Finite Group Databasebeta

← #6508793

Subgroups — #6508793

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic
3#35311C211yesmaximal, characteristic
4#34370C211yesmaximal, characteristic
5#28132C214yesmaximal, complemented
6#27657C214yesmaximal, complemented
7#25229C214yesmaximal, complemented
8#25219C214yesmaximal, complemented
9#24590C214yesmaximal, complemented
10#24540C214yesmaximal, complemented
11#21935C211yesmaximal, characteristic
12#21768C211yesmaximal, characteristic
13#20701C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3195—28no—
18#3095—28no—
19C2×#484C2212yes—
20C2×#485C2214yes—
21C2×#483C2212yes—
22C2×#483C2212yes—
23C2×#483C2212yes—
24C2×#470C2214yes—
25C2×#451C2212yes—
26C2×#451C2212yes—
27C2×#428C2212yes—
28C2×#419C2212yes—
29C2×#407C2211yescharacteristic
30C2×#407C2211yescharacteristic
31#2855—22no—
32#2625—22no—
33#2503C2212yes—
34#2502C2212yes—
35#2499C2214yes—
36#2484—22no—
37#2483—21no—
38#2483—21no—
39#2378—22no—
40#2362C2214yes—
41#2345—22no—
42#2345C2212yes—
43Q8×C22⋊C4C2212yes—
44#2332—22no—
45#2323C2214yes—
46#2317C2212yes—
47#2312—21no—
48#2312—21no—
49#2310C2212yes—
50#2280C2211yescharacteristic
51#2280C2211yescharacteristic
52#2085C2214yescomplemented
53#2084C2214yescomplemented
54#2074C2214yescomplemented
55#2072C2214yescomplemented
56#2068C2214yescomplemented
57#2066C2214yescomplemented
58#2056C2214yescomplemented
59#2056C2214yescomplemented
60#1949C2214yescomplemented
61#1948C2214yescomplemented
62#1928C2214yescomplemented
63#1927C2214yescomplemented
64#1894C2214yes—
65#1878C2214yescomplemented
66#1878C2214yescomplemented
67#1873C2214yes—
68#1851C2214yes—
69#1851C2214yes—
70#1825C2214yes—
71#1825C2214yes—
72#1824C2212yes—
73#1805C2214yescomplemented
74#1803C2214yescomplemented
75#1798C2214yes—
76#1787C2212yes—
77#1736C2212yes—
78#1736C2212yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×Q16C2314yes—
83C22×M16C2312yes—
84C22×M16C2312yes—
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—24no—
101C2×#117—24no—
102C2×#117C2312yes—
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×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42—21no—
116#485—216no—
117#484—28no—
118#483—28no—
119#483—28no—
120#483—28no—
121#470—216no—
122#451—28no—
123#451—28no—
124#428—28no—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2314yes—
130#419—28no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137#407—24no—
138#407—24no—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143#400—24no—
144#398—24no—
145#397—24no—
146#397—24no—
147#391—24no—
148#391—24no—
149#391C2314yes—
150#391C2314yes—
151#391C2314yes—
152#389—24no—
153#389—24no—
154#389C2314yes—
155#388—24no—
156#388—24no—
157#388—24no—
158#388—24no—
159#387—24no—
160#387—24no—
161#387—24no—
162#387—24no—
163#387—24no—
164#385—22no—
165#385—22no—
166#385—22no—
167#385—22no—
168#384C2314yes—
169#382C2314yes—
170#381—21no—
171#381—22no—
172#381—22no—
173#381—22no—
174#381—22no—
175#381D412yes—
176#380—22no—
177#380—22no—
178#380D412yes—
179#380D412yes—
180#380C2312yes—
181#380C2312yes—
182#379—21no—
183#379D411yescharacteristic
184#379D411yescharacteristic
185C4×C4⋊C4—24no—
186C4×C4⋊C4C2314yes—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4C2312yes—
192C4×C22⋊C4C2312yes—
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—21no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D414yes—
