Finite Group Databasebeta

← #4422456

Subgroups — #4422456

Order 512.

2718 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic, complemented
2C2×#2598C211yesmaximal, characteristic
3C2×#2621C211yesmaximal, characteristic, complemented
4C2×#2343C211yesmaximal, characteristic, complemented
5#16800C212yesmaximal, complemented
6#16705C212yesmaximal, complemented
7#16566C212yesmaximal, complemented
8#16263C212yesmaximal, complemented
9#15512C214yesmaximal, complemented
10#15390C214yesmaximal, complemented
11#15160C212yesmaximal
12#15016C212yesmaximal, complemented
13#14902C212yesmaximal, complemented
14#14759C212yesmaximal, complemented
15#14663C211yesmaximal, characteristic, complemented
16#14356C211yesmaximal, characteristic, complemented
17#14345C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2621—28no—
23#2598—28no—
24#2563—28no—
25C2×#400C2211yescharacteristic
26C2×#399C2211yescharacteristic
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#397C2212yes—
30C2×#397C2212yes—
31C2×#394C2211yescharacteristic
32C2×#394C2214yes—
33C2×#393C2212yes—
34C2×#393C2212yes—
35C2×#392C2211yescharacteristic
36#2343—28no—
37C2×#391C2212yes—
38C2×#390C2212yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2214yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#383C2211yescharacteristic
48C2×#383C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#382C2212yes—
52C2×#381C2211yescharacteristic
53C2×#380C2214yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56C2×#376C2211yescharacteristic
57C22×#95C2211yescharacteristic
58#1446C2212yes—
59#1446C2212yes—
60#1445C2212yes—
61#1445C2214yes—
62#1445C2212yes—
63#1445C2212yes—
64#1444C2212yes—
65#1443C2214yes—
66#1442C2212yes—
67#1442C2214yes—
68#1442C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1438C2212yes—
81#1437C2214yes—
82#1436C2214yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85C22⋊#95C2214yes—
86C22⋊#95C2214yes—
87C23×Q8C2312yes—
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2312yes—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—48no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117D418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#400—24no—
133#399—24no—
134#398—28no—
135#398—28no—
136#397—28no—
137#397—28no—
138#394—24no—
139#394—48no—
140#393—28no—
141#393—28no—
142#392—24no—
143#391—28no—
144#390—28no—
145#388—28no—
146#387—28no—
147#387—28no—
148#386—28no—
149#386—28no—
150#386—48no—
151#385—28no—
152#385—28no—
153#383—24no—
154#383—24no—
155#382—28no—
156#382—24no—
157#382—24no—
158#381—24no—
159#380—216no—
160C4×C4⋊C4D418yescomplemented
161C4×C22⋊C4—28no—
162#376—24no—
163C2×#95—24no—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C25C2412yes—
196C25C2411yescharacteristic
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—28no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—42no—
210C22×D4—44no—
211C22×D4—42no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4C2×D414yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C4⋊Q8—416no—
243C22⋊Q8—432no—
244C4×D4—416no—
245#117—232no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
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⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D418yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
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⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D414yes—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42C2×D414yes—
317C2×C42C2×D418yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42#4114yes—
321#95—22no—
322#95—24no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95C2×D414yes—
351#95C2×D414yes—
352C24—21no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24C22×D412yes—
369C24H(5,𝔽2)14yes—
370C242−1+414yes—
371C24C2511yescharacteristic
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—88no—
376C2×Q8—88no—
377C2×D4—28no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—44no—
381C2×D4—48no—
382C2×D4—416no—
383C2×D4—48no—
384C2×D4—44no—
385C2×D4—84no—
386C2×D4—88no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
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—22no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—88no—
462C22×C4—84no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—416no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—88no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C42—28no—
504C42—48no—
505C42—48no—
506C42#121116yes—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—42no—
532C23—44no—
533C23—42no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23#53514yes—
546C23C23×D411yescharacteristic
547C23C23×D411yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+412yes—
551C23#58512yes—
552C23#58514yes—
553C23#58512yes—
554Q8—816no—
555D4—416no—
556D4—816no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
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—816no—
633C2×C4—816no—
634C2×C4C2×#12114yes—
635C2×C4C2×#12114yes—
636C2×C4C2×#12114yes—
637C22—28no—
638C22—24no—
639C22—44no—
640C22—42no—
641C22—44no—
642C22—48no—
643C22—44no—
644C22—42no—
645C22—48no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—48no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—84no—
654C22—84no—
655C22—82no—
656C22C22×#12111yescharacteristic
657C22C2×#53511yescharacteristic
658C22#344912yes—
659C22#344911yescharacteristic
660C22#348511yescharacteristic
661C22#348611yescharacteristic
662C22#348812yes—
663C22#348912yes—
664C22#349412yes—
665C22#349512yes—
666C22#350012yes—
667C22#351012yes—
668C22#353514yes—
669C22#354514yes—
670C22#355314yes—
671C22#356614yes—
672C4—24no—
673C4—24no—
674C4—24no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—44no—
679C4—42no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4—82no—
684C4—84no—
685C4—82no—
686C4—82no—
687C2—44no—
688C2—84no—
689C2#3525211yescharacteristic
690C2#3532711yescharacteristic
691C2#3563311yescharacteristic
692C2#4142312yes—
693C2#4199112yes—
694C2#4229612yes—
695C2#4239112yes—
696C2#5088214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2563C2semidirect
3. C2×#2621C2semidirect
4. C2×#2343C2semidirect
5. #16800C2semidirect
6. #16705C2semidirect
7. #16566C2semidirect
8. #16263C2semidirect
9. #15512C2semidirect
10. #15390C2semidirect
12. #15016C2semidirect
13. #14902C2semidirect
14. #14759C2semidirect
15. #14663C2semidirect
16. #14356C2semidirect
17. #14345C2semidirect
160. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25982
C2×#26212
C2×#23432
#168002
#167052
#165662
#162632
#155122
#153902
#151602
#150162
#149022
#147592
#146632
#143562
#143452