Finite Group Databasebeta

← #6479535

Subgroups — #6479535

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#33100C214yesmaximal, complemented
4#29909C212yesmaximal, complemented
5#29908C212yesmaximal, complemented
6#27864C214yesmaximal, complemented
7#27861C214yesmaximal, complemented
8#27613C212yesmaximal, complemented
9#27610C212yesmaximal, complemented
10#27586C212yesmaximal, complemented
11#27583C212yesmaximal, complemented
12#21942C212yesmaximal
13#21765C212yesmaximal
14#20682C211yesmaximal, characteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#117—21no—
17C23×C42C2211yescharacteristic
18C2×#407C2212yes—
19#2855—21no—
20#2855—21no—
21#2855—21no—
22#2855—21no—
23#2855C2212yes—
24#2855C2212yes—
25#2855C2212yes—
26#2855C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#387—21no—
32C2×#387—21no—
33C2×#385—21no—
34#2285—21no—
35#2285—21no—
36#2285—21no—
37#2285—21no—
38#2285C2212yes—
39#2285C2212yes—
40#2285C2212yes—
41#2285C2212yes—
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#382—21no—
45C2×#381—21no—
46C2×#380—21no—
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51#2100C2214yescomplemented
52#2099C2214yescomplemented
53#2096C2214yescomplemented
54#2095C2214yescomplemented
55#2084C2214yescomplemented
56#2083C2214yescomplemented
57#2072C2214yescomplemented
58#2072C2214yescomplemented
59#1928C2218yescomplemented
60#1927C2218yescomplemented
61#1919C2218yescomplemented
62#1918C2218yescomplemented
63#1894C2218yes—
64#1872C2214yes—
65#1872C2214yes—
66#1853C2214yes—
67#1853C2214yes—
68#1852C2214yes—
69#1852C2214yes—
70#1823C2214yes—
71#1817C2212yes—
72#1817C2212yes—
73#1817C2212yes—
74#1817C2212yes—
75#1809C2218yes—
76#1787C2214yes—
77#1728C2212yes—
78#1728C2212yes—
79C23×Q8C2311yescharacteristic
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16C2314yes—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8C2312yes—
88C2×C22⋊Q8C2312yes—
89C2×#117—22no—
90C2×#117—22no—
91C2×#117—22no—
92C2×#117—22no—
93C2×#117—48no—
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⋊C4—21no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
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
123C2×#107C2318yes—
124C2×#106C2318yes—
125C2×#103C2318yes—
126C2×#103C2318yes—
127#407—28no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130#391—21no—
131#391—21no—
132#391—21no—
133#391—21no—
134#391C2312yes—
135#391C2312yes—
136#391C2312yes—
137#391C2312yes—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389C2312yes—
143#389C2312yes—
144#389C2312yes—
145#389C2312yes—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#385—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#384—21no—
159#384—21no—
160#384—21no—
161#384—21no—
162#384C2312yes—
163#384C2312yes—
164#384C2312yes—
165#384C2312yes—
166#382—22no—
167#382—22no—
168#382—22no—
169#382—22no—
170#381—22no—
171#381—22no—
172#381—22no—
173#381—22no—
174#380—21no—
175#380—21no—
176#380—22no—
177#380—22no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—22no—
182#380C2312yes—
183#380C2314yes—
184#380C2314yes—
185#380C2312yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2314yes—
194C2×#95—22no—
195C2×#95—22no—
196C2×#95—21no—
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#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210#336C2314yes—
211C25C2×D411yescharacteristic
212C22×Q8—21no—
213C22×Q8—24no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8C2412yes—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
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×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4#4118yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×M16—28no—
251C2×M16—44no—
252C2×M16—44no—
253C22×C8C2418yes—
254C22⋊Q8—28no—
255C22⋊Q8—28no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258#117—48no—
259#117—48no—
260#117—48no—
261#117—48no—
262C2×C4⋊C4—24no—
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—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—44no—
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—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—48no—
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×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—48no—
339#106—48no—
340#103—48no—
341#103—48no—
342C22⋊C8—28no—
343C22⋊C8—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
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—
359#95—22no—
360#95—22no—
361#95—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—41no—
367C24—41no—
368C24#12314yes—
369C24C22×D411yescharacteristic
370C24C22×D411yescharacteristic
371C24C22×D411yescharacteristic
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—88no—
379C22×C4—232no—
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—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—416no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—416no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4#12314yes—
452C22×C4#12314yes—
453C22×C4#12314yes—
454C22×C4AGL(1,ℤ/8)14yes—
455C22×C4#13714yes—
456C22×C4#13714yes—
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×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2511yescharacteristic
472M16—88no—
473C2×C8—28no—
474C2×C8—28no—
475C2×C8—28no—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—48no—
523C42—48no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—28no—
533C23—28no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23C2×#12312yes—
546C23C2×#12312yes—
547C23C2×#12312yes—
548C23C2×AGL(1,ℤ/8)12yes—
549C23C2×#13712yes—
550C23C2×#13712yes—
551C23#57512yes—
552C23C23×D411yescharacteristic
553Q8—48no—
554Q8—84no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
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—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—88no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—88no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4C2×#12312yes—
622C2×C4C2×#12312yes—
623C2×C4C2×#12312yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4C2×#13712yes—
629C2×C4C2×#13712yes—
630C2×C4C22×#4112yes—
631C2×C4#58512yes—
632C2×C4#58512yes—
633C8—48no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—416no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22#309214yes—
655C22#309514yes—
656C22C22×#12311yescharacteristic
657C22#344911yescharacteristic
658C22#344911yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C22×#13711yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57511yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—41no—
673C4—41no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—84no—
679C4—82no—
680C4#349114yes—
681C2—24no—
682C2—24no—
683C2—42no—
684C2#3509711yescharacteristic
685C2C2×#309211yescharacteristic
686C2C2×#309511yescharacteristic
687C2#6094911yescharacteristic
688C2#6095011yescharacteristic
689C2#6096911yescharacteristic
690C2#6097111yescharacteristic
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. #33100C2semidirect
4. #29909C2semidirect
5. #29908C2semidirect
6. #27864C2semidirect
7. #27861C2semidirect
8. #27613C2semidirect
9. #27610C2semidirect
10. #27586C2semidirect
11. #27583C2semidirect
51. #2100C22semidirect
52. #2099C22semidirect
53. #2096C22semidirect
54. #2095C22semidirect
55. #2084C22semidirect
56. #2083C22semidirect
57. #2072C22semidirect
58. #2072C22semidirect
59. #1928C22semidirect
60. #1927C22semidirect
61. #1919C22semidirect
62. #1918C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#22852
#331002
#299092
#299082
#278642
#278612
#276132
#276102
#275862
#275832
#219422
#217652
#206822