Finite Group Databasebeta

← #6500856

Subgroups — #6500856

Order 512.

1921 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35223C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic, complemented
3#27621C212yesmaximal, complemented
4#27609C212yesmaximal, complemented
5#27597C212yesmaximal, complemented
6#26212C214yesmaximal, complemented
7#26206C214yesmaximal, complemented
8#26130C214yesmaximal, complemented
9#26128C214yesmaximal, complemented
10#24012C212yesmaximal, complemented
11#21763C212yesmaximal, complemented
12#21739C211yesmaximal, characteristic, complemented
13C2×#1853C212yesmaximal, complemented
14C2×#537C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C2×#525C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#481C2212yescomplemented
19C2×#483C2212yescomplemented
20C22×#106C2214yes—
21C22×C22⋊C8C2212yes—
22#2847—22no—
23#2618—22no—
24#2417—22no—
25#2417C2214yes—
26#2416—22no—
27#2416C2214yes—
28#2376—22no—
29#2358—22no—
30#2352C2214yes—
31D4×C4⋊C4C2214yes—
32#2329—22no—
33#2325—22no—
34#2316C2211yescharacteristic
35#2314C2212yes—
36#2311C2211yescharacteristic
37#2311C2212yes—
38#2311C2211yescharacteristic
39C2×#380C2212yes—
40#2101C2214yescomplemented
41#2096C2212yescomplemented
42#2094C2212yescomplemented
43#2084C2214yescomplemented
44#2077C2214yescomplemented
45#2077C2214yescomplemented
46#2074C2212yescomplemented
47#2073C2212yescomplemented
48#2035C2214yescomplemented
49#2034C2214yescomplemented
50#1927C2214yescomplemented
51#1926C2214yescomplemented
52#1920C2214yescomplemented
53#1919C2214yescomplemented
54#1893C2214yes—
55#1892C2214yes—
56#1875C2212yescomplemented
57#1874C2212yes—
58#1874C2212yes—
59#1873C2212yescomplemented
60#1872C2212yes—
61#1872C2212yes—
62#1869C2214yes—
63#1869C2214yes—
64#1853C2214yes—
65#1853C2214yes—
66#1853C2214yes—
67#1853C2214yes—
68#1853C2214yescomplemented
69#1825C2214yes—
70#1825C2214yescomplemented
71#1818C2212yes—
72#1788C2214yes—
73#1780C2212yes—
74C2×#340C2214yes—
75C22×#41C2311yescharacteristic
76C22×#41C2311yescharacteristic
77C24×C4—21no—
78C24×C4C2311yescharacteristic
79C23×C8C2314yes—
80#537—44no—
81#525—28no—
82#525—28no—
83C2×#123—21no—
84C2×#123D412yes—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8D412yes—
87C2×C22⋊Q8C2312yes—
88C2×C22⋊Q8C2312yes—
89C2×#121—22no—
90C2×#121—21no—
91C2×#121—24no—
92C2×#121D412yes—
93C2×#121C2312yes—
94C2×#121C2312yes—
95C2×#120—21no—
96C2×#120D412yes—
97C2×C4×D4—22no—
98C2×C4×D4—24no—
99C2×C4×D4C2314yes—
100C2×#117C2311yescharacteristic
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C22×C4⋊C4—21no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4—44no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—22no—
113#483—44no—
114#481—44no—
115C2×#106—48no—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×C22⋊C8—28no—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2314yes—
131#398—24no—
132#396—24no—
133#396—24no—
134#394—24no—
135#391—22no—
136#391—21no—
137#391C2312yes—
138#390—22no—
139#390—21no—
140#390C2312yes—
141#389—22no—
142#389—24no—
143#389C2314yes—
144#388—24no—
145#388—24no—
146#387—24no—
147#387—24no—
148#386—22no—
149#386—22no—
150#386—22no—
151#386—24no—
152#386—24no—
153#386—22no—
154#386—24no—
155#385—24no—
156#385—24no—
157#384C2312yes—
158#382C2312yes—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—24no—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167#380C2314yes—
168C4×C4⋊C4C2314yes—
169C4×C22⋊C4—24no—
170C2×#95—22no—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#336C2312yes—
180#336C2312yes—
181#336C2314yes—
182C25C2×D411yescharacteristic
183C2×#41—24no—
184C2×#41—24no—
185C2×#41—42no—
186C2×#41—42no—
187C2×#41—42no—
188C2×#41—42no—
189C22×Q8C2×D411yescharacteristic
190C22×Q8C2411yescharacteristic
191C22×D4—22no—
192C22×D4—21no—
193C22×D4—22no—
194C22×D4C2×D414yes—
195C22×D4C2×D412yes—
196C22×D4C2×D411yescharacteristic
197C22×D4C2411yescharacteristic
198C23×C4—21no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—21no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—84no—
209C23×C4C2×D412yes—
210C23×C4C2×D411yescharacteristic
211C23×C4#4118yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C22×C8—28no—
216C22×C8—44no—
217C22×C8—44no—
218C22×C8C2414yes—
219C22×C8C2414yes—
220#123—28no—
221#123—44no—
222C22⋊Q8—28no—
223C22⋊Q8—24no—
224C22⋊Q8—24no—
225C22⋊Q8—24no—
226C22⋊Q8—42no—
