Finite Group Databasebeta

← #512662

Subgroups — #512662

Order 512.

818 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60797C211yesmaximal, characteristic
2#33358C212yesmaximal, complemented
3#13610C214yesmaximal, complemented
4#13609C214yesmaximal, complemented
5#13594C212yesmaximal, complemented
6#13583C212yesmaximal, complemented
7#3455C411yescharacteristic, complemented
8#3449C2211yescharacteristic
9#3225—22no—
10#3224—22no—
11#3223C412yescomplemented
12#3222C2212yes—
13#3221C412yescomplemented
14#3220C2212yes—
15#3150C411yescharacteristic, complemented
16#3147C2211yescharacteristic
17#3135C2211yescharacteristic
18#3131C411yescharacteristic, complemented
19#3035—22no—
20#3002—22no—
21C2×#433C411yescharacteristic, complemented
22C2×#410—22no—
23#2129—22no—
24#2128C2214yes—
25#2127—22no—
26#2127C2214yes—
27#2126—22no—
28#2126C2214yes—
29#1412C2218yescomplemented
30#1409C2214yescomplemented
31#1408C2214yescomplemented
32C22×#41D412yes—
33C2×#135—22no—
34C2×#135—22no—
35#542—22no—
36#541C2×C412yescomplemented
37#540C2×C412yescomplemented
38#539—41no—
39#538—41no—
40#537—41no—
41#536C2×C412yes—
42#536D412yes—
43#535D412yes—
44#535C2312yes—
45#534—41no—
46#533C2×C411yescharacteristic, complemented
47#532C2×C411yescharacteristic
48#532C2×C411yescharacteristic, complemented
49#532D412yes—
50#532C2311yescharacteristic
51#529—42no—
52#525—21no—
53#525—42no—
54C4×#41—21no—
55C2×#117—42no—
56C2×#117C2×C411yescharacteristic
57#497—21no—
58#497C2×C412yescomplemented
59#496—21no—
60#496C2×C412yescomplemented
61#495—22no—
62#494C2×C411yescharacteristic, complemented
63#493C2311yescharacteristic
64#492C2×C411yescharacteristic, complemented
65#492C2311yescharacteristic
66#491C2×C412yescomplemented
67#490C2312yes—
68#489—22no—
69#488C2×C412yescomplemented
70#488C2×C412yescomplemented
71#487—21no—
72#486—21no—
73#464—22no—
74#464—22no—
75#463—22no—
76#463—22no—
77#462—22no—
78#461—22no—
79#460—22no—
80#459—22no—
81#452—22no—
82#452—22no—
83#451—22no—
84#450—22no—
85#449—22no—
86#449—22no—
87#448—22no—
88#447—22no—
89#433—22no—
90#433C2×C412yescomplemented
91#433C2×C412yescomplemented
92C2×C4⋊C8—21no—
93#416—22no—
94#416—22no—
95#410—24no—
96#410—24no—
97#410—42no—
98#410—42no—
99C2×#99—24no—
100C2×#99—42no—
101C2×C22⋊C8—21no—
102C2×#97C2×C411yescharacteristic, complemented
103C2×C4×C8C2×C411yescharacteristic, complemented
104#354—24no—
105#354—24no—
106#353—24no—
107#353C2318yes—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—21no—
111C2×#41—82no—
112C2×#41—82no—
113C2×#41C22⋊C412yes—
114C2×#41C22⋊C412yes—
115C2×#41C2×D412yes—
116C2×#41C2×D412yes—
117C22×Q8C22⋊C412yes—
118C22×D4—22no—
119C22×D4C2×D412yes—
120C23×C4—22no—
121C23×C4C22⋊C411yescharacteristic
122#135—82no—
123#135—82no—
124#135—82no—
125#135—82no—
126C2×Q16—22no—
127C2×Q16—22no—
128C2×SD16—22no—
129C2×SD16—22no—
130C2×SD16—22no—
131C2×SD16—22no—
132C2×D8—22no—
133C2×D8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C4⋊Q8—21no—
137C4⋊Q8C22⋊C411yescharacteristic
138C4⋊Q8C22⋊C412yes—
139C4⋊Q8C22×C411yescharacteristic
140C4⋊Q8C22×C411yescharacteristic
141#127C22⋊C411yescharacteristic
142#127C22×C411yescharacteristic
143#127C2×D412yes—
144#127C2411yescharacteristic
145#126—42no—
146#126—42no—
147#125—42no—
148#125C22×C412yes—
149#124—21no—
150#124—42no—
151#124C22⋊C412yes—
152#124C22×C412yes—
153#124C2×D412yes—
154#124C2×D412yes—
155#123—42no—
156#123—42no—
157#123—42no—
158#123—41no—
159#123—42no—
160C22⋊Q8—22no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C22⋊Q8—41no—
165#121—22no—
166#121—42no—
167#121—42no—
168#121—42no—
169#121—41no—
170#121—42no—
171#121—41no—
172#121—41no—
173#120—42no—
174#120—42no—
175#120—41no—
176C4×Q8—22no—
177C4×Q8—41no—
178C4×D4—22no—
179C4×D4—41no—
