Finite Group Databasebeta

← #6458925

Subgroups — #6458925

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34180C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#27622C214yesmaximal, complemented
4#27619C214yesmaximal, complemented
5#27616C212yesmaximal, complemented
6#27616C212yesmaximal, complemented
7#27613C212yesmaximal, complemented
8#27610C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23662C212yesmaximal, complemented
11#21039C212yesmaximal, complemented
12#21035C212yesmaximal, complemented
13#20846C211yesmaximal, characteristic
14#20843C211yesmaximal, characteristic
15#20700C211yesmaximal, characteristic
16C2×#1823C211yesmaximal, characteristic
17#20685C211yesmaximal, characteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#407C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2347—21no—
26#2346—21no—
27#2345—21no—
28#2344—21no—
29#2343—22no—
30#2342—22no—
31#2317—22no—
32#2317—21no—
33#2317—21no—
34#2317C2212yes—
35#2317C2212yes—
36#2317C2214yes—
37#2317C2211yescharacteristic
38#2317C2211yescharacteristic
39#2314C2212yes—
40#2314C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#385—22no—
44#2285C2212yes—
45C2×#384—21no—
46#2283C2212yes—
47C2×#382—21no—
48#2280C2211yescharacteristic
49C2×#380C2211yescharacteristic
50#2274C2211yescharacteristic
51#2274C2211yescharacteristic
52#2101C2214yescomplemented
53#2101C2214yescomplemented
54#2100C2214yescomplemented
55#2099C2214yescomplemented
56#2085C2214yescomplemented
57#2085C2214yescomplemented
58#2084C2214yescomplemented
59#2083C2214yescomplemented
60#1921C2214yescomplemented
61#1921C2214yescomplemented
62#1920C2214yescomplemented
63#1920C2214yescomplemented
64#1919C2214yescomplemented
65#1919C2214yescomplemented
66#1918C2214yescomplemented
67#1918C2214yescomplemented
68#1854C2212yes—
69#1854C2212yes—
70#1854C2214yes—
71#1853C2211yescharacteristic
72#1853C2211yescharacteristic
73#1853C2212yes—
74#1852C2211yescharacteristic
75#1852C2211yescharacteristic
76#1852C2212yes—
77#1824C2212yes—
78#1824C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1823C2214yes—
82#1823C2212yes—
83#1823C2212yes—
84#1818C2214yes—
85#1817C2212yes—
86#1817C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1729C2212yes—
95C24×C4—21no—
96C24×C4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×#117—21no—
100C2×#117—21no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—28no—
104C2×#117—22no—
105C2×#117D412yes—
106C2×#117D411yescharacteristic
107C2×#117D411yescharacteristic
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D412yes—
134C22×C42—21no—
135C22×C42C2312yes—
136#428—28no—
137#427—28no—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144#407—24no—
145C2×C22⋊C8—24no—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#385—24no—
172#385—24no—
173#385—24no—
174#385—24no—
175#385—22no—
176#385—22no—
177#384—22no—
178#384—24no—
179#384—22no—
180#384C2312yes—
181#384C2312yes—
182#383—22no—
183#382—22no—
184#382—24no—
185#382—22no—
186#382—22no—
187#382C2312yes—
188#382C2312yes—
189#381—22no—
190#381—21no—
191#380—22no—
192#380—21no—
193#380—21no—
194#380C2312yes—
195#380C2311yescharacteristic
196#380C2311yescharacteristic
197#380C2312yes—
198#380C2311yescharacteristic
199#380C2311yescharacteristic
200#379—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215#376—22no—
216C2×#95—21no—
217C2×#95—22no—
218C2×#95—22no—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237C25C2×D411yescharacteristic
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4—82no—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4#4114yes—
256C23×C4#4114yes—
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2414yes—
269#117—28no—
270#117—28no—
271#117—28no—
272#117—24no—
273#117—24no—
274#117—24no—
275#117—24no—
