Finite Group Databasebeta

← #6468752

Subgroups — #6468752

Order 512.

2201 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C212yesmaximal, complemented
2#29418C214yesmaximal, complemented
3#29249C214yesmaximal, complemented
4#28126C214yesmaximal, complemented
5#27609C211yesmaximal, characteristic, complemented
6#24015C212yesmaximal, complemented
7#23925C211yesmaximal, characteristic
8#23708C214yesmaximal, complemented
9#21582C214yesmaximal, complemented
10#21452C214yesmaximal, complemented
11#20719C211yesmaximal, characteristic, complemented
12C2×#535C2212yes—
13C2×C4×#41C2211yescharacteristic
14C22×C4×D4C2212yescomplemented
15C2×#482C2214yes—
16C2×#447C2214yes—
17C2×#442C2214yes—
18C2×#432C2212yescomplemented
19#2328—22no—
20#2326—22no—
21#2324C2214yes—
22#2324C2214yescomplemented
23#2321—22no—
24#2321—22no—
25#2317C2212yescomplemented
26#2311C2212yes—
27#2309C2212yescomplemented
28C4×#127—22no—
29C4×#124—22no—
30C4×#121C2214yes—
31C4×#121C2214yescomplemented
32C4×#120—22no—
33C4×#120—22no—
34C4×#117C2212yescomplemented
35#2088C2214yescomplemented
36#2077C2214yescomplemented
37#2067C2214yescomplemented
38#2065C2214yescomplemented
39#2054C2214yes—
40#2034C2214yescomplemented
41#2033C2214yescomplemented
42#1999C2214yes—
43#1926C2214yescomplemented
44#1925C2214yescomplemented
45#1919C2214yescomplemented
46#1918C2214yescomplemented
47#1879C2214yes—
48#1879C2214yes—
49#1875C2214yes—
50#1874C2214yes—
51#1864C2214yescomplemented
52#1864C2214yescomplemented
53#1854C2214yescomplemented
54#1853C2214yescomplemented
55#1825C2214yescomplemented
56#1818C2212yescomplemented
57#1798C2214yescomplemented
58#1786C2214yescomplemented
59#1780C2211yescharacteristic
60#1780C2211yescharacteristic
61#1776C2214yescomplemented
62#1762C2214yescomplemented
63#1762C2214yescomplemented
64#1749C2214yescomplemented
65C22×#41C2311yescharacteristic
66C23×D4C2312yes—
67C24×C4—22no—
68C22×D8C2314yes—
69#535—216no—
70C2×#127—22no—
71C2×#124—22no—
72C2×#121C2314yes—
73C2×#121C2314yes—
74C2×#120—22no—
75C2×#120—22no—
76C4×#41—44no—
77C2×C4×Q8D412yes—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—44no—
83C2×C4×D4—44no—
84C2×C4×D4D412yes—
85C2×C4×D4D412yes—
86C2×C4×D4D412yes—
87C2×C4×D4C2314yes—
88C2×C4×D4C2314yes—
89C2×#117C2312yes—
90C2×#117C2312yes—
91C2×#117C2312yes—
92C2×#117C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—22no—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97#482—216no—
98#447—216no—
99#442—48no—
100#432—44no—
101C2×#107C2314yes—
102C2×#106C2314yes—
103C2×#106C2314yes—
104C2×C4⋊C8C2314yes—
105C2×#102C2314yes—
106C2×#102C2314yes—
107C2×#102C2314yes—
108C2×#102C2314yes—
109C2×#102C2314yes—
110C2×#102C2314yes—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×#97C2314yes—
116#391—22no—
117#390—22no—
118#390—22no—
119#390—22no—
120#390—24no—
121#390—24no—
122#390C2314yes—
123#390C2314yes—
124#389C2314yes—
125#389C2314yes—
126#388—24no—
127#388—24no—
128#388—24no—
129#388—24no—
130#386—22no—
131#386—22no—
132#386—24no—
133#386—24no—
134#386—24no—
135#386—24no—
136#385—24no—
137#385—24no—
138#385—22no—
139#385—22no—
140#384—22no—
141#382—22no—
142#380C2314yes—
143#380C2314yes—
144#379—22no—
145#379—22no—
146C4×C4⋊C4C2314yes—
147C4×C4⋊C4C2314yes—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152#376—22no—
153C43—22no—
154#340C2314yes—
155#340C2314yes—
156#340C2314yes—
157#340C2314yes—
158#340C2314yes—
159#340C2314yes—
160#336C2314yes—
161#336C2314yes—
162C25—22no—
163C2×#41—28no—
164C2×#41—42no—
165C22×Q8C2×D411yescharacteristic
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—28no—
169C22×D4—22no—
170C22×D4—28no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4C2×D411yescharacteristic
174C22×D4C2×D412yes—
175C22×D4C2414yes—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—48no—
180C23×C4—88no—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2411yescharacteristic
185C23×C4C2412yes—
186C2×D8—28no—
187C2×D8—28no—
188C2×D8—44no—
189C22×C8C2414yes—
190C22×C8C2414yes—
191#127—216no—
192#124—216no—
193#121—28no—
194#121—28no—
195#121—28no—
196#121#41116yes—
197#120—216no—
198#120—216no—
199C4×Q8—44no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—48no—
208C4×D4—48no—
209C4×D4—88no—
210C4×D4—88no—
