Finite Group Databasebeta

← #206413

Subgroups — #206413

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60700C211yesmaximal, characteristic
2#21415C211yesmaximal, characteristic, complemented
3C2×#1962C211yesmaximal, characteristic, complemented
4#20364C211yesmaximal, characteristic, complemented
5#17796C211yesmaximal, characteristic, complemented
6#17504C211yesmaximal, characteristic, complemented
7#17500C211yesmaximal, characteristic, complemented
8#9854C212yesmaximal, complemented
9#9852C212yesmaximal, complemented
10#9847C212yesmaximal, complemented
11#9846C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3184C412yescomplemented
15#3182C412yescomplemented
16#3180—22no—
17#3171—22no—
18#3169C2212yes—
19#3167C2212yes—
20#3165C411yescharacteristic, complemented
21#3151C411yescharacteristic, complemented
22#3141C411yescharacteristic, complemented
23#3138C411yescharacteristic, complemented
24#3055—22no—
25C4×AGL(1,ℤ/8)—22no—
26#2920C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C22×#104—22no—
29C2×C4×M16C2211yescharacteristic
30C2×C4×M16C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1994C2211yescharacteristic, complemented
33#1994C2211yescharacteristic, complemented
34#1962—22no—
35#1962C2212yes—
36#1962C2212yes—
37#1962C2211yescharacteristic, complemented
38#1962C2211yescharacteristic, complemented
39#1896—22no—
40C4×#104—22no—
41#1745—22no—
42#1744C2212yescomplemented
43#1744C2212yescomplemented
44#1728—22no—
45#1569—22no—
46#1567C2212yescomplemented
47#1566C2212yescomplemented
48#1565—22no—
49#1493C2212yescomplemented
50#1492C2212yes—
51#1492C2212yes—
52#1492C2212yescomplemented
53#1490—22no—
54#1489—22no—
55#1484—22no—
56#1483—22no—
57C22×#41—21no—
58C2×AGL(1,ℤ/8)—22no—
59C22×M16—21no—
60C22×M16—21no—
61C22×M16D412yes—
62#533C2×C411yescharacteristic
63#533C2×C411yescharacteristic
64#533C2311yescharacteristic
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532—21no—
70#532—21no—
71#532C2×C411yescharacteristic
72#532C2×C412yes—
73#532C2×C411yescharacteristic
74#532C2312yes—
75#532C2311yescharacteristic
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×#127C2×C411yescharacteristic
78#529—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82#529—21no—
83#529—21no—
84C2×#124—21no—
85#525—21no—
86#525—21no—
87#525—21no—
88C2×#121—21no—
89C2×#121—21no—
90C4×#41—21no—
91C4×#41—21no—
92C4×#41—21no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C22×C42—21no—
96C22×C42—21no—
97C22×C42D412yes—
98C22×C42C2311yescharacteristic
99#501—22no—
100#500C2×C412yescomplemented
101#498C2×C412yescomplemented
102#496—22no—
103#493C2×C412yescomplemented
104#492C2×C412yescomplemented
105#490—22no—
106#489—22no—
107#487C2×C412yescomplemented
108#486C2×C412yescomplemented
109#486C2×C412yescomplemented
110#486C2×C412yescomplemented
111#469—24no—
112#468—24no—
113#466—24no—
114#465—24no—
115#442—24no—
116#440—24no—
117C4×SD16—24no—
118C4×D8—24no—
119#432—21no—
120#432—21no—
121#431—21no—
122#431—21no—
123#428—22no—
124#423—21no—
125#423—21no—
126#423—21no—
127#423C2311yescharacteristic
128#423C2312yes—
129#423C2311yescharacteristic
130C2×C4⋊C8—21no—
131C2×C4⋊C8—21no—
132C2×#104—24no—
133C2×#104—24no—
134C2×#104—22no—
135C2×#104—22no—
136C2×#104—48no—
137#418—22no—
138#417—22no—
139#407—21no—
140C4×M16—21no—
141C4×M16—21no—
142C4×M16—21no—
143C4×M16C2×C411yescharacteristic
144C4×M16C2×C411yescharacteristic
145C4×M16C2×C412yes—
146C4×M16D412yes—
147C4×M16D412yes—
148C4×M16D412yes—
149C4×M16D412yes—
150C2×#97—21no—
151C2×#97D412yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2312yes—
154C43—21no—
155C43—21no—
156C43D412yes—
