Finite Group Databasebeta

← #4867129

Subgroups — #4867129

Order 512.

3974 conjugacy classes of subgroups.

The subgroup lattice has 776 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2564C211yesmaximal, characteristic, complemented
2C2×#2408C211yesmaximal, characteristic
3#17141C212yesmaximal, complemented
4#17084C212yesmaximal, complemented
5#16986C212yesmaximal, complemented
6#16664C212yesmaximal, complemented
7#16519C212yesmaximal, complemented
8#16298C212yesmaximal, complemented
9#16297C212yesmaximal, complemented
10#16256C214yesmaximal, complemented
11#16253C212yesmaximal, complemented
12#15948C214yesmaximal, complemented
13#14926C212yesmaximal, complemented
14#14724C212yesmaximal, complemented
15#14345C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×#123C2214yescomplemented
18C22×C22⋊Q8C2211yescharacteristic, complemented
19C22×#121C2212yes—
20C22×#120C2212yes—
21C22×C4×D4C2211yescharacteristic
22#2564—28no—
23#2408—28no—
24C2×#400C2211yescharacteristic, complemented
25C2×#399C2211yescharacteristic
26C2×#396C2214yescomplemented
27C2×#395C2211yescharacteristic, complemented
28C2×#394C2212yes—
29C2×#394C2212yes—
30C2×#394C2212yes—
31C2×#394C2214yes—
32C2×#393C2211yescharacteristic
33C2×#393C2214yescomplemented
34C2×#392C2212yes—
35C2×#390C2212yes—
36C2×#387C2212yescomplemented
37C2×#386C2212yes—
38C2×#386C2214yescomplemented
39C2×#385C2212yes—
40C2×#383C2211yescharacteristic, complemented
41C2×#382C2211yescharacteristic, complemented
42C2×#380C2211yescharacteristic
43C2×#380C2212yescomplemented
44C2×#379C2212yes—
45C2×#379C2214yes—
46C2×#379C2214yes—
47C22×#95C2211yescharacteristic, complemented
48#1445C2212yescomplemented
49#1444C2212yescomplemented
50#1444C2212yescomplemented
51#1442C2214yescomplemented
52#1441C2212yescomplemented
53#1441C2212yescomplemented
54#1441C2214yescomplemented
55#1440C2214yescomplemented
56#1440C2214yescomplemented
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2212yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1439C2214yescomplemented
66#1439C2214yescomplemented
67#1439C2214yescomplemented
68#1439C2214yescomplemented
69#1438C2212yes—
70#1438C2212yes—
71#1438C2212yes—
72#1438C2214yes—
73#1438C2214yescomplemented
74#1438C2214yescomplemented
75C22⋊#95C2212yes—
76C22⋊#95C2212yescomplemented
77C26C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4—24no—
80C23×D4—42no—
81C23×D4—42no—
82C23×D4—48no—
83C23×D4C2312yes—
84C23×D4C2314yes—
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#123—48no—
91C2×#123—48no—
92C2×#123—48no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—42no—
95C2×C22⋊Q8—42no—
96C2×#121—28no—
97C2×#121—48no—
98C2×#120—28no—
99C2×#120—44no—
100C2×#120—44no—
101C2×C4×D4—42no—
102C2×C4×D4—42no—
103C2×C4×D4D418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#400—24no—
134#399—24no—
135#396—48no—
136#395—24no—
137#394—28no—
138#394—44no—
139#394—44no—
140#394—48no—
141#393—24no—
142#393—48no—
143#392—28no—
144#390—28no—
145#387—28no—
146#386—28no—
147#386—48no—
148#385—28no—
149#383—42no—
150#382—42no—
151#380—24no—
152#380—28no—
153#379—28no—
154#379—48no—
155#379—48no—
156C2×#95—24no—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C25—24no—
179C25—28no—
180C25—24no—
181C25—24no—
182C25—44no—
183C25—42no—
184C25—42no—
185C25—42no—
186C25—42no—
187C25—44no—
188C25—44no—
189C25C2414yes—
190C25C2411yescharacteristic
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—41no—
194C22×Q8—41no—
195C22×Q8—44no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—24no—
199C22×D4—28no—
200C22×D4—28no—
201C22×D4—28no—
202C22×D4—28no—
203C22×D4—28no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—41no—
209C22×D4—41no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—416no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—432no—
219C22×D4—432no—
220C22×D4—432no—
221C22×D4C2×D414yes—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—28no—
228C23×C4—24no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—42no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247#123—464no—
248C22⋊Q8—416no—
249#121—432no—
250#120—432no—
251C4×D4—416no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D414yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—42no—
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—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
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—44no—
312C2×C22⋊C4—44no—
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⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4C2×D414yes—
331C2×C22⋊C4C2×D414yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42C2×D414yes—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—44no—
346#95—42no—
347#95—42no—
348#95—42no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356C24—22no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—24no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—416no—
383C24—48no—
