Finite Group Databasebeta

← #6470780

Subgroups — #6470780

Order 512.

2336 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3199C212yesmaximal, complemented
2#37297C211yesmaximal, characteristic, complemented
3#34275C211yesmaximal, characteristic, complemented
4#33113C212yesmaximal, complemented
5#32911C212yesmaximal, complemented
6#29765C212yesmaximal, complemented
7#29712C212yesmaximal, complemented
8#25228C214yesmaximal, complemented
9#25209C214yesmaximal, complemented
10#25097C214yesmaximal, complemented
11#25076C214yesmaximal, complemented
12#21936C211yesmaximal, characteristic, complemented
13#20914C212yesmaximal, complemented
14C2×#535C2212yes—
15C2×#535C2211yescharacteristic
16C2×#520C2211yescharacteristic
17C2×#518C2211yescharacteristic, complemented
18#3199—216no—
19C2×#482C2214yescomplemented
20C2×#482C2214yescomplemented
21C2×#469C2214yes—
22C2×#466C2214yes—
23C2×#459C2212yes—
24C2×#459C2212yes—
25C2×#447C2214yescomplemented
26C2×#447C2214yescomplemented
27C2×#418C2212yescomplemented
28C22×#102C2212yescomplemented
29C2×#408C2212yes—
30#2856C2212yes—
31#2733—21no—
32#2733—21no—
33#2679—21no—
34#2679—21no—
35#2669—24no—
36#2619C2212yes—
37#2500C2214yescomplemented
38#2433C2214yescomplemented
39#2363C2214yescomplemented
40#2361C2214yescomplemented
41#2358—24no—
42#2328—21no—
43#2326—21no—
44#2317C2211yescharacteristic, complemented
45C4×#127—21no—
46C4×#124—21no—
47#2297C2211yescharacteristic
48C2×#390C2211yescharacteristic, complemented
49C4×#117C2211yescharacteristic, complemented
50#2085C2214yescomplemented
51#2071C2214yescomplemented
52#2067C2214yescomplemented
53#2065C2214yescomplemented
54#2054C2214yescomplemented
55#2054C2214yescomplemented
56#2041C2214yescomplemented
57#2011C2214yes—
58#1982C2212yescomplemented
59#1982C2212yescomplemented
60#1948C2212yescomplemented
61#1947C2212yescomplemented
62#1926C2214yescomplemented
63#1925C2214yescomplemented
64#1879C2214yescomplemented
65#1879C2214yescomplemented
66#1875C2214yescomplemented
67#1875C2214yescomplemented
68#1874C2214yescomplemented
69#1874C2214yescomplemented
70#1851C2214yescomplemented
71#1850C2214yescomplemented
72#1804C2212yescomplemented
73#1804C2212yescomplemented
74#1798C2212yescomplemented
75#1797C2212yescomplemented
76#1736C2212yescomplemented
77#1735C2212yescomplemented
78C22×#41C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C22×D8C2314yes—
82C22×D8C2314yes—
83C22×M16C2312yes—
84C23×C8C2312yes—
85#535—24no—
86#535—28no—
87#535—28no—
88#535D418yescomplemented
89C2×#127—21no—
90C2×#127—22no—
91C2×#127C2312yes—
92C2×#127C2312yes—
93C2×#124—21no—
94C2×#124—22no—
95C2×#121—22no—
96C2×#121—22no—
97C2×#121C2314yes—
98C2×#121C2314yes—
99C2×#121C2314yes—
100C2×#120—22no—
101C2×#120—22no—
102C2×#120—22no—
103C2×#120—24no—
104C2×#120—24no—
105#520—42no—
106#520—42no—
107#518—42no—
108#518—42no—
109C2×C4×Q8—21no—
110C2×C4×D4—21no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—22no—
125C22×C22⋊C4—22no—
126C22×C42C2311yescharacteristic
127#482—216no—
128#482—216no—
129#469—216no—
130#466—216no—
131#459—44no—
132#459—44no—
133#447—216no—
134#447—216no—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141#418—44no—
142C2×#102—44no—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151#408—28no—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154#394—22no—
155#394—22no—
156#394—24no—
157#394—24no—
158#392—22no—
159#392—22no—
160#392—24no—
