Finite Group Databasebeta

← #6458978

Subgroups — #6458978

Order 512.

1743 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34203C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic
3#27686C212yesmaximal, complemented
4#27647C212yesmaximal, complemented
5#27638C214yesmaximal, complemented
6#27621C212yesmaximal, complemented
7#27602C212yesmaximal
8#27592C212yesmaximal, complemented
9#24003C212yesmaximal
10#23769C214yesmaximal, complemented
11#21579C212yesmaximal, complemented
12C4×#483C212yesmaximal, complemented
13#21369C212yesmaximal, complemented
14#20745C212yesmaximal, complemented
15#20716C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C2×#519C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#483C2212yes—
20C2×#483C2212yes—
21C2×#433C2212yes—
22#2362—22no—
23#2360—22no—
24#2344—22no—
25#2342—22no—
26#2323C2212yes—
27#2323C2212yes—
28#2316C2211yescharacteristic
29#2314—22no—
30#2314—22no—
31#2314C2212yes—
32#2314C2212yes—
33#2314C2212yes—
34#2314C2212yes—
35#2310C2211yescharacteristic
36C4×C22⋊Q8C2212yes—
37C4×C22⋊Q8C2212yes—
38C2×#388—22no—
39C2×#387—22no—
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42#2280C2211yescharacteristic
43C2×#380C2212yes—
44#2274C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46#2101C2214yescomplemented
47#2097C2214yescomplemented
48#2084C2214yescomplemented
49#2077C2214yes—
50#2076C2214yes—
51#2072C2214yescomplemented
52#2035C2214yes—
53#2035C2214yes—
54#1949C2214yescomplemented
55#1948C2214yescomplemented
56#1928C2214yescomplemented
57#1927C2214yescomplemented
58#1920C2214yescomplemented
59#1919C2214yescomplemented
60#1872C2212yes—
61#1872C2212yes—
62#1854C2214yes—
63#1853C2212yes—
64#1853C2214yes—
65#1852C2212yes—
66#1825C2214yes—
67#1823C2212yes—
68#1823C2212yes—
69#1818C2212yes—
70#1805C2214yescomplemented
71#1803C2214yescomplemented
72#1798C2212yes—
73#1797C2212yes—
74#1787C2214yes—
75#1779C2212yes—
76#1776C2214yescomplemented
77C4×#106C2214yescomplemented
78#1763C2214yescomplemented
79C4×#103C2214yescomplemented
80#1750C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×Q8C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8C2312yes—
89C2×C22⋊Q8C2312yes—
90#519—28no—
91C2×C4×Q8—22no—
92C2×C4×Q8C2312yes—
93C2×C4×Q8C2312yes—
94C2×#117—28no—
95C2×#117—22no—
96C2×#117D412yes—
97C2×#117D412yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C22×C4⋊C4—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4D412yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—22no—
112C22×C22⋊C4D412yes—
113C22×C42—22no—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#483—28no—
117#483—28no—
118#433—28no—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2312yes—
133C2×C4×C8C2312yes—
134#391—22no—
135#391C2312yes—
136#391C2312yes—
137#389—22no—
138#389—22no—
139#389—22no—
140#389C2312yes—
141#389C2312yes—
142#388—24no—
143#388—24no—
144#388—24no—
145#388—24no—
146#388—22no—
147#388—22no—
148#388—22no—
149#388—22no—
150#387—24no—
151#387—24no—
152#387—24no—
153#387—24no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#385—22no—
159#385—22no—
160#384—22no—
161#384C2312yes—
162#384C2312yes—
163#382—22no—
164#382C2312yes—
165#382C2312yes—
166#381—21no—
167#381—22no—
168#380—22no—
169#380—22no—
170#380—22no—
171#380C2314yes—
172#380C2314yes—
173#380C2312yes—
174#380C2312yes—
175#379—21no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—22no—
190C2×#95—22no—
191C2×#95—22no—
192C2×#95—22no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204C25C2×D411yescharacteristic
205C22×Q8—24no—
206C22×Q8—21no—
207C22×Q8—24no—
208C22×Q8—42no—
209C22×Q8C2412yes—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—42no—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C22×C8C2414yes—
226C22×C8C2414yes—
227C22⋊Q8—28no—
228C22⋊Q8—28no—
229C22⋊Q8—24no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8#4118yes—
