Finite Group Databasebeta

← #6640635

Subgroups — #6640635

Order 512.

1710 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2C2×#2499C211yesmaximal, characteristic, complemented
3#32980C212yesmaximal, complemented
4#32281C214yesmaximal, complemented
5#32053C214yesmaximal, complemented
6#31937C214yesmaximal, complemented
7#31839C214yesmaximal, complemented
8#29595C212yesmaximal, complemented
9#29105C211yesmaximal, characteristic, complemented
10#28305C211yesmaximal, characteristic, complemented
11#28118C212yesmaximal, complemented
12#27960C211yesmaximal, characteristic, complemented
13#27909C211yesmaximal, characteristic, complemented
14#27651C212yesmaximal, complemented
15#21902C211yesmaximal, characteristic, complemented
16C2×#538—21no—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#481C2212yes—
20C2×#480C2212yes—
21C2×#480C2212yes—
22C2×#448C2212yes—
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2847C2212yes—
26#2639—21no—
27#2617—21no—
28#2559—22no—
29#2557—22no—
30#2499—22no—
31#2499—22no—
32#2499C2214yes—
33#2499C2214yes—
34#2496C2212yes—
35#2491C2212yes—
36#2428C2212yes—
37#2422C2212yes—
38C2×#397—21no—
39C2×#397—21no—
40#2325—21no—
41C4×#121C2212yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#387—21no—
45C2×#387—21no—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2099C2214yescomplemented
50#2096C2212yescomplemented
51#2093C2214yescomplemented
52#2085C2214yescomplemented
53#2073C2214yescomplemented
54#2072C2212yescomplemented
55#2068C2214yescomplemented
56#2055C2214yescomplemented
57#2036C2214yes—
58#2036C2214yes—
59#2024C2212yescomplemented
60#2014C2212yescomplemented
61#2000C2214yes—
62#2000C2214yes—
63#1990C2212yescomplemented
64#1968C2214yescomplemented
65#1957C2212yescomplemented
66#1949C2214yescomplemented
67#1933C2214yescomplemented
68#1928C2214yescomplemented
69#1923C2214yes—
70#1923C2214yes—
71#1877C2214yescomplemented
72#1875C2214yes—
73#1875C2214yes—
74#1854C2214yes—
75#1854C2214yes—
76#1845C2212yescomplemented
77#1834C2212yescomplemented
78#1824C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1804C2214yescomplemented
82#1797C2212yes—
83#1787C2212yes—
84#1779C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×SD16C2314yes—
90#538—48no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#126—22no—
94C2×#124—22no—
95C2×#123—22no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—44no—
99C2×C22⋊Q8C2314yes—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121—22no—
103C2×#121C2312yes—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#120—22no—
107C2×C4×D4—22no—
108C2×C4×D4C2312yes—
109C2×#117—22no—
110C2×#117—44no—
111C2×#117C2314yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#481—44no—
126#480—44no—
127#480—44no—
128#448—44no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#397—22no—
147#397—24no—
148#397—24no—
149#397—24no—
150#397—24no—
151#397—22no—
152#394—22no—
153#394—22no—
154#394—22no—
155#393—22no—
156#392—22no—
157#391—21no—
158#391—22no—
159#391—21no—
160#391—21no—
161#391C2314yes—
162#391C2312yes—
163#391C2312yes—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2312yes—
168#389—21no—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—24no—
175#387—24no—
176#387—24no—
177#387—24no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#382—22no—
191#382C2314yes—
192#380—21no—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C22⋊C4—22no—
198C4×C22⋊C4D412yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4C2312yes—
201C4×C22⋊C4C2312yes—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C25C2×D411yescharacteristic
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8C2412yes—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2412yes—
