Finite Group Databasebeta

← #7470776

Subgroups — #7470776

Order 512.

1713 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45996C211yesmaximal, characteristic
2#39744C211yesmaximal, characteristic, complemented
3#32646C212yesmaximal, complemented
4#31741C214yesmaximal, complemented
5#31271C212yesmaximal, complemented
6#30988C214yesmaximal, complemented
7#30203C212yesmaximal, complemented
8#29555C212yesmaximal, complemented
9#29005C214yesmaximal, complemented
10#27279C214yesmaximal, complemented
11#25944C212yesmaximal, complemented
12#23979C212yesmaximal, complemented
13#23272C211yesmaximal, characteristic, complemented
14C2×H(3,𝔽4)—21no—
15C2×#557C2211yescharacteristic, complemented
16C2×#556C2211yescharacteristic
17C2×#490C2212yes—
18C2×#460C2212yes—
19C2×#450C2212yescomplemented
20C2×#418C2212yescomplemented
21#2868C2211yescharacteristic, complemented
22#2858C2212yes—
23#2819—21no—
24#2733—21no—
25#2721—24no—
26#2679—21no—
27#2670—24no—
28#2658—21no—
29#2548C2212yes—
30#2547C2211yescharacteristic
31#2546—21no—
32#2546C2211yescharacteristic, complemented
33#2545—21no—
34#2491C2214yes—
35#2441C2214yescomplemented
36#2433C2214yes—
37#2423C2214yescomplemented
38#2408C2212yes—
39#2327—21no—
40C4×#124C2211yescharacteristic
41#2295C2211yescharacteristic, complemented
42#2077C2214yescomplemented
43#2071C2214yescomplemented
44#2071C2214yescomplemented
45#2065C2214yescomplemented
46#2058C2212yescomplemented
47#2041C2214yes—
48#2041C2212yes—
49#2034C2214yes—
50#2025C2214yescomplemented
51#2024C2212yescomplemented
52#2023C2214yescomplemented
53#2023C2214yescomplemented
54#2014C2212yes—
55#2000C2214yes—
56#1990C2214yescomplemented
57#1989C2212yescomplemented
58#1985C2212yescomplemented
59#1968C2214yescomplemented
60#1957C2214yes—
61#1936C2214yescomplemented
62#1934C2214yescomplemented
63#1926C2214yescomplemented
64#1923C2214yescomplemented
65#1919C2214yescomplemented
66#1895C2214yes—
67#1894C2212yescomplemented
68#1880C2214yescomplemented
69#1880C2214yescomplemented
70#1875C2214yes—
71#1870C2212yes—
72#1825C2212yescomplemented
73#1786C2214yescomplemented
74#1763C2212yescomplemented
75#1753C2214yescomplemented
76#1744C2212yescomplemented
77C23×D4C2312yes—
78C22×SD16C2314yes—
79C22×M16C2312yes—
80H(3,𝔽4)—48no—
81#557—42no—
82#557—42no—
83#556—42no—
84#556—42no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2312yes—
93C2×#127C2312yes—
94C2×#126—22no—
95C2×#125—22no—
96C2×#125C2312yes—
97C2×#124—22no—
98C2×#124—21no—
99C2×#124—21no—
100C2×#124—24no—
101C2×#124—22no—
102C2×#124—22no—
103C2×#124D411yescharacteristic
104C2×C22⋊Q8C2312yes—
105C2×#121C2314yes—
106C2×#120—22no—
107C2×#120—24no—
108C2×#120—24no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117C2312yes—
116#490—44no—
117#460—44no—
118#450—44no—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121#418—44no—
122C2×#103C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2312yes—
132C2×#97C2312yes—
133C2×C4×C8C2314yes—
134#399—24no—
135#399—24no—
136#398—22no—
137#398—24no—
138#397—22no—
139#396—24no—
140#394—22no—
141#394—22no—
142#394—22no—
143#394—24no—
144#393—24no—
145#392—22no—
146#392—24no—
147#392—24no—
148#391—21no—
149#391—21no—
150#391D411yescharacteristic
151#391C2312yes—
152#390—22no—
153#390—22no—
154#390—22no—
155#390D411yescharacteristic
156#390C2314yes—
157#390C2314yes—
158#390C2314yes—
159#389—24no—
160#389C2312yes—
161#389C2314yes—
162#388—24no—
163#388—24no—
164#388D414yes—
165#387—22no—
166#387—24no—
167#387—24no—
168#384—22no—
169#384C2314yes—
170#382D411yescharacteristic
171#380C2312yes—
172#380C2314yes—
173#379—22no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2311yescharacteristic
177C4×C22⋊C4C2314yes—
178#376—21no—
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—21no—
193C22×Q8—22no—
194C22×Q8C2×D411yescharacteristic
195C22×Q8C2411yescharacteristic
196C22×D4—24no—
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—
208C23×C4C2412yes—
209C2×SD16—28no—
210C2×SD16—44no—
211C2×SD16—44no—
212C2×M16—44no—
213C2×M16—42no—
214C22×C8C2414yes—
215C22×C8C2414yes—
216C4⋊Q8—41no—
217C4⋊Q8—41no—
218C4⋊Q8—41no—
219C4⋊Q8—41no—
220C4⋊Q8—44no—
221C4⋊Q8—44no—
222C4⋊Q8—44no—
223C4⋊Q8—82no—
224C4⋊Q8—82no—
225#127—44no—
226#127—42no—
227#127—42no—
228#126—48no—
229#125—24no—
230#125—42no—
231#125—48no—
232#124—24no—
233#124—44no—
234#124—416no—
235#124—48no—
236#124—48no—
237#124—84no—
238#124—82no—
239C22⋊Q8—24no—
240C22⋊Q8—42no—
241#121—44no—
242#121—44no—
