Finite Group Databasebeta

← #6501945

Subgroups — #6501945

Order 512.

2000 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2421C211yesmaximal, characteristic, complemented
2#34205C211yesmaximal, characteristic
3#28118C212yesmaximal, complemented
4#27684C212yesmaximal, complemented
5#27600C211yesmaximal, characteristic, complemented
6#27176C214yesmaximal, complemented
7#26674C214yesmaximal, complemented
8#26570C214yesmaximal, complemented
9#24917C214yesmaximal, complemented
10#24003C211yesmaximal, characteristic, complemented
11#24002C211yesmaximal, characteristic
12#23919C211yesmaximal, characteristic
13#22081C212yesmaximal, complemented
14#21906C211yesmaximal, characteristic, complemented
15#21454C212yesmaximal, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C2×#518—21no—
19C22×C4×Q8C2211yescharacteristic, complemented
20C2×#483C2212yescomplemented
21C2×#480C2212yes—
22C2×#448C2212yes—
23C2×#450C2212yes—
24C2×#440C2214yescomplemented
25#2421—22no—
26#2421—22no—
27#2421C2214yes—
28#2421C2214yescomplemented
29C2×#400—21no—
30C2×#397—21no—
31#2364—22no—
32#2361C2212yes—
33#2361C2212yes—
34#2344C2212yes—
35#2342—22no—
36D4×C22⋊C4C2212yes—
37#2325—21no—
38#2325—21no—
39#2308—21no—
40C4×#121C2212yes—
41C4×#121C2212yes—
42C2×#391C2211yescharacteristic, complemented
43C2×#389C2211yescharacteristic, complemented
44C2×#388—21no—
45C2×#387—21no—
46C2×#382C2211yescharacteristic, complemented
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2214yescomplemented
51#2092C2214yescomplemented
52#2075C2214yes—
53#2068C2214yescomplemented
54#2055C2214yescomplemented
55#2035C2214yes—
56#2033C2214yes—
57#2000C2214yes—
58#1949C2214yescomplemented
59#1943C2214yescomplemented
60#1936C2214yescomplemented
61#1928C2214yescomplemented
62#1924C2214yes—
63#1924C2214yescomplemented
64#1921C2214yescomplemented
65#1880C2214yescomplemented
66#1877C2214yescomplemented
67#1874C2212yes—
68#1874C2212yes—
69#1872C2212yes—
70#1872C2212yescomplemented
71#1864C2214yescomplemented
72#1863C2214yescomplemented
73#1853C2212yescomplemented
74#1852C2212yes—
75#1852C2212yescomplemented
76#1852C2212yescomplemented
77#1823C2212yescomplemented
78#1823C2212yescomplemented
79#1804C2214yescomplemented
80#1797C2212yescomplemented
81#1787C2212yescomplemented
82#1786C2212yes—
83#1779C2211yescharacteristic
84#1779C2211yescharacteristic
85#1763C2214yescomplemented
86#1749C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×SD16C2314yes—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—44no—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×#121—21no—
98C2×#121—21no—
99C2×#121—28no—
100C2×#121C2312yes—
101C2×#121C2312yes—
102#518—48no—
103C2×C4×Q8—22no—
104C2×C4×Q8—44no—
105C2×C4×Q8C2314yes—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—22no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#483—44no—
133#480—28no—
134#450—44no—
135#448—28no—
136#440—48no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2314yes—
154#400—24no—
155#400—24no—
156#397—24no—
157#397—24no—
158#391—22no—
159#391C2314yes—
160#390C2312yes—
161#390C2312yes—
162#389—22no—
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#388—24no—
167#388—24no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#387—22no—
173#387—22no—
174#387—24no—
175#387—24no—
176#386—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#385—22no—
183#385—22no—
184#382—22no—
185#382C2314yes—
186#381—22no—
187#380—21no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#379—22no—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4C2312yes—
203C4×C22⋊C4C2312yes—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25—21no—
222C25C2×D411yescharacteristic
223C22×Q8—24no—
224C22×Q8—21no—
225C22×Q8—24no—
226C22×Q8—42no—
227C22×Q8C2412yes—
228C22×D4—28no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—24no—
232C22×D4—21no—
233C22×D4—42no—
234C22×D4—42no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—24no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×SD16—28no—
261C2×SD16—28no—
262C2×SD16—44no—
263C22×C8C2414yes—
264C22×C8C2414yes—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—42no—
268C22⋊Q8—42no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271#121—28no—
272#121—28no—
273#121—24no—
274#121—24no—
275#121—24no—
276#121#4118yes—
277C4×Q8—28no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—48no—
283C4×D4—48no—
284C4×D4—48no—
285C4×D4—48no—
286#117—48no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—48no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349#107—28no—
