Finite Group Databasebeta

← #6488988

Subgroups — #6488988

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#32956C212yesmaximal, complemented
4#32905C212yesmaximal, complemented
5#30127C214yesmaximal, complemented
6#29121C212yesmaximal, complemented
7#28370C212yesmaximal, complemented
8#28280C212yesmaximal, complemented
9#28252C212yesmaximal, complemented
10#27680C212yesmaximal, complemented
11#24501C214yesmaximal, complemented
12#24211C212yesmaximal
13#22147C214yesmaximal, complemented
14#20808C211yesmaximal, characteristic, complemented
15C22×#127C2211yescharacteristic
16C22×C4×D4—21no—
17C22×C4×D4—21no—
18C22×C4×D4C2212yes—
19C2×#476C2214yescomplemented
20C2×#460C2214yes—
21C2×#461C2214yescomplemented
22C2×#450C2214yes—
23C2×#431C2212yes—
24#2864—21no—
25#2863—21no—
26#2863C2212yescomplemented
27#2862C2211yescharacteristic, complemented
28#2859C2211yescharacteristic, complemented
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2855—21no—
32#2855C2212yes—
33#2851—21no—
34C4×C4⋊Q8C2211yescharacteristic, complemented
35C4×C4⋊Q8C2211yescharacteristic, complemented
36C4×#127—21no—
37C4×#127—21no—
38C4×#127—21no—
39C4×#127—21no—
40C4×#127C2212yes—
41C4×#127C2212yes—
42C4×#127C2212yes—
43C4×#127C2212yes—
44C4×#125—21no—
45C4×C22⋊Q8—21no—
46C4×C22⋊Q8C2212yes—
47#2295—21no—
48#2295C2212yescomplemented
49C2×#390—21no—
50C2×#390—21no—
51C2×#390C2212yes—
52C2×#384C2211yescharacteristic
53#2283—21no—
54C2×C43C2211yescharacteristic
55#2099C2214yescomplemented
56#2096C2214yescomplemented
57#2065C2218yescomplemented
58#2055C2218yescomplemented
59#2041C2214yes—
60#2014C2214yes—
61#1982C2214yes—
62#1968C2218yescomplemented
63#1948C2214yescomplemented
64#1934C2218yescomplemented
65#1877C2218yescomplemented
66#1872C2214yes—
67#1864C2218yescomplemented
68#1853C2214yes—
69#1842C2212yescomplemented
70#1831C2212yescomplemented
71#1803C2214yescomplemented
72#1802C2214yescomplemented
73#1802C2214yescomplemented
74#1797C2212yes—
75#1797C2212yes—
76#1786C2214yes—
77C4×C4⋊C8C2212yescomplemented
78C4×C4⋊C8C2212yescomplemented
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2312yes—
82C24×C4—22no—
83C22×SD16C2318yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127—28no—
87C2×#127—21no—
88C2×#127—21no—
89C2×#127C2312yes—
90C2×#127C2312yes—
91C2×#125—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8—21no—
95C2×C4×Q8C2312yes—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—48no—
107C2×C4×D4—48no—
108C2×C4×D4—48no—
109C2×C4×D4C2314yes—
110C2×C4×D4C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#476—48no—
125#461—48no—
126#460—216no—
127#450—48no—
128#431—44no—
129C2×#106C2318yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2318yes—
134C2×#102C2318yes—
135C2×#102C2318yes—
136C2×#102C2318yes—
137C2×C22⋊C8C2314yes—
138C2×C4×C8C2314yes—
139#391—21no—
140#391—21no—
141#391—21no—
142#391—21no—
143#391C2312yes—
144#391C2312yes—
145#391C2312yes—
146#391C2312yes—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390C2314yes—
158#390C2314yes—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#385—22no—
180#385—22no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#383—22no—
188#382—21no—
189#382—21no—
190#382—21no—
191#381—22no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380C2312yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203#376—22no—
204C43D411yescharacteristic
205C43D411yescharacteristic
206C43D411yescharacteristic
207C43D411yescharacteristic
208C43C2311yescharacteristic
209C43C2311yescharacteristic
210C43C2311yescharacteristic
211C43C2311yescharacteristic
212#340C2318yes—
213#340C2318yes—
214#336C2314yes—
215#336C2314yes—
216C25—22no—
217C22×Q8—21no—
218C22×Q8C2412yes—
219C22×D4—28no—
220C22×D4—28no—
221C22×D4—28no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4—44no—
234C22×D4C2414yes—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—48no—
239C23×C4—48no—
240C23×C4—48no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×SD16—216no—
249C2×SD16—48no—
250C2×SD16—48no—
251C22×C8C2418yes—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254#127—28no—
255#127—28no—
256#127—28no—
257#127—24no—
258#127#4118yes—
259#125—44no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—44no—
263C4×Q8—28no—
264C4×Q8—44no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—44no—