157C43C2311yescharacteristic
158C43C2311yescharacteristic
159#339—22no—
160#339—22no—
161#336—24no—
162#336—24no—
163#335—24no—
164#334—24no—
165Q8⋊C8—24no—
166D4⋊C8—24no—
167D4⋊C8—24no—
168D4⋊C8—24no—
169C2×#41—21no—
170C2×#41—22no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—22no—
174C2×#41—44no—
175C2×#41—44no—
176C2×#41—41no—
177C2×#41—41no—
178C22×Q8—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—41no—
183C23×C4—22no—
184C23×C4—41no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4#4111yescharacteristic
188AGL(1,ℤ/8)—48no—
189C2×SD16—24no—
190C2×D8—24no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—44no—
197C2×M16—44no—
198C2×M16—41no—
199C2×M16—44no—
200C2×M16C22⋊C412yes—
201C2×M16C22⋊C412yes—
202C2×M16C22⋊C414yes—
203C22×C8—22no—
204C22×C8—41no—
205C22×C8C2×D414yes—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8C22×C412yes—
210C4⋊Q8C22×C411yescharacteristic
211C4⋊Q8C22×C411yescharacteristic
212C4⋊Q8C22×C411yescharacteristic
213#127—21no—
214#127—21no—
215#127—21no—
216#127C22×C412yes—
217#127C22×C411yescharacteristic
218#126—41no—
219#126—41no—
220#125—21no—
221#125—21no—
222#125—21no—
223#124—21no—
224#124—22no—
225#124—21no—
226#124—22no—
227#124—21no—
228#124—41no—
229#124—41no—
230#123—41no—
231#123—41no—
232#123—41no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239#121—22no—
240#121—21no—
241#121—22no—
242#121—21no—
243#121—21no—
244#121—21no—
245#121—22no—
246#121—21no—
247#121—22no—
248#121—21no—
249#121—41no—
250#121—41no—
251#121—41no—
252#121—41no—
253#120—41no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×D4—22no—
261C4×D4—21no—
262C4×D4—22no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—41no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4—41no—
276C4×D4—41no—
277C4×D4—41no—
278C4×D4—41no—
279C4×D4—41no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—41no—
284#117—41no—
285#117—41no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C42—21no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—41no—
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42#4111yescharacteristic
313C2×C42#4111yescharacteristic
314C2×C42#4111yescharacteristic
315C2×C42C2411yescharacteristic
316#107—24no—
317#106—24no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—22no—
321C4⋊C8—21no—
322C4⋊C8—22no—
323C4⋊C8—21no—
324C4⋊C8—41no—
325C4⋊C8—41no—
326#104—44no—
327#104—48no—
328#104—44no—
329#104—48no—
330#103—24no—
331#102—24no—
332#102—24no—
333#102—24no—
334C22⋊C8—41no—
335C22⋊C8—41no—
336#97—22no—
337#97—22no—
338#97—41no—
339#97C22⋊C414yes—
340#97C22⋊C412yes—
341#97C22⋊C412yes—
342C4×C8—22no—
343C4×C8—41no—
344C4×C8C22×C412yes—
345C4×C8C22×C412yes—
346C4×C8C2×D414yes—
347C4×C8C2×D414yes—
348C24—41no—
349C24#12111yescharacteristic
350#41—48no—
351#41—44no—
352#41—42no—
353#41—42no—
354#41—44no—
355#41—44no—
356#41—82no—
357#41—84no—
358#41—82no—
359#41—82no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—44no—
365C2×Q8—41no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—44no—
373C2×D4—41no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—42no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—44no—
382C2×D4—82no—
383C2×D4—84no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12411yescharacteristic
413C22×C4#12411yescharacteristic
414C22×C4#12711yescharacteristic
415C22×C4#12711yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419SD16—48no—
420D8—48no—
421M16—24no—
422M16—24no—
423M16—44no—
424M16—44no—
425M16—44no—
426M16—44no—
427M16—42no—
428M16—42no—
429M16—82no—
430C2×C8—24no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—42no—
436C2×C8—82no—
