Finite Group Databasebeta

← #6500747

Subgroups — #6500747

Order 512.

2374 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35242C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic
3#27596C212yesmaximal, complemented
4#27592C212yesmaximal, complemented
5#24875C214yesmaximal, complemented
6#24871C214yesmaximal, complemented
7#24867C214yesmaximal, complemented
8#24863C214yesmaximal, complemented
9#24016C212yesmaximal, complemented
10#24014C212yesmaximal, complemented
11#21941C212yesmaximal, complemented
12#21739C211yesmaximal, characteristic
13C2×#1787C212yesmaximal, complemented
14C24×Q8C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#536C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#484C2212yescomplemented
19C2×#485C2214yescomplemented
20C2×#483C2212yescomplemented
21C22×#103C2214yes—
22C22×#103C2214yes—
23C22×C22⋊C8C2212yes—
24#2849—22no—
25#2614—22no—
26#2436—22no—
27#2434—22no—
28#2409C2214yes—
29#2409C2214yes—
30#2376—22no—
31#2358—22no—
32#2352C2214yes—
33D4×C4⋊C4C2214yes—
34#2329—22no—
35#2325—22no—
36#2314C2212yes—
37#2311C2212yes—
38#2310C2211yescharacteristic
39#2310C2211yescharacteristic
40#2280C2211yescharacteristic
41C2×#380C2212yes—
42#2097C2214yescomplemented
43#2096C2212yescomplemented
44#2095C2212yescomplemented
45#2074C2212yescomplemented
46#2074C2212yescomplemented
47#2072C2214yescomplemented
48#2036C2214yescomplemented
49#2036C2214yescomplemented
50#2035C2214yescomplemented
51#2035C2214yescomplemented
52#1928C2214yescomplemented
53#1928C2214yescomplemented
54#1927C2214yescomplemented
55#1927C2214yescomplemented
56#1873C2212yescomplemented
57#1873C2212yescomplemented
58#1872C2214yescomplemented
59#1869C2214yes—
60#1869C2214yes—
61#1854C2214yes—
62#1854C2214yes—
63#1853C2212yes—
64#1853C2212yes—
65#1852C2212yes—
66#1852C2212yes—
67#1818C2212yes—
68#1808C2214yes—
69#1807C2214yes—
70#1787C2214yes—
71#1787C2214yes—
72#1787C2214yes—
73#1787C2214yes—
74#1780C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8—21no—
77C23×Q8—22no—
78C23×Q8—28no—
79C23×Q8—44no—
80C23×Q8—44no—
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×Q8C2314yes—
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C23×C8C2314yes—
87#536—44no—
88#536—44no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×#124—21no—
92C2×#124—21no—
93#525—28no—
94C2×#123—22no—
95C2×#123—22no—
96C2×#123D412yes—
97C2×C22⋊Q8D412yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2314yes—
100C2×C22⋊Q8C2312yes—
101C2×#121D412yes—
102C2×#120D412yes—
103C2×C4×D4—24no—
104C2×C4×D4C2314yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—22no—
113#485—48no—
114#484—44no—
115#483—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#103—48no—
121C2×#103—48no—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×C22⋊C8—28no—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2312yes—
137#395—24no—
138#395—24no—
139#393—24no—
140#393—24no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391C2314yes—
145#391C2312yes—
146#391C2312yes—
147#390—22no—
148#389—24no—
149#388—24no—
150#388—24no—
151#387—24no—
152#387—24no—
153#386—22no—
154#386—22no—
155#386—24no—
156#386—24no—
157#386—24no—
158#385—24no—
159#385—24no—
160#385—24no—
161#385—24no—
162#384C2312yes—
163#382C2312yes—
164#381—21no—
165#381—22no—
166#381—22no—
167#380—22no—
168#380—22no—
169#380C2314yes—
170#380C2314yes—
171#380C2312yes—
172#380C2314yes—
173#380C2312yes—
174#379—21no—
175C4×C4⋊C4C2314yes—
176C4×C22⋊C4—22no—
177C4×C22⋊C4—24no—
178C2×#95—22no—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2314yes—
184#336C2312yes—
185#336C2312yes—
186C25C2×D411yescharacteristic
187C2×#41—24no—
188C2×#41—42no—
189C2×#41—42no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8—24no—
195C22×Q8—28no—
196C22×Q8—28no—
197C22×Q8—28no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—28no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×Q8—464no—
208C22×Q8—48no—
209C22×Q8—48no—
210C22×Q8—48no—
211C22×Q8—44no—
212C22×Q8C2×D411yescharacteristic
213C22×Q8C2412yes—
214C22×Q8C2412yes—
215C22×D4C2×D414yes—
216C22×D4C2×D412yes—
217C22×D4C2×D411yescharacteristic
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D412yes—
230C23×C4C2×D411yescharacteristic
231C23×C4#4118yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C22×C8—28no—
236C22×C8—44no—
237C22×C8—44no—
238C22×C8C2414yes—
239C22×C8C2414yes—
240C4⋊Q8—44no—
241C4⋊Q8—44no—
242#124—44no—
243#124—44no—
244#123—28no—
245#123—48no—
246#123—48no—
247C22⋊Q8—28no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—28no—
251C22⋊Q8—42no—
