Finite Group Databasebeta

← #482080

Subgroups — #482080

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60458C211yesmaximal, characteristic
2#33334C211yesmaximal, characteristic, complemented
3#20664C211yesmaximal, characteristic, complemented
4#11890C212yesmaximal, complemented
5#11888C212yesmaximal, complemented
6#11886C212yesmaximal, complemented
7#11885C212yesmaximal, complemented
8#11705C211yesmaximal, characteristic, complemented
9#11704C211yesmaximal, characteristic, complemented
10#11703C211yesmaximal, characteristic, complemented
11C2×#1347C211yesmaximal, characteristic, complemented
12C2×#576C411yescharacteristic, complemented
13C22×#131C2211yescharacteristic
14#3488—21no—
15C2×#504—21no—
16#3062—21no—
17#3062—22no—
18#3061C411yescharacteristic, complemented
19#3061C412yescomplemented
20#3058C411yescharacteristic, complemented
21#3058C412yescomplemented
22#3055—21no—
23#3055—22no—
24#3050C2211yescharacteristic
25#3050C2212yes—
26C2×#466C2211yescharacteristic
27C2×#466C2212yes—
28#2913C411yescharacteristic, complemented
29C2×#428—21no—
30#2897—21no—
31#2311—21no—
32#2113—21no—
33#2113—22no—
34#2112C2212yescomplemented
35#2112C2211yescharacteristic, complemented
36C2×C22⋊C16C2212yescomplemented
37C2×C22⋊C16C2211yescharacteristic, complemented
38#1809—22no—
39#1808C2212yescomplemented
40#1807C2212yescomplemented
41#1735C2211yescharacteristic, complemented
42C2×#340C2211yescharacteristic, complemented
43#1348C2211yescharacteristic, complemented
44#1348C2212yescomplemented
45#1348C2211yescharacteristic, complemented
46#1347C2212yescomplemented
47#1347C2212yescomplemented
48#1347C2211yescharacteristic, complemented
49#1347C2212yescomplemented
50#1347C2212yescomplemented
51#1347C2211yescharacteristic, complemented
52#1347C2212yescomplemented
53C2×H(5,𝔽2)—21no—
54C22×#41D411yescharacteristic
55#576—44no—
56C2×AGL(1,ℤ/8)—22no—
57C2×AGL(1,ℤ/8)—21no—
58C2×#135C2×C411yescharacteristic
59C2×#135C2×C412yescomplemented
60C22×D8C2×C411yescharacteristic
61C22×D8C2×C412yescomplemented
62C2×#131—44no—
63C2×#131C2×C411yescharacteristic
64C2×#131C2×C411yescharacteristic
65C2×#131C2×C412yes—
66C2×#131D411yescharacteristic
67C2×#131D411yescharacteristic
68C2×#131D412yes—
69C22×M16D411yescharacteristic
70C22×M16D412yes—
71C23×C8C2312yes—
72C23×C8C2311yescharacteristic
73#550—41no—
74#548—41no—
75#546—42no—
76#546—41no—
77D42—42no—
78D42—41no—
79#537—21no—
80#537—22no—
81#535—21no—
82#535—22no—
83#525—21no—
84#525—22no—
85C2×#121—21no—
86C2×#121—22no—
87#520—21no—
88C2×#117—21no—
89C22×C4⋊C4—21no—
90#504—44no—
91C2×M32—21no—
92C2×M32—22no—
93C22×C16—21no—
94C22×C16—22no—
95#469—22no—
96#469—22no—
97#469—22no—
98#468—22no—
99#468—22no—
100#468—22no—
101#467C2×C411yescharacteristic
102#467C2×C412yescomplemented
103#467C2311yescharacteristic
104#466—21no—
105#466—22no—
106#466—21no—
107#466—22no—
108#466C2×C412yes—
109#466C2×C411yescharacteristic
110#466C2×C412yescomplemented
111#466C2×C412yescomplemented
112#466C2×C412yescomplemented
113#466C2312yes—
114#466C2311yescharacteristic
115#465—21no—
116#465—22no—
117#465—21no—
118#465—22no—
119#428—22no—
120#428—21no—
121#428—44no—
122#427C2×C411yescharacteristic
123#427C2×C412yescomplemented
124C2×#107—22no—
125C2×#107C2×C411yescharacteristic
126C2×#107C2×C412yescomplemented
127C2×#106—22no—
128#418—22no—
129#418—21no—
130#417—22no—
131#417—21no—
132#416—21no—
133#416—22no—
134C2×#102—21no—
135C2×#102—22no—
136#391—41no—
137#390—41no—
138#386—42no—
139#380—22no—
140#380—22no—
141C22⋊C16—22no—
142C22⋊C16—21no—
143C22⋊C16—22no—
144C22⋊C16—21no—
145C22⋊C16C2311yescharacteristic
146C22⋊C16C2312yes—
147C22⋊C16C2311yescharacteristic
148#340—21no—
149#340—21no—
150#340C2311yescharacteristic
151#340C2312yes—
