Finite Group Databasebeta

← #6644845

Subgroups — #6644845

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35084C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic
3#33100C212yesmaximal, complemented
4#32270C212yesmaximal, complemented
5#32259C212yesmaximal, complemented
6#31347C214yesmaximal, complemented
7#31329C214yesmaximal, complemented
8#31272C212yesmaximal, complemented
9#31263C212yesmaximal, complemented
10#29641C212yesmaximal, complemented
11#27602C212yesmaximal, complemented
12#24559C212yesmaximal, complemented
13#24551C212yesmaximal, complemented
14#24003C212yesmaximal, complemented
15#21740C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#483C2212yes—
20C2×#483C2212yes—
21C2×#407C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2855C2212yes—
24#2848C2212yes—
25#2638—21no—
26#2638—21no—
27#2583—21no—
28#2563—21no—
29#2563—21no—
30#2563—21no—
31#2563—21no—
32#2563C2212yes—
33#2563C2212yes—
34#2563C2212yes—
35#2563C2212yes—
36#2561—21no—
37#2486C2211yescharacteristic
38#2486C2211yescharacteristic
39#2483—21no—
40#2415C2212yes—
41#2415C2212yes—
42#2409C2211yescharacteristic
43#2409C2211yescharacteristic
44#2405—21no—
45C2×#393—21no—
46C2×#393—21no—
47#2322—21no—
48#2322—21no—
49C2×#390—21no—
50C2×#382—21no—
51#2280C2211yescharacteristic
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2077C2214yescomplemented
56#2077C2214yescomplemented
57#2077C2214yescomplemented
58#2076C2214yescomplemented
59#2058C2214yescomplemented
60#2058C2214yescomplemented
61#2036C2214yescomplemented
62#2035C2214yescomplemented
63#2035C2214yescomplemented
64#2035C2214yescomplemented
65#2024C2214yescomplemented
66#2024C2214yescomplemented
67#1928C2214yescomplemented
68#1927C2214yescomplemented
69#1919C2214yescomplemented
70#1918C2214yescomplemented
71#1894C2214yescomplemented
72#1892C2214yescomplemented
73#1872C2212yes—
74#1872C2212yes—
75#1870C2212yes—
76#1870C2212yes—
77#1869C2212yes—
78#1869C2212yes—
79#1854C2214yescomplemented
80#1853C2212yes—
81#1853C2214yescomplemented
82#1852C2212yes—
83#1825C2214yescomplemented
84#1823C2212yes—
85#1823C2212yes—
86#1817C2212yes—
87#1809C2214yescomplemented
88#1807C2214yescomplemented
89#1787C2214yescomplemented
90#1779C2212yes—
91C23×Q8—21no—
92C23×Q8C2311yescharacteristic
93C23×D4D411yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#123—21no—
105C2×#123—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—42no—
109C2×C22⋊Q8—42no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C4×D4—42no—
115C2×C4×D4—42no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4D411yescharacteristic
135C22×C22⋊C4D411yescharacteristic
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#483—44no—
139#483—44no—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148#407—42no—
149C2×C22⋊C8—42no—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2314yes—
156#398—21no—
157#398—21no—
158#397—21no—
159#397—21no—
160#397—22no—
161#397—22no—
162#395—21no—
163#395—21no—
164#394—22no—
165#394—22no—
166#393—21no—
167#393—21no—
168#393—22no—
169#393—22no—
170#393—22no—
171#393—22no—
172#393—22no—
173#393—22no—
174#393—22no—
175#393—22no—
176#391C2312yes—
177#391C2312yes—
178#390—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#389—22no—
183#389—22no—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#386—21no—
191#386—21no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—21no—
197#385—21no—
198#382—22no—
199#382—22no—
200#382—22no—
201#382—22no—
202#381—22no—
203#381—21no—
204#381—21no—
205#381—21no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380C2314yes—
211#380C2314yes—
212#380C2314yes—
213#380C2314yes—
214#380C2312yes—
215#380C2312yes—
216#380C2311yescharacteristic
217#380C2311yescharacteristic
218#380C2312yes—
219#379—22no—
220#379—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—21no—
227C2×#95—21no—
228C2×#95—22no—
229C2×#95—22no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#336C2314yes—
239#336C2314yes—
240C25C2×D411yescharacteristic
241C25C2×D411yescharacteristic
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—41no—
246C22×Q8—41no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8—84no—
250C22×Q8C2412yes—
251C22×D4—24no—
252C22×D4—42no—
253C22×D4—42no—
254C22×D4C2×D411yescharacteristic
255C22×D4C2×D411yescharacteristic
256C22×D4C2×D411yescharacteristic
257C22×D4C2×D411yescharacteristic
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—44no—
267C23×C4—44no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×M16—44no—
282C2×M16—42no—
283C22×C8—44no—
284C22×C8—42no—
285C22×C8C2414yes—
286C22×C8C2414yes—
287C4⋊Q8—42no—
288C4⋊Q8—42no—
289C4⋊Q8—42no—
290C4⋊Q8—42no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293#123—44no—
294#123—44no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—42no—
298C22⋊Q8—42no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C4×D4—42no—
304C4×D4—42no—
305C4×D4—42no—
306C4×D4—42no—
307C4×D4—41no—
308C4×D4—41no—
309C4×D4—41no—
310C4×D4—41no—
311C4×D4—41no—
