Finite Group Databasebeta

← #4866536

Subgroups — #4866536

Order 512.

3792 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic, complemented
2C2×#2408C211yesmaximal, characteristic
3#17101C214yesmaximal, complemented
4#16986C212yesmaximal, complemented
5#16684C212yesmaximal, complemented
6#16633C212yesmaximal, complemented
7#16514C212yesmaximal, complemented
8#16298C214yesmaximal, complemented
9#16261C212yesmaximal, complemented
10#16259C212yesmaximal, complemented
11#15281C212yesmaximal, complemented
12#15281C212yesmaximal, complemented
13#15084C212yesmaximal, complemented
14#14724C212yesmaximal, complemented
15#14344C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×#125C2211yescharacteristic, complemented
18C22×#121C2212yes—
19C22×#120C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2566—28no—
22#2408—28no—
23C2×#399C2214yescomplemented
24C2×#396C2212yes—
25C2×#396C2212yescomplemented
26C2×#396C2214yescomplemented
27C2×#394C2212yes—
28C2×#394C2214yes—
29C2×#394C2212yes—
30C2×#394C2212yes—
31C2×#394C2214yescomplemented
32C2×#394C2214yescomplemented
33C2×#392C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#390C2212yes—
37C2×#388C2214yescomplemented
38C2×#388C2214yescomplemented
39C2×#385C2212yes—
40C2×#385C2212yescomplemented
41C2×#384C2211yescharacteristic, complemented
42C2×#382C2211yescharacteristic, complemented
43C2×#381C2211yescharacteristic, complemented
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic, complemented
46C2×#379C2212yes—
47C2×#379C2214yes—
48C2×#379C2214yes—
49C2×C4×C4⋊C4C2211yescharacteristic, complemented
50#1444C2212yescomplemented
51#1444C2212yescomplemented
52#1442C2214yescomplemented
53#1441C2212yescomplemented
54#1441C2214yescomplemented
55#1441C2212yescomplemented
56#1441C2212yescomplemented
57#1440C2212yescomplemented
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1439C2214yescomplemented
66#1439C2214yescomplemented
67#1439C2214yescomplemented
68#1439C2214yescomplemented
69#1438C2212yes—
70#1438C2214yes—
71#1438C2212yes—
72#1438C2212yes—
73#1438C2214yescomplemented
74#1438C2214yescomplemented
75#1434C2212yescomplemented
76C22⋊#95C2212yes—
77C26C2312yes—
78C23×D4—24no—
79C23×D4—42no—
80C23×D4—42no—
81C23×D4—48no—
82C23×D4C2312yes—
83C23×D4C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2314yes—
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⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#399—48no—
135#396—28no—
136#396—28no—
137#396—48no—
138#394—28no—
139#394—44no—
140#394—44no—
141#394—48no—
142#394—48no—
143#394—48no—
144#392—28no—
145#390—24no—
146#390—24no—
147#390—28no—
148#388—48no—
149#388—48no—
150#385—28no—
151#385—28no—
152#384—24no—
153#382—24no—
154#381—42no—
155#380—24no—
156#380—42no—
157#379—28no—
158#379—48no—
159#379—48no—
160C4×C4⋊C4—24no—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25—24no—
178C25—28no—
179C25—28no—
180C25—44no—
181C25—42no—
182C25—42no—
183C25—44no—
184C25—44no—
185C25—44no—
186C25C2411yescharacteristic
187C25C2414yes—
188C25C2411yescharacteristic
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—28no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—41no—
200C22×D4—41no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—416no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—432no—
212C22×D4—432no—
213C22×D4—432no—
214C22×D4C2×D414yes—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235#125—416no—
236#121—432no—
237#120—432no—
238C4×D4—416no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D414yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
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—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4C2×D414yes—
328C2×C22⋊C4C2×D414yes—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42C2×D414yes—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—44no—
344#95—42no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352C24—22no—
353C24—24no—
354C24—22no—
355C24—21no—
356C24—21no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—21no—
365C24—21no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—22no—
378C24—24no—
379C24—416no—
380C24—48no—
381C24—416no—
382C24—416no—
383C24—48no—
384C24—416no—
385C24—48no—
386C24—48no—
387C24—48no—
388C24—88no—
389C24—88no—
390C24—88no—
391C24—88no—
392C24—88no—
393C24—88no—
394C24—88no—
395C24—88no—
