Finite Group Databasebeta

← #6434404

Subgroups — #6434404

Order 512.

1676 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2C2×#2860C211yesmaximal, characteristic
3#30386C214yesmaximal, complemented
4#30384C212yesmaximal, complemented
5#30383C212yesmaximal, complemented
6#30381C214yesmaximal, complemented
7#30379C212yesmaximal, complemented
8#30376C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10#23731C211yesmaximal, characteristic
11#23670C212yesmaximal, complemented
12#23657C211yesmaximal, characteristic, complemented
13#23628C211yesmaximal, characteristic
14#23618C212yesmaximal
15#21159C212yesmaximal, complemented
16#21095C212yesmaximal
17C4×#423C211yesmaximal, characteristic, complemented
18C22×#126—21no—
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#423C2212yes—
22C2×#418C2212yes—
23C2×#418C2212yes—
24C2×C4×M16C2212yes—
25#2864C2212yescomplemented
26#2864C2212yescomplemented
27#2860—21no—
28#2860—21no—
29#2860—21no—
30#2860—21no—
31#2860C2212yes—
32#2860C2212yes—
33#2860C2212yes—
34#2860C2212yes—
35#2859C2211yescharacteristic, complemented
36#2859C2211yescharacteristic, complemented
37#2859C2211yescharacteristic, complemented
38#2859C2211yescharacteristic, complemented
39#2851—21no—
40#2851—21no—
41#2851—21no—
42#2851—21no—
43C4×#125—21no—
44C4×#125—21no—
45C4×#125—21no—
46C4×#125—21no—
47#2296C2212yes—
48#2296C2212yes—
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#388—21no—
56C2×#388—21no—
57C2×#388—21no—
58#2285C2211yescharacteristic
59#2285C2211yescharacteristic
60C2×C43C2211yescharacteristic
61#2101C2214yescomplemented
62#2100C2212yescomplemented
63#2099C2212yescomplemented
64#2094C2212yes—
65#2093C2212yes—
66#2092C2214yes—
67#2088C2214yescomplemented
68#2088C2214yescomplemented
69#2087C2214yescomplemented
70#2086C2214yescomplemented
71#2085C2214yescomplemented
72#2084C2212yescomplemented
73#2083C2212yescomplemented
74#2073C2214yes—
75#2071C2212yes—
76#2071C2212yes—
77#2067C2214yes—
78#2067C2214yes—
79#2065C2214yes—
80#2065C2214yes—
81#1944C2214yescomplemented
82#1943C2214yescomplemented
83#1942C2214yescomplemented
84#1941C2214yescomplemented
85#1936C2214yes—
86#1936C2214yes—
87#1934C2214yes—
88#1934C2214yes—
89#1767C2212yescomplemented
90#1767C2212yescomplemented
91C4×C4⋊C8C2212yescomplemented
92C4×C4⋊C8C2212yescomplemented
93#1736C2212yes—
94#1736C2212yes—
95#1736C2212yes—
96#1736C2212yes—
97#1728C2211yescharacteristic
98#1728C2211yescharacteristic
99C23×D4C2311yescharacteristic
100C22×M16C2312yes—
101C22×M16C2312yes—
102C2×#126—22no—
103C2×#126—22no—
104C2×#126—22no—
105C2×#126—22no—
106C2×#126—48no—
107C2×#125—21no—
108C2×#125—21no—
109C2×#125—21no—
110C2×#125—21no—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#428—28no—
133#428—44no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138#423—28no—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141#418—28no—
142#418—44no—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C4×M16—44no—
148C2×#97C2314yes—
149C2×C4×C8C2314yes—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#390C2312yes—
161#390C2312yes—
162#389—21no—
163#389—21no—
164#389—21no—
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#389—21no—
174#389—21no—
175#389—21no—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#388—22no—
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#388—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#385—22no—
211#385—22no—
212#385—22no—
213#385—22no—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#382—21no—
219#382—21no—
220#382—21no—
221#382—21no—
222#382C2312yes—
223#382C2312yes—
224#380—21no—
225#380—21no—
226#380—21no—
227#380—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—22no—
236C2×#95—21no—
237C2×#95—21no—
238C2×#95—21no—
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#336C2312yes—
256#336C2312yes—
257#336C2312yes—
258#336C2312yes—
259C25—21no—
260C22×D4—21no—
261C22×D4—24no—
262C22×D4—22no—
263C22×D4—24no—
264C22×D4—22no—
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2411yescharacteristic
276C2×M16—24no—
277C2×M16—24no—
278C2×M16—24no—
279C2×M16—24no—
280C2×M16—24no—
