Finite Group Databasebeta

← #7623777

Subgroups — #7623777

Order 512.

1562 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#61284C211yesmaximal, characteristic, complemented
2#61084C211yesmaximal, characteristic, complemented
3#33886C214yesmaximal, complemented
4#33885C214yesmaximal, complemented
5#33874C214yesmaximal, complemented
6#33873C214yesmaximal, complemented
7#33701C211yesmaximal, characteristic, complemented
8#33653C212yesmaximal, complemented
9#33627C211yesmaximal, characteristic, complemented
10#33612C212yesmaximal, complemented
11#33580C211yesmaximal, characteristic, complemented
12#33522C212yesmaximal, complemented
13C2×#2177C211yesmaximal, characteristic, complemented
14#33435C211yesmaximal, characteristic, complemented
15#33381C212yesmaximal, complemented
16C2×#576C2212yes—
17#3506—21no—
18#3496—21no—
19C2×#509C2212yescomplemented
20C22×D16C2211yescharacteristic
21C2×#504C2212yescomplemented
22#3355—22no—
23#3354—22no—
24#3341C2212yescomplemented
25#3339C2214yescomplemented
26#3334C2212yescomplemented
27#3283C2212yescomplemented
28#3280C2214yescomplemented
29D4×D8C2212yescomplemented
30#3273—22no—
31#3272—22no—
32#3160C2211yescharacteristic, complemented
33#3147C2211yescharacteristic
34#3096—22no—
35#3077—21no—
36#3061C2212yes—
37#3058C2212yescomplemented
38#3036—21no—
39#3012—22no—
40C2×C4×D8C2211yescharacteristic
41C2×C4×D8C2211yescharacteristic, complemented
42#2931C2211yescharacteristic
43#2913C2212yescomplemented
44#2245C2212yescomplemented
45#2244C2212yescomplemented
46#2243C2212yescomplemented
47#2243C2212yescomplemented
48#2237C2214yescomplemented
49#2235C2214yescomplemented
50#2234C2214yescomplemented
51#2233C2214yescomplemented
52#2228C2214yescomplemented
53#2226C2212yescomplemented
54#2224C2212yescomplemented
55#2220C2214yescomplemented
56#2218C2214yescomplemented
57#2217C2214yescomplemented
58#2215C2214yescomplemented
59#2210C2214yescomplemented
60#2208C2212yescomplemented
61#2206C2212yescomplemented
62#2186C2214yescomplemented
63#2186C2214yescomplemented
64#2184C2214yescomplemented
65#2184C2214yescomplemented
66#2177C2214yescomplemented
67#2177C2214yescomplemented
68#2168C2214yescomplemented
69#2168C2214yescomplemented
70#2159C2212yescomplemented
71C2×#367C2211yescharacteristic, complemented
72#2150C2211yescharacteristic, complemented
73#2140C2212yes—
74#2140C2214yescomplemented
75#2139C2212yescomplemented
76#2139C2214yescomplemented
77#2138C2212yescomplemented
78#2138C2214yescomplemented
79#2138C2212yescomplemented
80C2×#357C2211yescharacteristic
81C2×#357C2211yescharacteristic, complemented
82C2×#357C2214yescomplemented
83#2113C2212yescomplemented
84C2×#346C2211yescharacteristic, complemented
85C2×H(5,𝔽2)—22no—
86#576—28no—
87#576—44no—
88C2×AGL(1,ℤ/8)—22no—
89C2×AGL(1,ℤ/8)—24no—
90C2×#135—21no—
91C2×#135C2312yes—
92C2×#135C2314yes—
93C22×D8C2311yescharacteristic
94C22×D8C2314yes—
95C22×D8C2311yescharacteristic
96C2×#131C2312yes—
97#559—42no—
98#556—41no—
99#555—22no—
100#553—24no—
101#550—41no—
102#547—22no—
103#547—22no—
104#547—41no—
105#546—24no—
106#546—42no—
107D42—22no—
108D42—41no—
109#538—42no—
110#537—22no—
111#535—22no—
112#534—42no—
113#533—21no—
