Finite Group Databasebeta

← #6482926

Subgroups — #6482926

Order 512.

2084 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#33096C214yesmaximal, complemented
4#29891C214yesmaximal, complemented
5#27891C212yesmaximal, complemented
6#27890C212yesmaximal, complemented
7#27878C212yesmaximal, complemented
8#27875C212yesmaximal, complemented
9#27627C214yesmaximal, complemented
10#27626C214yesmaximal, complemented
11#21982C211yesmaximal, characteristic, complemented
12#21981C211yesmaximal, characteristic, complemented
13#21859C211yesmaximal, characteristic, complemented
14#21859C211yesmaximal, characteristic, complemented
15#20714C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#525—21no—
18C22×#117—21no—
19C23×C42C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2856C2211yescharacteristic
26#2854—21no—
27#2854C2212yes—
28#2850—21no—
29#2850—21no—
30#2847—21no—
31#2847C2212yes—
32C4×#121—21no—
33C4×#121—21no—
34C4×#121C2212yes—
35C4×#121C2212yes—
36#2297—21no—
37#2285—21no—
38#2285—21no—
39#2285—21no—
40#2285—21no—
41#2285C2211yescharacteristic
42#2285C2212yes—
43#2285C2212yes—
44#2285C2212yes—
45#2285C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#382—21no—
49C2×#380—21no—
50C2×#380C2212yes—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2212yes—
53#2077C2218yes—
54#2076C2218yes—
55#2034C2218yes—
56#2033C2218yes—
57#1990C2212yescomplemented
58#1990C2212yescomplemented
59#1957C2212yescomplemented
60#1957C2212yescomplemented
61#1948C2214yescomplemented
62#1947C2214yescomplemented
63#1926C2218yescomplemented
64#1925C2218yescomplemented
65#1921C2218yescomplemented
66#1920C2218yescomplemented
67#1874C2214yes—
68#1874C2214yes—
69#1853C2214yes—
70#1852C2214yes—
71#1845C2212yescomplemented
72#1844C2212yescomplemented
73#1834C2212yescomplemented
74#1834C2212yescomplemented
75#1817C2212yes—
76#1802C2214yescomplemented
77#1802C2214yescomplemented
78#1797C2212yes—
79#1797C2212yes—
80#1779C2212yes—
81C4×C22⋊C8C2212yes—
82C4×C22⋊C8C2212yes—
83C22×#41—21no—
84C24×C4D411yescharacteristic
85C24×C4D411yescharacteristic
86C24×C4C2311yescharacteristic
87#532—24no—
88#532—24no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127C2311yescharacteristic
91C2×#124—21no—
92#525—48no—
93C2×#123—22no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×#121—21no—
97C2×#121—21no—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121—21no—
101C2×#121C2312yes—
102C2×#121C2312yes—
103C2×#121C2312yes—
104C2×#120—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4C2312yes—
109C2×#117—22no—
110C2×#117—22no—
111C2×#117—22no—
112C2×#117—22no—
113C2×#117—21no—
114C2×#117—48no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4—22no—
122C22×C42—24no—
123C22×C42—24no—
124C22×C42—22no—
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#431—24no—
138#431—24no—
139C2×#107C2318yes—
140C2×#106C2318yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#102C2318yes—
144C2×#102C2318yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2312yes—
151#391—21no—
152#391—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#389—21no—
160#389—21no—
161#389C2312yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#386—22no—
168#386—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#384—21no—
174#384—21no—
175#384—21no—
176#384—21no—
177#384C2312yes—
178#384C2312yes—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#382—21no—
184#382—22no—
185#382—22no—
186#382—22no—
187#382—22no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C2×#95—22no—
215C2×#95—22no—
216#340C2318yes—
217#340C2318yes—
218#340C2318yes—
219#340C2318yes—
220#336C2314yes—
221#336C2314yes—
222C25C2×D411yescharacteristic
223C2×#41—44no—
224C2×#41—44no—
225C2×#41—44no—
226C22×Q8—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2412yes—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C22×C8C2414yes—
265C22×C8C2414yes—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268#127—22no—
269#127—22no—
270#124—44no—
271#123—48no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274#121—24no—
275#121—24no—
276#121—44no—
277#121—44no—
278#121—44no—
279#121—44no—
280#121—44no—
281#121—44no—
282#121—44no—
