Finite Group Databasebeta

← #481688

Subgroups — #481688

Order 512.

1129 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36033C211yesmaximal, characteristic
2#20354C211yesmaximal, characteristic
3#17369C211yesmaximal, characteristic
4#11759C214yesmaximal, complemented
5#11755C214yesmaximal, complemented
6#11715C212yesmaximal, complemented
7#11694C212yesmaximal, complemented
8C2×#556C411yescharacteristic, complemented
9C2×#518—21no—
10C2×#518C2211yescharacteristic
11#2975—21no—
12#2638C412yescomplemented
13#2621C2211yescharacteristic
14#2580C411yescharacteristic, complemented
15#2561C2212yes—
16#2547C411yescharacteristic, complemented
17#2482C2212yes—
18#2480—22no—
19#2468—22no—
20#2447—21no—
21#2435—21no—
22#2396C2211yescharacteristic
23#2395C412yescomplemented
24#2367—21no—
25#2362—21no—
26#2357C411yescharacteristic, complemented
27#1740—21no—
28#1740C2212yes—
29C2×#342—22no—
30C2×#342C2214yes—
31#1465—22no—
32#1462C2214yes—
33#1460—21no—
34#1458C2212yes—
35#1352C2214yes—
36#1351C2214yes—
37#1344C2218yes—
38C23×D4D411yescharacteristic
39#556—44no—
40C2×C4⋊Q8C2×C411yescharacteristic, complemented
41C2×#126—22no—
42C2×#124—21no—
43C2×#124—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yescomplemented
46#518—28no—
47#518—44no—
48#518—44no—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C411yescharacteristic, complemented
51C2×C4×D4—44no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×C4×D4D411yescharacteristic
57C2×C4×D4D411yescharacteristic
58C2×#117—22no—
59C2×#117C2×C412yescomplemented
60C2×#117D411yescharacteristic
61C2×#117C2311yescharacteristic
62C22×C22⋊C4—24no—
63C22×C22⋊C4D412yes—
64C22×C22⋊C4C2312yes—
65#436—44no—
66#431—22no—
67#407—24no—
68#400C2×C411yescharacteristic, complemented
69#399C2×C412yescomplemented
70#399C2311yescharacteristic
71#398—21no—
72#398—21no—
73#398C2×C412yescomplemented
74#397—22no—
75#397—22no—
76#397—21no—
77#397C2×C412yescomplemented
78#397C2×C412yescomplemented
79#397C2312yes—
80#396—21no—
81#395C2×C411yescharacteristic, complemented
82#394—22no—
83#394—22no—
84#393C2×C412yescomplemented
85#393C2311yescharacteristic
86#391—21no—
87#391—21no—
88#390—21no—
89#390—21no—
90#389—21no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#387—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#382C2×C411yescharacteristic, complemented
101#381—22no—
102#381C2×C412yescomplemented
103#380—22no—
104#379—22no—
105#379—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C411yescharacteristic, complemented
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4C2×C412yescomplemented
113#342—22no—
114#342—24no—
115#342C2314yes—
116#324—22no—
117#323—24no—
118#323C2314yes—
119C25C2×D412yes—
120C22×Q8—21no—
121C22×Q8C22×C411yescharacteristic
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C22⋊C412yes—
127C22×D4C22⋊C411yescharacteristic
128C22×D4C22⋊C411yescharacteristic
129C23×C4—44no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22×C412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C2×M16—44no—
136C4⋊Q8—42no—
137#126—48no—
138#124—44no—
139#124—44no—
140C22⋊Q8—42no—
141C22⋊Q8—42no—
142C22⋊Q8—44no—
143C4×Q8—42no—
144C4×Q8—44no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—44no—
150C4×D4—44no—
151C4×D4—432no—
152#117—28no—
153#117—42no—
154#117—42no—
155#117—44no—
156#117—44no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C22⋊C4—216no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22×C412yes—
193C2×C22⋊C4C2×D411yescharacteristic
194C2×C22⋊C4C2×D412yes—
195C2×C22⋊C4C2×D412yes—
196C2×C22⋊C4C2411yescharacteristic
