Finite Group Databasebeta

← #6500284

Subgroups — #6500284

Order 512.

3356 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic, complemented
2C22×#450C211yesmaximal, characteristic
3C2×#2435C211yesmaximal, characteristic
4#34212C211yesmaximal, characteristic, complemented
5#28123C214yesmaximal, complemented
6#25084C218yesmaximal, complemented
7#24479C214yesmaximal, complemented
8#24357C214yesmaximal, complemented
9#24217C214yesmaximal, complemented
10#21930C211yesmaximal, characteristic, complemented
11#21759C211yesmaximal, characteristic, complemented
12C2×#1786C211yesmaximal, characteristic
13C24×D4C2211yescharacteristic
14C23×SD16C2212yes—
15C22×#124C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18#3189—28no—
19C2×#484C2212yescomplemented
20C2×#484C2212yescomplemented
21C2×#467C2214yescomplemented
22C2×#464C2218yescomplemented
23C2×#451C2212yescomplemented
24C2×#451C2212yescomplemented
25C2×#450—28no—
26C2×#450C2214yes—
27C2×#450C2214yes—
28C22×#102C2212yes—
29C22×#102C2214yes—
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32#2435—22no—
33#2435—22no—
34#2435C2214yes—
35#2435C2214yes—
36C2×#394—22no—
37#2368—22no—
38#2360C2214yescomplemented
39#2345—22no—
40#2345C2212yescomplemented
41Q8×C22⋊C4C2212yescomplemented
42#2329—22no—
43#2315—21no—
44C4×C22⋊Q8C2214yescomplemented
45C2×#390C2212yes—
46C2×#380C2211yescharacteristic
47C2×#379—21no—
48C2×#379—21no—
49#2273—21no—
50C2×C4×C22⋊C4C2211yescharacteristic
51#2067C2218yescomplemented
52#2055C2218yescomplemented
53#2042C2218yescomplemented
54#2015C2218yescomplemented
55#1983C2218yescomplemented
56#1925C2218yescomplemented
57#1892C2214yescomplemented
58#1880C2218yescomplemented
59#1872C2214yescomplemented
60#1850C2214yescomplemented
61#1850C2214yescomplemented
62#1824C2212yescomplemented
63#1797C2214yescomplemented
64#1787C2212yescomplemented
65#1786C2214yes—
66#1786C2214yes—
67C26—21no—
68C23×Q8C2311yescharacteristic
69C23×Q8C2311yescharacteristic
70C23×D4—24no—
71C23×D4—21no—
72C23×D4—21no—
73C23×D4—21no—
74C23×D4—24no—
75C23×D4—48no—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C24×C4C2311yescharacteristic
80C22×Q16C2314yescomplemented
81C22×SD16—28no—
82C22×SD16—48no—
83C22×SD16C2318yes—
84C23×C8C2312yes—
85C23×C8C2312yes—
86C2×#124—24no—
87C2×#124—48no—
88C2×#124C2318yes—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8D418yes—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C4×Q8C2314yescomplemented
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—22no—
112C22×C22⋊C4—22no—
113C22×C22⋊C4—22no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42C2312yes—
117#484—28no—
118#484—28no—
119#467—216no—
120#464—416no—
121#451—28no—
122#451—28no—
123#450—216no—
124#450—216no—
125C2×#107C2314yescomplemented
126C2×C4⋊C8C2318yescomplemented
127C2×#103C2314yescomplemented
128C2×#103C2314yescomplemented
129C2×#103C2318yescomplemented
130C2×#102—28no—
131C2×#102—48no—
132C2×#102C2318yes—
133C2×#102C2318yes—
134C2×#102C2318yes—
135C2×C22⋊C8—24no—
136C2×C22⋊C8—24no—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141#394—28no—
142#394—28no—
143#391—24no—
144#391C2314yescomplemented
145#390—24no—
146#390C2318yes—
147#389—24no—
148#389C2314yescomplemented
149#387—24no—
150#387—24no—
151#387—24no—
152#385—24no—
153#385—24no—