396C24—88no—
397C24—88no—
398C24—84no—
399C24—84no—
400C24C22×D412yes—
401C24C22×D412yes—
402C24H(5,𝔽2)18yes—
403C242−1+414yes—
404C24C2511yescharacteristic
405C2×D4—216no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—44no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—416no—
412C2×D4—416no—
413C2×D4—416no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—416no—
418C2×D4—4128no—
419C2×D4—48no—
420C2×D4—416no—
421C2×D4—416no—
422C2×D4—48no—
423C2×D4—88no—
424C2×D4—88no—
425C2×D4—88no—
426C2×D4—88no—
427C2×D4—88no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4H(5,𝔽2)14yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4H(5,𝔽2)14yes—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—88no—
511C4⋊C4—84no—
512C4⋊C4—88no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—416no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C22⋊C4—48no—
521C22⋊C4—416no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C22⋊C4—84no—
530C22⋊C4—88no—
531C22⋊C4—88no—
532C22⋊C4—84no—
533C22⋊C4—88no—
534C42—28no—
535C42—44no—
536C42—44no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—28no—
542C23—28no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—28no—
549C23—24no—
550C23—24no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—432no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—42no—
580C23—42no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—42no—
599C23—44no—
600C23—41no—
601C23—44no—
602C23—41no—
603C23—44no—
604C23—832no—
605C23—816no—
606C23—832no—
607C23—832no—
608C23—832no—
609C23—832no—
610C23—816no—
611C23—832no—
612C23#53614yes—
613C23C23×D411yescharacteristic
614C23C2×H(5,𝔽2)11yescharacteristic
615C23C2×H(5,𝔽2)11yescharacteristic
616C23C2×H(5,𝔽2)12yes—
617C23C2×H(5,𝔽2)14yes—
618C23C2×H(5,𝔽2)12yes—
619C23C2×2−1+412yes—
620C23#58512yes—
621D4—416no—
622D4—432no—
623D4—432no—
624D4—832no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—28no—
640C2×C4—416no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—88no—
690C2×C4#53414yes—
691C2×C4#53514yes—
692C22—24no—
693C22—28no—
694C22—44no—
695C22—42no—
696C22—42no—
697C22—44no—
698C22—48no—
699C22—44no—
700C22—48no—
701C22—42no—
702C22—42no—
703C22—48no—
704C22—44no—
705C22—48no—
706C22—42no—
707C22—48no—
708C22—48no—
709C22—48no—
710C22—44no—
711C22—48no—
712C22—864no—
713C22—82no—
714C22—84no—
715C22—84no—
716C22—84no—
717C22—84no—
718C22—82no—
719C22—84no—
720C22—84no—
721C22—84no—
722C22—84no—
723C22—84no—
724C22C2×#53411yescharacteristic
725C22C2×#53511yescharacteristic
726C22C2×#53611yescharacteristic
727C22#348412yes—
728C22#348812yes—
729C22#353212yes—
730C22#353212yes—
731C22#353514yes—
732C22#353612yes—
733C22#353614yes—
734C22#353612yes—
735C22#353714yes—
736C22#354214yes—
737C22#354612yes—
738C22#354911yescharacteristic
739C22#355111yescharacteristic
740C4—24no—
741C4—24no—
742C4—42no—
743C4—42no—
744C4—42no—
745C4—44no—
746C4—84no—
747C4—82no—
748C4—82no—
749C4—84no—
750C4—82no—
751C4—84no—
752C2—44no—
753C2—82no—
754C2—88no—
755C2#3670211yescharacteristic
756C2#3727311yescharacteristic
757C2#3745111yescharacteristic
758C2#4059212yes—
759C2#4248012yes—
760C2#5504312yes—
761C2#5630214yes—
762C2#5826712yes—
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
1. C2×#2566C2semidirect
3. #17101C2semidirect
4. #16986C2semidirect
5. #16684C2semidirect
6. #16633C2semidirect
7. #16514C2semidirect
8. #16298C2semidirect
9. #16261C2semidirect
10. #16259C2semidirect
11. #15281C2semidirect
12. #15281C2semidirect
13. #15084C2semidirect
14. #14724C2semidirect
15. #14344C2semidirect
17. C22×#125C22semidirect
23. C2×#399C22semidirect
25. C2×#396C22semidirect
26. C2×#396C22semidirect
31. C2×#394C22semidirect
32. C2×#394C22semidirect
37. C2×#388C22semidirect
38. C2×#388C22semidirect
40. C2×#385C22semidirect
41. C2×#384C22semidirect
42. C2×#382C22semidirect
43. C2×#381C22semidirect
45. C2×#380C22semidirect
49. C2×C4×C4⋊C4C22semidirect
50. #1444C22semidirect
51. #1444C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1440C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
66. #1439C22semidirect
67. #1439C22semidirect
68. #1439C22semidirect
73. #1438C22semidirect
74. #1438C22semidirect
75. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#24082
#171012
#169862
#166842
#166332
#165142
#162982
#162612
#162592
#152812
#150842
#147242
#143442