Finite Group Databasebeta

← #6593802

Subgroups — #6593802

Order 512.

2161 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic, complemented
2C2×#2366C211yesmaximal, characteristic
3#33146C211yesmaximal, characteristic, complemented
4#33007C212yesmaximal, complemented
5#27588C212yesmaximal, complemented
6#27078C214yesmaximal, complemented
7#26770C214yesmaximal, complemented
8#26529C212yesmaximal, complemented
9#25855C211yesmaximal, characteristic
10#25325C212yesmaximal, complemented
11#24507C212yesmaximal, complemented
12#24211C211yesmaximal, characteristic
13#23220C212yesmaximal, complemented
14#23101C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#483C211yesmaximal, characteristic, complemented
17#20739C212yesmaximal, complemented
18C22×#127C2211yescharacteristic
19C22×C4×D4—21no—
20C22×C4×D4C2212yes—
21C2×#486C2212yes—
22C2×#483C2211yescharacteristic
23C2×#460C2212yes—
24C2×#450C2212yes—
25C2×#432C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2571C2211yescharacteristic, complemented
29#2568C2211yescharacteristic, complemented
30#2533—22no—
31#2514—21no—
32#2514C2212yescomplemented
33#2509C2212yescomplemented
34#2503—21no—
35#2493—22no—
36#2366—21no—
37#2366—21no—
38#2366—22no—
39#2366C2212yes—
40#2366C2214yes—
41#2366C2212yes—
42#2365C2211yescharacteristic, complemented
43#2360—21no—
44#2356—21no—
45#2355C2211yescharacteristic, complemented
46C4×C22⋊Q8C2211yescharacteristic
47C2×#390—21no—
48C2×#390C2212yes—
49C2×#386—22no—
50#2283C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2092C2214yescomplemented
54#2073C2214yescomplemented
55#2067C2214yescomplemented
56#2055C2214yescomplemented
57#2041C2212yes—
58#2041C2214yes—
59#2014C2212yes—
60#1988C2212yescomplemented
61#1982C2212yes—
62#1973C2214yescomplemented
63#1957C2212yes—
64#1947C2212yescomplemented
65#1939C2212yescomplemented
66#1936C2214yescomplemented
67#1927C2212yescomplemented
68#1926C2214yescomplemented
69#1925C2214yescomplemented
70#1920C2212yescomplemented
71#1918C2214yescomplemented
72#1917C2214yescomplemented
73#1877C2214yescomplemented
74#1877C2214yescomplemented
75#1872C2211yescharacteristic
76#1864C2214yescomplemented
77#1864C2214yescomplemented
78#1853C2211yescharacteristic
79#1852C2212yes—
80#1823C2211yescharacteristic
81#1823C2212yes—
82#1817C2212yes—
83#1804C2214yescomplemented
84#1804C2214yescomplemented
85#1797C2211yescharacteristic
86#1786C2212yes—
87#1786C2212yes—
88#1786C2212yes—
89C4×#106C2212yescomplemented
90#1770C2212yescomplemented
91#1762C2214yescomplemented
92#1762C2214yescomplemented
93C4×#103C2212yescomplemented
94C4×C22⋊C8C2211yescharacteristic
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C24×C4—22no—
99C22×SD16C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127—28no—
102C2×#127—21no—
103C2×#127—21no—
104C2×#127C2312yes—
105C2×#127C2312yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8C2311yescharacteristic
108C2×C4×Q8—21no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—24no—
113C2×C4×D4—28no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—44no—
117C2×C4×D4—48no—
118C2×C4×D4C2314yes—
119C2×C4×D4C2314yes—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—22no—
125C22×C22⋊C4—22no—
126C22×C22⋊C4—22no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#486—28no—
131#483—24no—
132#460—28no—
133#450—44no—
134#432—44no—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2311yescharacteristic
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#401—22no—
155#400—22no—
156#400—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391C2312yes—
165#391C2312yes—
166#391C2311yescharacteristic
167#390—22no—
168#390—22no—
169#390—24no—
170#390—22no—
171#390—22no—
172#390C2314yes—
173#390C2314yes—
174#389—21no—
175#389—22no—
176#389—21no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2311yescharacteristic
181#387—22no—
182#387—22no—
183#387—22no—
184#387—21no—
185#387—22no—
186#387—22no—
187#387—22no—
188#386—24no—
189#386—24no—
190#386—24no—
191#386—24no—
192#385—22no—
193#385—21no—
194#384C2311yescharacteristic
195#383—22no—
196#383—22no—
197#382—21no—
198#382—21no—
199#382C2312yes—
200#382C2312yes—
201#381—22no—
202#381—22no—
203#380—21no—
204#380C2312yes—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223#376—22no—
224C2×#95—21no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C25—22no—
238C22×Q8—21no—
239C22×Q8C2411yescharacteristic
240C22×D4—28no—
241C22×D4—28no—
242C22×D4—28no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—24no—
249C22×D4—24no—
250C22×D4—44no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—82no—
254C22×D4—82no—
255C22×D4C2414yes—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—48no—
261C23×C4—48no—
