Finite Group Databasebeta

← #6660949

Subgroups — #6660949

Order 512.

1884 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2855C211yesmaximal, characteristic
3#32741C214yesmaximal, complemented
4#30366C214yesmaximal, complemented
5#29966C212yesmaximal, complemented
6#29951C212yesmaximal, complemented
7#29908C212yesmaximal, complemented
8#29642C212yesmaximal, complemented
9#29302C212yesmaximal, complemented
10#29111C212yesmaximal, complemented
11#27601C212yesmaximal
12#24606C214yesmaximal, complemented
13#24006C212yesmaximal
14#20715C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8—21no—
17C22×C22⋊Q8C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#484C2214yes—
20C2×#432C2212yes—
21#2857—21no—
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2855—21no—
30#2855—21no—
31#2855—21no—
32#2855—21no—
33#2855C2212yes—
34#2855C2212yes—
35#2855C2212yes—
36#2855C2212yes—
37C2×#391—21no—
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#389—21no—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2212yes—
44C2×#387—21no—
45C2×#387—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#381—21no—
49C2×#380—21no—
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52#2099C2214yescomplemented
53#2096C2214yescomplemented
54#2086C2218yescomplemented
55#2084C2214yescomplemented
56#2077C2218yes—
57#2072C2214yescomplemented
58#2066C2218yescomplemented
59#2057C2214yescomplemented
60#2057C2214yescomplemented
61#2036C2218yes—
62#2024C2214yescomplemented
63#2024C2214yescomplemented
64#1944C2218yescomplemented
65#1935C2218yescomplemented
66#1872C2214yes—
67#1853C2214yes—
68#1853C2214yes—
69#1852C2214yes—
70#1824C2214yes—
71#1823C2214yes—
72#1817C2212yes—
73#1797C2212yes—
74#1797C2212yes—
75#1787C2214yes—
76#1779C2212yes—
77#1749C2212yes—
78#1749C2212yes—
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—44no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8—48no—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C42—44no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#484—48no—
128#432—44no—
129C2×#107C2318yes—
130C2×#106C2318yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2318yes—
133C2×#103C2318yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2314yes—
138#391—21no—
139#391—21no—
140#391—21no—
141#391—21no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391C2312yes—
147#391C2312yes—
148#391C2312yes—
149#391C2312yes—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2312yes—
162#389C2314yes—
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384C2312yes—
181#384C2312yes—
182#381—22no—
183#381—22no—
184#381—22no—
185#381—22no—
186#380—21no—
187#380—21no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380C2312yes—
195#380C2314yes—
196#380C2314yes—
197#380C2312yes—
198C2×#95—22no—
199C2×#95—21no—
200C2×#95—22no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207#340C2318yes—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213C25C2×D411yescharacteristic
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—21no—
217C22×Q8—24no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8—44no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8C2412yes—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
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×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C22×C8C2418yes—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C22⋊Q8—28no—
263C22⋊Q8—44no—
264C22⋊Q8—48no—
265C22⋊Q8—44no—
266C22⋊Q8—48no—
267C22⋊Q8—44no—
268C22⋊Q8—48no—
269C22⋊Q8—48no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—48no—
314C2×C22⋊C4—44no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—48no—
344#106—48no—
345C4⋊C8—44no—
346#103—48no—
347#103—48no—
348C22⋊C8—28no—
349C22⋊C8—28no—
350C22⋊C8—44no—
351#97—44no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—41no—
375C24—41no—
376C24#13714yes—
377C24C22×D411yescharacteristic
378C24C22×D411yescharacteristic
379C24C22×D411yescharacteristic
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—88no—
393C2×Q8—816no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4AGL(1,ℤ/8)14yes—
472C22×C4#13714yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
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×C4H(5,𝔽2)14yes—
485C22×C42−1+414yes—
486C22×C4C2511yescharacteristic
487C2×C8—28no—
488C2×C8—28no—
489C2×C8—28no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—28no—
548C23—28no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23#53412yes—
558C23#53612yes—
559C23C2×AGL(1,ℤ/8)12yes—
560C23C2×#13712yes—
561C23C2×#13712yes—
562C23C2×#13712yes—
563C23C2×#13712yes—
564C23C23×D411yescharacteristic
565Q8—48no—
566Q8—84no—
567Q8—88no—
568Q8—88no—
569C2×C4—28no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—28no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—88no—
636C2×C4—84no—
637C2×C4#53412yes—
638C2×C4#53612yes—
639C2×C4C2×AGL(1,ℤ/8)12yes—
640C2×C4C2×AGL(1,ℤ/8)12yes—
641C2×C4C2×#13712yes—
642C2×C4C2×#13712yes—
643C2×C4#57512yes—
644C2×C4C2×H(5,𝔽2)12yes—
645C2×C4C2×2−1+412yes—
646C2×C4#58512yes—
647C8—48no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—416no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22#319514yes—
669C22#323914yes—
670C22C2×#53411yescharacteristic
671C22C2×#53611yescharacteristic
672C22#344911yescharacteristic
673C22C22×AGL(1,ℤ/8)11yescharacteristic
674C22C22×#13711yescharacteristic
675C22C22×#13711yescharacteristic
676C22C2×#57511yescharacteristic
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—82no—
689C4—84no—
690C4—82no—
691C4—82no—
692C4—84no—
693C4—82no—
694C4#353514yes—
695C2—24no—
696C2—24no—
697C2—42no—
698C2#3744111yescharacteristic
699C2C2×#319511yescharacteristic
700C2#6075011yescharacteristic
701C2C2×#323911yescharacteristic
702C2#6086511yescharacteristic
703C2#6096011yescharacteristic
704C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #32741C2semidirect
4. #30366C2semidirect
5. #29966C2semidirect
6. #29951C2semidirect
7. #29908C2semidirect
8. #29642C2semidirect
9. #29302C2semidirect
10. #29111C2semidirect
12. #24606C2semidirect
52. #2099C22semidirect
53. #2096C22semidirect
54. #2086C22semidirect
55. #2084C22semidirect
57. #2072C22semidirect
58. #2066C22semidirect
59. #2057C22semidirect
60. #2057C22semidirect
62. #2024C22semidirect
63. #2024C22semidirect
64. #1944C22semidirect
65. #1935C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#28552
#327412
#303662
#299662
#299512
#299082
#296422
#293022
#291112
#276012
#246062
#240062
#207152