Finite Group Databasebeta

← #35420050

Subgroups — #35420050

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×#19127779C211yesmaximal, characteristic, complemented
3C2×#19127700C212yesmaximal, complemented
4#24908386C211yesmaximal, characteristic, complemented
5#24908264C211yesmaximal, characteristic, complemented
6#24907894C212yesmaximal, complemented
7#24907770C212yesmaximal, complemented
8#24907460C211yesmaximal, characteristic, complemented
9#24907401C212yesmaximal, complemented
10C4×#14581712C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C83C211yesmaximal, characteristic, complemented
12C22×#13050590C2212yes—
13C22×#13050574C2212yes—
14#19128356—24no—
15#19128356C418yescomplemented
16C2×#14581792C2211yescharacteristic
17C2×#14581774—22no—
18C2×#14581771C2212yes—
19C2×#14581763—21no—
20C2×C4×D332C2211yescharacteristic
21C2×C4⋊C4×D83—21no—
22#19127779—24no—
23C2×C4⋊(C83⋊C8)C2211yescharacteristic
24#19127700—28no—
25C2×#14581715C2212yes—
26C2×#14581715C2212yes—
27C2×#14581715C2212yes—
28C2×#14581715C2212yes—
29C2×#14581712C2212yes—
30C2×#14581712C2211yescharacteristic
31C2×#14581712C2211yescharacteristic
32C2×C22⋊(C83⋊C8)C2212yes—
33C2×C4×C83⋊C8C2211yescharacteristic
34#19127463C2211yescharacteristic
35#19127463C2211yescharacteristic
36#19127463C2212yes—
37C2×C4×D4×C83—21no—
38C2×C4×D4×C83C2211yescharacteristic
39C22×C4⋊C4×C83C2212yes—
40#389×C83—21no—
41#385×C83—22no—
42#384×C83C2211yescharacteristic
43#380×C83C2212yes—
44C4×C4⋊C4×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—
54C22×C4×D83—21no—
55C22×C4×D83—22no—
56C2×#13050574—24no—
57C2×#13050574—24no—
58C2×#13050574—42no—
59C22×C83⋊C8C2311yescharacteristic
60C22×C83⋊C8C2312yes—
61C22×C83⋊C8C2311yescharacteristic
62#14581792—22no—
63#14581792C2×C414yescomplemented
64#14581774—216no—
65#14581771—24no—
66#14581771C2×C418yescomplemented
67#14581763—28no—
68C4×D332—22no—
69C4×D332C2×C414yescomplemented
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83C2311yescharacteristic
73C2×Dic83⋊C4—22no—
74C4⋊C4×D83—28no—
75C2×C22⋊Dic83—22no—
76C2×(C2×D83)⋊C4—21no—
77C4⋊(C83⋊C8)—22no—
78#14581715—24no—
79#14581715—24no—
80#14581715—42no—
81#14581715—42no—
82#14581712—24no—
83#14581712—22no—
84#14581712D414yescomplemented
85C22⋊(C83⋊C8)—24no—
86C4×C83⋊C8—22no—
87C22×D4×C83D411yescharacteristic
88C23×C4×C83—22no—
89C23×C4×C83C2312yes—
90C4×D4×C83—44no—
91C4×D4×C83C2×C414yes—
92C4×D4×C83D414yes—
93C2×C4⋊C4×C83—24no—
94C2×C4⋊C4×C83—22no—
95C2×C4⋊C4×C83—21no—
96C2×C4⋊C4×C83—21no—
97C2×C4⋊C4×C83—42no—
98C2×C4⋊C4×C83D411yescharacteristic
99C2×C4⋊C4×C83C2312yes—
100C2×C4⋊C4×C83C2311yescharacteristic
101C2×C4⋊C4×C83C2311yescharacteristic
102C2×C22⋊C4×C83—22no—
103C2×C22⋊C4×C83D412yes—
104C2×C42×C83—21no—
105C2×C42×C83C2311yescharacteristic
106#95×C83—21no—
107#95×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—
130C2×C4×D83—24no—
131C2×C4×D83—28no—
132C2×C4×D83—44no—
133C2×C4×D83—44no—
134C2×C4×D83—42no—
135C2×C4×D83—42no—
136#13050574—48no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—22no—
140C2×C83⋊C8—22no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—22no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8C2×D412yes—
146C4⋊Dic83—24no—
147C4⋊Dic83—24no—
148C4⋊Dic83—21no—
149C4⋊Dic83C22×C412yes—
150Dic83⋊C4—28no—
151C22⋊Dic83—28no—
152(C2×D83)⋊C4—24no—
153C24×C83C2×D412yes—
154C2×D4×C83—22no—
155C2×D4×C83—22no—
156C2×D4×C83C22⋊C414yes—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—22no—
164C22×C4×C83—44no—
165C22×C4×C83—44no—
