Finite Group Databasebeta

← #6470320

Subgroups — #6470320

Order 512.

2946 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3009C212yesmaximal
2#37386C211yesmaximal, characteristic, complemented
3#34256C211yesmaximal, characteristic
4#32914C212yesmaximal, complemented
5#32905C212yesmaximal, complemented
6#29726C212yesmaximal, complemented
7#29717C212yesmaximal, complemented
8#24538C214yesmaximal, complemented
9#24536C214yesmaximal, complemented
10#24376C214yesmaximal, complemented
11#24358C214yesmaximal, complemented
12#21937C211yesmaximal, characteristic, complemented
13#21771C212yesmaximal, complemented
14C22×H(5,𝔽2)C2211yescharacteristic
15C22×#137C2212yes—
16C23×SD16C2212yes—
17C2×#536C2212yes—
18C2×#518C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#461C2214yescomplemented
21C2×#461C2214yescomplemented
22#3009—216no—
23C2×#452C2214yes—
24C2×#452C2214yes—
25C2×#450C2214yes—
26C2×#450C2214yes—
27C2×#418C2212yes—
28C2×#416C2212yes—
29C2×#408C2212yes—
30#2857C2212yescomplemented
31#2734—22no—
32#2724—22no—
33#2717—22no—
34#2717—22no—
35#2621C2212yescomplemented
36#2499C2214yescomplemented
37#2434C2214yescomplemented
38#2339—22no—
39#2339C2214yes—
40D4×C22⋊C4—22no—
41D4×C22⋊C4C2214yes—
42#2311C2211yescharacteristic
43#2311C2211yescharacteristic
44#2308—22no—
45#2297C2211yescharacteristic
46#2297C2211yescharacteristic
47C2×#386—22no—
48#2074C2212yescomplemented
49#2073C2212yescomplemented
50#2072C2212yescomplemented
51#2071C2212yescomplemented
52#2056C2214yescomplemented
53#2056C2214yescomplemented
54#2055C2214yescomplemented
55#2055C2214yescomplemented
56#2016C2212yescomplemented
57#2016C2212yescomplemented
58#2013C2212yescomplemented
59#2013C2212yescomplemented
60#1894C2214yescomplemented
61#1892C2214yescomplemented
62#1877C2214yescomplemented
63#1877C2214yescomplemented
64#1877C2214yescomplemented
65#1877C2214yescomplemented
66#1873C2214yescomplemented
67#1872C2214yescomplemented
68#1851C2214yescomplemented
69#1850C2214yescomplemented
70#1825C2214yescomplemented
71#1804C2214yescomplemented
72#1802C2214yescomplemented
73#1798C2212yescomplemented
74#1797C2212yescomplemented
75#1787C2214yescomplemented
76C2×H(5,𝔽2)—24no—
77C2×H(5,𝔽2)—44no—
78C2×H(5,𝔽2)—44no—
79C2×H(5,𝔽2)D418yes—
80C22×#41—21no—
81C22×#41C2312yes—
82C22×#41C2311yescharacteristic
83C22×#41C2311yescharacteristic
84C23×Q8C2312yes—
85C23×D4—22no—
86C23×D4—21no—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4—21no—
90C2×#137—28no—
91C2×#137—44no—
92C2×#137—44no—
93C22×Q16C2314yes—
94C22×SD16—28no—
95C22×SD16—44no—
96C22×SD16—44no—
97C22×SD16C2314yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100C22×M16C2312yes—
101C23×C8C2312yes—
102#536—28no—
103#536—28no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#124—22no—
108C2×#123—22no—
109C2×#123—22no—
110C2×C22⋊Q8C2314yes—
111C2×C22⋊Q8C2314yes—
112#518—44no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—44no—
116C2×C4×D4C2314yes—
117C2×C4×D4C2314yes—
118C2×#117C2312yes—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4—24no—
125C22×C42C2311yescharacteristic
126#461—48no—
127#461—48no—
128#452—216no—
129#452—216no—
130#450—216no—
131#450—216no—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134#418—28no—
135#416—28no—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144#408—28no—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147#398—24no—
148#397—24no—
149#397—24no—
150#397—24no—
151#395—24no—
152#395—24no—
153#391—21no—
154#391D412yes—
155#391C2314yes—
156#391C2314yes—
157#391C2312yes—
158#391C2312yes—
159#390—21no—
160#390D412yes—
161#390C2312yes—
