Finite Group Databasebeta

← #6624342

Subgroups — #6624342

Order 512.

1855 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#36551C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#33151C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic, complemented
5#29110C212yesmaximal, complemented
6#28114C212yesmaximal
7#27960C211yesmaximal, characteristic, complemented
8#27644C211yesmaximal, characteristic, complemented
9#27438C214yesmaximal, complemented
10#27376C214yesmaximal, complemented
11#26025C212yesmaximal, complemented
12#24677C212yesmaximal, complemented
13#23067C212yesmaximal, complemented
14#22758C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21578C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#480C2212yes—
21C2×#476C2212yes—
22C2×#447C2211yescharacteristic
23C2×C4×D8C2212yes—
24#2857C2211yescharacteristic
25#2851C2212yes—
26#2589—22no—
27#2563C2211yescharacteristic, complemented
28#2552C2211yescharacteristic, complemented
29#2509C2212yescomplemented
30#2497—21no—
31#2487—21no—
32#2454C2212yescomplemented
33#2450—22no—
34#2369C2211yescharacteristic, complemented
35#2354—22no—
36#2354—21no—
37#2354—21no—
38#2354C2212yes—
39#2354C2214yes—
40#2354C2212yes—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2211yescharacteristic, complemented
43#2345—21no—
44C4×#121C2212yes—
45C4×#121C2211yescharacteristic
46C2×#390C2212yes—
47C2×#388—22no—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2214yescomplemented
53#2083C2212yescomplemented
54#2073C2214yes—
55#2071C2212yescomplemented
56#2067C2214yes—
57#2054C2214yes—
58#2025C2214yes—
59#1990C2212yescomplemented
60#1989C2214yescomplemented
61#1949C2214yescomplemented
62#1947C2212yescomplemented
63#1939C2212yescomplemented
64#1936C2214yes—
65#1936C2214yes—
66#1926C2214yescomplemented
67#1925C2214yes—
68#1921C2214yescomplemented
69#1921C2214yescomplemented
70#1919C2214yescomplemented
71#1917C2214yescomplemented
72#1879C2212yes—
73#1879C2214yes—
74#1874C2211yescharacteristic
75#1864C2214yes—
76#1862C2212yes—
77#1853C2212yes—
78#1852C2211yescharacteristic
79#1834C2212yescomplemented
80#1823C2212yes—
81#1823C2212yes—
82#1823C2212yes—
83#1802C2212yes—
84#1802C2212yescomplemented
85#1797C2212yes—
86#1797C2211yescharacteristic
87#1786C2211yescharacteristic
88#1786C2212yes—
89#1776C2214yescomplemented
90#1770C2212yescomplemented
91#1762C2214yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×D8C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125C2312yes—
100C2×C22⋊Q8—21no—
101C2×#121C2311yescharacteristic
102C2×C4×D4—28no—
103C2×C4×D4—21no—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4C2311yescharacteristic
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C2×#117—22no—
113C2×#117—24no—
114C2×#117—48no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4—22no—
125C22×C42—22no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#480—28no—
129#476—28no—
130#447—24no—
131C4×D8—28no—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#400—22no—
151#399—22no—
152#398—22no—
153#396—22no—
154#393—22no—
155#391—21no—
156#391—22no—
157#390—22no—
158#390—22no—
159#390C2312yes—
160#390C2314yes—
161#390C2311yescharacteristic
162#390C2314yes—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389C2312yes—
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#388—21no—
173#388—22no—
174#388—24no—
175#388—24no—
176#388—24no—
177#388—24no—
178#387—22no—
179#387—22no—
180#387—22no—
181#386—21no—
182#386—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#382C2312yes—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—24no—
201#380—22no—
202#380C2311yescharacteristic
203#380C2312yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2314yes—
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—21no—
237C22×D4—24no—
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—22no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C2×D8—24no—
259C2×D8—24no—
260C2×D8—42no—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C4⋊Q8—24no—
265#125—44no—
266C22⋊Q8—82no—
267#121—22no—
268#121#4114yes—
269C4×D4—28no—
270C4×D4—28no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—84no—
278C4×D4#4118yes—
279#117—416no—
280#117—48no—
281#117—48no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—28no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—48no—