243#120—416no—
244#120—48no—
245#120—88no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—24no—
249C4×Q8—41no—
250C4×Q8—41no—
251C4×Q8—42no—
252C4×Q8—44no—
253C4×Q8—44no—
254C4×D4—41no—
255C4×D4—41no—
256#117—24no—
257#117—42no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2414yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2411yescharacteristic
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—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C22⋊C4C2×D412yes—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2412yes—
302C2×C42C2412yes—
303#106—44no—
304C4⋊C8—42no—
305#103—44no—
306#102—28no—
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—42no—
316C4×C8—44no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—21no—
321#95—22no—
322#95—24no—
323#95—22no—
324#95—24no—
325#95C2×D414yes—
326#95C2×D414yes—
327C24—24no—
328C24—44no—
329C24—82no—
330C24—84no—
331C24—84no—
332C24—82no—
333C24C22×D412yes—
334C2×Q8—22no—
335C2×Q8—21no—
336C2×Q8—24no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—44no—
346C2×Q8—82no—
347C2×Q8—82no—
348C2×Q8H(5,𝔽2)12yes—
349C2×D4—22no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—82no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—84no—
374C2×D4—88no—
375C2×D4—84no—
376C2×D4—84no—
377C22×C4—21no—
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×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C2511yescharacteristic
395SD16—88no—
396M16—84no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C4⋊C4—24no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—48no—
422C4⋊C4—44no—
423C4⋊C4—82no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4H(5,𝔽2)12yes—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C42—24no—
442C42—24no—
443C42—22no—
444C42—22no—
445C42—21no—
446C42—21no—
447C42—24no—
448C42—41no—
449C42—44no—
450C42—44no—
451C42—44no—
452C42—42no—
453C42AGL(1,ℤ/8)14yes—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—24no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—88no—
467C23—168no—
468C23—164no—
469C23—168no—
470C23—164no—
471C23C23×D411yescharacteristic
472Q8—24no—
473Q8—42no—
474Q8—82no—
475Q8—84no—
476Q8—84no—
477Q8—164no—
478D4—44no—
479D4—44no—
480D4—44no—
481D4—44no—
482D4—82no—
483D4—88no—
484D4—164no—
485D4—164no—
486D4—164no—
487D4—164no—
488D4—168no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4C2×AGL(1,ℤ/8)14yes—
524C2×C4C2×AGL(1,ℤ/8)12yes—
525C2×C4#57714yes—
526C2×C4C2×H(5,𝔽2)11yescharacteristic
527C2×C4C2×H(5,𝔽2)11yescharacteristic
528C2×C4#58514yes—
529C8—44no—
530C8—44no—
531C22—42no—
532C22—44no—
533C22—44no—
534C22—42no—
535C22—42no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22—1616no—
544C22#348411yescharacteristic
545C22#350012yes—
546C22C22×AGL(1,ℤ/8)11yescharacteristic
547C22C2×#57712yes—
548C22#358511yescharacteristic
549C4—24no—
550C4—24no—
551C4—22no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—84no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4#354012yes—
563C4#354512yes—
564C2—82no—
565C2—164no—
566C2#5174911yescharacteristic
567C2#6158711yescharacteristic
568C2#6166411yescharacteristic
569C2#6253912yes—
570C2#6260212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#45996C2
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
2. #39744C2semidirect
3. #32646C2semidirect
4. #31741C2semidirect
5. #31271C2semidirect
6. #30988C2semidirect
7. #30203C2semidirect
8. #29555C2semidirect
9. #29005C2semidirect
10. #27279C2semidirect
11. #25944C2semidirect
12. #23979C2semidirect
13. #23272C2semidirect
15. C2×#557C22semidirect
19. C2×#450C22semidirect
20. C2×#418C22semidirect
21. #2868C22semidirect
32. #2546C22semidirect
35. #2441C22semidirect
37. #2423C22semidirect
41. #2295C22semidirect
42. #2077C22semidirect
43. #2071C22semidirect
44. #2071C22semidirect
45. #2065C22semidirect
46. #2058C22semidirect
50. #2025C22semidirect
51. #2024C22semidirect
52. #2023C22semidirect
53. #2023C22semidirect
56. #1990C22semidirect
57. #1989C22semidirect
58. #1985C22semidirect
59. #1968C22semidirect
61. #1936C22semidirect
62. #1934C22semidirect
63. #1926C22semidirect
64. #1923C22semidirect
65. #1919C22semidirect
67. #1894C22semidirect
68. #1880C22semidirect
69. #1880C22semidirect
72. #1825C22semidirect
73. #1786C22semidirect
74. #1763C22semidirect
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
#397442
#326462
#317412
#312712
#309882
#302032
#295552
#290052
#272792
#259442
#239792
#232722