Finite Group Databasebeta

← #6506436

Subgroups — #6506436

Order 512.

2315 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37477C211yesmaximal, characteristic, complemented
2#35242C211yesmaximal, characteristic
3#29118C212yesmaximal, complemented
4#29117C212yesmaximal, complemented
5#27268C214yesmaximal, complemented
6#27266C214yesmaximal, complemented
7#27260C214yesmaximal, complemented
8#27258C214yesmaximal, complemented
9#25868C212yesmaximal, complemented
10#25866C212yesmaximal, complemented
11#21763C212yesmaximal
12C2×#1787C212yesmaximal, complemented
13#20528C211yesmaximal, characteristic, complemented
14C24×Q8C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#536C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#491C2212yescomplemented
19C2×#489C2212yescomplemented
20C2×#488C2212yescomplemented
21C2×#487C2212yescomplemented
22C22×#103C2214yes—
23C22×#103C2214yes—
24C22×C22⋊C8C2212yes—
25#2849—22no—
26#2841—22no—
27#2659—22no—
28#2646—22no—
29#2629C2212yes—
30#2628C2212yes—
31#2614—22no—
32#2442C2214yes—
33#2440C2214yes—
34#2437—22no—
35#2436C2214yes—
36#2434C2214yes—
37#2409—22no—
38#2409—22no—
39#2310C2211yescharacteristic
40#2310C2211yescharacteristic
41#2280C2211yescharacteristic
42C2×#380C2212yes—
43#2074C2214yescomplemented
44#2072C2214yescomplemented
45#2043C2214yescomplemented
46#2043C2214yescomplemented
47#2042C2214yescomplemented
48#2042C2214yescomplemented
49#2026C2214yescomplemented
50#2024C2214yescomplemented
51#1960C2212yescomplemented
52#1959C2212yescomplemented
53#1958C2212yescomplemented
54#1956C2212yescomplemented
55#1935C2214yescomplemented
56#1935C2214yescomplemented
57#1933C2214yescomplemented
58#1933C2214yescomplemented
59#1893C2214yes—
60#1892C2214yes—
61#1874C2212yes—
62#1874C2212yes—
63#1872C2212yes—
64#1872C2212yes—
65#1825C2214yes—
66#1788C2214yes—
67#1787C2214yes—
68#1787C2214yes—
69#1787C2214yes—
70#1787C2214yes—
71#1752C2214yes—
72C8×C22⋊C4C2214yes—
73#1750C2212yes—
74#1750C2212yes—
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⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#125C2312yes—
93C2×#124C2312yes—
94C2×#124C2312yes—
95#525—28no—
96C2×#123—21no—
97C2×#123—22no—
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8C2312yes—
104C2×#121C2312yes—
105C2×#120—21no—
106C2×C4×D4—22no—
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—22no—
113#491—44no—
114#489—44no—
115#488—44no—
116#487—44no—
117C2×#103—48no—
118C2×#103—48no—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8—28no—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×#97C2312yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139#400—22no—
140#400—24no—
141#399—22no—
142#398—24no—
143#398—24no—
144#398—24no—
145#396—24no—
146#395—22no—
147#395—24no—
148#395—24no—
149#395—24no—
150#393—22no—
151#393—24no—
152#393—24no—
153#391—21no—
154#391—22no—
155#391—21no—
156#391C2314yes—
157#391C2314yes—
158#389C2314yes—
159#389C2314yes—
160#386—24no—
161#386—24no—
162#385—24no—
163#385—24no—
164#385—24no—
165#385—24no—
166#384C2312yes—
167#382C2312yes—
168#381—22no—
169#381—22no—
170#381D412yes—
171#380—22no—
172#380—22no—
173#380—24no—
174#380D412yes—
175#380D412yes—
176#380C2314yes—
177#380C2314yes—
178#379D412yes—
179C4×C22⋊C4C2314yes—
180C2×#95—22no—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2314yes—
186#336C2314yes—
187C25C2×D411yescharacteristic
188C2×#41—24no—
189C2×#41—42no—
190C2×#41—42no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—28no—
198C22×Q8—28no—
199C22×Q8—28no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—28no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8—464no—
210C22×Q8—48no—
211C22×Q8—48no—
212C22×Q8—48no—
213C22×Q8—44no—
214C22×Q8C2411yescharacteristic
215C22×Q8C2412yes—
216C22×Q8C2412yes—
217C22×D4—22no—
218C22×D4—21no—
219C22×D4C2411yescharacteristic
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4#4118yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C22×C8—28no—
237C22×C8—44no—
238C22×C8—44no—
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—42no—
244C4⋊Q8—42no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247#125—42no—
248#125—42no—
249#124—24no—
250#124—24no—
251#124—42no—
252#124—42no—
253#123—48no—
254#123—48no—
255#123—44no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—48no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C22⋊Q8—88no—