114#532—21no—
115#532D411yescharacteristic
116#525—42no—
117C2×#121—42no—
118#520—22no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121#509—44no—
122C2×SD32C2314yes—
123C2×D16—24no—
124C2×D16C2314yes—
125C2×D16C2314yes—
126#504—44no—
127C2×M32C2312yes—
128C2×M32C2314yes—
129C22×C16C2311yescharacteristic
130C22×C16C2311yescharacteristic
131C22×C16C2314yes—
132#500C2311yescharacteristic
133#500C2311yescharacteristic
134#499—21no—
135#499C2312yes—
136#498—21no—
137#495—21no—
138#495C2312yes—
139#494C2311yescharacteristic
140#493C2311yescharacteristic
141#492—21no—
142#482—24no—
143#482—24no—
144#481—24no—
145#480—24no—
146#478—24no—
147#476—22no—
148#474—22no—
149#469—24no—
150#468—24no—
151#467C2314yes—
152#466C2314yes—
153#466C2314yes—
154#466C2314yes—
155#463—24no—
156#461—22no—
157#459—22no—
158#452—24no—
159#450—24no—
160#449—24no—
161#447—24no—
162C4×D8—21no—
163C4×D8—21no—
164C4×D8—21no—
165C4×D8—21no—
166C4×D8C2312yes—
167C4×D8C2312yes—
168C4×D8C2312yes—
169C4×D8C2312yes—
170#436D412yes—
171#436D412yes—
172C8×D4C2312yes—
173C8×D4C2312yes—
174#428—22no—
175#428—24no—
176#427—21no—
177#427C2314yes—
178#427C2312yes—
179C2×#107C2314yes—
180C2×#107C2311yescharacteristic
181C2×#107C2311yescharacteristic
182#423D411yescharacteristic
183#418—22no—
184#417—22no—
185C2×#102—21no—
186C2×#102—21no—
187#408D412yes—
188C2×C4×C8C2311yescharacteristic
189#367C2314yes—
190#367C2314yes—
191#366C2314yes—
192C4⋊C16C2312yes—
193C4⋊C16C2312yes—
194#358C2312yes—
195#358C2314yes—
196#358C2312yes—
197#358C2312yes—
198#358C2312yes—
199#357C2314yes—
200#357C2312yes—
201#357C2314yes—
202#357C2314yes—
203#357C2314yes—
204#357C2314yes—
205#357C2312yes—
206#357C2312yes—
207#357C2312yes—
208C22⋊C16C2314yes—
209C22⋊C16C2314yes—
210#346C2312yes—
211#346C2312yes—
212H(5,𝔽2)—48no—
213H(5,𝔽2)—84no—
214C2×#41—22no—
215C2×#41—24no—
216C2×#41—42no—
217C2×#41—42no—
218C2×#41C2×D412yes—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4—24no—
222C22×D4—44no—
223C22×D4—42no—
224C22×D4—42no—
225C23×C4—41no—
226C23×C4—41no—
227AGL(1,ℤ/8)—416no—
228AGL(1,ℤ/8)—48no—
229#135—28no—
230#135—24no—
231#135—24no—
232#135—44no—
233#135—82no—
234C2×Q16—21no—
235C2×Q16C2412yes—
236C2×SD16—24no—
237C2×SD16—22no—
238C2×SD16—22no—
239C2×D8—28no—
240C2×D8—21no—
241C2×D8—24no—
242C2×D8—21no—
243C2×D8—21no—
244C2×D8—22no—
245C2×D8—22no—
246C2×D8—22no—
247C2×D8—44no—
248C2×D8#4114yescomplemented
249C2×D8C2412yes—
250C2×D8C2412yes—
251C2×D8C2412yes—
252#131—24no—
253#131—24no—
254C2×M16C2×D412yes—
255C2×M16C2×D414yes—
256C22×C8C2×D411yescharacteristic
257C22×C8C2414yes—
258C22×C8C2411yescharacteristic
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8D811yescharacteristic
263#127—21no—
264#127—42no—
265#127D811yescharacteristic
266#126—44no—
267#125—22no—
268#124—22no—
269#124—42no—
270#124—44no—
271#124D812yes—
272#123—44no—
273#123—44no—
274#123—44no—
275#123—44no—
276C22⋊Q8—24no—
277C22⋊Q8—42no—
278C22⋊Q8—41no—
279C22⋊Q8—41no—
280C22⋊Q8—42no—
281#121—24no—
282#121—24no—
283#121—24no—
284#121—44no—
285#121—42no—
286#121—42no—
287#121—42no—
288#121—44no—
289#121—41no—
290#121—41no—
291#121—42no—
292#121—44no—
293#121D812yes—
