Finite Group Databasebeta

← #6508270

Subgroups — #6508270

Order 512.

1980 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×#2445C211yesmaximal, characteristic
3#29329C211yesmaximal, characteristic, complemented
4#29296C211yesmaximal, characteristic, complemented
5#29032C214yesmaximal, complemented
6#28898C214yesmaximal, complemented
7#26832C214yesmaximal, complemented
8#26260C214yesmaximal, complemented
9#25855C211yesmaximal, characteristic
10#25717C211yesmaximal, characteristic
11#24507C212yesmaximal, complemented
12#24507C212yesmaximal, complemented
13#22049C212yesmaximal, complemented
14#21453C212yesmaximal, complemented
15#20534C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#123—21no—
18C22×C4×D4C2211yescharacteristic
19C2×#492C2212yes—
20C2×#486C2212yes—
21C2×#450C2212yes—
22C2×#450C2212yes—
23C2×#440C2214yescomplemented
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2646—21no—
29#2627—21no—
30#2590—22no—
31#2588—22no—
32#2511C2212yescomplemented
33#2507C2212yescomplemented
34#2451C2212yescomplemented
35#2446C2212yescomplemented
36#2445—22no—
37#2445—22no—
38#2445C2214yes—
39#2445C2214yes—
40C2×#394—21no—
41C2×#394—21no—
42#2329—21no—
43#2313—21no—
44C4×C22⋊Q8C2212yes—
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#386—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2100C2212yescomplemented
52#2083C2212yescomplemented
53#2067C2214yescomplemented
54#2067C2214yescomplemented
55#2057C2212yescomplemented
56#2055C2214yescomplemented
57#2055C2214yescomplemented
58#2043C2212yescomplemented
59#2041C2214yes—
60#2014C2214yes—
61#1973C2214yescomplemented
62#1973C2214yescomplemented
63#1973C2214yescomplemented
64#1957C2212yes—
65#1957C2212yes—
66#1942C2214yescomplemented
67#1936C2214yescomplemented
68#1936C2214yescomplemented
69#1936C2214yescomplemented
70#1933C2214yescomplemented
71#1925C2214yescomplemented
72#1880C2214yescomplemented
73#1877C2214yescomplemented
74#1872C2212yes—
75#1872C2212yes—
76#1862C2214yescomplemented
77#1852C2212yes—
78#1852C2212yes—
79#1786C2212yes—
80#1786C2212yes—
81#1771C2214yescomplemented
82#1771C2214yescomplemented
83#1762C2214yescomplemented
84#1749C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×D4—21no—
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C22×SD16C2314yes—
93C22×SD16C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127—21no—
97C2×#127—21no—
98C2×#127—44no—
99C2×#127C2312yes—
100C2×#127C2312yes—
101C2×#123—24no—
102C2×#123—24no—
103C2×#123—48no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4—44no—
109C2×C4×D4C2314yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#492—44no—
126#486—44no—
127#450—28no—
128#450—28no—
129#440—48no—
130#432—28no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×#97C2314yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#400—22no—
150#400—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#395—22no—
156#394—24no—
157#394—24no—
158#394—24no—
159#394—24no—
160#391—21no—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#390—22no—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2314yes—
171#390C2314yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#386—24no—
194#386—24no—
195#385—22no—
196#385—22no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#381—22no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380C2312yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25—21no—
222C25—21no—
223C22×Q8—21no—
224C22×Q8C2412yes—
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—24no—
233C22×D4—24no—
234C22×D4—42no—
235C22×D4—42no—
236C22×D4—42no—
237C22×D4—44no—
238C22×D4—44no—
239C22×D4—44no—
240C22×D4—44no—
241C22×D4—82no—
242C22×D4—82no—
243C22×D4C2412yes—
244C22×D4C2412yes—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—24no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—82no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×SD16—28no—
259C2×SD16—28no—
260C2×SD16—28no—
261C2×SD16—44no—
262C2×SD16—44no—
263C2×SD16—44no—
264C22×C8C2414yes—
