Finite Group Databasebeta

← #6795027

Subgroups — #6795027

Order 512.

1875 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3#32953C211yesmaximal, characteristic
4#32448C214yesmaximal, complemented
5#32362C214yesmaximal, complemented
6#29909C212yesmaximal, complemented
7#28540C212yesmaximal, complemented
8#27736C212yesmaximal, complemented
9#27588C212yesmaximal
10#25813C211yesmaximal, characteristic, complemented
11#25449C212yesmaximal, complemented
12#24678C212yesmaximal, complemented
13#24507C212yesmaximal
14#24301C211yesmaximal, characteristic, complemented
15#22603C211yesmaximal, characteristic, complemented
16#21987C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#500C2212yescomplemented
23C2×#475C2212yescomplemented
24C2×#474C2212yes—
25C2×#450C2211yescharacteristic
26C2×C4×SD16C2212yes—
27C2×#432C2212yes—
28#2857C2211yescharacteristic
29#2855C2212yes—
30#2571C2211yescharacteristic, complemented
31#2568C2211yescharacteristic, complemented
32#2542—21no—
33#2542—21no—
34#2542—22no—
35#2542C2212yes—
36#2542C2212yes—
37#2542C2214yes—
38#2533—22no—
39#2514—21no—
40#2514C2212yescomplemented
41#2509C2212yescomplemented
42#2503—21no—
43#2493—22no—
44C2×#399—22no—
45#2365C2211yescharacteristic, complemented
46#2360—21no—
47#2356—21no—
48#2355C2211yescharacteristic, complemented
49C4×C22⋊Q8C2211yescharacteristic
50C2×#390C2212yes—
51C2×#384C2212yes—
52#2283C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2100C2214yescomplemented
55#2095C2214yescomplemented
56#2092C2214yes—
57#2083C2214yescomplemented
58#2073C2214yes—
59#2072C2214yescomplemented
60#2067C2214yes—
61#2058C2212yescomplemented
62#2057C2214yescomplemented
63#2055C2214yes—
64#2024C2212yescomplemented
65#1990C2212yescomplemented
66#1988C2212yescomplemented
67#1973C2214yes—
68#1948C2214yescomplemented
69#1947C2214yescomplemented
70#1939C2212yescomplemented
71#1936C2214yes—
72#1926C2214yes—
73#1925C2214yes—
74#1917C2214yescomplemented
75#1877C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1864C2212yes—
79#1862C2214yes—
80#1853C2212yes—
81#1853C2211yescharacteristic
82#1852C2212yes—
83#1852C2212yes—
84#1852C2212yes—
85#1847C2214yescomplemented
86#1836C2214yescomplemented
87#1817C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90#1786C2212yes—
91#1786C2211yescharacteristic
92#1770C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—48no—
113C2×C22⋊Q8—48no—
114C2×C22⋊Q8C2311yescharacteristic
115C2×C4×Q8—21no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—44no—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#500—44no—
135#475—44no—
136#474—44no—
137#450—24no—
138C4×SD16—44no—
139#432—28no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2311yescharacteristic
157C2×#97C2314yes—
158C2×C4×C8C2312yes—
159#401—22no—
160#400—22no—
161#400—22no—
162#400—22no—
163#400—22no—
164#400—22no—
165#399—24no—
166#399—24no—
167#399—24no—
168#399—24no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391C2312yes—
174#391C2312yes—
175#391C2311yescharacteristic
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2314yes—
180#389—21no—
181#389—22no—
182#389—21no—
183#389C2312yes—
184#389C2311yescharacteristic
185#389C2312yes—
186#389C2312yes—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—21no—
191#387—22no—
192#387—22no—
193#387—22no—
194#385—22no—
195#385—21no—
196#384—22no—
197#384—22no—
198#384C2311yescharacteristic
199#384C2314yes—
200#384C2314yes—
201#383—22no—
202#383—22no—
203#382—21no—
204#382—21no—
205#382C2312yes—
206#382C2312yes—
207#381—22no—
208#381—22no—
209#380—21no—
210#380C2312yes—
211#380C2311yescharacteristic
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225#376—22no—
226C2×#95D412yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C25—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2411yescharacteristic
246C22×D4—24no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—42no—
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278C22⋊Q8—416no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C22⋊Q8—48no—
282C22⋊Q8—416no—
283C22⋊Q8—44no—
284C22⋊Q8—48no—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—42no—