294#121D812yes—
295#120—44no—
296#120—44no—
297#120—44no—
298C4×Q8—42no—
299C4×Q8—41no—
300C4×Q8—41no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—44no—
306C4×D4—42no—
307C4×D4—44no—
308C4×D4—41no—
309C4×D4—41no—
310C4×D4—41no—
311C4×D4—41no—
312C4×D4—41no—
313C4×D4—41no—
314C4×D4C2×D412yes—
315C4×D4C2×D412yes—
316#117—22no—
317#117—24no—
318#117—42no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—42no—
323C2×C22⋊C4—41no—
324C2×C22⋊C4—41no—
325C2×C22⋊C4—44no—
326C2×C42C2×D411yescharacteristic
327SD32—44no—
328D16—28no—
329D16—44no—
330M32—24no—
331M32—44no—
332C2×C16—22no—
333C2×C16—22no—
334C2×C16—44no—
335C2×C16C2412yes—
336C2×C16C2412yes—
337C2×C16C2414yes—
338#107—21no—
339#107—24no—
340#107—21no—
341#107—21no—
342#107—21no—
343#107C2412yes—
344#107C2412yes—
345#107C2412yes—
346#107C2412yes—
347#106—22no—
348#106—22no—
349#106—24no—
350C4⋊C8C2×D412yes—
351C4⋊C8C2×D412yes—
352#103—24no—
353#102—22no—
354#102—22no—
355#102—22no—
356#102—22no—
357#102—24no—
358#102—24no—
359#102—24no—
360C22⋊C8C2×D414yes—
361C22⋊C8C2×D414yes—
362C4×C8C2×D411yescharacteristic
363C4×C8C2×D411yescharacteristic
364C4×C8C2411yescharacteristic
365C4×C8C2411yescharacteristic
366C24—41no—
367C24—41no—
368C24—44no—
369#41—24no—
370#41—24no—
371#41—48no—
372#41—44no—
373#41—44no—
374#41—84no—
375#41—88no—
376#41—164no—
377C2×Q8—22no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8C2×D812yes—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—44no—
396C2×D4—41no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—84no—
400C2×D4—88no—
401C2×D4—816no—
402C2×D4—164no—
403C2×D4C2×D814yes—
404C2×D4C2×D812yes—
405C2×D4C2×D812yes—
406C2×D4C2×D812yes—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—162no—
417C22×C4—162no—
418C22×C4C2×D811yescharacteristic
419C22×C4C2×D811yescharacteristic
420C22×C4C22×D414yes—
421C22×C4C22×D411yescharacteristic
422Q16—22no—
423Q16—22no—
424Q16—81no—
425SD16—48no—
426SD16—44no—
427SD16—82no—
428D8—24no—
429D8—22no—
430D8—22no—
431D8—22no—
432D8—22no—
433D8—22no—
434D8—48no—
435D8—42no—
436D8—81no—
437D8C2×#4114yes—
438M16—24no—
439M16—28no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8C22×D411yescharacteristic
444C2×C8C22×D411yescharacteristic
445C2×C8C22×D414yes—
446C2×C8C22×D411yescharacteristic
447C2×C8C2×#4112yes—
448C2×C8H(5,𝔽2)12yes—
449C2×C8C2511yescharacteristic
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—44no—
457C4⋊C4—41no—
458C4⋊C4—44no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4—41no—
464C4⋊C4C2×D812yes—
465C4⋊C4C2×D812yes—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—44no—
478C22⋊C4C2×D814yes—
479C22⋊C4C2×D814yes—
480C42—44no—
481C42—42no—
482C42C2×D811yescharacteristic
483C42C2×D811yescharacteristic
484C42C22×D411yescharacteristic
485C42C22×D411yescharacteristic
486C16—22no—
487C16—22no—
488C16—24no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—88no—
494C23—82no—
495C23—161no—
496C23—161no—
497C23—161no—
498C23—161no—
499C23—164no—
500C23—164no—
501C23C22×D814yes—
502C23C22×D811yescharacteristic
503Q8—24no—
504Q8—42no—
505Q8—42no—
506Q8—82no—
507Q8—161no—
508D4—28no—
509D4—24no—
