Finite Group Databasebeta

← #6561792

Subgroups — #6561792

Order 512.

1841 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36274C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#29922C211yesmaximal, characteristic, complemented
4#29908C212yesmaximal, complemented
5#27972C211yesmaximal, characteristic, complemented
6#27855C212yesmaximal, complemented
7#27647C211yesmaximal, characteristic, complemented
8#27583C212yesmaximal, complemented
9#26320C212yesmaximal, complemented
10#26200C214yesmaximal, complemented
11#24994C212yesmaximal, complemented
12#24863C214yesmaximal, complemented
13#23756C211yesmaximal, characteristic, complemented
14C2×#2072C212yesmaximal, complemented
15#22703C212yesmaximal, complemented
16#22469C211yesmaximal, characteristic
17D4×#103C212yesmaximal, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#483C2212yes—
22C2×C8×D4C2212yes—
23C22×#103C2214yes—
24#2857C2211yescharacteristic
25#2855C2212yes—
26#2582C2211yescharacteristic
27#2568C2211yescharacteristic
28#2451—21no—
29#2450—22no—
30#2439—21no—
31#2438—22no—
32#2417C2212yes—
33#2409C2212yes—
34Q8×C4⋊C4—21no—
35Q8×C4⋊C4C2212yes—
36#2349—21no—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—22no—
39D4×C4⋊C4C2214yes—
40D4×C4⋊C4C2212yes—
41D4×C4⋊C4C2211yescharacteristic
42#2339—21no—
43#2317—22no—
44#2317—21no—
45C4×C22⋊Q8C2211yescharacteristic
46C2×#391C2212yes—
47#2285C2212yes—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51#2099C2214yes—
52#2099C2211yescharacteristic
53#2096C2214yes—
54#2096C2211yescharacteristic
55#2096C2212yes—
56#2084C2211yescharacteristic
57#2084C2212yes—
58#2077C2212yes—
59#2072C2214yes—
60#2072C2212yes—
61#2072C2212yes—
62#2072C2211yescharacteristic
63#2072C2214yes—
64#2072C2212yes—
65#2072C2212yes—
66#2035C2212yes—
67#2035C2214yes—
68#1984C2212yes—
69#1981C2212yes—
70#1956C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1927C2214yes—
74#1927C2214yes—
75#1927C2214yes—
76#1920C2214yes—
77#1892C2214yes—
78#1872C2214yes—
79#1872C2214yes—
80#1869C2214yes—
81#1853C2214yes—
82#1853C2214yes—
83#1834C2212yes—
84#1823C2212yes—
85#1817C2212yes—
86#1807C2214yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1787C2214yes—
90C4×#106C2212yes—
91C4×#103C2212yes—
92C4×C22⋊C8C2211yescharacteristic
93C2×#340C2214yes—
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8C2312yes—
97C23×C8C2314yes—
98D4×Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C22⋊Q8C2312yes—
105#520—48no—
106C4×#41—28no—
107C2×C4×Q8—22no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4—22no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D412yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C2×#117—21no—
121C2×#117D412yes—
122C2×#117D411yescharacteristic
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#483—44no—
133C8×D4—28no—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#103—48no—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C4×C8C2312yes—
151#400—22no—
152#398—22no—
153#398—22no—
154#396—22no—
155#395—22no—
156#393—22no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2311yescharacteristic
166#389—22no—
167#389—24no—
168#389—24no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389C2312yes—
175#389C2311yescharacteristic
176#387—22no—
177#387—22no—
178#387—21no—
179#386—22no—
180#386—22no—
181#385—22no—
182#385—21no—
183#385—24no—
184#385—24no—
185#385—24no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—22no—
190#384—21no—
191#384—22no—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2314yes—
197#384C2311yescharacteristic
198#382—21no—
199#382—21no—
200#382—22no—
201#381—22no—
202#380—21no—
203#380—21no—
204#380—22no—
205#380—21no—
206#380—22no—
207#380C2312yes—
208#380C2314yes—
209#380C2311yescharacteristic
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41SD1618yes—
237C22×Q8—22no—
238C22×Q8—24no—
239C22×Q8—28no—
240C22×Q8—42no—
241C22×Q8—44no—
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2412yes—
244C22×Q8C2411yescharacteristic
245C22×D4C2×D412yes—
246C22×D4C2×D411yescharacteristic
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2412yes—
257C22×C8—28no—
