Finite Group Databasebeta

← #6480853

Subgroups — #6480853

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#33081C214yesmaximal, complemented
4#30150C212yesmaximal, complemented
5#30130C212yesmaximal, complemented
6#29514C212yesmaximal, complemented
7#29494C212yesmaximal, complemented
8#29435C214yesmaximal, complemented
9#29243C212yesmaximal, complemented
10#28123C212yesmaximal
11#27611C211yesmaximal, characteristic, complemented
12#27582C211yesmaximal, characteristic
13#24064C211yesmaximal, characteristic, complemented
14#23933C211yesmaximal, characteristic
15#22079C212yesmaximal, complemented
16#22043C212yesmaximal
17#20777C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×#126—21no—
20C2×#484C2212yes—
21C2×#450C2212yes—
22#2860—21no—
23#2860—21no—
24#2860—21no—
25#2860—21no—
26#2860C2212yes—
27#2860C2212yes—
28#2860C2212yes—
29#2860C2212yes—
30C4×C4⋊Q8C2211yescharacteristic, complemented
31C4×C4⋊Q8C2211yescharacteristic, complemented
32C4×C4⋊Q8C2211yescharacteristic, complemented
33C4×C4⋊Q8C2211yescharacteristic, complemented
34C4×#125—21no—
35C4×#125—21no—
36C4×C22⋊Q8—21no—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8C2212yes—
39C4×C22⋊Q8C2212yes—
40#2296—21no—
41#2295—21no—
42#2295C2212yescomplemented
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#388—21no—
50C2×#388—21no—
51C2×#388—21no—
52#2285C2211yescharacteristic
53C42×Q8—21no—
54C42×Q8C2212yescomplemented
55C4×#117—21no—
56C2×C43C2211yescharacteristic
57#2099C2212yescomplemented
58#2093C2212yes—
59#2088C2214yescomplemented
60#2084C2212yescomplemented
61#2071C2212yes—
62#2068C2214yescomplemented
63#2067C2214yes—
64#2057C2212yescomplemented
65#2055C2214yes—
66#2043C2212yescomplemented
67#2023C2212yes—
68#2013C2212yes—
69#1968C2214yes—
70#1942C2214yescomplemented
71#1934C2214yes—
72#1933C2214yescomplemented
73#1926C2214yes—
74#1925C2214yes—
75#1925C2214yes—
76#1921C2214yescomplemented
77#1920C2214yescomplemented
78#1918C2214yescomplemented
79#1880C2214yes—
80#1880C2214yes—
81#1880C2214yes—
82#1872C2212yes—
83#1872C2212yes—
84#1865C2214yescomplemented
85#1863C2214yescomplemented
86#1863C2214yescomplemented
87#1853C2212yes—
88#1852C2212yes—
89#1839C2212yescomplemented
90#1831C2212yescomplemented
91#1824C2212yes—
92#1817C2211yescharacteristic
93#1817C2211yescharacteristic
94#1797C2212yes—
95#1786C2212yes—
96C4×C4⋊C8C2212yescomplemented
97C4×C22⋊C8C2212yes—
98#1744C2212yescomplemented
99C23×D4C2311yescharacteristic
100C22×SD16C2314yes—
101#533—28no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#126—22no—
105C2×#126—22no—
106C2×#126—22no—
107C2×#126—22no—
108C2×#126—48no—
109C2×#125—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×#117—21no—
119C2×#117—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#484—28no—
140#450—28no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C4×C8C2314yes—
154#391—21no—
155#391—21no—
156#391C2312yes—
157#391C2312yes—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#385—22no—
207#385—22no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#382—21no—
211#380—21no—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231#376—22no—
232C2×#95—21no—
233C2×#95—21no—
234C2×#95—21no—
235C43—22no—
236C43D411yescharacteristic
237C43D411yescharacteristic
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43C2311yescharacteristic
241C43C2311yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2314yes—
251#336C2312yes—
252#336C2312yes—
253#336C2312yes—
254#336C2312yes—
255#336C2314yes—
256C25—21no—
257C22×Q8—21no—
258C22×Q8C2412yes—
259C22×D4—21no—
260C22×D4—24no—
261C22×D4—42no—
262C22×D4—42no—
263C22×D4C2412yes—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2411yescharacteristic
274C2×SD16—28no—
275C2×SD16—44no—
276C2×SD16—44no—
277C22×C8C2414yes—
278C22×C8C2414yes—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281#126—48no—
282#126—48no—
283#126—48no—
284#126—48no—
285#125—28no—
286C22⋊Q8—28no—
287C22⋊Q8—24no—
288C22⋊Q8#4118yes—
289C4×Q8—28no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8—44no—
296#117—28no—
297#117—44no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
