Finite Group Databasebeta

← #6644673

Subgroups — #6644673

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35087C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic
3#33007C212yesmaximal
4#32438C214yesmaximal, complemented
5#32368C214yesmaximal, complemented
6#31660C214yesmaximal, complemented
7#31532C214yesmaximal, complemented
8#29643C212yesmaximal, complemented
9#29110C211yesmaximal, characteristic
10#28302C211yesmaximal, characteristic
11#28123C212yesmaximal
12#27964C211yesmaximal, characteristic, complemented
13#27919C211yesmaximal, characteristic, complemented
14#27647C212yesmaximal, complemented
15#21914C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#484C2212yes—
19C2×#483C2212yes—
20C2×#483C2212yes—
21C2×#450C2212yes—
22#2857C2211yescharacteristic
23#2855C2212yes—
24#2851C2211yescharacteristic
25#2655—22no—
26#2566—22no—
27#2566—22no—
28#2566C2214yes—
29#2566C2214yes—
30#2514C2212yescomplemented
31#2506C2212yescomplemented
32#2493—22no—
33#2473—22no—
34#2453C2212yescomplemented
35#2446C2212yescomplemented
36C2×#396—22no—
37#2336—22no—
38C4×C22⋊Q8C2212yes—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#385—22no—
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45#2101C2214yescomplemented
46#2097C2214yescomplemented
47#2094C2212yes—
48#2084C2214yescomplemented
49#2073C2212yes—
50#2072C2214yescomplemented
51#2067C2214yes—
52#2059C2214yescomplemented
53#2059C2214yescomplemented
54#2055C2214yes—
55#2026C2214yescomplemented
56#2026C2214yescomplemented
57#2025C2212yes—
58#2013C2212yes—
59#1991C2212yescomplemented
60#1968C2214yes—
61#1960C2212yescomplemented
62#1948C2214yescomplemented
63#1936C2214yes—
64#1925C2214yes—
65#1917C2214yescomplemented
66#1916C2214yescomplemented
67#1880C2214yes—
68#1877C2214yes—
69#1877C2214yes—
70#1864C2214yes—
71#1864C2214yes—
72#1847C2212yescomplemented
73#1835C2212yescomplemented
74#1824C2212yes—
75#1823C2212yes—
76#1823C2212yes—
77#1817C2212yes—
78#1803C2214yescomplemented
79#1797C2212yes—
80#1786C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C22×SD16C2314yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#125—22no—
88C2×#125—22no—
89C2×#125—44no—
90C2×#125C2314yes—
91C2×#125C2311yescharacteristic
92C2×C22⋊Q8C2312yes—
93C2×C4×Q8C2312yes—
94C2×C4×D4—22no—
95C2×C4×D4—44no—
96C2×C4×D4C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4D412yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4—21no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—22no—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#484—44no—
111#483—44no—
112#483—44no—
113#450—44no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×#97C2312yes—
127C2×C4×C8C2312yes—
128#401—22no—
129#400—22no—
130#400—22no—
131#400—22no—
132#400—22no—
133#398—22no—
134#398—22no—
135#398—22no—
136#396—24no—
137#396—24no—
138#396—24no—
139#396—24no—
140#391C2312yes—
141#391C2312yes—
142#391C2312yes—
143#391C2312yes—
144#390—21no—
145#390—22no—
146#390—21no—
147#390C2314yes—
148#390C2312yes—
149#390C2312yes—
150#389—22no—
151#389—22no—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#389C2312yes—
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#385—22no—
166#385—22no—
167#385—22no—
168#385—24no—
169#385—24no—
170#385—24no—
171#385—24no—
172#384—21no—
173#384—21no—
174#384C2314yes—
175#384C2311yescharacteristic
176#383—22no—
177#382—22no—
178#382C2312yes—
179#382C2312yes—
180#382C2311yescharacteristic
181#381—22no—
182#380—22no—
183#380C2312yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4D412yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C2×#95—22no—
192C2×#95D412yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204C25—21no—
205C22×Q8C2412yes—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—24no—
209C22×D4—42no—
210C22×D4C2412yes—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—24no—
215C23×C4—44no—
