Finite Group Databasebeta

← #7224169

Subgroups — #7224169

Order 512.

1863 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46004C211yesmaximal, characteristic, complemented
2#36289C211yesmaximal, characteristic
3#33032C212yesmaximal, complemented
4#32807C212yesmaximal, complemented
5#32680C212yesmaximal, complemented
6#32118C214yesmaximal, complemented
7#31263C212yesmaximal, complemented
8#30542C214yesmaximal, complemented
9#27245C214yesmaximal, complemented
10#25945C212yesmaximal, complemented
11#24358C214yesmaximal, complemented
12#23849C212yesmaximal
13#23352C211yesmaximal, characteristic, complemented
14C23×SD16C2212yes—
15C2×#554C2211yescharacteristic
16C2×D42C2211yescharacteristic
17C2×#491C2212yescomplemented
18C2×#474C2212yes—
19C2×#461C2212yes—
20C2×#461C2212yes—
21C2×#448C2212yes—
22C2×#450C2214yes—
23#2862C2212yescomplemented
24#2858C2211yescharacteristic
25#2850—21no—
26#2844—21no—
27#2811—21no—
28#2804—21no—
29#2763—24no—
30#2568C2212yescomplemented
31#2563C2211yescharacteristic
32#2548C2211yescharacteristic
33#2504—21no—
34#2492C2214yescomplemented
35#2490C2214yes—
36#2472—24no—
37#2435—21no—
38#2434C2214yescomplemented
39#2409C2212yes—
40#2379—21no—
41D4×C22⋊C4C2214yes—
42C4×#124C2211yescharacteristic
43C2×#390—21no—
44C42×D4C2211yescharacteristic
45#2076C2214yescomplemented
46#2055C2214yescomplemented
47#2055C2214yescomplemented
48#2055C2214yescomplemented
49#2036C2214yescomplemented
50#2026C2214yescomplemented
51#2024C2214yescomplemented
52#2023C2214yescomplemented
53#2014C2214yes—
54#2013C2214yescomplemented
55#2013C2214yescomplemented
56#2000C2214yes—
57#1989C2212yescomplemented
58#1988C2212yescomplemented
59#1985C2212yescomplemented
60#1981C2212yescomplemented
61#1973C2214yescomplemented
62#1928C2214yescomplemented
63#1928C2214yescomplemented
64#1925C2214yescomplemented
65#1924C2214yescomplemented
66#1893C2212yes—
67#1892C2214yescomplemented
68#1877C2214yes—
69#1877C2214yes—
70#1877C2214yes—
71#1869C2212yes—
72#1850C2214yescomplemented
73#1831C2212yescomplemented
74#1787C2214yescomplemented
75#1786C2212yes—
76#1762C2212yes—
77C23×Q8C2312yes—
78C23×D4—21no—
79C23×D4C2312yes—
80C22×SD16—48no—
81C22×SD16—44no—
82C22×SD16C2314yes—
83C22×SD16C2314yes—
84C22×SD16C2314yes—
85C23×C8C2312yes—
86#554—42no—
87#554—42no—
88D42—42no—
89D42—42no—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2312yes—
94C2×#127C2311yescharacteristic
95C2×#125C2312yes—
96C2×#124—21no—
97C2×#124—22no—
98C2×#124D411yescharacteristic
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8C2314yes—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×#120—22no—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2311yescharacteristic
110C22×C4⋊C4—21no—
111C22×C22⋊C4—24no—
112C22×C42—21no—
113#491—44no—
114#474—44no—
115#461—44no—
116#461—44no—
117#450—48no—
118#448—44no—
119C2×#107C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2312yes—
133C2×#97C2312yes—
134#400—22no—
135#400—22no—
136#400—24no—
137#397—24no—
138#395—22no—
139#395—24no—
140#395—24no—
141#394—22no—
142#394—24no—
143#393—22no—
144#393—22no—
145#393—24no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—24no—
150#391D411yescharacteristic
151#391C2314yes—
152#391C2314yes—
153#391C2314yes—
154#390—21no—
155#390—21no—
156#390—24no—
157#390—21no—
158#390—21no—
159#390D411yescharacteristic
160#390C2314yes—
161#390C2314yes—
162#388—24no—
163#388—24no—
164#388D414yes—
165#386—22no—
166#386—24no—
167#386—24no—
168#385—24no—
169#385—22no—
170#385—24no—
171#382—21no—
172#382—22no—
173#382—22no—
174#382—22no—
175#382D411yescharacteristic
176#380C2314yes—
177#380C2312yes—
178#379—24no—
179C4×C4⋊C4C2311yescharacteristic
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4C2314yes—
185C43C2311yescharacteristic
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193C25—22no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—44no—
198C22×Q8—42no—
199C22×Q8—44no—
200C22×Q8—42no—
201C22×Q8C2×D411yescharacteristic
202C22×Q8C2414yes—
203C22×D4—24no—
204C22×D4—28no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—42no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—82no—
214C22×D4C2×D411yescharacteristic
215C22×D4C2412yes—
216C22×D4C2411yescharacteristic
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4C2412yes—
222C2×SD16—28no—
223C2×SD16—28no—
224C2×SD16—28no—
225C2×SD16—44no—
226C2×SD16—44no—
227C2×SD16—44no—
228C2×SD16—44no—
229C2×SD16—44no—
230C2×SD16—44no—
231C2×SD16—432no—
232C22×C8—44no—
233C22×C8—42no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—41no—
237C4⋊Q8—41no—
238C4⋊Q8—42no—
239C4⋊Q8—42no—
240C4⋊Q8—42no—
241C4⋊Q8—42no—
242C4⋊Q8—42no—
243C4⋊Q8—42no—
244#127—41no—
245#127—41no—
246#125—24no—
247#125—42no—
248#124—24no—
249#124—84no—
250#124—82no—
