Finite Group Databasebeta

← #6470545

Subgroups — #6470545

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#2352C211yesmaximal, characteristic
3#33100C212yesmaximal, complemented
4#32983C212yesmaximal, complemented
5#29913C212yesmaximal, complemented
6#29893C212yesmaximal, complemented
7#26217C214yesmaximal, complemented
8#26145C214yesmaximal, complemented
9#24952C214yesmaximal, complemented
10#24880C214yesmaximal, complemented
11#21942C211yesmaximal, characteristic, complemented
12#21765C211yesmaximal, characteristic, complemented
13#21742C211yesmaximal, characteristic
14#20911C211yesmaximal, characteristic, complemented
15#20846C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#484C2214yescomplemented
21C2×#428C2212yescomplemented
22C2×#419C2212yescomplemented
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25#2855C2212yes—
26#2848C2212yes—
27#2627—21no—
28#2626—21no—
29#2579—22no—
30#2564—22no—
31#2487C2212yes—
32#2417C2212yes—
33#2417C2212yes—
34#2409C2212yes—
35#2352—22no—
36#2352—22no—
37#2352C2214yes—
38#2352C2214yes—
39#2315—21no—
40#2308—21no—
41C2×#388—21no—
42C2×#387—21no—
43C2×#386—21no—
44C2×#382C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50#2101C2214yescomplemented
51#2097C2214yescomplemented
52#2085C2214yescomplemented
53#2077C2214yescomplemented
54#2077C2214yescomplemented
55#2074C2214yescomplemented
56#2035C2214yescomplemented
57#2035C2214yescomplemented
58#1928C2214yescomplemented
59#1927C2214yescomplemented
60#1919C2214yescomplemented
61#1918C2214yescomplemented
62#1894C2214yescomplemented
63#1873C2214yescomplemented
64#1872C2212yes—
65#1872C2212yes—
66#1872C2212yes—
67#1872C2212yes—
68#1872C2214yescomplemented
69#1870C2214yes—
70#1870C2214yes—
71#1854C2214yescomplemented
72#1853C2212yes—
73#1853C2212yes—
74#1853C2212yescomplemented
75#1852C2212yes—
76#1852C2212yes—
77#1852C2212yescomplemented
78#1852C2214yescomplemented
79#1824C2214yescomplemented
80#1823C2212yescomplemented
81#1817C2212yes—
82#1809C2214yescomplemented
83#1787C2212yescomplemented
84#1779C2212yes—
85#1736C2212yescomplemented
86#1736C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4D411yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×M16C2312yes—
93C22×M16C2312yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C2×#117—21no—
110C2×#117—44no—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—44no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4D411yescharacteristic
132C22×C22⋊C4D411yescharacteristic
133C22×C42—21no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136#484—48no—
137#428—44no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142#419—44no—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147#407—24no—
148#407—24no—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155#400—22no—
156#399—22no—
157#398—22no—
158#398—22no—
159#396—22no—
160#396—22no—
161#395—22no—
162#393—22no—
163#391C2312yes—
164#391C2312yes—
165#389C2312yes—
166#389C2312yes—
167#388—24no—
168#388—24no—
169#387—24no—
170#387—24no—
171#387—22no—
172#387—22no—
173#386—22no—
174#386—22no—
175#386—22no—
176#386—22no—
177#386—24no—
178#386—24no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#382—22no—
186#382C2314yes—
187#381—22no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—22no—
192#380—21no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198#380C2312yes—
199#380C2314yes—
200#380C2312yes—
201#380C2312yes—
202#379—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4C2314yes—
205C4×C22⋊C4—24no—
206C4×C22⋊C4—24no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95—21no—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25C2×D411yescharacteristic
224C25C2×D411yescharacteristic
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8C2412yes—
231C22×D4—24no—
232C22×D4—24no—
233C22×D4—42no—
234C22×D4C2×D412yes—
235C22×D4C2×D412yes—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—24no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—42no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4—42no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—82no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4#4114yes—
262C23×C4#4114yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—42no—
270C2×M16—42no—
271C2×M16—42no—
272C2×M16—42no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C4×D4—28no—
284C4×D4—28no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289#117—24no—
290#117—24no—
291#117—42no—
292#117—42no—
293#117—44no—
294#117—44no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—82no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—24no—
361C2×C42—44no—
362C2×C42—82no—