227C22⋊Q8—44no—
228#121—28no—
229#121—24no—
230#121—24no—
231#121—24no—
232#121—42no—
233#121—44no—
234#121—48no—
235#121—88no—
236#120—28no—
237#120—44no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—48no—
241C4×D4—88no—
242#117—24no—
243#117—24no—
244#117—44no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—84no—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D414yes—
286C2×C22⋊C4C2×D414yes—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42C2412yes—
291#106—28no—
292#106—28no—
293#106—44no—
294#106—44no—
295#103—44no—
296#103—44no—
297#102—44no—
298#102—44no—
299C22⋊C8—28no—
300C22⋊C8—24no—
301C22⋊C8—24no—
302C22⋊C8—44no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313C24—24no—
314C24—24no—
315C24—21no—
316C24—41no—
317C24—41no—
318C24#12314yes—
319C24C22×D412yes—
320C24C22×D411yescharacteristic
321C24C22×D412yes—
322#41—88no—
323#41—88no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—44no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—44no—
346C2×D4—42no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—84no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—416no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—48no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—82no—
379C22×C4—84no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—84no—
387C22×C4—816no—
388C22×C4#12318yes—
389C22×C4#12314yes—
390C22×C4C22×D414yes—
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C2×#4114yes—
395C22×C4C2×#4114yes—
396C22×C4C2×#4114yes—
397C22×C4C2511yescharacteristic
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—416no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—28no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4H(5,𝔽2)14yes—
426C22⋊C4—24no—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—48no—
434C22⋊C4—44no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—82no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C42—24no—
441C42—42no—
442C42—82no—
443C42—84no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—216no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—44no—
459C23—44no—
460C23—48no—
461C23—44no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23C2×#12312yes—
468C23C2×#12314yes—
469C23#57714yes—
470C23#57714yes—
471C23C23×D411yescharacteristic
472Q8—44no—
473Q8—82no—
474D4—416no—
475D4—44no—
476D4—82no—
477D4—84no—
478D4—84no—
479D4—88no—
480D4—164no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—816no—
517C2×C4—84no—
518C2×C4—164no—
519C2×C4—164no—
520C2×C4—164no—
521C2×C4C2×#12312yes—
522C2×C4C2×#12314yes—
523C2×C4C22×#4112yes—
524C2×C4C2×H(5,𝔽2)12yes—
525C2×C4#58512yes—
526C8—44no—
527C8—44no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—22no—
535C22—24no—
536C22—416no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—81no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22#309918yes—
551C22C22×#12311yescharacteristic
552C22#348411yescharacteristic
553C22#348511yescharacteristic
554C22C2×#57712yes—
555C22C2×#57712yes—
556C4—22no—
557C4—24no—
558C4—42no—
559C4—44no—
560C4—44no—
561C4—81no—
562C4—84no—
563C4—82no—
564C4—82no—
565C4—84no—
566C4#350514yes—
567C2—28no—
568C2—42no—
569C2—82no—
570C2—161no—
571C2#3535111yescharacteristic
572C2C2×#309912yes—
573C2#6163111yescharacteristic
574C2#6163311yescharacteristic
575C2#6176512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35223C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34360C2semidirect
3. #27621C2semidirect
4. #27609C2semidirect
5. #27597C2semidirect
6. #26212C2semidirect
7. #26206C2semidirect
8. #26130C2semidirect
9. #26128C2semidirect
10. #24012C2semidirect
11. #21763C2semidirect
12. #21739C2semidirect
13. C2×#1853C2semidirect
18. C2×#481C22semidirect
19. C2×#483C22semidirect
40. #2101C22semidirect
41. #2096C22semidirect
42. #2094C22semidirect
43. #2084C22semidirect
44. #2077C22semidirect
45. #2077C22semidirect
46. #2074C22semidirect
47. #2073C22semidirect
48. #2035C22semidirect
49. #2034C22semidirect
50. #1927C22semidirect
51. #1926C22semidirect
52. #1920C22semidirect
53. #1919C22semidirect
56. #1875C22semidirect
59. #1873C22semidirect
68. #1853C22semidirect
70. #1825C22semidirect

Maximal subgroups

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

subgroupindex
#352232
#343602
#276212
#276092
#275972
#262122
#262062
#261302
#261282
#240122
#217632
#217392
C2×#18532