166C22×C4×C83—82no—
167C22×C4×C83C22×C414yes—
168C22×C4×C83C2×D412yes—
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D414yes—
172C22×C4×C83C2411yescharacteristic
173C4⋊C4×C83—21no—
174C4⋊C4×C83—22no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—22no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—44no—
179C4⋊C4×C83—42no—
180C4⋊C4×C83—81no—
181C4⋊C4×C83C22⋊C414yes—
182C4⋊C4×C83C2×D412yes—
183C22⋊C4×C83—44no—
184C22⋊C4×C83C22⋊C418yes—
185C42×C83—42no—
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×C83—22no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—42no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83—42no—
219C2×C4×C83—42no—
220C2×C4×C83—41no—
221C2×C4×C83—88no—
222C2×C4×C83C2×C22⋊C412yes—
223C2×C4×C83C2×C22⋊C414yes—
224C2×C4×C83C2×C22⋊C412yes—
225C2×C4×C83AGL(1,ℤ/8)12yes—
226C2×C4×C83#13712yes—
227C2×C4×C83C23×C411yescharacteristic
228C2×C4×C83C22×D411yescharacteristic
229C2×C4×C83C22×D411yescharacteristic
230C2×C4×C83C2×#4111yescharacteristic
231C2×C4×C83H(5,𝔽2)11yescharacteristic
232C2×C4×C83H(5,𝔽2)11yescharacteristic
233C22⋊D83—416no—
234C2×Dic83—22no—
235C2×Dic83—24no—
236C2×Dic83—42no—
237C2×Dic83—42no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240D332—22no—
241D332—22no—
242D332—22no—
243D332C4×D414yescomplemented
244C4×D83—416no—
245C4×D83—48no—
246Dic166—24no—
247Dic166—24no—
248Dic166—42no—
249Dic166—42no—
250C83⋊C8—22no—
251C83⋊C8—42no—
252C83⋊C8—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—42no—
277C4×C83—81no—
278C4×C83—82no—
279C4×C83#41714yescomplemented
280C4×C83C2×C4×D411yescharacteristic
281C4×C83#51811yescharacteristic
282C4×C83D4211yescharacteristic
283C4×C83#54611yescharacteristic
284Dic83—44no—
285Dic83—42no—
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×#436—831no—
298C2×#417—832no—
299D4×C4⋊C4D8311yescharacteristic, complemented
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×M16—832no—
310#547—838no—
311#547—1664no—
312C2×C4⋊Q8—831no—
313C2×#123—1662no—
314C2×C22⋊Q8—832no—
315C2×#121—1661no—
316C2×C4×D4—21no—
317C2×C4×D4—831no—
318C2×C4×D4C2×D8311yescharacteristic
319C22×C4⋊C4—1661no—
320C22×C4⋊C4C2×D8312yes—
321#436—1664no—
322C2×C4⋊C8—831no—
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⋊C8—832no—
332C2×C4×C8—831no—
333#389—21no—
334#385—22no—
335#384C2×D8311yescharacteristic
336#380C2×D8312yes—
337C4×C4⋊C4C2×D8311yescharacteristic
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×D4C22⋊D8311yescharacteristic
351C23×C4—22no—
352C23×C4—1661no—
353C23×C4—1662no—
354C23×C4C22×D8312yes—
355C2×M16—3322no—
356C2×M16—1664no—
357C2×M16—1664no—
358C22×C8—831no—
359C22×C8—832no—
360C22×C8—831no—
361C4⋊Q8—834no—
362C4⋊Q8—1662no—
363#123—16616no—
364C22⋊Q8—838no—
365C22⋊Q8—1664no—
366#121—1668no—
367C4×D4—44no—
368C4×D4—834no—
369C4×D4—1662no—
370C4×D4C4×D8314yes—
371C4×D4D33214yes—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—42no—
377C2×C4⋊C4—831no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1662no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1668no—
382C2×C4⋊C4C22⋊D8311yescharacteristic
383C2×C4⋊C4C22×D8312yes—
384C2×C4⋊C4C22×D8311yescharacteristic
385C2×C4⋊C4C22×D8311yescharacteristic
386C2×C22⋊C4—22no—
387C2×C22⋊C4—1662no—
388C2×C22⋊C4—1661no—
389C2×C22⋊C4C22⋊D8312yes—
390C2×C42—21no—
391C2×C42C22×D8311yescharacteristic
392C4⋊C8—1662no—
393#103—3322no—
394#103—3322no—
395#103—1664no—
396#103—1664no—
397#102—834no—
398#102—1664no—
399#102—1662no—
400C22⋊C8—1664no—
401C4×C8—1662no—
402#95—21no—
403#95—22no—