437C2×C8C2×C22⋊C414yes—
438C2×C8C2×C22⋊C414yes—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—42no—
459C22⋊C4—41no—
460C22⋊C4—42no—
461C22⋊C4—41no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C42—24no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—44no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—44no—
477C42—44no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42C2×C22⋊C411yescharacteristic
482C42C2×C22⋊C411yescharacteristic
483C42C4×D411yescharacteristic
484C42C4×D411yescharacteristic
485C42C4×D411yescharacteristic
486C42C4×D411yescharacteristic
487C42#12111yescharacteristic
488C42#12111yescharacteristic
489C42#12411yescharacteristic
490C42#12411yescharacteristic
491C42#12711yescharacteristic
492C42#12711yescharacteristic
493C42AGL(1,ℤ/8)14yescomplemented
494C42C23×C411yescharacteristic
495C42C22×D411yescharacteristic
496C42C22×D411yescharacteristic
497C42C2×#4111yescharacteristic
498C23—22no—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23#39011yescharacteristic
507C23#39011yescharacteristic
508C23C2×#12111yescharacteristic
509Q8—42no—
510Q8—44no—
511Q8—42no—
512Q8—42no—
513Q8—82no—
514D4—42no—
515D4—44no—
516D4—44no—
517D4—42no—
518D4—44no—
519D4—42no—
520D4—82no—
521D4—82no—
522D4—84no—
523D4—82no—
524D4—82no—
525D4—84no—
526D4—82no—
527D4—82no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—416no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4#39011yescharacteristic
566C2×C4#39011yescharacteristic
567C2×C4#39011yescharacteristic
568C2×C4#39011yescharacteristic
569C2×C4#39011yescharacteristic
570C2×C4#39011yescharacteristic
571C2×C4#42114yes—
572C2×C4C22×C22⋊C411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C2×C4×D411yescharacteristic
575C2×C4C2×#12111yescharacteristic
576C2×C4C2×#12411yescharacteristic
577C2×C4C2×#12711yescharacteristic
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4#57512yes—
580C8—24no—
581C8—24no—
582C8—44no—
583C8—82no—
584C8—81no—
585C8—81no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—44no—
594C22—81no—
595C22—82no—
596C22—82no—
597C22—81no—
598C22—81no—
599C22—82no—
600C22—164no—
601C22#196312yes—
602C22C2×#39011yescharacteristic
603C22#305512yes—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—44no—
612C4—44no—
613C4—81no—
614C4—82no—
615C4—81no—
616C4—81no—
617C4#196312yes—
618C4C2×#42112yes—
619C4C4×AGL(1,ℤ/8)12yes—
620C4#295612yes—
621C4#315111yescharacteristic
622C4#315311yescharacteristic
623C2—22no—
624C2—22no—
625C2—41no—
626C2—161no—
627C2—162no—
628C2C2×#196311yescharacteristic
629C2#2123611yescharacteristic
630C2#2124111yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60700C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21415C2semidirect
3. C2×#1962C2semidirect
4. #20364C2semidirect
5. #17796C2semidirect
6. #17504C2semidirect
7. #17500C2semidirect
8. #9854C2semidirect
9. #9852C2semidirect
10. #9847C2semidirect
11. #9846C2semidirect
14. #3184C4semidirect
15. #3182C4semidirect
20. #3165C4semidirect
21. #3151C4semidirect
22. #3141C4semidirect
23. #3138C4semidirect
32. #1994C22semidirect
33. #1994C22semidirect
37. #1962C22semidirect
38. #1962C22semidirect
42. #1744C22semidirect
43. #1744C22semidirect
46. #1567C22semidirect
47. #1566C22semidirect
49. #1493C22semidirect
52. #1492C22semidirect
100. #500C2×C4semidirect
101. #498C2×C4semidirect
103. #493C2×C4semidirect
104. #492C2×C4semidirect
107. #487C2×C4semidirect
108. #486C2×C4semidirect
109. #486C2×C4semidirect
110. #486C2×C4semidirect
493. C42AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#607002
#214152
C2×#19622
#203642
#177962
#175042
#175002
#98542
#98522
#98472
#98462