Finite Group Databasebeta

← #6624501

Subgroups — #6624501

Order 512.

1877 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#33108C212yesmaximal, complemented
4#32905C211yesmaximal, characteristic
5#28302C212yesmaximal
6#28120C212yesmaximal
7#27905C211yesmaximal, characteristic, complemented
8#27647C211yesmaximal, characteristic, complemented
9#27488C214yesmaximal, complemented
10#27192C214yesmaximal, complemented
11#26090C212yesmaximal, complemented
12#24630C212yesmaximal, complemented
13#23094C212yesmaximal, complemented
14#22805C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16C4×#483C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#483C2212yes—
21C2×#483C2212yes—
22C2×#483C2212yes—
23C2×#461C2212yes—
24C2×#450C2211yescharacteristic
25C2×#450C2212yes—
26C2×C4×SD16C2212yes—
27#2857C2211yescharacteristic
28#2857C2212yes—
29#2585—22no—
30#2569—22no—
31#2552C2211yescharacteristic, complemented
32#2546C2211yescharacteristic, complemented
33#2511C2212yescomplemented
34#2509—21no—
35#2446C2212yescomplemented
36#2442—21no—
37#2370—21no—
38#2365C2211yescharacteristic, complemented
39#2355C2211yescharacteristic, complemented
40#2354—22no—
41#2354—21no—
42#2354—21no—
43#2354C2212yes—
44#2354C2212yes—
45#2354C2214yes—
46#2345—21no—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49C2×#390C2212yes—
50C2×#388—22no—
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2096C2214yescomplemented
56#2094C2214yes—
57#2084C2212yescomplemented
58#2072C2212yescomplemented
59#2065C2214yes—
60#2055C2214yes—
61#2013C2214yes—
62#1981C2214yescomplemented
63#1968C2214yes—
64#1968C2214yes—
65#1956C2212yescomplemented
66#1948C2212yescomplemented
67#1948C2214yescomplemented
68#1939C2212yescomplemented
69#1927C2214yescomplemented
70#1927C2214yescomplemented
71#1927C2214yescomplemented
72#1926C2214yes—
73#1920C2214yescomplemented
74#1916C2214yescomplemented
75#1880C2214yes—
76#1877C2212yes—
77#1877C2214yes—
78#1872C2212yes—
79#1872C2211yescharacteristic
80#1864C2212yes—
81#1853C2211yescharacteristic
82#1844C2212yescomplemented
83#1823C2212yes—
84#1803C2212yescomplemented
85#1802C2212yes—
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1786C2211yescharacteristic
89C4×#106C2214yescomplemented
90C8×C4⋊C4C2212yescomplemented
91C4×#103C2214yescomplemented
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#125—21no—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C22⋊Q8C2312yes—
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2312yes—
106C2×C4×D4—28no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C2×#117—24no—
113C2×#117—22no—
114C2×#117—48no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42—22no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#483—28no—
128#483—44no—
129#483—44no—
130#461—28no—
131#450—24no—
132#450—44no—
133C4×SD16—28no—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#400—22no—
156#400—22no—
157#398—22no—
158#398—22no—
159#398—22no—
160#391C2312yes—
161#391C2311yescharacteristic
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2314yes—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389C2312yes—
178#389C2311yescharacteristic
179#388—24no—
180#388—24no—
181#388—24no—
182#388—24no—
183#387—22no—
184#387—22no—
185#387—21no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#385—21no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#384C2312yes—
199#382—21no—
200#380—21no—
201#380—22no—
202#380—22no—
203#380—21no—
204#380—22no—
205#380—22no—
206#380—24no—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C2×#95—22no—
227C2×#95D412yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C25—21no—
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4—24no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—42no—
245C22×D4C2412yes—
246C23×C4—28no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C2×SD16—24no—
259C2×SD16—28no—
260C2×SD16—24no—
261C2×SD16—42no—
262C2×SD16—44no—
263C2×SD16—44no—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269#125—44no—
270C22⋊Q8—22no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—22no—
274C4×Q8—24no—
275C4×Q8—44no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4#4118yes—
282#117—416no—
283#117—48no—
