Finite Group Databasebeta

← #6357459

Subgroups — #6357459

Order 512.

2072 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60971C211yesmaximal, characteristic, complemented
2#60846C212yesmaximal, complemented
3#60798C211yesmaximal, characteristic
4C2×#3208C212yesmaximal, complemented
5#19717C214yesmaximal, complemented
6#19713C214yesmaximal, complemented
7#19705C214yesmaximal, complemented
8#19697C214yesmaximal, complemented
9#19656C214yesmaximal, complemented
10#19651C214yesmaximal, complemented
11#17325C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13C2×#533C2212yes—
14C22×#127C2211yescharacteristic
15#3255—22no—
16#3233—24no—
17#3232—24no—
18#3231C2214yescomplemented
19#3230C2212yescomplemented
20#3229C2214yescomplemented
21#3229C2212yescomplemented
22#3228C2214yescomplemented
23#3227C2212yescomplemented
24#3226C2212yescomplemented
25#3217—24no—
26#3209C2214yescomplemented
27#3208—24no—
28#3208—24no—
29#3208C2218yes—
30#3208C2214yes—
31#3208C2214yes—
32#3208C2214yes—
33#3205C2214yes—
34#3195—22no—
35#3185C2214yescomplemented
36#3160C2214yescomplemented
37#3151C2212yes—
38C2×#492C2214yes—
39#3138C2212yes—
40#3131C2214yescomplemented
41C2×#486C2214yes—
42#3113—24no—
43#3091—24no—
44#3076—24no—
45#3038—22no—
46C2×#460—24no—
47#3003—22no—
48C2×#450—24no—
49#2920C2211yescharacteristic
50C2×#431C2212yes—
51#1722C2214yescomplemented
52#1720C2218yescomplemented
53#1719C2218yescomplemented
54#1717C2218yescomplemented
55#1715C2214yescomplemented
56#1714C2218yescomplemented
57#1714C2218yescomplemented
58#1709C2214yescomplemented
59#1708C2218yescomplemented
60#1705C2214yescomplemented
61#1453C2214yescomplemented
62#1452C2214yescomplemented
63#1450C2214yescomplemented
64C23×D4—22no—
65C23×D4—21no—
66C2×AGL(1,ℤ/8)—24no—
67C22×SD16—28no—
68#533—22no—
69#533—22no—
70#533—22no—
71#533—22no—
72#533—22no—
73#533—22no—
74#533C2314yes—
75#533C2311yescharacteristic
76#533C2312yes—
77#533C2314yes—
78#532C2312yes—
79#532C2311yescharacteristic
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×#127—22no—
83C2×#127—22no—
84C2×#127—22no—
85C2×#127—22no—
86C2×#127—21no—
87C2×#127C2314yes—
88C2×#127C2312yes—
89#529—22no—
90#529—22no—
91#529—22no—
92C2×#125—22no—
93#525—22no—
94#525—21no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C4×#41—22no—
98C4×#41—21no—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×#117—22no—
102C22×C42C2311yescharacteristic
103#500C2314yescomplemented
104#500C2314yescomplemented
105#499—28no—
106#498C2314yescomplemented
107#498C2314yescomplemented
108#498C2314yescomplemented
109#496—24no—
110#494C2314yescomplemented
111#493C2314yes—
112#492—28no—
113#492C2318yes—
114#492C2318yes—
115#492C2314yes—
116#490—24no—
117#488—28no—
118#487C2314yescomplemented
119#487C2314yescomplemented
120#487C2318yescomplemented
121#486—28no—
122#486C2318yes—
123#486C2318yes—
124#486C2318yes—
125#486C2314yescomplemented
126#486C2314yescomplemented
127#485—28no—
128#484—24no—
129#483—24no—
130#483—28no—
131#482—28no—
132#481—24no—
133#480—24no—
134#479—28no—
135#478—28no—
136#477—24no—
137#476—24no—
138#475—24no—
139#475—28no—
140#474—24no—
141#470—28no—
142#462—28no—
143#460—28no—
144#460—28no—
145#460—216no—
