Finite Group Databasebeta

← #6640557

Subgroups — #6640557

Order 512.

1839 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35240C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3#32983C212yesmaximal, complemented
4#32271C212yesmaximal, complemented
5#32258C212yesmaximal, complemented
6#32039C212yesmaximal, complemented
7#32027C212yesmaximal, complemented
8#32024C214yesmaximal, complemented
9#32006C214yesmaximal, complemented
10#29894C212yesmaximal, complemented
11#27617C212yesmaximal, complemented
12#24556C212yesmaximal, complemented
13#24552C212yesmaximal, complemented
14#24064C212yesmaximal, complemented
15#21740C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#123—21no—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#484C2212yes—
21C2×#484C2212yes—
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2848C2212yes—
25#2848C2212yes—
26#2633—21no—
27#2633—21no—
28#2547C2211yescharacteristic
29#2547C2211yescharacteristic
30#2542C2211yescharacteristic
31#2542C2211yescharacteristic
32#2536C2212yes—
33#2536C2212yes—
34#2502—21no—
35#2486—21no—
36#2486—21no—
37#2486—21no—
38#2486—21no—
39#2486C2212yes—
40#2486C2212yes—
41#2486C2212yes—
42#2486C2212yes—
43#2482—21no—
44#2482—21no—
45#2435—21no—
46C2×#398—21no—
47C2×#397—21no—
48#2326—21no—
49#2326—21no—
50C2×#389C2212yes—
51#2283C2211yescharacteristic
52C2×#381—21no—
53C2×#380C2211yescharacteristic
54C2×#380C2212yes—
55#2101C2214yescomplemented
56#2085C2214yescomplemented
57#2077C2214yescomplemented
58#2076C2214yescomplemented
59#2075C2214yescomplemented
60#2075C2214yescomplemented
61#2057C2214yescomplemented
62#2043C2214yescomplemented
63#2036C2214yescomplemented
64#2036C2214yescomplemented
65#2035C2214yescomplemented
66#2035C2214yescomplemented
67#1919C2214yescomplemented
68#1918C2214yescomplemented
69#1894C2214yescomplemented
70#1892C2214yescomplemented
71#1872C2212yes—
72#1872C2212yes—
73#1872C2214yescomplemented
74#1872C2214yescomplemented
75#1870C2212yes—
76#1870C2212yes—
77#1869C2212yes—
78#1869C2212yes—
79#1863C2214yescomplemented
80#1863C2214yescomplemented
81#1853C2212yes—
82#1852C2212yes—
83#1852C2214yescomplemented
84#1852C2214yescomplemented
85#1824C2212yes—
86#1824C2212yes—
87#1817C2212yes—
88#1809C2214yescomplemented
89#1808C2214yescomplemented
90#1779C2212yes—
91C23×D4D411yescharacteristic
92C24×C4—21no—
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C22×M16C2312yes—
96C23×C8C2312yes—
97#533—42no—
98#533—42no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125—21no—
102C2×#124—21no—
103C2×#124—21no—
104C2×#123—22no—
105C2×#123—22no—
106C2×#123—22no—
107C2×#123—22no—
108C2×#123—48no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×D4—42no—
118C2×C4×D4—42no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—42no—
133C22×C4⋊C4—42no—
134C22×C4⋊C4D411yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C22⋊C4—22no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4D411yescharacteristic
142C22×C22⋊C4D411yescharacteristic
143C22×C42C2311yescharacteristic
144#484—44no—
145#484—44no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154#407—42no—
155C2×C22⋊C8—42no—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2314yes—
162#399—21no—
163#399—21no—
164#399—21no—
165#399—21no—
166#398—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#398—22no—
172#397—21no—
173#397—21no—
174#397—21no—
175#397—21no—
176#397—22no—
177#397—22no—
178#397—22no—
179#397—22no—
180#394—22no—
181#394—22no—
182#391—21no—
183#391—21no—
184#390—21no—
185#390—21no—
186#389—22no—
187#389—22no—
188#389C2314yes—
189#389C2314yes—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—21no—
201#384—21no—
202#383—21no—
203#382C2312yes—
204#381—21no—
205#381—22no—
206#381—22no—
207#381—22no—
208#381—22no—
209#380—22no—
210#380—22no—
211#380—21no—
212#380—21no—
213#380C2312yes—
214#380C2314yes—
215#380C2312yes—
216#380C2314yes—
217#380C2312yes—
218#380C2312yes—
219#380C2312yes—
220#380C2312yes—
221#380C2312yes—
222#379—22no—
223#379—22no—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—21no—
227#376—21no—
228C2×#95—21no—
229C2×#95—22no—
230C2×#95—22no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C25C2×D411yescharacteristic
242C25C2×D411yescharacteristic
243C22×Q8—21no—
244C22×Q8C2412yes—
245C22×D4—24no—
246C22×D4—42no—
247C22×D4—42no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2×D411yescharacteristic
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D411yescharacteristic
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4—44no—
263C23×C4—44no—
264C23×C4—44no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2411yescharacteristic
274C23×C4C2411yescharacteristic
275C23×C4C2411yescharacteristic
276C23×C4C2411yescharacteristic
277C23×C4C2411yescharacteristic
278C2×M16—44no—
279C2×M16—42no—
280C22×C8—44no—
281C22×C8—42no—
282C22×C8C2414yes—
283C22×C8C2414yes—
284C4⋊Q8—41no—
285C4⋊Q8—41no—
286C4⋊Q8—41no—
287C4⋊Q8—41no—
288#125—44no—
289#124—44no—
290#124—44no—
291#123—48no—
292#123—48no—
293#123—48no—
294#123—48no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C4×Q8—42no—
