Finite Group Databasebeta

← #6498106

Subgroups — #6498106

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#33151C212yesmaximal, complemented
4#32959C212yesmaximal
5#29987C212yesmaximal, complemented
6#29870C212yesmaximal, complemented
7#29580C212yesmaximal, complemented
8#29578C212yesmaximal, complemented
9#29396C214yesmaximal, complemented
10#29303C212yesmaximal, complemented
11#29110C212yesmaximal
12#27613C211yesmaximal, characteristic, complemented
13#27610C211yesmaximal, characteristic, complemented
14#27585C211yesmaximal, characteristic
15#27582C211yesmaximal, characteristic
16#22120C214yesmaximal, complemented
17#20778C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#478C2214yes—
21#2864—21no—
22#2864—21no—
23#2859C2211yescharacteristic, complemented
24#2859C2211yescharacteristic, complemented
25#2859C2211yescharacteristic, complemented
26#2859C2211yescharacteristic, complemented
27#2851—21no—
28#2851—21no—
29#2851C2212yes—
30#2851C2212yes—
31C4×#125—21no—
32C4×#125—21no—
33C4×#125C2212yescomplemented
34C4×#125C2212yescomplemented
35#2296—21no—
36#2296—21no—
37#2293—21no—
38#2293—21no—
39#2293—21no—
40#2293—21no—
41#2293C2212yes—
42#2293C2212yes—
43#2293C2212yes—
44#2293C2212yes—
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#388—21no—
50C2×#388—21no—
51#2285C2211yescharacteristic
52#2285C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4—21no—
55C2×C43C2211yescharacteristic
56#2100C2212yescomplemented
57#2099C2212yescomplemented
58#2094C2212yes—
59#2093C2212yes—
60#2085C2214yescomplemented
61#2084C2212yescomplemented
62#2083C2212yescomplemented
63#2073C2214yes—
64#2071C2212yes—
65#2071C2212yes—
66#2059C2214yescomplemented
67#2025C2214yes—
68#1989C2214yescomplemented
69#1949C2214yescomplemented
70#1944C2214yescomplemented
71#1943C2214yescomplemented
72#1936C2214yes—
73#1936C2214yes—
74#1926C2214yes—
75#1925C2214yes—
76#1921C2214yescomplemented
77#1919C2214yescomplemented
78#1919C2214yescomplemented
79#1918C2214yescomplemented
80#1864C2214yes—
81#1862C2214yes—
82#1853C2212yes—
83#1853C2212yes—
84#1852C2212yes—
85#1852C2212yes—
86#1839C2212yescomplemented
87#1831C2212yescomplemented
88#1823C2212yes—
89#1823C2212yes—
90#1817C2211yescharacteristic
91#1817C2211yescharacteristic
92#1804C2214yes—
93#1797C2212yes—
94#1786C2212yes—
95#1786C2212yes—
96#1767C2212yescomplemented
97#1749C2212yes—
98#1745C2212yescomplemented
99C23×D4C2311yescharacteristic
100C24×C4—21no—
101C2×#125—21no—
102C2×#125—21no—
103C2×#125C2312yes—
104C2×#125C2312yes—
105C2×C4×D4—28no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117—21no—
111C2×#117—21no—
112C2×#117—22no—
113C2×#117—22no—
114C2×#117—22no—
115C2×#117—22no—
116C2×#117—48no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42—21no—
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#478—216no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×#97C2314yes—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
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#389—21no—
170#389—21no—
171#389—21no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#382—21no—
205#382—21no—
206#382—21no—
207#382—21no—
208#382C2312yes—
209#382C2312yes—
210#380—21no—
211#380—21no—
212#380C2312yes—
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C2×#95—21no—
231C2×#95—21no—
232C2×#95—21no—
233C43D411yescharacteristic
234C43D411yescharacteristic
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43C2311yescharacteristic
238C43C2311yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2314yes—
250#336C2314yes—
251#336C2312yes—
252#336C2312yes—
253#336C2312yes—
254#336C2312yes—
255C25—21no—
256C22×D4—24no—
257C22×D4—21no—
258C22×D4—24no—
259C22×D4—42no—
260C22×D4C2412yes—
261C23×C4—28no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—44no—
268C23×C4—44no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2411yescharacteristic
276C22×C8C2414yes—
