Finite Group Databasebeta

← #6508791

Subgroups — #6508791

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C211yesmaximal, characteristic
2C2×#3094C211yesmaximal, characteristic
3#35311C211yesmaximal, characteristic, complemented
4#34363C211yesmaximal, characteristic
5#28128C214yesmaximal, complemented
6#27655C214yesmaximal, complemented
7#25225C214yesmaximal, complemented
8#25223C214yesmaximal, complemented
9#24588C214yesmaximal, complemented
10#24538C214yesmaximal, complemented
11#21935C211yesmaximal, characteristic, complemented
12#21768C211yesmaximal, characteristic, complemented
13#20701C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3193—28no—
18#3094—28no—
19C2×#484C2212yes—
20C2×#481C2212yes—
21C2×#482C2214yes—
22C2×#480C2212yes—
23C2×#480C2212yes—
24C2×#468C2214yes—
25C2×#448C2212yes—
26C2×#450C2212yes—
27C2×#428C2212yes—
28C2×#418C2212yes—
29C2×#407C2211yescharacteristic
30C2×#407C2211yescharacteristic
31#2855—22no—
32#2625—22no—
33#2503C2212yescomplemented
34#2502C2212yescomplemented
35#2499C2214yes—
36#2484—22no—
37#2483—21no—
38#2483—21no—
39#2377—22no—
40#2363C2214yes—
41#2344C2212yes—
42#2342—22no—
43D4×C22⋊C4C2212yes—
44#2333—22no—
45#2324C2214yes—
46#2317C2212yescomplemented
47#2310C2212yescomplemented
48#2309—21no—
49#2308—21no—
50#2280C2211yescharacteristic
51#2280C2211yescharacteristic
52#2085C2214yescomplemented
53#2083C2214yescomplemented
54#2073C2214yescomplemented
55#2071C2214yescomplemented
56#2068C2214yescomplemented
57#2066C2214yescomplemented
58#2055C2214yescomplemented
59#2055C2214yescomplemented
60#1949C2214yescomplemented
61#1947C2214yescomplemented
62#1928C2214yescomplemented
63#1927C2214yescomplemented
64#1894C2214yes—
65#1877C2214yescomplemented
66#1877C2214yescomplemented
67#1873C2214yescomplemented
68#1851C2214yes—
69#1851C2214yescomplemented
70#1825C2214yes—
71#1825C2214yescomplemented
72#1824C2212yescomplemented
73#1804C2214yescomplemented
74#1802C2214yescomplemented
75#1798C2214yescomplemented
76#1787C2212yescomplemented
77#1736C2212yescomplemented
78#1736C2212yescomplemented
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×SD16C2314yes—
84C22×M16C2312yes—
85C22×M16C2312yes—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—24no—
95C2×#121D418yes—
96C2×#121C2312yes—
97C2×#121C2312yes—
98C2×C4×D4C2314yes—
99C2×#117—21no—
100C2×#117—21no—
101C2×#117—22no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—22no—
117C22×C42—21no—
118#484—28no—
119#482—216no—
120#481—28no—
121#480—28no—
122#480—28no—
123#468—216no—
124#450—28no—
125#448—28no—
126#428—28no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132#418—28no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139#407—24no—
140#407—24no—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145#400—24no—
146#398—24no—
147#397—24no—
148#397—24no—
149#391—24no—
150#391C2314yes—
151#391C2314yes—
152#390—24no—
153#390—24no—
154#390C2314yes—
155#389—24no—
156#389C2314yes—
157#388—24no—
158#388—24no—
159#387—24no—
160#387—24no—
161#387—24no—
162#386—24no—
163#386—24no—
164#386—24no—
165#385—24no—
166#385—22no—
167#385—22no—
168#385—22no—
169#385—22no—
170#384C2314yes—
171#382C2314yes—
172#381—21no—
173#381—22no—
174#381—22no—
175#381—22no—
176#381—22no—
177#381—22no—
178#381D412yes—
179#380D412yes—
180#380D412yes—
181#380C2312yes—
182#380C2312yes—
183#379—21no—
184#379—22no—
185#379D411yescharacteristic
186#379D411yescharacteristic
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—24no—
191C4×C22⋊C4C2312yes—
192C4×C22⋊C4C2312yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25C2×D411yescharacteristic
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—42no—
207C22×Q8—42no—
208C22×Q8C2412yes—
209C22×D4—28no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4—84no—
217C22×D4C2×D414yes—
218C22×D4C2412yes—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D414yes—
231C23×C4#4114yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×SD16—28no—
236C2×SD16—44no—
237C2×SD16—44no—
238C2×M16—24no—
239C2×M16—24no—
240C2×M16—42no—
241C2×M16—42no—
242C2×M16—42no—
243C2×M16—42no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—44no—