404C24—1661no—
405C24C2×C22⋊D8312yes—
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—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—3324no—
439C22×C4—3324no—
440C22×C4—3322no—
441C22×C4—3322no—
442C22×C4—1661no—
443C22×C4—1662no—
444C22×C4—1664no—
445C22×C4—1668no—
446C22×C4C2×C4×D8314yes—
447C22×C4C2×C22⋊D8311yescharacteristic
448C22×C4C2×C22⋊D8312yes—
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C2×D33214yes—
451C22×C4C23×D8311yescharacteristic
452M16—3328no—
453C2×C8—832no—
454C2×C8—1662no—
455C2×C8—1661no—
456C2×C8—1662no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8—1662no—
460C2×C8—1661no—
461C2×C8—1661no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—832no—
471C4⋊C4—1661no—
472C4⋊C4—1664no—
473C4⋊C4—1668no—
474C4⋊C4—1664no—
475C4⋊C4(C2×D83)⋊C414yes—
476C4⋊C4D4×D8312yes—
477C22⋊C4—44no—
478C22⋊C4—1668no—
479C22⋊C4—1664no—
480C22⋊C4(C2×D83)⋊C418yes—
481C42—42no—
482C42C2×C4×D8312yes—
483C42C2×D33212yes—
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×C22⋊D8311yescharacteristic
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—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—88no—
527C2×C4—33216no—
528C2×C4—3322no—
529C2×C4—3322no—
530C2×C4—3321no—
531C2×C4—3321no—
532C2×C4—3328no—
533C2×C4—1662no—
534C2×C4—1664no—
535C2×C4C2×(C2×D83)⋊C412yes—
536C2×C4C2×(C2×D83)⋊C414yes—
537C2×C4C2×(C2×D83)⋊C412yes—
538C2×C4#1458181812yes—
539C2×C4#1458182312yes—
540C2×C4C22×C4×D8311yescharacteristic
541C2×C4C2×D4×D8311yescharacteristic
542C2×C4C22×D33211yescharacteristic
543C2×C4#41×D8311yescharacteristic
544C2×C4#1458183711yescharacteristic
545C2×C4#1458183811yescharacteristic
546C8—3322no—
547C8—3321no—
548C8—1662no—
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×C22⋊D8311yescharacteristic
565C22#1912833911yescharacteristic
566C22C2×#1458181811yescharacteristic
567C22C2×#1458182311yescharacteristic
568C22#1912851211yescharacteristic
569C22#1912851611yescharacteristic
570C4—22no—
571C4—24no—
572C4—42no—
573C4—82no—
574C4—81no—
575C4—3322no—
576C4—3324no—
577C4#1912770014yescomplemented
578C4C4×D4×D8311yescharacteristic
579C4#1912817311yescharacteristic
580C4D4×D33211yescharacteristic
581C4#1912834611yescharacteristic
582C2—44no—
583C2—3324no—
584C2D4×(C2×D83)⋊C411yescharacteristic
585C2C2×#1912770011yescharacteristic
586C2#2491599311yescharacteristic
587C2#2491905911yescharacteristic
588C2#2491910911yescharacteristic
589C2#2491976211yescharacteristic
590C2#2491977311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingD4×C4⋊C4×C83C2
socleC23×C83C22×D4
prime radicalC2×#19128356C2
cosocleC22×C4×C83C24
hypercenterD4×C4⋊C4D83
hypocenterC83#22431
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#19127779C2semidirect
3. C2×#19127700C2semidirect
4. #24908386C2semidirect
5. #24908264C2semidirect
6. #24907894C2semidirect
7. #24907770C2semidirect
8. #24907460C2semidirect
9. #24907401C2semidirect
10. C4×#14581712C2semidirect
11. D4×C4⋊C4×C83C2semidirect
15. #19128356C4semidirect
50. C2×#13050590C2×C4semidirect
63. #14581792C2×C4semidirect
66. #14581771C2×C4semidirect
69. C4×D332C2×C4semidirect
84. #14581712D4semidirect
243. D332C4×D4semidirect
279. C4×C83#417semidirect
299. D4×C4⋊C4D83semidirect
307. C83#22431semidirect
577. C4#19127700semidirect

Maximal subgroups

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

subgroupindex
C2×#191283562
C2×#191277792
C2×#191277002
#249083862
#249082642
#249078942
#249077702
#249074602
#249074012
C4×#145817122
D4×C4⋊C4×C832
#2243183