Finite Group Databasebeta

← #6529865

Subgroups — #6529865

Order 512.

1938 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3109C211yesmaximal, characteristic
2C2×#2987C211yesmaximal, characteristic, complemented
3#37619C211yesmaximal, characteristic, complemented
4#35172C211yesmaximal, characteristic, complemented
5#28276C214yesmaximal, complemented
6#28197C214yesmaximal, complemented
7#25485C214yesmaximal, complemented
8#25304C214yesmaximal, complemented
9#24618C214yesmaximal, complemented
10#24512C214yesmaximal, complemented
11#21989C211yesmaximal, characteristic, complemented
12#21869C211yesmaximal, characteristic, complemented
13#20739C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17#3109—28no—
18C2×#481C2212yes—
19C2×#476C2212yes—
20C2×#459C2212yes—
21C2×#447C2212yes—
22#2987—28no—
23C2×#446C2214yescomplemented
24C2×#446C2214yescomplemented
25C2×#442C2214yes—
26C2×#433C2214yes—
27C2×#432C2211yescharacteristic
28C2×#432C2211yescharacteristic
29#2848—22no—
30#2816—22no—
31#2658—21no—
32#2640—21no—
33#2633—22no—
34#2626—22no—
35#2624C2214yes—
36#2580C2214yes—
37#2580C2212yescomplemented
38#2580C2212yescomplemented
39#2566—22no—
40#2563—21no—
41#2548C2212yescomplemented
42#2546C2212yescomplemented
43#2542—21no—
44#2538C2214yes—
45#2432—22no—
46#2317C2212yes—
47#2309C2212yes—
48#2285C2211yescharacteristic
49#2283C2211yescharacteristic
50#2088C2214yescomplemented
51#2088C2214yescomplemented
52#2067C2214yes—
53#2067C2214yescomplemented
54#2065C2214yes—
55#2065C2214yescomplemented
56#2054C2214yes—
57#2054C2214yes—
58#1993C2212yescomplemented
59#1972C2214yes—
60#1970C2214yes—
61#1961C2212yescomplemented
62#1949C2214yescomplemented
63#1943C2214yescomplemented
64#1941C2214yescomplemented
65#1936C2214yes—
66#1936C2214yescomplemented
67#1934C2214yes—
68#1934C2214yescomplemented
69#1875C2214yes—
70#1854C2214yes—
71#1847C2212yescomplemented
72#1836C2212yescomplemented
73#1823C2212yes—
74#1804C2212yes—
75#1804C2212yes—
76#1802C2214yescomplemented
77#1786C2212yes—
78#1762C2214yes—
79C22×#41—21no—
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C22×D8C2314yes—
83#533—24no—
84#533—24no—
85#532—24no—
86#532—24no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8D411yescharacteristic
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#125D412yes—
93C2×#124—22no—
94C2×#124—21no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8D412yes—
97C2×C22⋊Q8D412yes—
98C2×#121—21no—
99C2×#121C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C42C2311yescharacteristic
115#481—28no—
116#476—28no—
117#459—28no—
118#447—28no—
119#446—216no—
120#446—216no—
121#442—216no—
122#433—216no—
123#432—24no—
124#432D418yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2312yes—
141C2×#97C2314yes—
142C2×#97C2312yes—
143C2×C4×C8C2314yes—
144#399—22no—
145#399—22no—
146#399—24no—
147#399—24no—
148#398—24no—
149#397—24no—
150#396—24no—
151#396—24no—
152#394—24no—
153#394—24no—
154#394—24no—
155#393—22no—
156#393—22no—
157#393—24no—
158#391—22no—
159#390—22no—
160#390—22no—
161#390—22no—
162#390C2314yes—
163#390C2314yes—
164#390C2314yes—
165#390C2314yes—
166#389C2314yes—
167#389C2314yes—
168#389C2314yes—
169#389C2314yes—
170#385—24no—
171#385—24no—
172#385—24no—
173#385—24no—
174#385—24no—
175#384—22no—
176#384C2312yes—
177#383—21no—
178#382—22no—
179#382C2311yescharacteristic
180#382C2311yescharacteristic
181#382C2312yes—
182#381—21no—
183#380—21no—
184#380—21no—
185#380—24no—
186#380C2312yes—
187#379—24no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—21no—
192#376—21no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199C25—21no—
200C2×#41—44no—
201C2×#41—84no—
202C22×Q8—21no—
203C22×Q8C2×D412yes—
204C22×Q8C2×D412yes—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—22no—
212C22×D4—42no—
213C22×D4C2412yes—
214C22×D4C2412yes—
215C23×C4—21no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—44no—
220C23×C4—84no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2411yescharacteristic
224C2×D8—28no—
225C2×D8—28no—
226C2×D8—44no—
227C22×C8C2414yes—
228C22×C8C2414yes—
229C4⋊Q8—24no—
230C4⋊Q8—24no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—84no—
234#127—22no—
235#127—22no—
236#125—28no—
