Finite Group Databasebeta

← #206419

Subgroups — #206419

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60946C211yesmaximal, characteristic
2#21415C211yesmaximal, characteristic, complemented
3C2×#1962C211yesmaximal, characteristic, complemented
4#20463C211yesmaximal, characteristic, complemented
5#17817C211yesmaximal, characteristic, complemented
6#17481C211yesmaximal, characteristic, complemented
7#17478C211yesmaximal, characteristic, complemented
8#9851C212yesmaximal, complemented
9#9849C212yesmaximal, complemented
10#9847C212yesmaximal, complemented
11#9846C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3247—22no—
15#3230C411yescharacteristic, complemented
16#3228C412yescomplemented
17#3227C411yescharacteristic, complemented
18#3208C411yescharacteristic, complemented
19#3207C412yescomplemented
20#3205C411yescharacteristic, complemented
21#3188—22no—
22#3169C2212yes—
23#3167C2212yes—
24#3106—22no—
25#3105—22no—
26#2920C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C22×#104—22no—
31C2×C43C2211yescharacteristic
32#1994C2211yescharacteristic, complemented
33#1994C2211yescharacteristic, complemented
34#1962—22no—
35#1962C2212yes—
36#1962C2212yes—
37#1962C2211yescharacteristic, complemented
38#1962C2211yescharacteristic, complemented
39#1896—22no—
40#1767—22no—
41C4×#104—22no—
42#1749—22no—
43#1744C2212yescomplemented
44C4×#97C2212yescomplemented
45#1574—22no—
46#1573—22no—
47#1567C2212yescomplemented
48#1566C2212yescomplemented
49#1496—22no—
50#1495—22no—
51#1493C2212yescomplemented
52#1492C2212yes—
53#1492C2212yes—
54#1492C2212yescomplemented
55#1478—22no—
56#1477—22no—
57C22×#41—21no—
58C22×M16—21no—
59#533C2×C411yescharacteristic
60#533C2×C411yescharacteristic
61#533C2311yescharacteristic
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67#532—21no—
68#532C2×C412yes—
69#532C2×C411yescharacteristic
70#532C2×C411yescharacteristic
71#532C2312yes—
72#532C2311yescharacteristic
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×#127C2×C411yescharacteristic
75#529—21no—
76#529—21no—
77#529—21no—
78#529—21no—
79#529—21no—
80#529—21no—
81C2×#124—21no—
82#525—21no—
83#525—21no—
84#525—21no—
85C2×#121—21no—
86C2×#121—21no—
87C4×#41—21no—
88C4×#41—21no—
89C4×#41—21no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C22×C42—21no—
93C22×C42—21no—
94C22×C42D412yes—
95C22×C42C2311yescharacteristic
96#500C2×C412yescomplemented
97#498C2×C412yescomplemented
98#493C2×C412yescomplemented
99#492C2×C412yescomplemented
100#487C2×C412yescomplemented
101#486C2×C412yescomplemented
102#486C2×C412yescomplemented
103#486C2×C412yescomplemented
104#484—22no—
105#482—24no—
106#481—22no—
107#478—24no—
108#477—22no—
109#476—22no—
110#466—24no—
111#465—24no—
112#463—24no—
113#460—22no—
114#459—22no—
115#450—22no—
116#449—24no—
117#447—22no—
118C4×SD16—24no—
119C4×D8—24no—
120#432—21no—
121#432—21no—
122#431—21no—
123#431—21no—
124#431—21no—
125#431—21no—
126#431C2×C411yescharacteristic
127#431C2×C412yes—
128#431C2×C411yescharacteristic
129#431D412yes—
130#431D412yes—
131#431D412yes—
132#431D412yes—
133#423—21no—
134#423—21no—
135#423—21no—
136#423C2311yescharacteristic
137#423C2312yes—
138#423C2311yescharacteristic
139C2×C4⋊C8—21no—
140C2×C4⋊C8—21no—
141C2×C4⋊C8—21no—
142C2×C4⋊C8D412yes—
143C2×#104—24no—
144C2×#104—24no—
145C2×#104—22no—
146C2×#104—22no—
147C2×#104—48no—
148#407—21no—
149C2×C22⋊C8—21no—
150C2×C22⋊C8D412yes—
