Finite Group Databasebeta

← #6490679

Subgroups — #6490679

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C2×C4×#125C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic
4#33148C211yesmaximal, characteristic
5#33147C211yesmaximal, characteristic
6#33146C211yesmaximal, characteristic
7#30097C214yesmaximal, complemented
8#29275C212yesmaximal, complemented
9#29274C212yesmaximal, complemented
10#29222C212yesmaximal, complemented
11#28691C212yesmaximal, complemented
12#27964C212yesmaximal
13#27617C212yesmaximal
14#27616C212yesmaximal
15#25815C212yesmaximal
16#22249C214yesmaximal, complemented
17#20777C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C2×#499C2214yes—
20#2864C2212yes—
21#2863—21no—
22#2863—21no—
23#2862C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2855—21no—
28#2855—21no—
29#2851C2212yes—
30C4×C4⋊Q8C2211yescharacteristic
31C4×C4⋊Q8C2211yescharacteristic
32C4×#125—21no—
33C4×#125—21no—
34C4×#125—21no—
35C4×#125—21no—
36C4×#125C2212yes—
37C4×#125C2212yes—
38C4×#125C2212yes—
39C4×#125C2212yes—
40C4×#125C2212yes—
41C4×C22⋊Q8—21no—
42C4×C22⋊Q8—21no—
43#2295—21no—
44#2295—21no—
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#387—21no—
48C2×#387—21no—
49#2283C2212yes—
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C43C2211yescharacteristic
56#2101C2214yes—
57#2101C2214yes—
58#2101C2214yes—
59#2101C2214yes—
60#2085C2214yes—
61#2085C2214yes—
62#2059C2214yes—
63#2059C2214yes—
64#1991C2214yes—
65#1991C2214yes—
66#1988C2212yes—
67#1988C2212yes—
68#1987C2212yes—
69#1987C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1921C2214yes—
75#1921C2214yes—
76#1920C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1918C2214yes—
82#1853C2212yes—
83#1853C2212yes—
84#1852C2212yes—
85#1852C2212yes—
86#1839C2212yes—
87#1835C2214yes—
88#1831C2212yes—
89#1824C2212yes—
90#1824C2212yes—
91#1823C2212yes—
92#1823C2212yes—
93#1817C2212yes—
94#1797C2211yescharacteristic
95#1797C2211yescharacteristic
96C4×C4⋊C8C2212yes—
97C4×C22⋊C8C2212yes—
98#1744C2212yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125—28no—
102C2×#125—21no—
103C2×#125—21no—
104C2×#125C2312yes—
105C2×#125C2312yes—
106C2×#125C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42—21no—
120C22×C42—21no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#499—216no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C4×C8C2314yes—
138#391—21no—
139#391—21no—
140#391—21no—
141#391—21no—
142#391—21no—
143#391—21no—
144#391—21no—
145#391—21no—
146#389—21no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#385—22no—
183#385—22no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#383—22no—
187#382—21no—
188#382—21no—
189#382C2312yes—
190#382C2312yes—
191#382C2312yes—
192#382C2312yes—
193#382C2312yes—
194#381—22no—
195#380—21no—
196#380—21no—
197#380C2312yes—
198#380C2312yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222#376—22no—
223C2×#95—21no—
224C2×#95—21no—
225C2×#95—21no—
226C2×#95—21no—
227C2×#95—21no—
228C2×#95—21no—
229C43D411yescharacteristic
230C43D411yescharacteristic
231C43D411yescharacteristic
232C43D411yescharacteristic
233C43C2311yescharacteristic
234C43C2311yescharacteristic
235C43C2311yescharacteristic
236C43C2311yescharacteristic
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2314yes—
252#336C2314yes—
253#336C2314yes—
254C22×Q8—21no—
255C22×Q8—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C22×C8C2414yes—
268C22×C8C2414yes—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271#125—28no—
272#125—28no—
273#125—24no—
274#125—24no—
275#125—24no—
276#125—28no—
277#125#4118yes—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C4×Q8—44no—
281C4×Q8—44no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—44no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—28no—
371#107—28no—
372#106—28no—
373#106—28no—
374C4⋊C8—24no—
375C4⋊C8—24no—
376C22⋊C8—24no—
377C22⋊C8—24no—
378C4×C8—28no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405C24C22×D411yescharacteristic
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4#13514yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C22×C4C2511yescharacteristic
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4C2×#4114yes—
509C4⋊C4C2×#4114yes—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—44no—
541C42—44no—
542C42—44no—
543C42—44no—
544C42—44no—
545C42—44no—
546C42C2×#4114yes—
547C422−1+414yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23#53612yes—
553C23C2×#13512yes—
554C23#57512yes—
555C23C23×D411yescharacteristic
556Q8—84no—
557Q8—84no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
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—84no—
623C2×C4—84no—
624C2×C4#52512yes—
625C2×C4#52512yes—
626C2×C4#53612yes—
627C2×C4#53612yes—
628C2×C4#53612yes—
629C2×C4C2×#13512yes—
630C2×C4C2×#13512yes—
631C2×C4#57512yes—
632C2×C4#57512yes—
633C2×C4C22×#4112yes—
634C2×C4C2×2−1+411yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4C2×2−1+412yes—
637C8—44no—
638C8—44no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22#323314yes—
646C22C2×#52511yescharacteristic
647C22C2×#53611yescharacteristic
648C22C2×#53611yescharacteristic
649C22C22×#13511yescharacteristic
650C22C2×#57511yescharacteristic
651C22C2×#57511yescharacteristic
652C22C2×#57511yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—41no—
660C4—44no—
661C4—41no—
662C4—44no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4#323214yes—
674C4#352312yes—
675C4#352312yes—
676C2—24no—
677C2#3524811yescharacteristic
678C2#6053611yescharacteristic
679C2#6054611yescharacteristic
680C2C2×#323211yescharacteristic
681C2C2×#323311yescharacteristic
682C2#6086611yescharacteristic
683C2#6086611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #30097C2semidirect
8. #29275C2semidirect
9. #29274C2semidirect
10. #29222C2semidirect
11. #28691C2semidirect
16. #22249C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×C4×#1252
#331492
#331482
#331472
#331462
#300972
#292752
#292742
#292222
#286912
#279642
#276172
#276162
#258152
#222492
#207772