Finite Group Databasebeta

← #6635577

Subgroups — #6635577

Order 512.

1832 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37296C211yesmaximal, characteristic, complemented
2#35541C211yesmaximal, characteristic, complemented
3#32917C212yesmaximal, complemented
4#31958C214yesmaximal, complemented
5#31934C214yesmaximal, complemented
6#31003C214yesmaximal, complemented
7#30985C214yesmaximal, complemented
8#29768C212yesmaximal, complemented
9#28310C211yesmaximal, characteristic, complemented
10#28302C211yesmaximal, characteristic, complemented
11#27980C211yesmaximal, characteristic, complemented
12#27974C211yesmaximal, characteristic, complemented
13#24012C212yesmaximal, complemented
14#23924C212yesmaximal, complemented
15#21905C211yesmaximal, characteristic, complemented
16C2×#537C2212yes—
17C2×#535C2211yescharacteristic
18C2×#518C2211yescharacteristic
19C2×#481C2212yes—
20C2×#483C2212yes—
21C2×#463C2214yes—
22C2×#449C2214yes—
23C2×#450C2212yes—
24C2×#447C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2732—22no—
28#2682—22no—
29#2671—24no—
30#2620C2212yes—
31#2502C2212yescomplemented
32#2501C2212yescomplemented
33#2447—22no—
34#2445—22no—
35#2432C2212yescomplemented
36#2431C2212yescomplemented
37#2422C2214yes—
38#2420C2214yes—
39#2412—24no—
40#2316C2211yescharacteristic
41#2311C2211yescharacteristic
42#2297C2211yescharacteristic
43C2×#390C2211yescharacteristic
44#2285C2211yescharacteristic
45#2094C2212yes—
46#2094C2212yes—
47#2084C2214yescomplemented
48#2073C2214yescomplemented
49#2072C2214yescomplemented
50#2066C2214yescomplemented
51#2065C2214yescomplemented
52#2042C2214yescomplemented
53#2035C2214yescomplemented
54#2034C2214yescomplemented
55#2013C2212yes—
56#2011C2212yes—
57#2000C2214yes—
58#1999C2214yes—
59#1972C2214yes—
60#1968C2214yes—
61#1960C2212yescomplemented
62#1957C2212yescomplemented
63#1924C2214yescomplemented
64#1923C2214yescomplemented
65#1879C2214yes—
66#1877C2214yes—
67#1875C2212yes—
68#1875C2212yes—
69#1874C2214yes—
70#1873C2212yes—
71#1873C2212yes—
72#1853C2214yescomplemented
73#1836C2212yescomplemented
74#1834C2212yescomplemented
75#1802C2214yes—
76#1798C2212yes—
77#1788C2214yescomplemented
78#1780C2212yes—
79#1749C2211yescharacteristic
80C4×C22⋊C8C2211yescharacteristic
81C22×#41C2312yes—
82C22×#41C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C22×SD16C2314yes—
85C22×D8C2314yes—
86#537—44no—
87#537—44no—
88#535—44no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127—21no—
91C2×#127C2311yescharacteristic
92C2×#124—21no—
93C2×#124C2312yes—
94C2×#123—24no—
95C2×#123—24no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×#121—22no—
100C2×#121—22no—
101C2×#121—22no—
102C2×#121—22no—
103C2×#121C2312yes—
104C2×#121C2314yes—
105C2×#121C2312yes—
106C2×#120—22no—
107#518—44no—
108C2×C4×D4—22no—
109C2×C4×D4C2314yes—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C42C2311yescharacteristic
118#483—44no—
119#481—44no—
120#463—48no—
121#450—44no—
122#449—48no—
123#447—44no—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2312yes—
140C2×C4×C8C2312yes—
141#399—24no—
142#398—24no—
143#397—24no—
144#397—24no—
145#396—24no—
146#394—24no—
147#394—24no—
148#394—24no—
149#394—24no—
150#392—24no—
151#391—21no—
152#391D412yes—
153#391C2312yes—
154#391C2312yes—
155#390—21no—
156#390—22no—
157#390—22no—
158#390D412yes—
159#390C2312yes—
160#390C2312yes—
161#390C2312yes—
162#390C2312yes—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#386—22no—
167#386—24no—
168#386—24no—
169#386D414yes—
170#385—24no—
171#385—24no—
172#385—24no—
173#384C2312yes—
174#384C2312yes—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#382C2312yes—
178#381—24no—
179#380—22no—
180#380—22no—
181#380—22no—
182#380C2314yes—
183C4×C4⋊C4—22no—
184C4×C22⋊C4C2312yes—
185C4×C22⋊C4C2312yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2312yes—
192#336C2312yes—
193C25—21no—
194C2×#41—24no—
195C2×#41—42no—
196C2×#41—42no—
197C2×#41—42no—
198C2×#41—42no—
199C2×#41—44no—
200C2×#41—44no—
201C22×Q8C2×D411yescharacteristic
202C22×Q8C2412yes—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4—42no—
212C22×D4C2×D412yes—
213C22×D4C2×D411yescharacteristic
214C22×D4C2412yes—
215C22×D4C2412yes—
216C23×C4—21no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4C2×D412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C2×SD16—28no—
