Finite Group Databasebeta

← #6493793

Subgroups — #6493793

Order 512.

1587 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35310C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic
3#33151C212yesmaximal, complemented
4#33108C212yesmaximal, complemented
5#29949C212yesmaximal, complemented
6#29930C212yesmaximal, complemented
7#27873C214yesmaximal, complemented
8#27686C212yesmaximal, complemented
9#27647C212yesmaximal, complemented
10#27642C214yesmaximal, complemented
11#24559C212yesmaximal
12#24551C212yesmaximal
13#21967C212yesmaximal, complemented
14#21898C212yesmaximal, complemented
15#20628C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#483C2212yes—
19C2×#483C2212yes—
20C2×#424C2212yes—
21C2×#407C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2857C2211yescharacteristic
24#2851C2211yescharacteristic
25#2848C2212yes—
26#2629C2211yescharacteristic
27#2625C2212yes—
28#2621C2211yescharacteristic
29#2480—22no—
30#2478—22no—
31#2475—22no—
32#2473—22no—
33#2323C2212yes—
34#2314—22no—
35#2314—22no—
36#2314C2212yes—
37#2314C2211yescharacteristic
38#2314C2212yes—
39#2314C2212yes—
40#2314C2212yes—
41C4×C22⋊Q8C2212yes—
42C2×#388—22no—
43C2×#387—22no—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48#2101C2214yescomplemented
49#2097C2214yescomplemented
50#2084C2214yescomplemented
51#2077C2214yes—
52#2077C2214yes—
53#2072C2214yescomplemented
54#2036C2214yes—
55#2035C2214yes—
56#1989C2214yescomplemented
57#1981C2214yescomplemented
58#1949C2214yescomplemented
59#1949C2214yescomplemented
60#1948C2214yescomplemented
61#1948C2214yescomplemented
62#1927C2214yescomplemented
63#1927C2214yescomplemented
64#1921C2214yescomplemented
65#1919C2214yescomplemented
66#1894C2214yes—
67#1892C2214yes—
68#1872C2212yes—
69#1872C2212yes—
70#1853C2212yes—
71#1852C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1809C2214yes—
75#1807C2214yes—
76#1805C2214yescomplemented
77#1803C2214yescomplemented
78#1798C2212yes—
79#1798C2212yes—
80#1797C2212yes—
81#1797C2212yes—
82#1729C2212yes—
83C24×C4C2311yescharacteristic
84C22×M16C2312yes—
85C23×C8C2312yes—
86#541—28no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#126—22no—
89C2×#125—22no—
90C2×#125C2311yescharacteristic
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8C2312yes—
95C2×#117—28no—
96C2×#117D412yes—
97C2×#117D412yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—22no—
103C22×C4⋊C4D412yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—22no—
107C22×C22⋊C4D412yes—
108C22×C42—22no—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#483—44no—
112#483—44no—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117#424—28no—
118C2×C4⋊C8C2314yes—
119C2×C4⋊C8C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124#407—42no—
125C2×C22⋊C8—42no—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130#400—22no—
131#398—22no—
132#398—22no—
133#398—22no—
134#397—22no—
135#397—22no—
136#397—22no—
137#396—22no—
138#391C2312yes—
139#391C2312yes—
140#389—22no—
141#389—22no—
142#389C2312yes—
143#389C2312yes—
144#388—24no—
145#388—24no—
146#388—24no—
147#388—24no—
148#388—22no—
149#388—22no—
150#388—22no—
151#387—24no—
152#387—24no—
153#387—24no—
154#387—24no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#385—22no—
160#385—22no—
161#385—22no—
162#385—22no—
163#385—22no—
164#384—22no—
165#384C2312yes—
166#384C2312yes—
167#384C2311yescharacteristic
168#382—22no—
169#382C2312yes—
170#382C2312yes—
171#382C2311yescharacteristic
172#380—22no—
173#380—22no—
174#380—22no—
175#380C2314yes—
176#380C2314yes—
177#380C2312yes—
178#380C2312yes—
179#380C2312yes—
180#379—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2312yes—
187C4×C22⋊C4—22no—
188C2×#95—22no—
189C2×#95—22no—
190C2×#95—22no—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25C2×D411yescharacteristic
202C22×Q8C2412yes—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—42no—
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C2×M16—24no—
