Finite Group Databasebeta

← #6500752

Subgroups — #6500752

Order 512.

2718 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2407C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#27583C212yesmaximal, complemented
4#27582C212yesmaximal, complemented
5#24883C218yesmaximal, complemented
6#24881C218yesmaximal, complemented
7#24006C212yesmaximal, complemented
8#24005C212yesmaximal, complemented
9#21942C212yesmaximal, complemented
10#21742C211yesmaximal, characteristic, complemented
11#20601C212yesmaximal, complemented
12C23×#41C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C2×#518—21no—
16C22×C4×D4C2211yescharacteristic
17C2×#484C2214yescomplemented
18C2×#481C2214yescomplemented
19C2×#417C2214yescomplemented
20C2×#417C2214yescomplemented
21C2×#407C2212yes—
22#2407—24no—
23#2407C2218yes—
24C2×#394—21no—
25C2×#392—21no—
26#2349—24no—
27D4×C4⋊C4C2214yescomplemented
28D4×C4⋊C4C2214yescomplemented
29#2316—21no—
30#2315—21no—
31#2315—21no—
32C2×#391C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#386—21no—
35C2×#385—21no—
36#2285C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40#2096C2214yescomplemented
41#2093C2214yescomplemented
42#2072C2214yescomplemented
43#2071C2214yescomplemented
44#2036C2218yescomplemented
45#2034C2218yescomplemented
46#1927C2218yescomplemented
47#1925C2218yescomplemented
48#1874C2214yescomplemented
49#1872C2214yescomplemented
50#1870C2218yescomplemented
51#1853C2214yes—
52#1853C2214yescomplemented
53#1852C2214yes—
54#1852C2214yescomplemented
55#1817C2212yes—
56#1809C2218yescomplemented
57#1788C2218yescomplemented
58#1787C2214yes—
59#1786C2214yes—
60#1779C2212yes—
61C22×#41—22no—
62C22×#41—28no—
63C22×#41—44no—
64C22×#41—44no—
65C22×#41C2314yes—
66C23×Q8C2311yescharacteristic
67C23×D4—21no—
68C23×D4—21no—
69C23×D4D411yescharacteristic
70C23×D4C2311yescharacteristic
71C24×C4—21no—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C22×M16C2314yes—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—44no—
77C2×C22⋊Q8C2314yes—
78C2×#121—22no—
79C2×#121—44no—
80C2×#121C2314yes—
81#518—48no—
82C2×C4×D4—28no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4D414yes—
86C2×C4×D4C2314yes—
87C2×#117—22no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4—22no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4D411yescharacteristic
104C22×C22⋊C4D411yescharacteristic
105C22×C42C2311yescharacteristic
106#484—48no—
107#481—48no—
108C2×#107C2318yes—
109C2×#106C2318yes—
110#417—48no—
111#417—48no—
112C2×#103C2318yes—
113C2×#103C2318yes—
114C2×#102C2318yes—
115C2×#102C2318yes—
116#407—28no—
117C2×C22⋊C8C2314yes—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120C2×C22⋊C8C2314yes—
121#394—28no—
122#392—28no—
123#391—22no—
124#391C2314yes—
125#390—22no—
126#390C2314yes—
127#389—24no—
128#389—24no—
129#386—28no—
130#385—28no—
131#385—24no—
132#385—24no—
133#385—24no—
134#385—24no—
135#384C2312yes—
136#384C2312yes—
137#382—22no—
138#380—22no—
139#380—22no—
140#380—22no—
141#380—22no—
142#380—22no—
143#380—22no—
144#380C2314yes—
145#380C2314yes—
146#380C2314yes—
147#380C2314yes—
148C4×C4⋊C4—22no—
149C4×C4⋊C4C2314yes—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—21no—
153C2×#95—21no—
154C2×#95—21no—
155#340C2318yes—
156#340C2318yes—
157#336C2314yes—
158#336C2314yes—
159C25—21no—
160C25C2×D411yescharacteristic
161C25C2×D411yescharacteristic
162C2×#41—28no—
163C2×#41—28no—
164C2×#41—28no—
165C2×#41—28no—
166C2×#41—44no—
167C2×#41—44no—
168C2×#41—48no—
169C2×#41—48no—
170C2×#41—464no—
171C2×#41—48no—
172C22×Q8—24no—
173C22×Q8—21no—
174C22×Q8—24no—
175C22×Q8—42no—
176C22×Q8C2412yes—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—21no—
184C22×D4—24no—
185C22×D4—42no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—82no—
192C22×D4C2×D414yes—
193C22×D4C2412yes—
194C23×C4—21no—
195C23×C4—24no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4#4118yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C2×M16—28no—
221C2×M16—44no—
222C2×M16—44no—
223C22×C8C2418yes—
224C22⋊Q8—28no—
225C22⋊Q8—44no—
226C22⋊Q8—48no—
227#121—28no—
228#121—44no—
229#121—48no—
230C4×D4—216no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—416no—
234C4×D4—416no—
