Finite Group Databasebeta

← #6644835

Subgroups — #6644835

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35000C211yesmaximal, characteristic
2C2×#2562C211yesmaximal, characteristic, complemented
3#33096C212yesmaximal, complemented
4#32264C214yesmaximal, complemented
5#31332C214yesmaximal, complemented
6#31328C214yesmaximal, complemented
7#31266C214yesmaximal, complemented
8#29640C212yesmaximal, complemented
9#27603C211yesmaximal, characteristic, complemented
10#27601C211yesmaximal, characteristic, complemented
11#24553C212yesmaximal, complemented
12#24550C212yesmaximal, complemented
13#24005C211yesmaximal, characteristic, complemented
14#24002C211yesmaximal, characteristic, complemented
15#21738C211yesmaximal, characteristic, complemented
16C2×#534—21no—
17C22×#125C2211yescharacteristic, complemented
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#481C2212yes—
21C2×#480C2212yes—
22C22×C22⋊C8C2212yescomplemented
23#2854C2212yes—
24#2847C2212yes—
25#2618—21no—
26#2617—21no—
27#2562—22no—
28#2562—22no—
29#2562C2214yes—
30#2562C2214yescomplemented
31#2543—22no—
32#2534—22no—
33#2486C2212yes—
34#2416C2212yes—
35#2416C2212yes—
36#2408C2212yes—
37C2×#400—21no—
38C2×#394—21no—
39#2308—21no—
40C2×#388—21no—
41C2×#388—21no—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic, complemented
46C2×#380C2211yescharacteristic, complemented
47C2×C4×C4⋊C4C2211yescharacteristic, complemented
48#2077C2214yescomplemented
49#2077C2214yescomplemented
50#2076C2214yescomplemented
51#2075C2214yescomplemented
52#2059C2214yescomplemented
53#2059C2214yescomplemented
54#2034C2214yescomplemented
55#2034C2214yescomplemented
56#2033C2214yescomplemented
57#2033C2214yescomplemented
58#2025C2214yescomplemented
59#2025C2214yescomplemented
60#1926C2214yescomplemented
61#1925C2214yescomplemented
62#1921C2214yescomplemented
63#1920C2214yescomplemented
64#1893C2214yescomplemented
65#1893C2214yescomplemented
66#1874C2212yes—
67#1874C2212yes—
68#1869C2214yes—
69#1869C2214yescomplemented
70#1853C2212yescomplemented
71#1853C2212yescomplemented
72#1853C2212yescomplemented
73#1852C2212yescomplemented
74#1852C2212yescomplemented
75#1852C2212yescomplemented
76#1824C2212yescomplemented
77#1824C2212yescomplemented
78#1824C2212yescomplemented
79#1823C2212yescomplemented
80#1817C2212yescomplemented
81#1808C2214yescomplemented
82#1807C2214yescomplemented
83#1786C2212yescomplemented
84#1786C2212yescomplemented
85#1779C2211yescharacteristic
86#1779C2211yescharacteristic
87C23×D4—21no—
88C23×D4D411yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C23×C8C2314yes—
93#534—48no—
94C2×#125—22no—
95C2×#125—44no—
96C2×#125C2314yes—
97C2×#123—22no—
98C2×#123—22no—
99C2×#121—22no—
100C2×#121—22no—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121—44no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#120—22no—
109C2×#120—22no—
110C2×C4×D4—44no—
111C2×C4×D4D412yes—
112C2×C4×D4D412yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4D411yescharacteristic
132C22×C22⋊C4D411yescharacteristic
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135#481—44no—
136#480—44no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8—44no—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153#400—24no—
154#400—24no—
155#398—22no—
156#397—22no—
157#396—22no—
158#396—22no—
159#394—24no—
160#394—24no—
161#394—22no—
162#394—22no—
163#394—22no—
164#392—22no—
165#390C2312yes—
166#390C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—24no—
170#388—24no—
171#388—24no—
172#388—24no—
173#388—22no—
174#388—22no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#381—22no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—22no—
187#380—22no—
188#380—21no—
189#380—21no—
190#380C2312yes—
191#380C2312yes—
192#380C2314yes—
193#380C2314yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198#380C2312yes—
199#380C2312yes—
200#379—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4C2314yes—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C25—21no—
220C25C2×D411yescharacteristic
221C25C2×D411yescharacteristic
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—42no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4—42no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4C2×D412yes—
237C22×D4C2×D412yes—
238C22×D4C2412yes—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4—42no—
251C23×C4—42no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2412yes—
262C23×C4C2411yescharacteristic
263C22×C8—44no—
264C22×C8—44no—
265C22×C8—44no—
266C22×C8C2414yes—
267C22×C8C2414yes—
268#125—44no—
269#125—44no—
270#125—48no—
271#123—48no—
272#123—48no—
273#121—24no—
274#121—24no—
275#121—42no—
276#121—42no—
277#121—44no—
278#121—44no—
279#121—48no—
280#121—48no—
281#121—44no—
282#121—44no—
283#120—48no—
284#120—48no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—42no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—42no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
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—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—42no—
340C2×C22⋊C4—42no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—82no—