350#107—44no—
351#106—44no—
352C4⋊C8—44no—
353#103—28no—
354#103—28no—
355#103—44no—
356#102—28no—
357#102—44no—
358#102—44no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363C22⋊C8—42no—
364C22⋊C8—42no—
365#97—44no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—22no—
384C24—22no—
385C24—24no—
386C24—22no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—42no—
392C24—42no—
393C24—81no—
394C24—81no—
395C24—81no—
396C24—81no—
397C24AGL(1,ℤ/8)14yes—
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C24C22×D411yescharacteristic
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—416no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8H(5,𝔽2)14yes—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—432no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—88no—
423C2×D4C2×#4114yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
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—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—88no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—81no—
485C22×C4—81no—
486C22×C4—81no—
487C22×C4—81no—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4114yes—
496C22×C4C2511yescharacteristic
497SD16—416no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—48no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—81no—
522C4⋊C4—81no—
523C4⋊C4C2×#4114yes—
524C22⋊C4—28no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—28no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C42—24no—
543C42—24no—
544C42—42no—
545C42—42no—
546C42—48no—
547C42—44no—
548C42—44no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—28no—
566C23—24no—
567C23—44no—
568C23—44no—
569C23—48no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—82no—
577C23—82no—
578C23—82no—
579C23—82no—
580C23—84no—
581C23—84no—
582C23—162no—
583C23—162no—
584C23—162no—
585C23—162no—
586C23#52512yes—
587C23C2×AGL(1,ℤ/8)12yes—
588C23C2×AGL(1,ℤ/8)12yes—
589C23C2×AGL(1,ℤ/8)12yes—
590C23#57512yes—
591C23C23×D411yescharacteristic
592Q8—28no—
593Q8—48no—
594Q8—44no—
595D4—28no—
596D4—416no—
597D4—416no—
598D4—44no—
599D4—84no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—24no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—164no—
657C2×C4—164no—
658C2×C4—162no—
659C2×C4—162no—
660C2×C4#52512yes—
661C2×C4C22×#4112yes—
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4#58512yes—
664C8—44no—
665C8—44no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—88no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—168no—
695C22#311414yes—
696C22C2×#52511yescharacteristic
697C22#348411yescharacteristic
698C22#348511yescharacteristic
699C22C22×AGL(1,ℤ/8)11yescharacteristic
700C22C2×#57511yescharacteristic
701C22#358611yescharacteristic
702C22#358611yescharacteristic
703C4—22no—
704C4—42no—
705C4—42no—
706C4—41no—
707C4—41no—
708C4—44no—
709C4—44no—
710C4—42no—
711C4—42no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4#351314yes—
718C2—24no—
719C2—42no—
720C2—42no—
721C2—84no—
722C2#3535411yescharacteristic
723C2C2×#311411yescharacteristic
724C2#6056911yescharacteristic
725C2#6157211yescharacteristic
726C2#6158311yescharacteristic
727C2#6170411yescharacteristic
728C2#6171211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2421C2semidirect
3. #28118C2semidirect
4. #27684C2semidirect
5. #27600C2semidirect
6. #27176C2semidirect
7. #26674C2semidirect
8. #26570C2semidirect
9. #24917C2semidirect
10. #24003C2semidirect
13. #22081C2semidirect
14. #21906C2semidirect
15. #21454C2semidirect
19. C22×C4×Q8C22semidirect
20. C2×#483C22semidirect
24. C2×#440C22semidirect
28. #2421C22semidirect
42. C2×#391C22semidirect
43. C2×#389C22semidirect
46. C2×#382C22semidirect
50. #2101C22semidirect
51. #2092C22semidirect
53. #2068C22semidirect
54. #2055C22semidirect
58. #1949C22semidirect
59. #1943C22semidirect
60. #1936C22semidirect
61. #1928C22semidirect
63. #1924C22semidirect
64. #1921C22semidirect
65. #1880C22semidirect
66. #1877C22semidirect
70. #1872C22semidirect
71. #1864C22semidirect
72. #1863C22semidirect
73. #1853C22semidirect
75. #1852C22semidirect
76. #1852C22semidirect
77. #1823C22semidirect
78. #1823C22semidirect
79. #1804C22semidirect
80. #1797C22semidirect
81. #1787C22semidirect
85. #1763C22semidirect
86. #1749C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24212
#342052
#281182
#276842
#276002
#271762
#266742
#265702
#249172
#240032
#240022
#239192
#220812
#219062
#214542