283#120—48no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—48no—
287C4×D4—48no—
288C4×D4—44no—
289#117—48no—
290#117—48no—
291#117—48no—
292#117—48no—
293#117—44no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—48no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—48no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42#4114yes—
369C2×C42#4114yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—48no—
375#106—48no—
376C4⋊C8—24no—
377C4⋊C8—24no—
378#102—48no—
379#102—48no—
380C22⋊C8—24no—
381C22⋊C8—24no—
382C22⋊C8—44no—
383C22⋊C8—44no—
384C4×C8—24no—
385C4×C8—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—41no—
403C24—41no—
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406C24C22×D411yescharacteristic
407#41—816no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C22×C4—232no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—416no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4#12314yes—
498C22×C4#12314yes—
499C22×C4#13514yes—
500C22×C4AGL(1,ℤ/8)14yes—
501C22×C4AGL(1,ℤ/8)14yes—
502C22×C4#13714yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4C2511yescharacteristic
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—28no—
526C4⋊C4—28no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—48no—
574C42—48no—
575C42—44no—
576C42#12314yes—
577C42#12314yes—
578C42C2×#4112yes—
579C42C2×#4112yes—
580C42C2×#4112yes—
581C42C2×#4112yes—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—28no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23C2×#12312yes—
613C23C2×#13512yes—
614C23C2×AGL(1,ℤ/8)12yes—
615C23C2×AGL(1,ℤ/8)12yes—
616C23C2×#13712yes—
617C23C23×D411yescharacteristic
618Q8—84no—
619D4—44no—
620D4—44no—
621D4—88no—
622D4—88no—
623D4—88no—
624D4—84no—
625D4—84no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—88no—
682C2×C4—88no—
683C2×C4—88no—
684C2×C4#482116yes—
685C2×C4C2×#12312yes—
686C2×C4C2×#12312yes—
687C2×C4C2×#12312yes—
688C2×C4C2×#12312yes—
689C2×C4C2×#12312yes—
690C2×C4C2×#13512yes—
691C2×C4C2×#13512yes—
692C2×C4C2×AGL(1,ℤ/8)12yes—
693C2×C4C2×AGL(1,ℤ/8)12yes—
694C2×C4C2×AGL(1,ℤ/8)12yes—
695C2×C4C2×AGL(1,ℤ/8)12yes—
696C2×C4C2×#13712yes—
697C2×C4C2×#13712yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4#58512yes—
701C2×C4#58512yes—
702C8—44no—
703C8—44no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—22no—
716C22—416no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—41no—
724C22—41no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22C2×#48214yes—
733C22C22×#12311yescharacteristic
734C22#344911yescharacteristic
735C22#344911yescharacteristic
736C22C22×#13511yescharacteristic
737C22C22×AGL(1,ℤ/8)11yescharacteristic
738C22C22×AGL(1,ℤ/8)11yescharacteristic
739C22C22×#13711yescharacteristic
740C4—22no—
741C4—22no—
742C4—22no—
743C4—22no—
744C4—41no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—84no—
749C4—84no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4C2×#48214yes—
754C4C2×#48214yes—
755C4#309214yes—
756C4#349312yes—
757C4#349312yes—
758C2—24no—
759C2—42no—
760C2—42no—
761C2—162no—
762C2#3515711yescharacteristic
763C2C22×#48211yescharacteristic
764C2C2×#309211yescharacteristic
765C2#6094611yescharacteristic
766C2#6094711yescharacteristic
767C2#6096811yescharacteristic
768C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36624C2
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
2. C2×#2285C2semidirect
3. #33096C2semidirect
4. #29891C2semidirect
5. #27891C2semidirect
6. #27890C2semidirect
7. #27878C2semidirect
8. #27875C2semidirect
9. #27627C2semidirect
10. #27626C2semidirect
11. #21982C2semidirect
12. #21981C2semidirect
13. #21859C2semidirect
14. #21859C2semidirect
15. #20714C2semidirect
57. #1990C22semidirect
58. #1990C22semidirect
59. #1957C22semidirect
60. #1957C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1926C22semidirect
64. #1925C22semidirect
65. #1921C22semidirect
66. #1920C22semidirect
71. #1845C22semidirect
72. #1844C22semidirect
73. #1834C22semidirect
74. #1834C22semidirect
76. #1802C22semidirect
77. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#22852
#330962
#298912
#278912
#278902
#278782
#278752
#276272
#276262
#219822
#219812
#218592
#207142