Finite Group Databasebeta

← #768742

Subgroups — #768742

Order 512.

4094 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2309C211yesmaximal, characteristic
2C2×#2308C211yesmaximal, characteristic
3#16074C212yesmaximal, complemented
4#15344C214yesmaximal, complemented
5#15342C212yesmaximal, complemented
6#15287C214yesmaximal, complemented
7#15285C214yesmaximal, complemented
8#14820C214yesmaximal, complemented
9#13958C212yesmaximal, complemented
10#13878C212yesmaximal, complemented
11#13877C212yesmaximal, complemented
12#13869C212yesmaximal, complemented
13C2×#1438C211yesmaximal, characteristic
14C24×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C2×#397C2214yescomplemented
22C2×#397C2214yescomplemented
23#2309—28no—
24#2308—28no—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#385C2214yescomplemented
34C2×#381C2212yes—
35C2×#381C2214yescomplemented
36C2×#380C2214yes—
37C2×#379C2212yes—
38C2×#379C2212yes—
39C2×#379C2212yes—
40C22×#95C2212yes—
41#1445C2214yescomplemented
42#1445C2214yescomplemented
43#1444C2214yescomplemented
44#1442C2214yescomplemented
45#1440C2214yescomplemented
46#1440C2214yescomplemented
47#1440C2214yescomplemented
48#1440C2214yescomplemented
49#1440C2214yescomplemented
50#1440C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1438C2212yes—
56#1438C2212yes—
57#1438C2211yescharacteristic
58#1438C2211yescharacteristic
59#1438C2212yes—
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1434C2214yescomplemented
63#1434C2214yescomplemented
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yescomplemented
68C22⋊#95C2214yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×D4—28no—
72C23×D4—28no—
73C23×D4—44no—
74C23×D4D418yes—
75C23×D4C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—28no—
86C2×#117—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—48no—
96C22×C22⋊C4—48no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119#397—48no—
120#397—48no—
121#390—28no—
122#390—28no—
123#390—28no—
124#390—28no—
125#386—28no—
126#386—28no—
127#386—28no—
128#386—28no—
129#385—48no—
130#381—28no—
131#381—48no—
132#380—48no—
133#379—28no—
134#379—28no—
135#379—28no—
136C2×#95—28no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25—24no—
157C25—24no—
158C25—24no—
159C25—24no—
160C25—24no—
161C25—44no—
162C25—44no—
163C25—44no—
164C25—44no—
165C25C2×D414yes—
166C25C2×D414yes—
167C25#4118yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C22×D4—232no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—232no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—416no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—464no—
189C22×D4C2×D414yes—
190C22×D4C2×D414yes—
191C22×D4C2×D414yes—
192C22×D4C2×D414yes—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C2×D414yes—
213C23×C4#4118yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#117—416no—
223#117—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—432no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—416no—
270C2×C22⋊C4—432no—
271C2×C22⋊C4—416no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—44no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—28no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—28no—
318C24—28no—
319C24—232no—
320C24—28no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—28no—
332C24—28no—
333C24—48no—
334C24—416no—
335C24—416no—
336C24—48no—
337C24—416no—
338C24—48no—
339C24—48no—
340C24—416no—
341C24—416no—
342C24—84no—
343C24—84no—
344C24—88no—
345C24—88no—
346C24—88no—
347C24—84no—
348C24—84no—
349C24—84no—
350C24—84no—
351C24—84no—
352C24—84no—
353C24—88no—
354C24#123116yes—
355C24C22×D412yes—
356C24C22×D412yes—
357C24C22×D412yes—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C2×#4114yes—
361C24C2×#4114yes—
362C24C2×#4114yes—
363C24C2×#4114yes—
364C24H(5,𝔽2)14yes—
365C24C2511yescharacteristic
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—416no—
381C2×D4—416no—
382C2×D4—4128no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—232no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
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—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—416no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—48no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—88no—
448C22×C4#123116yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)18yes—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—88no—
457C4⋊C4—88no—
458C22⋊C4—216no—
459C22⋊C4—48no—
460C22⋊C4—416no—
461C22⋊C4—416no—
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—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C42—48no—
477C42—48no—
478C23—24no—
479C23—24no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—232no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—28no—
510C23—28no—
511C23—432no—
512C23—432no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—416no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—816no—
547C23—816no—
548C23—832no—
549C23—832no—
550C23—816no—
551C23—816no—
552C23—832no—
553C23—832no—
554C23C2×#12314yes—
555C23C2×#12314yes—
556C23C2×#12314yes—
557C23C2×#12314yes—
558C23#53414yes—
559C23#551116yes—
560C23C23×D411yescharacteristic
561C23C22×#4112yes—
562C23C22×#4112yes—
563C23C2×H(5,𝔽2)12yes—
564C23C2×H(5,𝔽2)14yes—
565C23#58514yes—
566D4—416no—
567D4—416no—
568D4—416no—
569D4—416no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—232no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
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—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—832no—
620C2×C4—816no—
621C2×C4—816no—
622C2×C4C2×#12314yes—
623C2×C4C2×#12314yes—
624C22—28no—
625C22—28no—
626C22—216no—
627C22—216no—
628C22—28no—
629C22—28no—
630C22—28no—
631C22—24no—
632C22—24no—
633C22—44no—
634C22—44no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—832no—
650C22—832no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22#248318yes—
656C22C22×#12311yescharacteristic
657C22C22×#12311yescharacteristic
658C22C2×#53411yescharacteristic
659C22C2×#55114yes—
660C22#349012yes—
661C22#349112yes—
662C22#349414yes—
663C22#349614yes—
664C22#350514yes—
665C22#350812yes—
666C22#351012yes—
667C22#353212yes—
668C22#353312yes—
669C22#354214yes—
670C4—24no—
671C4—24no—
672C4—48no—
673C4—84no—
674C4—84no—
675C4—84no—
676C4—84no—
677C4—84no—
678C2—28no—
679C2—44no—
680C2—44no—
681C2—82no—
682C2—82no—
683C2C2×#248311yescharacteristic
684C2#3536012yes—
685C2#3554112yes—
686C2#3673712yes—
687C2#4393114yes—
688C2#4393314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16074C2semidirect
4. #15344C2semidirect
5. #15342C2semidirect
6. #15287C2semidirect
7. #15285C2semidirect
8. #14820C2semidirect
9. #13958C2semidirect
10. #13878C2semidirect
11. #13877C2semidirect
12. #13869C2semidirect
21. C2×#397C22semidirect
22. C2×#397C22semidirect
33. C2×#385C22semidirect
35. C2×#381C22semidirect
41. #1445C22semidirect
42. #1445C22semidirect
43. #1444C22semidirect
44. #1442C22semidirect
45. #1440C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23092
C2×#23082
#160742
#153442
#153422
#152872
#152852
#148202
#139582
#138782
#138772
#138692
C2×#14382