Finite Group Databasebeta

← #6624510

Subgroups — #6624510

Order 512.

1931 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#33111C212yesmaximal, complemented
4#32902C211yesmaximal, characteristic
5#28312C212yesmaximal, complemented
6#28114C212yesmaximal, complemented
7#27909C211yesmaximal, characteristic, complemented
8#27647C211yesmaximal, characteristic, complemented
9#27504C214yesmaximal, complemented
10#27206C214yesmaximal, complemented
11#26087C212yesmaximal, complemented
12#24631C212yesmaximal, complemented
13#23099C212yesmaximal, complemented
14#22811C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21578C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#483C2212yes—
21C2×#480C2212yes—
22C2×#480C2212yes—
23C2×#459C2212yes—
24C2×#450C2211yescharacteristic
25C2×#447C2212yes—
26C2×C4×SD16C2212yes—
27#2856C2211yescharacteristic
28#2850C2212yes—
29#2597—22no—
30#2571C2211yescharacteristic, complemented
31#2548C2211yescharacteristic, complemented
32#2513—21no—
33#2513C2212yescomplemented
34#2504—21no—
35#2447C2212yescomplemented
36#2444—22no—
37#2366C2211yescharacteristic, complemented
38#2361—21no—
39#2357—21no—
40#2355C2211yescharacteristic, complemented
41#2354—22no—
42#2354—21no—
43#2354—21no—
44#2354C2212yes—
45#2354C2212yes—
46#2354C2214yes—
47C4×C22⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49C2×#390C2212yes—
50C2×#388—22no—
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2093C2214yescomplemented
56#2092C2214yes—
57#2084C2212yescomplemented
58#2072C2212yescomplemented
59#2067C2214yes—
60#2054C2214yes—
61#2012C2214yes—
62#1983C2214yescomplemented
63#1973C2214yes—
64#1972C2214yes—
65#1957C2212yescomplemented
66#1949C2214yescomplemented
67#1948C2212yescomplemented
68#1939C2212yescomplemented
69#1928C2214yescomplemented
70#1926C2214yescomplemented
71#1926C2214yescomplemented
72#1925C2214yes—
73#1921C2214yescomplemented
74#1917C2214yescomplemented
75#1879C2214yes—
76#1879C2214yes—
77#1877C2212yes—
78#1874C2212yes—
79#1872C2211yescharacteristic
80#1864C2212yes—
81#1853C2211yescharacteristic
82#1845C2212yescomplemented
83#1823C2212yes—
84#1803C2212yescomplemented
85#1802C2212yes—
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1786C2211yescharacteristic
89#1776C2214yescomplemented
90#1770C2212yescomplemented
91#1762C2214yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×SD16C2312yes—
98C22×D8C2314yes—
99C2×#127C2311yescharacteristic
100C2×#124C2312yes—
101C2×C22⋊Q8C2311yescharacteristic
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×D4—28no—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—22no—
115C2×#117—22no—
116C2×#117—24no—
117C2×#117—48no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—22no—
126C22×C22⋊C4—22no—
127C22×C42—22no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#483—44no—
131#480—28no—
132#480—44no—
133#459—28no—
134#450—24no—
135#447—44no—
136C4×SD16—28no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#397—22no—
159#397—22no—
160#397—22no—
161#397—22no—
162#394—22no—
163#391C2311yescharacteristic
164#391C2312yes—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—22no—
171#390—22no—
172#390C2312yes—
173#390C2314yes—
174#390C2312yes—
175#390C2312yes—
176#390C2314yes—
177#389—21no—
178#389C2312yes—
179#389C2311yescharacteristic
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—24no—
185#388—24no—
186#388—24no—
187#388—24no—
188#388—22no—
189#387—21no—
190#387—22no—
191#386—22no—
192#386—22no—
193#386—22no—
194#386—22no—
195#385—21no—
196#385—22no—
197#384—21no—
198#384—21no—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#382—21no—
203#382—21no—
204#382C2312yes—
205#381—22no—
206#380—22no—
207#380—22no—
208#380—24no—
209#380C2311yescharacteristic
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C2×#95—22no—
227C2×#95D412yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C25—21no—
238C22×Q8C2411yescharacteristic
239C22×D4—24no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—42no—
247C22×D4C2412yes—
248C22×D4C2412yes—
249C23×C4—28no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—42no—
265C2×D8—28no—
266C2×D8—44no—
267C2×D8—44no—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271#127—24no—
272#124—44no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275#121—24no—
276#121—24no—
277#121—44no—
278#121—82no—