146#460—28no—
147#459—28no—
148#451—28no—
149#450—28no—
150#450—28no—
151#450—216no—
152#450—28no—
153#447—28no—
154#441—28no—
155#432D414yes—
156#431D414yes—
157#431D414yes—
158#431C2312yes—
159#431C2312yes—
160#431C2314yes—
161#431C2314yes—
162#427—28no—
163#423D412yes—
164C2×C4⋊C8D414yes—
165#418—24no—
166#416—28no—
167C2×#102—28no—
168#407D412yes—
169C2×C22⋊C8D414yes—
170C4×M16C2312yes—
171C2×C4×C8C2314yes—
172#322C2318yescomplemented
173#322C2314yescomplemented
174C82C2314yescomplemented
175C25—41no—
176C2×#41—21no—
177C2×#41—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—41no—
190C22×D4—88no—
191C22×D4—84no—
192C23×C4C2×D412yes—
193C23×C4C2×D411yescharacteristic
194AGL(1,ℤ/8)—88no—
195C2×Q16—28no—
196C2×SD16—216no—
197C2×SD16—216no—
198C2×SD16—28no—
199C2×SD16—816no—
200C2×D8—28no—
201C2×M16C2×D414yes—
202C22×C8C2×D418yes—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209C4⋊Q8C2414yes—
210C4⋊Q8C2414yes—
211C4⋊Q8C2412yes—
212C4⋊Q8C2411yescharacteristic
213#127—24no—
214#127—24no—
215#127—24no—
216#127—24no—
217#127—22no—
218#127—41no—
219#127—42no—
220#127—41no—
221#127—42no—
222#127C2414yes—
223#127C2411yescharacteristic
224#126—42no—
225#125—24no—
226#125—24no—
227#125—21no—
228#125—22no—
229#125—42no—
230#125—42no—
231#124—22no—
232#124—21no—
233#123—42no—
234#123—41no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—42no—
243C22⋊Q8—42no—
244C22⋊Q8—42no—
245C22⋊Q8—42no—
246#121—22no—
247#121—22no—
248#121—22no—
249#120—41no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—42no—
258C4×Q8—42no—
259C4×Q8—42no—
260C4×Q8—42no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—42no—
265C4×D4—42no—
266C4×D4—42no—
267#117—24no—
268#117—24no—
269#117—21no—
270#117—22no—
271#117—42no—
272#117—42no—
273#117—42no—
274#117—41no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C42—42no—
283C2×C42—41no—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2412yes—
290C2×C42C2411yescharacteristic
291#107—28no—
292#107—28no—
293#106—28no—
294#106—28no—
295#106—28no—
296C4⋊C8C2×D414yes—
297C4⋊C8C2×D414yes—
298C4⋊C8C2×D418yes—
299C4⋊C8C2×D418yes—
300#103—28no—
301#103—28no—
302#103—216no—
303#102—216no—
304#102—216no—
305#102—28no—
306#102—28no—
307#102—216no—
308C22⋊C8C2×D414yes—
309C22⋊C8C2×D414yes—
310C22⋊C8C2×D418yes—
311C22⋊C8C2×D418yes—
312#97C2×D414yes—
313C4×C8C2×D418yes—
314C4×C8C2414yes—
315C4×C8C2414yes—
316C4×C8C2414yes—
317C24—42no—
318C24—44no—
319C24—88no—
320C24—84no—
321C24#12711yescharacteristic
322#41—44no—
323#41—48no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—21no—
327C2×Q8—24no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—48no—
331C2×Q8—84no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—28no—
335C2×D4—21no—
336C2×D4—48no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—44no—
340C2×D4—42no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—416no—
345C2×D4—88no—
346C2×D4—88no—
347C2×D4—88no—
348C2×D4—88no—
349C2×D4—88no—
350C2×D4—88no—
351C2×D4—88no—
352C2×D4—84no—
353C22×C4—28no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4#12712yes—
360C22×C4#12712yes—
361C22×C4#12712yes—
362C22×C4#12712yes—
363C22×C4#12711yescharacteristic