346C2×C22⋊C4—82no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—82no—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356#107—44no—
357#107—44no—
358#106—44no—
359#106—44no—
360#102—44no—
361#102—44no—
362#102—44no—
363#102—44no—
364C22⋊C8—24no—
365C22⋊C8—24no—
366C22⋊C8—44no—
367C22⋊C8—42no—
368C22⋊C8—42no—
369C22⋊C8—42no—
370C22⋊C8—42no—
371#95—24no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24—41no—
395C24—41no—
396C24—41no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—81no—
401C24—81no—
402C24—81no—
403C24—81no—
404C24#13514yes—
405C24AGL(1,ℤ/8)14yes—
406C24C22×D411yescharacteristic
407C24C22×D411yescharacteristic
408C24C22×D412yes—
409C24C22×D411yescharacteristic
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—416no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—81no—
431C2×D4—81no—
432C2×D4—88no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—816no—
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—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—88no—
496C22×C4—88no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4H(5,𝔽2)14yes—
508C22×C4C2511yescharacteristic
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—88no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—81no—
536C4⋊C4—81no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—84no—
558C22⋊C4—82no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—84no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—21no—
570C23—21no—
571C23—22no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—21no—
577C23—21no—
578C23—22no—
579C23—22no—
580C23—21no—
581C23—21no—
582C23—22no—
583C23—21no—
584C23—21no—
585C23—28no—
586C23—28no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—82no—
599C23—82no—
600C23—82no—
601C23—82no—
602C23—82no—
603C23—82no—
604C23—82no—
605C23—82no—
606C23—84no—
607C23—84no—
608C23—84no—
609C23—162no—
610C23—162no—
611C23—162no—
612C23—162no—
613C23#53412yes—
614C23C2×#13512yes—
615C23C2×#13512yes—
616C23C2×#13512yes—
617C23C2×AGL(1,ℤ/8)14yes—
618C23C2×AGL(1,ℤ/8)12yes—
619C23C23×D411yescharacteristic
620D4—48no—
621D4—48no—
622D4—44no—
623D4—44no—
624D4—84no—
625D4—88no—
626D4—88no—
627D4—164no—
628D4—164no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—28no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—84no—
675C2×C4—82no—
676C2×C4—82no—
677C2×C4—82no—
678C2×C4—82no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—84no—
684C2×C4—162no—
685C2×C4—162no—
686C2×C4#53412yes—
687C2×C4C2×H(5,𝔽2)12yes—
688C2×C4#58512yes—
689C2×C4#58512yes—
690C8—44no—
691C8—44no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—24no—
703C22—24no—
704C22—22no—
705C22—22no—
706C22—24no—
707C22—42no—
708C22—42no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—42no—
715C22—41no—
716C22—42no—
717C22—41no—
718C22—41no—
719C22—88no—
720C22—88no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—168no—
729C22#318814yes—
730C22C2×#53411yescharacteristic
731C22#348511yescharacteristic
732C22#348511yescharacteristic
733C22C22×#13511yescharacteristic
734C22C22×AGL(1,ℤ/8)11yescharacteristic
735C22#358511yescharacteristic
736C22#358611yescharacteristic
737C4—22no—
738C4—42no—
739C4—42no—
740C4—41no—
741C4—42no—
742C4—41no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4#354514yes—
751C2—24no—
752C2—42no—
753C2—44no—
754C2—42no—
755C2—162no—
756C2#3711911yescharacteristic
757C2C2×#318811yescharacteristic
758C2#6075511yescharacteristic
759C2#6169611yescharacteristic
760C2#6169911yescharacteristic
761C2#6170411yescharacteristic
762C2#6170711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35000C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2562C2semidirect
3. #33096C2semidirect
4. #32264C2semidirect
5. #31332C2semidirect
6. #31328C2semidirect
7. #31266C2semidirect
8. #29640C2semidirect
9. #27603C2semidirect
10. #27601C2semidirect
11. #24553C2semidirect
12. #24550C2semidirect
13. #24005C2semidirect
14. #24002C2semidirect
15. #21738C2semidirect
17. C22×#125C22semidirect
22. C22×C22⋊C8C22semidirect
30. #2562C22semidirect
45. C2×#380C22semidirect
46. C2×#380C22semidirect
47. C2×C4×C4⋊C4C22semidirect
48. #2077C22semidirect
49. #2077C22semidirect
50. #2076C22semidirect
51. #2075C22semidirect
52. #2059C22semidirect
53. #2059C22semidirect
54. #2034C22semidirect
55. #2034C22semidirect
56. #2033C22semidirect
57. #2033C22semidirect
58. #2025C22semidirect
59. #2025C22semidirect
60. #1926C22semidirect
61. #1925C22semidirect
62. #1921C22semidirect
63. #1920C22semidirect
64. #1893C22semidirect
65. #1893C22semidirect
69. #1869C22semidirect
70. #1853C22semidirect
71. #1853C22semidirect
72. #1853C22semidirect
73. #1852C22semidirect
74. #1852C22semidirect
75. #1852C22semidirect
76. #1824C22semidirect
77. #1824C22semidirect
78. #1824C22semidirect
79. #1823C22semidirect
80. #1817C22semidirect
81. #1808C22semidirect
82. #1807C22semidirect
83. #1786C22semidirect
84. #1786C22semidirect

Maximal subgroups

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

subgroupindex
#350002
C2×#25622
#330962
#322642
#313322
#313282
#312662
#296402
#276032
#276012
#245532
#245502
#240052
#240022
#217382