154#380—21no—
155#380—21no—
156#380—22no—
157#380—22no—
158#380C2312yes—
159#380C2312yes—
160#379—24no—
161#379—24no—
162#379—24no—
163#379—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4C2314yescomplemented
166C4×C22⋊C4—24no—
167C4×C22⋊C4—22no—
168C4×C22⋊C4—22no—
169C4×C22⋊C4—24no—
170C4×C22⋊C4D412yes—
171C4×C22⋊C4D412yes—
172C4×C22⋊C4C2312yes—
173C4×C22⋊C4C2312yes—
174C2×#95—21no—
175C2×#95D412yes—
176#340C2318yescomplemented
177#336C2318yescomplemented
178C25—22no—
179C25—22no—
180C25—22no—
181C25—48no—
182C25—44no—
183C25—44no—
184C25—44no—
185C25—84no—
186C25C2×D411yescharacteristic
187C22×Q8—21no—
188C22×Q8—21no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8C2×D414yes—
195C22×Q8C2412yes—
196C22×Q8C2412yes—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—28no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—432no—
213C22×D4—432no—
214C22×D4—48no—
215C22×D4—48no—
216C22×D4—48no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4C2414yes—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—44no—
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D414yes—
233C23×C4#4114yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×Q16—28no—
238C2×Q16—48no—
239C2×SD16—216no—
240C2×SD16—216no—
241C2×SD16—48no—
242C2×SD16—432no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—44no—
246C22×C8—44no—
247C22×C8C2414yes—
248C22×C8C2414yes—
249#124—216no—
250#124—48no—
251#124—416no—
252C22⋊Q8—28no—
253C22⋊Q8—28no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C22⋊Q8C2×D418yes—
263C22⋊Q8C2×D418yes—
264C4×Q8—48no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—84no—
302C2×C22⋊C4—84no—
303C2×C22⋊C4C2×D412yes—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—44no—
310C2×C42C2×D414yes—
311C2×C42C2414yes—
312#107—28no—
313C4⋊C8—48no—
314#103—28no—
315#103—28no—
316#103—48no—
317#102—216no—
318#102—216no—
319#102—48no—
320C22⋊C8—24no—
321C22⋊C8—24no—
322C22⋊C8—24no—
323C22⋊C8—24no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95C2×D414yes—
331#95C2×D414yes—
332C24—28no—
333C24—24no—
334C24—42no—
335C24—42no—
336C24—48no—
337C24—48no—
338C24—44no—
339C24—44no—
340C24—44no—
341C24—432no—
342C24—44no—
343C24—44no—
344C24—44no—
345C24—44no—
346C24—44no—
347C24—44no—
348C24—44no—
349C24—44no—
350C24—44no—
351C24—84no—
352C24—84no—
353C24—832no—
354C24—832no—
355C24—816no—
356C24—816no—
357C24#12114yes—
358C24#13514yes—
359C24#13714yes—
360C24C22×D411yescharacteristic
361C24C22×D411yescharacteristic
362C24C22×D411yescharacteristic
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—44no—
370C2×Q8—48no—
371C2×Q8—42no—
372C2×Q8—82no—
373C2×Q8—84no—
374C2×Q8—84no—
375C2×Q8—84no—
376C2×Q8C22×D414yes—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4—28no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—432no—
395C2×D4—432no—
396C2×D4—432no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4H(5,𝔽2)18yes—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4#12114yes—
432C22×C4#12114yes—
433C22×C4#12114yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2511yescharacteristic
446Q16—88no—
447SD16—432no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—28no—
451C2×C8—28no—
452C2×C8—48no—
453C2×C8—84no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—28no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—84no—
