Finite Group Databasebeta

← #7624343

Subgroups — #7624343

Order 512.

1299 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62328C211yesmaximal, characteristic, complemented
2#61284C211yesmaximal, characteristic, complemented
3#33921C212yesmaximal, complemented
4#33914C214yesmaximal, complemented
5#33908C212yesmaximal, complemented
6#33892C212yesmaximal, complemented
7#33885C214yesmaximal, complemented
8#33861C212yesmaximal, complemented
9#33801C211yesmaximal, characteristic, complemented
10#33794C211yesmaximal, characteristic, complemented
11#33765C214yesmaximal, complemented
12#33669C211yesmaximal, characteristic, complemented
13#33660C211yesmaximal, characteristic, complemented
14#33650C214yesmaximal, complemented
15#33495C211yesmaximal, characteristic, complemented
16#3549—21no—
17#3506—21no—
18#3355—22no—
19#3354—22no—
20#3341C2212yescomplemented
21#3341C2212yescomplemented
22#3341C2212yescomplemented
23#3339C2214yescomplemented
24#3334C2212yescomplemented
25#3325—22no—
26#3325—22no—
27D4×D8C2212yescomplemented
28D4×D8C2212yescomplemented
29#3253—21no—
30#3199—22no—
31#3169C2211yescharacteristic
32#3169C2211yescharacteristic
33#3160C2211yescharacteristic, complemented
34#3096—22no—
35#3077—21no—
36#3058C2214yes—
37#3058C2212yescomplemented
38C2×C4×D8C2211yescharacteristic, complemented
39#2931C2211yescharacteristic
40#2913C2212yescomplemented
41#2255C2212yescomplemented
42#2254C2211yescharacteristic, complemented
43#2252C2211yescharacteristic, complemented
44#2244C2212yescomplemented
45#2240C2211yescharacteristic, complemented
46#2240C2211yescharacteristic, complemented
47#2234C2214yescomplemented
48#2234C2214yescomplemented
49#2234C2214yescomplemented
50#2234C2214yescomplemented
51#2234C2214yescomplemented
52#2234C2214yescomplemented
53#2233C2214yescomplemented
54#2233C2212yescomplemented
55#2232C2214yescomplemented
56#2232C2212yescomplemented
57#2228C2214yescomplemented
58#2226C2212yescomplemented
59#2226C2212yescomplemented
60#2224C2212yescomplemented
61#2224C2212yescomplemented
62#2218C2214yescomplemented
63#2218C2214yescomplemented
64#2215C2214yescomplemented
65#2215C2214yescomplemented
66#2206C2212yes—
67#2206C2212yes—
68#2184C2214yescomplemented
69#2184C2214yescomplemented
70#2184C2212yescomplemented
71#2184C2212yescomplemented
72#2177C2214yescomplemented
73C4×D16C2212yescomplemented
74C4×D16C2212yescomplemented
75#2169C2212yescomplemented
76#2168C2214yescomplemented
77D4×C16C2212yescomplemented
78#2163C2211yescharacteristic, complemented
79#2162C2211yescharacteristic, complemented
80#2159C2214yescomplemented
81#2158C2214yescomplemented
82#2140C2214yescomplemented
83C2×#357C2214yescomplemented
84#2113C2214yescomplemented
85C22×D8C2314yes—
86C22×D8C2311yescharacteristic
87C2×#131C2312yes—
88H(3,𝔽4)—42no—
89#559—42no—
90#556—41no—
91#555—22no—
92#555—22no—
93#555—22no—
94#555—41no—
95#553—24no—
96#550—41no—
97#547—22no—
98D42—22no—
99D42—22no—
100D42—41no—
101#538—42no—
102#535—22no—
103#535—24no—
104#535—42no—
105#535—42no—
106#533—21no—
107#532—21no—
108#532—21no—
109#532D411yescharacteristic
110#525—42no—
111#520—22no—
112C2×C4×D4—21no—
113C2×D16C2312yes—
114C2×D16C2312yes—
115C2×D16C2314yes—
116C2×M32C2314yes—
117C22×C16C2314yes—
118#500C2311yescharacteristic
119#500C2311yescharacteristic
120#500C2311yescharacteristic
121#500C2311yescharacteristic
122#499—21no—
123#499C2312yes—
124#498—21no—
125#493C2311yescharacteristic
126#493C2311yescharacteristic
127#486—21no—
128#486—21no—
129#482—24no—
130#482—24no—
131#482—24no—
132#482—24no—
133#481—24no—
134#480—24no—
135#478—24no—
136#476—22no—
137#476—22no—
138#476—22no—
139#474—22no—
140#469—24no—
141#469—24no—
142#469—24no—
143#466—21no—
144#466—21no—
145#466C2314yes—
146#466C2314yes—
147#466C2314yes—
148#466C2314yes—
149#466C2312yes—
150#466C2312yes—
151#459—22no—
152#459—22no—
153#447—24no—
154#447—24no—
155C4×D8—21no—
156C4×D8—21no—