226C23×C4#4114yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C2×Q16—28no—
231C2×Q16—44no—
232C2×Q16—44no—
233C2×M16—24no—
234C2×M16—24no—
235C2×M16—42no—
236C2×M16—42no—
237C2×M16—42no—
238C2×M16—42no—
239C22×C8C2414yes—
240C22×C8C2414yes—
241C22⋊Q8—28no—
242C22⋊Q8—28no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—44no—
250C22⋊Q8—44no—
251C22⋊Q8C2×D418yes—
252C22⋊Q8C2×D418yes—
253C4×Q8—44no—
254C4×Q8—44no—
255#117—24no—
256#117—24no—
257#117—24no—
258#117—24no—
259#117—42no—
260#117—42no—
261#117—44no—
262#117—44no—
263#117—42no—
264#117—42no—
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—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42C2414yes—
318#107—28no—
319#106—28no—
320#106—28no—
321C4⋊C8—44no—
322C4⋊C8—44no—
323#103—28no—
324#103—28no—
325#103—28no—
326#103—28no—
327#103—44no—
328#103—44no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331C22⋊C8—24no—
332C22⋊C8—24no—
333#95—22no—
334#95—22no—
335#95—24no—
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#95C2×D414yes—
346#95C2×D414yes—
347C24—42no—
348C24—42no—
349C24—42no—
350C24#12114yes—
351C24C22×D411yescharacteristic
352C24C22×D411yescharacteristic
353C24C22×D411yescharacteristic
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—44no—
363C2×Q8—48no—
364C2×Q8—42no—
365C2×Q8—82no—
366C2×Q8—84no—
367C2×Q8—82no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×Q8—82no—
371C2×Q8C22×D414yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4#12114yes—
404C22×C4#12114yes—
405C22×C4#12114yes—
406C22×C4C22×D414yes—
407C22×C4C22×D414yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2511yescharacteristic
418Q16—88no—
419M16—48no—
420M16—84no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4C22×D414yes—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C42—24no—
466C42—24no—
467C42—44no—
468C42—44no—
469C42—84no—
470C42—82no—
471C42—84no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—88no—
485C23C2×#12112yes—
486C23C2×#12112yes—
487C23C2×#12112yes—
488C23C23×D411yescharacteristic
489Q8—28no—
490Q8—42no—
491Q8—42no—
492Q8—44no—
493Q8—42no—
494Q8—42no—
495Q8—84no—
496Q8—84no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4C2×#12112yes—
540C2×C4C2×#12112yes—
541C2×C4C2×#12112yes—
542C2×C4C23×D411yescharacteristic
543C2×C4C22×#4111yescharacteristic
544C2×C4#58512yes—
545C2×C4#58512yes—
546C8—44no—
547C8—44no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—88no—
561C22C22×#12111yescharacteristic
562C22#348512yes—
563C22#358612yes—
564C22#358612yes—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—44no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—84no—
576C4—82no—
577C4#348512yes—
578C4#349111yescharacteristic
579C4#349111yescharacteristic
580C2—42no—
581C2—42no—
582C2—42no—
583C2#3544711yescharacteristic
584C2#6042411yescharacteristic
585C2#6042711yescharacteristic
586C2#6178114yes—
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. #27657C2semidirect
7. #25229C2semidirect
8. #25219C2semidirect
9. #24590C2semidirect
10. #24540C2semidirect
52. #2085C22semidirect
53. #2084C22semidirect
54. #2074C22semidirect
55. #2072C22semidirect
56. #2068C22semidirect
57. #2066C22semidirect
58. #2056C22semidirect
59. #2056C22semidirect
60. #1949C22semidirect
61. #1948C22semidirect
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×#31952
C2×#30952
#353112
#343702
#281322
#276572
#252292
#252192
#245902
#245402
#219352
#217682
#207012