251C22⋊Q8—44no—
252#121—28no—
253#121—28no—
254#121—24no—
255#121—24no—
256#121C2×D418yes—
257#121C2×D418yes—
258C4×D4—44no—
259C4×D4—44no—
260#117—24no—
261#117—24no—
262#117—24no—
263#117—24no—
264#117—42no—
265#117—42no—
266#117—44no—
267#117—44no—
268#117—42no—
269#117—42no—
270#117—84no—
271C2×C4⋊C4—21no—
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—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—28no—
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—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—84no—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—44no—
322C2×C42C2414yes—
323#107—28no—
324#107—28no—
325#106—28no—
326C4⋊C8—44no—
327C4⋊C8—44no—
328#103—28no—
329#103—44no—
330#103—44no—
331#102—28no—
332#102—28no—
333#102—28no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337C22⋊C8—24no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95C2×D414yes—
349#95C2×D414yes—
350C24—22no—
351C24—24no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—44no—
356C24—82no—
357C24—82no—
358C24#12114yes—
359C24C22×D411yescharacteristic
360C24C22×D411yescharacteristic
361C24C22×D411yescharacteristic
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—44no—
367C2×Q8—42no—
368C2×Q8—84no—
369C2×Q8—82no—
370C2×Q8—82no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—48no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4—816no—
386C2×D4C22×D414yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4#12114yes—
419C22×C4#12114yes—
420C22×C4#12114yes—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2511yescharacteristic
433SD16—88no—
434M16—48no—
435M16—84no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4C22×D414yes—
462C22⋊C4—28no—
463C22⋊C4—28no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C42—24no—
481C42—24no—
482C42—44no—
483C42—44no—
484C42—82no—
485C42—84no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—48no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—84no—
505C23—84no—
506C23—88no—
507C23—168no—
508C23C2×#12112yes—
509C23C2×#12112yes—
510C23C2×#12112yes—
511C23C23×D411yescharacteristic
512Q8—44no—
513Q8—42no—
514Q8—42no—
515Q8—84no—
516D4—28no—
517D4—42no—
518D4—42no—
519D4—84no—
520D4—88no—
521D4—88no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—168no—
563C2×C4C2×#12112yes—
564C2×C4C2×#12112yes—
565C2×C4C2×#12112yes—
566C2×C4C23×D411yescharacteristic
567C2×C4C22×#4111yescharacteristic
568C2×C4#58512yes—
569C2×C4#58512yes—
570C8—44no—
571C8—44no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—88no—
588C22—84no—
589C22—84no—
590C22—168no—
591C22C22×#12111yescharacteristic
592C22#348512yes—
593C22#358512yes—
594C22#358612yes—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—44no—
599C4—44no—
600C4—44no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—84no—
605C4—82no—
606C4#348512yes—
607C4#349111yescharacteristic
608C4#349111yescharacteristic
609C2—42no—
610C2—42no—
611C2—42no—
612C2—84no—
613C2#3544711yescharacteristic
614C2#6042311yescharacteristic
615C2#6042711yescharacteristic
616C2#6178014yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
3. #35311C2semidirect
5. #28128C2semidirect
6. #27655C2semidirect
7. #25225C2semidirect
8. #25223C2semidirect
9. #24588C2semidirect
10. #24538C2semidirect
11. #21935C2semidirect
12. #21768C2semidirect
13. #20701C2semidirect
33. #2503C22semidirect
34. #2502C22semidirect
46. #2317C22semidirect
47. #2310C22semidirect
52. #2085C22semidirect
53. #2083C22semidirect
54. #2073C22semidirect
55. #2071C22semidirect
56. #2068C22semidirect
57. #2066C22semidirect
58. #2055C22semidirect
59. #2055C22semidirect
60. #1949C22semidirect
61. #1947C22semidirect
62. #1928C22semidirect
63. #1927C22semidirect
65. #1877C22semidirect
66. #1877C22semidirect
67. #1873C22semidirect
69. #1851C22semidirect
71. #1825C22semidirect
72. #1824C22semidirect
73. #1804C22semidirect
74. #1802C22semidirect
75. #1798C22semidirect
76. #1787C22semidirect
77. #1736C22semidirect
78. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C2×#30942
#353112
#343632
#281282
#276552
#252252
#252232
#245882
#245382
#219352
#217682
#207012