Finite Group Databasebeta

← #4929373

Subgroups — #4929373

Order 512.

2978 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C2×#2273C211yesmaximal, characteristic
3#15043C214yesmaximal, complemented
4#14830C214yesmaximal, complemented
5#14797C214yesmaximal, complemented
6#14727C214yesmaximal
7#14292C211yesmaximal, characteristic, complemented
8#14197C214yesmaximal, complemented
9#14177C214yesmaximal, complemented
10#14165C211yesmaximal, characteristic, complemented
11#14138C211yesmaximal, characteristic, complemented
12#13868C211yesmaximal, characteristic
13#13852C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C22⋊C4C2212yes—
17C23×C22⋊C4C2211yescharacteristic
18#2561—28no—
19C2×#397C2212yes—
20C2×#397C2214yes—
21C2×#397C2214yes—
22C2×#393C2212yes—
23C2×#391C2212yescomplemented
24C2×#388C2214yes—
25C2×#385C2214yes—
26C2×#381C2212yes—
27C2×#381C2211yescharacteristic
28C2×#379C2212yes—
29C2×#379C2212yes—
30#2273—28no—
31C2×C4×C4⋊C4C2212yescomplemented
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2214yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2214yes—
38#1446C2212yescomplemented
39#1445C2214yescomplemented
40#1445C2212yescomplemented
41#1443C2214yescomplemented
42#1443C2214yescomplemented
43#1442C2214yescomplemented
44#1441C2214yescomplemented
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1437C2212yes—
53#1437C2212yes—
54#1436C2212yescomplemented
55#1436C2212yescomplemented
56#1436C2214yescomplemented
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61C4⋊#95C2212yescomplemented
62#1434C2214yescomplemented
63#1434C2214yescomplemented
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C4×#95C2212yescomplemented
69C26C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C24×C4C2314yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—24no—
77C2×#117—24no—
78C2×#117—44no—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—216no—
88C22×C22⋊C4D418yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2314yes—
107C22×C42C2312yes—
108#397—28no—
109#397—216no—
110#397—216no—
111#393—28no—
112#391—44no—
113#388—216no—
114#385—48no—
115#381—28no—
116#381—24no—
117#379—28no—
118#379—44no—
119C4×C4⋊C4—44no—
120C4×C22⋊C4—24no—
121C4×C22⋊C4—28no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—24no—
124C4×C22⋊C4—216no—
125C4×C22⋊C4—216no—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C25—22no—
151C25—22no—
152C25—28no—
153C25—44no—
154C25—42no—
155C25—42no—
156C25C2×D414yes—
157C25C2411yescharacteristic
158C25C2412yes—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—42no—
163C22×Q8—42no—
164C23×C4—28no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—28no—
168C23×C4—28no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4C2×D414yes—
177C23×C4C2×D414yes—
178C23×C4#4114yes—
179C23×C4#4114yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C22⋊Q8—416no—
192#117—416no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C22⋊C4—264no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—232no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4C2×D418yes—
244C2×C22⋊C4C2×D418yes—
245C2×C22⋊C4#41116yes—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—44no—
261C2×C42—44no—
262#95—28no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—24no—
269#95—24no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286C24—216no—
287C24—216no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—22no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—22no—
298C24—24no—
299C24—28no—
300C24—216no—
301C24—48no—
302C24—48no—
303C24—82no—
304C24—82no—
305C24—84no—
306C24—84no—
307C24—84no—
308C24—88no—
309C24—88no—
310C24—88no—
311C24—88no—
312C24C22×D412yes—
313C24C22×D414yes—
314C24C2×#4112yes—
315C24C2×#4112yes—
316C24C2×#4118yes—
317C24C2511yescharacteristic
318C2×Q8—48no—
319C2×Q8—44no—
320C2×Q8—82no—
321C2×Q8—82no—
322C2×Q8—84no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—232no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—44no—
379C22×C4—48no—
380C22×C4—44no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4#123116yes—
392C22×C4C22×D414yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C2×#4118yes—
396C22×C4C2×#4118yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4H(5,𝔽2)14yes—
402C4⋊C4—416no—
403C4⋊C4—48no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—88no—
407C22⋊C4—216no—
408C22⋊C4—216no—
409C22⋊C4—232no—
410C22⋊C4—216no—
411C22⋊C4—216no—
412C22⋊C4—416no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—416no—
417C22⋊C4—88no—
418C22⋊C4—84no—
419C22⋊C4—88no—
420C42—28no—
421C42—44no—
422C42—84no—
423C42—84no—
424C42—84no—
425C42—88no—
426C23—232no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—28no—
433C23—28no—
434C23—24no—
435C23—28no—
436C23—22no—
437C23—22no—
438C23—28no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—28no—
444C23—28no—
445C23—28no—
446C23—28no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—88no—
462C23—88no—
463C23—832no—
464C23—816no—
465C23—832no—
466C23—816no—
467C23C2×#12314yes—
468C23#52518yes—
469C23C23×D411yescharacteristic
470C23C22×#4111yescharacteristic
471C23C22×#4111yescharacteristic
472C23C22×#4114yes—
473C23C2×H(5,𝔽2)12yes—
474C23#58514yes—
475C23#58512yes—
476Q8—88no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—416no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—88no—
540C2×C4—832no—
541C2×C4C2×#12314yes—
542C2×C4C2×#12314yes—
543C22—24no—
544C22—28no—
545C22—24no—
546C22—24no—
547C22—216no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—48no—
552C22—48no—
553C22—48no—
554C22—48no—
555C22—44no—
556C22—48no—
557C22—48no—
558C22—48no—
559C22—832no—
560C22C22×#12311yescharacteristic
561C22C2×#52512yes—
562C22#344412yes—
563C22#344412yes—
564C22#348512yes—
565C22#349011yescharacteristic
566C22#349111yescharacteristic
567C22#349612yes—
568C22#350512yes—
569C22#351012yes—
570C22#351012yes—
571C22#351314yes—
572C22#351414yes—
573C22#352114yes—
574C22#352114yes—
575C4—24no—
576C4—24no—
577C4—44no—
578C4—44no—
579C4—48no—
580C4—48no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—84no—
585C4—84no—
586C2—44no—
587C2—48no—
588C2#3536011yescharacteristic
589C2#3540812yes—
590C2#3548812yes—
591C2#3562611yescharacteristic
592C2#3579214yes—
593C2#3596714yes—
594C2#3662611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #15043C2semidirect
4. #14830C2semidirect
5. #14797C2semidirect
7. #14292C2semidirect
8. #14197C2semidirect
9. #14177C2semidirect
10. #14165C2semidirect
11. #14138C2semidirect
13. #13852C2semidirect
23. C2×#391C22semidirect
31. C2×C4×C4⋊C4C22semidirect
38. #1446C22semidirect
39. #1445C22semidirect
40. #1445C22semidirect
41. #1443C22semidirect
42. #1443C22semidirect
43. #1442C22semidirect
44. #1441C22semidirect
54. #1436C22semidirect
55. #1436C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. C4⋊#95C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect
68. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#22732
#150432
#148302
#147972
#147272
#142922
#141972
#141772
#141652
#141382
#138682
#138522