Finite Group Databasebeta

← #6132472

Subgroups — #6132472

Order 512.

2404 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic, complemented
2C2×#2850C211yesmaximal, characteristic
3#17048C212yesmaximal, complemented
4#17009C212yesmaximal, complemented
5#16901C212yesmaximal, complemented
6#16821C212yesmaximal, complemented
7#16812C212yesmaximal, complemented
8#16690C214yesmaximal, complemented
9#16681C212yesmaximal, complemented
10#16565C212yesmaximal, complemented
11#16562C214yesmaximal, complemented
12#16519C212yesmaximal, complemented
13#16365C211yesmaximal, characteristic, complemented
14#16362C211yesmaximal, characteristic
15#16254C211yesmaximal, characteristic, complemented
16#16253C211yesmaximal, characteristic
17#14682C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2855—28no—
24#2850—28no—
25C2×#398C2212yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#394C2212yes—
31C2×#393C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#388C2212yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#385C2212yes—
45C2×#382C2211yescharacteristic
46C2×#382C2212yes—
47C2×#381C2212yes—
48C2×#380C2211yescharacteristic
49C2×#380C2212yes—
50#1446C2212yes—
51#1445C2214yes—
52#1445C2214yes—
53#1444C2212yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2212yes—
76#1438C2212yes—
77#1438C2214yes—
78#1438C2214yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2211yescharacteristic
84C22⋊#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#124—24no—
93C2×#124—44no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8—44no—
98C2×#121—24no—
99C2×#121—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C42—44no—
127C22×C42D418yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#398—44no—
133#397—44no—
134#396—48no—
135#395—44no—
136#395—44no—
137#394—44no—
138#393—44no—
139#391—24no—
140#391—24no—
141#391—24no—
142#390—24no—
143#389—24no—
144#389—24no—
145#388—28no—
146#388—28no—
147#387—28no—
148#387—28no—
149#386—28no—
150#386—28no—
151#385—28no—
152#382—24no—
153#382—44no—
154#381—28no—
155#380—24no—
156#380—44no—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×D4—24no—
196C22×D4—28no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—44no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#124—416no—
233C22⋊Q8—416no—
234C22⋊Q8—416no—
235#121—416no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
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—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
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—42no—
299C2×C22⋊C4—42no—
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×C42—24no—
307C2×C42—24no—
308C2×C42—416no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24C22×D412yes—
362C242−1+414yes—
363C24C2511yescharacteristic
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—88no—
371C2×Q8—88no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—44no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
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—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C42—28no—
500C42—28no—
501C42—44no—
502C42—44no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—28no—
510C23—24no—
511C23—28no—
512C23—24no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23#53614yes—
543C23C23×D411yescharacteristic
544C23C2×H(5,𝔽2)12yes—
545C23C2×H(5,𝔽2)12yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×2−1+412yes—
548C23C2×2−1+411yescharacteristic
549C23#58512yes—
550C23#58512yes—
551C23#58512yes—
552Q8—88no—
553Q8—88no—
554D4—88no—
555D4—816no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—416no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4#53414yes—
636C2×C4#53514yes—
637C2×C4#53714yes—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—48no—
645C22—48no—
646C22—42no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—82no—
658C22C2×#53411yescharacteristic
659C22C2×#53511yescharacteristic
660C22C2×#53611yescharacteristic
661C22C2×#53711yescharacteristic
662C22#344911yescharacteristic
663C22#344911yescharacteristic
664C22#344911yescharacteristic
665C22#353414yes—
666C22#354112yes—
667C22#354114yes—
668C22#354212yes—
669C22#354612yes—
670C22#354712yes—
671C22#356114yes—
672C22#356112yes—
673C22#356312yes—
674C22#356614yes—
675C4—24no—
676C4—24no—
677C4—24no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—84no—
687C4—84no—
688C4—82no—
689C4—82no—
690C2—44no—
691C2—162no—
692C2#3744011yescharacteristic
693C2#3744411yescharacteristic
694C2#3744911yescharacteristic
695C2#3753711yescharacteristic
696C2#3755211yescharacteristic
697C2#3755511yescharacteristic
698C2#3762111yescharacteristic
699C2#5323314yes—
700C2#5323514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2855C2semidirect
3. #17048C2semidirect
4. #17009C2semidirect
5. #16901C2semidirect
6. #16821C2semidirect
7. #16812C2semidirect
8. #16690C2semidirect
9. #16681C2semidirect
10. #16565C2semidirect
11. #16562C2semidirect
12. #16519C2semidirect
13. #16365C2semidirect
15. #16254C2semidirect
17. #14682C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#28502
#170482
#170092
#169012
#168212
#168122
#166902
#166812
#165652
#165622
#165192
#163652
#163622
#162542
#162532
#146822