Finite Group Databasebeta

← #6470763

Subgroups — #6470763

Order 512.

2220 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3200C212yesmaximal, complemented
2#37297C211yesmaximal, characteristic
3#34266C211yesmaximal, characteristic, complemented
4#33116C212yesmaximal, complemented
5#32903C212yesmaximal, complemented
6#29758C212yesmaximal, complemented
7#29722C212yesmaximal, complemented
8#25222C214yesmaximal, complemented
9#25203C214yesmaximal, complemented
10#25093C214yesmaximal, complemented
11#25080C214yesmaximal, complemented
12#21937C211yesmaximal, characteristic, complemented
13#20914C212yesmaximal, complemented
14C2×#536C2211yescharacteristic
15C2×#535C2212yes—
16C2×#520C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic, complemented
18#3200—216no—
19C2×#485C2214yescomplemented
20C2×#482C2214yescomplemented
21C2×#468C2214yes—
22C2×#465C2214yes—
23C2×#460C2214yes—
24C2×#448C2214yescomplemented
25C2×#450C2214yescomplemented
26C2×#418C2212yescomplemented
27C22×#102C2212yescomplemented
28C2×#408C2212yes—
29#2856C2212yes—
30#2733—22no—
31#2679—22no—
32#2669—22no—
33#2669—22no—
34#2619C2212yes—
35#2500C2214yes—
36#2433C2214yes—
37#2367—22no—
38#2365—22no—
39#2362C2214yescomplemented
40#2360C2214yescomplemented
41#2322—22no—
42#2310C2211yescharacteristic, complemented
43C4×#123—22no—
44#2297C2211yescharacteristic
45#2296C2211yescharacteristic, complemented
46#2274C2211yescharacteristic, complemented
47#2084C2212yescomplemented
48#2083C2212yescomplemented
49#2074C2212yescomplemented
50#2074C2212yescomplemented
51#2067C2214yescomplemented
52#2065C2214yescomplemented
53#2055C2214yescomplemented
54#2055C2214yescomplemented
55#2041C2212yes—
56#2041C2212yes—
57#2014C2212yes—
58#2014C2212yes—
59#1982C2214yes—
60#1949C2214yescomplemented
61#1926C2214yescomplemented
62#1925C2214yescomplemented
63#1880C2214yescomplemented
64#1880C2214yescomplemented
65#1875C2214yes—
66#1875C2214yescomplemented
67#1874C2214yes—
68#1874C2214yescomplemented
69#1851C2214yescomplemented
70#1850C2214yescomplemented
71#1803C2214yescomplemented
72#1798C2212yescomplemented
73#1797C2212yescomplemented
74#1736C2212yescomplemented
75#1735C2212yescomplemented
76C22×#41C2311yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2312yes—
79C22×SD16C2314yes—
80C22×SD16C2314yes—
81C22×M16C2312yes—
82C23×C8C2312yes—
83#536—24no—
84#536D418yescomplemented
85#535—28no—
86#535—28no—
87C2×C4⋊Q8—21no—
88C2×#127—22no—
89C2×#127C2312yes—
90C2×#127C2312yes—
91C2×#124—21no—
92C2×#124—22no—
93C2×#123—22no—
94C2×C22⋊Q8C2314yes—
95C2×#121—22no—
96C2×#121—22no—
97C2×#121C2314yes—
98C2×#121C2314yes—
99C2×#120—22no—
100C2×#120—22no—
101C2×#120—24no—
102C2×#120—24no—
103#520—44no—
104C2×C4×Q8—21no—
105C2×C4×Q8—22no—
106C2×C4×Q8—44no—
107C2×C4×Q8C2314yes—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120#485—216no—
121#482—216no—
122#468—216no—
123#465—216no—
124#460—48no—
125#450—216no—
126#448—216no—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131#418—44no—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102—44no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141#408—28no—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144#394—24no—
145#394—24no—
146#394—24no—
147#392—24no—
148#392—24no—
149#392—24no—
150#391—22no—
151#391D412yes—
152#391C2312yes—
153#391C2312yes—
154#390—22no—
155#390—22no—
156#390D412yes—
157#390C2314yes—
158#390C2314yes—
159#389—24no—
160#389C2312yes—
161#389C2312yes—
162#388—24no—
163#388—24no—
164#388—24no—
165#388—24no—
166#387—24no—
167#387—24no—
168#386D412yes—
169#386D412yes—
170#385—21no—
171#385—21no—
172#385—24no—
173#384C2312yes—
174#382C2312yes—
175#381—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—24no—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—24no—
182C4×C22⋊C4C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2312yes—
188#336C2312yes—
189#336C2312yes—
190#336C2312yes—
191C25—22no—
192C2×#41—42no—
193C2×#41—42no—
194C2×#41C2×D414yes—
195C2×#41#4114yes—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—42no—
199C22×Q8—42no—
200C22×Q8C2×D414yes—
201C22×Q8C2×D411yescharacteristic
202C22×Q8C2412yes—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—42no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4C2×D411yescharacteristic
216C22×D4C2×D412yes—
217C22×D4C2414yes—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—22no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C2×SD16—28no—
226C2×SD16—28no—
227C2×SD16—44no—
228C2×SD16—44no—
229C2×SD16—44no—
230C2×SD16—44no—
231C2×M16—24no—
232C2×M16—42no—
233C2×M16—42no—
234C22×C8—24no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8C2414yes—
238C22×C8C2414yes—
239C4⋊Q8—28no—
240#127—44no—
241#127—44no—
242#127—48no—
243#124—28no—
244#124—48no—
245#123—216no—
246C22⋊Q8—28no—
247C22⋊Q8C2×D4116yes—
