Finite Group Databasebeta

← #6506557

Subgroups — #6506557

Order 512.

2012 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35032C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic, complemented
3#29336C212yesmaximal, complemented
4#29329C211yesmaximal, characteristic, complemented
5#29328C211yesmaximal, characteristic, complemented
6#27973C211yesmaximal, characteristic, complemented
7#27972C211yesmaximal, characteristic, complemented
8#27706C214yesmaximal, complemented
9#27704C214yesmaximal, complemented
10#27282C214yesmaximal, complemented
11#27278C214yesmaximal, complemented
12#25908C212yesmaximal, complemented
13#21769C212yesmaximal, complemented
14#20909C212yesmaximal, complemented
15#20524C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#525C2212yes—
18C22×C22⋊Q8—21no—
19C23×C4⋊C4C2211yescharacteristic
20C2×#489C2214yescomplemented
21C2×#417C2214yes—
22C2×#407C2212yes—
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2850C2212yes—
26#2617—21no—
27#2617—21no—
28#2602—22no—
29#2602—22no—
30#2500C2212yes—
31#2499C2212yes—
32#2440—22no—
33#2440—22no—
34#2440C2214yes—
35#2440C2214yes—
36#2432C2212yes—
37#2431C2212yes—
38C2×#398—21no—
39C2×#395—21no—
40#2325—21no—
41#2325—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#385—21no—
45C2×#384C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48#2101C2214yescomplemented
49#2100C2212yescomplemented
50#2099C2212yescomplemented
51#2072C2214yescomplemented
52#2071C2214yescomplemented
53#2043C2212yescomplemented
54#2042C2214yescomplemented
55#2041C2212yescomplemented
56#2024C2214yescomplemented
57#2023C2214yescomplemented
58#1993C2214yescomplemented
59#1957C2212yescomplemented
60#1956C2212yescomplemented
61#1936C2214yescomplemented
62#1935C2214yescomplemented
63#1934C2214yescomplemented
64#1933C2214yescomplemented
65#1895C2214yes—
66#1894C2214yes—
67#1874C2212yes—
68#1874C2212yes—
69#1874C2212yes—
70#1874C2214yes—
71#1874C2212yes—
72#1872C2212yes—
73#1872C2212yes—
74#1872C2212yes—
75#1872C2214yes—
76#1872C2212yes—
77#1834C2212yescomplemented
78#1834C2212yescomplemented
79#1825C2214yes—
80#1788C2214yes—
81#1753C2214yes—
82#1753C2214yes—
83#1749C2212yes—
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2211yescharacteristic
86#1736C2214yes—
87C22×#41C2312yes—
88C23×Q8—21no—
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×M16C2314yes—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2314yes—
96C2×#127C2311yescharacteristic
97C2×#124C2312yes—
98#525—28no—
99#525—28no—
100C2×#123—22no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—48no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×#121—21no—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×#120—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×#117C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—44no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4D411yescharacteristic
129C22×C22⋊C4D411yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#489—48no—
133#417—48no—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142#407—28no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149#398—24no—
150#398—24no—
151#397—22no—
152#397—22no—
153#395—24no—
154#395—24no—
155#394—22no—
156#394—22no—
157#394—22no—
158#394—22no—
159#392—22no—
160#392—22no—
161#391C2312yes—
162#391C2312yes—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2312yes—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#385—24no—
186#385—24no—
187#385—22no—
188#385—22no—
189#384—21no—
190#384—21no—
191#384C2312yes—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382C2312yes—
196#380—21no—
197#380—21no—
198#380—22no—
199#380C2314yes—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4C2312yes—
203C4×C22⋊C4C2312yes—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25C2×D411yescharacteristic
221C2×#41—24no—
222C2×#41—24no—
223C2×#41—44no—
224C2×#41—44no—
225C22×Q8—24no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8—82no—
229C22×Q8C2412yes—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—24no—
241C23×C4—22no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—82no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4#4118yes—
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C2×M16—28no—
