Finite Group Databasebeta

← #6653183

Subgroups — #6653183

Order 512.

1688 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#34946C211yesmaximal, characteristic
3#32905C212yesmaximal
4#31857C214yesmaximal, complemented
5#31663C214yesmaximal, complemented
6#30754C214yesmaximal, complemented
7#30464C214yesmaximal
8#29664C212yesmaximal, complemented
9#29131C212yesmaximal, complemented
10#28449C212yesmaximal, complemented
11#28254C212yesmaximal, complemented
12#28159C212yesmaximal, complemented
13#21926C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#497C2214yes—
17C2×#464C2214yes—
18C2×#462C2212yescomplemented
19C2×#462C2212yescomplemented
20C2×#461C2212yes—
21C2×#461C2212yes—
22C2×#450C2214yes—
23#2868C2211yescharacteristic, complemented
24#2859C2211yescharacteristic, complemented
25#2857C2212yes—
26#2647—21no—
27#2629—21no—
28#2563C2212yes—
29#2562—22no—
30#2561—22no—
31#2547C2212yes—
32#2507—22no—
33#2503C2212yes—
34#2497—22no—
35#2446C2214yescomplemented
36#2435C2212yes—
37#2429C2214yes—
38#2398—21no—
39#2398—21no—
40#2330—21no—
41#2314—21no—
42C4×C4⋊Q8C2212yescomplemented
43C4×#124C2212yes—
44C4×#124C2212yes—
45#2280—21no—
46#2280—21no—
47C4×#117C2211yescharacteristic
48#2101C2214yescomplemented
49#2074C2214yescomplemented
50#2056C2214yescomplemented
51#2055C2214yes—
52#2042C2214yes—
53#2024C2214yescomplemented
54#2016C2214yescomplemented
55#2016C2214yescomplemented
56#2015C2214yes—
57#2013C2214yes—
58#2001C2214yes—
59#2001C2214yes—
60#1960C2214yescomplemented
61#1936C2214yes—
62#1935C2214yescomplemented
63#1924C2214yes—
64#1917C2214yescomplemented
65#1880C2214yes—
66#1878C2214yescomplemented
67#1877C2214yes—
68#1877C2214yes—
69#1872C2214yes—
70#1842C2211yescharacteristic, complemented
71#1831C2211yescharacteristic, complemented
72#1805C2214yescomplemented
73#1803C2212yescomplemented
74#1803C2212yescomplemented
75#1802C2212yes—
76#1802C2212yes—
77#1797C2212yes—
78#1767C2212yescomplemented
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×Q16C2314yes—
82C22×Q16C2314yes—
83C22×SD16C2314yes—
84C22×SD16C2314yes—
85#533—42no—
86#533—42no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2314yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#125—21no—
92C2×#125—21no—
93C2×#124C2312yes—
94C2×#124C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2312yes—
98C2×C4×Q8—21no—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2314yes—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—42no—
104C2×C4×D4—42no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×#117—21no—
108C2×#117D411yescharacteristic
109C2×#117D411yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C42C2311yescharacteristic
113#497—48no—
114#464—48no—
115#462—44no—
116#462—44no—
117#461—44no—
118#461—44no—
119#450—48no—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2314yes—
133#400—22no—
134#400—22no—
135#400—24no—
136#398—24no—
137#398—24no—
138#397—22no—
139#397—22no—
140#396—24no—
141#394—22no—
142#394—22no—
143#393—22no—
144#393—22no—
145#391C2314yes—
146#391C2312yes—
147#391C2314yes—
148#391C2312yes—
149#390C2312yes—
150#390C2312yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—24no—
157#389C2314yes—
158#388—24no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#387—24no—
164#387—24no—
165#384—21no—
166#384—21no—
167#384—22no—
168#384C2312yes—
169#382—21no—
170#382C2312yes—
171#382C2312yes—
172#381—22no—
173#381—22no—
174#380—22no—
175#380—22no—
176#380—22no—
177#380—22no—
178#380—22no—
179#380—22no—
180#379—22no—
181#379—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4C2312yes—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4D412yes—
191C4×C22⋊C4C2312yes—
192#376D411yescharacteristic
193#376D411yescharacteristic
194C43C2311yescharacteristic
195C43C2311yescharacteristic
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25—21no—
204C22×Q8—21no—
205C22×Q8C2412yes—
206C22×Q8C2414yes—
207C22×D4—24no—
208C22×D4—21no—
209C22×D4—41no—
210C22×D4—41no—
211C22×D4—42no—
212C22×D4C2412yes—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2411yescharacteristic
222C2×Q16—44no—
223C2×Q16—44no—
224C2×Q16—44no—
225C2×Q16—44no—
226C2×Q16—44no—
227C2×Q16—44no—
228C2×SD16—28no—
229C2×SD16—28no—
230C2×SD16—44no—
231C2×SD16—44no—
232C2×SD16—44no—
233C2×SD16—44no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238C4⋊Q8—41no—