265C22×C8C2414yes—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268#127—24no—
269#127—24no—
270#127—42no—
271#127—42no—
272#127—44no—
273#127—44no—
274#123—416no—
275#123—416no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—44no—
279C4×Q8—44no—
280C4×D4—28no—
281C4×D4—44no—
282C4×D4—48no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4—82no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343#107—44no—
344#107—44no—
345C4⋊C8—28no—
346#103—44no—
347#103—44no—
348#102—28no—
349#102—28no—
350#102—28no—
351#102—44no—
352#102—44no—
353#102—44no—
354C22⋊C8—24no—
355C22⋊C8#4118yes—
356#97—28no—
357#97—44no—
358C4×C8—42no—
359C4×C8—42no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C22×D411yescharacteristic
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—42no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—416no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—88no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—816no—
432C2×D4—816no—
433C2×D4H(5,𝔽2)14yes—
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—48no—
459C22×C4—48no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—84no—
467C22×C4—816no—
468C22×C4AGL(1,ℤ/8)14yes—
469C22×C4#13714yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4114yes—
480C22×C4C2511yescharacteristic
481SD16—416no—
482SD16—88no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8C2×#4118yes—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—24no—
532C42—42no—
533C42—41no—
534C42—41no—
535C42—82no—
536C42—82no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—24no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23#53212yes—
570C23C2×AGL(1,ℤ/8)12yes—
571C23C2×#13712yes—
572C23C23×D411yescharacteristic
573Q8—44no—
574Q8—44no—
575Q8—162no—
576D4—28no—
577D4—48no—
578D4—44no—
579D4—44no—
580D4—44no—
581D4—88no—
582D4—88no—
583D4—88no—
584D4—88no—
585D4—84no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—816no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4#53212yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4C2×AGL(1,ℤ/8)12yes—
639C2×C4C2×#13712yes—
640C2×C4C2×#13712yes—
641C2×C4C22×#4112yes—
642C2×C4C2×H(5,𝔽2)12yes—
643C2×C4#58512yes—
644C8—28no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—816no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—168no—
668C22#318214yes—
669C22C2×#53211yescharacteristic
670C22#348411yescharacteristic
671C22#348711yescharacteristic
672C22C22×AGL(1,ℤ/8)11yescharacteristic
673C22C22×#13711yescharacteristic
674C22#358511yescharacteristic
675C22#358511yescharacteristic
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4#351314yes—
692C2—24no—
693C2—84no—
694C2—162no—
695C2#3539611yescharacteristic
696C2C2×#318211yescharacteristic
697C2#6070411yescharacteristic
698C2#6157011yescharacteristic
699C2#6157111yescharacteristic
700C2#6189011yescharacteristic
701C2#6189211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #35015C2semidirect
3. #29329C2semidirect
4. #29296C2semidirect
5. #29032C2semidirect
6. #28898C2semidirect
7. #26832C2semidirect
8. #26260C2semidirect
11. #24507C2semidirect
12. #24507C2semidirect
13. #22049C2semidirect
14. #21453C2semidirect
15. #20534C2semidirect
23. C2×#440C22semidirect
32. #2511C22semidirect
33. #2507C22semidirect
34. #2451C22semidirect
35. #2446C22semidirect
51. #2100C22semidirect
52. #2083C22semidirect
53. #2067C22semidirect
54. #2067C22semidirect
55. #2057C22semidirect
56. #2055C22semidirect
57. #2055C22semidirect
58. #2043C22semidirect
61. #1973C22semidirect
62. #1973C22semidirect
63. #1973C22semidirect
66. #1942C22semidirect
67. #1936C22semidirect
68. #1936C22semidirect
69. #1936C22semidirect
70. #1933C22semidirect
71. #1925C22semidirect
72. #1880C22semidirect
73. #1877C22semidirect
76. #1862C22semidirect
81. #1771C22semidirect
82. #1771C22semidirect
83. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#24452
#293292
#292962
#290322
#288982
#268322
#262602
#258552
#257172
#245072
#220492
#214532
#205342