257C2×M16—44no—
258C2×M16—44no—
259C22×C8C2414yes—
260C22×C8C2414yes—
261C4⋊Q8—28no—
262C4⋊Q8—44no—
263C4⋊Q8—42no—
264C4⋊Q8—48no—
265#127—42no—
266#124—44no—
267#123—48no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—416no—
273C22⋊Q8—416no—
274#121—24no—
275#121—24no—
276#121—24no—
277#121—24no—
278#121—44no—
279#121—44no—
280#120—48no—
281C4×D4—48no—
282C4×D4—48no—
283#117—24no—
284#117—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—22no—
345C2×C42—42no—
346C2×C42—42no—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#103—28no—
354#103—44no—
355#103—44no—
356#103—44no—
357#102—28no—
358#102—44no—
359#102—44no—
360#102—44no—
361C22⋊C8—28no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364#97—44no—
365C4×C8—42no—
366C4×C8—42no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24#12414yes—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391#41—416no—
392#41—88no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—84no—
400C2×Q8—816no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
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—416no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—48no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4—816no—
465C22×C4#12414yes—
466C22×C4#12418yes—
467C22×C4AGL(1,ℤ/8)14yes—
468C22×C4#13714yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C2×#4114yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2511yescharacteristic
481M16—88no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4H(5,𝔽2)14yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—41no—
538C42—41no—
539C42—42no—
540C42—44no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—28no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23C2×#12414yes—
566C23C2×#12412yes—
567C23C2×AGL(1,ℤ/8)12yes—
568C23C2×#13712yes—
569C23C23×D411yescharacteristic
570Q8—44no—
571Q8—44no—
572Q8—816no—
573D4—44no—
574D4—44no—
575D4—88no—
576D4—88no—
577D4—88no—
578D4—162no—
579D4—162no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
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—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—88no—
630C2×C4—816no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4C2×#12414yes—
635C2×C4C2×#12412yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4C2×#13712yes—
639C2×C4C2×#13712yes—
640C2×C4C22×#4112yes—
641C2×C4C2×H(5,𝔽2)12yes—
642C2×C4#58512yes—
643C8—44no—
644C8—44no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—416no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22#314114yes—
667C22C22×#12411yescharacteristic
668C22#348411yescharacteristic
669C22#348511yescharacteristic
670C22C22×AGL(1,ℤ/8)11yescharacteristic
671C22C22×#13711yescharacteristic
672C22#358511yescharacteristic
673C22#358611yescharacteristic
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—24no—
679C4—22no—
680C4—42no—
681C4—44no—
682C4—42no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#350514yes—
690C2—24no—
691C2—44no—
692C2—162no—
693C2#3537711yescharacteristic
694C2C2×#314111yescharacteristic
695C2#6062111yescharacteristic
696C2#6158211yescharacteristic
697C2#6158411yescharacteristic
698C2#6170611yescharacteristic
699C2#6170911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35032C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2440C2semidirect
3. #29336C2semidirect
4. #29329C2semidirect
5. #29328C2semidirect
6. #27973C2semidirect
7. #27972C2semidirect
8. #27706C2semidirect
9. #27704C2semidirect
10. #27282C2semidirect
11. #27278C2semidirect
12. #25908C2semidirect
13. #21769C2semidirect
14. #20909C2semidirect
15. #20524C2semidirect
20. C2×#489C22semidirect
48. #2101C22semidirect
49. #2100C22semidirect
50. #2099C22semidirect
51. #2072C22semidirect
52. #2071C22semidirect
53. #2043C22semidirect
54. #2042C22semidirect
55. #2041C22semidirect
56. #2024C22semidirect
57. #2023C22semidirect
58. #1993C22semidirect
59. #1957C22semidirect
60. #1956C22semidirect
61. #1936C22semidirect
62. #1935C22semidirect
63. #1934C22semidirect
64. #1933C22semidirect
77. #1834C22semidirect
78. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#350322
C2×#24402
#293362
#293292
#293282
#279732
#279722
#277062
#277042
#272822
#272782
#259082
#217692
#209092
#205242