Finite Group Databasebeta

← #6795029

Subgroups — #6795029

Order 512.

1699 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36193C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3#32956C211yesmaximal, characteristic
4#32455C214yesmaximal, complemented
5#32373C214yesmaximal, complemented
6#29913C212yesmaximal, complemented
7#28539C212yesmaximal, complemented
8#27735C212yesmaximal, complemented
9#27582C212yesmaximal
10#25813C211yesmaximal, characteristic, complemented
11#25450C212yesmaximal, complemented
12#24679C212yesmaximal, complemented
13#24507C212yesmaximal
14#24302C211yesmaximal, characteristic, complemented
15#22603C211yesmaximal, characteristic, complemented
16#21986C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#500C2212yescomplemented
21C2×#477C2212yescomplemented
22C2×#476C2212yes—
23C2×#450C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#432C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2567C2211yescharacteristic, complemented
29#2563—21no—
30#2563—21no—
31#2563—22no—
32#2563C2212yes—
33#2563C2212yes—
34#2563C2214yes—
35#2552C2211yescharacteristic, complemented
36#2520—22no—
37#2511C2212yescomplemented
38#2508—21no—
39#2507C2212yescomplemented
40#2442—21no—
41#2438—22no—
42C2×#393—22no—
43#2370—21no—
44#2369C2211yescharacteristic, complemented
45#2355C2211yescharacteristic, complemented
46#2344—21no—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#390C2212yes—
49#2285C2212yes—
50C2×#382C2212yes—
51C2×#380—21no—
52C2×#380—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yescomplemented
55#2097C2214yescomplemented
56#2093C2214yes—
57#2085C2214yescomplemented
58#2074C2214yescomplemented
59#2071C2214yes—
60#2067C2214yes—
61#2057C2212yescomplemented
62#2057C2214yescomplemented
63#2055C2214yes—
64#2024C2212yescomplemented
65#1990C2212yescomplemented
66#1987C2212yescomplemented
67#1973C2214yes—
68#1949C2214yescomplemented
69#1949C2214yescomplemented
70#1939C2212yescomplemented
71#1936C2214yes—
72#1925C2214yes—
73#1925C2214yes—
74#1917C2214yescomplemented
75#1877C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1864C2212yes—
79#1864C2214yes—
80#1853C2212yes—
81#1853C2211yescharacteristic
82#1852C2212yes—
83#1852C2212yes—
84#1852C2212yes—
85#1847C2214yescomplemented
86#1836C2214yescomplemented
87#1817C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90#1786C2212yes—
91#1786C2211yescharacteristic
92#1770C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—44no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—22no—
122C22×C22⋊C4—22no—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#500—44no—
126#477—44no—
127#476—44no—
128#450—24no—
129C4×SD16—44no—
130#432—28no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2311yescharacteristic
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#400—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#395—22no—
156#393—24no—
157#393—24no—
158#393—24no—
159#393—24no—
160#391—21no—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#391C2311yescharacteristic
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2314yes—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389C2312yes—
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2312yes—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—21no—
184#387—22no—
185#387—22no—
186#385—22no—
187#385—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#382—21no—
192#382—22no—
193#382—22no—
194#382—22no—
195#382C2314yes—
196#382C2314yes—
197#381—22no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—24no—
204#380—22no—
205#380—24no—
206#380C2312yes—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C25—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—82no—
242C22×Q8—82no—
243C22×Q8C2411yescharacteristic
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—42no—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274#125—44no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277C22⋊Q8—82no—
278C4×Q8—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—44no—
284C4×D4—44no—
285#117—84no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—84no—
332C2×C22⋊C4—84no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348#107—44no—
349#107—44no—
350#107—44no—
351#106—42no—
352#106—44no—
353C4⋊C8—28no—
354C4⋊C8—24no—
355#103—42no—
356#103—44no—
357#102—28no—
358#102—24no—
359#102—24no—
360#102—42no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—22no—
364C22⋊C8#4114yes—
365#97—28no—
366C4×C8—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—24no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24C22×D411yescharacteristic
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×Q8—816no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D42−1+414yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—816no—
458C22×C4#13714yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4112yes—
469C22×C42−1+412yes—
470C22×C4C2511yescharacteristic
471SD16—48no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8C2×#4114yes—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—42no—
534C42—82no—
535C42—82no—
536C42H(5,𝔽2)14yes—
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—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23#54712yes—
556C23C2×#13712yes—
557C23#57612yes—
558C23C23×D411yescharacteristic
559Q8—42no—
560Q8—42no—
561Q8—88no—
562Q8—88no—
563Q8—162no—
564D4—24no—
565D4—24no—
566D4—48no—
567D4—42no—
568D4—42no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—88no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—88no—
623C2×C4—88no—
624C2×C4#53512yes—
625C2×C4#53512yes—
626C2×C4#54712yes—
627C2×C4C2×#13712yes—
628C2×C4C2×#13712yes—
629C2×C4#57512yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)12yes—
632C2×C4C2×2−1+412yes—
633C2×C4C2×2−1+411yescharacteristic
634C8—24no—
635C8—44no—
636C8—42no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—816no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22#335714yes—
652C22C2×#53511yescharacteristic
653C22C2×#54711yescharacteristic
654C22#348611yescharacteristic
655C22C22×#13711yescharacteristic
656C22C2×#57511yescharacteristic
657C22C2×#57611yescharacteristic
658C22#358511yescharacteristic
659C4—22no—
660C4—22no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—81no—
668C4—81no—
669C4—82no—
670C4—81no—
671C4—84no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4#321514yes—
676C4#352012yes—
677C4#353812yes—
678C2—24no—
679C2—84no—
680C2#3891711yescharacteristic
681C2C2×#321511yescharacteristic
682C2#6081111yescharacteristic
683C2C2×#335711yescharacteristic
684C2#6125611yescharacteristic
685C2#6182611yescharacteristic
686C2#6183511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #36193C2semidirect
4. #32455C2semidirect
5. #32373C2semidirect
6. #29913C2semidirect
7. #28539C2semidirect
8. #27735C2semidirect
10. #25813C2semidirect
11. #25450C2semidirect
12. #24679C2semidirect
14. #24302C2semidirect
15. #22603C2semidirect
16. #21986C2semidirect
20. C2×#500C22semidirect
21. C2×#477C22semidirect
28. #2567C22semidirect
35. #2552C22semidirect
37. #2511C22semidirect
39. #2507C22semidirect
44. #2369C22semidirect
45. #2355C22semidirect
54. #2101C22semidirect
55. #2097C22semidirect
57. #2085C22semidirect
58. #2074C22semidirect
61. #2057C22semidirect
62. #2057C22semidirect
64. #2024C22semidirect
65. #1990C22semidirect
66. #1987C22semidirect
68. #1949C22semidirect
69. #1949C22semidirect
70. #1939C22semidirect
74. #1917C22semidirect
85. #1847C22semidirect
86. #1836C22semidirect
92. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#361932
C2×#25632
#329562
#324552
#323732
#299132
#285392
#277352
#275822
#258132
#254502
#246792
#245072
#243022
#226032
#219862
C4×#4502