251C22⋊Q8—48no—
252C22⋊Q8—44no—
253C22⋊Q8—44no—
254C22⋊Q8—44no—
255#121—24no—
256#121—42no—
257#121—44no—
258#120—48no—
259C4×Q8—41no—
260C4×Q8—41no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—41no—
268C4×D4—41no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2411yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—88no—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D412yes—
301C2×C42—21no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—82no—
305C2×C42C2×D412yes—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2412yes—
310C2×C42C2412yes—
311#107—44no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314C4⋊C8—42no—
315#103—28no—
316#103—44no—
317#103—44no—
318#103—44no—
319#103—44no—
320#102—28no—
321#102—44no—
322#102—44no—
323C22⋊C8—42no—
324C22⋊C8—44no—
325#97—42no—
326#95—21no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95C2×D414yes—
338#95C2×D414yes—
339C24—21no—
340C24—22no—
341C24—44no—
342C24—48no—
343C24—44no—
344C24—84no—
345C24—82no—
346C24C22×D412yes—
347C2×Q8—22no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—42no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—82no—
360C2×Q8—84no—
361C2×Q8—82no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×Q8—84no—
365C2×Q8—88no—
366C2×D4—21no—
367C2×D4—24no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—24no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—48no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—82no—
384C2×D4—84no—
385C2×D4—88no—
386C2×D4—816no—
387C2×D4—82no—
388C2×D4#13714yes—
389C2×D4H(5,𝔽2)14yes—
390C2×D4H(5,𝔽2)12yes—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—82no—
407C22×C4—84no—
408C22×C4—82no—
409C22×C4—84no—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C2511yescharacteristic
418SD16—416no—
419SD16—416no—
420SD16—88no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—48no—
428C4⋊C4—21no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—84no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4H(5,𝔽2)12yes—
448C22⋊C4—24no—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—48no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—82no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C42—22no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—21no—
468C42—21no—
469C42—22no—
470C42—41no—
471C42—42no—
472C42—82no—
473C42—82no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—44no—
488C23—88no—
489C23—88no—
490C23—816no—
491C23—168no—
492C23—164no—
493C23C2×#13714yes—
494C23C23×D411yescharacteristic
495Q8—44no—
496Q8—44no—
497Q8—44no—
498Q8—44no—
499Q8—82no—
500Q8—88no—
501Q8—164no—
502Q8—162no—
503D4—22no—
504D4—24no—
505D4—44no—
506D4—42no—
507D4—42no—
508D4—44no—
509D4—84no—
510D4—88no—
511D4—88no—
512D4—162no—
513D4H(3,𝔽4)14yes—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—164no—
551C2×C4C2×#13712yes—
552C2×C4#57714yes—
553C2×C4C2×H(5,𝔽2)11yescharacteristic
554C2×C4C2×H(5,𝔽2)12yes—
555C2×C4C2×H(5,𝔽2)11yescharacteristic
556C2×C4#58512yes—
557C8—44no—
558C8—44no—
559C22—28no—
560C22—24no—
561C22—24no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—84no—
570C22—84no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—1616no—
575C22#348411yescharacteristic
576C22#348812yes—
577C22C22×#13711yescharacteristic
578C22C2×#57712yes—
579C22#358511yescharacteristic
580C4—24no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—82no—
589C4—84no—
590C4—84no—
591C4—82no—
592C4C2×H(3,𝔽4)11yescharacteristic
593C4#354511yescharacteristic
594C4#355812yes—
595C2—44no—
596C2—44no—
597C2—162no—
598C2#4597711yescharacteristic
599C2#6157011yescharacteristic
600C2#6166211yescharacteristic
601C2#6198012yes—
602C2#6201612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #46004C2semidirect
3. #33032C2semidirect
4. #32807C2semidirect
5. #32680C2semidirect
6. #32118C2semidirect
7. #31263C2semidirect
8. #30542C2semidirect
9. #27245C2semidirect
10. #25945C2semidirect
11. #24358C2semidirect
13. #23352C2semidirect
17. C2×#491C22semidirect
23. #2862C22semidirect
30. #2568C22semidirect
34. #2492C22semidirect
38. #2434C22semidirect
45. #2076C22semidirect
46. #2055C22semidirect
47. #2055C22semidirect
48. #2055C22semidirect
49. #2036C22semidirect
50. #2026C22semidirect
51. #2024C22semidirect
52. #2023C22semidirect
54. #2013C22semidirect
55. #2013C22semidirect
57. #1989C22semidirect
58. #1988C22semidirect
59. #1985C22semidirect
60. #1981C22semidirect
61. #1973C22semidirect
62. #1928C22semidirect
63. #1928C22semidirect
64. #1925C22semidirect
65. #1924C22semidirect
67. #1892C22semidirect
72. #1850C22semidirect
73. #1831C22semidirect
74. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#460042
#362892
#330322
#328072
#326802
#321182
#312632
#305422
#272452
#259452
#243582
#238492
#233522