Finite Group Databasebeta

← #6508785

Subgroups — #6508785

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic
3#35311C211yesmaximal, characteristic
4#34380C211yesmaximal, characteristic
5#28132C214yesmaximal, complemented
6#27656C214yesmaximal, complemented
7#25208C214yesmaximal, complemented
8#25206C214yesmaximal, complemented
9#24588C214yesmaximal, complemented
10#24540C214yesmaximal, complemented
11#21933C211yesmaximal, characteristic
12#21765C211yesmaximal, characteristic
13C2×#1825C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3189—28no—
18#3095—28no—
19C2×#484C2212yes—
20C2×#484C2212yes—
21C2×#484C2212yes—
22C2×#485C2214yes—
23C2×#483C2212yes—
24C2×#467C2214yes—
25C2×#451C2212yes—
26C2×#451C2212yes—
27C2×#428C2212yes—
28C2×#419C2212yes—
29C2×#407C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2855—22no—
32#2625—21no—
33#2625—21no—
34#2503C2212yes—
35#2502C2212yes—
36#2499C2214yes—
37#2484—22no—
38#2483—22no—
39#2378—22no—
40#2360C2214yes—
41#2344C2212yes—
42#2342—22no—
43Q8×C22⋊C4C2212yes—
44#2329—22no—
45#2323C2214yes—
46#2317C2212yes—
47#2316—21no—
48#2313—21no—
49#2310C2212yes—
50#2280C2211yescharacteristic
51C2×#380C2211yescharacteristic
52#2085C2214yescomplemented
53#2083C2214yescomplemented
54#2074C2214yescomplemented
55#2072C2214yescomplemented
56#2068C2214yescomplemented
57#2066C2214yescomplemented
58#2056C2214yescomplemented
59#2056C2214yescomplemented
60#1949C2214yescomplemented
61#1947C2214yescomplemented
62#1928C2214yescomplemented
63#1927C2214yescomplemented
64#1894C2214yes—
65#1878C2214yescomplemented
66#1878C2214yescomplemented
67#1872C2214yes—
68#1851C2214yes—
69#1850C2214yes—
70#1825C2214yes—
71#1825C2214yes—
72#1823C2212yes—
73#1805C2214yescomplemented
74#1803C2214yescomplemented
75#1798C2214yes—
76#1787C2212yes—
77C2×#340C2212yes—
78C2×#340C2212yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×Q16C2314yes—
83C22×M16C2312yes—
84C23×C8C2312yes—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8D418yes—
93C2×C22⋊Q8C2312yes—
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×C4×Q8C2314yes—
98C2×#117—21no—
99C2×#117—21no—
100C2×#117—22no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C42—21no—
118#485—216no—
119#484—28no—
120#484—28no—
121#484—28no—
122#483—28no—
123#467—216no—
124#451—28no—
125#451—28no—
126#428—28no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132#419—28no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139#407—24no—
140C2×C22⋊C8—24no—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145#400—22no—
146#400—22no—
147#398—22no—
148#398—22no—
149#397—24no—
150#397—24no—
151#391—22no—
152#391—22no—
153#391—24no—
154#391C2314yes—
155#391C2314yes—
156#391C2314yes—
157#389—24no—
158#389—22no—
159#389—22no—
160#389—24no—
161#389C2314yes—
162#388—24no—
163#388—24no—
164#387—24no—
165#387—24no—
166#387—24no—
167#387—24no—
168#387—24no—
169#385—24no—
170#385—24no—
171#385—24no—
172#384C2314yes—
173#382C2314yes—
174#381—24no—
175#381—22no—
176#381—24no—
177#381D412yes—
178#380—21no—
179#380—21no—
180#380—22no—
181#380D411yescharacteristic
182#380D411yescharacteristic
183#380D412yes—
184#380C2312yes—
185#380C2312yes—
186#379D412yes—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—24no—
190C4×C22⋊C4C2312yes—
191C4×C22⋊C4C2312yes—
192C2×#95—21no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25C2×D411yescharacteristic
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8C2×D414yes—
212C22×Q8C2412yes—
213C22×Q8C2412yes—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—21no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D414yes—
227C23×C4#4114yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C2×Q16—28no—
232C2×Q16—44no—
233C2×Q16—44no—
234C2×M16—24no—
235C2×M16—42no—
236C2×M16—42no—
237C22×C8—24no—