152#340C2311yescharacteristic
153H(5,𝔽2)—44no—
154H(5,𝔽2)—44no—
155C2×#41—21no—
156C2×#41—22no—
157C2×#41—42no—
158C2×#41—42no—
159C2×#41—41no—
160C2×#41C22⋊C411yescharacteristic
161C2×#41C22⋊C411yescharacteristic
162C2×#41C22⋊C412yes—
163C2×#41SD1612yes—
164C2×#41SD1611yescharacteristic
165C2×#41SD1611yescharacteristic
166C22×Q8D811yescharacteristic
167C22×D4—21no—
168C22×D4—22no—
169C22×D4—41no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—41no—
173C22×D4D812yes—
174C22×D4D811yescharacteristic
175C23×C4—41no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178AGL(1,ℤ/8)—48no—
179AGL(1,ℤ/8)—44no—
180#135—42no—
181#135—44no—
182C2×Q16C22×C411yescharacteristic
183C2×SD16—21no—
184C2×SD16—22no—
185C2×D8—21no—
186C2×D8—22no—
187C2×D8—42no—
188C2×D8—44no—
189C2×D8C22×C412yes—
190C2×D8C22×C411yescharacteristic
191#131—24no—
192#131—24no—
193#131—44no—
194#131—44no—
195#131—42no—
196#131—42no—
197C2×M16—24no—
198C2×M16—44no—
199C2×M16C22⋊C412yes—
200C2×M16C22⋊C411yescharacteristic
201C2×M16C22⋊C412yes—
202C2×M16C22⋊C412yes—
203C2×M16C22⋊C412yes—
204C2×M16C22⋊C412yes—
205C2×M16C22⋊C411yescharacteristic
206C22×C8—44no—
207C22×C8—42no—
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C412yes—
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D412yes—
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D412yes—
217C22×C8C2×D411yescharacteristic
218C22×C8C2411yescharacteristic
219#127—41no—
220#127—42no—
221#124—41no—
222#124—42no—
223#123—41no—
224#123—42no—
225C22⋊Q8—21no—
226C22⋊Q8—22no—
227C22⋊Q8—21no—
228C22⋊Q8—41no—
229C22⋊Q8—42no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—21no—
234#121—22no—
235#121—22no—
236#121—21no—
237#121—42no—
238#121—42no—
239#121—42no—
240#121—41no—
241#121—42no—
242#121—41no—
243#121—42no—
244#121—41no—
245#121—42no—
246#120—41no—
247#120—42no—
248#120—42no—
249#120—42no—
250#120—42no—
251C4×Q8—41no—
252C4×D4—42no—
253C4×D4—41no—
254C4×D4—41no—
255C4×D4—42no—
256C4×D4—41no—
257C4×D4—42no—
258#117—21no—
259#117—22no—
260#117—42no—
261#117—42no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C42—41no—
276M32—42no—
277M32—44no—
278C2×C16—21no—
279C2×C16—22no—
280C2×C16—21no—
281C2×C16—42no—
282C2×C16—44no—
283#107—21no—
284#107—22no—
285#107—44no—
286#107C22×C412yes—
287#107C22×C411yescharacteristic
288#107C22×C411yescharacteristic
289#106—21no—
290#106—22no—
291#106—44no—
292#103—21no—
293#103—22no—
294#103—21no—
295#102—22no—
296#102—22no—
297#102—22no—
298#102—21no—
299#102—22no—
300#102—22no—
301#102—21no—
302#95—42no—
303#95—42no—
304C24—42no—
305C24—41no—
306C24C2×D811yescharacteristic
307C24C2×D812yes—
308#41—22no—
309#41—22no—
310#41—22no—
311#41—22no—
312#41—42no—
313#41—42no—
314#41—44no—
315#41—42no—
316#41—42no—
317#41—44no—
318#41—44no—
319#41—42no—
320#41—42no—
321#41—84no—
322C2×Q8—21no—
323C2×Q8—42no—
324C2×Q8#10211yescharacteristic
325C2×Q8#10211yescharacteristic
326C2×Q8#10212yes—
327C2×D4—22no—
328C2×D4—21no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—41no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—88no—
347C2×D4—84no—
348C2×D4—161no—
349C2×D4—162no—
350C2×D4—161no—
351C2×D4—162no—
352C2×D4#10212yes—
353C2×D4#10211yescharacteristic
354C2×D4#10212yes—
355C2×D4#10211yescharacteristic
356C2×D4#10212yes—
357C2×D4#10212yes—
358C2×D4#10212yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—82no—
371C22×C4—161no—
372C22×C4—161no—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×D811yescharacteristic
378C22×C4C2×D812yes—
379C22×C4C2×SD1611yescharacteristic
380C22×C4C2×SD1611yescharacteristic
381C22×C4C2×SD1612yes—
382C22×C4C2×SD1612yes—
383C22×C4C22×D411yescharacteristic
384Q16—41no—
385SD16—42no—
386SD16—44no—
387D8—42no—
388D8—41no—
389D8—42no—
390D8—44no—
391M16—22no—
