Finite Group Databasebeta

← #6497282

Subgroups — #6497282

Order 512.

1872 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#33150C214yesmaximal, complemented
4#29943C214yesmaximal, complemented
5#27965C212yesmaximal, complemented
6#27953C212yesmaximal, complemented
7#27881C214yesmaximal, complemented
8#27611C212yesmaximal, complemented
9#27610C212yesmaximal, complemented
10#27603C212yesmaximal
11#27601C212yesmaximal
12#21986C212yesmaximal, complemented
13#21838C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic
15C22×#125C2211yescharacteristic
16C22×#117—21no—
17C23×C42C2211yescharacteristic
18C2×#432C2212yes—
19#2851—21no—
20#2851—21no—
21#2851—21no—
22#2851—21no—
23#2851C2212yes—
24#2851C2212yes—
25#2851C2212yes—
26#2851C2212yes—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#387—21no—
30C2×#387—21no—
31C2×#385—21no—
32C2×#385—21no—
33#2285—21no—
34#2285—21no—
35#2285—21no—
36#2285—21no—
37#2285C2212yes—
38#2285C2212yes—
39#2285C2212yes—
40#2285C2212yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#382—21no—
44C2×#382C2211yescharacteristic
45C2×#380—21no—
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2212yes—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51#2099C2214yescomplemented
52#2099C2214yescomplemented
53#2084C2214yescomplemented
54#2083C2214yescomplemented
55#2077C2218yes—
56#2076C2218yes—
57#1992C2214yescomplemented
58#1992C2214yescomplemented
59#1989C2218yescomplemented
60#1949C2218yescomplemented
61#1920C2218yescomplemented
62#1918C2218yescomplemented
63#1853C2214yes—
64#1853C2214yes—
65#1852C2214yes—
66#1852C2214yes—
67#1847C2214yescomplemented
68#1836C2214yescomplemented
69#1824C2214yes—
70#1824C2214yes—
71#1824C2214yes—
72#1823C2214yes—
73#1817C2212yes—
74#1797C2212yes—
75#1797C2212yes—
76#1779C2212yes—
77#1749C2212yes—
78#1749C2212yes—
79C24×C4D411yescharacteristic
80C24×C4D411yescharacteristic
81C24×C4C2311yescharacteristic
82C2×#125—21no—
83C2×#125—21no—
84C2×#125—28no—
85C2×#125C2312yes—
86C2×#125C2312yes—
87C2×#117—22no—
88C2×#117—22no—
89C2×#117—22no—
90C2×#117—22no—
91C2×#117—48no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—22no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C42—28no—
110C22×C42—22no—
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#432—28no—
124C2×#107C2318yes—
125C2×#106C2318yes—
126C2×C4⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130C2×#97C2314yes—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389C2312yes—
136#389C2312yes—
137#389C2312yes—
138#389C2312yes—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#385—22no—
148#385—22no—
149#385—22no—
150#385—22no—
151#385—22no—
152#385—22no—
153#385—22no—
154#385—22no—
155#384—21no—
156#384—21no—
157#384—21no—
158#384—21no—
159#384C2312yes—
160#384C2312yes—
161#384C2312yes—
162#384C2312yes—
163#382—21no—
164#382—21no—
165#382—22no—
166#382—22no—
167#382—22no—
168#382—22no—
169#382C2312yes—
170#382C2312yes—
171#380—21no—
172#380—21no—
173#380—21no—
174#380—21no—
175#380—22no—
176#380—22no—
177#380—22no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380C2312yes—
182#380C2314yes—
183#380C2312yes—
184#380C2314yes—
185#380C2312yes—
186#380C2312yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2314yes—
195C2×#95—22no—
196C2×#95—22no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203#340C2318yes—
204#340C2318yes—
205#340C2318yes—
206#340C2318yes—
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211C25C2×D411yescharacteristic
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C22×C8C2418yes—
246#125—28no—
247#125—28no—
248#125—44no—
249#125—44no—
250#117—48no—
251#117—48no—
252#117—48no—
253#117—48no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—44no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—48no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42#4118yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—48no—
337#106—48no—
338C4⋊C8—28no—
339C22⋊C8—28no—
340C22⋊C8—28no—
341C22⋊C8—44no—
342#97—28no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—41no—
364C24—41no—
365C24#13714yes—
366C24C22×D411yescharacteristic
367C24C22×D411yescharacteristic
368C24C22×D411yescharacteristic
369C22×C4—232no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4#12314yes—
446C22×C4#12314yes—
447C22×C4AGL(1,ℤ/8)14yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4114yes—
460C22×C42−1+414yes—
461C22×C4C2511yescharacteristic
462C2×C8—28no—
463C2×C8—28no—
464C2×C8—28no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—48no—
512C42—48no—
513C42#12314yes—
514C42#12314yes—
515C42C2×#4114yes—
516C42C2×#4114yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—28no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—28no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23C2×#12312yes—
539C23#53612yes—
540C23C2×AGL(1,ℤ/8)12yes—
541C23C2×#13712yes—
542C23C2×#13712yes—
543C23C2×#13712yes—
544C23#57512yes—
545C23C23×D411yescharacteristic
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
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—21no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—88no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—88no—
611C2×C4—84no—
612C2×C4C2×#12312yes—
613C2×C4C2×#12312yes—
614C2×C4C2×#12312yes—
615C2×C4C2×#12312yes—
616C2×C4C2×#12312yes—
617C2×C4#53612yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4#57512yes—
621C2×C4C22×#4112yes—
622C2×C4C2×2−1+412yes—
623C2×C4#58512yes—
624C8—48no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—416no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22#309214yes—
646C22#323614yes—
647C22C22×#12311yescharacteristic
648C22C2×#53611yescharacteristic
649C22#344911yescharacteristic
650C22C22×AGL(1,ℤ/8)11yescharacteristic
651C22C22×#13711yescharacteristic
652C22C2×#57511yescharacteristic
653C22C2×#57511yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—41no—
664C4—41no—
665C4—84no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—84no—
671C4#349514yes—
672C2—24no—
673C2—24no—
674C2—42no—
675C2#3531111yescharacteristic
676C2C2×#309211yescharacteristic
677C2#6051111yescharacteristic
678C2C2×#323611yescharacteristic
679C2#6086411yescharacteristic
680C2#6094911yescharacteristic
681C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33150C2semidirect
4. #29943C2semidirect
5. #27965C2semidirect
6. #27953C2semidirect
7. #27881C2semidirect
8. #27611C2semidirect
9. #27610C2semidirect
12. #21986C2semidirect
13. #21838C2semidirect
51. #2099C22semidirect
52. #2099C22semidirect
53. #2084C22semidirect
54. #2083C22semidirect
57. #1992C22semidirect
58. #1992C22semidirect
59. #1989C22semidirect
60. #1949C22semidirect
61. #1920C22semidirect
62. #1918C22semidirect
67. #1847C22semidirect
68. #1836C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#22852
#331502
#299432
#279652
#279532
#278812
#276112
#276102
#276032
#276012
#219862
#218382
#207152