Finite Group Databasebeta

← #6489341

Subgroups — #6489341

Order 512.

2334 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#32959C214yesmaximal, complemented
4#30077C214yesmaximal, complemented
5#28718C212yesmaximal, complemented
6#28514C212yesmaximal, complemented
7#28207C214yesmaximal, complemented
8#27950C211yesmaximal, characteristic, complemented
9#27948C211yesmaximal, characteristic, complemented
10#27588C214yesmaximal, complemented
11#25855C211yesmaximal, characteristic
12#25855C211yesmaximal, characteristic
13#22220C212yesmaximal, complemented
14#22219C212yesmaximal, complemented
15#20777C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4—21no—
18C22×C4×D4—21no—
19C22×C4×D4C2212yes—
20C2×#486C2212yes—
21C2×#486C2212yes—
22C2×#478C2218yescomplemented
23#2864C2212yescomplemented
24#2863—21no—
25#2863—21no—
26#2862C2211yescharacteristic, complemented
27#2859C2211yescharacteristic, complemented
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2855—21no—
31#2855—21no—
32#2851C2212yes—
33C4×C4⋊Q8C2211yescharacteristic, complemented
34C4×C4⋊Q8C2211yescharacteristic, complemented
35C4×#127—21no—
36C4×#127—21no—
37C4×#127—21no—
38C4×#127—21no—
39C4×#127C2212yes—
40C4×#127C2212yes—
41C4×#127C2212yes—
42C4×#127C2212yes—
43C4×#125C2212yescomplemented
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8—21no—
46#2295—21no—
47#2295—21no—
48C2×#390—21no—
49C2×#390—21no—
50C2×#390C2212yes—
51C2×#384C2211yescharacteristic
52#2283C2212yes—
53C2×C43C2211yescharacteristic
54#2092C2218yescomplemented
55#2084C2214yescomplemented
56#2083C2214yescomplemented
57#2073C2218yescomplemented
58#2041C2214yes—
59#2041C2214yes—
60#1990C2212yescomplemented
61#1990C2212yescomplemented
62#1957C2212yes—
63#1957C2212yes—
64#1926C2218yescomplemented
65#1925C2218yescomplemented
66#1864C2218yescomplemented
67#1862C2218yescomplemented
68#1853C2214yes—
69#1852C2214yes—
70#1845C2212yescomplemented
71#1844C2212yescomplemented
72#1839C2212yescomplemented
73#1831C2212yescomplemented
74#1817C2212yes—
75#1804C2218yescomplemented
76#1797C2212yes—
77#1786C2214yes—
78#1786C2214yes—
79C4×C4⋊C8C2212yescomplemented
80C4×C22⋊C8C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1744C2212yescomplemented
83C23×D4—21no—
84C23×D4—21no—
85C23×D4C2312yes—
86C24×C4—22no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×#127—28no—
90C2×#127—21no—
91C2×#127—21no—
92C2×#127C2312yes—
93C2×#127C2312yes—
94C2×#125C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—48no—
110C2×C4×D4—48no—
111C2×C4×D4—48no—
112C2×C4×D4C2314yes—
113C2×C4×D4C2314yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4—22no—
118C22×C22⋊C4—22no—
119C22×C22⋊C4—22no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#486—28no—
128#486—28no—
129#478—416no—
130C2×#107C2318yes—
131C2×#106C2318yes—
132C2×C4⋊C8C2314yes—
133C2×#102C2318yes—
134C2×#102C2318yes—
135C2×#102C2318yes—
136C2×#102C2318yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141#391—21no—
142#391—21no—
143#391—21no—
144#391—21no—
145#391—21no—
146#391—21no—
147#391—21no—
148#391—21no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390—22no—
159#390C2314yes—
160#390C2314yes—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
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#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#383—22no—
190#382C2312yes—
191#382C2312yes—
192#382C2312yes—
193#381—22no—
194#380—21no—
195#380—21no—
196#380C2312yes—
197#380C2312yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205#376—22no—
206C43D411yescharacteristic
207C43D411yescharacteristic
208C43D411yescharacteristic
209C43D411yescharacteristic
210C43C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213C43C2311yescharacteristic
214#340C2318yes—
215#340C2318yes—
216#336C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2312yes—
220#336C2312yes—
221C25—22no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×D4—28no—
225C22×D4—28no—
226C22×D4—28no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—28no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—44no—
238C22×D4C2414yes—
239C23×C4—22no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—48no—
243C23×C4—48no—
244C23×C4—48no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C22×C8C2414yes—
253C22×C8C2414yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256#127—28no—
257#127—28no—
258#127—28no—
259#127—24no—
260#127#4118yes—