284#117—48no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4SD1618yescomplemented
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—48no—
335C2×C22⋊C4—84no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—44no—
344C2×C42—48no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353#106—28no—
354#106—24no—
355#106—44no—
356#106—44no—
357C4⋊C8—24no—
358C4⋊C8—44no—
359#103—28no—
360#103—24no—
361#103—28no—
362#103—44no—
363#103—44no—
364#102—24no—
365#102—24no—
366#102—42no—
367#102—44no—
368C22⋊C8—24no—
369C22⋊C8—22no—
370C22⋊C8#4114yes—
371C4×C8—24no—
372C4×C8—44no—
373#95—24no—
374#95—24no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—24no—
395C24—22no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24C22×D411yescharacteristic
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4C2×#4114yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—432no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4C2×SD1614yes—
456C22×C4C2×SD1618yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4112yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4C2511yescharacteristic
469SD16—48no—
470SD16—88no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8C2×#4114yes—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4C2×SD1618yes—
513C4⋊C4C2×SD1614yes—
514C4⋊C4C2×SD1614yes—
515C4⋊C4C2×#4114yes—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—48no—
534C42—48no—
535C42—42no—
536C42—82no—
537C42—82no—
538C42C2×#4114yes—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—24no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23#55012yes—
558C23C22×SD1612yes—
559C23#57712yes—
560C23C23×D411yescharacteristic
561Q8—42no—
562Q8—44no—
563Q8—44no—
564Q8—42no—
565D4—24no—
566D4—24no—
567D4—48no—
568D4—42no—
569D4—42no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—416no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—416no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—84no—
622C2×C4—88no—
623C2×C4#52512yes—
624C2×C4#52512yes—
625C2×C4#55012yes—
626C2×C4C22×SD1614yes—
627C2×C4C22×SD1614yes—
628C2×C4C22×SD1612yes—
629C2×C4C22×SD1612yes—
630C2×C4#57512yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4112yes—
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4#58512yes—
635C8—24no—
636C8—42no—
637C8—44no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—816no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22#339214yes—
653C22C2×#52511yescharacteristic
654C22C2×#55011yescharacteristic
655C22#348511yescharacteristic
656C22C23×SD1611yescharacteristic
657C22C2×#57511yescharacteristic
658C22C2×#57711yescharacteristic
659C22#358611yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—24no—
664C4—24no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—41no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4—81no—
673C4—84no—
674C4—82no—
675C4—81no—
676C4#310114yes—
677C4#347512yes—
678C4#349412yes—
679C2—24no—
680C2—84no—
681C2#3651711yescharacteristic
682C2C2×#310111yescharacteristic
683C2#6058511yescharacteristic
684C2C2×#339211yescharacteristic
685C2#6140311yescharacteristic
686C2#6169511yescharacteristic
687C2#6173111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36599C2semidirect
3. #33108C2semidirect
7. #27905C2semidirect
8. #27647C2semidirect
9. #27488C2semidirect
10. #27192C2semidirect
11. #26090C2semidirect
12. #24630C2semidirect
13. #23094C2semidirect
14. #22805C2semidirect
15. #22625C2semidirect
16. C4×#483C2semidirect
31. #2552C22semidirect
32. #2546C22semidirect
33. #2511C22semidirect
35. #2446C22semidirect
38. #2365C22semidirect
39. #2355C22semidirect
55. #2096C22semidirect
57. #2084C22semidirect
58. #2072C22semidirect
62. #1981C22semidirect
65. #1956C22semidirect
66. #1948C22semidirect
67. #1948C22semidirect
68. #1939C22semidirect
69. #1927C22semidirect
70. #1927C22semidirect
71. #1927C22semidirect
73. #1920C22semidirect
74. #1916C22semidirect
82. #1844C22semidirect
84. #1803C22semidirect
89. C4×#106C22semidirect
90. C8×C4⋊C4C22semidirect
91. C4×#103C22semidirect
311. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#365992
C2×#23542
#331082
#329052
#283022
#281202
#279052
#276472
#274882
#271922
#260902
#246302
#230942
#228052
#226252
C4×#4832
C4×#4502