218C2×M16—42no—
219C2×M16—42no—
220C22×C8—24no—
221C22×C8—42no—
222C22×C8—42no—
223C22×C8C2414yes—
224C22×C8C2414yes—
225C4⋊Q8—24no—
226#126—48no—
227#125—24no—
228#125—48no—
229C22⋊Q8—28no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—44no—
233C4×Q8—28no—
234#117—24no—
235#117—28no—
236#117—28no—
237#117—24no—
238#117—24no—
239#117#4118yes—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
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—84no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—84no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C22⋊C4C2×D412yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—84no—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294#107—44no—
295#106—44no—
296#106—44no—
297#106—44no—
298C4⋊C8—28no—
299C4⋊C8—28no—
300#103—44no—
301#103—44no—
302#103—44no—
303#103—44no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
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#95—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—41no—
329C24—42no—
330C24C22×D411yescharacteristic
331C24C22×D411yescharacteristic
332C24C22×D411yescharacteristic
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—24no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C2×#4114yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C42−1+412yes—
385C22×C4C2511yescharacteristic
386M16—84no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—84no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4#12318yes—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C22⋊C4#12318yes—
431C42—24no—
432C42—24no—
433C42—22no—
434C42—22no—
435C42—44no—
436C42—82no—
437C42—84no—
438C42—84no—
439C42C2×#4114yes—
440C42C2×#4114yes—
441C23—22no—
442C23—21no—
443C23—21no—
444C23—22no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—22no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—22no—
453C23—44no—
454C23—48no—
455C23—44no—
456C23—84no—
457C23—82no—
458C23C2×#12312yes—
459C23#53712yes—
460C23#57714yes—
461C23C23×D411yescharacteristic
462Q8—44no—
463Q8—44no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—44no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—88no—
511C2×C4—88no—
512C2×C4C2×#12314yes—
513C2×C4C2×#12314yes—
514C2×C4C2×#12312yes—
515C2×C4#53712yes—
516C2×C4C22×#4112yes—
517C2×C4C2×H(5,𝔽2)11yescharacteristic
518C2×C4C2×2−1+411yescharacteristic
519C2×C4#58512yes—
520C8—44no—
521C8—44no—
522C22—24no—
523C22—24no—
524C22—22no—
525C22—22no—
526C22—24no—
527C22—22no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—42no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—88no—
537C22#326014yes—
538C22C22×#12311yescharacteristic
539C22C2×#53711yescharacteristic
540C22#344911yescharacteristic
541C22C2×#57712yes—
542C22#358612yes—
543C4—24no—
544C4—24no—
545C4—22no—
546C4—22no—
547C4—42no—
548C4—42no—
549C4—41no—
550C4—82no—
551C4—84no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4—82no—
556C4#349412yes—
557C4#349512yes—
558C2—24no—
559C2—42no—
560C2—81no—
561C2#3526311yescharacteristic
562C2#6052212yes—
563C2C2×#326011yescharacteristic
564C2#6092111yescharacteristic
565C2#6097712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33151C2semidirect
4. #33108C2semidirect
5. #29949C2semidirect
6. #29930C2semidirect
7. #27873C2semidirect
8. #27686C2semidirect
9. #27647C2semidirect
10. #27642C2semidirect
13. #21967C2semidirect
14. #21898C2semidirect
48. #2101C22semidirect
49. #2097C22semidirect
50. #2084C22semidirect
53. #2072C22semidirect
56. #1989C22semidirect
57. #1981C22semidirect
58. #1949C22semidirect
59. #1949C22semidirect
60. #1948C22semidirect
61. #1948C22semidirect
62. #1927C22semidirect
63. #1927C22semidirect
64. #1921C22semidirect
65. #1919C22semidirect
76. #1805C22semidirect
77. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#353102
C2×#23142
#331512
#331082
#299492
#299302
#278732
#276862
#276472
#276422
#245592
#245512
#219672
#218982
#206282