Finite Group Databasebeta

← #6506598

Subgroups — #6506598

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37297C211yesmaximal, characteristic
2#35532C211yesmaximal, characteristic, complemented
3#29343C212yesmaximal, complemented
4#29331C212yesmaximal, complemented
5#27973C212yesmaximal, complemented
6#27704C214yesmaximal, complemented
7#27702C214yesmaximal, complemented
8#27279C214yesmaximal, complemented
9#27254C214yesmaximal, complemented
10#25915C212yesmaximal, complemented
11#21770C212yesmaximal, complemented
12#20914C212yesmaximal, complemented
13#20530C211yesmaximal, characteristic, complemented
14C2×#544C2211yescharacteristic, complemented
15C2×#535C2212yes—
16C2×#520C2211yescharacteristic
17C2×#490C2212yes—
18C2×#486C2212yes—
19C2×#418C2212yescomplemented
20C22×#102C2212yescomplemented
21C2×#408C2212yes—
22#2856C2212yes—
23#2733—21no—
24#2733—21no—
25#2679—21no—
26#2679—21no—
27#2669—24no—
28#2619C2212yes—
29#2567—21no—
30#2566—21no—
31#2565—21no—
32#2542—21no—
33#2500C2214yes—
34#2449—24no—
35#2441C2214yescomplemented
36#2440C2214yescomplemented
37#2433C2214yes—
38#2317C2211yescharacteristic, complemented
39#2311C2211yescharacteristic
40#2297C2211yescharacteristic
41#2296C2211yescharacteristic, complemented
42C2×#384C2211yescharacteristic, complemented
43#2101C2214yescomplemented
44#2100C2212yescomplemented
45#2100C2212yescomplemented
46#2071C2214yescomplemented
47#2071C2214yescomplemented
48#2041C2212yes—
49#2041C2214yes—
50#2041C2212yes—
51#2025C2214yescomplemented
52#2023C2214yescomplemented
53#1993C2212yescomplemented
54#1991C2212yescomplemented
55#1957C2214yes—
56#1936C2214yescomplemented
57#1934C2214yescomplemented
58#1934C2214yescomplemented
59#1934C2214yescomplemented
60#1895C2214yes—
61#1893C2214yes—
62#1875C2214yes—
63#1875C2214yescomplemented
64#1875C2214yescomplemented
65#1874C2214yes—
66#1874C2214yescomplemented
67#1874C2214yescomplemented
68#1834C2214yescomplemented
69#1825C2214yescomplemented
70#1786C2214yescomplemented
71#1753C2214yescomplemented
72C8×C22⋊C4C2214yescomplemented
73#1750C2212yescomplemented
74C4×C22⋊C8C2212yescomplemented
75#1736C2212yescomplemented
76#1735C2212yescomplemented
77C22×#41C2311yescharacteristic
78C23×D4C2312yes—
79C22×M16C2312yes—
80C23×C8C2312yes—
81#544—42no—
82#544—42no—
83#535—28no—
84#535—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2312yes—
87C2×#127—22no—
88C2×#127C2312yes—
89C2×#127C2312yes—
90C2×#125—21no—
91C2×#125C2312yes—
92C2×#124—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×#121—22no—
96C2×#121—22no—
97C2×#121C2314yes—
98C2×#121C2314yes—
99C2×#120—22no—
100C2×#120—22no—
101C2×#120—24no—
102C2×#120—24no—
103#520—42no—
104#520—42no—
105C2×C4×Q8—21no—
106C2×C4×D4—21no—
107C2×C4×D4—22no—
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4—22no—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#490—44no—
119#486—44no—
120#418—44no—
121C2×#102—44no—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130#408—28no—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2314yes—
133C2×#97C2312yes—
134C2×C4×C8C2314yes—
135C2×C4×C8C2312yes—
136#399—24no—
137#398—24no—
138#396—24no—
139#395—24no—
140#394—22no—
141#394—22no—
142#394—24no—
143#394—24no—
144#392—22no—
145#392—22no—
146#392—24no—
147#392—24no—
148#391—21no—
149#391D411yescharacteristic
150#391D411yescharacteristic
151#390—21no—
152#390—22no—
153#390—22no—
154#390D411yescharacteristic
155#390D411yescharacteristic
156#390C2314yes—
157#390C2314yes—
158#389—24no—
159#389C2314yes—
160#389C2312yes—
161#389C2312yes—
162#388—24no—
163#388—24no—
164#387—24no—
165#386—22no—
166#386D414yes—
167#385—22no—
168#385—24no—
169#385—22no—
170#384—21no—
171#384—21no—
172#384—21no—
173#384C2314yes—
174#384C2312yes—
175#382—21no—
176#382C2312yes—
177#380C2314yes—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C22⋊C4C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2312yes—
190#336C2312yes—
191#336C2314yes—
192C25—22no—
193C2×#41—41no—
194C2×#41—41no—
195C2×#41—42no—
196C2×#41#4118yes—
197C22×Q8—22no—
198C22×Q8C2×D411yescharacteristic
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4C2×D411yescharacteristic
213C22×D4C2×D412yes—
214C22×D4C2414yes—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—22no—
218C23×C4C2×D412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C2×M16—24no—
222C2×M16—42no—
223C2×M16—42no—
224C22×C8—24no—
225C22×C8—42no—
226C22×C8—42no—
227C22×C8C2414yes—
228C22×C8C2414yes—
229C4⋊Q8—24no—
230C4⋊Q8—42no—
231C4⋊Q8—44no—
232#127—44no—
233#127—42no—
234#127—42no—
235#127—48no—
236#125—24no—
237#125—42no—
238#125—44no—
239#124—48no—
240C22⋊Q8—48no—
241C22⋊Q8—88no—
242#121—28no—
243#121—28no—
244#121—28no—
245#121—28no—
246#121—84no—
247#121—84no—
248#120—48no—
249#120—48no—
250#120—88no—
251#120—88no—
252C4×Q8—44no—