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D412yes—
369Q16—88no—
370SD16—816no—
371SD16—816no—
372SD16—88no—
373D8—88no—
374M16—44no—
375C2×C8—216no—
376C2×C8C22×D418yes—
377C2×C8C22×D418yes—
378C2×C8C22×D418yes—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—41no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C42—41no—
405C42—42no—
406C42—44no—
407C42—44no—
408C42#12711yescharacteristic
409C42#12712yes—
410C42#12711yescharacteristic
411C42#12712yes—
412C42C22×D412yes—
413C42C22×D411yescharacteristic
414C42C22×D412yes—
415C42C22×D412yes—
416C42C22×D412yes—
417C42C2511yescharacteristic
418C23—24no—
419C23—41no—
420C23—41no—
421C23—44no—
422C23—84no—
423C23—88no—
424C23—88no—
425C23—88no—
426C23—88no—
427C23—1616no—
428C23C2×#12711yescharacteristic
429C23C2×#12712yes—
430Q8—42no—
431Q8—48no—
432Q8—48no—
433Q8—44no—
434Q8—84no—
435Q8—84no—
436D4—42no—
437D4—48no—
438D4—416no—
439D4—44no—
440D4—84no—
441D4—88no—
442D4—88no—
443D4—88no—
444D4—82no—
445D4—84no—
446D4—88no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—82no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4C2×#12712yes—
465C2×C4C2×#12711yescharacteristic
466C2×C4C2×#12712yes—
467C2×C4C2×#12712yes—
468C2×C4C2×#12712yes—
469C2×C4#57718yes—
470C2×C4C23×D411yescharacteristic
471C2×C4C23×D412yes—
472C8—28no—
473C8—28no—
474C8—28no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—21no—
479C22—44no—
480C22—82no—
481C22—88no—
482C22—81no—
483C22—84no—
484C22—1616no—
485C22—168no—
486C22#315414yes—
487C22C22×#12711yescharacteristic
488C4—22no—
489C4—22no—
490C4—21no—
491C4—22no—
492C4—82no—
493C4—84no—
494C4—81no—
495C4—84no—
496C4#315414yes—
497C4C2×#57714yes—
498C4#358512yes—
499C2—24no—
500C2—41no—
501C2—161no—
502C2—164no—
503C2C2×#315412yes—
504C2#6064411yescharacteristic
subgroupG/H
center, socleC22C22×#127
commutator, Frattini, cosocleC42C25
prime radical#60798C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60971C2semidirect
2. #60846C2semidirect
4. C2×#3208C2semidirect
5. #19717C2semidirect
6. #19713C2semidirect
7. #19705C2semidirect
8. #19697C2semidirect
9. #19656C2semidirect
10. #19651C2semidirect
11. #17325C2semidirect
18. #3231C22semidirect
19. #3230C22semidirect
20. #3229C22semidirect
21. #3229C22semidirect
22. #3228C22semidirect
23. #3227C22semidirect
24. #3226C22semidirect
26. #3209C22semidirect
35. #3185C22semidirect
36. #3160C22semidirect
40. #3131C22semidirect
51. #1722C22semidirect
52. #1720C22semidirect
53. #1719C22semidirect
54. #1717C22semidirect
55. #1715C22semidirect
56. #1714C22semidirect
57. #1714C22semidirect
58. #1709C22semidirect
59. #1708C22semidirect
60. #1705C22semidirect
61. #1453C22semidirect
62. #1452C22semidirect
63. #1450C22semidirect
103. #500C23semidirect
104. #500C23semidirect
106. #498C23semidirect
107. #498C23semidirect
108. #498C23semidirect
110. #494C23semidirect
118. #487C23semidirect
119. #487C23semidirect
120. #487C23semidirect
125. #486C23semidirect
126. #486C23semidirect
172. #322C23semidirect
173. #322C23semidirect
174. C82C23semidirect

Maximal subgroups

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

subgroupindex
#609712
#608462
#607982
C2×#32082
#197172
#197132
#197052
#196972
#196562
#196512
#173252