162#390C2312yes—
163#389—22no—
164#389—22no—
165#387—24no—
166#387—24no—
167#386—22no—
168#386—21no—
169#386—21no—
170#386—22no—
171#386—24no—
172#386—24no—
173#386—24no—
174#386—24no—
175#386D412yes—
176#386D412yes—
177#385—24no—
178#385—24no—
179#384C2312yes—
180#382C2312yes—
181#381—24no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380C2314yes—
186#379—24no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4C2314yes—
189C2×#95—24no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2312yes—
195#336C2312yes—
196#336C2312yes—
197#336C2312yes—
198C25—21no—
199C25—22no—
200H(5,𝔽2)—432no—
201H(5,𝔽2)—432no—
202C2×#41—28no—
203C2×#41—24no—
204C2×#41—22no—
205C2×#41—24no—
206C2×#41—24no—
207C2×#41—42no—
208C2×#41—42no—
209C2×#41—42no—
210C2×#41—42no—
211C2×#41—42no—
212C2×#41—42no—
213C2×#41—44no—
214C2×#41—44no—
215C2×#41C2×D418yes—
216C2×#41C2×D414yes—
217C2×#41C2×D414yes—
218C2×#41#4114yes—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—42no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2×D411yescharacteristic
227C22×Q8C2414yes—
228C22×Q8C2411yescharacteristic
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—24no—
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—28no—
236C22×D4—216no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—24no—
241C22×D4—42no—
242C22×D4—42no—
243C22×D4—44no—
244C22×D4—44no—
245C22×D4—48no—
246C22×D4—48no—
247C22×D4C2×D414yes—
248C22×D4C2×D418yes—
249C22×D4C2×D411yescharacteristic
250C22×D4C2×D412yes—
251C22×D4C2411yescharacteristic
252C22×D4C2412yes—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4—44no—
260C23×C4—84no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264C23×C4C2412yes—
265#137—432no—
266C2×Q16—28no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×SD16—28no—
270C2×SD16—28no—
271C2×SD16—28no—
272C2×SD16—28no—
273C2×SD16—28no—
274C2×SD16—28no—
275C2×SD16—28no—
276C2×SD16—44no—
277C2×SD16—44no—
278C2×SD16—432no—
279C2×M16—24no—
280C2×M16—42no—
281C2×M16—42no—
282C22×C8—24no—
283C22×C8—42no—
284C22×C8—42no—
285C22×C8C2414yes—
286C22×C8C2414yes—
287C4⋊Q8—44no—
288C4⋊Q8—44no—
289C4⋊Q8—48no—
290#124—48no—
291#123—48no—
292#123—48no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C4×D4—28no—
298C4×D4—28no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—48no—
302C4×D4—48no—
303#117—24no—
304#117—24no—
305#117—42no—
306#117—42no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—48no—
332C2×C22⋊C4—88no—
333C2×C22⋊C4C2×D414yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D414yes—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—42no—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342C4⋊C8—44no—
343C4⋊C8—44no—
344#103—28no—
345#103—44no—
346#103—44no—
347#103—44no—
348#103—44no—
349#102—28no—
350#102—28no—
351#102—28no—
352C22⋊C8—28no—
353C22⋊C8—28no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—24no—
366C24—24no—
367C24—22no—
368C24—24no—
369C24—21no—
370C24—28no—
371C24—44no—
372C24—44no—
373C24—82no—
374C24—82no—
375C24—84no—
376C24—84no—
377C24C22×D412yes—
378C24C22×D411yescharacteristic
379#41—216no—
380#41—48no—
381#41—48no—
382#41—48no—
383#41—48no—
384#41—48no—
385#41—48no—
386#41—48no—
387#41—432no—
388C2×Q8—21no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8—84no—
405C2×Q8—84no—
406C2×Q8#12114yes—
407C2×Q8C22×D412yes—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—216no—
411C2×D4—216no—
412C2×D4—216no—
413C2×D4—28no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—432no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—464no—
438C2×D4#12114yes—
439C2×D4#12118yes—
