#22431semidirect
577. C4#19127703semidirect

Maximal subgroups

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

subgroupindex
C2×#191283562
C2×#191277032
D4×C4⋊Dic832
#249083852
#249082652
#249078962
#249077682
#249074642
#249074042
C4×#145817142
C2×#436×C832
#2243183