277C22×C8C2414yes—
278#125—28no—
279#125—44no—
280#125—44no—
281#125—44no—
282C4×D4—28no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4#4118yes—
288#117—48no—
289#117—48no—
290#117—48no—
291#117—48no—
292#117—44no—
293#117—44no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—42no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42—44no—
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×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—28no—
387#107—44no—
388#106—28no—
389#106—44no—
390C4⋊C8—28no—
391#102—28no—
392#102—28no—
393#102—44no—
394#102—44no—
395C22⋊C8—24no—
396C22⋊C8—24no—
397#97—44no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416C24—24no—
417C24—22no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24C22×D411yescharacteristic
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—416no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4C2×#4114yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—432no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4AGL(1,ℤ/8)14yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C42−1+414yes—
490C22×C4C2511yescharacteristic
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4C2×#4114yes—
528C22⋊C4—28no—
529C22⋊C4—28no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—44no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42C2×#4114yes—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—24no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23#53612yes—
591C23C2×AGL(1,ℤ/8)12yes—
592C23#57512yes—
593C23C23×D411yescharacteristic
594D4—28no—
595D4—48no—
596D4—44no—
597C2×C4—28no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—28no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—28no—
614C2×C4—21no—
615C2×C4—24no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—416no—
636C2×C4—416no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4#52512yes—
663C2×C4#52512yes—
664C2×C4#53612yes—
665C2×C4C2×AGL(1,ℤ/8)12yes—
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4#57512yes—
668C2×C4#57512yes—
669C2×C4C22×#4112yes—
670C2×C4C2×2−1+412yes—
671C2×C4#58512yes—
672C8—44no—
673C8—44no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—816no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22#323414yes—
689C22C2×#52511yescharacteristic
690C22C2×#53611yescharacteristic
691C22#344911yescharacteristic
692C22C22×AGL(1,ℤ/8)11yescharacteristic
693C22C2×#57511yescharacteristic
694C22C2×#57511yescharacteristic
695C22C2×#57511yescharacteristic
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—44no—
709C4—41no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4#349514yes—
717C2—24no—
718C2—84no—
719C2#3532511yescharacteristic
720C2#6054611yescharacteristic
721C2#6055811yescharacteristic
722C2C2×#323411yescharacteristic
723C2#6086611yescharacteristic
724C2#6095411yescharacteristic
725C2#6096511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #36640C2semidirect
3. #33151C2semidirect
5. #29987C2semidirect
6. #29870C2semidirect
7. #29580C2semidirect
8. #29578C2semidirect
9. #29396C2semidirect
10. #29303C2semidirect
12. #27613C2semidirect
13. #27610C2semidirect
16. #22120C2semidirect
17. #20778C2semidirect
23. #2859C22semidirect
24. #2859C22semidirect
25. #2859C22semidirect
26. #2859C22semidirect
33. C4×#125C22semidirect
34. C4×#125C22semidirect
56. #2100C22semidirect
57. #2099C22semidirect
60. #2085C22semidirect
61. #2084C22semidirect
62. #2083C22semidirect
66. #2059C22semidirect
68. #1989C22semidirect
69. #1949C22semidirect
70. #1944C22semidirect
71. #1943C22semidirect
76. #1921C22semidirect
77. #1919C22semidirect
78. #1919C22semidirect
79. #1918C22semidirect
86. #1839C22semidirect
87. #1831C22semidirect
96. #1767C22semidirect
98. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#366402
C2×#22932
#331512
#329592
#299872
#298702
#295802
#295782
#293962
#293032
#291102
#276132
#276102
#275852
#275822
#221202
#207782