281C2×M16#4118yes—
282C22×C8C2414yes—
283C22×C8C2414yes—
284#126—48no—
285#126—48no—
286#126—48no—
287#126—48no—
288#125—44no—
289#125—44no—
290#125—44no—
291#125—44no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—22no—
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—22no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—44no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—42no—
363C2×C42—42no—
364C2×C42—42no—
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×C42C2×D411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393#107—28no—
394#107—44no—
395#106—28no—
396#106—44no—
397C4⋊C8—28no—
398C4⋊C8—28no—
399#102—28no—
400#102—28no—
401#102—44no—
402#102—44no—
403#97—44no—
404C4×C8—44no—
405#95—22no—
406#95—22no—
407#95—22no—
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—
423C24—22no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24—42no—
431C24C22×D411yescharacteristic
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—48no—
440C2×D4—48no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4114yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+414yes—
491C22×C42−1+412yes—
492C22×C4C2511yescharacteristic
493M16—216no—
494M16—48no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8C2×#4118yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
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—21no—
570C42—21no—
571C42—22no—
572C42—21no—
573C42—22no—
574C42—21no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—44no—
582C42—44no—
583C42AGL(1,ℤ/8)14yes—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23—84no—
601C23—84no—
602C23#53212yes—
603C23#53612yes—
604C23#53712yes—
605C23C23×D411yescharacteristic
606D4—44no—
607D4—44no—
608D4—84no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
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—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
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—84no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4#53212yes—
670C2×C4#53612yes—
671C2×C4#53712yes—
672C2×C4#53918yes—
673C2×C4#54118yes—
674C2×C4C2×AGL(1,ℤ/8)12yes—
675C2×C4C2×AGL(1,ℤ/8)12yes—
676C2×C4C2×AGL(1,ℤ/8)12yes—
677C2×C4#57512yes—
678C2×C4#57512yes—
679C2×C4#57512yes—
680C2×C4C22×#4112yes—
681C2×C4C2×H(5,𝔽2)11yescharacteristic
682C2×C4C2×2−1+412yes—
683C2×C4C2×2−1+411yescharacteristic
684C8—28no—
685C8—44no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—168no—
700C22#266414yes—
701C22C2×#53211yescharacteristic
702C22C2×#53611yescharacteristic
703C22C2×#53711yescharacteristic
704C22C22×AGL(1,ℤ/8)11yescharacteristic
705C22C2×#57511yescharacteristic
706C22C2×#57511yescharacteristic
707C22C2×#57511yescharacteristic
708C4—22no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4#266414yes—
728C4C2×#53912yes—
729C4C2×#54112yes—
730C2—24no—
731C2—162no—
732C2C2×#266411yescharacteristic
733C2#6070011yescharacteristic
734C2#6070211yescharacteristic
735C2#6086411yescharacteristic
736C2#6086611yescharacteristic
737C2#6091611yescharacteristic
738C2#6091711yescharacteristic
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. #36640C2semidirect
3. #30386C2semidirect
4. #30384C2semidirect
5. #30383C2semidirect
6. #30381C2semidirect
7. #30379C2semidirect
8. #30376C2semidirect
9. #23747C2semidirect
11. #23670C2semidirect
12. #23657C2semidirect
15. #21159C2semidirect
17. C4×#423C2semidirect
25. #2864C22semidirect
26. #2864C22semidirect
35. #2859C22semidirect
36. #2859C22semidirect
37. #2859C22semidirect
38. #2859C22semidirect
61. #2101C22semidirect
62. #2100C22semidirect
63. #2099C22semidirect
67. #2088C22semidirect
68. #2088C22semidirect
69. #2087C22semidirect
70. #2086C22semidirect
71. #2085C22semidirect
72. #2084C22semidirect
73. #2083C22semidirect
81. #1944C22semidirect
82. #1943C22semidirect
83. #1942C22semidirect
84. #1941C22semidirect
89. #1767C22semidirect
90. #1767C22semidirect
91. C4×C4⋊C8C22semidirect
92. C4×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#366402
C2×#28602
#303862
#303842
#303832
#303812
#303792
#303762
#237472
#237312
#236702
#236572
#236282
#236182
#211592
#210952
C4×#4232