Finite Group Databasebeta

← #4453318

Subgroups — #4453318

Order 512.

3288 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2579C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic
3C22×#395C211yesmaximal, characteristic, complemented
4C2×#2318C211yesmaximal, characteristic
5#16663C212yesmaximal, complemented
6#16565C212yesmaximal, complemented
7#16295C214yesmaximal, complemented
8#16293C214yesmaximal, complemented
9#16264C212yesmaximal, complemented
10#16262C212yesmaximal, complemented
11#14955C212yesmaximal, complemented
12#14945C212yesmaximal
13#14786C212yesmaximal, complemented
14#14776C212yesmaximal
15#14347C211yesmaximal, characteristic, complemented
16#13923C211yesmaximal, characteristic, complemented
17#13922C211yesmaximal, characteristic, complemented
18C24×Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23#2579—28no—
24#2409—28no—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#397C2214yes—
28C2×#396C2212yes—
29C2×#395—28no—
30C2×#395C2212yes—
31C2×#395C2212yes—
32C2×#395C2212yes—
33C2×#395C2214yes—
34C2×#393C2212yes—
35#2318—28no—
36C2×#391C2212yes—
37C2×#391C2214yes—
38C2×#388C2212yes—
39C2×#387C2212yes—
40C2×#387C2212yes—
41C2×#386C2212yes—
42C2×#386C2211yescharacteristic
43C2×#386C2211yescharacteristic
44C2×#386C2212yes—
45C2×#385C2212yes—
46C2×#385C2211yescharacteristic
47C2×#385C2212yes—
48C2×#385C2211yescharacteristic
49C2×#385C2212yes—
50C2×#383C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#381C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×#380C2211yescharacteristic
55C2×#380C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57C22×#95C2212yes—
58C22×#95C2211yescharacteristic
59#1446C2212yes—
60#1446C2212yes—
61#1445C2214yes—
62#1445C2214yes—
63#1444C2214yes—
64#1443C2214yes—
65#1443C2214yes—
66#1442C2212yes—
67#1442C2212yes—
68#1442C2214yes—
69#1442C2214yes—
70#1442C2214yes—
71#1442C2214yes—
72#1441C2212yes—
73#1441C2212yes—
74#1441C2214yes—
75#1441C2214yes—
76#1441C2214yes—
77#1440C2214yes—
78#1440C2214yes—
79#1439C2214yes—
80#1436C2214yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×Q8—24no—
88C23×Q8—24no—
89C23×Q8—24no—
90C23×Q8—48no—
91C23×Q8C2312yes—
92C23×Q8C2312yes—
93C23×D4C2311yescharacteristic
94C24×C4C2312yes—
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2312yes—
98C24×C4C2311yescharacteristic
99C24×C4C2311yescharacteristic
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—48no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4D418yes—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—48no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#398—28no—
136#398—48no—
137#397—48no—
138#396—28no—
139#395—28no—
140#395—28no—
141#395—28no—
142#395—216no—
143#393—28no—
144#391—28no—
145#391—48no—
146#388—28no—
147#387—28no—
148#387—28no—
149#386—24no—
150#386—24no—
151#386—28no—
152#386—28no—
153#385—28no—
154#385—28no—
155#385—24no—
156#385—24no—
157#385—28no—
158#383—24no—
159#382—24no—
160#381—24no—
161#380—24no—
162#380—24no—
163#380—24no—
164C4×C22⋊C4—24no—
165C2×#95—28no—
166C2×#95D418yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C2×#95C2312yes—
198C25C2411yescharacteristic
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—28no—
207C22×Q8—28no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×Q8—432no—
211C22×Q8—432no—
212C22×Q8—432no—
213C22×Q8—416no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4C2×D414yes—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—28no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—44no—
235C23×C4—42no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4C2×D418yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4#4114yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C23×C4C2414yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C22⋊Q8—432no—
262C4×D4—232no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—432no—
290C2×C4⋊C4—416no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2×D414yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D414yes—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—44no—
328C2×C42C2×D414yes—
329#95—28no—
330#95—28no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95C2×D414yes—
358#95C2×D414yes—
359#95C2×D418yes—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24C22×D412yes—
371C24C22×D412yes—
372C24C22×D412yes—
373C24C2×#4112yes—
374C242−1+414yes—
375C242−1+414yes—
376C24C2511yescharacteristic
377C2×Q8—216no—
378C2×Q8—216no—
379C2×Q8—416no—
380C2×Q8—416no—
381C2×Q8—416no—
382C2×Q8—416no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—4128no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—48no—
392C2×D4—48no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—216no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
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—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
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—416no—
471C22×C4—416no—
472C22×C4—44no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—44no—
480C22×C4—84no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4#121116yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)18yes—
497C22×C42−1+414yes—
498C4⋊C4—28no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—416no—
502C4⋊C4—416no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—48no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—28no—
522C42—84no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23C2×#12114yes—
551C23#53614yes—
552C23#53614yes—
553C23#53618yes—
554C23#54118yes—
555C23C23×D411yescharacteristic
556C23C23×D411yescharacteristic
557C23C22×#4111yescharacteristic
558C23C2×H(5,𝔽2)14yes—
559C23C2×2−1+412yes—
560C23C2×2−1+412yes—
561C23C2×2−1+412yes—
562C23#58512yes—
563Q8—432no—
564Q8—432no—
565D4—416no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—432no—
594C2×C4—432no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
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—88no—
647C2×C4—88no—
648C2×C4C2×#12114yes—
649C2×C4C2×#12114yes—
650C22—24no—
651C22—24no—
652C22—28no—
653C22—28no—
654C22—28no—
655C22—28no—
656C22—216no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—82no—
663C22—82no—
664C22—84no—
665C22#262118yes—
666C22C22×#12111yescharacteristic
667C22C2×#53612yes—
668C22C2×#53611yescharacteristic
669C22C2×#53611yescharacteristic
670C22C2×#54112yes—
671C22#348412yes—
672C22#348412yes—
673C22#348511yescharacteristic
674C22#348611yescharacteristic
675C22#348912yes—
676C22#349012yes—
677C22#349512yes—
678C22#354114yes—
679C22#354114yes—
680C22#355214yes—
681C22#355214yes—
682C4—24no—
683C4—24no—
684C4—48no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—42no—
689C4—44no—
690C4—44no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—84no—
695C4—82no—
696C4—82no—
697C2—28no—
698C2—82no—
699C2C2×#262111yescharacteristic
700C2#3534811yescharacteristic
701C2#3563311yescharacteristic
702C2#3671612yes—
703C2#3723312yes—
704C2#4058912yes—
705C2#4059112yes—
706C2#4064314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#395C2semidirect
5. #16663C2semidirect
6. #16565C2semidirect
7. #16295C2semidirect
8. #16293C2semidirect
9. #16264C2semidirect
10. #16262C2semidirect
11. #14955C2semidirect
13. #14786C2semidirect
15. #14347C2semidirect
16. #13923C2semidirect
17. #13922C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25792
C2×#24092
C22×#3952
C2×#23182
#166632
#165652
#162952
#162932
#162642
#162622
#149552
#149452
#147862
#147762
#143472
#139232
#139222