Finite Group Databasebeta

← #6563533

Subgroups — #6563533

Order 512.

1745 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36405C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#29936C211yesmaximal, characteristic, complemented
4#29909C212yesmaximal, complemented
5#27950C211yesmaximal, characteristic, complemented
6#27861C212yesmaximal, complemented
7#27647C211yesmaximal, characteristic, complemented
8#27615C212yesmaximal, complemented
9#26278C212yesmaximal, complemented
10#26144C214yesmaximal, complemented
11#25062C212yesmaximal, complemented
12#24951C214yesmaximal, complemented
13#23756C211yesmaximal, characteristic, complemented
14#23658C212yesmaximal, complemented
15#22665C212yesmaximal, complemented
16#22504C212yesmaximal, complemented
17#22473C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#483C2212yes—
22C2×#436C2212yes—
23C2×#428C2214yes—
24#2857C2211yescharacteristic
25#2855C2212yes—
26#2582C2211yescharacteristic
27#2546C2211yescharacteristic
28#2485C2212yes—
29#2478—22no—
30#2468—22no—
31#2447—21no—
32#2432—21no—
33#2415C2212yes—
34#2365—21no—
35#2362—21no—
36#2354C2211yescharacteristic
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—22no—
41D4×C4⋊C4C2214yes—
42D4×C4⋊C4C2212yes—
43#2317—21no—
44#2317—22no—
45C4×C22⋊Q8C2211yescharacteristic
46#2296C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51#2100C2214yes—
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2096C2211yescharacteristic
55#2095C2214yes—
56#2084C2214yes—
57#2084C2211yescharacteristic
58#2084C2212yes—
59#2083C2212yes—
60#2083C2212yes—
61#2083C2214yes—
62#2083C2212yes—
63#2077C2212yes—
64#2075C2214yes—
65#2072C2211yescharacteristic
66#2072C2212yes—
67#2035C2212yes—
68#1990C2212yes—
69#1988C2212yes—
70#1984C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1928C2214yes—
74#1921C2214yes—
75#1919C2214yes—
76#1919C2214yes—
77#1894C2214yes—
78#1872C2214yes—
79#1870C2214yes—
80#1854C2214yes—
81#1852C2214yes—
82#1852C2214yes—
83#1845C2212yes—
84#1825C2214yes—
85#1823C2212yes—
86#1817C2212yes—
87#1809C2214yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91C4×#103C2212yes—
92C4×C22⋊C8C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C22×M16C2314yes—
97#556—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#126—22no—
101C2×#124—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2311yescharacteristic
105#520—48no—
106C4×#41—28no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—24no—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D412yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D412yes—
122C2×#117C2312yes—
123C2×#117C2312yes—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#483—44no—
133#436—28no—
134#428—48no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C4×C8C2312yes—
149#398—22no—
150#398—22no—
151#397—22no—
152#397—22no—
153#394—22no—
154#394—22no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391C2312yes—
159#391C2311yescharacteristic
160#390—21no—
161#389—22no—
162#389—24no—
163#389—24no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—21no—
168#389C2312yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—21no—
180#385—22no—
181#385—22no—
182#385—22no—
183#385—21no—
184#385—24no—
185#385—24no—
186#385—24no—
187#384—22no—
188#384—21no—
189#384—22no—
190#384—21no—
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2314yes—
196#384C2311yescharacteristic
197#382—22no—
198#382—21no—
199#381—22no—
200#380—21no—
201#380—21no—
202#380—22no—
203#380C2312yes—
204#380C2314yes—
205#380C2312yes—
206#380C2312yes—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—24no—
237C22×Q8—21no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×D4C2×D412yes—
241C22×D4C2×D411yescharacteristic
242C23×C4—24no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C2×M16—28no—
251C2×M16—28no—
252C2×M16—44no—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258#126—48no—
259#124—44no—
260C22⋊Q8—28no—
261C22⋊Q8—22no—
262C22⋊Q8—41no—
263C22⋊Q8—44no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—48no—
268C4×Q8—44no—
269C4×D4—24no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—416no—
273C4×D4—48no—
274C4×D4#4118yes—
275#117—24no—
276#117—28no—
277#117—28no—
278#117—24no—
279#117—42no—
280#117—48no—
281#117—44no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—28no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—44no—
353#107—44no—
354#107—44no—
355#106—24no—
356#106—24no—
357#106—28no—
358C4⋊C8—24no—
359#103—24no—
360#103—42no—
361#103—44no—
362C22⋊C8—28no—
363C22⋊C8—24no—
364C22⋊C8—42no—
365C4×C8—24no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385#41—232no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8#13714yes—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4#12314yes—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—82no—
432C22×C4—84no—
433C22×C4—82no—
434C22×C4—84no—
435C22×C4AGL(1,ℤ/8)14yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4118yes—
445C22×C4C2511yescharacteristic
446M16—416no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C82−1+414yes—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—42no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—84no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4#12314yes—
495C4⋊C4H(5,𝔽2)12yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—81no—
506C22⋊C4—82no—
507C22⋊C4#12318yes—
508C42—24no—
509C42—22no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—44no—
515C42—48no—
516C42—44no—
517C42—42no—
518C42—44no—
519C42—42no—
520C42C2×#4114yes—
521C23—21no—
522C23—22no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23C2×#12314yes—
534C23C2×AGL(1,ℤ/8)12yes—
535C23C23×D411yescharacteristic
536Q8—28no—
537Q8—42no—
538Q8—42no—
539Q8—84no—
540D4—216no—
541D4—28no—
542C2×C4—216no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—164no—
594C2×C4—168no—
595C2×C4C2×#12314yes—
596C2×C4C2×#12312yes—
597C2×C4C2×#12312yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×#13714yes—
601C2×C4C2×#13712yes—
602C2×C4#57812yes—
603C2×C4C22×#4112yes—
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+412yes—
606C2×C4#58511yescharacteristic
607C8—24no—
608C8—42no—
609C8—44no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—22no—
614C22—24no—
615C22—24no—
616C22—42no—
617C22—42no—
618C22—41no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—41no—
623C22—41no—
624C22C22×#12311yescharacteristic
625C22#348811yescharacteristic
626C22#349911yescharacteristic
627C22C22×AGL(1,ℤ/8)11yescharacteristic
628C22C22×#13711yescharacteristic
629C22C2×#57811yescharacteristic
630C22#358511yescharacteristic
631C4—22no—
632C4—24no—
633C4—22no—
634C4—22no—
635C4—24no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—82no—
640C4—84no—
641C4—82no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4—81no—
647C4#309214yes—
648C4#351512yes—
649C4#352012yes—
650C2—42no—
651C2—44no—
652C2#3599811yescharacteristic
653C2C2×#309211yescharacteristic
654C2#6052611yescharacteristic
655C2#6196311yescharacteristic
656C2#6200011yescharacteristic
657C2#6247911yescharacteristic
658C2#6251011yescharacteristic
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. #29936C2semidirect
4. #29909C2semidirect
5. #27950C2semidirect
6. #27861C2semidirect
7. #27647C2semidirect
8. #27615C2semidirect
9. #26278C2semidirect
10. #26144C2semidirect
11. #25062C2semidirect
12. #24951C2semidirect
13. #23756C2semidirect
14. #23658C2semidirect
15. #22665C2semidirect
16. #22504C2semidirect

Maximal subgroups

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

subgroupindex
#364052
#341832
#299362
#299092
#279502
#278612
#276472
#276152
#262782
#261442
#250622
#249512
#237562
#236582
#226652
#225042
#224732