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—42no—
202C2×C42C22⋊C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C4⋊C8—42no—
206C22⋊C8—24no—
207C22⋊C8—44no—
208C4×C8—42no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95—21no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—21no—
218#95—21no—
219#95—22no—
220#95C22×C411yescharacteristic
221#95C22×C412yes—
222#95C22×C411yescharacteristic
223C24—24no—
224C24—22no—
225C24—22no—
226C24—24no—
227C24—82no—
228C24—82no—
229C24C2×C22⋊C412yes—
230C24C2×C22⋊C412yes—
231C24C22×D411yescharacteristic
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—81no—
237C2×D4—216no—
238C2×D4—216no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4#9914yes—
247C2×D4#10414yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—416no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—164no—
267C22×C4—164no—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4AGL(1,ℤ/8)12yes—
275C22×C4#13712yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278M16—164no—
279C2×C8—44no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—48no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—82no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—28no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—416no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—81no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—81no—
318C42—22no—
319C42—22no—
320C42—42no—
321C42—44no—
322C42—48no—
323C42—42no—
324C42—42no—
325C42—82no—
326C23—28no—
327C23—28no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—22no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—84no—
341C23—84no—
342C23—162no—
343C23—162no—
344C23—164no—
345C23C2×#9914yes—
346C23C2×#10414yes—
347C23C22×C22⋊C411yescharacteristic
348C23C2×AGL(1,ℤ/8)11yescharacteristic
349C23C2×#13711yescharacteristic
350Q8—82no—
351Q8—84no—
352D4—216no—
353D4—28no—
354D4—28no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—48no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—162no—
395C2×C4—162no—
396C2×C4—1616no—
397C2×C4C2×#9912yes—
398C2×C4#41712yes—
399C2×C4#41712yes—
400C2×C4C2×#10412yes—
401C8—84no—
402C22—21no—
403C22—21no—
404C22—24no—
405C22—24no—
406C22—21no—
407C22—21no—
408C22—24no—
409C22—24no—
410C22—41no—
411C22—41no—
412C22—44no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—42no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—81no—
421C22—88no—
422C22—88no—
423C22C22×#9911yescharacteristic
424C22C2×#41711yescharacteristic
425C22C22×#10411yescharacteristic
426C4—22no—
427C4—41no—
428C4—42no—
429C4—41no—
430C4—41no—
431C4—42no—
432C4—41no—
433C4—81no—
434C4—81no—
435C4—82no—
436C4—88no—
437C4—81no—
438C4—81no—
439C4—82no—
440C4#150414yes—
441C2—21no—
442C2—21no—
443C2—41no—
444C2—44no—
445C2—44no—
446C2—41no—
447C2C2×#150411yescharacteristic
448C2#3335011yescharacteristic
449C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11759C2semidirect
5. #11755C2semidirect
6. #11715C2semidirect
7. #11694C2semidirect
8. C2×#556C4semidirect
12. #2638C4semidirect
14. #2580C4semidirect
16. #2547C4semidirect
23. #2395C4semidirect
26. #2357C4semidirect
40. C2×C4⋊Q8C2×C4semidirect
45. C2×C22⋊Q8C2×C4semidirect
50. C2×C4×Q8C2×C4semidirect
59. C2×#117C2×C4semidirect
68. #400C2×C4semidirect
69. #399C2×C4semidirect
73. #398C2×C4semidirect
77. #397C2×C4semidirect
78. #397C2×C4semidirect
81. #395C2×C4semidirect
84. #393C2×C4semidirect
100. #382C2×C4semidirect
102. #381C2×C4semidirect
109. C4×C4⋊C4C2×C4semidirect
112. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#360332
#203542
#173692
#117592
#117552
#117152
#116942