Finite Group Databasebeta

← #6488642

Subgroups — #6488642

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35314C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#33018C214yesmaximal, complemented
4#30077C212yesmaximal, complemented
5#30039C212yesmaximal, complemented
6#29498C212yesmaximal, complemented
7#29492C212yesmaximal, complemented
8#29263C212yesmaximal, complemented
9#29116C212yesmaximal, complemented
10#28320C212yesmaximal, complemented
11#28169C212yesmaximal, complemented
12#27652C212yesmaximal, complemented
13#24211C212yesmaximal
14#22160C214yesmaximal, complemented
15#20785C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C2×#518C2212yes—
18C2×#485C2214yes—
19C2×#460C2214yes—
20C2×#450C2214yes—
21#2864C2211yescharacteristic, complemented
22#2863C2212yescomplemented
23#2862C2211yescharacteristic, complemented
24#2628C2211yescharacteristic
25#2625C2212yes—
26#2621C2211yescharacteristic
27#2524—22no—
28#2523—22no—
29#2520—22no—
30#2517—22no—
31#2366—22no—
32#2366—22no—
33D4×C22⋊C4—22no—
34D4×C22⋊C4—22no—
35#2331C2211yescharacteristic, complemented
36#2328C2212yes—
37#2328C2212yes—
38#2327C2211yescharacteristic, complemented
39#2320C2212yescomplemented
40#2309C2212yes—
41C4×#127C2212yes—
42C4×#127C2212yes—
43C4×C22⋊Q8C2212yes—
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2092C2214yescomplemented
47#2084C2212yescomplemented
48#2083C2212yescomplemented
49#2073C2214yescomplemented
50#2072C2212yescomplemented
51#2072C2212yescomplemented
52#2041C2212yes—
53#2041C2212yes—
54#2023C2214yescomplemented
55#2014C2212yes—
56#2014C2212yes—
57#2013C2214yescomplemented
58#1982C2214yes—
59#1973C2214yescomplemented
60#1968C2214yescomplemented
61#1949C2214yescomplemented
62#1936C2214yescomplemented
63#1934C2214yescomplemented
64#1877C2214yescomplemented
65#1877C2214yescomplemented
66#1877C2214yescomplemented
67#1877C2214yescomplemented
68#1864C2214yescomplemented
69#1864C2214yescomplemented
70#1862C2214yescomplemented
71#1862C2214yescomplemented
72#1839C2212yescomplemented
73#1832C2212yescomplemented
74#1825C2214yes—
75#1818C2212yes—
76#1805C2214yescomplemented
77#1797C2212yes—
78#1786C2214yes—
79#1768C2212yescomplemented
80#1750C2212yes—
81#1745C2211yescharacteristic, complemented
82#1744C2211yescharacteristic, complemented
83C23×D4—22no—
84C23×D4C2312yes—
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×#127—28no—
90C2×#127—22no—
91C2×#127C2312yes—
92C2×#127C2312yes—
93C2×C22⋊Q8C2312yes—
94#518—48no—
95C2×C4×Q8C2312yes—
96C2×C4×D4—24no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4C2314yes—
102C2×C4×D4C2314yes—
103C2×#117C2312yes—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C22×C22⋊C4—24no—
107C22×C42C2311yescharacteristic
108#485—48no—
109#460—216no—
110#450—48no—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2314yes—
123C2×#97C2312yes—
124C2×C4×C8C2312yes—
125#400—22no—
126#400—21no—
127#399—21no—
128#398—21no—
129#398—21no—
130#398—22no—
131#397—21no—
132#397—21no—
133#397—22no—
134#397—22no—
135#395—21no—
136#393—21no—
137#391—22no—
138#391—22no—
139#391C2312yes—
140#391C2312yes—
141#391C2312yes—
142#391C2312yes—
143#390—22no—
144#390—22no—
145#390—22no—
146#390—22no—
147#390—22no—
148#390—22no—
149#390C2314yes—
150#390C2314yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#386—24no—
167#386—24no—
168#386—24no—
169#386—24no—
170#385—22no—
171#384C2312yes—
172#383—22no—
173#383—22no—
174#382—22no—
175#382—22no—
176#382C2312yes—
177#382C2311yescharacteristic
178#382C2311yescharacteristic
179#380—22no—
180#380C2312yes—
181#379—22no—
182#379—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4D412yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4D412yes—
190C4×C22⋊C4D412yes—
191#376C2311yescharacteristic
192#376C2311yescharacteristic
193C43C2311yescharacteristic
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204#336C2314yes—
205C25—22no—
206C22×Q8—21no—
207C22×Q8C2412yes—
208C22×D4—216no—
209C22×D4—28no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—28no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—84no—
218C22×D4—84no—
219C22×D4C2414yes—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4C2×D412yes—
