Finite Group Databasebeta

← #636792

Subgroups — #636792

Order 512.

4962 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#126C211yesmaximal, characteristic, complemented
2C2×#2406C211yesmaximal, characteristic, complemented
3C2×#2404C211yesmaximal, characteristic
4C2×#2297C211yesmaximal, characteristic, complemented
5#16656C214yesmaximal, complemented
6#16655C214yesmaximal, complemented
7#15091C212yesmaximal, complemented
8#14917C218yesmaximal, complemented
9#14911C212yesmaximal, complemented
10#14684C211yesmaximal, characteristic, complemented
11#14256C212yesmaximal, complemented
12#14255C212yesmaximal, complemented
13#13870C211yesmaximal, characteristic, complemented
14#13868C211yesmaximal, characteristic, complemented
15C23×#41C2211yescharacteristic
16C22×#126—28no—
17C22×#126—28no—
18C22×#126—28no—
19C22×#126C2214yes—
20C22×#126C2214yes—
21C22×#123C2212yes—
22C22×#120C2212yes—
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29C23×C22⋊C4C2211yescharacteristic
30C23×C42C2211yescharacteristic
31#2406—28no—
32#2404—28no—
33C2×#395C2212yescomplemented
34C2×#394C2212yescomplemented
35C2×#393C2212yes—
36C2×#392C2212yes—
37#2297—28no—
38C2×#391C2212yescomplemented
39C2×#390C2212yescomplemented
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#385C2214yescomplemented
46C2×#381C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#381C2212yes—
49C2×#379C2212yes—
50C2×C4×C22⋊C4C2214yescomplemented
51#1442C2218yescomplemented
52#1441C2214yescomplemented
53#1441C2218yescomplemented
54#1440C2218yes—
55#1439C2214yes—
56#1439C2218yes—
57#1439C2218yescomplemented
58#1439C2218yescomplemented
59#1438C2218yes—
60#1438C2218yes—
61#1437C2214yes—
62#1436C2214yescomplemented
63#1434C2214yescomplemented
64#1434C2214yescomplemented
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C22×#41—28no—
71C22×#41—24no—
72C22×#41—24no—
73C22×#41—24no—
74C22×#41—44no—
75C23×Q8C2311yescharacteristic
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C23×D4C2314yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#126—2128no—
87C2×#126—216no—
88C2×#126—216no—
89C2×#126—216no—
90C2×#126—216no—
91C2×#126—216no—
92C2×#126—216no—
93C2×#123—28no—
94C2×#123—28no—
95C2×#123—44no—
96C2×#120—28no—
97C2×#120—28no—
98C2×#120—44no—
99C2×#117—24no—
100C2×#117—44no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4D418yescomplemented
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4—24no—
120C22×C22⋊C4—24no—
121C22×C22⋊C4—24no—
122C22×C22⋊C4—24no—
123C22×C22⋊C4—24no—
124C22×C22⋊C4D418yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2318yes—
139C22×C42—24no—
140C22×C42—24no—
141C22×C42D418yescomplemented
142C22×C42C2312yes—
143C22×C42C2312yes—
144C22×C42C2312yes—
145#395—28no—
146#394—28no—
147#393—28no—
148#392—28no—
149#391—28no—
150#390—28no—
151#386—28no—
152#386—28no—
153#386—28no—
154#386—28no—
155#386—216no—
156#385—216no—
157#381—24no—
158#381—24no—
159#381—44no—
160#379—44no—
161C4×C22⋊C4—216no—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C25—22no—
176C25—22no—
177C25—24no—
178C25—24no—
179C25—44no—
180C25—44no—
181C25C2×D414yes—
182C25C2412yes—
183C25C2412yes—
184C25C2411yescharacteristic
185C2×#41—232no—
186C2×#41—432no—
187C2×#41—432no—
188C2×#41—416no—
189C2×#41—416no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—42no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—28no—
203C22×D4—28no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—42no—
208C22×D4—44no—
209C22×D4—48no—
210C22×D4—48no—
211C22×D4#4118yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2414yes—
246C23×C4C2414yes—
247C23×C4C2414yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256#126—2128no—
257#126—2128no—
258#123—432no—
259#120—432no—
260#117—416no—
261C2×C4⋊C4—232no—
262C2×C4⋊C4—232no—
263C2×C4⋊C4—232no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—232no—
282C2×C22⋊C4—232no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—232no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4—48no—
313C2×C22⋊C4—48no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4C2×D418yes—
321C2×C22⋊C4C2×D418yes—
322C2×C42—232no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42C2×D418yes—
330C2×C42C2×D418yes—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—48no—
340#95—48no—
