Finite Group Databasebeta

← #7534968

Subgroups — #7534968

Order 512.

2417 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59816C211yesmaximal, characteristic
2#36559C211yesmaximal, characteristic, complemented
3#31261C214yesmaximal, complemented
4#29139C212yesmaximal, complemented
5#28141C214yesmaximal, complemented
6#24967C214yesmaximal, complemented
7#24240C214yesmaximal, complemented
8#23969C214yesmaximal, complemented
9#23173C211yesmaximal, characteristic, complemented
10D4×#102C214yesmaximal, complemented
11C4×#460C212yesmaximal, complemented
12C2×#554C2212yescomplemented
13C2×D42—21no—
14C2×D42C2214yes—
15C2×#460C2212yes—
16C2×#450C2214yescomplemented
17C2×C4×SD16C2214yescomplemented
18C2×C8×D4C2214yescomplemented
19C22×#102C2214yescomplemented
20#2868C2211yescharacteristic, complemented
21#2859—21no—
22#2679—22no—
23#2566—22no—
24#2563C2212yescomplemented
25#2538C2214yescomplemented
26#2537—21no—
27#2537C2214yes—
28#2452—24no—
29#2408—24no—
30#2408C2214yes—
31D4×C4⋊C4C2214yescomplemented
32C4×#127—21no—
33C4×#127C2212yes—
34C4×#127C2211yescharacteristic
35C42×Q8C2211yescharacteristic, complemented
36#2077C2218yescomplemented
37#2034C2218yescomplemented
38#2023C2218yescomplemented
39#2000C2218yescomplemented
40#1982C2212yes—
41#1957C2214yes—
42#1954C2214yescomplemented
43#1948C2212yescomplemented
44#1926C2214yescomplemented
45#1893C2218yescomplemented
46#1892C2214yescomplemented
47#1880C2214yescomplemented
48#1874C2218yescomplemented
49#1869C2214yescomplemented
50#1834C2214yescomplemented
51#1824C2214yescomplemented
52#1803C2212yescomplemented
53#1802C2214yescomplemented
54#1802C2214yescomplemented
55#1786C2218yescomplemented
56C4×C4⋊C8C2212yescomplemented
57C4×#102C2214yescomplemented
58C23×D4—24no—
59C23×D4C2314yes—
60C22×SD16C2314yes—
61C23×C8C2314yes—
62#554—44no—
63#554—44no—
64D42—216no—
65D42—48no—
66D42—48no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8C2312yes—
69C2×#127D411yescharacteristic
70C2×#127C2312yes—
71C2×#125—22no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8C2314yes—
74C2×#121—24no—
75C2×#121—24no—
76C2×#121C2318yes—
77C2×#120—24no—
78C2×#120—28no—
79C2×C4×Q8C2312yes—
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—22no—
82C2×C4×D4—24no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4C2314yes—
86C2×C4×D4C2314yes—
87C2×C4×D4C2312yes—
88C22×C4⋊C4C2314yes—
89#460—28no—
90#450—48no—
91C4×SD16—48no—
92C8×D4—48no—
93C2×#106C2314yes—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×#103C2314yes—
97C2×#102—48no—
98C2×#102C2314yes—
99C2×#102C2314yes—
100C2×#102C2318yes—
101C2×#102C2318yes—
102C2×#102C2314yes—
103C2×C22⋊C8C2318yes—
104C2×C22⋊C8C2314yes—
105C2×C4×C8C2314yes—
106#396—28no—
107#394—24no—
108#394—28no—
109#393—24no—
110#392—24no—
111#390—22no—
112#390—24no—
113#390—22no—
114#390—22no—
115#390D412yes—
116#390D414yes—
117#390C2314yes—
118#389—24no—
119#389—24no—
120#385—28no—
121#384—21no—
122#384—21no—
123#384D411yescharacteristic
124#384C2312yes—
125#384C2314yes—
126#382—22no—
127#380—24no—
128#380C2318yes—
129#380C2314yes—
130#379—24no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2312yes—
135C43—21no—
136C43C2311yescharacteristic
137#340C2318yes—
138#340C2318yes—
139#340C2314yes—
140#340C2314yes—
141#336C2314yes—
142C25—24no—
143C22×Q8—22no—
144C22×Q8C2411yescharacteristic
145C22×D4—216no—
146C22×D4—28no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—24no—
150C22×D4—44no—
151C22×D4—48no—
152C22×D4—44no—
153C22×D4—416no—
154C22×D4—416no—
155C22×D4—88no—
156C22×D4C2×D414yes—
157C22×D4C2×D412yes—
158C22×D4C2414yes—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4C2414yes—
162C2×SD16—28no—
163C2×SD16—28no—
164C2×SD16—44no—
165C22×C8—44no—
166C22×C8—48no—
167C22×C8C2414yes—
168C22×C8C2414yes—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—84no—
172#127—24no—
173#127SD1618yescomplemented
174#127#4118yes—
175#125—48no—
176C22⋊Q8—28no—
177C22⋊Q8—44no—
178C22⋊Q8—416no—
179#121—216no—
180#121—48no—
181#121—416no—
182#121—416no—
183#120—432no—
184#120—416no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×D4—28no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—28no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—48no—
196C4×D4—48no—
197C4×D4—84no—
198C4×D4—88no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2414yes—
214C2×C4⋊C4C2412yes—
215C2×C4⋊C4C2411yescharacteristic
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4C2×D418yes—
224C2×C22⋊C4C2×D414yes—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2411yescharacteristic
233C2×C42C2412yes—
