Finite Group Databasebeta

← #7470774

Subgroups — #7470774

Order 512.

1761 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45996C211yesmaximal, characteristic, complemented
2#39489C211yesmaximal, characteristic, complemented
3#32647C212yesmaximal, complemented
4#31739C214yesmaximal, complemented
5#31270C212yesmaximal, complemented
6#30984C214yesmaximal, complemented
7#30203C212yesmaximal, complemented
8#29552C212yesmaximal, complemented
9#28999C214yesmaximal, complemented
10#27279C214yesmaximal, complemented
11#25946C212yesmaximal, complemented
12#23976C212yesmaximal, complemented
13#23272C211yesmaximal, characteristic, complemented
14C2×H(3,𝔽4)—21no—
15C2×#555C2211yescharacteristic
16C2×#554C2211yescharacteristic, complemented
17C2×#490C2212yes—
18C2×#459C2212yes—
19C2×#447C2212yescomplemented
20C2×#418C2212yescomplemented
21#2868C2211yescharacteristic, complemented
22#2858C2212yes—
23#2819—21no—
24#2733—21no—
25#2679—21no—
26#2674—24no—
27#2670—24no—
28#2620—21no—
29#2567C2211yescharacteristic, complemented
30#2563C2211yescharacteristic
31#2548C2212yes—
32#2501—21no—
33#2499—21no—
34#2491C2214yes—
35#2441C2214yescomplemented
36#2433C2214yes—
37#2422C2214yescomplemented
38#2408C2212yes—
39#2317—21no—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic, complemented
42#2076C2214yescomplemented
43#2071C2214yescomplemented
44#2071C2214yescomplemented
45#2067C2214yescomplemented
46#2057C2212yescomplemented
47#2041C2214yes—
48#2041C2212yes—
49#2033C2214yes—
50#2025C2214yescomplemented
51#2023C2214yescomplemented
52#2023C2212yescomplemented
53#2023C2214yescomplemented
54#2011C2212yes—
55#1999C2214yes—
56#1990C2214yescomplemented
57#1989C2212yescomplemented
58#1985C2212yescomplemented
59#1970C2214yescomplemented
60#1957C2214yes—
61#1936C2214yescomplemented
62#1934C2214yescomplemented
63#1925C2214yescomplemented
64#1923C2214yescomplemented
65#1921C2214yescomplemented
66#1895C2214yes—
67#1895C2212yescomplemented
68#1879C2214yescomplemented
69#1879C2214yescomplemented
70#1875C2214yes—
71#1870C2212yes—
72#1825C2212yescomplemented
73#1786C2214yescomplemented
74#1762C2212yescomplemented
75#1753C2214yescomplemented
76#1744C2212yescomplemented
77C23×D4C2312yes—
78C22×D8C2314yes—
79C22×M16C2312yes—
80H(3,𝔽4)—48no—
81#555—42no—
82#555—42no—
83#554—42no—
84#554—42no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×#127C2312yes—
90C2×#127C2312yes—
91C2×#125C2312yes—
92C2×#124—22no—
93C2×#124—24no—
94C2×#124—22no—
95C2×#124—22no—
96C2×#124—22no—
97C2×#124D411yescharacteristic
98C2×#123—24no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×#121—21no—
102C2×#121C2312yes—
103C2×#121C2314yes—
104C2×#120—22no—
105C2×#120—24no—
106C2×#120—24no—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×D4C2311yescharacteristic
109C2×C4×D4C2312yes—
110C2×C4×D4C2311yescharacteristic
111C2×#117—21no—
112C2×#117—21no—
113C2×#117C2312yes—
114C22×C4⋊C4—21no—
115#490—44no—
116#459—44no—
117#447—44no—
118C2×#107C2314yes—
119C2×C4⋊C8C2312yes—
120#418—44no—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2312yes—
131C2×#97C2312yes—
132C2×C4×C8C2314yes—
133#399—24no—
134#397—24no—
135#396—24no—
136#395—22no—
137#394—22no—
138#394—22no—
139#394—22no—
140#394—24no—
141#394—24no—
142#393—22no—
143#393—24no—
144#392—22no—
145#392—24no—
146#392—24no—
147#391—21no—
148#391D411yescharacteristic
149#390—21no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390D411yescharacteristic
154#390C2314yes—
155#390C2314yes—
156#390C2314yes—
157#390C2312yes—
158#389—24no—
159#389C2314yes—
160#389C2312yes—
161#388—22no—
162#388—24no—
163#388D414yes—
164#387—24no—
165#385—24no—
166#384—21no—
167#384—21no—
168#384—21no—
169#384C2314yes—
170#382—21no—
171#382D411yescharacteristic
172#380C2314yes—
173#380C2312yes—
174#379—22no—
175C4×C4⋊C4C2311yescharacteristic
176C4×C4⋊C4C2312yes—
177C4×C22⋊C4—21no—
178C4×C22⋊C4C2314yes—
179C43C2311yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2312yes—
187#336C2312yes—
188#336C2314yes—
189#336C2314yes—
190C25—22no—
191C22×Q8—21no—
192C22×Q8—22no—
193C22×Q8—21no—
194C22×Q8C2×D411yescharacteristic
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—44no—
203C22×D4—42no—
204C22×D4—44no—
205C22×D4—42no—
206C22×D4C2×D411yescharacteristic
207C22×D4C2414yes—
208C22×D4C2411yescharacteristic
209C23×C4—21no—
210C23×C4—22no—
211C23×C4C2412yes—
212C2×D8—28no—
213C2×D8—44no—
214C2×D8—44no—
215C2×M16—44no—
216C2×M16—42no—
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—41no—
220C4⋊Q8—41no—
221C4⋊Q8—41no—
222C4⋊Q8—41no—
223C4⋊Q8—44no—
224C4⋊Q8—84no—
225#127—44no—
226#127—42no—
227#127—42no—
228#125—24no—
229#125—42no—
230#124—24no—
231#124—48no—
232#124—416no—
233#124—48no—
234#124—48no—
235#124—84no—
236#123—88no—