341#95—48no—
342#95—48no—
343#95—48no—
344C24—24no—
345C24—24no—
346C24—28no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—28no—
358C24—28no—
359C24—24no—
360C24—28no—
361C24—24no—
362C24—216no—
363C24—48no—
364C24—48no—
365C24—48no—
366C24—48no—
367C24—48no—
368C24—48no—
369C24—44no—
370C24—44no—
371C24—48no—
372C24—816no—
373C24—816no—
374C24C22×D414yes—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C2×#4114yes—
379C24C2×#4114yes—
380C24H(5,𝔽2)14yes—
381C24C2511yescharacteristic
382#41—4128no—
383C2×Q8—28no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×D4—28no—
389C2×D4—216no—
390C2×D4—216no—
391C2×D4—216no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—44no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—416no—
400C2×D4—416no—
401C2×D4—44no—
402C2×D4—88no—
403C2×D4—816no—
404C2×D4—816no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—216no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—216no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—28no—
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—28no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4#120116yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C2×#4114yes—
494C4⋊C4—232no—
495C4⋊C4—232no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C22⋊C4—232no—
501C22⋊C4—232no—
502C22⋊C4—216no—
503C22⋊C4—216no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—88no—
513C22⋊C4—816no—
514C42—216no—
515C42—216no—
516C42—48no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—216no—
524C23—216no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—24no—
532C23—28no—
533C23—216no—
534C23—416no—
535C23—416no—
536C23—416no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—48no—
561C23—832no—
562C23—816no—
563C23—832no—
564C23—816no—
565C23C2×#12014yes—
566C23#52514yes—
567C23#53514yes—
568C23#54618yes—
569C23#54618yes—
570C23C23×D411yescharacteristic
571C23C23×D411yescharacteristic
572C23C23×D411yescharacteristic
573C23C22×#4112yes—
574C23C2×H(5,𝔽2)12yes—
575C23#58514yes—
576C23#58514yes—
577Q8—416no—
578D4—416no—
579D4—432no—
580D4—832no—
581C2×C4—232no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—432no—
607C2×C4—416no—
608C2×C4—416no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
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—44no—
636C2×C4—48no—
637C2×C4—48no—
638C2×C4—48no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—48no—
642C2×C4C2×#12014yes—
643C2×C4C2×#12014yes—
644C22—28no—
645C22—24no—
646C22—24no—
647C22—216no—
648C22—216no—
649C22—28no—
650C22—48no—
651C22—44no—
652C22—44no—
653C22—48no—
654C22—48no—
655C22—48no—
656C22—48no—
657C22—44no—
658C22—44no—
659C22—832no—
660C22—84no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—84no—
665C22—88no—
666C22—84no—
667C22—84no—
668C22#240418yes—
669C22C22×#12011yescharacteristic
670C22C2×#52511yescharacteristic
671C22C2×#53511yescharacteristic
672C22#344912yes—
673C22#344912yes—
674C22#344912yes—
675C22#344912yes—
676C22C2×#54612yes—
677C22C2×#54612yes—
678C22#348512yes—
679C22#348712yes—
680C22#352118yes—
681C22#356018yes—
682C4—24no—
683C4—24no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—44no—
690C4—84no—
691C4—84no—
692C4—84no—
693C2—28no—
694C2—84no—
695C2—84no—
696C2#3463211yescharacteristic
697C2C2×#240411yescharacteristic
698C2#3514312yes—
699C2#3514911yescharacteristic
700C2#3754712yes—
701C2#3803014yes—
702C2#3803114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2404C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#126C2semidirect
2. C2×#2406C2semidirect
4. C2×#2297C2semidirect
5. #16656C2semidirect
6. #16655C2semidirect
7. #15091C2semidirect
8. #14917C2semidirect
9. #14911C2semidirect
10. #14684C2semidirect
11. #14256C2semidirect
12. #14255C2semidirect
13. #13870C2semidirect
14. #13868C2semidirect
33. C2×#395C22semidirect
34. C2×#394C22semidirect
38. C2×#391C22semidirect
39. C2×#390C22semidirect
45. C2×#385C22semidirect
50. C2×C4×C22⋊C4C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
62. #1436C22semidirect
63. #1434C22semidirect
64. #1434C22semidirect
109. C22×C4⋊C4D4semidirect
141. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C23×#1262
C2×#24062
C2×#24042
C2×#22972
#166562
#166552
#150912
#149172
#149112
#146842
#142562
#142552
#138702
#138682