Finite Group Databasebeta

← #200876

Subgroups — #200876

Order 512.

1497 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 1497 conjugacy classes of subgroups — too many to draw. Only its 246 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60798C211yesmaximal, characteristic
2C2×#2380C211yesmaximal, characteristic, complemented
3#17741C211yesmaximal, characteristic, complemented
4#17593C212yesmaximal, complemented
5#9428C214yesmaximal, complemented
6#9425C214yesmaximal, complemented
7#7975C212yesmaximal, complemented
8#3444C2211yescharacteristic
9C22×#127C2211yescharacteristic
10#3217—24no—
11#3208C414yescomplemented
12#3205C414yescomplemented
13#3151C2212yes—
14#3138C2212yes—
15#3038—22no—
16#3003—22no—
17#2920C2212yes—
18#2920C2211yescharacteristic
19#2380—24no—
20#2380—22no—
21#2380C2214yes—
22C2×#386—24no—
23C2×#386—22no—
24C2×#383C2212yes—
25#1555C2214yes—
26#1545—24no—
27#1521—24no—
28#1515C2214yescomplemented
29#1514C2214yescomplemented
30#1504—24no—
31#1288C2218yescomplemented
32C23×D4—22no—
33C23×D4D412yes—
34C24×C4—22no—
35C2×AGL(1,ℤ/8)—24no—
36#533C2×C412yescomplemented
37#533C2311yescharacteristic
38#532C2×C412yescomplemented
39#532C2311yescharacteristic
40C2×#127—22no—
41C2×#127—22no—
42C2×#127—22no—
43C2×#127C2×C414yes—
44C2×#127D412yes—
45C2×#127D414yes—
46C2×#127C2312yes—
47#529—22no—
48#529—22no—
49#529—22no—
50#525—21no—
51#525—22no—
52C4×#41—21no—
53C4×#41—22no—
54C22×C22⋊C4—24no—
55C22×C22⋊C4—22no—
56C22×C42C2311yescharacteristic
57#496—24no—
58#493C2×C414yescomplemented
59#492C2×C414yescomplemented
60#490—24no—
61#486C2×C418yescomplemented
62#460—28no—
63#459—28no—
64#450—28no—
65#447—28no—
66#432—22no—
67#432—22no—
68#432C2314yes—
69#431—22no—
70#431—22no—
71#431C2×C414yescomplemented
72#423—21no—
73#423—22no—
74#418—24no—
75#407—21no—
76#407—22no—
77C4×M16—22no—
78C4×M16C2312yes—
79#386—28no—
80#386—216no—
81#386—28no—
82#386—216no—
83#383—22no—
84#383—22no—
85#383D414yes—
86#383C2314yes—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D414yes—
90#331—28no—
91#328—28no—
92#327—28no—
93C25—22no—
94C2×#41—22no—
95C2×#41—21no—
96C22×D4—24no—
97C22×D4—24no—
98C22×D4—22no—
99C22×D4—24no—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—24no—
104C22×D4—82no—
105C22×D4—82no—
106C22×D4—84no—
107C22×D4—84no—
108C22×D4C22⋊C418yes—
109C22×D4C2×D414yes—
110C23×C4—24no—
111C23×C4—28no—
112C23×C4—88no—
113C23×C4—84no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116AGL(1,ℤ/8)—84no—
117AGL(1,ℤ/8)—84no—
118C2×SD16—28no—
119C2×D8—28no—
120C2×M16—22no—
121C2×M16—42no—
122C4⋊Q8C22×C412yescomplemented
123C4⋊Q8C22×C411yescharacteristic, complemented
124#127—24no—
125#127—24no—
126#127—24no—
127#127—24no—
128#127—22no—
129#127—22no—
130#127—24no—
131#127C22⋊C418yes—
132#127C22×C414yes—
133#127C22×C411yescharacteristic, complemented
134#127C2×D414yes—
135#126—42no—
136#125—22no—
137#125—21no—
138#124—22no—
139#124—21no—
140#123—41no—
141#123—42no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145#121—22no—
146#121—22no—
147#121—22no—
148#120—41no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—42no—
158#117—22no—
159#117—21no—
160#117—42no—
161#117—41no—
162C2×C22⋊C4—216no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—88no—
165C2×C22⋊C4—84no—
166C2×C42—41no—
167C2×C42C22×C412yes—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42#4112yes—
172C2×C42#4112yes—