151C4×M16—21no—
152C2×#97—24no—
153C2×C4×C8C2312yes—
154C2×C4×C8C2312yes—
155C43—21no—
156C43—21no—
157C43D412yes—
158C43C2311yescharacteristic
159C43C2311yescharacteristic
160#339—22no—
161#339—22no—
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—21no—
171C2×#41—22no—
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
188C2×SD16—24no—
189C2×D8—24no—
190C2×M16—22no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—41no—
194C2×M16—44no—
195C22×C8—22no—
196C22×C8—41no—
197C22×C8C2×D414yes—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8C22×C412yes—
202C4⋊Q8C22×C411yescharacteristic
203C4⋊Q8C22×C411yescharacteristic
204C4⋊Q8C22×C411yescharacteristic
205#127—21no—
206#127—21no—
207#127—21no—
208#127C22×C412yes—
209#127C22×C411yescharacteristic
210#126—41no—
211#126—41no—
212#125—21no—
213#125—21no—
214#125—21no—
215#124—21no—
216#124—22no—
217#124—21no—
218#124—22no—
219#124—21no—
220#124—41no—
221#124—41no—
222#123—41no—
223#123—41no—
224#123—41no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231#121—22no—
232#121—21no—
233#121—22no—
234#121—21no—
235#121—21no—
236#121—21no—
237#121—22no—
238#121—21no—
239#121—22no—
240#121—21no—
241#121—41no—
242#121—41no—
243#121—41no—
244#121—41no—
245#120—41no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×D4—22no—
253C4×D4—21no—
254C4×D4—22no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—41no—
263C4×D4—41no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4—41no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—41no—
276#117—41no—
277#117—41no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C42—21no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—41no—
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42#4111yescharacteristic
305C2×C42#4111yescharacteristic
306C2×C42#4111yescharacteristic
307C2×C42C2411yescharacteristic
308#107—24no—
309#106—24no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—22no—
315C4⋊C8—21no—
316C4⋊C8—22no—
317C4⋊C8—21no—
318C4⋊C8—41no—
319C4⋊C8—41no—
320C4⋊C8C22⋊C412yes—
321C4⋊C8C22⋊C412yes—
322C4⋊C8C22⋊C414yes—
323#104—44no—
324#104—48no—
325#104—44no—
326#104—48no—
327#103—24no—
328#102—24no—
329#102—24no—
330#102—24no—
331C22⋊C8—22no—
332C22⋊C8—22no—
333C22⋊C8—41no—
334C22⋊C8—41no—
335C22⋊C8C22⋊C412yes—
336C22⋊C8C22⋊C414yes—
337C22⋊C8C22⋊C412yes—
338#97—48no—
339#97—41no—
340C4×C8—22no—
341C4×C8—41no—
342C4×C8C22×C412yes—
343C4×C8C22×C412yes—
344C4×C8C2×D414yes—
345C4×C8C2×D414yes—
346C24—41no—
347C24#12111yescharacteristic
348#41—48no—
349#41—44no—
350#41—42no—
351#41—42no—
352#41—44no—
353#41—44no—
354#41—82no—
355#41—84no—
356#41—82no—
357#41—82no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—22no—
362C2×Q8—44no—
363C2×Q8—41no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—44no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—42no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—44no—
380C2×D4—82no—
381C2×D4—84no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12411yescharacteristic
411C22×C4#12411yescharacteristic
412C22×C4#12711yescharacteristic
413C22×C4#12711yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4111yescharacteristic
417SD16—84no—
418D8—84no—
419M16—44no—
420M16—42no—
421M16—42no—
422M16—82no—
423C2×C8—24no—
424C2×C8—44no—
425C2×C8—44no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—82no—
430C2×C8C2×C22⋊C414yes—