238C22×C8—42no—
239C22×C8—42no—
240C22×C8C2414yes—
241C22×C8C2414yes—
242C22⋊Q8—28no—
243C22⋊Q8—28no—
244C22⋊Q8—24no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—44no—
251C22⋊Q8—44no—
252C22⋊Q8C2×D418yes—
253C22⋊Q8C2×D418yes—
254C4×Q8—44no—
255C4×Q8—44no—
256#117—24no—
257#117—24no—
258#117—24no—
259#117—24no—
260#117—42no—
261#117—42no—
262#117—44no—
263#117—44no—
264#117—84no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D412yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42C2414yes—
319#107—28no—
320#107—28no—
321#106—28no—
322C4⋊C8—44no—
323C4⋊C8—44no—
324#103—28no—
325#103—28no—
326#103—28no—
327#103—28no—
328#103—44no—
329#103—44no—
330C22⋊C8—24no—
331C22⋊C8—24no—
332C22⋊C8—24no—
333C22⋊C8—24no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95C2×D414yes—
347#95C2×D412yes—
348#95C2×D412yes—
349C24—24no—
350C24—41no—
351C24—42no—
352C24—41no—
353C24#12114yes—
354C24C22×D411yescharacteristic
355C24C22×D411yescharacteristic
356C24C22×D411yescharacteristic
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—44no—
366C2×Q8—48no—
367C2×Q8—42no—
368C2×Q8—82no—
369C2×Q8—84no—
370C2×Q8—82no—
371C2×Q8—82no—
372C2×Q8—82no—
373C2×Q8—82no—
374C2×Q8C22×D414yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4#12114yes—
413C22×C4#12114yes—
414C22×C4#12114yes—
415C22×C4C22×D414yes—
416C22×C4C22×D414yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2511yescharacteristic
427Q16—88no—
428M16—48no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—84no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—82no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4C22×D414yes—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—28no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—42no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—82no—
473C42—24no—
474C42—24no—
475C42—44no—
476C42—44no—
477C42—84no—
478C42—84no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—28no—
491C23—44no—
492C23—44no—
493C23—84no—
494C23—84no—
495C23C2×#12112yes—
496C23C2×#12112yes—
497C23C2×#12112yes—
498C23#57814yes—
499C23C23×D411yescharacteristic
500Q8—28no—
501Q8—42no—
502Q8—42no—
503Q8—44no—
504Q8—42no—
505Q8—42no—
506Q8—84no—
507Q8—84no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—164no—
554C2×C4C2×#12112yes—
555C2×C4C2×#12112yes—
556C2×C4C2×#12112yes—
557C2×C4C23×D411yescharacteristic
558C2×C4C22×#4111yescharacteristic
559C2×C4#58512yes—
560C2×C4#58512yes—
561C8—44no—
562C8—44no—
563C22—22no—
564C22—24no—
565C22—24no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—88no—
578C22#304414yes—
579C22C22×#12111yescharacteristic
580C22#348512yes—
581C22C2×#57812yes—
582C22#358511yescharacteristic
583C22#358611yescharacteristic
584C4—22no—
585C4—22no—
586C4—22no—
587C4—44no—
588C4—44no—
589C4—44no—
590C4—44no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—84no—
595C4—82no—
596C4#348512yes—
597C4#349111yescharacteristic
598C4#349111yescharacteristic
599C2—24no—
600C2—42no—
601C2—42no—
602C2#3544711yescharacteristic
603C2C2×#304411yescharacteristic
604C2#6042511yescharacteristic
605C2#6175412yes—
606C2#6175712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #28132C2semidirect
6. #27656C2semidirect
7. #25208C2semidirect
8. #25206C2semidirect
9. #24588C2semidirect
10. #24540C2semidirect
52. #2085C22semidirect
53. #2083C22semidirect
54. #2074C22semidirect
55. #2072C22semidirect
56. #2068C22semidirect
57. #2066C22semidirect
58. #2056C22semidirect
59. #2056C22semidirect
60. #1949C22semidirect
61. #1947C22semidirect
62. #1928C22semidirect
63. #1927C22semidirect
65. #1878C22semidirect
66. #1878C22semidirect
73. #1805C22semidirect
74. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C2×#30952
#353112
#343802
#281322
#276562
#252082
#252062
#245882
#245402
#219332
#217652
C2×#18252