465C4⋊C4—82no—
466C4⋊C4—84no—
467C4⋊C4C22×D414yes—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C42—28no—
483C42—44no—
484C42—84no—
485C42—84no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—28no—
493C23—44no—
494C23—44no—
495C23—48no—
496C23—48no—
497C23—416no—
498C23—416no—
499C23—416no—
500C23—416no—
501C23—416no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—816no—
509C23—816no—
510C23—816no—
511C23—816no—
512C23—88no—
513C23—88no—
514C23—88no—
515C23—88no—
516C23—88no—
517C23—8128no—
518C23—88no—
519C23—88no—
520C23—88no—
521C23#464116yescomplemented
522C23C2×#12112yes—
523C23C2×#12112yes—
524C23C2×#12112yes—
525C23C2×#13512yes—
526C23C2×#13512yes—
527C23C2×#13512yes—
528C23C2×#13712yes—
529C23C2×#13712yes—
530C23C2×#13712yes—
531C23C23×D411yescharacteristic
532Q8—28no—
533Q8—44no—
534Q8—44no—
535Q8—44no—
536Q8—84no—
537Q8—84no—
538D4—28no—
539D4—28no—
540D4—416no—
541D4—416no—
542D4—416no—
543D4—44no—
544D4—44no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—164no—
581C2×C4C2×#12112yes—
582C2×C4C2×#12112yes—
583C2×C4C2×#12112yes—
584C2×C4C23×D411yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4C2×H(5,𝔽2)14yes—
587C8—44no—
588C8—44no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—416no—
602C22—48no—
603C22—48no—
604C22—48no—
605C22—832no—
606C22—832no—
607C22—832no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22C2×#46414yes—
613C22C2×#46414yes—
614C22C2×#46414yes—
615C22C22×#12111yescharacteristic
616C22#348412yes—
617C22C22×#13511yescharacteristic
618C22C22×#13711yescharacteristic
619C22#358512yes—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—82no—
628C4—82no—
629C4—84no—
630C4#348412yes—
631C4#349012yes—
632C2—24no—
633C2—24no—
634C2—24no—
635C2—88no—
636C2#3534811yescharacteristic
637C2C22×#46411yescharacteristic
638C2#6042611yescharacteristic
639C2#6156112yes—
640C2#6157012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24×D4C22
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
1. C2×#3189C2semidirect
4. #34212C2semidirect
5. #28123C2semidirect
6. #25084C2semidirect
7. #24479C2semidirect
8. #24357C2semidirect
9. #24217C2semidirect
10. #21930C2semidirect
11. #21759C2semidirect
19. C2×#484C22semidirect
20. C2×#484C22semidirect
21. C2×#467C22semidirect
22. C2×#464C22semidirect
23. C2×#451C22semidirect
24. C2×#451C22semidirect
38. #2360C22semidirect
40. #2345C22semidirect
41. Q8×C22⋊C4C22semidirect
44. C4×C22⋊Q8C22semidirect
51. #2067C22semidirect
52. #2055C22semidirect
53. #2042C22semidirect
54. #2015C22semidirect
55. #1983C22semidirect
56. #1925C22semidirect
57. #1892C22semidirect
58. #1880C22semidirect
59. #1872C22semidirect
60. #1850C22semidirect
61. #1850C22semidirect
62. #1824C22semidirect
63. #1797C22semidirect
64. #1787C22semidirect
80. C22×Q16C23semidirect
101. C2×C4×Q8C23semidirect
125. C2×#107C23semidirect
126. C2×C4⋊C8C23semidirect
127. C2×#103C23semidirect
128. C2×#103C23semidirect
129. C2×#103C23semidirect
144. #391C23semidirect
148. #389C23semidirect
165. C4×C4⋊C4C23semidirect
176. #340C23semidirect
177. #336C23semidirect
521. C23#464semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C22×#4502
C2×#24352
#342122
#281232
#250842
#244792
#243572
#242172
#219302
#217592
C2×#17862