216C23×C4—84no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D412yes—
221C23×C4C2411yescharacteristic
222C2×SD16—28no—
223C2×SD16—44no—
224C2×SD16—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—42no—
228#125—48no—
229#125—42no—
230#125—48no—
231#125—48no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C4×Q8—44no—
235C4×D4—48no—
236C4×D4—48no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—84no—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—84no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42C2×D412yes—
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D412yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2412yes—
290C2×C42C2412yes—
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293#107—44no—
294#106—44no—
295#106—44no—
296C4⋊C8—44no—
297#103—44no—
298#103—44no—
299#103—44no—
300#102—28no—
301#102—44no—
302#102—44no—
303C22⋊C8—24no—
304C22⋊C8—24no—
305#97—42no—
306C4×C8—42no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95C2×D412yes—
325#95C2×D412yes—
326#95C2×D412yes—
327#95C2×D412yes—
328C24—24no—
329C24—22no—
330C24—44no—
331C24—42no—
332C24—42no—
333C24—44no—
334C24C22×D411yescharacteristic
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—416no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D42−1+414yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—48no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—88no—
374C22×C4—816no—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D412yes—
382C22×C4H(5,𝔽2)14yes—
383C22×C4C2511yescharacteristic
384SD16—88no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—48no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C42—24no—
426C42—24no—
427C42—22no—
428C42—22no—
429C42—24no—
430C42—41no—
431C42—42no—
432C42—41no—
433C42—44no—
434C42—82no—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—24no—
439C23—48no—
440C23—416no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—88no—
446C23—84no—
447C23—84no—
448C23#53512yes—
449C23#57714yes—
450C23C23×D411yescharacteristic
451Q8—44no—
452Q8—44no—
453D4—28no—
454D4—48no—
455D4—44no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—28no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—48no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—84no—
497C2×C4—816no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4#53512yes—
502C2×C4#57514yes—
503C2×C4C2×H(5,𝔽2)12yes—
504C2×C4C2×2−1+412yes—
505C2×C4#58512yes—
506C8—44no—
507C8—44no—
508C22—24no—
509C22—22no—
510C22—24no—
511C22—22no—
512C22—48no—
513C22—44no—
514C22—816no—
515C22—84no—
516C22—82no—
517C22—82no—
518C22#322214yes—
519C22C2×#53511yescharacteristic
520C22#348511yescharacteristic
521C22#348611yescharacteristic
522C22C2×#57512yes—
523C22C2×#57712yes—
524C4—24no—
525C4—22no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4#353514yes—
539C2—24no—
540C2—84no—
541C2#3711811yescharacteristic
542C2C2×#322211yescharacteristic
543C2#6080211yescharacteristic
544C2#6173612yes—
545C2#6184112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35087C2semidirect
4. #32438C2semidirect
5. #32368C2semidirect
6. #31660C2semidirect
7. #31532C2semidirect
8. #29643C2semidirect
12. #27964C2semidirect
13. #27919C2semidirect
14. #27647C2semidirect
15. #21914C2semidirect
30. #2514C22semidirect
31. #2506C22semidirect
34. #2453C22semidirect
35. #2446C22semidirect
45. #2101C22semidirect
46. #2097C22semidirect
48. #2084C22semidirect
50. #2072C22semidirect
52. #2059C22semidirect
53. #2059C22semidirect
55. #2026C22semidirect
56. #2026C22semidirect
59. #1991C22semidirect
61. #1960C22semidirect
62. #1948C22semidirect
65. #1917C22semidirect
66. #1916C22semidirect
72. #1847C22semidirect
73. #1835C22semidirect
78. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#350872
C2×#25662
#330072
#324382
#323682
#316602
#315322
#296432
#291102
#283022
#281232
#279642
#279192
#276472
#219142