262#121—24no—
263#121—24no—
264#120—44no—
265C4×D4—48no—
266#117—24no—
267#117—42no—
268#117—42no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—84no—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C42—21no—
302C2×C42C2412yes—
303C2×C42C2412yes—
304#103—28no—
305#103—28no—
306#103—28no—
307#103—28no—
308#103—44no—
309#103—44no—
310#103—44no—
311#103—44no—
312#103—44no—
313#103—44no—
314#102—44no—
315#102—44no—
316C22⋊C8—28no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319#97—42no—
320#97—42no—
321C4×C8—42no—
322C4×C8—42no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—24no—
333#95C2×D414yes—
334#95C2×D414yes—
335C24—24no—
336C24—24no—
337C24—21no—
338C24—41no—
339C24—41no—
340C24#12414yes—
341C24C22×D411yescharacteristic
342C24C22×D412yes—
343#41—88no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—232no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—416no—
359C2×Q8—416no—
360C2×Q8—416no—
361C2×Q8—416no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—416no—
365C2×Q8—432no—
366C2×Q8—464no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—48no—
370C2×Q8—84no—
371C2×Q8—82no—
372C2×Q8H(5,𝔽2)14yes—
373C2×Q8H(5,𝔽2)14yes—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—44no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—416no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—432no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4#12418yes—
418C22×C4#12414yes—
419C22×C4C22×D412yes—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4114yes—
426C22×C4C2×#4114yes—
427C22×C4C2511yescharacteristic
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—88no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—42no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—82no—
455C22⋊C4—28no—
456C22⋊C4—28no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—48no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—44no—
466C22⋊C4—84no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—42no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—216no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—22no—
481C23—44no—
482C23—48no—
483C23—44no—
484C23—44no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23C2×#12412yes—
491C23C2×#12414yes—
492C23#57714yes—
493C23#57814yes—
494C23C23×D411yescharacteristic
495Q8—28no—
496Q8—28no—
497Q8—48no—
498Q8—48no—
499Q8—432no—
500Q8—44no—
501Q8—44no—
502Q8—44no—
503Q8—164no—
504D4—44no—
505D4—84no—
506D4—88no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—432no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—84no—
545C2×C4—164no—
546C2×C4—164no—
547C2×C4C2×#12412yes—
548C2×C4C2×#12414yes—
549C2×C4C22×#4112yes—
550C2×C4C2×H(5,𝔽2)12yes—
551C2×C4C2×H(5,𝔽2)12yes—
552C8—44no—
553C8—44no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—22no—
561C22—24no—
562C22—416no—
563C22—42no—
564C22—41no—
565C22—41no—
566C22—81no—
567C22—82no—
568C22—81no—
569C22—81no—
570C22—82no—
571C22#313718yes—
572C22C22×#12411yescharacteristic
573C22#348411yescharacteristic
574C22#348411yescharacteristic
575C22C2×#57712yes—
576C22C2×#57812yes—
577C4—22no—
578C4—24no—
579C4—44no—
580C4—44no—
581C4—44no—
582C4—42no—
583C4—42no—
584C4—81no—
585C4—84no—
586C4—84no—
587C4—82no—
588C4#349014yes—
589C2—28no—
590C2—42no—
591C2—161no—
592C2#3536711yescharacteristic
593C2C2×#313712yes—
594C2#6163212yes—
595C2#6163311yescharacteristic
596C2#6163411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
1. #37477C2semidirect
3. #29118C2semidirect
4. #29117C2semidirect
5. #27268C2semidirect
6. #27266C2semidirect
7. #27260C2semidirect
8. #27258C2semidirect
9. #25868C2semidirect
10. #25866C2semidirect
12. C2×#1787C2semidirect
13. #20528C2semidirect
18. C2×#491C22semidirect
19. C2×#489C22semidirect
20. C2×#488C22semidirect
21. C2×#487C22semidirect
43. #2074C22semidirect
44. #2072C22semidirect
45. #2043C22semidirect
46. #2043C22semidirect
47. #2042C22semidirect
48. #2042C22semidirect
49. #2026C22semidirect
50. #2024C22semidirect
51. #1960C22semidirect
52. #1959C22semidirect
53. #1958C22semidirect
54. #1956C22semidirect
55. #1935C22semidirect
56. #1935C22semidirect
57. #1933C22semidirect
58. #1933C22semidirect

Maximal subgroups

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

subgroupindex
#374772
#352422
#291182
#291172
#272682
#272662
#272602
#272582
#258682
#258662
#217632
C2×#17872
#205282