258C22×C8—44no—
259C22×C8—44no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C22⋊Q8—22no—
266C22⋊Q8—28no—
267C22⋊Q8—41no—
268C22⋊Q8—48no—
269C22⋊Q8—44no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—48no—
274C4×D4—24no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—44no—
278C4×D4—416no—
279C4×D4—48no—
280C4×D4—44no—
281C4×D4#4118yes—
282#117—24no—
283#117—28no—
284#117—48no—
285#117—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4—84no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—28no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#106—42no—
353#106—44no—
354#106—42no—
355#106—44no—
356C4⋊C8—24no—
357#103—24no—
358#103—28no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—42no—
363#103—44no—
364C22⋊C8—28no—
365C22⋊C8—42no—
366C22⋊C8—42no—
367C4×C8—24no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387C24C22×D411yescharacteristic
388C24C22×D412yes—
389#41—232no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—44no—
400C2×Q8—416no—
401C2×Q8—44no—
402C2×Q8—82no—
403C2×Q8—88no—
404C2×Q8C2×SD1614yes—
405C2×Q8H(5,𝔽2)12yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4#12314yes—
409C2×D4C2×SD1618yes—
410C2×D4C2×SD1614yes—
411C2×D4#13714yes—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—82no—
444C22×C4—84no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—84no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4C2×SD1618yes—
451C22×C4C2×SD1614yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4118yes—
461C22×C4C2511yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—88no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—84no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—84no—
508C4⋊C4—82no—
509C4⋊C4#12314yes—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—81no—
518C22⋊C4—82no—
519C22⋊C4#12318yes—
520C42—24no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—42no—
527C42—48no—
528C42—44no—
529C42—42no—
530C42C2×#4114yes—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23C2×#12314yes—
542C23C22×SD1612yes—
543C23C22×SD1614yes—
544C23C2×#13714yes—
545C23C23×D411yescharacteristic
546Q8—28no—
547Q8—24no—
548Q8—42no—
549Q8—44no—
550Q8—44no—
551Q8—88no—
552D4—216no—
553D4#483116yescomplemented
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—216no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—88no—
606C2×C4—164no—
607C2×C4—168no—
608C2×C4—164no—
609C2×C4C2×#12314yes—
610C2×C4C2×#12312yes—
611C2×C4C2×#12312yes—
612C2×C4C22×SD1614yes—
613C2×C4C22×SD1612yes—
614C2×C4C22×SD1612yes—
615C2×C4C2×#13712yes—
616C2×C4#57812yes—
617C2×C4C22×#4112yes—
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4#58512yes—
620C2×C4#58511yescharacteristic
621C8—44no—
622C8—42no—
623C8—42no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×#48318yes—
638C22C22×#12311yescharacteristic
639C22#348811yescharacteristic
640C22#350011yescharacteristic
641C22C23×SD1611yescharacteristic
642C22C22×#13711yescharacteristic
643C22C2×#57811yescharacteristic
644C22#358511yescharacteristic
645C4—22no—
646C4—24no—
647C4—22no—
648C4—22no—
649C4—24no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—81no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4—82no—
659C4—82no—
660C4—84no—
661C4C2×#48314yes—
662C4#351312yes—
663C4#352112yes—
664C2—28no—
665C2—42no—
666C2#3597611yescharacteristic
667C2C22×#48311yescharacteristic
668C2#6052611yescharacteristic
669C2#6195111yescharacteristic
670C2#6198411yescharacteristic
671C2#6252611yescharacteristic
672C2#6258211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29922C2semidirect
4. #29908C2semidirect
5. #27972C2semidirect
6. #27855C2semidirect
7. #27647C2semidirect
8. #27583C2semidirect
9. #26320C2semidirect
10. #26200C2semidirect
11. #24994C2semidirect
12. #24863C2semidirect
13. #23756C2semidirect
14. C2×#2072C2semidirect
15. #22703C2semidirect
17. D4×#103C2semidirect
553. D4#483semidirect

Maximal subgroups

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

subgroupindex
#362742
#341832
#299222
#299082
#279722
#278552
#276472
#275832
#263202
#262002
#249942
#248632
#237562
C2×#20722
#227032
#224692
D4×#1032