248#121—28no—
249#121—28no—
250#121—28no—
251#121—28no—
252#121—84no—
253#121—84no—
254#120—48no—
255#120—48no—
256#120—88no—
257#120—88no—
258C4×Q8—28no—
259C4×Q8—44no—
260C4×Q8—48no—
261C4×Q8—44no—
262C4×D4—44no—
263C4×D4—48no—
264#117—22no—
265#117—24no—
266#117—24no—
267#117—24no—
268#117—42no—
269#117—42no—
270#117—42no—
271#117C2×D414yes—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2411yescharacteristic
286C2×C4⋊C4C2411yescharacteristic
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D414yes—
304C2×C42—21no—
305C2×C42—22no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42C2×D412yes—
313C2×C42C2412yes—
314C2×C42C2412yes—
315#107—28no—
316#106—28no—
317C4⋊C8—44no—
318C4⋊C8—44no—
319#103—28no—
320#103—28no—
321#102—28no—
322#102—28no—
323#102—28no—
324#102—28no—
325#102—44no—
326#102—44no—
327C22⋊C8—28no—
328C22⋊C8—28no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—24no—
335#95C2×D414yes—
336#95C2×D414yes—
337C24—24no—
338C24—44no—
339C24—82no—
340C24—82no—
341C24—84no—
342C24—84no—
343C24C22×D412yes—
344C24C22×D411yescharacteristic
345#41—48no—
346#41—48no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—48no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—48no—
357C2×Q8#12114yes—
358C2×Q8C22×D414yes—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—42no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4#12114yes—
392C2×D4#12118yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—82no—
415C22×C4—84no—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4114yes—
425C22×C4C2511yescharacteristic
426SD16—88no—
427SD16—88no—
428M16—84no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—84no—
436C4⋊C4—24no—
437C4⋊C4—216no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4C22×D414yes—
452C4⋊C4H(5,𝔽2)14yes—
453C22⋊C4—24no—
454C22⋊C4—216no—
455C22⋊C4—28no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—48no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—42no—
471C42—48no—
472C42—44no—
473C42—42no—
474C42—44no—
475C42—84no—
476C23—24no—
477C23—22no—
478C23—21no—
479C23—22no—
480C23—21no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—88no—
492C23—168no—
493C23—168no—
494C23—164no—
495C23—164no—
496C23C2×#12114yes—
497C23C2×#12112yes—
498C23C23×D411yescharacteristic
499Q8—28no—
500Q8—44no—
501Q8—48no—
502Q8—44no—
503D4—48no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—88no—
510D4—164no—
511D4—164no—
512D4—168no—
513D4—168no—
514D4—168no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—44no—
537C2×C4—48no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4—82no—
552C2×C4—164no—
553C2×C4C2×#12112yes—
554C2×C4C2×#12114yes—
555C2×C4#57714yes—
556C2×C4C23×D411yescharacteristic
557C2×C4C22×#4111yescharacteristic
558C2×C4C2×H(5,𝔽2)12yes—
559C2×C4#58512yes—
560C8—44no—
561C8—44no—
562C22—24no—
563C22—28no—
564C22—22no—
565C22—24no—
566C22—42no—
567C22—44no—
568C22—41no—
569C22—42no—
570C22—41no—
571C22—42no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—1616no—
580C22C22×#12111yescharacteristic
581C22#348411yescharacteristic
582C22#348511yescharacteristic
583C22C2×#57712yes—
584C22#358512yes—
585C4—24no—
586C4—22no—
587C4—24no—
588C4—44no—
589C4—44no—
590C4—44no—
591C4—42no—
592C4—44no—
593C4—84no—
594C4—81no—
595C4—81no—
596C4#345912yes—
597C4#349612yes—
598C2—42no—
599C2—44no—
600C2—164no—
601C2#3498411yescharacteristic
602C2#6043112yes—
603C2#6163311yescharacteristic
604C2#6164811yescharacteristic
605C2#6180512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37297C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3200C2semidirect
3. #34266C2semidirect
4. #33116C2semidirect
5. #32903C2semidirect
6. #29758C2semidirect
7. #29722C2semidirect
8. #25222C2semidirect
9. #25203C2semidirect
10. #25093C2semidirect
11. #25080C2semidirect
12. #21937C2semidirect
13. #20914C2semidirect
17. C22×C4×Q8C22semidirect
19. C2×#485C22semidirect
20. C2×#482C22semidirect
24. C2×#448C22semidirect
25. C2×#450C22semidirect
26. C2×#418C22semidirect
27. C22×#102C22semidirect
39. #2362C22semidirect
40. #2360C22semidirect
42. #2310C22semidirect
45. #2296C22semidirect
46. #2274C22semidirect
47. #2084C22semidirect
48. #2083C22semidirect
49. #2074C22semidirect
50. #2074C22semidirect
51. #2067C22semidirect
52. #2065C22semidirect
53. #2055C22semidirect
54. #2055C22semidirect
60. #1949C22semidirect
61. #1926C22semidirect
62. #1925C22semidirect
63. #1880C22semidirect
64. #1880C22semidirect
66. #1875C22semidirect
68. #1874C22semidirect
69. #1851C22semidirect
70. #1850C22semidirect
71. #1803C22semidirect
72. #1798C22semidirect
73. #1797C22semidirect
74. #1736C22semidirect
75. #1735C22semidirect
84. #536D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32002
#372972
#342662
#331162
#329032
#297582
#297222
#252222
#252032
#250932
#250802
#219372
#209142