Finite Group Databasebeta

← #6462375

Subgroups — #6462375

Order 512.

1914 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#29580C212yesmaximal, complemented
4#29560C212yesmaximal, complemented
5#29433C214yesmaximal, complemented
6#29329C211yesmaximal, characteristic, complemented
7#29296C211yesmaximal, characteristic, complemented
8#29103C211yesmaximal, characteristic
9#28302C211yesmaximal, characteristic
10#24609C212yesmaximal, complemented
11#24507C212yesmaximal
12#23780C212yesmaximal, complemented
13#23683C212yesmaximal, complemented
14#21579C212yesmaximal, complemented
15#21453C212yesmaximal
16#21288C214yesmaximal, complemented
17C4×#432C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#483C2212yes—
21C2×#450C2212yes—
22C2×#440C2214yes—
23C2×#432C2212yes—
24#2864—21no—
25#2863—21no—
26#2863C2212yescomplemented
27#2862C2211yescharacteristic, complemented
28#2859C2211yescharacteristic, complemented
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2855—21no—
32#2855C2212yes—
33#2851—21no—
34C4×C4⋊Q8C2211yescharacteristic, complemented
35C4×C4⋊Q8C2211yescharacteristic, complemented
36C4×#125—21no—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8C2212yes—
39#2295—21no—
40#2295C2212yescomplemented
41#2293—21no—
42#2293—21no—
43#2293—21no—
44#2293—21no—
45#2293C2212yes—
46#2293C2212yes—
47#2293C2212yes—
48#2293C2212yes—
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#388—21no—
55#2283—21no—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4—21no—
58C2×C43C2211yescharacteristic
59#2100C2212yescomplemented
60#2094C2212yes—
61#2088C2214yescomplemented
62#2083C2212yescomplemented
63#2071C2212yes—
64#2068C2214yescomplemented
65#2067C2214yes—
66#2057C2212yescomplemented
67#2055C2214yes—
68#2043C2212yescomplemented
69#2023C2212yes—
70#2013C2212yes—
71#1973C2214yes—
72#1968C2214yes—
73#1943C2214yescomplemented
74#1942C2214yescomplemented
75#1936C2214yes—
76#1935C2214yescomplemented
77#1934C2214yes—
78#1933C2214yescomplemented
79#1928C2214yescomplemented
80#1919C2214yescomplemented
81#1877C2214yes—
82#1872C2212yes—
83#1872C2212yes—
84#1862C2214yes—
85#1852C2212yes—
86#1852C2212yes—
87#1823C2212yes—
88#1786C2212yes—
89#1776C2214yescomplemented
90#1767C2212yescomplemented
91C4×C4⋊C8C2212yescomplemented
92#1763C2214yescomplemented
93#1762C2214yes—
94#1749C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C4×C22⋊C8C2211yescharacteristic
97#1745C2212yescomplemented
98C4×#97C2212yescomplemented
99C23×D4C2311yescharacteristic
100C24×C4—21no—
101C22×SD16C2314yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2312yes—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×D4—28no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—22no—
115C2×#117—22no—
116C2×#117—22no—
117C2×#117—22no—
118C2×#117—48no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C42—21no—
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#483—28no—
138#450—28no—
139#440—216no—
140#432—28no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2314yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2312yes—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#385—22no—
210#385—22no—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#383—22no—
214#382—21no—
215#382—21no—
216#382—21no—
217#381—22no—
218#380—21no—
219#380—21no—
220#380—21no—
221#380C2312yes—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237#376—22no—
238C2×#95—21no—
239C2×#95—21no—
240C2×#95—21no—
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43D411yescharacteristic
244C43D411yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247C43C2311yescharacteristic
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#336C2312yes—
256#336C2312yes—
257#336C2314yes—
258C25—21no—
259C22×Q8—21no—
260C22×Q8C2412yes—
261C22×D4—24no—
262C22×D4—21no—
263C22×D4—24no—
264C22×D4—42no—
265C22×D4C2412yes—
266C23×C4—28no—
267C23×C4—21no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—44no—
273C23×C4—44no—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×SD16—28no—
282C2×SD16—28no—
283C2×SD16—44no—
284C22×C8C2414yes—
285C22×C8C2414yes—
286C4⋊Q8—42no—
287C4⋊Q8—42no—
288#125—44no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—44no—