440C2×D4C22×D412yes—
441C2×D4C22×D414yes—
442C2×D4H(5,𝔽2)14yes—
443C2×D4H(5,𝔽2)14yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—816no—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4112yes—
486C22×C4C2511yescharacteristic
487Q16—416no—
488SD16—416no—
489SD16—416no—
490SD16—416no—
491M16—48no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—48no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C22⋊C4—28no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—24no—
526C42—24no—
527C42—42no—
528C42—42no—
529C42—44no—
530C23—216no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—21no—
535C23—24no—
536C23—24no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—416no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—88no—
551C23—816no—
552C23—84no—
553C23—84no—
554C23—88no—
555C23—164no—
556C23—164no—
557C23—168no—
558C23C2×#12112yes—
559C23C2×#12114yes—
560C23C23×D411yescharacteristic
561Q8—24no—
562Q8—44no—
563Q8—42no—
564Q8—44no—
565Q8—44no—
566Q8—44no—
567Q8—44no—
568Q8—88no—
569D4—216no—
570D4—24no—
571D4—24no—
572D4—24no—
573D4—24no—
574D4—44no—
575D4—48no—
576D4—416no—
577D4—416no—
578D4—48no—
579D4—42no—
580D4#53418yes—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—416no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—816no—
617C2×C4—82no—
618C2×C4—88no—
619C2×C4—84no—
620C2×C4—168no—
621C2×C4C2×#12114yes—
622C2×C4C2×#12112yes—
623C2×C4#57714yes—
624C2×C4C23×D411yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4C2×H(5,𝔽2)12yes—
628C8—44no—
629C8—44no—
630C22—24no—
631C22—28no—
632C22—24no—
633C22—22no—
634C22—28no—
635C22—48no—
636C22—41no—
637C22—44no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22—44no—
642C22—48no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—816no—
648C22—81no—
649C22—81no—
650C22—82no—
651C22—1616no—
652C22C22×#12111yescharacteristic
653C22#348411yescharacteristic
654C22#348411yescharacteristic
655C22C2×#57712yes—
656C22#358612yes—
657C4—22no—
658C4—24no—
659C4—24no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—84no—
664C4—84no—
665C4—82no—
666C4—82no—
667C4C2×#53412yes—
668C4#349012yes—
669C2—42no—
670C2—44no—
671C2—48no—
672C2—82no—
673C2#3498111yescharacteristic
674C2#6042912yes—
675C2#6163111yescharacteristic
676C2#6164911yescharacteristic
677C2#6168012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×H(5,𝔽2)C22
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
2. #37386C2semidirect
4. #32914C2semidirect
5. #32905C2semidirect
6. #29726C2semidirect
7. #29717C2semidirect
8. #24538C2semidirect
9. #24536C2semidirect
10. #24376C2semidirect
11. #24358C2semidirect
12. #21937C2semidirect
13. #21771C2semidirect
20. C2×#461C22semidirect
21. C2×#461C22semidirect
30. #2857C22semidirect
35. #2621C22semidirect
36. #2499C22semidirect
37. #2434C22semidirect
48. #2074C22semidirect
49. #2073C22semidirect
50. #2072C22semidirect
51. #2071C22semidirect
52. #2056C22semidirect
53. #2056C22semidirect
54. #2055C22semidirect
55. #2055C22semidirect
56. #2016C22semidirect
57. #2016C22semidirect
58. #2013C22semidirect
59. #2013C22semidirect
60. #1894C22semidirect
61. #1892C22semidirect
62. #1877C22semidirect
63. #1877C22semidirect
64. #1877C22semidirect
65. #1877C22semidirect
66. #1873C22semidirect
67. #1872C22semidirect
68. #1851C22semidirect
69. #1850C22semidirect
70. #1825C22semidirect
71. #1804C22semidirect
72. #1802C22semidirect
73. #1798C22semidirect
74. #1797C22semidirect
75. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30092
#373862
#342562
#329142
#329052
#297262
#297172
#245382
#245362
#243762
#243582
#219372
#217712