Finite Group Databasebeta

← #4466154

Subgroups — #4466154

Order 512.

3708 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2679C211yesmaximal, characteristic, complemented
2C2×#2620C211yesmaximal, characteristic, complemented
3#17151C214yesmaximal, complemented
4#16887C212yesmaximal, complemented
5#16537C214yesmaximal, complemented
6#16249C214yesmaximal, complemented
7#15643C212yesmaximal, complemented
8#15453C212yesmaximal, complemented
9#15268C214yesmaximal, complemented
10#15259C212yesmaximal, complemented
11#15257C212yesmaximal, complemented
12#14886C212yesmaximal, complemented
13#14503C211yesmaximal, characteristic, complemented
14C22×#124C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#120C2214yes—
18C22×#120C2212yes—
19C22×#120C2212yescomplemented
20C22×#117C2211yescharacteristic, complemented
21#2679—28no—
22#2620—28no—
23C2×#396C2214yescomplemented
24C2×#394C2212yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2212yes—
28C2×#394C2214yescomplemented
29C2×#393C2214yescomplemented
30C2×#392C2212yes—
31C2×#392C2212yes—
32C2×#392C2214yes—
33C2×#391C2211yescharacteristic, complemented
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2211yescharacteristic
37C2×#388C2214yescomplemented
38C2×#386C2214yescomplemented
39C2×#384C2211yescharacteristic, complemented
40C2×#379C2214yescomplemented
41C2×#379C2212yescomplemented
42#1445C2212yescomplemented
43#1445C2214yescomplemented
44#1442C2214yescomplemented
45#1441C2212yescomplemented
46#1440C2214yescomplemented
47#1440C2214yescomplemented
48#1440C2212yescomplemented
49#1440C2214yescomplemented
50#1440C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2212yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yescomplemented
62#1438C2214yescomplemented
63#1438C2212yescomplemented
64#1437C2214yescomplemented
65#1437C2212yescomplemented
66#1437C2214yescomplemented
67#1436C2212yescomplemented
68#1434C2212yescomplemented
69C26C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2314yes—
74C23×D4C2312yes—
75C23×D4C2314yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C2×#124—24no—
79C2×#124—44no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—42no—
82C2×C22⋊Q8—42no—
83C2×#121—24no—
84C2×#121—42no—
85C2×#121—42no—
86C2×#120—28no—
87C2×#120—216no—
88C2×#120—48no—
89C2×#120—48no—
90C2×#120—48no—
91C2×#120—48no—
92C2×#120—44no—
93C2×#117—24no—
94C2×#117—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124#396—216no—
125#394—28no—
126#394—216no—
127#394—216no—
128#394—44no—
129#394—48no—
130#393—48no—
131#392—44no—
132#392—44no—
133#392—48no—
134#391—24no—
135#390—28no—
136#390—44no—
137#390D418yescomplemented
138#388—48no—
139#386—48no—
140#384—24no—
141#379—44no—
142#379—48no—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C25—24no—
159C25—28no—
160C25—44no—
161C25—44no—
162C25—42no—
163C25—44no—
164C25—44no—
165C25—44no—
166C25—42no—
167C25C2412yes—
168C25C2414yes—
169C25C2411yescharacteristic
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—42no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—24no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—42no—
208C22×D4C2×D414yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—42no—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224#124—416no—
225C22⋊Q8—416no—
226#121—416no—
227#120—464no—
228#120—432no—
229#120—432no—
230#117—416no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2×D414yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
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⋊C4C2×D4116yes—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42—42no—
310C2×C42C2×D414yes—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—42no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—44no—
326C24—21no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—416no—
355C24—48no—
356C24—48no—
357C24—416no—
358C24—48no—
359C24—84no—
360C24—84no—
361C24—88no—
362C24—88no—
363C24—88no—
364C24—88no—
365C24—88no—
366C24—84no—
367C24—88no—
368C24—88no—
369C24—88no—
370C24—84no—
371C24—88no—
372C24—88no—
373C24—88no—
374C24—88no—
375C24C22×D414yes—
376C24H(5,𝔽2)18yes—