253C4×D4—44no—
254C4×D4—48no—
255#117—22no—
256#117—22no—
257#117—24no—
258#117—24no—
259#117—24no—
260#117—41no—
261#117—41no—
262#117—42no—
263#117—42no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—84no—
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2411yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4C2×D414yes—
304C2×C22⋊C4C2×D414yes—
305C2×C22⋊C4C2×D412yes—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—21no—
310C2×C42—42no—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2412yes—
314C2×C42C2412yes—
315#102—28no—
316#102—28no—
317#102—44no—
318#102—44no—
319#102—44no—
320#102—44no—
321#102—44no—
322#102—44no—
323C22⋊C8—28no—
324C22⋊C8—28no—
325#97—42no—
326C4×C8—44no—
327C4×C8—42no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95C2×D414yes—
335#95C2×D414yes—
336C24—24no—
337C24—44no—
338C24—82no—
339C24—82no—
340C24—84no—
341C24—84no—
342C24C22×D412yes—
343C24C22×D411yescharacteristic
344#41—48no—
345#41—48no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—44no—
349C2×Q8—84no—
350C2×Q8#12414yes—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—41no—
358C2×D4—42no—
359C2×D4—41no—
360C2×D4—42no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4#12414yes—
385C2×D4#12418yes—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—84no—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4114yes—
417C22×C4C2511yescharacteristic
418M16—84no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—84no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—84no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—84no—
450C4⋊C4H(5,𝔽2)12yes—
451C4⋊C4H(5,𝔽2)12yes—
452C22⋊C4—24no—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—42no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—84no—
464C22⋊C4—84no—
465C22⋊C4—84no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—24no—
470C42—41no—
471C42—41no—
472C42—44no—
473C42—42no—
474C42—44no—
475C23—21no—
476C23—22no—
477C23—21no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—24no—
482C23—22no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—88no—
491C23—168no—
492C23—168no—
493C23—164no—
494C23—164no—
495C23C2×#12414yes—
496C23C2×#12412yes—
497C23C23×D411yescharacteristic
498Q8—44no—
499Q8—164no—
500D4—48no—
501D4—44no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—88no—
507D4—164no—
508D4—164no—
509D4—164no—
510D4—164no—
511D4—168no—
512D4—168no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—48no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—164no—
547C2×C4—162no—
548C2×C4—162no—
549C2×C4C2×#12414yes—
550C2×C4C2×#12412yes—
551C2×C4#57614yes—
552C2×C4#57714yes—
553C2×C4C22×#4112yes—
554C2×C4C2×H(5,𝔽2)11yescharacteristic
555C2×C4C2×H(5,𝔽2)11yescharacteristic
556C2×C4#58512yes—
557C8—44no—
558C8—44no—
559C22—24no—
560C22—22no—
561C22—24no—
562C22—28no—
563C22—42no—
564C22—44no—
565C22—41no—
566C22—42no—
567C22—41no—
568C22—42no—
569C22—84no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—1616no—
577C22C22×#12411yescharacteristic
578C22#348411yescharacteristic
579C22#348511yescharacteristic
580C22C2×#57612yes—
581C22C2×#57712yes—
582C4—24no—
583C4—22no—
584C4—24no—
585C4—24no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—82no—
591C4—84no—
592C4—82no—
593C4—82no—
594C4#350512yes—
595C4#350512yes—
596C2—42no—
597C2—44no—
598C2—164no—
599C2#3537711yescharacteristic
600C2#6062212yes—
601C2#6166111yescharacteristic
602C2#6166411yescharacteristic
603C2#6176612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37297C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35532C2semidirect
3. #29343C2semidirect
4. #29331C2semidirect
5. #27973C2semidirect
6. #27704C2semidirect
7. #27702C2semidirect
8. #27279C2semidirect
9. #27254C2semidirect
10. #25915C2semidirect
11. #21770C2semidirect
12. #20914C2semidirect
13. #20530C2semidirect
14. C2×#544C22semidirect
19. C2×#418C22semidirect
20. C22×#102C22semidirect
35. #2441C22semidirect
36. #2440C22semidirect
38. #2317C22semidirect
41. #2296C22semidirect
42. C2×#384C22semidirect
43. #2101C22semidirect
44. #2100C22semidirect
45. #2100C22semidirect
46. #2071C22semidirect
47. #2071C22semidirect
51. #2025C22semidirect
52. #2023C22semidirect
53. #1993C22semidirect
54. #1991C22semidirect
56. #1936C22semidirect
57. #1934C22semidirect
58. #1934C22semidirect
59. #1934C22semidirect
63. #1875C22semidirect
64. #1875C22semidirect
66. #1874C22semidirect
67. #1874C22semidirect
68. #1834C22semidirect
69. #1825C22semidirect
70. #1786C22semidirect
71. #1753C22semidirect
72. C8×C22⋊C4C22semidirect
73. #1750C22semidirect
74. C4×C22⋊C8C22semidirect
75. #1736C22semidirect
76. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#372972
#355322
#293432
#293312
#279732
#277042
#277022
#272792
#272542
#259152
#217702
#209142
#205302