Finite Group Databasebeta

← #4930613

Subgroups — #4930613

Order 512.

5336 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic, complemented
3C2×#2431C211yesmaximal, characteristic
4C2×#2624C211yesmaximal, characteristic, complemented
5#14905C214yesmaximal, complemented
6#14777C214yesmaximal, complemented
7#14721C218yesmaximal, complemented
8#14026C212yesmaximal, complemented
9#14025C212yesmaximal, complemented
10#13977C212yesmaximal, complemented
11#13976C212yesmaximal, complemented
12#13937C211yesmaximal, characteristic, complemented
13#13924C211yesmaximal, characteristic, complemented
14#13853C211yesmaximal, characteristic, complemented
15C23×#41C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×#123C2214yes—
19C22×#121C2211yescharacteristic
20C2×#520—28no—
21C2×#520—28no—
22C2×#520—28no—
23C22×C4×Q8C2212yes—
24C22×C4×D4C2214yes—
25C22×C4×D4C2212yes—
26C22×#117C2212yes—
27C22×#117C2211yescharacteristic
28C23×C4⋊C4C2212yes—
29C23×C4⋊C4C2211yescharacteristic
30#2624—28no—
31#2501—28no—
32#2431—28no—
33C2×#399C2212yes—
34C2×#399C2214yes—
35C2×#397C2212yes—
36C2×#394C2214yes—
37C2×#392C2214yes—
38C2×#391C2212yes—
39C2×#391C2211yescharacteristic
40C2×#390C2212yes—
41C2×#390C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2212yes—
44C2×#389C2212yes—
45C2×#386C2214yes—
46C2×#386C2218yes—
47C2×#385C2218yes—
48C2×#384C2212yes—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1445C2214yes—
55#1444C2218yes—
56#1440C2214yes—
57#1439C2218yes—
58#1439C2218yes—
59#1438C2218yes—
60#1438C2214yes—
61#1437C2218yes—
62#1436C2218yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C4×#95C2212yes—
68C4×#95C2212yes—
69C22×#41—28no—
70C22×#41—28no—
71C22×#41—24no—
72C22×#41—24no—
73C22×#41D418yes—
74C23×Q8C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C23×D4C2314yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2314yes—
83C24×C4C2312yes—
84C2×#127—24no—
85C2×#127—28no—
86C2×#124—24no—
87C2×#124—28no—
88C2×#123—216no—
89C2×#123—416no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121—24no—
93#520—2128no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×D4—216no—
98C2×C4×D4—28no—
99C2×C4×D4—216no—
100C2×C4×D4—28no—
101C2×C4×D4—216no—
102C2×C4×D4—28no—
103C2×#117—28no—
104C2×#117—24no—
105C2×#117—24no—
106C2×#117—28no—
107C2×#117—28no—
108C2×#117D418yescomplemented
109C22×C4⋊C4—28no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4—28no—
114C22×C4⋊C4D418yescomplemented
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2318yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2318yes—
130C22×C22⋊C4C2318yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2312yes—
141#399—28no—
142#399—216no—
143#397—28no—
144#394—216no—
145#392—216no—
146#391—24no—
147#391—28no—
148#390—24no—
149#390—28no—
150#390—28no—
151#390—28no—
152#389—28no—
153#386—216no—
154#386—416no—
155#385—416no—
156#384—28no—
157#380—24no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—24no—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2318yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C25C2412yes—
173C25C2411yescharacteristic
174C25C2412yes—
175C2×#41—264no—
176C2×#41—264no—
177C2×#41—232no—
178C2×#41—232no—
179C2×#41—232no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8C2×D414yes—
184C22×Q8#4118yes—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—24no—
189C22×D4—28no—
190C22×D4—216no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—28no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—28no—
203C22×D4—48no—
204C22×D4—44no—
205C22×D4—48no—
206C22×D4C2×D418yes—
207C22×D4C2×D414yes—
208C22×D4#4118yes—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—48no—
221C23×C4—48no—
222C23×C4—48no—
223C23×C4C2×D418yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239#127—416no—
240#124—416no—
241#123—464no—
242#121—416no—
243C4×Q8—264no—
244C4×D4—2128no—
245C4×D4—264no—
246#117—264no—
247#117—232no—
248C2×C4⋊C4—264no—
249C2×C4⋊C4—232no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D414yes—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—216no—
292C2×C22⋊C4—216no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4C2×D418yes—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—28no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42C2×D418yes—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—216no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—44no—