237C22⋊Q8—48no—
238C22⋊Q8—44no—
239#121—24no—
240#121—42no—
241#121—44no—
242#121—44no—
243#121—44no—
244#120—416no—
245#120—48no—
246#120—88no—
247C4×Q8—41no—
248C4×Q8—41no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—24no—
252C4×D4—42no—
253C4×D4—41no—
254C4×D4—41no—
255#117—24no—
256#117—42no—
257#117—44no—
258#117—44no—
259C2×C4⋊C4—22no—
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—22no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2411yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4C2×D414yes—
293C2×C22⋊C4C2×D414yes—
294C2×C22⋊C4C2×D412yes—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42C2×D412yes—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2412yes—
303C2×C42C2412yes—
304#107—44no—
305C4⋊C8—42no—
306#102—28no—
307#102—44no—
308#102—44no—
309#102—44no—
310#102—44no—
311#102—44no—
312#102—44no—
313#102—44no—
314C22⋊C8—44no—
315C22⋊C8—42no—
316#97—42no—
317C4×C8—44no—
318#95—22no—
319#95—21no—
320#95—22no—
321#95—24no—
322#95—22no—
323#95—24no—
324#95C2×D414yes—
325#95C2×D414yes—
326C24—21no—
327C24—24no—
328C24—44no—
329C24—82no—
330C24—84no—
331C24—84no—
332C24—82no—
333C24C22×D412yes—
334C2×Q8—22no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—44no—
340C2×Q8—82no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×Q8—82no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—44no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—82no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—84no—
372C2×D4—88no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4H(5,𝔽2)12yes—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—84no—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C2511yescharacteristic
399D8—88no—
400M16—84no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4—84no—
425C4⋊C4—82no—
426C4⋊C4—84no—
427C4⋊C4H(5,𝔽2)12yes—
428C22⋊C4—24no—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—44no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—21no—
450C42—21no—
451C42—24no—
452C42—41no—
453C42—44no—
454C42—44no—
455C42—42no—
456C42AGL(1,ℤ/8)14yes—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—41no—
464C23—42no—
465C23—41no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—88no—
475C23—168no—
476C23—164no—
477C23—168no—
478C23—164no—
479C23C23×D411yescharacteristic
480Q8—82no—
481Q8—164no—
482Q8—162no—
483Q8—162no—
484D4—24no—
485D4—42no—
486D4—44no—
487D4—44no—
488D4—44no—
489D4—44no—
490D4—82no—
491D4—88no—
492D4—164no—
493D4—164no—
494D4—164no—
495D4—164no—
496D4—164no—
497D4—168no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—48no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—164no—
531C2×C4C2×AGL(1,ℤ/8)12yes—
532C2×C4C2×AGL(1,ℤ/8)14yes—
533C2×C4#57614yes—
534C2×C4C2×H(5,𝔽2)11yescharacteristic
535C2×C4C2×H(5,𝔽2)11yescharacteristic
536C2×C4#58514yes—
537C8—44no—
538C8—44no—
539C22—42no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—82no—
548C22—82no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22—1616no—
557C22#348411yescharacteristic
558C22#350012yes—
559C22C22×AGL(1,ℤ/8)11yescharacteristic
560C22C2×#57612yes—
561C22#358611yescharacteristic
562C4—24no—
563C4—24no—
564C4—22no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—82no—
571C4—84no—
572C4—82no—
573C4—82no—
574C4#354012yes—
575C4#354512yes—
576C2—82no—
577C2—82no—
578C2—164no—
579C2#5174911yescharacteristic
580C2#6158511yescharacteristic
581C2#6166111yescharacteristic
582C2#6253512yes—
583C2#6260512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #45996C2semidirect
2. #39489C2semidirect
3. #32647C2semidirect
4. #31739C2semidirect
5. #31270C2semidirect
6. #30984C2semidirect
7. #30203C2semidirect
8. #29552C2semidirect
9. #28999C2semidirect
10. #27279C2semidirect
11. #25946C2semidirect
12. #23976C2semidirect
13. #23272C2semidirect
16. C2×#554C22semidirect
19. C2×#447C22semidirect
20. C2×#418C22semidirect
21. #2868C22semidirect
29. #2567C22semidirect
35. #2441C22semidirect
37. #2422C22semidirect
41. #2293C22semidirect
42. #2076C22semidirect
43. #2071C22semidirect
44. #2071C22semidirect
45. #2067C22semidirect
46. #2057C22semidirect
50. #2025C22semidirect
51. #2023C22semidirect
52. #2023C22semidirect
53. #2023C22semidirect
56. #1990C22semidirect
57. #1989C22semidirect
58. #1985C22semidirect
59. #1970C22semidirect
61. #1936C22semidirect
62. #1934C22semidirect
63. #1925C22semidirect
64. #1923C22semidirect
65. #1921C22semidirect
67. #1895C22semidirect
68. #1879C22semidirect
69. #1879C22semidirect
72. #1825C22semidirect
73. #1786C22semidirect
74. #1762C22semidirect
75. #1753C22semidirect
76. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#459962
#394892
#326472
#317392
#312702
#309842
#302032
#295522
#289992
#272792
#259462
#239762
#232722