363C2×C42C2×D412yes—
364C2×C42C2412yes—
365#107—28no—
366#107—44no—
367#106—44no—
368#106—44no—
369#103—28no—
370#103—44no—
371#103—44no—
372#103—44no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377C22⋊C8—44no—
378C22⋊C8—44no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—24no—
385#95—22no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—41no—
400C24—41no—
401C24—41no—
402C24—41no—
403C24—41no—
404C24—41no—
405C24—41no—
406C24—41no—
407C24#12314yes—
408C24#13714yes—
409C24C22×D411yescharacteristic
410C24C22×D411yescharacteristic
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—48no—
418C2×Q8—42no—
419C2×Q8—81no—
420C2×Q8—81no—
421C2×Q8—84no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—416no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—42no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
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—48no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—84no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—816no—
497C22×C4#12314yes—
498C22×C4#12314yes—
499C22×C4#12314yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2511yescharacteristic
513M16—48no—
514M16—84no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—81no—
543C4⋊C4—81no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4H(5,𝔽2)14yes—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—48no—
557C22⋊C4—44no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C42—24no—
565C42—22no—
566C42—22no—
567C42—44no—
568C42—44no—
569C42—84no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—22no—
578C23—21no—
579C23—21no—
580C23—28no—
581C23—22no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—22no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—22no—
593C23—28no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—82no—
603C23—82no—
604C23—82no—
605C23—82no—
606C23C2×#12312yes—
607C23C2×#12312yes—
608C23C2×#12312yes—
609C23C2×#13714yes—
610C23C2×#13712yes—
611C23#57512yes—
612C23C23×D411yescharacteristic
613Q8—44no—
614Q8—44no—
615Q8—84no—
616D4—48no—
617D4—48no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—48no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—48no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—816no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4—162no—
680C2×C4—162no—
681C2×C4C2×#12312yes—
682C2×C4C2×#12312yes—
683C2×C4C2×#12312yes—
684C2×C4C22×#4111yescharacteristic
685C2×C4C22×#4111yescharacteristic
686C2×C4C2×H(5,𝔽2)12yes—
687C2×C4#58512yes—
688C8—44no—
689C8—44no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—24no—
697C22—22no—
698C22—22no—
699C22—24no—
700C22—22no—
701C22—22no—
702C22—24no—
703C22—22no—
704C22—22no—
705C22—416no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—42no—
710C22—41no—
711C22—42no—
712C22—41no—
713C22—41no—
714C22—42no—
715C22—41no—
716C22—41no—
717C22—42no—
718C22—88no—
719C22#309514yes—
720C22C22×#12311yescharacteristic
721C22#348411yescharacteristic
722C22#348511yescharacteristic
723C22C22×#13711yescharacteristic
724C22C2×#57511yescharacteristic
725C22#358511yescharacteristic
726C22#358611yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—42no—
731C4—42no—
732C4—44no—
733C4—42no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4#345912yes—
742C4#350512yes—
743C2—24no—
744C2—44no—
745C2—42no—
746C2—42no—
747C2#3498311yescharacteristic
748C2C2×#309511yescharacteristic
749C2#6051711yescharacteristic
750C2#6157311yescharacteristic
751C2#6158111yescharacteristic
752C2#6170811yescharacteristic
753C2#6171611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33100C2semidirect
4. #32983C2semidirect
5. #29913C2semidirect
6. #29893C2semidirect
7. #26217C2semidirect
8. #26145C2semidirect
9. #24952C2semidirect
10. #24880C2semidirect
11. #21942C2semidirect
12. #21765C2semidirect
14. #20911C2semidirect
15. #20846C2semidirect
20. C2×#484C22semidirect
21. C2×#428C22semidirect
22. C2×#419C22semidirect
50. #2101C22semidirect
51. #2097C22semidirect
52. #2085C22semidirect
53. #2077C22semidirect
54. #2077C22semidirect
55. #2074C22semidirect
56. #2035C22semidirect
57. #2035C22semidirect
58. #1928C22semidirect
59. #1927C22semidirect
60. #1919C22semidirect
61. #1918C22semidirect
62. #1894C22semidirect
63. #1873C22semidirect
68. #1872C22semidirect
71. #1854C22semidirect
74. #1853C22semidirect
77. #1852C22semidirect
78. #1852C22semidirect
79. #1824C22semidirect
80. #1823C22semidirect
82. #1809C22semidirect
83. #1787C22semidirect
85. #1736C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#23522
#331002
#329832
#299132
#298932
#262172
#261452
#249522
#248802
#219422
#217652
#217422
#209112
#208462