Finite Group Databasebeta

← #4457783

Subgroups — #4457783

Order 512.

3608 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#542C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3C2×#2563C211yesmaximal, characteristic, complemented
4C2×#2628C211yesmaximal, characteristic, complemented
5#16656C214yesmaximal, complemented
6#16559C214yesmaximal, complemented
7#16366C214yesmaximal, complemented
8#16360C214yesmaximal, complemented
9#15236C212yesmaximal, complemented
10#15230C212yesmaximal, complemented
11#15226C212yesmaximal, complemented
12#15219C212yesmaximal
13#13942C211yesmaximal, characteristic, complemented
14#13941C211yesmaximal, characteristic, complemented
15#13852C211yesmaximal, characteristic, complemented
16C2×#542—28no—
17C2×#542—28no—
18C2×#542—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2212yes—
22C22×#124C2211yescharacteristic
23C22×#124C2212yes—
24C22×#123C2212yes—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2212yes—
27C22×#121C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117C2211yescharacteristic
31#2628—28no—
32#2568—28no—
33#2563—28no—
34C2×#400C2212yes—
35C2×#400C2214yes—
36C2×#399C2212yes—
37C2×#395C2212yes—
38C2×#393C2214yes—
39C2×#393C2212yes—
40C2×#391C2214yes—
41C2×#391C2212yes—
42C2×#391C2212yes—
43C2×#390C2212yes—
44C2×#390C2212yes—
45C2×#389C2212yes—
46C2×#382C2211yescharacteristic
47C2×#382C2212yes—
48C2×#382C2212yes—
49C2×#380C2212yes—
50C2×#380C2214yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54#1446C2218yes—
55#1442C2218yes—
56#1441C2212yes—
57#1441C2218yes—
58#1441C2212yes—
59#1441C2214yes—
60#1440C2218yes—
61#1440C2218yes—
62#1440C2218yes—
63#1439C2214yes—
64#1439C2218yes—
65#1439C2212yes—
66#1439C2212yes—
67#1438C2218yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C4×#95C2212yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C23×Q8C2312yes—
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80#542—2128no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×#126—28no—
88C2×#126—28no—
89C2×#126—28no—
90C2×#124—24no—
91C2×#124—24no—
92C2×#124—28no—
93C2×#124—28no—
94C2×#124—24no—
95C2×#124—28no—
96C2×#123—28no—
97C2×#123—28no—
98C2×#123—28no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—24no—
105C2×#121—24no—
106C2×#121—24no—
107C2×#121—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8D418yes—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C2×C4×D4D418yes—
114C2×#117—24no—
115C2×#117—24no—
116C2×#117D418yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2311yescharacteristic
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2318yes—
139C22×C22⋊C4C2318yes—
140C22×C22⋊C4C2314yes—
141C22×C42C2312yes—
142C22×C42C2312yes—
143C22×C42C2311yescharacteristic
144C22×C42C2312yes—
145C22×C42C2311yescharacteristic
146#400—216no—
147#400—28no—
148#399—28no—
149#395—28no—
150#393—216no—
151#393—28no—
152#391—28no—
153#391—216no—
154#391—44no—
155#390—28no—
156#390—44no—
157#389—28no—
158#382—28no—
159#382—28no—
160#382—24no—
161#380—216no—
162#380—28no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—24no—
165C4×C22⋊C4—44no—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2318yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2318yes—
174C2×#95C2318yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2318yes—
178C2×#95C2318yes—
179C2×#95C2318yes—
180C2×#95C2314yes—
181C25C2411yescharacteristic
182C25C2412yes—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—48no—
196C22×Q8—48no—
197C22×Q8C2×D414yes—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—48no—
208C22×D4C2×D414yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—48no—
218C23×C4C2×D414yes—
219C23×C4C2×D418yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C4⋊Q8—232no—
234C4⋊Q8—416no—
235#126—264no—
236#124—232no—
237#124—264no—
238#123—264no—
239C22⋊Q8—264no—
240C22⋊Q8—232no—
241#121—232no—
242C4×Q8—232no—
243C4×D4—232no—
244#117—232no—
245C2×C4⋊C4—22no—
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—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