268C4×D4—44no—
269C4×D4—48no—
270C4×D4—48no—
271C4×D4—48no—
272C4×D4—48no—
273C4×D4—48no—
274C4×D4—48no—
275C4×D4—48no—
276C4×D4—48no—
277C4×D4—48no—
278C4×D4—48no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—48no—
313C2×C22⋊C4—48no—
314C2×C22⋊C4—48no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42SD1618yescomplemented
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#106—48no—
354C4⋊C8—28no—
355C4⋊C8—28no—
356C4⋊C8—44no—
357#103—48no—
358#102—216no—
359#102—48no—
360#102—48no—
361C22⋊C8—28no—
362C4×C8—44no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377C24—28no—
378C24—24no—
379C24—44no—
380C24—44no—
381C24—44no—
382C24—44no—
383C24—44no—
384C24—44no—
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—416no—
404C2×D4—416no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—432no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—432no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4C2×#4118yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—832no—
450C22×C4C2×SD1614yes—
451C22×C4C2×SD1614yes—
452C22×C4C2×SD1614yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469SD16—816no—
470C2×C8—28no—
471C2×C8—28no—
472C2×C8—28no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—21no—
519C42—22no—
520C42—21no—
521C42—21no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—44no—
529C42—44no—
530C42C2×SD1614yes—
531C42C2×SD1614yes—
532C42C2×SD1614yes—
533C42C2×SD1614yes—
534C42AGL(1,ℤ/8)14yes—
535C42C2×#4114yes—
536C42H(5,𝔽2)14yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—28no—
542C23—416no—
543C23—416no—
544C23—416no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—88no—
552C23—88no—
553C23—88no—
554C23—88no—
555C23#53712yes—
556C23C22×SD1612yes—
557C23#57512yes—
558C23C23×D411yescharacteristic
559Q8—44no—
560Q8—44no—
561Q8—84no—
562D4—28no—
563D4—28no—
564D4—416no—
565D4—416no—
566D4—416no—
567D4—416no—
568D4—416no—
569D4—44no—
570D4—44no—
571C2×C4—28no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—816no—
619C2×C4—88no—
620C2×C4—816no—
621C2×C4#52512yes—
622C2×C4#52512yes—
623C2×C4#53512yes—
624C2×C4#53512yes—
625C2×C4#53712yes—
626C2×C4C22×SD1612yes—
627C2×C4C22×SD1612yes—
628C2×C4C22×SD1612yes—
629C2×C4C22×SD1612yes—
630C2×C4C22×SD1612yes—
631C2×C4C22×SD1612yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4#57512yes—
636C2×C4C22×#4112yes—
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×H(5,𝔽2)12yes—
639C2×C4C2×2−1+411yescharacteristic
640C8—48no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—48no—
648C22—48no—
649C22—48no—
650C22—832no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22#324614yes—
656C22C2×#52511yescharacteristic
657C22C2×#53511yescharacteristic
658C22C2×#53711yescharacteristic
659C22C23×SD1611yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57511yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—41no—
673C4—41no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#310114yes—
680C4#320814yes—
681C4#351912yes—
682C4#352012yes—
683C2—24no—
684C2—88no—
685C2#3523611yescharacteristic
686C2C2×#310111yescharacteristic
687C2#6055811yescharacteristic
688C2C2×#320811yescharacteristic
689C2#6080511yescharacteristic
690C2C2×#324611yescharacteristic
691C2#6091611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36635C2semidirect
3. #32956C2semidirect
4. #32905C2semidirect
5. #30127C2semidirect
6. #29121C2semidirect
7. #28370C2semidirect
8. #28280C2semidirect
9. #28252C2semidirect
10. #27680C2semidirect
11. #24501C2semidirect
13. #22147C2semidirect
14. #20808C2semidirect
19. C2×#476C22semidirect
21. C2×#461C22semidirect
26. #2863C22semidirect
27. #2862C22semidirect
28. #2859C22semidirect
34. C4×C4⋊Q8C22semidirect
35. C4×C4⋊Q8C22semidirect
48. #2295C22semidirect
55. #2099C22semidirect
56. #2096C22semidirect
57. #2065C22semidirect
58. #2055C22semidirect
62. #1968C22semidirect
63. #1948C22semidirect
64. #1934C22semidirect
65. #1877C22semidirect
67. #1864C22semidirect
69. #1842C22semidirect
70. #1831C22semidirect
71. #1803C22semidirect
72. #1802C22semidirect
73. #1802C22semidirect
77. C4×C4⋊C8C22semidirect
78. C4×C4⋊C8C22semidirect
324. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×C4×#1272
#329562
#329052
#301272
#291212
#283702
#282802
#282522
#276802
#245012
#242112
#221472
#208082