224C2×SD16—44no—
225C2×SD16—44no—
226C2×D8—28no—
227C2×D8—44no—
228C2×D8—44no—
229C22×C8C2414yes—
230C22×C8C2414yes—
231C4⋊Q8—42no—
232#127—42no—
233#127—44no—
234#124—44no—
235#124—44no—
236#123—88no—
237#123—88no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—42no—
241C22⋊Q8—42no—
242C22⋊Q8—48no—
243#121—28no—
244#121—24no—
245#121—24no—
246#121—44no—
247#121—42no—
248#121—42no—
249#121—48no—
250#121—48no—
251#121—48no—
252#121—84no—
253#120—48no—
254C4×D4—28no—
255C4×D4—44no—
256C4×D4—48no—
257#117—42no—
258#117—42no—
259#117—42no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—84no—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D414yes—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—42no—
302C2×C42C2×D412yes—
303C2×C42C2412yes—
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306#106—44no—
307#106—44no—
308C4⋊C8—44no—
309#103—44no—
310#103—44no—
311#103—44no—
312#102—44no—
313#102—44no—
314#102—44no—
315#102—44no—
316#102—44no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319C22⋊C8—44no—
320C22⋊C8—44no—
321#97—42no—
322C4×C8—42no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—42no—
335C24—42no—
336C24—82no—
337C24—82no—
338C24C22×D412yes—
339C24C22×D411yescharacteristic
340#41—88no—
341#41—88no—
342#41—88no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—416no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—48no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4H(5,𝔽2)14yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—82no—
405C22×C4—84no—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D414yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4H(5,𝔽2)14yes—
412C22×C4C2511yescharacteristic
413SD16—88no—
414D8—88no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—82no—
440C22⋊C4—28no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C42—24no—
458C42—22no—
459C42—22no—
460C42—42no—
461C42—42no—
462C42—42no—
463C23—22no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—48no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—88no—
484C23—162no—
485C23—162no—
486C23—164no—
487C23#53512yes—
488C23C23×D411yescharacteristic
489Q8—44no—
490Q8—44no—
491Q8—82no—
492D4—28no—
493D4—48no—
494D4—44no—
495D4—44no—
496D4—44no—
497D4—82no—
498D4—84no—
499D4—88no—
500D4—164no—
501D4—164no—
502D4—168no—
503D4—164no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—84no—
536C2×C4—88no—
537C2×C4—168no—
538C2×C4—162no—
539C2×C4—164no—
540C2×C4#53512yes—
541C2×C4#57714yes—
542C2×C4C2×H(5,𝔽2)12yes—
543C2×C4C2×H(5,𝔽2)12yes—
544C2×C4#58512yes—
545C8—44no—
546C8—44no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—48no—
558C22—44no—
559C22—84no—
560C22—82no—
561C22—82no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—84no—
567C22—168no—
568C22C2×#53511yescharacteristic
569C22#348411yescharacteristic
570C22#348511yescharacteristic
571C22C2×#57712yes—
572C22#358512yes—
573C4—24no—
574C4—22no—
575C4—42no—
576C4—42no—
577C4—44no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4#353414yes—
585C2—42no—
586C2—82no—
587C2—84no—
588C2—162no—
589C2#3672111yescharacteristic
590C2#6082011yescharacteristic
591C2#6082711yescharacteristic
592C2#6168512yes—
593C2#6175812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37296C2semidirect
2. #35541C2semidirect
3. #32917C2semidirect
4. #31958C2semidirect
5. #31934C2semidirect
6. #31003C2semidirect
7. #30985C2semidirect
8. #29768C2semidirect
9. #28310C2semidirect
10. #28302C2semidirect
11. #27980C2semidirect
12. #27974C2semidirect
13. #24012C2semidirect
14. #23924C2semidirect
15. #21905C2semidirect
31. #2502C22semidirect
32. #2501C22semidirect
35. #2432C22semidirect
36. #2431C22semidirect
47. #2084C22semidirect
48. #2073C22semidirect
49. #2072C22semidirect
50. #2066C22semidirect
51. #2065C22semidirect
52. #2042C22semidirect
53. #2035C22semidirect
54. #2034C22semidirect
61. #1960C22semidirect
62. #1957C22semidirect
63. #1924C22semidirect
64. #1923C22semidirect
72. #1853C22semidirect
73. #1836C22semidirect
74. #1834C22semidirect
77. #1788C22semidirect

Maximal subgroups

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

subgroupindex
#372962
#355412
#329172
#319582
#319342
#310032
#309852
#297682
#283102
#283022
#279802
#279742
#240122
#239242
#219052