292C4×Q8—28no—
293C4×Q8—44no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4#4118yes—
300#117—48no—
301#117—48no—
302#117—48no—
303#117—48no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—28no—
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—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—42no—
361C2×C42—42no—
362C2×C42—42no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#107—28no—
393#107—44no—
394#106—28no—
395C4⋊C8—28no—
396#103—28no—
397#103—28no—
398#103—44no—
399#102—28no—
400#102—28no—
401#102—44no—
402C22⋊C8—24no—
403C22⋊C8#4118yes—
404#97—28no—
405#97—28no—
406C4×C8—42no—
407C4×C8—42no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428C24—24no—
429C24—22no—
430C24—42no—
431C24—42no—
432C24—42no—
433C24—42no—
434C24—42no—
435C24—42no—
436C24C22×D411yescharacteristic
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—416no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4C2×#4114yes—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—28no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—432no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4AGL(1,ℤ/8)14yes—
494C22×C4#13714yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C2×#4114yes—
509C22×C4C2511yescharacteristic
510SD16—416no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8C2×#4118yes—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4C2×#4114yes—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42—44no—
585C42—44no—
586C42—44no—
587C42—44no—
588C42C2×#4114yes—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—24no—
594C23—48no—
595C23—48no—
596C23—48no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—84no—
604C23—84no—
605C23—84no—
606C23—84no—
607C23#53212yes—
608C23C2×AGL(1,ℤ/8)12yes—
609C23C2×#13712yes—
610C23C23×D411yescharacteristic
611Q8—44no—
612Q8—44no—
613Q8—84no—
614D4—28no—
615D4—48no—
616D4—44no—
617C2×C4—28no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—28no—
637C2×C4—21no—
638C2×C4—24no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—416no—
655C2×C4—416no—
656C2×C4—48no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4#52512yes—
681C2×C4#52512yes—
682C2×C4#53212yes—
683C2×C4C2×AGL(1,ℤ/8)12yes—
684C2×C4C2×AGL(1,ℤ/8)12yes—
685C2×C4C2×#13712yes—
686C2×C4C2×#13712yes—
687C2×C4#57512yes—
688C2×C4#57512yes—
689C2×C4C22×#4112yes—
690C2×C4C22×#4112yes—
691C2×C4#58512yes—
692C8—28no—
693C8—44no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—816no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22#318214yes—
709C22C2×#52511yescharacteristic
710C22C2×#53211yescharacteristic
711C22#344911yescharacteristic
712C22C22×AGL(1,ℤ/8)11yescharacteristic
713C22C22×#13711yescharacteristic
714C22C2×#57511yescharacteristic
715C22C2×#57511yescharacteristic
716C4—22no—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—44no—
729C4—41no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4#347514yes—
737C2—24no—
738C2—84no—
739C2#3487711yescharacteristic
740C2#6055811yescharacteristic
741C2#6056211yescharacteristic
742C2C2×#318211yescharacteristic
743C2#6070211yescharacteristic
744C2#6095811yescharacteristic
745C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #29580C2semidirect
4. #29560C2semidirect
5. #29433C2semidirect
6. #29329C2semidirect
7. #29296C2semidirect
10. #24609C2semidirect
12. #23780C2semidirect
13. #23683C2semidirect
14. #21579C2semidirect
16. #21288C2semidirect
17. C4×#432C2semidirect
26. #2863C22semidirect
27. #2862C22semidirect
28. #2859C22semidirect
34. C4×C4⋊Q8C22semidirect
35. C4×C4⋊Q8C22semidirect
40. #2295C22semidirect
59. #2100C22semidirect
61. #2088C22semidirect
62. #2083C22semidirect
64. #2068C22semidirect
66. #2057C22semidirect
68. #2043C22semidirect
73. #1943C22semidirect
74. #1942C22semidirect
76. #1935C22semidirect
78. #1933C22semidirect
79. #1928C22semidirect
80. #1919C22semidirect
89. #1776C22semidirect
90. #1767C22semidirect
91. C4×C4⋊C8C22semidirect
92. #1763C22semidirect
97. #1745C22semidirect
98. C4×#97C22semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×#22932
#295802
#295602
#294332
#293292
#292962
#291032
#283022
#246092
#245072
#237802
#236832
#215792
#214532
#212882
C4×#4322