Finite Group Databasebeta

← #4861911

Subgroups — #4861911

Order 512.

3672 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2408C211yesmaximal, characteristic
2C2×#2633C211yesmaximal, characteristic, complemented
3#17147C212yesmaximal, complemented
4#17088C212yesmaximal, complemented
5#16815C212yesmaximal, complemented
6#16693C212yesmaximal, complemented
7#16633C212yesmaximal, complemented
8#16541C214yesmaximal, complemented
9#16259C212yesmaximal, complemented
10#16255C214yesmaximal, complemented
11#15480C212yesmaximal, complemented
12#15284C212yesmaximal, complemented
13#14764C212yesmaximal, complemented
14#14724C212yesmaximal, complemented
15#13940C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×#125C2211yescharacteristic, complemented
18C22×#121C2212yes—
19C22×#120C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2633—28no—
22#2408—28no—
23C2×#399C2211yescharacteristic
24C2×#399C2211yescharacteristic, complemented
25C2×#398C2212yescomplemented
26C2×#397C2211yescharacteristic
27C2×#397C2214yes—
28C2×#397C2211yescharacteristic, complemented
29C2×#394C2212yes—
30C2×#394C2212yes—
31C2×#394C2212yes—
32C2×#394C2214yes—
33C2×#394C2212yes—
34C2×#394C2214yescomplemented
35C2×#392C2212yes—
36C2×#390C2212yes—
37C2×#388C2214yescomplemented
38C2×#385C2212yes—
39C2×#385C2212yescomplemented
40C2×#383C2211yescharacteristic, complemented
41C2×#381C2211yescharacteristic, complemented
42C2×#380C2211yescharacteristic
43C2×#379C2212yes—
44C2×#379C2214yes—
45C2×#379C2214yes—
46C2×C4×C22⋊C4C2211yescharacteristic, complemented
47#1444C2212yescomplemented
48#1443C2214yescomplemented
49#1442C2212yescomplemented
50#1441C2212yescomplemented
51#1441C2214yescomplemented
52#1441C2212yescomplemented
53#1440C2214yes—
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1440C2212yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1438C2212yes—
66#1438C2212yes—
67#1438C2212yes—
68#1438C2214yes—
69#1438C2214yescomplemented
70#1438C2214yescomplemented
71#1438C2214yescomplemented
72#1437C2212yes—
73#1437C2214yescomplemented
74#1436C2212yescomplemented
75#1434C2212yescomplemented
76C22⋊#95C2212yes—
77C26C2312yes—
78C23×D4—24no—
79C23×D4—42no—
80C23×D4—42no—
81C23×D4—48no—
82C23×D4C2312yes—
83C23×D4C2312yes—
84C23×D4C2314yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#125—24no—
89C2×#125—42no—
90C2×#125—42no—
91C2×#121—28no—
92C2×#121—48no—
93C2×#120—28no—
94C2×#120—48no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#399—24no—
131#399—24no—
132#398—28no—
133#397—24no—
134#397—24no—
135#397—48no—
136#394—28no—
137#394—28no—
138#394—44no—
139#394—44no—
140#394—48no—
141#394—48no—
142#392—28no—
143#390—28no—
144#388—48no—
145#385—28no—
146#385—28no—
147#383—24no—
148#381—42no—
149#380—24no—
150#379—28no—
151#379—48no—
152#379—48no—
153C4×C22⋊C4—42no—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C25—24no—
175C25—28no—
176C25—28no—
177C25—44no—
178C25—42no—
179C25—42no—
180C25—44no—
181C25—44no—
182C25—44no—
183C25C2411yescharacteristic
184C25C2414yes—
185C25C2411yescharacteristic
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—28no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—416no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—432no—
209C22×D4—432no—
210C22×D4—432no—
211C22×D4C2×D414yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#125—416no—
233#121—432no—
234#120—432no—
235C4×D4—416no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D414yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4C2×D414yes—
317C2×C22⋊C4C2×D414yes—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—42no—
321C2×C42—44no—
322C2×C42C2×D414yes—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343C24—24no—
344C24—21no—
345C24—21no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—21no—
350C24—21no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—416no—
369C24—48no—
370C24—416no—