276#117—24no—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—22no—
281#117—24no—
282#117—24no—
283#117—24no—
284#117—24no—
285#117—84no—
286#117#4118yes—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—84no—
318C2×C4⋊C4—84no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—82no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—28no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2412yes—
365C2×C42C2412yes—
366#107—24no—
367#107—24no—
368#107—28no—
369#106—24no—
370#106—24no—
371#106—28no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—41no—
400C24—41no—
401C24#12314yes—
402C24C22×D411yescharacteristic
403C24C22×D411yescharacteristic
404C24C22×D411yescharacteristic
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—816no—
454C22×C4#12314yes—
455C22×C4#12314yes—
456C22×C4#12314yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4114yes—
471C22×C42−1+414yes—
472C22×C4C2511yescharacteristic
473M16—48no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—48no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4C2×#4114yes—
516C4⋊C4C2×#4114yes—
517C22⋊C4—28no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—28no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—81no—
529C22⋊C4—81no—
530C22⋊C4—82no—
531C42—28no—
532C42—24no—
533C42—24no—
534C42—28no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—42no—
540C42—42no—
541C42—82no—
542C42—82no—
543C42—82no—
544C42—82no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—28no—
556C23—21no—
557C23—21no—
558C23—28no—
559C23—21no—
560C23—21no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23C2×#12312yes—
567C23C2×#12312yes—
568C23C2×#12312yes—
569C23#52512yes—
570C23#53612yes—
571C23#57612yes—
572C23#57712yes—
573C23C23×D411yescharacteristic
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—84no—
628C2×C4—88no—
629C2×C4—88no—
630C2×C4—88no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4C2×#12312yes—
634C2×C4C2×#12312yes—
635C2×C4C2×#12312yes—
636C2×C4#52512yes—
637C2×C4#53612yes—
638C2×C4#55218yes—
639C2×C4#55218yes—
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4112yes—
643C2×C4C2×2−1+412yes—
644C8—42no—
645C8—42no—
646C8—44no—
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—22no—
662C22—416no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22#248414yes—
668C22#309614yes—
669C22C22×#12311yescharacteristic
670C22C2×#52511yescharacteristic
671C22C2×#53611yescharacteristic
672C22C2×#57611yescharacteristic
673C22C2×#57711yescharacteristic
674C22#358611yescharacteristic
675C22#358611yescharacteristic
676C4—22no—
677C4—22no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—84no—
685C4—81no—
686C4—82no—
687C4—81no—
688C4—81no—
689C4—82no—
690C4—81no—
691C4#248414yes—
692C4C2×#55212yes—
693C4C2×#55212yes—
694C2—24no—
695C2—24no—
696C2—42no—
697C2C2×#248411yescharacteristic
698C2C2×#309611yescharacteristic
699C2#6051911yescharacteristic
700C2#6058411yescharacteristic
701C2#6058511yescharacteristic
702C2#6087111yescharacteristic
703C2#6087211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #27622C2semidirect
4. #27619C2semidirect
5. #27616C2semidirect
6. #27616C2semidirect
7. #27613C2semidirect
8. #27610C2semidirect
9. #23748C2semidirect
10. #23662C2semidirect
11. #21039C2semidirect
12. #21035C2semidirect
52. #2101C22semidirect
53. #2101C22semidirect
54. #2100C22semidirect
55. #2099C22semidirect
56. #2085C22semidirect
57. #2085C22semidirect
58. #2084C22semidirect
59. #2083C22semidirect
60. #1921C22semidirect
61. #1921C22semidirect
62. #1920C22semidirect
63. #1920C22semidirect
64. #1919C22semidirect
65. #1919C22semidirect
66. #1918C22semidirect
67. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#341802
C2×#23172
#276222
#276192
#276162
#276132
#276102
#237482
#236622
#210392
#210352
#208462
#208432
#207002
C2×#18232
#206852