261#125—28no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×D4—216no—
267C4×D4—48no—
268C4×D4—48no—
269C4×D4—48no—
270C4×D4—48no—
271C4×D4—48no—
272C4×D4—48no—
273C4×D4—48no—
274C4×D4—48no—
275C4×D4—48no—
276C4×D4—48no—
277C4×D4—48no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4—48no—
313C2×C22⋊C4—48no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—42no—
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×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—48no—
352#106—48no—
353C4⋊C8—28no—
354#102—216no—
355#102—216no—
356#102—48no—
357#102—48no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C4×C8—24no—
361C4×C8—24no—
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—
376C24—28no—
377C24—24no—
378C24—44no—
379C24—44no—
380C24—44no—
381C24—44no—
382C24—44no—
383C24—44no—
384C24C22×D411yescharacteristic
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—432no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—432no—
404C2×D4—432no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4C2×#4118yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
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—24no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—832no—
446C22×C4#13514yes—
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×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C42−1+414yes—
462C22×C4C2511yescharacteristic
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—48no—
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—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—41no—
523C42—41no—
524C42—44no—
525C42—44no—
526C42AGL(1,ℤ/8)14yes—
527C42C2×#4114yes—
528C42H(5,𝔽2)14yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—28no—
534C23—416no—
535C23—416no—
536C23—416no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—88no—
544C23—88no—
545C23—88no—
546C23—88no—
547C23#53612yes—
548C23C2×#13512yes—
549C23C2×AGL(1,ℤ/8)12yes—
550C23C23×D411yescharacteristic
551Q8—84no—
552Q8—84no—
553D4—216no—
554D4—416no—
555D4—416no—
556D4—416no—
557D4—416no—
558D4—416no—
559D4—48no—
560C2×C4—28no—
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—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
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—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
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—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—816no—
609C2×C4—816no—
610C2×C4—88no—
611C2×C4#52512yes—
612C2×C4#52512yes—
613C2×C4#53512yes—
614C2×C4#53512yes—
615C2×C4#53612yes—
616C2×C4C2×#13512yes—
617C2×C4C2×#13512yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4#57512yes—
624C2×C4C22×#4112yes—
625C2×C4C2×H(5,𝔽2)12yes—
626C2×C4C2×2−1+412yes—
627C8—44no—
628C8—44no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—832no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22#322814yes—
644C22C2×#52511yescharacteristic
645C22C2×#53511yescharacteristic
646C22C2×#53611yescharacteristic
647C22C22×#13511yescharacteristic
648C22C22×AGL(1,ℤ/8)11yescharacteristic
649C22C22×AGL(1,ℤ/8)11yescharacteristic
650C22C2×#57511yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4#310514yes—
668C4#311414yes—
669C4#352014yes—
670C2—24no—
671C2—88no—
672C2#3524011yescharacteristic
673C2C2×#310511yescharacteristic
674C2C2×#311411yescharacteristic
675C2#6078211yescharacteristic
676C2#6079811yescharacteristic
677C2C2×#322811yescharacteristic
678C2#6086411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
1. #36635C2semidirect
3. #32959C2semidirect
4. #30077C2semidirect
5. #28718C2semidirect
6. #28514C2semidirect
7. #28207C2semidirect
8. #27950C2semidirect
9. #27948C2semidirect
10. #27588C2semidirect
13. #22220C2semidirect
14. #22219C2semidirect
15. #20777C2semidirect
22. C2×#478C22semidirect
23. #2864C22semidirect
26. #2862C22semidirect
27. #2859C22semidirect
33. C4×C4⋊Q8C22semidirect
34. C4×C4⋊Q8C22semidirect
43. C4×#125C22semidirect
54. #2092C22semidirect
55. #2084C22semidirect
56. #2083C22semidirect
57. #2073C22semidirect
60. #1990C22semidirect
61. #1990C22semidirect
64. #1926C22semidirect
65. #1925C22semidirect
66. #1864C22semidirect
67. #1862C22semidirect
70. #1845C22semidirect
71. #1844C22semidirect
72. #1839C22semidirect
73. #1831C22semidirect
75. #1804C22semidirect
79. C4×C4⋊C8C22semidirect
82. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×C4×#1272
#329592
#300772
#287182
#285142
#282072
#279502
#279482
#275882
#258552
#222202
#222192
#207772