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—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—216no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4C2×D414yes—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D418yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42C2×D414yes—
324C2×C42C2×D418yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328#95—24no—
329#95—24no—
330#95—216no—
331#95—28no—
332#95—216no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—216no—
337#95—44no—
338#95—48no—
339#95—48no—
340#95—44no—
341#95—44no—
342#95—48no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—28no—
357C24—28no—
358C24—24no—
359C24C22×D414yes—
360C24C22×D412yes—
361C24C2511yescharacteristic
362C2×Q8—216no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—416no—
367C2×Q8—48no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—88no—
373C2×Q8—88no—
374C2×Q8—88no—
375C2×Q8—88no—
376C2×Q8—88no—
377C2×D4—216no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—88no—
386C2×D4—88no—
387C2×D4—88no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—216no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—416no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C22×C42−1+412yes—
468C4⋊C4—216no—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—216no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—88no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—216no—
484C22⋊C4—216no—
485C22⋊C4—28no—
486C22⋊C4—28no—
487C22⋊C4—216no—
488C22⋊C4—416no—
489C22⋊C4—816no—
490C22⋊C4—88no—
491C22⋊C4—816no—
492C42—28no—
493C42—216no—
494C42—48no—
495C42—48no—
496C42#120116yes—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—28no—
506C23—24no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—44no—
524C23—48no—
525C23—44no—
526C23—48no—
527C23—44no—
528C23—44no—
529C23—48no—
530C23—48no—
531C23C23×D411yescharacteristic
532C23C23×D411yescharacteristic
533C23C23×D411yescharacteristic
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×2−1+411yescharacteristic
536C23C2×2−1+411yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23#58514yes—
539C23#58514yes—
540Q8—416no—
541Q8—816no—
542Q8—816no—
543D4—416no—
544D4—816no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—416no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—816no—
607C2×C4C2×#12014yes—
608C2×C4C2×#12014yes—
609C2×C4C2×#12014yes—
610C2×C4#53614yes—
611C2×C4#53614yes—
612C2×C4#53614yes—
613C2×C4#53714yes—
614C2×C4#53714yes—
615C2×C4#53714yes—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—48no—
625C22—48no—
626C22—84no—
627C22—84no—
628C22—82no—
629C22—84no—
630C22—82no—
631C22—84no—
632C22—82no—
633C22—82no—
634C22C22×#12011yescharacteristic
635C22C2×#53611yescharacteristic
636C22C2×#53611yescharacteristic
637C22C2×#53611yescharacteristic
638C22C2×#53711yescharacteristic
639C22C2×#53711yescharacteristic
640C22C2×#53711yescharacteristic
641C22#348512yes—
642C22#348712yes—
643C22#350012yes—
644C22#350012yes—
645C22#350012yes—
646C22#350012yes—
647C22#354014yes—
648C22#355714yes—
649C22#356514yes—
650C22#356514yes—
651C4—24no—
652C4—24no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—44no—
658C4—42no—
659C4—84no—
660C4—84no—
661C4—84no—
662C4—84no—
663C4#261618yes—
664C2—84no—
665C2—82no—
666C2C2×#261611yescharacteristic
667C2#3561611yescharacteristic
668C2#3567611yescharacteristic
669C2#3742112yes—
670C2#3742312yes—
671C2#4264712yes—
672C2#4265012yes—
673C2#4287512yes—
674C2#4288212yes—
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
2. C2×#2568C2semidirect
3. C2×#2563C2semidirect
4. C2×#2628C2semidirect
5. #16656C2semidirect
6. #16559C2semidirect
7. #16366C2semidirect
8. #16360C2semidirect
9. #15236C2semidirect
10. #15230C2semidirect
11. #15226C2semidirect
13. #13942C2semidirect
14. #13941C2semidirect
15. #13852C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5422
C2×#25682
C2×#25632
C2×#26282
#166562
#165592
#163662
#163602
#152362
#152302
#152262
#152192
#139422
#139412
#138522