Finite Group Databasebeta

← #7470784

Subgroups — #7470784

Order 512.

1769 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45996C211yesmaximal, characteristic, complemented
2#38958C211yesmaximal, characteristic, complemented
3#32629C212yesmaximal, complemented
4#31767C214yesmaximal, complemented
5#31271C212yesmaximal, complemented
6#31000C214yesmaximal, complemented
7#30213C212yesmaximal, complemented
8#29558C212yesmaximal, complemented
9#28992C214yesmaximal, complemented
10#27273C214yesmaximal, complemented
11#25933C212yesmaximal, complemented
12#23976C212yesmaximal, complemented
13#23271C211yesmaximal, characteristic, complemented
14C2×H(3,𝔽4)—21no—
15C2×#555C2211yescharacteristic
16C2×D4×Q8C2211yescharacteristic, complemented
17C2×#486C2212yes—
18C2×#459C2212yes—
19C2×#447C2212yescomplemented
20C2×#418C2212yescomplemented
21#2862C2211yescharacteristic, complemented
22#2858C2212yes—
23#2819—21no—
24#2733—21no—
25#2700—24no—
26#2679—21no—
27#2670—24no—
28#2620—21no—
29#2567C2211yescharacteristic, complemented
30#2548C2212yes—
31#2547C2211yescharacteristic
32#2499—21no—
33#2491C2214yes—
34#2439C2214yescomplemented
35#2436—21no—
36#2433C2214yes—
37#2420C2214yescomplemented
38#2408C2212yes—
39#2310—21no—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic, complemented
42#2077C2214yescomplemented
43#2073C2214yescomplemented
44#2073C2214yescomplemented
45#2065C2214yescomplemented
46#2058C2212yescomplemented
47#2041C2214yes—
48#2041C2212yes—
49#2034C2214yes—
50#2023C2214yescomplemented
51#2023C2214yescomplemented
52#2023C2214yescomplemented
53#2023C2212yescomplemented
54#2011C2212yes—
55#1999C2214yes—
56#1993C2214yescomplemented
57#1988C2212yescomplemented
58#1984C2212yescomplemented
59#1972C2214yescomplemented
60#1961C2214yes—
61#1936C2214yescomplemented
62#1934C2214yescomplemented
63#1926C2214yescomplemented
64#1924C2214yescomplemented
65#1920C2214yescomplemented
66#1895C2214yes—
67#1895C2212yescomplemented
68#1879C2214yescomplemented
69#1879C2214yescomplemented
70#1875C2214yes—
71#1870C2212yes—
72#1825C2212yescomplemented
73#1786C2214yescomplemented
74C4×#102C2212yescomplemented
75#1753C2214yescomplemented
76#1745C2212yescomplemented
77C23×Q8—21no—
78C23×D4C2312yes—
79C22×D8C2314yes—
80C22×M16C2312yes—
81H(3,𝔽4)—48no—
82#555—42no—
83#555—42no—
84D4×Q8—42no—
85D4×Q8—42no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127C2312yes—
91C2×#127C2312yes—
92C2×#124—22no—
93C2×#124—21no—
94C2×#124—24no—
95C2×#124—22no—
96C2×#124—22no—
97C2×#124—22no—
98C2×#124D411yescharacteristic
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—
114#486—44no—
115#459—44no—
116#447—44no—
117C2×#106C2314yes—
118C2×C4⋊C8C2312yes—
119#418—44no—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2312yes—
130C2×#97C2314yes—
131C2×C4×C8C2312yes—
132#397—22no—
133#397—24no—
134#396—24no—
135#395—22no—
136#394—22no—
137#394—22no—
138#394—22no—
139#394—24no—
140#394—24no—
141#393—24no—
142#393—24no—
143#392—22no—
144#392—24no—
145#392—24no—
146#391—21no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391—24no—
152#391D411yescharacteristic
153#390—22no—
154#390—22no—
155#390—22no—
156#390D411yescharacteristic
157#390C2314yes—
158#390C2314yes—
159#390C2314yes—
160#390C2312yes—
161#389C2314yes—
162#389C2312yes—
163#388—22no—
164#388—24no—
165#388D414yes—
166#387—24no—
167#386—24no—
168#384—21no—
169#382D411yescharacteristic
170#382C2314yes—
171#381—24no—
172#380C2312yes—
173#380C2314yes—
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—22no—
194C22×Q8—21no—
195C22×Q8—44no—
196C22×Q8—44no—
197C22×Q8—82no—
198C22×Q8C2×D411yescharacteristic
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—44no—
207C22×D4—42no—
208C22×D4—44no—
209C22×D4—42no—
210C22×D4C2×D411yescharacteristic
211C22×D4C2414yes—
212C22×D4C2411yescharacteristic
213C23×C4—21no—
214C23×C4—22no—
215C23×C4C2412yes—
216C2×D8—28no—
217C2×D8—44no—
218C2×D8—44no—
219C2×M16—44no—
220C2×M16—42no—
221C22×C8C2414yes—
222C22×C8C2414yes—
223C4⋊Q8—24no—
224C4⋊Q8—42no—
225C4⋊Q8—41no—
226C4⋊Q8—41no—
227C4⋊Q8—41no—
228C4⋊Q8—41no—
229C4⋊Q8—44no—
230#127—44no—
231#127—42no—
232#127—42no—
233#124—24no—
234#124—48no—
235#124—48no—
236#124—416no—
237#124—48no—
238#124—84no—
239#124—82no—
240C22⋊Q8—48no—
241C22⋊Q8—44no—
242#121—24no—
243#121—42no—
244#121—44no—
245#121—44no—
246#121—44no—
247#120—416no—
248#120—48no—
249#120—88no—
250C4×Q8—41no—
251C4×Q8—41no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—24no—
255C4×D4—42no—
256C4×D4—41no—
257C4×D4—41no—
258#117—24no—
259#117—42no—