377C24H(5,𝔽2)12yes—
378C24C2511yescharacteristic
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—84no—
383C2×Q8—84no—
384C2×D4—216no—
385C2×D4—28no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—416no—
389C2×D4—416no—
390C2×D4—416no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—416no—
394C2×D4—48no—
395C2×D4—416no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—84no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4—88no—
406C2×D4—84no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C2×D4—88no—
412C2×D4—84no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4—88no—
417C2×D4—88no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
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—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)14yes—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—88no—
484C4⋊C4—84no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—48no—
494C22⋊C4—416no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—84no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C42—28no—
506C42—44no—
507C42—84no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—28no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—28no—
521C23—28no—
522C23—24no—
523C23—28no—
524C23—44no—
525C23—41no—
526C23—41no—
527C23—44no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—42no—
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—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—816no—
580C23—832no—
581C23—832no—
582C23—832no—
583C23—816no—
584C23—816no—
585C23—832no—
586C23—832no—
587C23—832no—
588C23C23×D411yescharacteristic
589C23C2×H(5,𝔽2)12yes—
590C23C2×H(5,𝔽2)11yescharacteristic
591C23C2×H(5,𝔽2)14yes—
592C23C2×H(5,𝔽2)12yes—
593C23C2×H(5,𝔽2)11yescharacteristic
594C23#58514yes—
595Q8—88no—
596D4—88no—
597D4—816no—
598D4—816no—
599D4—816no—
600D4—832no—
601D4—832no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—416no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
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#53514yes—
659C22—48no—
660C22—48no—
661C22—42no—
662C22—48no—
663C22—44no—
664C22—42no—
665C22—44no—
666C22—42no—
667C22—44no—
668C22—48no—
669C22—48no—
670C22—48no—
671C22—48no—
672C22—48no—
673C22—48no—
674C22—48no—
675C22—864no—
676C22—82no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—82no—
690C22—84no—
691C22C2×#53511yescharacteristic
692C22#348411yescharacteristic
693C22#348412yes—
694C22#348411yescharacteristic
695C22#350012yes—
696C22#353212yes—
697C22#353414yes—
698C22#353712yes—
699C22#353712yes—
700C22#353712yes—
701C22#354012yes—
702C22#354214yes—
703C22#354214yes—
704C22#354512yes—
705C22#355814yes—
706C4—24no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—42no—
711C4—82no—
712C4—82no—
713C4—84no—
714C4—84no—
715C4—84no—
716C2—82no—
717C2—88no—
718C2—84no—
719C2#3671711yescharacteristic
720C2#4242612yes—
721C2#4635412yes—
722C2#5174911yescharacteristic
723C2#5200411yescharacteristic
724C2#5491914yes—
725C2#5798814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2679C2semidirect
2. C2×#2620C2semidirect
3. #17151C2semidirect
4. #16887C2semidirect
5. #16537C2semidirect
6. #16249C2semidirect
7. #15643C2semidirect
8. #15453C2semidirect
9. #15268C2semidirect
10. #15259C2semidirect
11. #15257C2semidirect
12. #14886C2semidirect
13. #14503C2semidirect
15. C22×C22⋊Q8C22semidirect
19. C22×#120C22semidirect
20. C22×#117C22semidirect
23. C2×#396C22semidirect
28. C2×#394C22semidirect
29. C2×#393C22semidirect
33. C2×#391C22semidirect
37. C2×#388C22semidirect
38. C2×#386C22semidirect
39. C2×#384C22semidirect
40. C2×#379C22semidirect
41. C2×#379C22semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1442C22semidirect
45. #1441C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
61. #1438C22semidirect
62. #1438C22semidirect
63. #1438C22semidirect
64. #1437C22semidirect
65. #1437C22semidirect
66. #1437C22semidirect
67. #1436C22semidirect
68. #1434C22semidirect
137. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26792
C2×#26202
#171512
#168872
#165372
#162492
#156432
#154532
#152682
#152592
#152572
#148862
#145032