303C4×Q8—42no—
304C4×Q8—44no—
305C4×D4—42no—
306C4×D4—42no—
307C4×D4—42no—
308C4×D4—42no—
309C4×D4—41no—
310C4×D4—41no—
311C4×D4—41no—
312C4×D4—41no—
313C4×D4—41no—
314C4×D4—41no—
315C4×D4—41no—
316C4×D4—41no—
317#117—44no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4—42no—
341C2×C4⋊C4—44no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4—48no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2414yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—42no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—48no—
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C22⋊C4C2×D411yescharacteristic
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#107—44no—
390#107—44no—
391#107—44no—
392#106—44no—
393#103—44no—
394#103—44no—
395#103—44no—
396#103—44no—
397C22⋊C8—24no—
398C22⋊C8—24no—
399C22⋊C8—42no—
400C22⋊C8—44no—
401C22⋊C8—44no—
402C22⋊C8—42no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—21no—
413#95—21no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—24no—
423C24—22no—
424C24—24no—
425C24—42no—
426C24—42no—
427C24—41no—
428C24C22×D411yescharacteristic
429C24C22×D411yescharacteristic
430C24C22×D411yescharacteristic
431C24C22×D411yescharacteristic
432C24C22×D411yescharacteristic
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—41no—
450C2×D4—41no—
451C2×D4—416no—
452C2×D4AGL(1,ℤ/8)14yes—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—41no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—84no—
504C22×C4—84no—
505C22×C4—816no—
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×C4C22×D411yescharacteristic
515C22×C4H(5,𝔽2)12yes—
516C22×C4H(5,𝔽2)12yes—
517C22×C4C2511yescharacteristic
518M16—84no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—84no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—48no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—48no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—41no—
556C22⋊C4—41no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4—44no—
562C22⋊C4—48no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—48no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C42—22no—
576C42—22no—
577C42—21no—
578C42—21no—
579C42—21no—
580C42—21no—
581C42—44no—
582C42—42no—
583C42—42no—
584C23—22no—
585C23—21no—
586C23—21no—
587C23—28no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—21no—
601C23—22no—
602C23—21no—
603C23—21no—
604C23—22no—
605C23—48no—
606C23—48no—
607C23—44no—
608C23—48no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—82no—
613C23—84no—
614C23#53412yes—
615C23C2×AGL(1,ℤ/8)12yes—
616C23C2×AGL(1,ℤ/8)12yes—
617C23#57512yes—
618C23#57814yes—
619C23C23×D411yescharacteristic
620Q8—44no—
621Q8—44no—
622Q8—84no—
623D4—44no—
624D4—44no—
625D4—44no—
626D4—44no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
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—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—82no—
679C2×C4—82no—
680C2×C4—82no—
681C2×C4—88no—
682C2×C4—88no—
683C2×C4—88no—
684C2×C4—88no—
685C2×C4#53412yes—
686C2×C4C2×AGL(1,ℤ/8)12yes—
687C2×C4C2×H(5,𝔽2)11yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4#58514yes—
690C8—44no—
691C8—44no—
692C22—24no—
693C22—24no—
694C22—22no—
695C22—22no—
696C22—24no—
697C22—24no—
698C22—24no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—416no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—42no—
713C22—42no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—88no—
718C22#320214yes—
719C22C2×#53411yescharacteristic
720C22#348512yes—
721C22C22×AGL(1,ℤ/8)11yescharacteristic
722C22C2×#57511yescharacteristic
723C22C2×#57812yes—
724C4—22no—
725C4—22no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—82no—
734C4—84no—
735C4—82no—
736C4—84no—
737C4—82no—
738C4—82no—
739C4#353312yes—
740C4#353312yes—
741C2—24no—
742C2—42no—
743C2—42no—
744C2—42no—
745C2—81no—
746C2#3695511yescharacteristic
747C2C2×#320211yescharacteristic
748C2#6075011yescharacteristic
749C2#6173712yes—
750C2#6173812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32983C2semidirect
4. #32271C2semidirect
5. #32258C2semidirect
6. #32039C2semidirect
7. #32027C2semidirect
8. #32024C2semidirect
9. #32006C2semidirect
10. #29894C2semidirect
11. #27617C2semidirect
12. #24556C2semidirect
13. #24552C2semidirect
14. #24064C2semidirect
55. #2101C22semidirect
56. #2085C22semidirect
57. #2077C22semidirect
58. #2076C22semidirect
59. #2075C22semidirect
60. #2075C22semidirect
61. #2057C22semidirect
62. #2043C22semidirect
63. #2036C22semidirect
64. #2036C22semidirect
65. #2035C22semidirect
66. #2035C22semidirect
67. #1919C22semidirect
68. #1918C22semidirect
69. #1894C22semidirect
70. #1892C22semidirect
73. #1872C22semidirect
74. #1872C22semidirect
79. #1863C22semidirect
80. #1863C22semidirect
83. #1852C22semidirect
84. #1852C22semidirect
88. #1809C22semidirect
89. #1808C22semidirect

Maximal subgroups

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

subgroupindex
#352402
C2×#24862
#329832
#322712
#322582
#320392
#320272
#320242
#320062
#298942
#276172
#245562
#245522
#240642
#217402