Finite Group Databasebeta

← #6617012

Subgroups — #6617012

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#33096C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic, complemented
5#27950C211yesmaximal, characteristic, complemented
6#27678C211yesmaximal, characteristic, complemented
7#27588C212yesmaximal, complemented
8#27493C214yesmaximal, complemented
9#27363C214yesmaximal, complemented
10#26087C212yesmaximal, complemented
11#24677C212yesmaximal, complemented
12#24500C212yesmaximal
13#23061C212yesmaximal, complemented
14#22807C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16C4×#447C211yesmaximal, characteristic
17#21361C212yesmaximal, complemented
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#476C2212yes—
21C2×#447C2211yescharacteristic
22C2×C4×D8C2212yes—
23C2×#431C2212yes—
24#2857C2211yescharacteristic
25#2854C2212yes—
26#2571C2211yescharacteristic, complemented
27#2563C2211yescharacteristic, complemented
28#2532—22no—
29#2513C2212yescomplemented
30#2508C2212yescomplemented
31#2495—21no—
32#2488—22no—
33#2486—21no—
34#2369C2211yescharacteristic, complemented
35#2354—21no—
36#2354—21no—
37#2354—22no—
38#2354C2212yes—
39#2354C2212yes—
40#2354C2214yes—
41#2354C2211yescharacteristic, complemented
42#2352—21no—
43#2346—21no—
44C4×#121C2211yescharacteristic
45C2×#390C2212yes—
46C2×#388—22no—
47C2×#384C2211yescharacteristic
48#2283C2212yes—
49C2×#380—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2212yescomplemented
53#2094C2212yescomplemented
54#2092C2214yes—
55#2083C2214yescomplemented
56#2073C2214yes—
57#2067C2214yes—
58#2054C2214yes—
59#1990C2212yescomplemented
60#1970C2214yes—
61#1949C2214yescomplemented
62#1949C2214yescomplemented
63#1948C2212yescomplemented
64#1939C2212yescomplemented
65#1934C2214yes—
66#1926C2214yes—
67#1926C2214yescomplemented
68#1925C2214yes—
69#1925C2214yescomplemented
70#1921C2214yescomplemented
71#1920C2214yescomplemented
72#1916C2214yescomplemented
73#1879C2212yes—
74#1874C2212yes—
75#1874C2211yescharacteristic
76#1864C2214yes—
77#1862C2212yes—
78#1853C2212yes—
79#1853C2212yes—
80#1852C2212yes—
81#1852C2212yes—
82#1852C2211yescharacteristic
83#1845C2212yescomplemented
84#1817C2212yes—
85#1802C2212yes—
86#1802C2212yescomplemented
87#1797C2211yescharacteristic
88#1786C2212yes—
89#1786C2211yescharacteristic
90C4×#107C2214yescomplemented
91C4×#106C2214yescomplemented
92C8×C4⋊C4C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×D8C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#123—22no—
100C2×C22⋊Q8—21no—
101C2×#121C2311yescharacteristic
102C2×C4×D4—28no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×#117—21no—
110C2×#117—22no—
111C2×#117—24no—
112C2×#117—22no—
113C2×#117—48no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—22no—
122C22×C22⋊C4—22no—
123C22×C42—22no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#476—28no—
127#447—24no—
128C4×D8—28no—
129#431—28no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×C4×C8C2312yes—
148C2×C4×C8C2314yes—
149#400—22no—
150#400—22no—
151#398—22no—
152#397—22no—
153#397—22no—
154#393—22no—
155#391—21no—
156#391—22no—
157#390—22no—
158#390—22no—
159#390C2312yes—
160#390C2314yes—
161#390C2312yes—
162#390C2311yescharacteristic
163#390C2314yes—
164#389—21no—
165#389—21no—
166#389—21no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#388—22no—
172#388—22no—
173#388—22no—
174#388—21no—
175#388—22no—
176#388—24no—
177#388—24no—
178#388—24no—
179#388—24no—
180#388—22no—
181#387—22no—
182#386—22no—
183#386—21no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#383—22no—
190#382—21no—
191#382—22no—
192#382C2312yes—
193#382C2312yes—
194#381—22no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—24no—
201#380—21no—
202#380C2312yes—
203#380C2311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220#376—22no—
221C2×#95—22no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C25—21no—
235C22×Q8—21no—
236C22×D4—24no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—42no—
242C22×D4C2411yescharacteristic
243C22×D4C2412yes—
244C23×C4—28no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×D8—24no—
258C2×D8—24no—
259C2×D8—42no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264#123—84no—
265C22⋊Q8—82no—
266#121—22no—
267#121—22no—
268C4×D4—28no—