262C23×C4—84no—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2411yescharacteristic
268C2×SD16—28no—
269C2×SD16—44no—
270C2×SD16—44no—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C4⋊Q8—24no—
275#127—28no—
276#127—28no—
277#127—28no—
278#127—24no—
279#127#4118yes—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282C22⋊Q8—44no—
283C4×Q8—24no—
284C4×Q8—44no—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—48no—
290C4×D4—48no—
291C4×D4—416no—
292C4×D4—48no—
293C4×D4—48no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4#4114yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—48no—
333C2×C22⋊C4—48no—
334C2×C22⋊C4—84no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—44no—
354#106—24no—
355#106—42no—
356C4⋊C8—24no—
357C4⋊C8—44no—
358#103—24no—
359#103—42no—
360#102—28no—
361#102—28no—
362#102—28no—
363#102—44no—
364#102—44no—
365#102—44no—
366#102—44no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—24no—
370#97—44no—
371C4×C8—24no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—28no—
398C24—24no—
399C24—44no—
400C24—44no—
401C24—44no—
402C24—44no—
403C24—44no—
404C24—44no—
405C24C22×D411yescharacteristic
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—416no—
424C2×D4—416no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—432no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4—816no—
437C2×D4C2×#4118yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—832no—
472C22×C4AGL(1,ℤ/8)14yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4C2511yescharacteristic
485SD16—88no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—81no—
521C4⋊C4C2×#4114yes—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—44no—
541C42—42no—
542C42—44no—
543C42—44no—
544C42—42no—
545C42—82no—
546C42C2×#4114yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—28no—
552C23—416no—
553C23—416no—
554C23—416no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—88no—
562C23—88no—
563C23—88no—
564C23—88no—
565C23#54612yes—
566C23C2×AGL(1,ℤ/8)12yes—
567C23#57712yes—
568C23C23×D411yescharacteristic
569Q8—42no—
570Q8—42no—
571Q8—84no—
572D4—28no—
573D4—28no—
574D4—416no—
575D4—416no—
576D4—416no—
577D4—44no—
578D4—44no—
579D4—88no—
580D4—88no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—84no—
633C2×C4—816no—
634C2×C4—816no—
635C2×C4—84no—
636C2×C4#52512yes—
637C2×C4#52512yes—
638C2×C4#54612yes—
639C2×C4C2×AGL(1,ℤ/8)12yes—
640C2×C4C2×AGL(1,ℤ/8)12yes—
641C2×C4#57512yes—
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4112yes—
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4#58512yes—
646C8—42no—
647C8—42no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—48no—
656C22—48no—
657C22—48no—
658C22—832no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22#332414yes—
664C22C2×#52511yescharacteristic
665C22C2×#54611yescharacteristic
666C22#348711yescharacteristic
667C22C22×AGL(1,ℤ/8)11yescharacteristic
668C22C2×#57511yescharacteristic
669C22C2×#57711yescharacteristic
670C22#358511yescharacteristic
671C4—22no—
672C4—22no—
673C4—22no—
674C4—41no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—81no—
681C4—82no—
682C4—81no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4—81no—
687C4#311414yes—
688C4#347512yes—
689C4#351312yes—
690C2—24no—
691C2—88no—
692C2#3616211yescharacteristic
693C2C2×#311411yescharacteristic
694C2#6057911yescharacteristic
695C2C2×#332411yescharacteristic
696C2#6116011yescharacteristic
697C2#6189011yescharacteristic
698C2#6190411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36177C2semidirect
3. #33146C2semidirect
4. #33007C2semidirect
5. #27588C2semidirect
6. #27078C2semidirect
7. #26770C2semidirect
8. #26529C2semidirect
10. #25325C2semidirect
11. #24507C2semidirect
13. #23220C2semidirect
14. #23101C2semidirect
15. #22603C2semidirect
16. C4×#483C2semidirect
17. #20739C2semidirect
28. #2571C22semidirect
29. #2568C22semidirect
32. #2514C22semidirect
33. #2509C22semidirect
42. #2365C22semidirect
45. #2355C22semidirect
53. #2092C22semidirect
54. #2073C22semidirect
55. #2067C22semidirect
56. #2055C22semidirect
60. #1988C22semidirect
62. #1973C22semidirect
64. #1947C22semidirect
65. #1939C22semidirect
66. #1936C22semidirect
67. #1927C22semidirect
68. #1926C22semidirect
69. #1925C22semidirect
70. #1920C22semidirect
71. #1918C22semidirect
72. #1917C22semidirect
73. #1877C22semidirect
74. #1877C22semidirect
76. #1864C22semidirect
77. #1864C22semidirect
83. #1804C22semidirect
84. #1804C22semidirect
89. C4×#106C22semidirect
90. #1770C22semidirect
91. #1762C22semidirect
92. #1762C22semidirect
93. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×#23662
#331462
#330072
#275882
#270782
#267702
#265292
#258552
#253252
#245072
#242112
#232202
#231012
#226032
C4×#4832
#207392