260#117—44no—
261#117—82no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2414yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2411yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4C2×D414yes—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D414yes—
294C2×C42—22no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D412yes—
300C2×C42C2411yescharacteristic
301C2×C42C2412yes—
302C2×C42C2412yes—
303#106—44no—
304C4⋊C8—42no—
305#102—28no—
306#102—44no—
307#102—44no—
308#102—44no—
309#102—44no—
310#102—44no—
311#102—44no—
312#102—44no—
313C22⋊C8—44no—
314C22⋊C8—42no—
315#97—44no—
316C4×C8—42no—
317#95—22no—
318#95—21no—
319#95—22no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95C2×D414yes—
326#95C2×D414yes—
327C24—21no—
328C24—24no—
329C24—44no—
330C24—82no—
331C24—84no—
332C24—84no—
333C24—82no—
334C24C22×D412yes—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—44no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—44no—
341C2×Q8—42no—
342C2×Q8—44no—
343C2×Q8—42no—
344C2×Q8—82no—
345C2×Q8—82no—
346C2×Q8—82no—
347C2×Q8—82no—
348C2×Q8—816no—
349C2×Q8AGL(1,ℤ/8)14yes—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—44no—
367C2×D4—82no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—84no—
378C2×D4—88no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4H(5,𝔽2)12yes—
383C22×C4—24no—
384C22×C4—21no—
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—42no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—82no—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C2511yescharacteristic
406D8—88no—
407M16—84no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—84no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4H(5,𝔽2)12yes—
432C22⋊C4—24no—
433C22⋊C4—28no—
434C22⋊C4—42no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—21no—
454C42—21no—
455C42—24no—
456C42—41no—
457C42—42no—
458C42—44no—
459C42—42no—
460C42—44no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—22no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—88no—
479C23—168no—
480C23—164no—
481C23—168no—
482C23—164no—
483C23C23×D411yescharacteristic
484Q8—44no—
485Q8—88no—
486Q8—84no—
487Q8—164no—
488Q8—162no—
489D4—24no—
490D4—42no—
491D4—44no—
492D4—44no—
493D4—44no—
494D4—44no—
495D4—82no—
496D4—88no—
497D4—164no—
498D4—164no—
499D4—164no—
500D4—164no—
501D4—164no—
502D4—168no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—48no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—88no—
538C2×C4C2×AGL(1,ℤ/8)12yes—
539C2×C4C2×AGL(1,ℤ/8)14yes—
540C2×C4C2×H(5,𝔽2)11yescharacteristic
541C2×C4C2×H(5,𝔽2)11yescharacteristic
542C2×C4#58514yes—
543C8—44no—
544C8—44no—
545C22—42no—
546C22—42no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22—84no—
556C22—84no—
557C22—84no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—1616no—
563C22#348411yescharacteristic
564C22#350012yes—
565C22C22×AGL(1,ℤ/8)11yescharacteristic
566C22#358511yescharacteristic
567C22#358512yes—
568C4—22no—
569C4—24no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—41no—
576C4—82no—
577C4—84no—
578C4—82no—
579C4—82no—
580C4#354012yes—
581C4#354512yes—
582C2—82no—
583C2—82no—
584C2—164no—
585C2#5174911yescharacteristic
586C2#6156911yescharacteristic
587C2#6167311yescharacteristic
588C2#6257012yes—
589C2#6258812yes—
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. #38958C2semidirect
3. #32629C2semidirect
4. #31767C2semidirect
5. #31271C2semidirect
6. #31000C2semidirect
7. #30213C2semidirect
8. #29558C2semidirect
9. #28992C2semidirect
10. #27273C2semidirect
11. #25933C2semidirect
12. #23976C2semidirect
13. #23271C2semidirect
16. C2×D4×Q8C22semidirect
19. C2×#447C22semidirect
20. C2×#418C22semidirect
21. #2862C22semidirect
29. #2567C22semidirect
34. #2439C22semidirect
37. #2420C22semidirect
41. #2293C22semidirect
42. #2077C22semidirect
43. #2073C22semidirect
44. #2073C22semidirect
45. #2065C22semidirect
46. #2058C22semidirect
50. #2023C22semidirect
51. #2023C22semidirect
52. #2023C22semidirect
53. #2023C22semidirect
56. #1993C22semidirect
57. #1988C22semidirect
58. #1984C22semidirect
59. #1972C22semidirect
61. #1936C22semidirect
62. #1934C22semidirect
63. #1926C22semidirect
64. #1924C22semidirect
65. #1920C22semidirect
67. #1895C22semidirect
68. #1879C22semidirect
69. #1879C22semidirect
72. #1825C22semidirect
73. #1786C22semidirect
74. C4×#102C22semidirect
75. #1753C22semidirect
76. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#459962
#389582
#326292
#317672
#312712
#310002
#302132
#295582
#289922
#272732
#259332
#239762
#232712