510D4—24no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—44no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—88no—
520D4—82no—
521D4—88no—
522D4—84no—
523D4—82no—
524D4—161no—
525D4—164no—
526D4—164no—
527D4—164no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—88no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—162no—
542C2×C4—162no—
543C2×C4—161no—
544C2×C4—161no—
545C2×C4—164no—
546C2×C4#54612yes—
547C2×C4C22×D814yes—
548C2×C4C22×D811yescharacteristic
549C2×C4C22×D811yescharacteristic
550C2×C4C22×D811yescharacteristic
551C2×C4C22×D811yescharacteristic
552C2×C4C22×D811yescharacteristic
553C2×C4#57712yes—
554C2×C4C23×D411yescharacteristic
555C8—24no—
556C8—21no—
557C8—21no—
558C8#54612yes—
559C8C22×#4111yescharacteristic
560C8C2×H(5,𝔽2)11yescharacteristic
561C22—21no—
562C22—21no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—44no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—164no—
576C22—164no—
577C22—328no—
578C22#329412yes—
579C22C23×D811yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—24no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4#329412yes—
591C4C2×#54611yescharacteristic
592C4C2×#57711yescharacteristic
593C2—22no—
594C2—44no—
595C2—84no—
596C2—162no—
597C2—162no—
598C2C2×#329411yescharacteristic
599C2#6263611yescharacteristic
600C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #532 ◁ #2931 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #61284C2semidirect
2. #61084C2semidirect
3. #33886C2semidirect
4. #33885C2semidirect
5. #33874C2semidirect
6. #33873C2semidirect
7. #33701C2semidirect
8. #33653C2semidirect
9. #33627C2semidirect
10. #33612C2semidirect
11. #33580C2semidirect
12. #33522C2semidirect
13. C2×#2177C2semidirect
14. #33435C2semidirect
15. #33381C2semidirect
19. C2×#509C22semidirect
21. C2×#504C22semidirect
24. #3341C22semidirect
25. #3339C22semidirect
26. #3334C22semidirect
27. #3283C22semidirect
28. #3280C22semidirect
29. D4×D8C22semidirect
32. #3160C22semidirect
37. #3058C22semidirect
41. C2×C4×D8C22semidirect
43. #2913C22semidirect
44. #2245C22semidirect
45. #2244C22semidirect
46. #2243C22semidirect
47. #2243C22semidirect
48. #2237C22semidirect
49. #2235C22semidirect
50. #2234C22semidirect
51. #2233C22semidirect
52. #2228C22semidirect
53. #2226C22semidirect
54. #2224C22semidirect
55. #2220C22semidirect
56. #2218C22semidirect
57. #2217C22semidirect
58. #2215C22semidirect
59. #2210C22semidirect
60. #2208C22semidirect
61. #2206C22semidirect
62. #2186C22semidirect
63. #2186C22semidirect
64. #2184C22semidirect
65. #2184C22semidirect
66. #2177C22semidirect
67. #2177C22semidirect
68. #2168C22semidirect
69. #2168C22semidirect
70. #2159C22semidirect
71. C2×#367C22semidirect
72. #2150C22semidirect
74. #2140C22semidirect
75. #2139C22semidirect
76. #2139C22semidirect
77. #2138C22semidirect
78. #2138C22semidirect
79. #2138C22semidirect
81. C2×#357C22semidirect
82. C2×#357C22semidirect
83. #2113C22semidirect
84. C2×#346C22semidirect
248. C2×D8#41semidirect

Maximal subgroups

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

subgroupindex
#612842
#610842
#338862
#338852
#338742
#338732
#337012
#336532
#336272
#336122
#335802
#335222
C2×#21772
#334352
#333812