161#392—24no—
162#391D411yescharacteristic
163#391D411yescharacteristic
164#390—24no—
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#390D411yescharacteristic
170#390D411yescharacteristic
171#390C2314yes—
172#390C2314yes—
173#390C2314yes—
174#389C2314yes—
175#388—24no—
176#388—24no—
177#388—24no—
178#388—24no—
179#386—24no—
180#386—24no—
181#386D414yes—
182#385—22no—
183#385—24no—
184#384—21no—
185#384C2312yes—
186#382—21no—
187#382C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—24no—
192C4×C22⋊C4C2314yes—
193#376—21no—
194C43—21no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25—22no—
203C2×#41—41no—
204C2×#41—41no—
205C2×#41—41no—
206C2×#41—41no—
207C2×#41C2×D414yes—
208C2×#41#4114yes—
209C22×Q8C2×D411yescharacteristic
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—24no—
222C22×D4—41no—
223C22×D4—41no—
224C22×D4—41no—
225C22×D4—41no—
226C22×D4—42no—
227C22×D4—42no—
228C22×D4—42no—
229C22×D4—42no—
230C22×D4C2×D414yes—
231C22×D4C2×D412yes—
232C22×D4C2×D411yescharacteristic
233C22×D4C2414yes—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4C2×D412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C2×D8—28no—
243C2×D8—28no—
244C2×D8—44no—
245C2×D8—44no—
246C2×D8—44no—
247C2×D8—44no—
248C2×M16—24no—
249C2×M16—42no—
250C2×M16—42no—
251C22×C8—24no—
252C22×C8—42no—
253C22×C8—42no—
254C22×C8C2414yes—
255C22×C8C2414yes—
256#127—28no—
257#127—42no—
258#127—42no—
259#127—42no—
260#127—42no—
261#127—48no—
262#124—28no—
263#124—48no—
264#121—28no—
265#121—28no—
266#121—28no—
267#121—28no—
268#121—28no—
269#121—84no—
270#121—84no—
271#121C2×D4116yes—
272#120—216no—
273#120—48no—
274#120—48no—
275#120—88no—
276#120—88no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—42no—
281C4×D4—42no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—88no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—41no—
294#117—41no—
295#117—42no—
296#117C2×D414yes—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—41no—
305C2×C4⋊C4—41no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—48no—
329C2×C22⋊C4—84no—
330C2×C22⋊C4C2×D414yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D414yes—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—41no—
340C2×C42—41no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2412yes—
345#107—28no—
346#106—28no—
347C4⋊C8—42no—
348C4⋊C8—42no—
349C4⋊C8—42no—
350C4⋊C8—42no—
351#102—28no—
352#102—28no—
353#102—28no—
354#102—28no—
355#102—28no—
356#102—28no—
357#102—44no—
358#102—44no—
359C22⋊C8—28no—
360C22⋊C8—28no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95C2×D414yes—
367#95C2×D414yes—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—42no—
372C24—42no—
373C24—44no—
374C24—82no—
375C24—82no—
376C24—82no—
377C24—82no—
378C24—84no—
379C24—84no—
380C24C22×D412yes—
381C24C22×D411yescharacteristic
382#41—44no—
383#41—44no—
384#41—44no—
385#41—44no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8#12114yes—
389C2×D4—216no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—41no—
405C2×D4—42no—
406C2×D4—41no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4#12118yes—
438C2×D4#12114yes—
439C2×D4C22×D414yes—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4114yes—
474C22×C4C2511yescharacteristic
475D8—88no—
476D8—88no—
477M16—84no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—84no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—84no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4C22×D414yes—
