Finite Group Databasebeta

← #6469270

Subgroups — #6469270

Order 512.

2462 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#29498C214yesmaximal, complemented
4#29192C212yesmaximal, complemented
5#28613C212yesmaximal, complemented
6#27972C211yesmaximal, characteristic, complemented
7#27950C211yesmaximal, characteristic, complemented
8#25855C211yesmaximal, characteristic
9#25717C211yesmaximal, characteristic
10#24507C214yesmaximal, complemented
11#23684C214yesmaximal, complemented
12#21692C212yesmaximal, complemented
13#21646C212yesmaximal, complemented
14#21453C214yesmaximal, complemented
15C4×#432C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4—21no—
18C22×C4×D4—21no—
19C22×C4×D4C2212yes—
20C2×#492C2212yes—
21C2×#486C2212yes—
22C2×#450C2214yes—
23C2×#440C2218yescomplemented
24C2×#432C2212yes—
25#2864—21no—
26#2863—21no—
27#2863C2212yescomplemented
28#2862C2211yescharacteristic, complemented
29#2859C2211yescharacteristic, complemented
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2855—21no—
33#2855C2212yes—
34#2851—21no—
35C4×C4⋊Q8C2211yescharacteristic, complemented
36C4×C4⋊Q8C2211yescharacteristic, complemented
37C4×#127—21no—
38C4×#127—21no—
39C4×#127—21no—
40C4×#127—21no—
41C4×#127C2212yes—
42C4×#127C2212yes—
43C4×#127C2212yes—
44C4×#127C2212yes—
45C4×#125—21no—
46C4×C22⋊Q8—21no—
47C4×C22⋊Q8C2212yes—
48#2295—21no—
49#2295C2212yescomplemented
50C2×#390—21no—
51C2×#390—21no—
52C2×#390C2212yes—
53C2×#384C2211yescharacteristic
54#2283—21no—
55C2×C43C2211yescharacteristic
56#2083C2214yescomplemented
57#2072C2214yescomplemented
58#2067C2218yescomplemented
59#2055C2218yescomplemented
60#2041C2214yes—
61#2014C2214yes—
62#1990C2212yescomplemented
63#1973C2218yescomplemented
64#1957C2212yes—
65#1957C2212yes—
66#1956C2212yescomplemented
67#1936C2218yescomplemented
68#1877C2218yescomplemented
69#1872C2214yes—
70#1862C2218yescomplemented
71#1852C2214yes—
72#1845C2212yescomplemented
73#1834C2212yescomplemented
74#1786C2214yes—
75#1767C2212yescomplemented
76C4×C4⋊C8C2212yescomplemented
77#1762C2218yescomplemented
78#1749C2212yes—
79C4×C22⋊C8C2211yescharacteristic
80C4×C22⋊C8C2211yescharacteristic
81#1745C2212yescomplemented
82C4×#97C2212yescomplemented
83C23×D4—21no—
84C23×D4—21no—
85C23×D4C2312yes—
86C24×C4—22no—
87C22×SD16C2318yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127—28no—
91C2×#127—21no—
92C2×#127—21no—
93C2×#127C2312yes—
94C2×#127C2312yes—
95C2×#125—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8C2312yes—
98C2×C4×Q8—21no—
99C2×C4×Q8C2312yes—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—48no—
111C2×C4×D4—48no—
112C2×C4×D4—48no—
113C2×C4×D4C2314yes—
114C2×C4×D4C2314yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4—22no—
119C22×C22⋊C4—22no—
120C22×C22⋊C4—22no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#492—28no—
129#486—28no—
130#450—216no—
131#440—416no—
132#432—28no—
133C2×#107C2318yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2318yes—
136C2×#102C2318yes—
137C2×#102C2318yes—
138C2×#102C2318yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×#97C2314yes—
142C2×#97C2314yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2312yes—
145#391—21no—
146#391—21no—
147#391—21no—
148#391—21no—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#391C2312yes—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390—22no—
161#390—22no—
162#390—22no—
163#390C2314yes—
164#390C2314yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#383—22no—
194#382—21no—
195#382—21no—
196#382—21no—
197#381—22no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380C2312yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—22no—
209#376—22no—
210C43D411yescharacteristic
211C43D411yescharacteristic
212C43D411yescharacteristic
213C43D411yescharacteristic
214C43C2311yescharacteristic
215C43C2311yescharacteristic
216C43C2311yescharacteristic
217C43C2311yescharacteristic
218#340C2318yes—
219#340C2318yes—
220#336C2312yes—
221#336C2312yes—
222#336C2314yes—
223C25—22no—
224C22×Q8—21no—
225C22×Q8C2412yes—
226C22×D4—28no—
227C22×D4—28no—
228C22×D4—28no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—28no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—44no—
238C22×D4—44no—
239C22×D4—44no—
240C22×D4C2414yes—
241C23×C4—22no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—48no—
245C23×C4—48no—
