Finite Group Databasebeta

← #6519133

Subgroups — #6519133

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35240C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#26314C214yesmaximal, complemented
4#26247C214yesmaximal, complemented
5#25021C214yesmaximal, complemented
6#24976C214yesmaximal, complemented
7#24561C214yesmaximal, complemented
8#24559C214yesmaximal, complemented
9#21965C211yesmaximal, characteristic, complemented
10#21892C211yesmaximal, characteristic, complemented
11#21404C211yesmaximal, characteristic, complemented
12#21190C211yesmaximal, characteristic
13#20825C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#485C2214yescomplemented
17C2×#483C2214yescomplemented
18C2×#423C2211yescharacteristic
19C2×#407C2212yes—
20C2×#407C2212yes—
21C2×C4×M16C2211yescharacteristic
22#2848C2212yes—
23#2848C2212yes—
24#2633—22no—
25#2547C2212yes—
26#2542C2212yes—
27#2536C2214yes—
28#2502—21no—
29#2482—21no—
30#2482—21no—
31#2435—21no—
32#2353—22no—
33#2353—22no—
34#2352C2214yes—
35#2352C2214yes—
36#2326—22no—
37#2313—21no—
38#2312—21no—
39#2308—21no—
40#2283C2211yescharacteristic
41#2283C2211yescharacteristic
42#2283C2211yescharacteristic
43#2280C2212yes—
44C2×#380C2212yes—
45#2273—21no—
46#2077C2214yescomplemented
47#2077C2214yescomplemented
48#2077C2214yescomplemented
49#2076C2214yescomplemented
50#2036C2214yescomplemented
51#2036C2214yescomplemented
52#2036C2214yescomplemented
53#2035C2214yescomplemented
54#1993C2212yescomplemented
55#1987C2212yescomplemented
56#1981C2212yescomplemented
57#1960C2212yescomplemented
58#1955C2214yes—
59#1954C2212yes—
60#1954C2212yes—
61#1949C2214yescomplemented
62#1949C2214yescomplemented
63#1894C2214yes—
64#1894C2214yes—
65#1873C2214yescomplemented
66#1872C2214yescomplemented
67#1854C2214yescomplemented
68#1852C2214yescomplemented
69#1846C2212yescomplemented
70#1835C2212yescomplemented
71#1825C2214yescomplemented
72#1823C2214yescomplemented
73#1809C2214yes—
74#1809C2214yes—
75#1805C2214yescomplemented
76#1776C2214yescomplemented
77C4×#103C2212yescomplemented
78C4×#103C2212yescomplemented
79C23×D4D411yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×M16C2312yes—
83C22×M16C2312yes—
84#533—24no—
85#533—24no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×#125—21no—
89C2×#124—22no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2314yes—
93C2×C22⋊Q8C2314yes—
94C2×C4×Q8—21no—
95C2×C4×Q8C2312yes—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4D412yes—
99C2×C4×D4D412yes—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117—21no—
103C2×#117C2314yes—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4—22no—
107C22×C22⋊C4D411yescharacteristic
108C22×C22⋊C4D411yescharacteristic
109C22×C42C2311yescharacteristic
110#485—48no—
111#483—48no—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116#423—24no—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123#407—44no—
124#407—44no—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129C4×M16—24no—
130C2×#97C2312yes—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#399—22no—
134#399—22no—
135#398—24no—
136#397—22no—
137#397—22no—
138#394—24no—
139#391—22no—
140#390—22no—
141#388—24no—
142#388—24no—
143#388—24no—
144#388—24no—
145#387—24no—
146#387—24no—
147#386—24no—
148#386—24no—
149#385—24no—
150#385—24no—
151#385—22no—
152#385—22no—
153#384—22no—
154#383—21no—
155#383—21no—
156#383—21no—
157#382C2312yes—
158#382C2312yes—
159#382C2312yes—
160#381—21no—
161#381—21no—
162#381—22no—
163#381—21no—
164#380—22no—
165#380—22no—
166#380C2312yes—
167#380C2312yes—
168#380C2312yes—
169#380C2314yes—
170#380C2314yes—
171#380C2314yes—
172#380C2314yes—
173#379—22no—
174#379—22no—
175#379—22no—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—24no—
185C4×C22⋊C4—24no—
186#376—21no—
187#376—21no—
188#376—21no—
189C2×#95—22no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198C25C2×D411yescharacteristic
199C25C2×D411yescharacteristic
200C22×Q8—21no—
201C22×Q8C2412yes—
202C22×D4—24no—
203C22×D4—44no—
204C22×D4C2×D412yes—
205C22×D4C2×D412yes—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C2×M16—24no—
