Finite Group Databasebeta

← #6567791

Subgroups — #6567791

Order 512.

1819 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#34308C211yesmaximal, characteristic
3#29981C211yesmaximal, characteristic, complemented
4#29968C212yesmaximal, complemented
5#29276C211yesmaximal, characteristic, complemented
6#29218C211yesmaximal, characteristic, complemented
7#29156C212yesmaximal, complemented
8#27678C212yesmaximal, complemented
9#26510C212yesmaximal, complemented
10#26410C214yesmaximal, complemented
11#26110C212yesmaximal, complemented
12#25990C214yesmaximal, complemented
13#23775C211yesmaximal, characteristic, complemented
14C2×#2100C212yesmaximal, complemented
15#22703C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22485C212yesmaximal
18C2×#554—21no—
19C2×#547C2211yescharacteristic
20C2×#520—21no—
21C2×C8×D4C2212yes—
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2567—21no—
25#2552—21no—
26#2552C2212yes—
27#2502—22no—
28#2440—22no—
29#2440C2214yes—
30#2365C2211yescharacteristic
31#2361C2212yes—
32#2360—21no—
33#2359—22no—
34D4×C4⋊C4—21no—
35D4×C4⋊C4C2211yescharacteristic
36D4×C4⋊C4C2211yescharacteristic
37D4×C4⋊C4C2212yes—
38#2344C2212yes—
39#2339—21no—
40#2323—21no—
41#2322—22no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#121C2212yes—
45#2293C2211yescharacteristic
46C2×#389C2212yes—
47C2×#384C2212yes—
48C42×D4C2211yescharacteristic
49#2100C2212yes—
50#2100C2212yes—
51#2100C2212yes—
52#2100C2212yes—
53#2100C2214yes—
54#2100C2214yes—
55#2100C2211yescharacteristic
56#2094C2212yes—
57#2094C2211yescharacteristic
58#2088C2214yes—
59#2088C2214yes—
60#2088C2212yes—
61#2065C2212yes—
62#2058C2211yescharacteristic
63#2058C2212yes—
64#2058C2212yes—
65#2023C2212yes—
66#2023C2211yescharacteristic
67#1990C2212yes—
68#1988C2212yes—
69#1987C2214yes—
70#1948C2212yes—
71#1948C2214yes—
72#1948C2212yes—
73#1939C2212yes—
74#1920C2214yes—
75#1920C2214yes—
76#1893C2214yes—
77#1874C2214yes—
78#1864C2214yes—
79#1853C2214yes—
80#1853C2214yes—
81#1850C2214yes—
82#1834C2212yes—
83#1831C2211yescharacteristic
84#1823C2214yes—
85#1807C2214yes—
86#1802C2212yes—
87#1802C2214yes—
88#1797C2212yes—
89C4×#106C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×#102C2212yes—
92C2×#340C2214yes—
93C2×#340C2214yes—
94C2×#336C2212yes—
95C22×#41—21no—
96C23×C8C2314yes—
97#554—48no—
98#547—24no—
99#547Q818yescomplemented
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—22no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—24no—
110C2×#121C2312yes—
111#520—48no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2314yes—
133C22×C42C2312yes—
134C8×D4—28no—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C4×C8C2312yes—
147#398—24no—
148#398—24no—
149#395—24no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—22no—
154#390C2311yescharacteristic
155#390C2311yescharacteristic
156#389—22no—
157#389—22no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2311yescharacteristic
167#389C2311yescharacteristic
168#389C2312yes—
169#389C2314yes—
170#389C2312yes—
171#388—21no—
172#388—22no—
173#388—21no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#386—22no—
179#386—22no—
180#385—22no—
181#385—24no—
182#385—24no—
183#385—22no—
184#385—22no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2314yes—
193#384C2312yes—
194#381—22no—
195#380—22no—
196#380—21no—
197#380C2312yes—
198#380C2314yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4—21no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2312yes—
213C2×#95—22no—
214C43C2311yescharacteristic
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C2×#41—28no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C22×C8—28no—
251C22×C8—44no—
252C22×C8—44no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—22no—
257C4⋊Q8—42no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—82no—
261C4⋊Q8C2×Q814yes—
262#125—48no—
263#123—216no—
264C22⋊Q8—28no—
265C22⋊Q8—416no—
266C22⋊Q8—48no—