180C4×D4—42no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—42no—
184C4×D4—42no—
185#117—22no—
186#117—21no—
187#117—41no—
188#117—44no—
189#117—44no—
190#117—82no—
191#117—82no—
192#117C22⋊C412yes—
193#117C22×C412yes—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—82no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—44no—
198C2×C42—41no—
199C2×C42—82no—
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—22no—
204#103—22no—
205#103—22no—
206#103—22no—
207#103—22no—
208#102—22no—
209#102—22no—
210#102—22no—
211#102—22no—
212#99—28no—
213#99—44no—
214#99—44no—
215#99—44no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218#97—21no—
219#97C22×C412yes—
220C4×C8—21no—
221C4×C8C22×C411yescharacteristic
222C4×C8C22×C411yescharacteristic
223C24—22no—
224C24#9911yescharacteristic
225#41—42no—
226#41—42no—
227#41—42no—
228#41—42no—
229#41—42no—
230#41—42no—
231#41—82no—
232#41—82no—
233#41—82no—
234#41—82no—
235#41—82no—
236#41—82no—
237C2×Q8—22no—
238C2×Q8—21no—
239C2×Q8—41no—
240C2×Q8—82no—
241C2×Q8C2×C22⋊C412yes—
242C2×Q8C2×C22⋊C412yes—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4C2×C22⋊C412yes—
259C2×D4C22×D412yes—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—84no—
276C22×C4#9912yes—
277C22×C4#9911yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280Q16—82no—
281Q16—82no—
282SD16—82no—
283SD16—82no—
284SD16—82no—
285SD16—82no—
286D8—82no—
287D8—82no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—48no—
320C22⋊C4—84no—
321C42—21no—
322C42—21no—
323C42—42no—
324C42—41no—
325C42—84no—
326C42—84no—
327C42C2×C22⋊C411yescharacteristic
328C42C2×C22⋊C412yes—
329C42C23×C411yescharacteristic
330C23—22no—
331C23—22no—
332C23—41no—
333C23—42no—
334C23—42no—
335C23—41no—
336C23—81no—
337C23—162no—
338C23C2×#9912yes—
339C23C2×#9911yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—42no—
343Q8—82no—
344Q8—161no—
345D4—42no—
346D4—42no—
347D4—42no—
348D4—82no—
349D4—82no—
350D4—82no—
351D4—82no—
352D4—82no—
353D4—82no—
354D4—162no—
355D4—161no—
356D4—161no—
357D4—161no—
358D4—162no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—88no—
387C2×C4—162no—
388C2×C4—168no—
389C2×C4C2×#9912yes—
390C2×C4C2×#9911yescharacteristic
391C2×C4C22×C22⋊C411yescharacteristic
392C8—41no—
393C8—42no—
394C8—41no—
395C22—22no—
396C22—21no—
397C22—41no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22—84no—
405C22—81no—
406C22—161no—
407C22—162no—
408C22C22×#9911yescharacteristic
409C4—21no—
410C4—22no—
411C4—42no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—82no—
417C4—81no—
418C4—168no—
419C4#212711yescharacteristic
420C4#212711yescharacteristic
421C2—21no—
422C2—42no—
423C2—41no—
424C2—82no—
425C2—82no—
426C2—161no—
427C2C2×#212711yescharacteristic
subgroupG/H
centerC4#2127
commutatorC42C23×C4
Frattini, cosocle#127C24
socleC2C2×#2127
prime radical#60797C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C23×C4 ◁ #3449 ◁ G
LECSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33358C2semidirect
3. #13610C2semidirect
4. #13609C2semidirect
5. #13594C2semidirect
6. #13583C2semidirect
7. #3455C4semidirect
11. #3223C4semidirect
13. #3221C4semidirect
15. #3150C4semidirect
18. #3131C4semidirect
21. C2×#433C4semidirect
29. #1412C22semidirect
30. #1409C22semidirect
31. #1408C22semidirect
36. #541C2×C4semidirect
37. #540C2×C4semidirect
46. #533C2×C4semidirect
48. #532C2×C4semidirect
58. #497C2×C4semidirect
60. #496C2×C4semidirect
62. #494C2×C4semidirect
64. #492C2×C4semidirect
66. #491C2×C4semidirect
69. #488C2×C4semidirect
70. #488C2×C4semidirect
90. #433C2×C4semidirect
91. #433C2×C4semidirect
102. C2×#97C2×C4semidirect
103. C2×C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#607972
#333582
#136102
#136092
#135942
#135832