239C4⋊Q8—41no—
240C4⋊Q8—41no—
241C4⋊Q8—41no—
242C4⋊Q8—44no—
243C4⋊Q8—44no—
244#125—44no—
245#125—44no—
246#124—24no—
247#124—24no—
248#124—42no—
249#124—42no—
250C22⋊Q8—42no—
251C22⋊Q8—42no—
252C22⋊Q8—44no—
253C22⋊Q8—44no—
254C22⋊Q8—44no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—44no—
258C4×Q8—44no—
259C4×Q8—44no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—42no—
263C4×D4—42no—
264C4×D4—44no—
265C4×D4—44no—
266#117—24no—
267#117—42no—
268#117—44no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2414yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D412yes—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2411yescharacteristic
319C2×C42C2412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C4⋊C8—24no—
323C4⋊C8—24no—
324C4⋊C8—44no—
325#103—44no—
326#103—44no—
327#103—44no—
328#103—44no—
329#103—44no—
330#103—44no—
331#102—28no—
332#102—44no—
333C22⋊C8—44no—
334#97—44no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D412yes—
351#95C2×D412yes—
352C24—22no—
353C24—44no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—82no—
358C24C22×D411yescharacteristic
359C2×Q8—22no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—42no—
372C2×Q8—82no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—44no—
379C2×D4—42no—
380C2×D4—44no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4H(5,𝔽2)14yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—816no—
407C22×C4#13714yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2511yescharacteristic
418Q16—88no—
419Q16—88no—
420SD16—88no—
421SD16—88no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C22⋊C4—24no—
452C22⋊C4—42no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—84no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—21no—
474C42—21no—
475C42—22no—
476C42—22no—
477C42—42no—
478C42—42no—
479C42—44no—
480C42—84no—
481C42H(5,𝔽2)12yes—
482C42H(5,𝔽2)12yes—
483C23—22no—
484C23—21no—
485C23—22no—
486C23—21no—
487C23—24no—
488C23—48no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—88no—
496C23—84no—
497C23—88no—
498C23—84no—
499C23C2×#13712yes—
500C23C23×D411yescharacteristic
501Q8—44no—
502Q8—42no—
503Q8—44no—
504Q8—42no—
505Q8—44no—
506Q8—42no—
507Q8—44no—
508Q8—42no—
509Q8—84no—
510D4—24no—
511D4—24no—
512D4—48no—
513D4—42no—
514D4—42no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—82no—
554C2×C4—88no—
555C2×C4—84no—
556C2×C4—88no—
557C2×C4#53512yes—
558C2×C4#53512yes—
559C2×C4C2×#13712yes—
560C2×C4C2×#13712yes—
561C2×C4#57512yes—
562C2×C4C2×H(5,𝔽2)11yescharacteristic
563C2×C4C2×H(5,𝔽2)12yes—
564C2×C4C2×H(5,𝔽2)11yescharacteristic
565C2×C4#58512yes—
566C8—44no—
567C8—44no—
568C22—24no—
569C22—22no—
570C22—42no—
571C22—41no—
572C22—41no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—816no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22C2×#53511yescharacteristic
582C22#348812yes—
583C22C22×#13711yescharacteristic
584C22C2×#57511yescharacteristic
585C22#358612yes—
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—84no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—84no—
599C4#321514yes—
600C4#353711yescharacteristic
601C4#353911yescharacteristic
602C4#354212yes—
603C2—42no—
604C2—84no—
605C2#3727011yescharacteristic
606C2C2×#321511yescharacteristic
607C2#6080911yescharacteristic
608C2#6196012yes—
609C2#6196712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35284C2semidirect
4. #31857C2semidirect
5. #31663C2semidirect
6. #30754C2semidirect
8. #29664C2semidirect
9. #29131C2semidirect
10. #28449C2semidirect
11. #28254C2semidirect
12. #28159C2semidirect
13. #21926C2semidirect
18. C2×#462C22semidirect
19. C2×#462C22semidirect
23. #2868C22semidirect
24. #2859C22semidirect
35. #2446C22semidirect
42. C4×C4⋊Q8C22semidirect
48. #2101C22semidirect
49. #2074C22semidirect
50. #2056C22semidirect
53. #2024C22semidirect
54. #2016C22semidirect
55. #2016C22semidirect
60. #1960C22semidirect
62. #1935C22semidirect
64. #1917C22semidirect
66. #1878C22semidirect
70. #1842C22semidirect
71. #1831C22semidirect
72. #1805C22semidirect
73. #1803C22semidirect
74. #1803C22semidirect
78. #1767C22semidirect

Maximal subgroups

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

subgroupindex
#352842
#349462
#329052
#318572
#316632
#307542
#304642
#296642
#291312
#284492
#282542
#281592
#219262