329#95—48no—
330#95—48no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—28no—
337C24—24no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—28no—
348C24—24no—
349C24—24no—
350C24C22×D412yes—
351C24C22×D412yes—
352C24C22×D412yes—
353C24C22×D414yes—
354C24C2×#4118yes—
355C24C2511yescharacteristic
356#41—2256no—
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—216no—
360C2×Q8—216no—
361C2×Q8#123116yescomplemented
362C2×D4—232no—
363C2×D4—232no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—28no—
367C2×D4—216no—
368C2×D4—28no—
369C2×D4—216no—
370C2×D4—216no—
371C2×D4—416no—
372C2×D4—432no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—416no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—88no—
382C2×D4—816no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4#123116yescomplemented
386C22×C4—232no—
387C22×C4—232no—
388C22×C4—216no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—232no—
423C22×C4—216no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—416no—
439C22×C4—816no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4—816no—
445C22×C4#120116yes—
446C22×C4C22×D414yes—
447C22×C4C22×D418yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4118yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4H(5,𝔽2)18yes—
460C22×C42−1+414yes—
461C4⋊C4—232no—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—28no—
466C4⋊C4—432no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—88no—
470C22⋊C4—232no—
471C22⋊C4—216no—
472C22⋊C4—216no—
473C22⋊C4—432no—
474C22⋊C4—48no—
475C22⋊C4—432no—
476C22⋊C4—48no—
477C22⋊C4—416no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—84no—
482C42—232no—
483C42—216no—
484C42—44no—
485C42—44no—
486C42—48no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—24no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—216no—
505C23—216no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—48no—
513C23—44no—
514C23—44no—
515C23—48no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23C2×#12014yes—
534C23C2×#12318yes—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C23×D411yescharacteristic
538C23C22×#4112yes—
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)14yes—
541C23C2×2−1+412yes—
542C23#58512yes—
543Q8—232no—
544D4—264no—
545D4—232no—
546D4—832no—
547D4—816no—
548D4—88no—
549C2×C4—264no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—232no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—216no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—816no—
601C2×C4—832no—
602C2×C4C2×#12014yes—
603C2×C4C2×#12014yes—
604C2×C4C2×#12318yes—
605C2×C4C2×#12314yes—
606C2×C4C2×#12314yes—
607C2×C4#52518yes—
608C2×C4#52518yes—
609C22—24no—
610C22—28no—
611C22—24no—
612C22—216no—
613C22—24no—
614C22—44no—
615C22—48no—
616C22—416no—
617C22—44no—
618C22—44no—
619C22—48no—
620C22—44no—
621C22—44no—
622C22—48no—
623C22—44no—
624C22—48no—
625C22—82no—
626C22—82no—
627C22—88no—
628C22—84no—
629C22—84no—
630C22—82no—
631C22—84no—
632C22—84no—
633C22C22×#12011yescharacteristic
634C22C22×#12311yescharacteristic
635C22C22×#12311yescharacteristic
636C22C2×#52512yes—
637C22C2×#52512yes—
638C22#348412yes—
639C22#348912yes—
640C22#349712yes—
641C22#349812yes—
642C22#349812yes—
643C22#349911yescharacteristic
644C22#350011yescharacteristic
645C22#351314yes—
646C22#351714yes—
647C22#352014yes—
648C22#352214yes—
649C4—24no—
650C4—28no—
651C4—44no—
652C4—48no—
653C4—44no—
654C4—48no—
655C4—44no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4#284918yes—
660C2—44no—
661C2—48no—
662C2—162no—
663C2#3552711yescharacteristic
664C2#3553412yes—
665C2#3553512yes—
666C2#3566011yescharacteristic
667C2#3571514yes—
668C2#3598712yes—
669C2#3600512yes—
670C2C2×#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2431C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#520C2semidirect
2. C2×#2501C2semidirect
4. C2×#2624C2semidirect
5. #14905C2semidirect
6. #14777C2semidirect
7. #14721C2semidirect
8. #14026C2semidirect
9. #14025C2semidirect
10. #13977C2semidirect
11. #13976C2semidirect
12. #13937C2semidirect
13. #13924C2semidirect
14. #13853C2semidirect
108. C2×#117D4semidirect
114. C22×C4⋊C4D4semidirect
361. C2×Q8#123semidirect
385. C2×D4#123semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#25012
C2×#24312
C2×#26242
#149052
#147772
#147212
#140262
#140252
#139772
#139762
#139372
#139242
#138532