312C4×D4—41no—
313C4×D4—41no—
314C4×D4—41no—
315#117—44no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4—41no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2414yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—42no—
359C2×C22⋊C4—42no—
360C2×C22⋊C4—44no—
361C2×C22⋊C4—84no—
362C2×C22⋊C4—84no—
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C22⋊C4C2×D411yescharacteristic
365C2×C22⋊C4C2×D411yescharacteristic
366C2×C22⋊C4C2×D411yescharacteristic
367C2×C22⋊C4C2×D411yescharacteristic
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—44no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—44no—
382#106—44no—
383#106—44no—
384#106—44no—
385#103—44no—
386#103—44no—
387#103—44no—
388#103—44no—
389C22⋊C8—24no—
390C22⋊C8—24no—
391C22⋊C8—42no—
392C22⋊C8—44no—
393C22⋊C8—44no—
394C22⋊C8—42no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—24no—
417C24—24no—
418C24—42no—
419C24—42no—
420C24—41no—
421C24—41no—
422C24C22×D411yescharacteristic
423C24C22×D411yescharacteristic
424C24C22×D411yescharacteristic
425C24C22×D411yescharacteristic
426C24C22×D411yescharacteristic
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—82no—
435C2×Q8—84no—
436C2×Q8—84no—
437C2×Q8—84no—
438C2×Q8—84no—
439C2×Q8—816no—
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×D4#13714yes—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—84no—
500C22×C4—84no—
501C22×C4—84no—
502C22×C4—84no—
503C22×C4—82no—
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×C4H(5,𝔽2)12yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4C2511yescharacteristic
516M16—84no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—84no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—44no—
556C22⋊C4—42no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—42no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—84no—
563C22⋊C4—84no—
564C22⋊C4—84no—
565C22⋊C4—84no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C42—22no—
570C42—22no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—44no—
576C42—44no—
577C42—82no—
578C23—22no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—22no—
585C23—21no—
586C23—21no—
587C23—22no—
588C23—22no—
589C23—21no—
590C23—21no—
591C23—22no—
592C23—22no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—22no—
598C23—22no—
599C23—44no—
600C23—44no—
601C23—48no—
602C23—48no—
603C23—44no—
604C23—44no—
605C23—82no—
606C23—82no—
607C23—84no—
608C23—84no—
609C23#53412yes—
610C23C2×#13712yes—
611C23C2×#13712yes—
612C23#57714yes—
613C23C23×D411yescharacteristic
614Q8—44no—
615Q8—44no—
616Q8—88no—
617Q8—88no—
618Q8—84no—
619D4—44no—
620D4—44no—
621D4—44no—
622D4—44no—
623C2×C4—28no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—44no—
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—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
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—44no—
669C2×C4—44no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4—82no—
674C2×C4—82no—
675C2×C4—82no—
676C2×C4—88no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4—88no—
680C2×C4#53412yes—
681C2×C4C2×#13712yes—
682C2×C4#57512yes—
683C2×C4C2×H(5,𝔽2)11yescharacteristic
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C2×C4#58514yes—
686C8—44no—
687C8—44no—
688C22—24no—
689C22—24no—
690C22—22no—
691C22—22no—
692C22—24no—
693C22—24no—
694C22—22no—
695C22—22no—
696C22—42no—
697C22—42no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—88no—
713C22—88no—
714C22#320014yes—
715C22C2×#53411yescharacteristic
716C22#348512yes—
717C22C22×#13711yescharacteristic
718C22C2×#57511yescharacteristic
719C22C2×#57712yes—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—82no—
730C4—84no—
731C4—82no—
732C4—84no—
733C4—82no—
734C4—82no—
735C4#354512yes—
736C4#354512yes—
737C2—24no—
738C2—42no—
739C2—42no—
740C2—81no—
741C2—81no—
742C2#3711911yescharacteristic
743C2C2×#320011yescharacteristic
744C2#6075111yescharacteristic
745C2#6173112yes—
746C2#6173512yes—
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. #33100C2semidirect
4. #32270C2semidirect
5. #32259C2semidirect
6. #31347C2semidirect
7. #31329C2semidirect
8. #31272C2semidirect
9. #31263C2semidirect
10. #29641C2semidirect
11. #27602C2semidirect
12. #24559C2semidirect
13. #24551C2semidirect
14. #24003C2semidirect
55. #2077C22semidirect
56. #2077C22semidirect
57. #2077C22semidirect
58. #2076C22semidirect
59. #2058C22semidirect
60. #2058C22semidirect
61. #2036C22semidirect
62. #2035C22semidirect
63. #2035C22semidirect
64. #2035C22semidirect
65. #2024C22semidirect
66. #2024C22semidirect
67. #1928C22semidirect
68. #1927C22semidirect
69. #1919C22semidirect
70. #1918C22semidirect
71. #1894C22semidirect
72. #1892C22semidirect
79. #1854C22semidirect
81. #1853C22semidirect
83. #1825C22semidirect
87. #1809C22semidirect
88. #1807C22semidirect
89. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#350842
C2×#25632
#331002
#322702
#322592
#313472
#313292
#312722
#312632
#296412
#276022
#245592
#245512
#240032
#217402