235#117—48no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4—82no—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D414yes—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42C2×D412yes—
282C2×C42C2412yes—
283#107—48no—
284#106—48no—
285#103—216no—
286#103—48no—
287#102—216no—
288#102—48no—
289C22⋊C8—28no—
290C22⋊C8—28no—
291C22⋊C8—44no—
292C22⋊C8—44no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—22no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24#12314yes—
321C24#13714yes—
322C24C22×D412yes—
323C24C22×D412yes—
324C24C22×D411yescharacteristic
325#41—232no—
326#41—416no—
327#41—416no—
328#41—416no—
329#41—464no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—416no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8H(5,𝔽2)14yes—
338C2×D4—232no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—24no—
342C2×D4—28no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—416no—
347C2×D4—48no—
348C2×D4—42no—
349C2×D4—48no—
350C2×D4—432no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—42no—
355C2×D4—84no—
356C2×D4—816no—
357C2×D4H(5,𝔽2)14yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—416no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—48no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—48no—
389C22×C4—42no—
390C22×C4—432no—
391C22×C4—42no—
392C22×C4—48no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—88no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4#12314yes—
413C22×C4#12318yes—
414C22×C4AGL(1,ℤ/8)14yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4114yes—
423C22×C4C2511yescharacteristic
424M16—88no—
425C2×C8—28no—
426C2×C8—28no—
427C2×C8—28no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—48no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C42—24no—
455C42—24no—
456C42—48no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—28no—
474C23—28no—
475C23—24no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—48no—
481C23—48no—
482C23—48no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23C2×#12314yes—
502C23C2×#12312yes—
503C23C2×AGL(1,ℤ/8)12yes—
504C23C2×#13714yes—
505C23C2×#13712yes—
506C23C23×D411yescharacteristic
507Q8—28no—
508Q8—48no—
509Q8—44no—
510D4—232no—
511D4—28no—
512D4—48no—
513D4—432no—
514D4—44no—
515D4—816no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—432no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—84no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—164no—
558C2×C4—164no—
559C2×C4—164no—
560C2×C4—164no—
561C2×C4C2×#12312yes—
562C2×C4C2×#12314yes—
563C2×C4C2×AGL(1,ℤ/8)14yes—
564C2×C4C22×#4112yes—
565C2×C4C2×H(5,𝔽2)12yes—
566C2×C4C2×H(5,𝔽2)12yes—
567C8—48no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—88no—
592C22—88no—
593C22—816no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22#309214yes—
599C22C22×#12311yescharacteristic
600C22#348411yescharacteristic
601C22#348411yescharacteristic
602C22C22×AGL(1,ℤ/8)11yescharacteristic
603C22C22×#13711yescharacteristic
604C22#358511yescharacteristic
605C22#358611yescharacteristic
606C4—24no—
607C4—22no—
608C4—24no—
609C4—42no—
610C4—44no—
611C4—42no—
612C4—82no—
613C4—84no—
614C4—82no—
615C4—84no—
616C4#349014yes—
617C2—24no—
618C2—44no—
619C2—44no—
620C2—84no—
621C2#3535011yescharacteristic
622C2C2×#309211yescharacteristic
623C2#6052011yescharacteristic
624C2#6157011yescharacteristic
625C2#6157311yescharacteristic
626C2#6158511yescharacteristic
627C2#6158711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2407C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34181C2semidirect
3. #27583C2semidirect
4. #27582C2semidirect
5. #24883C2semidirect
6. #24881C2semidirect
7. #24006C2semidirect
8. #24005C2semidirect
9. #21942C2semidirect
10. #21742C2semidirect
11. #20601C2semidirect
17. C2×#484C22semidirect
18. C2×#481C22semidirect
19. C2×#417C22semidirect
20. C2×#417C22semidirect
27. D4×C4⋊C4C22semidirect
28. D4×C4⋊C4C22semidirect
40. #2096C22semidirect
41. #2093C22semidirect
42. #2072C22semidirect
43. #2071C22semidirect
44. #2036C22semidirect
45. #2034C22semidirect
46. #1927C22semidirect
47. #1925C22semidirect
48. #1874C22semidirect
49. #1872C22semidirect
50. #1870C22semidirect
52. #1853C22semidirect
54. #1852C22semidirect
56. #1809C22semidirect
57. #1788C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24072
#341812
#275832
#275822
#248832
#248812
#240062
#240052
#219422
#217422
#206012