232C23×C4—24no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C2×SD16—44no—
249C2×SD16—44no—
250C2×SD16—44no—
251C22×C8C2414yes—
252C22×C8C2414yes—
253C4⋊Q8—42no—
254#127—42no—
255#126—48no—
256#124—48no—
257#123—48no—
258C22⋊Q8—48no—
259C22⋊Q8—48no—
260C22⋊Q8—44no—
261#121—24no—
262#121—24no—
263#121—48no—
264#121—44no—
265#121—44no—
266#121—44no—
267#121—44no—
268#120—48no—
269C4×D4—44no—
270C4×D4—48no—
271#117—48no—
272#117—48no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—44no—
326C2×C42C2×D412yes—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—44no—
333#107—44no—
334#106—44no—
335C4⋊C8—44no—
336#103—44no—
337#103—44no—
338#103—44no—
339#102—44no—
340#102—44no—
341#102—44no—
342C22⋊C8—24no—
343C22⋊C8—24no—
344C22⋊C8—44no—
345C22⋊C8—44no—
346C4×C8—42no—
347C4×C8—42no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—41no—
371C24—41no—
372C24—41no—
373C24—41no—
374C24C22×D412yes—
375C24C22×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—81no—
385C2×Q8—81no—
386C2×Q8—88no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—81no—
399C2×D4—81no—
400C2×D4—81no—
401C2×D4—81no—
402C2×D4—82no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4—82no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—416no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4#13514yes—
451C22×C4AGL(1,ℤ/8)14yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4H(5,𝔽2)14yes—
461C22×C4C2511yescharacteristic
462SD16—88no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—41no—
512C42—41no—
513C42—42no—
514C42—48no—
515C42—44no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23#53512yes—
542C23C2×#13512yes—
543C23C2×AGL(1,ℤ/8)12yes—
544C23C23×D411yescharacteristic
545Q8—44no—
546Q8—44no—
547Q8—84no—
548Q8—162no—
549D4—44no—
550D4—44no—
551D4—88no—
552D4—162no—
553D4—162no—
554D4—164no—
555D4—164no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—164no—
609C2×C4#53512yes—
610C2×C4C2×#13512yes—
611C2×C4C2×#13512yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4C2×H(5,𝔽2)12yes—
615C2×C4#58512yes—
616C2×C4#58512yes—
617C8—44no—
618C8—44no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—88no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22#320714yes—
641C22C2×#53511yescharacteristic
642C22#348511yescharacteristic
643C22#348511yescharacteristic
644C22C22×#13511yescharacteristic
645C22C22×AGL(1,ℤ/8)11yescharacteristic
646C22#358611yescharacteristic
647C22#358611yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4#353314yes—
664C2—24no—
665C2—82no—
666C2—162no—
667C2#3695611yescharacteristic
668C2C2×#320711yescharacteristic
669C2#6080711yescharacteristic
670C2#6169611yescharacteristic
671C2#6170111yescharacteristic
672C2#6170411yescharacteristic
673C2#6172211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35005C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2499C2semidirect
3. #32980C2semidirect
4. #32281C2semidirect
5. #32053C2semidirect
6. #31937C2semidirect
7. #31839C2semidirect
8. #29595C2semidirect
9. #29105C2semidirect
10. #28305C2semidirect
11. #28118C2semidirect
12. #27960C2semidirect
13. #27909C2semidirect
14. #27651C2semidirect
15. #21902C2semidirect
49. #2099C22semidirect
50. #2096C22semidirect
51. #2093C22semidirect
52. #2085C22semidirect
53. #2073C22semidirect
54. #2072C22semidirect
55. #2068C22semidirect
56. #2055C22semidirect
59. #2024C22semidirect
60. #2014C22semidirect
63. #1990C22semidirect
64. #1968C22semidirect
65. #1957C22semidirect
66. #1949C22semidirect
67. #1933C22semidirect
68. #1928C22semidirect
71. #1877C22semidirect
76. #1845C22semidirect
77. #1834C22semidirect
81. #1804C22semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#24992
#329802
#322812
#320532
#319372
#318392
#295952
#291052
#283052
#281182
#279602
#279092
#276512
#219022