371C24—416no—
372C24—48no—
373C24—416no—
374C24—48no—
375C24—48no—
376C24—48no—
377C24—88no—
378C24—88no—
379C24—88no—
380C24—88no—
381C24—88no—
382C24—88no—
383C24—88no—
384C24—88no—
385C24—88no—
386C24—84no—
387C24—84no—
388C24—88no—
389C24C22×D412yes—
390C24C22×D412yes—
391C24H(5,𝔽2)18yes—
392C24H(5,𝔽2)14yes—
393C24C2511yescharacteristic
394C2×D4—216no—
395C2×D4—28no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—416no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—416no—
407C2×D4—4128no—
408C2×D4—48no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—48no—
412C2×D4—88no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4—88no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—84no—
481C22×C4—84no—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—88no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—48no—
503C22⋊C4—416no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—88no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C22⋊C4—84no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C42—28no—
520C42—84no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—432no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—42no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—832no—
587C23—816no—
588C23—832no—
589C23—832no—
590C23—832no—
591C23—832no—
592C23—816no—
593C23—832no—
594C23#53514yes—
595C23C23×D411yescharacteristic
596C23C2×H(5,𝔽2)11yescharacteristic
597C23C2×H(5,𝔽2)11yescharacteristic
598C23C2×H(5,𝔽2)12yes—
599C23C2×H(5,𝔽2)14yes—
600C23#58512yes—
601C23#58514yes—
602D4—432no—
603D4—416no—
604D4—432no—
605D4—832no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—416no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—88no—
671C2×C4#53414yes—
672C22—28no—
673C22—24no—
674C22—42no—
675C22—44no—
676C22—42no—
677C22—48no—
678C22—48no—
679C22—42no—
680C22—44no—
681C22—48no—
682C22—44no—
683C22—48no—
684C22—42no—
685C22—42no—
686C22—48no—
687C22—48no—
688C22—48no—
689C22—48no—
690C22—48no—
691C22—864no—
692C22—84no—
693C22—84no—
694C22—84no—
695C22—84no—
696C22—84no—
697C22—84no—
698C22—84no—
699C22—84no—
700C22—84no—
701C22—84no—
702C22C2×#53411yescharacteristic
703C22C2×#53511yescharacteristic
704C22#344911yescharacteristic
705C22#348412yes—
706C22#350012yes—
707C22#353212yes—
708C22#353212yes—
709C22#353314yes—
710C22#353414yes—
711C22#353414yes—
712C22#354012yes—
713C22#354212yes—
714C22#354212yes—
715C22#355112yes—
716C22#356014yes—
717C4—24no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—82no—
723C4—84no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—84no—
728C4—84no—
729C2—44no—
730C2—82no—
731C2—88no—
732C2#3670211yescharacteristic
733C2#3741911yescharacteristic
734C2#3743711yescharacteristic
735C2#4049412yes—
736C2#4050212yes—
737C2#5580614yes—
738C2#5747414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2408C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2633C2semidirect
3. #17147C2semidirect
4. #17088C2semidirect
5. #16815C2semidirect
6. #16693C2semidirect
7. #16633C2semidirect
8. #16541C2semidirect
9. #16259C2semidirect
10. #16255C2semidirect
11. #15480C2semidirect
12. #15284C2semidirect
13. #14764C2semidirect
14. #14724C2semidirect
15. #13940C2semidirect
17. C22×#125C22semidirect
24. C2×#399C22semidirect
25. C2×#398C22semidirect
28. C2×#397C22semidirect
34. C2×#394C22semidirect
37. C2×#388C22semidirect
39. C2×#385C22semidirect
40. C2×#383C22semidirect
41. C2×#381C22semidirect
46. C2×C4×C22⋊C4C22semidirect
47. #1444C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1440C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
69. #1438C22semidirect
70. #1438C22semidirect
71. #1438C22semidirect
73. #1437C22semidirect
74. #1436C22semidirect
75. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24082
C2×#26332
#171472
#170882
#168152
#166932
#166332
#165412
#162592
#162552
#154802
#152842
#147642
#147242
#139402