252C22⋊Q8—44no—
253C22⋊Q8—42no—
254#121—28no—
255#120—28no—
256C4×D4—28no—
257C4×D4—28no—
258C4×D4—88no—
259#117—24no—
260#117—42no—
261#117—42no—
262#117—44no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2411yescharacteristic
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2411yescharacteristic
281C2×C22⋊C4—24no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—84no—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×D414yes—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42C2412yes—
301#107—44no—
302#107—44no—
303#106—44no—
304#106—44no—
305#103—28no—
306#103—28no—
307#103—28no—
308#103—28no—
309#103—44no—
310#103—44no—
311#103—44no—
312#103—44no—
313C22⋊C8—28no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C22⋊C8—42no—
317C22⋊C8—42no—
318C22⋊C8—44no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330C24—24no—
331C24—24no—
332C24—41no—
333C24—41no—
334C24#12314yes—
335C24C22×D412yes—
336C24C22×D411yescharacteristic
337C24C22×D412yes—
338#41—88no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—232no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—416no—
354C2×Q8—416no—
355C2×Q8—416no—
356C2×Q8—416no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—416no—
360C2×Q8—432no—
361C2×Q8—464no—
362C2×Q8—48no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8H(5,𝔽2)14yes—
366C2×Q8H(5,𝔽2)14yes—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—41no—
375C2×D4—41no—
376C22×C4—24no—
377C22×C4—24no—
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—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—416no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—432no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—84no—
414C22×C4#12318yes—
415C22×C4#12314yes—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4114yes—
423C22×C4C2511yescharacteristic
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—88no—
431C4⋊C4—24no—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—48no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—84no—
447C22⋊C4—24no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C22⋊C4—48no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—84no—
459C22⋊C4—82no—
460C22⋊C4—84no—
461C42—24no—
462C42—42no—
463C42—42no—
464C42—82no—
465C42—84no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—216no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—44no—
481C23—48no—
482C23—44no—
483C23—44no—
484C23C2×#12312yes—
485C23C2×#12314yes—
486C23#57714yes—
487C23#57814yes—
488C23C23×D411yescharacteristic
489Q8—28no—
490Q8—28no—
491Q8—48no—
492Q8—48no—
493Q8—432no—
494Q8—44no—
495Q8—44no—
496Q8—82no—
497D4—416no—
498D4—82no—
499D4—84no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—48no—
516C2×C4—44no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—432no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—164no—
540C2×C4—164no—
541C2×C4—164no—
542C2×C4C2×#12314yes—
543C2×C4C2×#12312yes—
544C2×C4C22×#4112yes—
545C2×C4C2×H(5,𝔽2)12yes—
546C2×C4C2×H(5,𝔽2)12yes—
547C8—44no—
548C8—44no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—22no—
556C22—24no—
557C22—416no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22#309718yes—
567C22C22×#12311yescharacteristic
568C22#348411yescharacteristic
569C22#348411yescharacteristic
570C22C2×#57712yes—
571C22C2×#57812yes—
572C4—22no—
573C4—24no—
574C4—42no—
575C4—44no—
576C4—44no—
577C4—44no—
578C4—82no—
579C4—84no—
580C4—81no—
581C4—84no—
582C4—81no—
583C4#349014yes—
584C2—28no—
585C2—42no—
586C2—82no—
587C2#3535011yescharacteristic
588C2C2×#309712yes—
589C2#6163212yes—
590C2#6163311yescharacteristic
591C2#6163411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27596C2semidirect
4. #27592C2semidirect
5. #24875C2semidirect
6. #24871C2semidirect
7. #24867C2semidirect
8. #24863C2semidirect
9. #24016C2semidirect
10. #24014C2semidirect
11. #21941C2semidirect
13. C2×#1787C2semidirect
18. C2×#484C22semidirect
19. C2×#485C22semidirect
20. C2×#483C22semidirect
42. #2097C22semidirect
43. #2096C22semidirect
44. #2095C22semidirect
45. #2074C22semidirect
46. #2074C22semidirect
47. #2072C22semidirect
48. #2036C22semidirect
49. #2036C22semidirect
50. #2035C22semidirect
51. #2035C22semidirect
52. #1928C22semidirect
53. #1928C22semidirect
54. #1927C22semidirect
55. #1927C22semidirect
56. #1873C22semidirect
57. #1873C22semidirect
58. #1872C22semidirect

Maximal subgroups

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

subgroupindex
#352422
#343602
#275962
#275922
#248752
#248712
#248672
#248632
#240162
#240142
#219412
#217392
C2×#17872