246C23×C4—48no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×SD16—216no—
255C2×SD16—216no—
256C2×SD16—48no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261#127—28no—
262#127—28no—
263#127—28no—
264#127—24no—
265#127#4118yes—
266#125—44no—
267C22⋊Q8—28no—
268C22⋊Q8—44no—
269C4×Q8—28no—
270C4×Q8—44no—
271C4×D4—216no—
272C4×D4—48no—
273C4×D4—48no—
274C4×D4—48no—
275C4×D4—48no—
276C4×D4—48no—
277C4×D4—48no—
278C4×D4—48no—
279C4×D4—48no—
280C4×D4—48no—
281C4×D4—48no—
282C4×D4—48no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4—48no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—42no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—48no—
357C4⋊C8—28no—
358#103—48no—
359#102—216no—
360#102—216no—
361#102—48no—
362C22⋊C8—24no—
363C22⋊C8#4118yes—
364#97—28no—
365#97—28no—
366C4×C8—24no—
367C4×C8—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382C24—28no—
383C24—24no—
384C24—44no—
385C24—44no—
386C24—44no—
387C24—44no—
388C24—44no—
389C24—44no—
390C24C22×D411yescharacteristic
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—24no—
404C2×D4—28no—
405C2×D4—432no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—432no—
410C2×D4—432no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4C2×#4118yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—832no—
452C22×C4AGL(1,ℤ/8)14yes—
453C22×C4#13714yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4114yes—
468C22×C4C2511yescharacteristic
469SD16—432no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8C2×#4118yes—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—28no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—41no—
530C42—41no—
531C42—44no—
532C42—44no—
533C42AGL(1,ℤ/8)14yes—
534C42C2×#4114yes—
535C42H(5,𝔽2)14yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—28no—
541C23—416no—
542C23—416no—
543C23—416no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—88no—
551C23—88no—
552C23—88no—
553C23—88no—
554C23#53212yes—
555C23C2×AGL(1,ℤ/8)12yes—
556C23C2×#13712yes—
557C23C23×D411yescharacteristic
558Q8—48no—
559Q8—84no—
560D4—216no—
561D4—416no—
562D4—416no—
563D4—416no—
564D4—416no—
565D4—416no—
566D4—48no—
567C2×C4—28no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—816no—
616C2×C4—816no—
617C2×C4—88no—
618C2×C4#52512yes—
619C2×C4#52512yes—
620C2×C4#53212yes—
621C2×C4#53512yes—
622C2×C4#53512yes—
623C2×C4C2×AGL(1,ℤ/8)12yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4C2×#13712yes—
629C2×C4C2×#13712yes—
630C2×C4#57512yes—
631C2×C4C22×#4112yes—
632C2×C4C22×#4112yes—
633C2×C4C2×H(5,𝔽2)12yes—
634C8—28no—
635C8—44no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—48no—
643C22—48no—
644C22—48no—
645C22—832no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22#318214yes—
651C22C2×#52511yescharacteristic
652C22C2×#53211yescharacteristic
653C22C2×#53511yescharacteristic
654C22C22×AGL(1,ℤ/8)11yescharacteristic
655C22C22×AGL(1,ℤ/8)11yescharacteristic
656C22C22×#13711yescharacteristic
657C22C2×#57511yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4#311014yes—
675C4#311414yes—
676C4#348114yes—
677C2—24no—
678C2—88no—
679C2#3496511yescharacteristic
680C2C2×#311411yescharacteristic
681C2C2×#311011yescharacteristic
682C2C2×#318211yescharacteristic
683C2#6070011yescharacteristic
684C2#6079811yescharacteristic
685C2#6080411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36635C2semidirect
3. #29498C2semidirect
4. #29192C2semidirect
5. #28613C2semidirect
6. #27972C2semidirect
7. #27950C2semidirect
10. #24507C2semidirect
11. #23684C2semidirect
12. #21692C2semidirect
13. #21646C2semidirect
14. #21453C2semidirect
15. C4×#432C2semidirect
23. C2×#440C22semidirect
27. #2863C22semidirect
28. #2862C22semidirect
29. #2859C22semidirect
35. C4×C4⋊Q8C22semidirect
36. C4×C4⋊Q8C22semidirect
49. #2295C22semidirect
56. #2083C22semidirect
57. #2072C22semidirect
58. #2067C22semidirect
59. #2055C22semidirect
62. #1990C22semidirect
63. #1973C22semidirect
66. #1956C22semidirect
67. #1936C22semidirect
68. #1877C22semidirect
70. #1862C22semidirect
72. #1845C22semidirect
73. #1834C22semidirect
75. #1767C22semidirect
76. C4×C4⋊C8C22semidirect
77. #1762C22semidirect
81. #1745C22semidirect
82. C4×#97C22semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×C4×#1272
#294982
#291922
#286132
#279722
#279502
#258552
#257172
#245072
#236842
#216922
#216462
#214532
C4×#4322