384C24—416no—
385C24—416no—
386C24—48no—
387C24—416no—
388C24—48no—
389C24—48no—
390C24—48no—
391C24—88no—
392C24—88no—
393C24—88no—
394C24—84no—
395C24—84no—
396C24—84no—
397C24—84no—
398C24—88no—
399C24—88no—
400C24—88no—
401C24—88no—
402C24—88no—
403C24—84no—
404C24—84no—
405C24C22×D412yes—
406C24C22×D412yes—
407C24H(5,𝔽2)18yes—
408C24H(5,𝔽2)14yes—
409C242−1+414yes—
410C24C2511yescharacteristic
411C2×Q8—44no—
412C2×Q8—84no—
413C2×Q8—84no—
414C2×Q8—84no—
415C2×D4—216no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—44no—
419C2×D4—416no—
420C2×D4—416no—
421C2×D4—416no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—416no—
428C2×D4—4128no—
429C2×D4—48no—
430C2×D4—416no—
431C2×D4—416no—
432C2×D4—416no—
433C2×D4—416no—
434C2×D4—48no—
435C2×D4—88no—
436C2×D4—88no—
437C2×D4—88no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—28no—
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—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—416no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—84no—
505C22×C4—84no—
506C22×C4—84no—
507C22×C4—88no—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4H(5,𝔽2)14yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4H(5,𝔽2)12yes—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—416no—
521C4⋊C4—416no—
522C4⋊C4—48no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C22⋊C4—416no—
529C22⋊C4—416no—
530C22⋊C4—416no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—416no—
534C22⋊C4—416no—
535C22⋊C4—88no—
536C22⋊C4—88no—
537C22⋊C4—84no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C22⋊C4—88no—
544C42—28no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—28no—
562C23—28no—
563C23—28no—
564C23—28no—
565C23—28no—
566C23—28no—
567C23—28no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—432no—
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—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—832no—
612C23—816no—
613C23—816no—
614C23—816no—
615C23—832no—
616C23—832no—
617C23—832no—
618C23—816no—
619C23—816no—
620C23—816no—
621C23#53514yes—
622C23#53614yes—
623C23C23×D411yescharacteristic
624C23C2×H(5,𝔽2)11yescharacteristic
625C23C2×H(5,𝔽2)11yescharacteristic
626C23C2×H(5,𝔽2)12yes—
627C23C2×H(5,𝔽2)14yes—
628C23C2×H(5,𝔽2)12yes—
629C23C2×2−1+412yes—
630C23#58512yes—
631Q8—88no—
632D4—432no—
633D4—416no—
634D4—432no—
635D4—832no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—28no—
650C2×C4—416no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
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—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—42no—
696C2×C4—44no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—44no—
700C2×C4—42no—
701C2×C4—88no—
702C2×C4—816no—
703C2×C4#53414yes—
704C22—24no—
705C22—24no—
706C22—28no—
707C22—28no—
708C22—42no—
709C22—42no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—42no—
717C22—42no—
718C22—48no—
719C22—48no—
720C22—48no—
721C22—48no—
722C22—48no—
723C22—48no—
724C22—48no—
725C22—48no—
726C22—48no—
727C22—864no—
728C22—84no—
729C22—84no—
730C22—84no—
731C22—84no—
732C22—84no—
733C22—84no—
734C22—84no—
735C22—84no—
736C22C2×#53411yescharacteristic
737C22C2×#53511yescharacteristic
738C22C2×#53611yescharacteristic
739C22#348412yes—
740C22#348812yes—
741C22#353212yes—
742C22#353212yes—
743C22#353214yes—
744C22#353414yes—
745C22#353514yes—
746C22#353612yes—
747C22#353612yes—
748C22#353614yes—
749C22#354612yes—
750C22#354911yescharacteristic
751C22#355111yescharacteristic
752C4—24no—
753C4—42no—
754C4—42no—
755C4—42no—
756C4—44no—
757C4—42no—
758C4—82no—
759C4—84no—
760C4—82no—
761C4—84no—
762C4—82no—
763C4—84no—
764C2—44no—
765C2—44no—
766C2—88no—
767C2#3671411yescharacteristic
768C2#3727311yescharacteristic
769C2#3745111yescharacteristic
770C2#4059012yes—
771C2#4059212yes—
772C2#5418314yes—
773C2#5796412yes—
774C2#5825912yes—
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×#2564C2semidirect
3. #17141C2semidirect
4. #17084C2semidirect
5. #16986C2semidirect
6. #16664C2semidirect
7. #16519C2semidirect
8. #16298C2semidirect
9. #16297C2semidirect
10. #16256C2semidirect
11. #16253C2semidirect
12. #15948C2semidirect
13. #14926C2semidirect
14. #14724C2semidirect
15. #14345C2semidirect
17. C22×#123C22semidirect
18. C22×C22⋊Q8C22semidirect
24. C2×#400C22semidirect
26. C2×#396C22semidirect
27. C2×#395C22semidirect
33. C2×#393C22semidirect
36. C2×#387C22semidirect
38. C2×#386C22semidirect
40. C2×#383C22semidirect
41. C2×#382C22semidirect
43. C2×#380C22semidirect
47. C22×#95C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1440C22semidirect
56. #1440C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
66. #1439C22semidirect
67. #1439C22semidirect
68. #1439C22semidirect
73. #1438C22semidirect
74. #1438C22semidirect
76. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25642
C2×#24082
#171412
#170842
#169862
#166642
#165192
#162982
#162972
#162562
#162532
#159482
#149262
#147242
#143452