225C2×M16—24no—
226C2×M16—24no—
227C2×M16—42no—
228C2×M16—42no—
229C2×M16—42no—
230C22×C8C2412yes—
231C22×C8C2414yes—
232C22×C8C2412yes—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237#125—44no—
238#124—48no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—44no—
242C22⋊Q8—44no—
243C22⋊Q8—44no—
244C22⋊Q8—44no—
245C22⋊Q8—44no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—44no—
249C4×D4—28no—
250C4×D4—28no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—24no—
255#117—44no—
256#117—44no—
257#117—44no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C22⋊C4C2×D412yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42#4114yes—
301C2×C42#4114yes—
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#107—44no—
305#107—44no—
306#106—28no—
307#106—44no—
308C4⋊C8—24no—
309C4⋊C8—24no—
310#103—28no—
311#103—44no—
312#103—44no—
313#103—44no—
314C22⋊C8—28no—
315C22⋊C8—28no—
316C22⋊C8—44no—
317C22⋊C8—44no—
318#97—24no—
319C4×C8—22no—
320C4×C8—22no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24C22×D412yes—
346C24C22×D411yescharacteristic
347C24C22×D412yes—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—44no—
353C2×Q8—42no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—416no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
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—22no—
379C22×C4—22no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—48no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4#12314yes—
396C22×C4#12314yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2511yescharacteristic
407M16—48no—
408M16—84no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8H(5,𝔽2)14yes—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C42—44no—
452C42—44no—
453C42—82no—
454C42—82no—
455C42—84no—
456C42#12314yes—
457C42#12314yes—
458C42C2×#4112yes—
459C42C2×#4112yes—
460C42C2×#4112yes—
461C42C2×#4112yes—
462C23—28no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—44no—
476C23—82no—
477C23—82no—
478C23—88no—
479C23—82no—
480C23—82no—
481C23—88no—
482C23C2×#12312yes—
483C23#57514yes—
484C23C23×D411yescharacteristic
485Q8—44no—
486Q8—42no—
487Q8—42no—
488Q8—84no—
489D4—48no—
490D4—48no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—168no—
529C2×C4C2×#12312yes—
530C2×C4C2×#12312yes—
531C2×C4C2×#12312yes—
532C2×C4C2×#12312yes—
533C2×C4C2×#12312yes—
534C2×C4C22×#4111yescharacteristic
535C2×C4C22×#4111yescharacteristic
536C2×C4C2×H(5,𝔽2)12yes—
537C2×C4#58512yes—
538C8—24no—
539C8—42no—
540C8—44no—
541C22—28no—
542C22—22no—
543C22—24no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—88no—
556C22—88no—
557C22C22×#12311yescharacteristic
558C22#349612yes—
559C22C2×#57512yes—
560C22#358612yes—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—84no—
569C4—84no—
570C4—84no—
571C4—82no—
572C4—82no—
573C4#349412yes—
574C4#350811yescharacteristic
575C4#351011yescharacteristic
576C2—44no—
577C2—42no—
578C2—82no—
579C2#3554111yescharacteristic
580C2#6051111yescharacteristic
581C2#6051911yescharacteristic
582C2#6229714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26314C2semidirect
4. #26247C2semidirect
5. #25021C2semidirect
6. #24976C2semidirect
7. #24561C2semidirect
8. #24559C2semidirect
9. #21965C2semidirect
10. #21892C2semidirect
11. #21404C2semidirect
13. #20825C2semidirect
16. C2×#485C22semidirect
17. C2×#483C22semidirect
46. #2077C22semidirect
47. #2077C22semidirect
48. #2077C22semidirect
49. #2076C22semidirect
50. #2036C22semidirect
51. #2036C22semidirect
52. #2036C22semidirect
53. #2035C22semidirect
54. #1993C22semidirect
55. #1987C22semidirect
56. #1981C22semidirect
57. #1960C22semidirect
61. #1949C22semidirect
62. #1949C22semidirect
65. #1873C22semidirect
66. #1872C22semidirect
67. #1854C22semidirect
68. #1852C22semidirect
69. #1846C22semidirect
70. #1835C22semidirect
71. #1825C22semidirect
72. #1823C22semidirect
75. #1805C22semidirect
76. #1776C22semidirect
77. C4×#103C22semidirect
78. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#352402
#341852
#263142
#262472
#250212
#249762
#245612
#245592
#219652
#218922
#214042
#211902
#208252