267#121—24no—
268#121C2×Q818yes—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×D4—24no—
273C4×D4—22no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—48no—
277C4×D4—48no—
278C4×D4—48no—
279C4×D4SD1618yes—
280C4×D4C2×Q814yes—
281C4×D4C2×Q814yes—
282C4×D4#4114yes—
283#117—48no—
284#117—44no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×Q818yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#106—42no—
356#106—42no—
357#106—44no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361C4⋊C8—42no—
362#102—42no—
363#102—42no—
364#102—44no—
365C22⋊C8—28no—
366C4×C8—24no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C24C22×D412yes—
385#41—432no—
386C2×Q8—24no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×D4—28no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4C22⋊Q814yes—
402C2×D4C2×SD1614yes—
403C2×D4AGL(1,ℤ/8)14yes—
404C2×D4C22×Q812yes—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—82no—
435C22×C4C2×SD1618yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×Q814yes—
446C22×C4C2×#4114yes—
447C22×C4C2511yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—88no—
458C4⋊C4—28no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—81no—
488C4⋊C4—82no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4C22⋊Q814yes—
494C4⋊C4C2×SD1614yes—
495C4⋊C4C22×Q812yes—
496C4⋊C4C22×Q814yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—28no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—82no—
507C22⋊C4C22⋊Q818yes—
508C22⋊C4C2×SD1618yes—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—42no—
515C42—42no—
516C42—41no—
517C42—44no—
518C42—44no—
519C42—42no—
520C42—44no—
521C42C2×SD1614yes—
522C42C22×Q812yes—
523C42C2×#4112yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C2×C22⋊Q814yes—
536C23C22×SD1614yes—
537C23C2×AGL(1,ℤ/8)14yes—
538C23C23×D411yescharacteristic
539Q8—48no—
540Q8—84no—
541Q8—164no—
542D4—24no—
543D4—416no—
544D4—42no—
545D4#476116yes—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—416no—
575C2×C4—48no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—164no—
595C2×C4—164no—
596C2×C4—168no—
597C2×C4C2×C22⋊Q812yes—
598C2×C4C2×C22⋊Q814yes—
599C2×C4C2×C22⋊Q812yes—
600C2×C4C22×SD1612yes—
601C2×C4C22×SD1614yes—
602C2×C4C22×SD1612yes—
603C2×C4C22×SD1612yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4#57812yes—
606C2×C4C23×Q811yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4#58512yes—
609C2×C4#58512yes—
610C8—42no—
611C8—44no—
612C8—42no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22C2×#47618yes—
627C22C22×C22⋊Q811yescharacteristic
628C22#348511yescharacteristic
629C22#350011yescharacteristic
630C22C23×SD1611yescharacteristic
631C22C22×AGL(1,ℤ/8)11yescharacteristic
632C22C2×#57811yescharacteristic
633C22#358511yescharacteristic
634C4—22no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—24no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—41no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4—81no—
648C4—82no—
649C4—84no—
650C4C2×#47614yes—
651C4#350412yes—
652C4#352112yes—
653C2—28no—
654C2—82no—
655C2#3603111yescharacteristic
656C2C22×#47611yescharacteristic
657C2#6049811yescharacteristic
658C2#6169711yescharacteristic
659C2#6173711yescharacteristic
660C2#6252611yescharacteristic
661C2#6257611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36180C2semidirect
3. #29981C2semidirect
4. #29968C2semidirect
5. #29276C2semidirect
6. #29218C2semidirect
7. #29156C2semidirect
8. #27678C2semidirect
9. #26510C2semidirect
10. #26410C2semidirect
11. #26110C2semidirect
12. #25990C2semidirect
13. #23775C2semidirect
14. C2×#2100C2semidirect
15. #22703C2semidirect
16. #22696C2semidirect
99. #547Q8semidirect

Maximal subgroups

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

subgroupindex
#361802
#343082
#299812
#299682
#292762
#292182
#291562
#276782
#265102
#264102
#261102
#259902
#237752
C2×#21002
#227032
#226962
#224852