392M16—24no—
393M16—24no—
394M16—24no—
395M16—42no—
396M16—42no—
397M16#9914yes—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8C2×C22⋊C411yescharacteristic
410C2×C8C2×C22⋊C412yes—
411C2×C8C2×C22⋊C411yescharacteristic
412C2×C8C2×C22⋊C412yes—
413C2×C8C23×C411yescharacteristic
414C2×C8C22×D411yescharacteristic
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—82no—
430C22⋊C4—41no—
431C22⋊C4—42no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—42no—
442C22⋊C4—82no—
443C22⋊C4—162no—
444C22⋊C4—162no—
445C42—42no—
446C42—41no—
447C42—82no—
448C16—41no—
449C16—41no—
450C16—42no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—84no—
462C23—82no—
463C23—161no—
464C23—162no—
465C23—161no—
466C23—162no—
467C23C2×#10212yes—
468C23C2×#10211yescharacteristic
469C23C2×#10212yes—
470C23C2×#10211yescharacteristic
471C23C22×D811yescharacteristic
472Q8—22no—
473Q8—22no—
474Q8—42no—
475Q8—41no—
476Q8—41no—
477D4—24no—
478D4—24no—
479D4—22no—
480D4—22no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—41no—
486D4—41no—
487D4—42no—
488D4—84no—
489D4—84no—
490D4—82no—
491D4—81no—
492D4—82no—
493D4—81no—
494D4—82no—
495D4—82no—
496D4—161no—
497D4—162no—
498D4—162no—
499D4—162no—
500D4—162no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—81no—
530C2×C4—161no—
531C2×C4—162no—
532C2×C4—164no—
533C2×C4C2×#9912yes—
534C2×C4C2×#10212yes—
535C2×C4C2×#10211yescharacteristic
536C2×C4C2×#10212yes—
537C2×C4C2×#10211yescharacteristic
538C2×C4#42112yes—
539C2×C4C22×C22⋊C411yescharacteristic
540C2×C4C22×SD1611yescharacteristic
541C8—21no—
542C8—21no—
543C8—21no—
544C8—22no—
545C8—21no—
546C8C2×#9914yes—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—21no—
553C22—22no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—21no—
558C22—22no—
559C22—22no—
560C22—42no—
561C22—42no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—81no—
566C22—82no—
567C22—81no—
568C22—82no—
569C22—81no—
570C22—322no—
571C22—324no—
572C22#151912yes—
573C22C22×#10211yescharacteristic
574C22#341511yescharacteristic
575C22#341611yescharacteristic
576C4—21no—
577C4—21no—
578C4—22no—
579C4—22no—
580C4—82no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—161no—
585C4—161no—
586C4—162no—
587C4#151912yes—
588C4C22×#9911yescharacteristic
589C4C2×#42111yescharacteristic
590C2—21no—
591C2—21no—
592C2—22no—
593C2—41no—
594C2—42no—
595C2—42no—
596C2—162no—
597C2—161no—
598C2C2×#151911yescharacteristic
599C2#3338111yescharacteristic
600C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#60458C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33334C2semidirect
3. #20664C2semidirect
4. #11890C2semidirect
5. #11888C2semidirect
6. #11886C2semidirect
7. #11885C2semidirect
8. #11705C2semidirect
9. #11704C2semidirect
10. #11703C2semidirect
11. C2×#1347C2semidirect
12. C2×#576C4semidirect
18. #3061C4semidirect
19. #3061C4semidirect
20. #3058C4semidirect
21. #3058C4semidirect
28. #2913C4semidirect
34. #2112C22semidirect
35. #2112C22semidirect
36. C2×C22⋊C16C22semidirect
37. C2×C22⋊C16C22semidirect
39. #1808C22semidirect
40. #1807C22semidirect
41. #1735C22semidirect
42. C2×#340C22semidirect
43. #1348C22semidirect
44. #1348C22semidirect
45. #1348C22semidirect
46. #1347C22semidirect
47. #1347C22semidirect
48. #1347C22semidirect
49. #1347C22semidirect
50. #1347C22semidirect
51. #1347C22semidirect
52. #1347C22semidirect
59. C2×#135C2×C4semidirect
61. C22×D8C2×C4semidirect
102. #467C2×C4semidirect
110. #466C2×C4semidirect
111. #466C2×C4semidirect
112. #466C2×C4semidirect
123. #427C2×C4semidirect
126. C2×#107C2×C4semidirect

Maximal subgroups

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

subgroupindex
#604582
#333342
#206642
#118902
#118882
#118862
#118852
#117052
#117042
#117032
C2×#13472