290C4×D4—42no—
291C4×D4—44no—
292C4×D4—44no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—48no—
341C2×C22⋊C4—48no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357#107—44no—
358#107—44no—
359#107—44no—
360#106—42no—
361#106—44no—
362C4⋊C8—28no—
363C4⋊C8—24no—
364#103—42no—
365#103—44no—
366#102—28no—
367#102—24no—
368#102—24no—
369#102—42no—
370#102—44no—
371C22⋊C8—24no—
372C22⋊C8—22no—
373C22⋊C8#4114yes—
374#97—28no—
375C4×C8—24no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—24no—
400C24—22no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24C22×D411yescharacteristic
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—816no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D42−1+414yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—41no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—816no—
467C22×C4AGL(1,ℤ/8)14yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C2×#4112yes—
478C22×C42−1+412yes—
479C22×C4C2511yescharacteristic
480SD16—48no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8C2×#4114yes—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—82no—
522C4⋊C4AGL(1,ℤ/8)14yes—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—48no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—42no—
542C42—42no—
543C42—44no—
544C42—82no—
545C42H(5,𝔽2)14yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—24no—
551C23—48no—
552C23—48no—
553C23—48no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23#54712yes—
565C23C2×AGL(1,ℤ/8)12yes—
566C23#57812yes—
567C23C23×D411yescharacteristic
568Q8—42no—
569Q8—42no—
570Q8—88no—
571Q8—84no—
572Q8—88no—
573D4—24no—
574D4—24no—
575D4—48no—
576D4—42no—
577D4—42no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
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—21no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—88no—
629C2×C4—84no—
630C2×C4—88no—
631C2×C4—88no—
632C2×C4#53512yes—
633C2×C4#53512yes—
634C2×C4#54712yes—
635C2×C4C2×AGL(1,ℤ/8)14yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4C2×AGL(1,ℤ/8)12yes—
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×H(5,𝔽2)12yes—
641C2×C4C2×2−1+412yes—
642C2×C4C2×2−1+411yescharacteristic
643C8—24no—
644C8—44no—
645C8—42no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—816no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22#335314yes—
661C22C2×#53511yescharacteristic
662C22C2×#54711yescharacteristic
663C22#348611yescharacteristic
664C22C22×AGL(1,ℤ/8)11yescharacteristic
665C22C22×AGL(1,ℤ/8)11yescharacteristic
666C22C2×#57811yescharacteristic
667C22#358611yescharacteristic
668C4—22no—
669C4—24no—
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—41no—
676C4—81no—
677C4—81no—
678C4—82no—
679C4—84no—
680C4—82no—
681C4—82no—
682C4—81no—
683C4—81no—
684C4#321714yes—
685C4#352012yes—
686C4#353812yes—
687C2—24no—
688C2—84no—
689C2#3891711yescharacteristic
690C2C2×#321711yescharacteristic
691C2#6081411yescharacteristic
692C2C2×#335311yescharacteristic
693C2#6125211yescharacteristic
694C2#6182711yescharacteristic
695C2#6183711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #36177C2semidirect
4. #32448C2semidirect
5. #32362C2semidirect
6. #29909C2semidirect
7. #28540C2semidirect
8. #27736C2semidirect
10. #25813C2semidirect
11. #25449C2semidirect
12. #24678C2semidirect
14. #24301C2semidirect
15. #22603C2semidirect
16. #21987C2semidirect
22. C2×#500C22semidirect
23. C2×#475C22semidirect
30. #2571C22semidirect
31. #2568C22semidirect
40. #2514C22semidirect
41. #2509C22semidirect
45. #2365C22semidirect
48. #2355C22semidirect
54. #2100C22semidirect
55. #2095C22semidirect
57. #2083C22semidirect
59. #2072C22semidirect
61. #2058C22semidirect
62. #2057C22semidirect
64. #2024C22semidirect
65. #1990C22semidirect
66. #1988C22semidirect
68. #1948C22semidirect
69. #1947C22semidirect
70. #1939C22semidirect
74. #1917C22semidirect
85. #1847C22semidirect
86. #1836C22semidirect
92. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×#25422
#329532
#324482
#323622
#299092
#285402
#277362
#275882
#258132
#254492
#246782
#245072
#243012
#226032
#219872
C4×#4502