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—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—42no—
363C2×C42—42no—
364C2×C42—42no—
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×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393#107—28no—
394#107—44no—
395#106—44no—
396C4⋊C8—44no—
397#103—28no—
398#103—44no—
399#103—44no—
400#102—28no—
401#102—44no—
402#102—44no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405C4×C8—44no—
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—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422C24—22no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24C22×D411yescharacteristic
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8C2×#4114yes—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—88no—
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—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4#13514yes—
481C22×C4#13714yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4114yes—
496C22×C4C2511yescharacteristic
497SD16—88no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—28no—
507C4⋊C4—24no—
508C4⋊C4—28no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4C2×#4114yes—
534C22⋊C4—28no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C42—28no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—44no—
577C42—44no—
578C42—44no—
579C42—44no—
580C42—44no—
581C42AGL(1,ℤ/8)14yes—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23#52512yes—
601C23C2×#13512yes—
602C23C2×#13712yes—
603C23C23×D411yescharacteristic
604Q8—28no—
605Q8—48no—
606Q8—44no—
607D4—44no—
608D4—44no—
609D4—84no—
610C2×C4—28no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
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—28no—
630C2×C4—21no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
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—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4#52512yes—
672C2×C4C2×#13512yes—
673C2×C4C2×#13512yes—
674C2×C4C2×AGL(1,ℤ/8)12yes—
675C2×C4C2×AGL(1,ℤ/8)12yes—
676C2×C4C2×AGL(1,ℤ/8)12yes—
677C2×C4C2×#13712yes—
678C2×C4C2×#13712yes—
679C2×C4#57512yes—
680C2×C4C22×#4112yes—
681C2×C4#58512yes—
682C2×C4#58512yes—
683C8—44no—
684C8—44no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—168no—
699C22#310814yes—
700C22C2×#52511yescharacteristic
701C22#344911yescharacteristic
702C22#344911yescharacteristic
703C22C22×#13511yescharacteristic
704C22C22×AGL(1,ℤ/8)11yescharacteristic
705C22C22×#13711yescharacteristic
706C22C2×#57511yescharacteristic
707C4—22no—
708C4—22no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—22no—
715C4—22no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—44no—
720C4—41no—
721C4—44no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4#352114yes—
728C2—24no—
729C2—162no—
730C2#3514311yescharacteristic
731C2C2×#310811yescharacteristic
732C2#6056111yescharacteristic
733C2#6094211yescharacteristic
734C2#6094611yescharacteristic
735C2#6096011yescharacteristic
736C2#6096911yescharacteristic
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
2. C4×#533C2semidirect
3. #33081C2semidirect
4. #30150C2semidirect
5. #30130C2semidirect
6. #29514C2semidirect
7. #29494C2semidirect
8. #29435C2semidirect
9. #29243C2semidirect
11. #27611C2semidirect
13. #24064C2semidirect
15. #22079C2semidirect
17. #20777C2semidirect
30. C4×C4⋊Q8C22semidirect
31. C4×C4⋊Q8C22semidirect
32. C4×C4⋊Q8C22semidirect
33. C4×C4⋊Q8C22semidirect
42. #2295C22semidirect
54. C42×Q8C22semidirect
57. #2099C22semidirect
59. #2088C22semidirect
60. #2084C22semidirect
62. #2068C22semidirect
64. #2057C22semidirect
66. #2043C22semidirect
70. #1942C22semidirect
72. #1933C22semidirect
76. #1921C22semidirect
77. #1920C22semidirect
78. #1918C22semidirect
84. #1865C22semidirect
85. #1863C22semidirect
86. #1863C22semidirect
89. #1839C22semidirect
90. #1831C22semidirect
96. C4×C4⋊C8C22semidirect
98. #1744C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28602
C4×#5332
#330812
#301502
#301302
#295142
#294942
#294352
#292432
#281232
#276112
#275822
#240642
#239332
#220792
#220432
#207772