Finite Group Databasebeta

← #6660599

Subgroups — #6660599

Order 512.

1850 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#32786C212yesmaximal, complemented
4#32771C212yesmaximal, complemented
5#30366C214yesmaximal, complemented
6#29938C212yesmaximal, complemented
7#29894C214yesmaximal
8#29691C212yesmaximal, complemented
9#29335C212yesmaximal, complemented
10#29302C212yesmaximal, complemented
11#27680C212yesmaximal, complemented
12#24711C214yesmaximal, complemented
13#24302C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C23×C42C2211yescharacteristic
19C2×#477C2214yescomplemented
20C2×#432C2212yes—
21#2857—21no—
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2855—21no—
34#2855C2212yes—
35#2851—21no—
36#2851—21no—
37#2848—21no—
38#2848C2212yes—
39C4×C22⋊Q8—21no—
40C4×C22⋊Q8—21no—
41C4×C22⋊Q8C2212yes—
42C4×C22⋊Q8C2212yes—
43#2296—21no—
44C2×#391—21no—
45C2×#389—21no—
46C2×#389C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49#2280—21no—
50C2×#380—21no—
51C2×#380C2212yes—
52#2099C2214yescomplemented
53#2099C2214yescomplemented
54#2096C2214yescomplemented
55#2084C2214yescomplemented
56#2077C2218yes—
57#2075C2218yes—
58#2057C2214yescomplemented
59#2057C2214yescomplemented
60#2043C2214yescomplemented
61#2036C2218yes—
62#2035C2218yes—
63#2024C2214yescomplemented
64#1987C2214yescomplemented
65#1948C2214yescomplemented
66#1944C2218yescomplemented
67#1935C2218yescomplemented
68#1872C2214yes—
69#1872C2214yes—
70#1853C2214yes—
71#1852C2214yes—
72#1817C2212yes—
73#1803C2214yescomplemented
74#1797C2212yes—
75#1797C2212yes—
76#1779C2212yes—
77#1749C2212yes—
78#1749C2212yes—
79C23×Q8—21no—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83#533—44no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—44no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×#125—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—48no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×#117—21no—
106C2×#117—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4—22no—
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#477—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—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391C2312yes—
145#391C2312yes—
146#389—21no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389C2312yes—
159#389C2314yes—
160#389C2312yes—
161#389C2314yes—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#385—22no—
167#385—22no—
168#385—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#384—21no—
173#384—21no—
174#384C2312yes—
175#384C2311yescharacteristic
176#384C2312yes—
177#384C2311yescharacteristic
178#382—21no—
179#381—22no—
180#380—21no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380C2312yes—
194#380C2314yes—
195#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198#379—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C2×#95—22no—
206C2×#95—22no—
207#340C2318yes—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213C25C2×D411yescharacteristic
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×Q8—44no—
220C22×Q8C2412yes—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
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×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C22×C8C2418yes—
251C4⋊Q8—28no—
252C4⋊Q8—44no—
253C4⋊Q8—42no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257#125—44no—
258C22⋊Q8—28no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C22⋊Q8—48no—
263C22⋊Q8—48no—
264C22⋊Q8—48no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C4×Q8—28no—
269C4×Q8—44no—
270#117—44no—
271#117—44no—
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—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
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—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—48no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
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×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—48no—
346#106—48no—
347C4⋊C8—28no—
348#103—48no—
349#103—48no—
350C22⋊C8—28no—
351C22⋊C8—44no—
352C22⋊C8—44no—
353#97—44no—
354#95—24no—
355#95—24no—
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—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—41no—
374C24—41no—
375C24C22×D411yescharacteristic
376C24C22×D411yescharacteristic
377C24C22×D411yescharacteristic
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—816no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
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—24no—
430C22×C4—24no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
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—416no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4AGL(1,ℤ/8)14yes—
457C22×C4#13714yes—
458C22×C4#13714yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C42−1+414yes—
471C22×C4C2511yescharacteristic
472C2×C8—28no—
473C2×C8—28no—
474C2×C8—28no—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—44no—
528C42—44no—
529C42H(5,𝔽2)14yes—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—28no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23#53612yes—
554C23C2×AGL(1,ℤ/8)12yes—
555C23C2×#13712yes—
556C23C2×#13712yes—
557C23#57512yes—
558C23C23×D411yescharacteristic
559Q8—48no—
560Q8—84no—
561Q8—88no—
562Q8—88no—
563C2×C4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—88no—
624C2×C4—88no—
625C2×C4#53412yes—
626C2×C4#53412yes—
627C2×C4#53612yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4C2×AGL(1,ℤ/8)12yes—
630C2×C4C2×#13712yes—
631C2×C4C2×#13712yes—
632C2×C4C2×#13712yes—
633C2×C4C2×#13712yes—
634C2×C4C2×H(5,𝔽2)12yes—
635C2×C4C2×2−1+412yes—
636C2×C4#58512yes—
637C8—48no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—416no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22#323714yes—
660C22C2×#53411yescharacteristic
661C22C2×#53611yescharacteristic
662C22#344911yescharacteristic
663C22C22×AGL(1,ℤ/8)11yescharacteristic
664C22C22×#13711yescharacteristic
665C22C22×#13711yescharacteristic
666C22C2×#57511yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—82no—
678C4—84no—
679C4—82no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4#319514yes—
684C4#354114yes—
685C2—24no—
686C2—42no—
687C2—42no—
688C2#3743811yescharacteristic
689C2C2×#319511yescharacteristic
690C2#6075011yescharacteristic
691C2C2×#323711yescharacteristic
692C2#6086011yescharacteristic
693C2#6096311yescharacteristic
694C2#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. #32786C2semidirect
4. #32771C2semidirect
5. #30366C2semidirect
6. #29938C2semidirect
8. #29691C2semidirect
9. #29335C2semidirect
10. #29302C2semidirect
11. #27680C2semidirect
12. #24711C2semidirect
13. #24302C2semidirect
19. C2×#477C22semidirect
52. #2099C22semidirect
53. #2099C22semidirect
54. #2096C22semidirect
55. #2084C22semidirect
58. #2057C22semidirect
59. #2057C22semidirect
60. #2043C22semidirect
63. #2024C22semidirect
64. #1987C22semidirect
65. #1948C22semidirect
66. #1944C22semidirect
67. #1935C22semidirect
73. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#28572
#327862
#327712
#303662
#299382
#298942
#296912
#293352
#293022
#276802
#247112
#243022
#207152