Finite Group Databasebeta

← #626402

Subgroups — #626402

Order 512.

3174 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C212yesmaximal, complemented
2C2×#2325C211yesmaximal, characteristic
3C22×#387C211yesmaximal, characteristic, complemented
4#15818C214yesmaximal, complemented
5#15812C214yesmaximal, complemented
6#15795C214yesmaximal, complemented
7#15786C214yesmaximal, complemented
8#15019C211yesmaximal, characteristic, complemented
9#15019C211yesmaximal, characteristic, complemented
10#14926C211yesmaximal, characteristic
11#14926C211yesmaximal, characteristic
12#14706C212yesmaximal, complemented
13#14347C211yesmaximal, characteristic, complemented
14#14256C212yesmaximal, complemented
15#13960C212yesmaximal, complemented
16C22×#123C2212yes—
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2486—216no—
24C2×#398C2214yes—
25C2×#397C2214yes—
26#2325—28no—
27C2×#389C2214yes—
28C2×#388C2212yes—
29C2×#387—28no—
30C2×#387C2212yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2214yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1445C2212yes—
48#1445C2212yes—
49#1444C2212yes—
50#1444C2212yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2212yes—
66#1438C2212yes—
67#1437C2214yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1434C2214yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2212yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—44no—
93C2×#123—44no—
94C2×#121—28no—
95C2×#121—24no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4D418yescomplemented
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C42—24no—
125C22×C42—44no—
126C22×C42C2312yes—
127C22×C42C2314yes—
128#398—216no—
129#397—216no—
130#389—216no—
131#388—28no—
132#387—28no—
133#387—216no—
134#387—216no—
135#386—28no—
136#386—28no—
137#386—44no—
138#386—44no—
139#385—28no—
140#381—28no—
141#381—28no—
142#380—24no—
143#380—28no—
144#380—216no—
145C2×#95—24no—
146C2×#95—24no—
147C2×#95—24no—
148C2×#95D418yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C25C2412yes—
177C25C2411yescharacteristic
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4#4118yes—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4#4114yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238#123—432no—
239#123—432no—
240#121—232no—
241C4×D4—432no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—416no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C2×D4116yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—48no—
303C2×C42—416no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95C2×D418yes—
330#95C2×D418yes—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24C22×D412yes—
346C24C22×D412yes—
347C24C2×#4114yes—
348C24C2×#4112yes—
349C24C2×#4114yes—
350C24H(5,𝔽2)14yes—
351C24C2511yescharacteristic
352C2×D4—216no—
353C2×D4—216no—
354C2×D4—216no—
355C2×D4—28no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—88no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—44no—
439C22×C4—416no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4#121116yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C42−1+414yes—
467C4⋊C4—28no—
468C4⋊C4—232no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—416no—
472C4⋊C4—416no—
473C4⋊C4—48no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C22⋊C4—216no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C42—416no—
488C42—48no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—28no—
505C23—28no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23C2×#12114yes—
525C23#52514yes—
526C23#53514yes—
527C23#54618yes—
528C23#55318yes—
529C23C23×D411yescharacteristic
530C23C23×D411yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4112yes—
533C23C2×H(5,𝔽2)12yes—
534C23C2×2−1+412yes—
535C23#58512yes—
536C23#58512yes—
537C23#58512yes—
538D4—432no—
539D4—88no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—416no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—816no—
613C2×C4—88no—
614C2×C4—88no—
615C2×C4C2×#12118yes—
616C22—24no—
617C22—24no—
618C22—216no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—216no—
623C22—24no—
624C22—44no—
625C22—48no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—82no—
630C22—82no—
631C22—84no—
632C22—82no—
633C22—84no—
634C22—82no—
635C22#253518yes—
636C22C22×#12111yescharacteristic
637C22C2×#52511yescharacteristic
638C22C2×#53511yescharacteristic
639C22#344912yes—
640C22#346112yes—
641C22C2×#54612yes—
642C22C2×#55312yes—
643C22#348511yescharacteristic
644C22#348511yescharacteristic
645C22#348511yescharacteristic
646C22#348611yescharacteristic
647C22#349112yes—
648C22#349712yes—
649C22#351514yes—
650C22#352114yes—
651C22#353314yes—
652C22#353514yes—
653C4—28no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4—84no—
666C2—28no—
667C2—84no—
668C2C2×#253511yescharacteristic
669C2#3508911yescharacteristic
670C2#3515311yescharacteristic
671C2#3545211yescharacteristic
672C2#3545311yescharacteristic
673C2#3695611yescharacteristic
674C2#3711811yescharacteristic
675C2#3808714yes—
676C2#3925414yes—
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
1. C2×#2486C2semidirect
3. C22×#387C2semidirect
4. #15818C2semidirect
5. #15812C2semidirect
6. #15795C2semidirect
7. #15786C2semidirect
8. #15019C2semidirect
9. #15019C2semidirect
12. #14706C2semidirect
13. #14347C2semidirect
14. #14256C2semidirect
15. #13960C2semidirect
103. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#23252
C22×#3872
#158182
#158122
#157952
#157862
#150192
#149262
#147062
#143472
#142562
#139602