Finite Group Databasebeta

← #764597

Subgroups — #764597

Order 512.

4114 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#542C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C2×#2501C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic, complemented
5#15949C214yesmaximal, complemented
6#15927C214yesmaximal, complemented
7#15430C214yesmaximal, complemented
8#15262C214yesmaximal, complemented
9#15226C212yesmaximal, complemented
10#15143C212yesmaximal, complemented
11#14138C211yesmaximal, characteristic, complemented
12#14136C211yesmaximal, characteristic
13#13977C212yesmaximal, complemented
14#13976C212yesmaximal, complemented
15#13853C211yesmaximal, characteristic, complemented
16C2×#542—28no—
17C2×#542—28no—
18C2×#542—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#126C2212yes—
21C22×#124C2211yescharacteristic
22C22×#124C2212yes—
23C22×#123C2212yes—
24C22×#123C2214yes—
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C22×#121C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#2502—28no—
34#2501—28no—
35C2×#399C2214yes—
36C2×#398C2214yes—
37C2×#395C2218yes—
38#2317—28no—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#390C2212yes—
42C2×#390C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#385C2214yes—
48C2×#385C2218yes—
49C2×#384C2212yes—
50C2×#384C2212yes—
51C2×#384C2212yes—
52C2×#382C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55#1445C2212yes—
56#1445C2212yes—
57#1444C2218yes—
58#1444C2218yes—
59#1440C2212yes—
60#1440C2212yes—
61#1440C2218yes—
62#1440C2214yes—
63#1439C2218yes—
64#1439C2218yes—
65#1438C2214yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2218yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4×#95C2212yes—
72C4×#95C2212yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80#542—2128no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×#126—28no—
85C2×#126—28no—
86C2×#126—28no—
87C2×#124—24no—
88C2×#124—24no—
89C2×#124—28no—
90C2×#124—28no—
91C2×#124—28no—
92C2×#124—24no—
93C2×#123—28no—
94C2×#123—216no—
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—416no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8D418yes—
104C2×#121—24no—
105C2×#121—24no—
106C2×#121D418yes—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×D4—24no—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C2×#117—24no—
114C2×#117—24no—
115C2×#117—24no—
116C2×#117—28no—
117C2×#117D418yes—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2318yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2318yes—
134C22×C4⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2318yes—
142C22×C22⋊C4C2318yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2311yescharacteristic
145C22×C22⋊C4C2311yescharacteristic
146C22×C22⋊C4C2312yes—
147C22×C42C2312yes—
148C22×C42C2312yes—
149C22×C42C2312yes—
150C22×C42C2312yes—
151C22×C42C2311yescharacteristic
152C22×C42C2311yescharacteristic
153#399—216no—
154#398—216no—
155#395—416no—
156#391—28no—
157#391—28no—
158#390—28no—
159#390—28no—
160#389—28no—
161#389—28no—
162#385—28no—
163#385—28no—
164#385—216no—
165#385—416no—
166#384—28no—
167#384—28no—
168#384—28no—
169#382—28no—
170C4×C22⋊C4—24no—
171C4×C22⋊C4—24no—
172C2×#95C2314yes—
173C2×#95C2318yes—
174C2×#95C2312yes—
175C2×#95C2318yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2318yes—
182C2×#95C2314yes—
183C2×#95C2318yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—48no—
194C22×Q8C2×D414yes—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—48no—
202C22×D4C2×D414yes—
203C22×D4C2×D418yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—48no—
213C23×C4—48no—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C4⋊Q8—232no—
231#126—264no—
232#124—264no—
233#124—232no—
234#123—264no—
235#123—464no—
236C22⋊Q8—264no—
237C22⋊Q8—232no—
238#121—232no—
239C4×Q8—232no—
240C4×D4—232no—
241#117—232no—
242#117—232no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
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—216no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—216no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—416no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4#4118yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
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—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—216no—
308C2×C22⋊C4—216no—
309C2×C22⋊C4—216no—
310C2×C22⋊C4—216no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4—48no—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4C2×D418yes—
316C2×C22⋊C4C2×D418yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—28no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—216no—
339#95—216no—
340#95—28no—
341#95—216no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—48no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—28no—
354C24—28no—
355C24—24no—
356C24C22×D414yes—
357C24C22×D412yes—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C2511yescharacteristic
361C2×Q8—216no—
362C2×Q8—28no—
363C2×Q8—216no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—48no—
367C2×Q8—88no—
368C2×Q8—88no—
369C2×Q8—88no—
370C2×D4—216no—
371C2×D4—28no—
372C2×D4—216no—
373C2×D4—416no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—48no—
377C2×D4—88no—
378C2×D4—88no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—216no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—416no—
430C22×C4—48no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4#120116yes—
437C22×C4C22×D412yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4114yes—
452C22×C4H(5,𝔽2)18yes—
453C22×C42−1+418yes—
454C22×C42−1+414yes—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—432no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—88no—
467C4⋊C4—816no—
468C4⋊C4—88no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—216no—
472C22⋊C4—216no—
473C22⋊C4—216no—
474C22⋊C4—216no—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—432no—
480C22⋊C4—432no—
481C42—216no—
482C42—216no—
483C42—28no—
484C42—28no—
485C42—48no—
486C42—48no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23C2×#12014yes—
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C23×D411yescharacteristic
520C23C22×#4112yes—
521C23C2×H(5,𝔽2)14yes—
522C23C2×2−1+412yes—
523C23C2×2−1+414yes—
524Q8—416no—
525Q8—816no—
526D4—416no—
527D4—816no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—232no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—216no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—816no—
587C2×C4—816no—
588C2×C4C2×#12014yes—
589C2×C4C2×#12014yes—
590C2×C4#52514yes—
591C2×C4#53614yes—
592C2×C4#54718yes—
593C2×C4#54718yes—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—48no—
598C22—44no—
599C22—48no—
600C22—44no—
601C22—416no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—84no—
606C22—88no—
607C22—84no—
608C22C22×#12011yescharacteristic
609C22C2×#52511yescharacteristic
610C22C2×#53611yescharacteristic
611C22C2×#54712yes—
612C22C2×#54712yes—
613C22#348412yes—
614C22#348612yes—
615C22#349812yes—
616C22#349812yes—
617C22#349912yes—
618C22#349912yes—
619C22#352014yes—
620C22#352214yes—
621C22#354414yes—
622C22#355214yes—
623C4—24no—
624C4—48no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—48no—
629C4—42no—
630C4—44no—
631C4—44no—
632C4—42no—
633C4—48no—
634C4—84no—
635C4#240618yes—
636C2—44no—
637C2—84no—
638C2C2×#240611yescharacteristic
639C2#3552711yescharacteristic
640C2#3553512yes—
641C2#3566611yescharacteristic
642C2#3730312yes—
643C2#4487514yes—
644C2#4498914yes—
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×#2502C2semidirect
4. C2×#2317C2semidirect
5. #15949C2semidirect
6. #15927C2semidirect
7. #15430C2semidirect
8. #15262C2semidirect
9. #15226C2semidirect
10. #15143C2semidirect
11. #14138C2semidirect
13. #13977C2semidirect
14. #13976C2semidirect
15. #13853C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5422
C2×#25022
C2×#25012
C2×#23172
#159492
#159272
#154302
#152622
#152262
#151432
#141382
#141362
#139772
#139762
#138532