Finite Group Databasebeta

← #6498231

Subgroups — #6498231

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33218C212yesmaximal, complemented
4#33058C212yesmaximal
5#30312C211yesmaximal, characteristic, complemented
6#30283C211yesmaximal, characteristic, complemented
7#30235C211yesmaximal, characteristic, complemented
8#30212C211yesmaximal, characteristic, complemented
9#29574C212yesmaximal, complemented
10#29572C212yesmaximal, complemented
11#29516C214yesmaximal, complemented
12#29167C214yesmaximal, complemented
13#27863C212yesmaximal, complemented
14#27862C212yesmaximal, complemented
15#22251C212yesmaximal, complemented
16#22226C212yesmaximal, complemented
17#20364C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#501C2212yes—
20C2×#490C2212yes—
21C2×C4×M16C2212yes—
22#2864—21no—
23#2864—21no—
24#2861—21no—
25#2860—21no—
26#2860C2212yes—
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2854—21no—
36#2854C2212yes—
37#2851—21no—
38#2851—21no—
39#2851C2212yes—
40#2851C2212yes—
41#2850—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#125—21no—
45C4×#125—21no—
46C4×#125C2212yes—
47C4×#125C2212yes—
48C4×#124—21no—
49C4×#121—21no—
50C4×#121C2212yes—
51#2296—21no—
52#2296—21no—
53#2293—21no—
54#2293C2212yes—
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57C2×C43C2211yescharacteristic
58#2101C2214yes—
59#2100C2212yes—
60#2099C2212yes—
61#2094C2212yes—
62#2093C2212yes—
63#2092C2214yes—
64#2084C2212yes—
65#2083C2212yes—
66#2071C2212yes—
67#2071C2212yes—
68#2059C2214yes—
69#2025C2214yes—
70#1993C2212yes—
71#1992C2212yes—
72#1992C2212yes—
73#1961C2212yes—
74#1942C2214yes—
75#1941C2214yes—
76#1934C2214yes—
77#1934C2214yes—
78#1926C2214yes—
79#1925C2214yes—
80#1921C2214yes—
81#1920C2214yes—
82#1920C2214yes—
83#1918C2214yes—
84#1895C2214yes—
85#1895C2214yes—
86#1864C2214yes—
87#1862C2214yes—
88#1847C2212yes—
89#1836C2212yes—
90#1809C2214yes—
91#1809C2214yes—
92#1745C2211yescharacteristic
93#1745C2211yescharacteristic
94#1745C2212yes—
95#1744C2212yes—
96#1744C2212yes—
97#1728C2212yes—
98#1728C2212yes—
99C22×M16C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127C2311yescharacteristic
102#529—48no—
103C2×#126—22no—
104C2×#125—21no—
105C2×#125—21no—
106C2×#125—21no—
107C2×#125C2312yes—
108C2×#125C2312yes—
109C2×#124—21no—
110C2×#124—21no—
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C2×#117—21no—
122C2×#117—21no—
123C2×#117—22no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#501—28no—
132#490—28no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C4×M16—28no—
142C2×#97C2312yes—
143C2×#97C2312yes—
144C2×C4×C8C2314yes—
145#391—21no—
146#391—21no—
147#390—21no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
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#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#386—22no—
198#386—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#382—21no—
210#382—21no—
211#382—21no—
212#382—21no—
213#382—21no—
214#382C2312yes—
215#382C2312yes—
216#380—21no—
217#380—21no—
218#380—21no—
219#380C2312yes—
220#380C2312yes—
221#380C2312yes—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C43D411yescharacteristic
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43D411yescharacteristic
238C43C2311yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241C43C2311yescharacteristic
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2312yes—
251#336C2312yes—
252#336C2312yes—
253#336C2312yes—
254#336C2314yes—
255#336C2314yes—
256C22×Q8—21no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4C2412yes—
261C23×C4—22no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×M16—28no—
270C2×M16—28no—
271C2×M16—44no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—24no—
275#127—24no—
276#126—48no—
277#125—24no—
278#125—24no—
279#125—44no—
280#125—44no—
281#125—44no—
282#125—44no—
283#124—44no—
284#124—44no—
285#123—48no—
286C22⋊Q8—44no—
287#121—28no—
288#121—44no—
289#121—44no—
290C4×Q8—44no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—48no—
296#117—44no—
297#117—44no—
298#117—48no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
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—21no—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
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×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382#107—28no—
383#107—44no—
384#106—28no—
385#106—44no—
386#102—28no—
387#102—28no—
388#102—44no—
389#102—44no—
390#97—24no—
391#97#4118yes—
392C4×C8—28no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411C24—22no—
412C24C22×D411yescharacteristic
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—42no—
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—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
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×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C42−1+414yes—
473C22×C4C2511yescharacteristic
474M16—416no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8C2×#4118yes—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—44no—
556C42—44no—
557C42—44no—
558C42AGL(1,ℤ/8)14yes—
559C42C2×#4114yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23#53612yes—
574C23C23×D411yescharacteristic
575Q8—84no—
576D4—44no—
577D4—44no—
578D4—84no—
579D4—84no—
580D4—88no—
581C2×C4—28no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
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—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—88no—
640C2×C4#53212yes—
641C2×C4#53212yes—
642C2×C4#53612yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4#57512yes—
647C2×C4#57512yes—
648C2×C4#57512yes—
649C2×C4C22×#4112yes—
650C2×C4C2×2−1+412yes—
651C2×C4#58512yes—
652C8—28no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22C2×#53211yescharacteristic
669C22C2×#53611yescharacteristic
670C22#344911yescharacteristic
671C22C22×AGL(1,ℤ/8)11yescharacteristic
672C22C2×#57511yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57511yescharacteristic
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4#349514yes—
696C2—42no—
697C2—162no—
698C2#3532611yescharacteristic
699C2#6070011yescharacteristic
700C2#6070211yescharacteristic
701C2#6086111yescharacteristic
702C2#6086411yescharacteristic
703C2#6095111yescharacteristic
704C2#6095311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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. #33218C2semidirect
5. #30312C2semidirect
6. #30283C2semidirect
7. #30235C2semidirect
8. #30212C2semidirect
9. #29574C2semidirect
10. #29572C2semidirect
11. #29516C2semidirect
12. #29167C2semidirect
13. #27863C2semidirect
14. #27862C2semidirect
15. #22251C2semidirect
16. #22226C2semidirect
17. #20364C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#366312
#332182
#330582
#303122
#302832
#302352
#302122
#295742
#295722
#295162
#291672
#278632
#278622
#222512
#222262
#203642