431C2×C8C2×C22⋊C414yes—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—42no—
452C22⋊C4—41no—
453C22⋊C4—42no—
454C22⋊C4—41no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C42—24no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—44no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—44no—
470C42—44no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42C2×C22⋊C411yescharacteristic
475C42C2×C22⋊C411yescharacteristic
476C42C4×D411yescharacteristic
477C42C4×D411yescharacteristic
478C42C4×D411yescharacteristic
479C42C4×D411yescharacteristic
480C42#12111yescharacteristic
481C42#12111yescharacteristic
482C42#12411yescharacteristic
483C42#12411yescharacteristic
484C42#12711yescharacteristic
485C42#12711yescharacteristic
486C42AGL(1,ℤ/8)14yes—
487C42C23×C411yescharacteristic
488C42C22×D411yescharacteristic
489C42C22×D411yescharacteristic
490C42C2×#4111yescharacteristic
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23#39011yescharacteristic
500C23#39011yescharacteristic
501C23C2×#12111yescharacteristic
502Q8—42no—
503Q8—44no—
504Q8—42no—
505Q8—42no—
506Q8—82no—
507D4—42no—
508D4—44no—
509D4—44no—
510D4—42no—
511D4—44no—
512D4—42no—
513D4—82no—
514D4—82no—
515D4—84no—
516D4—82no—
517D4—82no—
518D4—84no—
519D4—82no—
520D4—82no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—416no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4#39011yescharacteristic
559C2×C4#39011yescharacteristic
560C2×C4#39011yescharacteristic
561C2×C4#39011yescharacteristic
562C2×C4#39011yescharacteristic
563C2×C4#39011yescharacteristic
564C2×C4#44414yes—
565C2×C4C22×C22⋊C411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×#12411yescharacteristic
570C2×C4C2×#12711yescharacteristic
571C2×C4C2×AGL(1,ℤ/8)12yes—
572C2×C4#57512yes—
573C8—24no—
574C8—24no—
575C8—44no—
576C8—82no—
577C8—81no—
578C8—81no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—44no—
587C22—81no—
588C22—82no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22—82no—
593C22—164no—
594C22#197512yes—
595C22C2×#39011yescharacteristic
596C22#305512yes—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—44no—
605C4—44no—
606C4—81no—
607C4—82no—
608C4—81no—
609C4—81no—
610C4#197512yes—
611C4#290112yes—
612C4C4×AGL(1,ℤ/8)12yes—
613C4C2×#44412yes—
614C4#315111yescharacteristic
615C4#315311yescharacteristic
616C2—22no—
617C2—22no—
618C2—41no—
619C2—161no—
620C2—162no—
621C2#2121011yescharacteristic
622C2#2123611yescharacteristic
623C2C2×#197511yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60946C2
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. #20463C2semidirect
5. #17817C2semidirect
6. #17481C2semidirect
7. #17478C2semidirect
8. #9851C2semidirect
9. #9849C2semidirect
10. #9847C2semidirect
11. #9846C2semidirect
15. #3230C4semidirect
16. #3228C4semidirect
17. #3227C4semidirect
18. #3208C4semidirect
19. #3207C4semidirect
20. #3205C4semidirect
32. #1994C22semidirect
33. #1994C22semidirect
37. #1962C22semidirect
38. #1962C22semidirect
43. #1744C22semidirect
44. C4×#97C22semidirect
47. #1567C22semidirect
48. #1566C22semidirect
51. #1493C22semidirect
54. #1492C22semidirect
96. #500C2×C4semidirect
97. #498C2×C4semidirect
98. #493C2×C4semidirect
99. #492C2×C4semidirect
100. #487C2×C4semidirect
101. #486C2×C4semidirect
102. #486C2×C4semidirect
103. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#609462
#214152
C2×#19622
#204632
#178172
#174812
#174782
#98512
#98492
#98472
#98462