500C4⋊C4H(5,𝔽2)12yes—
501C4⋊C4H(5,𝔽2)12yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—216no—
505C22⋊C4—28no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—41no—
527C42—41no—
528C42—44no—
529C42—44no—
530C42—44no—
531C42—82no—
532C23—24no—
533C23—22no—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—24no—
541C23—48no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—88no—
554C23—88no—
555C23—168no—
556C23—164no—
557C23—162no—
558C23—162no—
559C23—168no—
560C23—164no—
561C23—164no—
562C23C2×#12112yes—
563C23C2×#12114yes—
564C23C23×D411yescharacteristic
565Q8—44no—
566D4—24no—
567D4—24no—
568D4—44no—
569D4—48no—
570D4—48no—
571D4—42no—
572D4—42no—
573D4—44no—
574D4—44no—
575D4—44no—
576D4—44no—
577D4—88no—
578D4—816no—
579D4—164no—
580D4—164no—
581D4—164no—
582D4—164no—
583D4—168no—
584D4—168no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—84no—
619C2×C4—164no—
620C2×C4—164no—
621C2×C4—164no—
622C2×C4—162no—
623C2×C4C2×#12112yes—
624C2×C4C2×#12114yes—
625C2×C4#57614yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4#58511yescharacteristic
631C2×C4#58511yescharacteristic
632C8—44no—
633C8—44no—
634C22—28no—
635C22—24no—
636C22—22no—
637C22—24no—
638C22—42no—
639C22—44no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—41no—
644C22—48no—
645C22—44no—
646C22—84no—
647C22—82no—
648C22—82no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—168no—
657C22—1616no—
658C22C22×#12111yescharacteristic
659C22#348411yescharacteristic
660C22#348511yescharacteristic
661C22C2×#57612yes—
662C22#358512yes—
663C4—24no—
664C4—22no—
665C4—24no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—84no—
673C4—82no—
674C4#345911yescharacteristic
675C4#345911yescharacteristic
676C4#349611yescharacteristic
677C4#349611yescharacteristic
678C2—44no—
679C2—42no—
680C2—84no—
681C2—164no—
682C2#3498411yescharacteristic
683C2#6042812yes—
684C2#6166111yescharacteristic
685C2#6167311yescharacteristic
686C2#6175212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3199C2semidirect
2. #37297C2semidirect
3. #34275C2semidirect
4. #33113C2semidirect
5. #32911C2semidirect
6. #29765C2semidirect
7. #29712C2semidirect
8. #25228C2semidirect
9. #25209C2semidirect
10. #25097C2semidirect
11. #25076C2semidirect
12. #21936C2semidirect
13. #20914C2semidirect
17. C2×#518C22semidirect
19. C2×#482C22semidirect
20. C2×#482C22semidirect
25. C2×#447C22semidirect
26. C2×#447C22semidirect
27. C2×#418C22semidirect
28. C22×#102C22semidirect
37. #2500C22semidirect
38. #2433C22semidirect
39. #2363C22semidirect
40. #2361C22semidirect
44. #2317C22semidirect
48. C2×#390C22semidirect
49. C4×#117C22semidirect
50. #2085C22semidirect
51. #2071C22semidirect
52. #2067C22semidirect
53. #2065C22semidirect
54. #2054C22semidirect
55. #2054C22semidirect
56. #2041C22semidirect
58. #1982C22semidirect
59. #1982C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
64. #1879C22semidirect
65. #1879C22semidirect
66. #1875C22semidirect
67. #1875C22semidirect
68. #1874C22semidirect
69. #1874C22semidirect
70. #1851C22semidirect
71. #1850C22semidirect
72. #1804C22semidirect
73. #1804C22semidirect
74. #1798C22semidirect
75. #1797C22semidirect
76. #1736C22semidirect
77. #1735C22semidirect
88. #535D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31992
#372972
#342752
#331132
#329112
#297652
#297122
#252282
#252092
#250972
#250762
#219362
#209142