237#124—44no—
238#124—84no—
239C22⋊Q8—28no—
240C22⋊Q8—28no—
241C22⋊Q8—88no—
242#121—24no—
243#121—24no—
244#121—44no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—44no—
256C4×D4—44no—
257C4×D4—44no—
258#117—24no—
259#117—24no—
260#117—42no—
261#117—42no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4C2×D414yes—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D414yes—
304C2×C42#4114yes—
305C2×C42C2414yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#107—44no—
309#106—28no—
310#106—44no—
311C4⋊C8—28no—
312C4⋊C8—28no—
313C4⋊C8—24no—
314C4⋊C8C2×D418yes—
315#102—28no—
316#102—28no—
317#102—28no—
318#102—44no—
319#102—44no—
320#102—44no—
321C22⋊C8—24no—
322C22⋊C8C2×D418yes—
323#97—28no—
324#97—24no—
325#97C2×D418yes—
326C4×C8—28no—
327#95—22no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—42no—
343C24—42no—
344C24—82no—
345C24—82no—
346C24C22×D411yescharacteristic
347#41—816no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—44no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—44no—
354C2×Q8—42no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—42no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—82no—
374C2×D4—84no—
375C2×D42−1+414yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—82no—
393C22×C4—816no—
394C22×C4#12114yes—
395C22×C4#12114yes—
396C22×C4C22×D414yes—
397C22×C4C22×D414yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C2×#4112yes—
405C22×C4C2511yescharacteristic
406D8—416no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8C22×D418yes—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—82no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C22⋊C4—28no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—82no—
450C42—24no—
451C42—24no—
452C42—44no—
453C42—44no—
454C42#12114yes—
455C42#12114yes—
456C42C22×D412yes—
457C42C22×D412yes—
458C42C2×#4112yes—
459C42C2×#4112yes—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—48no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—84no—
476C23—84no—
477C23—88no—
478C23—88no—
479C23C2×#12112yes—
480C23C23×D411yescharacteristic
481Q8—84no—
482Q8—84no—
483Q8—84no—
484D4—24no—
485D4—24no—
486D4—48no—
487D4—42no—
488D4—44no—
489D4—42no—
490D4—44no—
491D4—84no—
492D4—84no—
493D4—88no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—88no—
524C2×C4—88no—
525C2×C4—88no—
526C2×C4—84no—
527C2×C4C2×#12112yes—
528C2×C4C2×#12112yes—
529C2×C4C2×#12112yes—
530C2×C4C2×#12112yes—
531C2×C4C2×#12112yes—
532C2×C4C23×D411yescharacteristic
533C2×C4C22×#4111yescharacteristic
534C2×C4C2×2−1+412yes—
535C2×C4#58512yes—
536C8—28no—
537C8—44no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—42no—
542C22—42no—
543C22—44no—
544C22—48no—
545C22—816no—
546C22—82no—
547C22—82no—
548C22—84no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—84no—
553C22—84no—
554C22C22×#12111yescharacteristic
555C22#348611yescharacteristic
556C22#348711yescharacteristic
557C22#358512yes—
558C22#358512yes—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—44no—
565C4—44no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—84no—
571C4#348912yes—
572C4#349512yes—
573C2—42no—
574C2—84no—
575C2—162no—
576C2#3566111yescharacteristic
577C2#6045411yescharacteristic
578C2#6045411yescharacteristic
579C2#6187212yes—
580C2#6193812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2987C2semidirect
3. #37619C2semidirect
4. #35172C2semidirect
5. #28276C2semidirect
6. #28197C2semidirect
7. #25485C2semidirect
8. #25304C2semidirect
9. #24618C2semidirect
10. #24512C2semidirect
11. #21989C2semidirect
12. #21869C2semidirect
13. #20739C2semidirect
23. C2×#446C22semidirect
24. C2×#446C22semidirect
37. #2580C22semidirect
38. #2580C22semidirect
41. #2548C22semidirect
42. #2546C22semidirect
50. #2088C22semidirect
51. #2088C22semidirect
53. #2067C22semidirect
55. #2065C22semidirect
58. #1993C22semidirect
61. #1961C22semidirect
62. #1949C22semidirect
63. #1943C22semidirect
64. #1941C22semidirect
66. #1936C22semidirect
68. #1934C22semidirect
71. #1847C22semidirect
72. #1836C22semidirect
76. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31092
C2×#29872
#376192
#351722
#282762
#281972
#254852
#253042
#246182
#245122
#219892
#218692
#207392