Finite Group Databasebeta

← #6475411

Subgroups — #6475411

Order 512.

1960 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic
3#33218C214yesmaximal, complemented
4#30295C214yesmaximal, complemented
5#29298C214yesmaximal, complemented
6#29297C212yesmaximal, complemented
7#29296C212yesmaximal, complemented
8#27956C212yesmaximal, complemented
9#27953C212yesmaximal, complemented
10#25819C212yesmaximal, complemented
11#25816C212yesmaximal, complemented
12#21942C212yesmaximal
13#21942C212yesmaximal
14C4×#407C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#125C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C23×C42C2211yescharacteristic
19C2×#501C2214yescomplemented
20C2×#407C2212yes—
21C2×C4×M16C2212yes—
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2851—21no—
31#2851—21no—
32#2851—21no—
33#2851—21no—
34#2851C2212yes—
35#2851C2212yes—
36#2851C2212yes—
37#2851C2212yes—
38C2×#391—21no—
39C2×#389—21no—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#387—21no—
44C2×#387—21no—
45C2×#385—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#380—21no—
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2212yes—
53#2085C2218yescomplemented
54#2084C2214yescomplemented
55#2083C2214yescomplemented
56#2059C2218yescomplemented
57#2058C2214yescomplemented
58#2057C2214yescomplemented
59#1993C2214yescomplemented
60#1992C2214yescomplemented
61#1942C2218yescomplemented
62#1941C2218yescomplemented
63#1853C2214yes—
64#1853C2214yes—
65#1853C2214yes—
66#1853C2214yes—
67#1852C2214yes—
68#1852C2214yes—
69#1852C2214yes—
70#1852C2214yes—
71#1847C2214yescomplemented
72#1809C2218yes—
73#1809C2218yes—
74#1749C2212yes—
75#1749C2212yes—
76C4×C22⋊C8C2212yes—
77C4×C22⋊C8C2212yes—
78#1728C2212yes—
79C23×Q8—21no—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16C2314yes—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—44no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#125—21no—
90C2×#125—21no—
91C2×#125—44no—
92C2×#125C2312yes—
93C2×#125C2312yes—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—48no—
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⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—22no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—44no—
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#501—48no—
130C2×#107C2318yes—
131C2×#107C2318yes—
132C2×#106C2318yes—
133C2×#106C2318yes—
134#407Q8116yescomplemented
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C4×M16—44no—
138C2×#97C2314yes—
139C2×C4×C8C2314yes—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#389—21no—
145#389—21no—
146#389—21no—
147#389—21no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2312yes—
155#389C2312yes—
156#389C2314yes—
157#389C2312yes—
158#389C2312yes—
159#389C2314yes—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#385—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#384—21no—
177#384—21no—
178#384C2312yes—
179#384C2312yes—
180#382—21no—
181#382—21no—
182#382C2312yes—
183#382C2312yes—
184#380—21no—
185#380—21no—
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#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198#380C2312yes—
199#380C2314yes—
200C2×#95—22no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#336C2314yes—
213#336C2314yes—
214C25C2×D411yescharacteristic
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×Q8—44no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
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×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×Q818yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×M16—44no—
254C2×M16—44no—
255C2×M16C2×Q8116yescomplemented
256C22×C8C2418yes—
257C4⋊Q8—28no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261#125—28no—
262#125—44no—
263#125—44no—
264#125—44no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
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—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—48no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
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×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—48no—
349#107—48no—
350#106—48no—
351#106—48no—
352C22⋊C8C2×Q8116yes—
353C22⋊C8C2×Q8116yescomplemented
354#97—28no—
355C4×C8—44no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—41no—
377C24—41no—
378C24C4⋊Q814yes—
379C24C22×D411yescharacteristic
380C24C22×D411yescharacteristic
381C24C22×D411yescharacteristic
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—816no—
389C22×C4—22no—
390C22×C4—22no—
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—24no—
420C22×C4—24no—
421C22×C4—416no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
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—416no—
443C22×C4—416no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4C4⋊Q814yes—
463C22×C4C4⋊Q814yes—
464C22×C4C4⋊Q814yes—
465C22×C4AGL(1,ℤ/8)14yes—
466C22×C4#13714yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×Q814yes—
479C22×C4C22×Q814yes—
480C22×C4C22×Q814yes—
481C22×C4C2511yescharacteristic
482M16—416no—
483C2×C8—28no—
484C2×C8—28no—
485C2×C8C22×Q8116yes—
486C4⋊C4—24no—
487C4⋊C4—28no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—48no—
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—22no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42H(5,𝔽2)14yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—28no—
545C23—28no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23C2×C4⋊Q812yes—
558C23C2×C4⋊Q812yes—
559C23C2×C4⋊Q812yes—
560C23C2×AGL(1,ℤ/8)12yes—
561C23C2×#13712yes—
562C23#57512yes—
563C23#57512yes—
564C23C23×D411yescharacteristic
565Q8—88no—
566Q8—88no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—21no—
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—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
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—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—88no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4C2×C4⋊Q812yes—
634C2×C4C2×C4⋊Q812yes—
635C2×C4C2×C4⋊Q812yes—
636C2×C4#53412yes—
637C2×C4#53412yes—
638C2×C4C2×AGL(1,ℤ/8)12yes—
639C2×C4C2×AGL(1,ℤ/8)12yes—
640C2×C4C2×#13712yes—
641C2×C4C2×#13712yes—
642C2×C4C23×Q812yes—
643C2×C4C2×H(5,𝔽2)12yes—
644C2×C4#58512yes—
645C8—216no—
646C22—22no—
647C22—22no—
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—416no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22#316514yes—
667C22#316614yes—
668C22C22×C4⋊Q811yescharacteristic
669C22C2×#53411yescharacteristic
670C22#344911yescharacteristic
671C22C22×AGL(1,ℤ/8)11yescharacteristic
672C22C22×#13711yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57511yescharacteristic
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—41no—
685C4—41no—
686C4—84no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—84no—
691C4—82no—
692C4#350114yes—
693C2—24no—
694C2—24no—
695C2—42no—
696C2#3501111yescharacteristic
697C2C2×#316511yescharacteristic
698C2C2×#316611yescharacteristic
699C2#6075011yescharacteristic
700C2#6075111yescharacteristic
701C2#6094911yescharacteristic
702C2#6095011yescharacteristic
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. #33218C2semidirect
4. #30295C2semidirect
5. #29298C2semidirect
6. #29297C2semidirect
7. #29296C2semidirect
8. #27956C2semidirect
9. #27953C2semidirect
10. #25819C2semidirect
11. #25816C2semidirect
19. C2×#501C22semidirect
53. #2085C22semidirect
54. #2084C22semidirect
55. #2083C22semidirect
56. #2059C22semidirect
57. #2058C22semidirect
58. #2057C22semidirect
59. #1993C22semidirect
60. #1992C22semidirect
61. #1942C22semidirect
62. #1941C22semidirect
71. #1847C22semidirect
134. #407Q8semidirect
255. C2×M16C2×Q8semidirect
353. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#28512
#332182
#302952
#292982
#292972
#292962
#279562
#279532
#258192
#258162
#219422
C4×#4072