223C23×C4C2411yescharacteristic
224C2×SD16—28no—
225C2×SD16—28no—
226C2×SD16—44no—
227C2×SD16—44no—
228C2×SD16—44no—
229C2×SD16—44no—
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—42no—
233C4⋊Q8—42no—
234#127—216no—
235#127—28no—
236#127—24no—
237#127#4118yes—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C4×Q8—28no—
241C4×D4—44no—
242C4×D4—44no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4—416no—
246C4×D4—84no—
247C4×D4—84no—
248C4×D4—84no—
249C4×D4—84no—
250#117—24no—
251#117—28no—
252#117—42no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—416no—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299#107—44no—
300#106—44no—
301C4⋊C8—28no—
302#103—44no—
303#103—44no—
304#102—28no—
305#102—28no—
306#102—44no—
307#102—44no—
308#102—44no—
309#102—44no—
310C22⋊C8—28no—
311#97—42no—
312C4×C8—42no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95C2×D412yes—
333#95C2×D412yes—
334#95C2×D412yes—
335#95C2×D412yes—
336C24—28no—
337C24—24no—
338C24—44no—
339C24—44no—
340C24—84no—
341C24—84no—
342C24C22×D411yescharacteristic
343C2×Q8—24no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—24no—
347C2×Q8—42no—
348C2×Q8—82no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—216no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—832no—
369C2×D4C2×#4118yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—42no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4H(5,𝔽2)14yes—
396C22×C4C2511yescharacteristic
397SD16—88no—
398SD16—88no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C22⋊C4—24no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C22⋊C4—82no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—42no—
446C42—42no—
447C42—44no—
448C42—44no—
449C42—82no—
450C42C2×#4114yes—
451C23—22no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—28no—
456C23—416no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—816no—
464C23—168no—
465C23—164no—
466C23—164no—
467C23#53512yes—
468C23C23×D411yescharacteristic
469Q8—44no—
470Q8—44no—
471Q8—162no—
472D4—28no—
473D4—28no—
474D4—416no—
475D4—416no—
476D4—416no—
477D4—44no—
478D4—44no—
479D4—816no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—168no—
518C2×C4—168no—
519C2×C4#52514yes—
520C2×C4#53512yes—
521C2×C4#57714yes—
522C2×C4C22×#4112yes—
523C2×C4C2×H(5,𝔽2)12yes—
524C2×C4#58512yes—
525C8—44no—
526C8—44no—
527C22—22no—
528C22—24no—
529C22—41no—
530C22—41no—
531C22—42no—
532C22—48no—
533C22—48no—
534C22—48no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—1616no—
540C22C2×#52511yescharacteristic
541C22C2×#53511yescharacteristic
542C22#344911yescharacteristic
543C22C2×#57712yes—
544C22#358512yes—
545C4—22no—
546C4—24no—
547C4—24no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—84no—
558C4—82no—
559C4#351314yes—
560C2—42no—
561C2—88no—
562C2#3523211yescharacteristic
563C2#6057912yes—
564C2#6083312yes—
565C2#6101211yescharacteristic
566C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35314C2semidirect
3. #33018C2semidirect
4. #30077C2semidirect
5. #30039C2semidirect
6. #29498C2semidirect
7. #29492C2semidirect
8. #29263C2semidirect
9. #29116C2semidirect
10. #28320C2semidirect
11. #28169C2semidirect
12. #27652C2semidirect
14. #22160C2semidirect
15. #20785C2semidirect
21. #2864C22semidirect
22. #2863C22semidirect
23. #2862C22semidirect
35. #2331C22semidirect
38. #2327C22semidirect
39. #2320C22semidirect
46. #2092C22semidirect
47. #2084C22semidirect
48. #2083C22semidirect
49. #2073C22semidirect
50. #2072C22semidirect
51. #2072C22semidirect
54. #2023C22semidirect
57. #2013C22semidirect
59. #1973C22semidirect
60. #1968C22semidirect
61. #1949C22semidirect
62. #1936C22semidirect
63. #1934C22semidirect
64. #1877C22semidirect
65. #1877C22semidirect
66. #1877C22semidirect
67. #1877C22semidirect
68. #1864C22semidirect
69. #1864C22semidirect
70. #1862C22semidirect
71. #1862C22semidirect
72. #1839C22semidirect
73. #1832C22semidirect
76. #1805C22semidirect
79. #1768C22semidirect
81. #1745C22semidirect
82. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#353142
#342102
#330182
#300772
#300392
#294982
#294922
#292632
#291162
#283202
#281692
#276522
#242112
#221602
#207852