234#106—44no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237#103—44no—
238#102—28no—
239#102—28no—
240#102—48no—
241#102—44no—
242#102—48no—
243C22⋊C8—48no—
244C22⋊C8—44no—
245C4×C8—28no—
246#95—22no—
247#95—21no—
248#95—24no—
249#95—28no—
250C24—24no—
251C24—416no—
252C24—48no—
253C24—84no—
254C24—88no—
255C24—84no—
256C24C22×D414yes—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8—84no—
262C2×Q8H(5,𝔽2)12yes—
263C2×D4—28no—
264C2×D4—28no—
265C2×D4—28no—
266C2×D4—28no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—416no—
275C2×D4—416no—
276C2×D4—416no—
277C2×D4—416no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—82no—
281C2×D4—82no—
282C2×D4—84no—
283C2×D4—816no—
284C2×D4—88no—
285C2×D4—88no—
286C2×D4—864no—
287C2×D4C2×SD16116yes—
288C2×D4C2×SD1618yes—
289C2×D4C2×#4118yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—28no—
293C22×C4—24no—
294C22×C4—28no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—48no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—48no—
303C22×C4—88no—
304C22×C4C22×D414yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D412yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D412yes—
309C22×C4C2511yescharacteristic
310SD16—416no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—88no—
318C4⋊C4—24no—
319C4⋊C4—28no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—42no—
328C4⋊C4—48no—
329C4⋊C4—48no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—48no—
334C4⋊C4—82no—
335C4⋊C4—84no—
336C4⋊C4H(5,𝔽2)12yes—
337C22⋊C4—216no—
338C22⋊C4—44no—
339C22⋊C4—48no—
340C22⋊C4—416no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—84no—
345C22⋊C4—88no—
346C42—24no—
347C42—24no—
348C42—21no—
349C42—21no—
350C42—41no—
351C42—42no—
352C42—44no—
353C42—44no—
354C42C2×SD1614yes—
355C42C2×#4114yes—
356C42H(5,𝔽2)14yes—
357C23—24no—
358C23—28no—
359C23—24no—
360C23—28no—
361C23—28no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—832no—
369C23—816no—
370C23—168no—
371C23—1616no—
372C23—168no—
373C23C22×SD1618yes—
374C23C23×D411yescharacteristic
375Q8—24no—
376Q8—41no—
377Q8—41no—
378Q8—164no—
379D4—28no—
380D4—28no—
381D4—232no—
382D4—48no—
383D4—44no—
384D4—44no—
385D4—416no—
386D4—88no—
387D4—816no—
388D4—832no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—28no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—48no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—816no—
413C2×C4—168no—
414C2×C4#54614yes—
415C2×C4#54614yes—
416C2×C4C22×SD1612yes—
417C2×C4C22×SD1614yes—
418C2×C4#57614yes—
419C2×C4C22×#4112yes—
420C2×C4C2×H(5,𝔽2)11yescharacteristic
421C2×C4C2×H(5,𝔽2)11yescharacteristic
422C2×C4C2×H(5,𝔽2)12yes—
423C8—44no—
424C8—44no—
425C22—28no—
426C22—216no—
427C22—48no—
428C22—44no—
429C22—44no—
430C22—48no—
431C22—48no—
432C22—48no—
433C22—84no—
434C22—84no—
435C22—84no—
436C22—84no—
437C22—1632no—
438C22C2×#54612yes—
439C22#348411yescharacteristic
440C22C23×SD1611yescharacteristic
441C22C2×#57612yes—
442C22#358611yescharacteristic
443C4—24no—
444C4—22no—
445C4—44no—
446C4—42no—
447C4—41no—
448C4—42no—
449C4—82no—
450C4—84no—
451C4—84no—
452C4#329518yes—
453C4#348111yescharacteristic
454C4#351912yes—
455C4#354911yescharacteristic
456C2—48no—
457C2—88no—
458C2#5904111yescharacteristic
459C2C2×#329512yes—
460C2#6119912yes—
461C2#6156211yescharacteristic
462C2#6166111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59816C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36559C2semidirect
3. #31261C2semidirect
4. #29139C2semidirect
5. #28141C2semidirect
6. #24967C2semidirect
7. #24240C2semidirect
8. #23969C2semidirect
9. #23173C2semidirect
10. D4×#102C2semidirect
11. C4×#460C2semidirect
12. C2×#554C22semidirect
16. C2×#450C22semidirect
17. C2×C4×SD16C22semidirect
18. C2×C8×D4C22semidirect
19. C22×#102C22semidirect
20. #2868C22semidirect
24. #2563C22semidirect
25. #2538C22semidirect
31. D4×C4⋊C4C22semidirect
35. C42×Q8C22semidirect
36. #2077C22semidirect
37. #2034C22semidirect
38. #2023C22semidirect
39. #2000C22semidirect
42. #1954C22semidirect
43. #1948C22semidirect
44. #1926C22semidirect
45. #1893C22semidirect
46. #1892C22semidirect
47. #1880C22semidirect
48. #1874C22semidirect
49. #1869C22semidirect
50. #1834C22semidirect
51. #1824C22semidirect
52. #1803C22semidirect
53. #1802C22semidirect
54. #1802C22semidirect
55. #1786C22semidirect
56. C4×C4⋊C8C22semidirect
57. C4×#102C22semidirect
173. #127SD16semidirect

Maximal subgroups

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

subgroupindex
#598162
#365592
#312612
#291392
#281412
#249672
#242402
#239692
#231732
D4×#1022
C4×#4602