Finite Group Databasebeta

← #6476966

Subgroups — #6476966

Order 512.

2050 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#33176C212yesmaximal, complemented
4#33049C212yesmaximal, complemented
5#30326C214yesmaximal, complemented
6#28581C214yesmaximal, complemented
7#27972C211yesmaximal, characteristic, complemented
8#27948C211yesmaximal, characteristic, complemented
9#27889C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic, complemented
11#25722C211yesmaximal, characteristic, complemented
12#25600C212yesmaximal, complemented
13#24386C214yesmaximal
14#22050C214yesmaximal, complemented
15C4×#407C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C23×C42C2211yescharacteristic
20C2×#494C2212yescomplemented
21C2×#487C2212yescomplemented
22C2×#470C2218yes—
23C2×#451C2214yes—
24C2×#407C2212yes—
25C2×C4×M16C2212yes—
26#2857—21no—
27#2857—21no—
28#2857—21no—
29#2857—21no—
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2212yes—
33#2857C2212yes—
34#2857C2212yes—
35#2857C2212yes—
36#2857C2211yescharacteristic
37#2857C2211yescharacteristic
38#2855—21no—
39#2855C2212yes—
40#2851—21no—
41#2851—21no—
42#2848—21no—
43#2848C2212yes—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8—21no—
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2212yes—
48#2296—21no—
49C2×#391C2212yes—
50C2×#389—21no—
51C2×#389—21no—
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54#2280—21no—
55C2×#380—21no—
56C2×#380—21no—
57#2084C2214yescomplemented
58#2072C2214yescomplemented
59#2043C2214yescomplemented
60#2035C2218yes—
61#2016C2214yescomplemented
62#2001C2218yes—
63#1990C2212yescomplemented
64#1971C2218yescomplemented
65#1956C2212yescomplemented
66#1956C2212yescomplemented
67#1956C2212yescomplemented
68#1935C2218yescomplemented
69#1927C2218yescomplemented
70#1894C2218yes—
71#1878C2218yescomplemented
72#1872C2214yes—
73#1853C2214yes—
74#1844C2212yescomplemented
75#1834C2212yescomplemented
76#1787C2214yes—
77#1749C2212yes—
78#1749C2212yes—
79C4×C22⋊C8C2212yes—
80C4×C22⋊C8C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1728C2212yes—
83C23×Q8C2312yes—
84C24×C4D411yescharacteristic
85C24×C4D411yescharacteristic
86C24×C4C2311yescharacteristic
87C22×Q16C2318yes—
88C22×M16C2314yes—
89#533—44no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#125—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—48no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2314yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2314yes—
109C2×C4×Q8—21no—
110C2×C4×Q8C2312yes—
111C2×#117—21no—
112C2×#117—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—22no—
120C22×C42—44no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#494—44no—
134#487—44no—
135#470—416no—
136#451—216no—
137C2×#106C2318yes—
138C2×#103C2318yes—
139C2×#103C2318yes—
140C2×#103C2318yes—
141C2×#103C2318yes—
142#407—28no—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2312yes—
146C4×M16—44no—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2312yes—
150#391—21no—
151#391—21no—
152#391—22no—
153#391—22no—
154#391C2312yes—
155#391C2312yes—
156#391C2314yes—
157#391C2314yes—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2311yescharacteristic
190#382—21no—
191#381—22no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380C2312yes—
208#380C2312yes—
209#380C2312yes—
210#379—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C2×#95—22no—
218C2×#95—22no—
219#340C2318yes—
220#340C2318yes—
221#336C2312yes—
222#336C2312yes—
223#336C2314yes—
224C25C2×D411yescharacteristic
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—28no—
228C22×Q8—44no—
229C22×Q8—44no—
230C22×Q8C2412yes—
231C22×Q8C2414yes—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
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×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×Q16—216no—
262C2×Q16—48no—
263C2×Q16—48no—
264C2×M16—28no—
265C2×M16—28no—
266C2×M16—44no—
267C22×C8C2414yes—
268C22×C8C2414yes—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276#125—44no—
277C22⋊Q8—28no—
278C22⋊Q8—216no—
279C22⋊Q8—28no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—48no—
284C22⋊Q8—48no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C4×Q8—44no—
288C4×Q8—44no—
289#117—44no—
290#117—44no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—48no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42—42no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#106—48no—
365#103—216no—
366#103—48no—
367#103—48no—
368#103—48no—
369C22⋊C8—24no—
370C22⋊C8—28no—
371C22⋊C8#4118yes—
372#97—44no—
373C4×C8—24no—
374C4×C8—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—41no—
395C24—41no—
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398C24C22×D411yescharacteristic
399C2×Q8—28no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—28no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—816no—
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—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4#13714yes—
478C22×C4#13714yes—
479C22×C4#13714yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4114yes—
492C22×C4H(5,𝔽2)14yes—
493C22×C4C2511yescharacteristic
494Q16—816no—
495M16—416no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8C2×#4118yes—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—28no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—48no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—44no—
554C42—44no—
555C42H(5,𝔽2)14yes—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—28no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23#53212yes—
580C23#53512yes—
581C23C2×#13712yes—
582C23C2×#13712yes—
583C23C2×#13712yes—
584C23C23×D411yescharacteristic
585Q8—48no—
586Q8—48no—
587Q8—48no—
588Q8—88no—
589Q8—84no—
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—21no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—88no—
649C2×C4—88no—
650C2×C4#53212yes—
651C2×C4#53412yes—
652C2×C4#53412yes—
653C2×C4#53512yes—
654C2×C4C2×#13712yes—
655C2×C4C2×#13712yes—
656C2×C4C2×#13712yes—
657C2×C4C2×#13712yes—
658C2×C4C2×#13712yes—
659C2×C4C2×#13712yes—
660C2×C4#57512yes—
661C2×C4C22×#4112yes—
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4C2×H(5,𝔽2)12yes—
664C8—28no—
665C8—44no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—416no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22#318514yes—
688C22C2×#53211yescharacteristic
689C22C2×#53411yescharacteristic
690C22C2×#53511yescharacteristic
691C22C22×#13711yescharacteristic
692C22C22×#13711yescharacteristic
693C22C22×#13711yescharacteristic
694C22C2×#57511yescharacteristic
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—84no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—84no—
710C4#319514yes—
711C4#319614yes—
712C4#351914yes—
713C2—24no—
714C2—42no—
715C2—42no—
716C2#3507811yescharacteristic
717C2C2×#318511yescharacteristic
718C2#6070111yescharacteristic
719C2C2×#319511yescharacteristic
720C2C2×#319611yescharacteristic
721C2#6080011yescharacteristic
722C2#6080411yescharacteristic
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. #33176C2semidirect
4. #33049C2semidirect
5. #30326C2semidirect
6. #28581C2semidirect
7. #27972C2semidirect
8. #27948C2semidirect
9. #27889C2semidirect
10. #25856C2semidirect
11. #25722C2semidirect
12. #25600C2semidirect
14. #22050C2semidirect
20. C2×#494C22semidirect
21. C2×#487C22semidirect
57. #2084C22semidirect
58. #2072C22semidirect
59. #2043C22semidirect
61. #2016C22semidirect
63. #1990C22semidirect
64. #1971C22semidirect
65. #1956C22semidirect
66. #1956C22semidirect
67. #1956C22semidirect
68. #1935C22semidirect
69. #1927C22semidirect
71. #1878C22semidirect
74. #1844C22semidirect
75. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#28572
#331762
#330492
#303262
#285812
#279722
#279482
#278892
#258562
#257222
#256002
#243862
#220502
C4×#4072