233C4×Q8—28no—
234C4×Q8—28no—
235C4×Q8—48no—
236#117—24no—
237#117—28no—
238#117—28no—
239#117—24no—
240#117—24no—
241#117—24no—
242#117—42no—
243#117—48no—
244#117#4118yes—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—84no—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—84no—
283C2×C22⋊C4C2×D412yes—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D412yes—
286C2×C22⋊C4C2×D412yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—84no—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#107—44no—
305#106—28no—
306#106—28no—
307#106—44no—
308C4⋊C8—28no—
309#103—28no—
310#103—28no—
311#103—44no—
312#103—44no—
313C22⋊C8—28no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C22⋊C8—44no—
317#97—24no—
318C4×C8—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—41no—
339C24—42no—
340C24C22×D411yescharacteristic
341C24C22×D411yescharacteristic
342C24C22×D411yescharacteristic
343C2×Q8—24no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—416no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8C2×#4114yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—416no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—82no—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4114yes—
399C22×C4C2×#4114yes—
400C22×C4C2511yescharacteristic
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C4⋊C4—24no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—28no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4#12318yes—
429C4⋊C4C2×#4114yes—
430C22⋊C4—28no—
431C22⋊C4—24no—
432C22⋊C4—28no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—82no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4#12318yes—
442C42—24no—
443C42—24no—
444C42—22no—
445C42—22no—
446C42—42no—
447C42—42no—
448C42—44no—
449C42—44no—
450C42—82no—
451C42—82no—
452C42—84no—
453C42—84no—
454C42C2×#4114yes—
455C42C2×#4114yes—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—22no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—44no—
469C23—48no—
470C23—44no—
471C23—84no—
472C23—82no—
473C23C2×#12312yes—
474C23#52512yes—
475C23#57714yes—
476C23C23×D411yescharacteristic
477Q8—28no—
478Q8—48no—
479Q8—44no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—84no—
525C2×C4—88no—
526C2×C4—88no—
527C2×C4C2×#12314yes—
528C2×C4C2×#12314yes—
529C2×C4C2×#12312yes—
530C2×C4#52512yes—
531C2×C4C22×#4112yes—
532C2×C4C22×#4112yes—
533C2×C4#58512yes—
534C8—44no—
535C8—44no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—24no—
541C22—22no—
542C22—42no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—88no—
551C22#312614yes—
552C22C22×#12311yescharacteristic
553C22C2×#52511yescharacteristic
554C22#344911yescharacteristic
555C22C2×#57712yes—
556C22#358612yes—
557C4—24no—
558C4—24no—
559C4—22no—
560C4—22no—
561C4—42no—
562C4—42no—
563C4—44no—
564C4—41no—
565C4—44no—
566C4—82no—
567C4—84no—
568C4—82no—
569C4—84no—
570C4#347514yes—
571C2—24no—
572C2—42no—
573C2—81no—
574C2#3480911yescharacteristic
575C2#6052212yes—
576C2C2×#312611yescharacteristic
577C2#6056911yescharacteristic
578C2#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. #27686C2semidirect
4. #27647C2semidirect
5. #27638C2semidirect
6. #27621C2semidirect
8. #27592C2semidirect
10. #23769C2semidirect
11. #21579C2semidirect
12. C4×#483C2semidirect
13. #21369C2semidirect
14. #20745C2semidirect
46. #2101C22semidirect
47. #2097C22semidirect
48. #2084C22semidirect
51. #2072C22semidirect
54. #1949C22semidirect
55. #1948C22semidirect
56. #1928C22semidirect
57. #1927C22semidirect
58. #1920C22semidirect
59. #1919C22semidirect
70. #1805C22semidirect
71. #1803C22semidirect
76. #1776C22semidirect
77. C4×#106C22semidirect
78. #1763C22semidirect
79. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#342032
C2×#23142
#276862
#276472
#276382
#276212
#276022
#275922
#240032
#237692
#215792
C4×#4832
#213692
#207452
#207162