269C4×D4—28no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—44no—
275C4×D4#4118yes—
276#117—48no—
277#117—416no—
278#117—44no—
279#117—48no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—48no—
330C2×C22⋊C4—84no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—48no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348#107—24no—
349#107—44no—
350#107—44no—
351#106—28no—
352#106—44no—
353C4⋊C8—28no—
354C4⋊C8—24no—
355#102—24no—
356#102—28no—
357#102—24no—
358#102—24no—
359#102—42no—
360#102—44no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—22no—
364C22⋊C8#4114yes—
365C4×C8—28no—
366C4×C8—24no—
367#95—24no—
368#95—24no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—24no—
387C24—22no—
388C24—21no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24C22×D411yescharacteristic
396C2×Q8—42no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4C2×#4114yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—44no—
441C22×C4—432no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4#13514yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4112yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466D8—48no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8C2×#4114yes—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4AGL(1,ℤ/8)14yes—
504C4⋊C4C2×#4114yes—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—48no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—81no—
516C22⋊C4—82no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C42—22no—
522C42—22no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—48no—
531C42—42no—
532C42—48no—
533C42—44no—
534C42—82no—
535C42C2×#4114yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—24no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23#54712yes—
562C23C2×#13512yes—
563C23#57612yes—
564C23C23×D411yescharacteristic
565Q8—162no—
566D4—24no—
567D4—24no—
568D4—48no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—84no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—416no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—416no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4#52512yes—
627C2×C4#52512yes—
628C2×C4#54712yes—
629C2×C4C2×#13512yes—
630C2×C4C2×#13512yes—
631C2×C4C2×AGL(1,ℤ/8)14yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4112yes—
635C2×C4C2×2−1+411yescharacteristic
636C2×C4#58512yes—
637C8—24no—
638C8—44no—
639C8—42no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—816no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22#334114yes—
662C22C2×#52511yescharacteristic
663C22C2×#54711yescharacteristic
664C22#348711yescharacteristic
665C22C22×#13511yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C2×#57611yescharacteristic
668C22#358611yescharacteristic
669C4—24no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—41no—
678C4—81no—
679C4—81no—
680C4—82no—
681C4—81no—
682C4—82no—
683C4—84no—
684C4#310514yes—
685C4#347512yes—
686C4#349512yes—
687C2—24no—
688C2—84no—
689C2—161no—
690C2#3642711yescharacteristic
691C2C2×#310511yescharacteristic
692C2#6058411yescharacteristic
693C2C2×#334111yescharacteristic
694C2#6125211yescharacteristic
695C2#6188211yescharacteristic
696C2#6190711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
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. #36176C2semidirect
2. C2×#2354C2semidirect
3. #33096C2semidirect
4. #32956C2semidirect
5. #27950C2semidirect
6. #27678C2semidirect
7. #27588C2semidirect
8. #27493C2semidirect
9. #27363C2semidirect
10. #26087C2semidirect
11. #24677C2semidirect
13. #23061C2semidirect
14. #22807C2semidirect
15. #22599C2semidirect
17. #21361C2semidirect
26. #2571C22semidirect
27. #2563C22semidirect
29. #2513C22semidirect
30. #2508C22semidirect
34. #2369C22semidirect
41. #2354C22semidirect
52. #2100C22semidirect
53. #2094C22semidirect
55. #2083C22semidirect
59. #1990C22semidirect
61. #1949C22semidirect
62. #1949C22semidirect
63. #1948C22semidirect
64. #1939C22semidirect
67. #1926C22semidirect
69. #1925C22semidirect
70. #1921C22semidirect
71. #1920C22semidirect
72. #1916C22semidirect
83. #1845C22semidirect
86. #1802C22semidirect
90. C4×#107C22semidirect
91. C4×#106C22semidirect
92. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#23542
#330962
#329562
#279502
#276782
#275882
#274932
#273632
#260872
#246772
#245002
#230612
#228072
#225992
C4×#4472
#213612