157C4×D8—21no—
158C4×D8—21no—
159C4×D8C2312yes—
160C4×D8C2312yes—
161C4×D8C2312yes—
162C4×D8C2312yes—
163#436D412yes—
164#436D412yes—
165C8×D4C2312yes—
166C8×D4C2312yes—
167#428—22no—
168#428—24no—
169#427—21no—
170#427C2314yes—
171#427C2314yes—
172#427C2312yes—
173C2×#107C2314yes—
174C2×#107C2311yescharacteristic
175#423D411yescharacteristic
176#418—22no—
177#418—24no—
178C2×#102—21no—
179#408D412yes—
180C2×C4×C8C2311yescharacteristic
181#367C2314yes—
182#367C2314yes—
183#367C2314yes—
184#366C2312yes—
185#366C2314yes—
186#366C2312yes—
187#366C2314yes—
188C4⋊C16C2312yes—
189C4⋊C16C2312yes—
190#357C2312yes—
191#357C2314yes—
192#357C2314yes—
193#357C2314yes—
194#357C2312yes—
195#357C2314yes—
196#357C2314yes—
197#357C2312yes—
198#357C2312yes—
199#357C2314yes—
200C22⋊C16C2312yes—
201C22⋊C16C2312yes—
202C22⋊C16C2314yes—
203C22⋊C16C2314yes—
204#346C2312yes—
205C4×C16C2311yescharacteristic
206C4×C16C2311yescharacteristic
207C2×#41—42no—
208C2×#41—41no—
209C2×#41—41no—
210C2×#41C2×D412yes—
211C22×D4—21no—
212C22×D4—24no—
213C22×D4—42no—
214C22×D4—42no—
215C23×C4—41no—
216C2×D8—21no—
217C2×D8—24no—
218C2×D8—21no—
219C2×D8—21no—
220C2×D8—42no—
221C2×D8—44no—
222C2×D8—44no—
223C2×D8C2412yes—
224C2×D8C2412yes—
225C2×D8C2412yes—
226#131—44no—
227C2×M16C2×D412yes—
228C2×M16C2×D414yes—
229C22×C8C2×D411yescharacteristic
230C22×C8C2411yescharacteristic
231C22×C8C2414yes—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8D811yescharacteristic
237#127—21no—
238#127—21no—
239#127—44no—
240#127—42no—
241#127D811yescharacteristic
242#126—44no—
243#125—22no—
244#124—44no—
245#124—42no—
246#124—44no—
247#124—44no—
248#124—44no—
249#124—41no—
250#124—41no—
251#124D812yes—
252#123—44no—
253#123—44no—
254C22⋊Q8—41no—
255C22⋊Q8—41no—
256C22⋊Q8—42no—
257#121—22no—
258#121—22no—
259#121—24no—
260#121—24no—
261#121—24no—
262#121—24no—
263#121—42no—
264#121—42no—
265#121—42no—
266#121—44no—
267#121—41no—
268#121—41no—
269#121—42no—
270#121—42no—
271#121D812yes—
272#121D812yes—
273#120—44no—
274#120—44no—
275#120—44no—
276#120—44no—
277#120—44no—
278C4×Q8—42no—
279C4×Q8—41no—
280C4×Q8—41no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—44no—
286C4×D4—42no—
287C4×D4—44no—
288C4×D4—41no—
289C4×D4—41no—
290C4×D4—41no—
291C4×D4—41no—
292C4×D4C2×D412yes—
293C4×D4C2×D412yes—
294#117—22no—
295#117—24no—
296#117—24no—
297#117—42no—
298#117—42no—
299#117—42no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C22⋊C4—41no—
303C2×C42C2×D411yescharacteristic
304D16—42no—
305D16—44no—
306D16—42no—
307M32—44no—
308C2×C16—44no—
309C2×C16C2412yes—
310C2×C16C2412yes—
311C2×C16C2414yes—
312#107—21no—
313#107—24no—
314#107—21no—
315#107—21no—
316#107—21no—
317#107—21no—
318#107C2412yes—
319#107C2412yes—
320#107C2412yes—
321#107C2412yes—
322#107C2412yes—
323#106—22no—
324#106—22no—
325#106—24no—
326#106—24no—
327C4⋊C8C2×D412yes—
328C4⋊C8C2×D412yes—
329#102—22no—
330#102—22no—
331#102—22no—
332#102—22no—
333#102—24no—
334#102—24no—
335#102—24no—
336#102—24no—
337#102—22no—
338#102—22no—
339C22⋊C8C2×D414yes—
340C22⋊C8C2×D414yes—
341C4×C8C2×D411yescharacteristic
342C4×C8C2×D411yescharacteristic
343C4×C8C2411yescharacteristic
344C4×C8C2411yescharacteristic
345C24—41no—
346C24—44no—
347#41—42no—
348#41—42no—
349#41—162no—
350#41—162no—
351#41—164no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8C2×D812yes—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—44no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—164no—
377C2×D4—164no—
378C2×D4C2×D812yes—
379C2×D4C2×D814yes—
380C2×D4C2×D812yes—
381C2×D4C2×D812yes—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—162no—
389C22×C4C2×D811yescharacteristic
390C22×C4C2×D811yescharacteristic
391C22×C4C22×D414yes—
392C22×C4C22×D411yescharacteristic
393D8—22no—
394D8—22no—
395D8—22no—
396D8—24no—
397D8—22no—
398D8—42no—
399D8—81no—
400D8—81no—
401D8—84no—
402M16—44no—