211#117—28no—
212#117—28no—
213#117—42no—
214#117—42no—
215#117—42no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2414yes—
225C2×C4⋊C4C2414yes—
226C2×C4⋊C4C2414yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4C2×D412yes—
241C2×C22⋊C4C2×D412yes—
242C2×C22⋊C4C2×D411yescharacteristic
243C2×C22⋊C4C2×D411yescharacteristic
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2412yes—
255C2×C42C2411yescharacteristic
256C2×C42C2411yescharacteristic
257#107—28no—
258#106—28no—
259#106—44no—
260C4⋊C8—44no—
261#102—28no—
262#102—28no—
263#102—28no—
264#102—28no—
265#102—44no—
266#102—44no—
267C22⋊C8—28no—
268C22⋊C8—28no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271#97—28no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278C24—28no—
279C24—24no—
280C24—44no—
281C24—44no—
282C24—84no—
283C24—84no—
284C24C22×D411yescharacteristic
285C24C22×D412yes—
286#41—416no—
287C2×Q8—24no—
288C2×Q8—41no—
289C2×D4—28no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—216no—
293C2×D4—216no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—416no—
301C2×D4—416no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—84no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4C2×#4118yes—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—44no—
329C22×C4—48no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—832no—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C2×#4118yes—
341C22×C4C2511yescharacteristic
342D8—416no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C4⋊C4C2×#4118yes—
361C22⋊C4—28no—
362C22⋊C4—216no—
363C22⋊C4—216no—
364C22⋊C4—41no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—42no—
368C22⋊C4—48no—
369C22⋊C4—44no—
370C22⋊C4—48no—
371C22⋊C4—84no—
372C22⋊C4—84no—
373C22⋊C4—84no—
374C22⋊C4—84no—
375C22⋊C4—84no—
376C22⋊C4—84no—
377C42—28no—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—48no—
385C42—44no—
386C42—84no—
387C42H(5,𝔽2)14yes—
388C23—24no—
389C23—24no—
390C23—22no—
391C23—22no—
392C23—24no—
393C23—28no—
394C23—22no—
395C23—21no—
396C23—416no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—816no—
404C23—816no—
405C23—88no—
406C23—88no—
407C23#52514yes—
408C23C23×D411yescharacteristic
409Q8—44no—
410D4—28no—
411D4—28no—
412D4—48no—
413D4—416no—
414D4—44no—
415D4—44no—
416D4—44no—
417D4—816no—
418D4—816no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—48no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—816no—
446C2×C4—816no—
447C2×C4—88no—
448C2×C4#52514yes—
449C2×C4#53414yes—
450C2×C4C22×#4114yes—
451C2×C4C2×H(5,𝔽2)12yes—
452C8—44no—
453C8—44no—
454C22—24no—
455C22—28no—
456C22—42no—
457C22—42no—
458C22—44no—
459C22—42no—
460C22—41no—
461C22—48no—
462C22—44no—
463C22—48no—
464C22—48no—
465C22—832no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22C2×#52512yes—
471C22C2×#53411yescharacteristic
472C22#358514yes—
473C4—24no—
474C4—24no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—41no—
479C4—48no—
480C4—84no—
481C4—84no—
482C4#348114yes—
483C2—44no—
484C2—81no—
485C2—88no—
486C2#3495611yescharacteristic
487C2#6057414yes—
488C2#6075312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23925C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#535C2semidirect
2. #29418C2semidirect
3. #29249C2semidirect
4. #28126C2semidirect
5. #27609C2semidirect
6. #24015C2semidirect
8. #23708C2semidirect
9. #21582C2semidirect
10. #21452C2semidirect
11. #20719C2semidirect
14. C22×C4×D4C22semidirect
18. C2×#432C22semidirect
22. #2324C22semidirect
25. #2317C22semidirect
27. #2309C22semidirect
31. C4×#121C22semidirect
34. C4×#117C22semidirect
35. #2088C22semidirect
36. #2077C22semidirect
37. #2067C22semidirect
38. #2065C22semidirect
40. #2034C22semidirect
41. #2033C22semidirect
43. #1926C22semidirect
44. #1925C22semidirect
45. #1919C22semidirect
46. #1918C22semidirect
51. #1864C22semidirect
52. #1864C22semidirect
53. #1854C22semidirect
54. #1853C22semidirect
55. #1825C22semidirect
56. #1818C22semidirect
57. #1798C22semidirect
58. #1786C22semidirect
61. #1776C22semidirect
62. #1762C22semidirect
63. #1762C22semidirect
64. #1749C22semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
#294182
#292492
#281262
#276092
#240152
#239252
#237082
#215822
#214522
#207192