279C4×Q8—24no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—84no—
288C4×D4#4118yes—
289#117—416no—
290#117—48no—
291#117—48no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—48no—
343C2×C22⋊C4—84no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42—48no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360#107—28no—
361#107—44no—
362#106—24no—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—24no—
367#103—44no—
368#102—24no—
369#102—24no—
370#102—28no—
371#102—28no—
372#102—42no—
373#102—44no—
374#102—44no—
375C22⋊C8—24no—
376C22⋊C8—22no—
377C22⋊C8#4114yes—
378#97—44no—
379C4×C8—24no—
380#95—24no—
381#95—24no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—24no—
398C24—22no—
399C24—22no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24C22×D411yescharacteristic
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—81no—
433C2×D4—81no—
434C2×D4C2×#4114yes—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—432no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4AGL(1,ℤ/8)14yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4C2511yescharacteristic
484SD16—48no—
485D8—88no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8C2×#4114yes—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—81no—
522C4⋊C4C2×#4114yes—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—24no—
545C42—24no—
546C42—42no—
547C42—42no—
548C42—48no—
549C42—48no—
550C42—42no—
551C42—42no—
552C42—82no—
553C42C2×#4114yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—24no—
559C23—48no—
560C23—48no—
561C23—48no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23#55012yes—
580C23C2×AGL(1,ℤ/8)12yes—
581C23#57612yes—
582C23C23×D411yescharacteristic
583Q8—42no—
584Q8—42no—
585D4—24no—
586D4—24no—
587D4—48no—
588D4—42no—
589D4—44no—
590D4—42no—
591D4—44no—
592D4—88no—
593D4—84no—
594D4—162no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—44no—
621C2×C4—416no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—416no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—44no—
645C2×C4—84no—
646C2×C4—88no—
647C2×C4—164no—
648C2×C4#52512yes—
649C2×C4#52512yes—
650C2×C4#55012yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4C2×AGL(1,ℤ/8)12yes—
653C2×C4#57512yes—
654C2×C4C22×#4111yescharacteristic
655C2×C4C22×#4112yes—
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4#58512yes—
658C8—24no—
659C8—42no—
660C8—44no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—816no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22#339914yes—
683C22C2×#52511yescharacteristic
684C22C2×#55011yescharacteristic
685C22#348511yescharacteristic
686C22C22×AGL(1,ℤ/8)11yescharacteristic
687C22C2×#57511yescharacteristic
688C22C2×#57611yescharacteristic
689C22#358611yescharacteristic
690C4—22no—
691C4—22no—
692C4—22no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—44no—
698C4—41no—
699C4—81no—
700C4—81no—
701C4—81no—
702C4—84no—
703C4—82no—
704C4—81no—
705C4#311414yes—
706C4#347512yes—
707C4#349412yes—
708C2—24no—
709C2—84no—
710C2—162no—
711C2#3651711yescharacteristic
712C2C2×#311411yescharacteristic
713C2#6058411yescharacteristic
714C2C2×#339911yescharacteristic
715C2#6140311yescharacteristic
716C2#6171711yescharacteristic
717C2#6174011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36587C2semidirect
2. C2×#2354C2semidirect
3. #33111C2semidirect
5. #28312C2semidirect
6. #28114C2semidirect
7. #27909C2semidirect
8. #27647C2semidirect
9. #27504C2semidirect
10. #27206C2semidirect
11. #26087C2semidirect
12. #24631C2semidirect
13. #23099C2semidirect
14. #22811C2semidirect
15. #22629C2semidirect
16. #21578C2semidirect
17. C4×#450C2semidirect
30. #2571C22semidirect
31. #2548C22semidirect
33. #2513C22semidirect
35. #2447C22semidirect
37. #2366C22semidirect
40. #2355C22semidirect
55. #2093C22semidirect
57. #2084C22semidirect
58. #2072C22semidirect
62. #1983C22semidirect
65. #1957C22semidirect
66. #1949C22semidirect
67. #1948C22semidirect
68. #1939C22semidirect
69. #1928C22semidirect
70. #1926C22semidirect
71. #1926C22semidirect
73. #1921C22semidirect
74. #1917C22semidirect
82. #1845C22semidirect
84. #1803C22semidirect
89. #1776C22semidirect
90. #1770C22semidirect
91. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#23542
#331112
#329022
#283122
#281142
#279092
#276472
#275042
#272062
#260872
#246312
#230992
#228112
#226292
#215782
C4×#4502