403M16—42no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—22no—
407C2×C8—41no—
408C2×C8C22×D411yescharacteristic
409C2×C8C22×D411yescharacteristic
410C2×C8C22×D411yescharacteristic
411C2×C8C22×D414yes—
412C2×C8C2511yescharacteristic
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—41no—
419C4⋊C4—44no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—41no—
426C4⋊C4C2×D812yes—
427C4⋊C4C2×D812yes—
428C22⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—44no—
441C22⋊C4C2×D814yes—
442C22⋊C4C2×D814yes—
443C42—44no—
444C42—44no—
445C42—42no—
446C42C2×D811yescharacteristic
447C42C2×D811yescharacteristic
448C42C22×D411yescharacteristic
449C42C22×D411yescharacteristic
450C16—22no—
451C16—24no—
452C16—22no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—82no—
457C23—161no—
458C23—161no—
459C23—164no—
460C23—164no—
461C23—164no—
462C23C22×D814yes—
463C23C22×D811yescharacteristic
464Q8—42no—
465Q8—82no—
466Q8—161no—
467Q8—161no—
468Q8—161no—
469D4—24no—
470D4—42no—
471D4—42no—
472D4—44no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—82no—
480D4—82no—
481D4—82no—
482D4—164no—
483D4—161no—
484D4—161no—
485D4—164no—
486D4—164no—
487D4—164no—
488D4—162no—
489D4—164no—
490D4—162no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—162no—
506C2×C4—161no—
507C2×C4—164no—
508C2×C4C22×D811yescharacteristic
509C2×C4C22×D814yes—
510C2×C4C22×D811yescharacteristic
511C2×C4C22×D811yescharacteristic
512C2×C4C22×D811yescharacteristic
513C2×C4C22×D811yescharacteristic
514C2×C4C23×D411yescharacteristic
515C8—21no—
516C8—21no—
517C8—21no—
518C8—24no—
519C8#58512yes—
520C22—21no—
521C22—24no—
522C22—21no—
523C22—24no—
524C22—21no—
525C22—24no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—164no—
536C22—328no—
537C22C23×D811yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—24no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4#341512yes—
549C4#348511yescharacteristic
550C4#358611yescharacteristic
551C2—41no—
552C2—44no—
553C2—162no—
554C2—162no—
555C2—162no—
556C2#6169411yescharacteristic
557C2C2×#341511yescharacteristic
558C2#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. #62328C2semidirect
2. #61284C2semidirect
3. #33921C2semidirect
4. #33914C2semidirect
5. #33908C2semidirect
6. #33892C2semidirect
7. #33885C2semidirect
8. #33861C2semidirect
9. #33801C2semidirect
10. #33794C2semidirect
11. #33765C2semidirect
12. #33669C2semidirect
13. #33660C2semidirect
14. #33650C2semidirect
15. #33495C2semidirect
20. #3341C22semidirect
21. #3341C22semidirect
22. #3341C22semidirect
23. #3339C22semidirect
24. #3334C22semidirect
27. D4×D8C22semidirect
28. D4×D8C22semidirect
33. #3160C22semidirect
37. #3058C22semidirect
38. C2×C4×D8C22semidirect
40. #2913C22semidirect
41. #2255C22semidirect
42. #2254C22semidirect
43. #2252C22semidirect
44. #2244C22semidirect
45. #2240C22semidirect
46. #2240C22semidirect
47. #2234C22semidirect
48. #2234C22semidirect
49. #2234C22semidirect
50. #2234C22semidirect
51. #2234C22semidirect
52. #2234C22semidirect
53. #2233C22semidirect
54. #2233C22semidirect
55. #2232C22semidirect
56. #2232C22semidirect
57. #2228C22semidirect
58. #2226C22semidirect
59. #2226C22semidirect
60. #2224C22semidirect
61. #2224C22semidirect
62. #2218C22semidirect
63. #2218C22semidirect
64. #2215C22semidirect
65. #2215C22semidirect
68. #2184C22semidirect
69. #2184C22semidirect
70. #2184C22semidirect
71. #2184C22semidirect
72. #2177C22semidirect
73. C4×D16C22semidirect
74. C4×D16C22semidirect
75. #2169C22semidirect
76. #2168C22semidirect
77. D4×C16C22semidirect
78. #2163C22semidirect
79. #2162C22semidirect
80. #2159C22semidirect
81. #2158C22semidirect
82. #2140C22semidirect
83. C2×#357C22semidirect
84. #2113C22semidirect

Maximal subgroups

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

subgroupindex
#623282
#612842
#339212
#339142
#339082
#338922
#338852
#338612
#338012
#337942
#337652
#336692
#336602
#336502
#334952