336C2×C22⋊C4—84no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—44no—
343C2×C42—48no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352#107—24no—
353#107—28no—
354#106—28no—
355C4⋊C8—24no—
356C4⋊C8—44no—
357#102—24no—
358#102—28no—
359#102—28no—
360#102—24no—
361#102—24no—
362#102—44no—
363#102—42no—
364C22⋊C8—22no—
365C22⋊C8—24no—
366C22⋊C8—22no—
367#97—44no—
368C4×C8—24no—
369#95—24no—
370#95—24no—
371#95—21no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—24no—
387C24—21no—
388C24—22no—
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—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
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×D4C2×#4112yes—
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—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—432no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4AGL(1,ℤ/8)14yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4112yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469D8—48no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4C2×#4112yes—
510C4⋊C4C2×#4114yes—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—24no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—81no—
521C22⋊C4—82no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—24no—
531C42—42no—
532C42—42no—
533C42—48no—
534C42—48no—
535C42—42no—
536C42—82no—
537C42C2×#4114yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—24no—
543C23—22no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—41no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23#54812yes—
564C23C2×AGL(1,ℤ/8)12yes—
565C23#57612yes—
566C23C23×D411yescharacteristic
567Q8—162no—
568D4—24no—
569D4—24no—
570D4—24no—
571D4—48no—
572D4—48no—
573D4—42no—
574D4—42no—
575D4—42no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
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—22no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—44no—
603C2×C4—416no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—416no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—84no—
627C2×C4—88no—
628C2×C4—84no—
629C2×C4—164no—
630C2×C4#52512yes—
631C2×C4#52512yes—
632C2×C4#54812yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4#57512yes—
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4112yes—
638C2×C4C2×2−1+411yescharacteristic
639C2×C4#58512yes—
640C8—44no—
641C8—42no—
642C8—42no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22—44no—
653C22—42no—
654C22—44no—
655C22—816no—
656C22—81no—
657C22—82no—
658C22—81no—
659C22—82no—
660C22—82no—
661C22—81no—
662C22—82no—
663C22—81no—
664C22#336614yes—
665C22C2×#52511yescharacteristic
666C22C2×#54811yescharacteristic
667C22#348511yescharacteristic
668C22C22×AGL(1,ℤ/8)11yescharacteristic
669C22C2×#57511yescharacteristic
670C22C2×#57611yescharacteristic
671C22#358611yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—44no—
681C4—41no—
682C4—81no—
683C4—82no—
684C4—84no—
685C4—81no—
686C4—82no—
687C4#311414yes—
688C4#347512yes—
689C4#349512yes—
690C2—24no—
691C2—82no—
692C2—84no—
693C2#3651611yescharacteristic
694C2C2×#311411yescharacteristic
695C2#6058411yescharacteristic
696C2C2×#336611yescharacteristic
697C2#6132311yescharacteristic
698C2#6171711yescharacteristic
699C2#6174011yescharacteristic
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. #36551C2semidirect
2. C2×#2354C2semidirect
3. #33151C2semidirect
4. #32956C2semidirect
5. #29110C2semidirect
7. #27960C2semidirect
8. #27644C2semidirect
9. #27438C2semidirect
10. #27376C2semidirect
11. #26025C2semidirect
12. #24677C2semidirect
13. #23067C2semidirect
14. #22758C2semidirect
15. #22629C2semidirect
16. #21578C2semidirect
27. #2563C22semidirect
28. #2552C22semidirect
29. #2509C22semidirect
32. #2454C22semidirect
34. #2369C22semidirect
42. D4×C4⋊C4C22semidirect
52. #2099C22semidirect
53. #2083C22semidirect
55. #2071C22semidirect
59. #1990C22semidirect
60. #1989C22semidirect
61. #1949C22semidirect
62. #1947C22semidirect
63. #1939C22semidirect
66. #1926C22semidirect
68. #1921C22semidirect
69. #1921C22semidirect
70. #1919C22semidirect
71. #1917C22semidirect
79. #1834C22semidirect
84. #1802C22semidirect
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
#365512
C2×#23542
#331512
#329562
#291102
#281142
#279602
#276442
#274382
#273762
#260252
#246772
#230672
#227582
#226292
#215782
C4×#4472