173C2×C42C2411yescharacteristic
174C4⋊C8—24no—
175C4⋊C8—24no—
176C4⋊C8—42no—
177#102—216no—
178C22⋊C8—24no—
179C22⋊C8—24no—
180C22⋊C8—42no—
181#97—22no—
182#97—24no—
183C4×C8—42no—
184C4×C8C22×C414yescomplemented
185#95—24no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95C2×D418yes—
191C24—24no—
192C24—28no—
193C24—48no—
194C24—44no—
195C24—48no—
196C24—44no—
197C24#12011yescharacteristic
198#41—44no—
199#41—84no—
200C2×Q8—21no—
201C2×Q8—22no—
202C2×D4—21no—
203C2×D4—28no—
204C2×D4—22no—
205C2×D4—28no—
206C2×D4—24no—
207C2×D4—28no—
208C2×D4—28no—
209C2×D4—28no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—82no—
215C2×D4—82no—
216C2×D4—88no—
217C2×D4—88no—
218C2×D4—88no—
219C2×D4—88no—
220C2×D4—84no—
221C2×D4—84no—
222C2×D4—88no—
223C2×D4—88no—
224C2×D4—84no—
225C2×D4—84no—
226C2×D4C2×C22⋊C418yes—
227C22×C4—28no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—416no—
231C22×C4—48no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—816no—
235C22×C4—88no—
236C22×C4—832no—
237C22×C4C2×C22⋊C412yes—
238C22×C4C4×D412yes—
239C22×C4C4×D412yes—
240C22×C4#12011yescharacteristic
241C22×C4#12012yes—
242C22×C4#12112yes—
243C22×C4#12112yes—
244C22×C4#12312yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D412yes—
247SD16—88no—
248D8—88no—
249M16—44no—
250M16—84no—
251C2×C8—24no—
252C2×C8—44no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—816no—
265C22⋊C4—88no—
266C42—41no—
267C42—42no—
268C42C2×C22⋊C412yes—
269C42C4×D412yes—
270C42C4×D412yes—
271C42#12112yes—
272C42#12112yes—
273C42#12311yescharacteristic
274C42#12311yescharacteristic
275C42C23×C411yescharacteristic
276C42C22×D411yescharacteristic
277C42C2×#4112yes—
278C23—24no—
279C23—22no—
280C23—28no—
281C23—48no—
282C23—48no—
283C23—416no—
284C23—416no—
285C23—416no—
286C23—44no—
287C23—44no—
288C23—41no—
289C23—88no—
290C23—88no—
291C23—816no—
292C23#38612yes—
293C23C2×#12011yescharacteristic
294Q8—42no—
295Q8—82no—
296D4—48no—
297D4—42no—
298D4—84no—
299D4—88no—
300D4—88no—
301D4—88no—
302D4—88no—
303D4—88no—
304D4—82no—
305D4—84no—
306D4—82no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—416no—
318C2×C4—48no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—816no—
324C2×C4—832no—
325C2×C4#38612yes—
326C2×C4#38612yes—
327C2×C4#38612yes—
328C2×C4C22×C22⋊C411yescharacteristic
329C2×C4C2×C4×D412yes—
330C2×C4C2×#12112yes—
331C2×C4C2×#12311yescharacteristic
332C8—44no—
333C8—84no—
334C22—28no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—416no—
339C22—48no—
340C22—84no—
341C22—84no—
342C22—88no—
343C22—832no—
344C22—82no—
345C22—81no—
346C22#190414yes—
347C22C2×#38611yescharacteristic
348C22#301912yes—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—22no—
356C4—88no—
357C4—82no—
358C4—81no—
359C2—22no—
360C2—24no—
361C2—88no—
362C2—161no—
363C2#2095311yescharacteristic
364C2C2×#190412yes—
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#60798C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2380C2semidirect
3. #17741C2semidirect
4. #17593C2semidirect
5. #9428C2semidirect
6. #9425C2semidirect
7. #7975C2semidirect
11. #3208C4semidirect
12. #3205C4semidirect
28. #1515C22semidirect
29. #1514C22semidirect
31. #1288C22semidirect
36. #533C2×C4semidirect
38. #532C2×C4semidirect
58. #493C2×C4semidirect
59. #492C2×C4semidirect
61. #486C2×C4semidirect
71. #431C2×C4semidirect
122. C4⋊Q8C22×C4semidirect
123. C4⋊Q8C22×C4semidirect
133. #127C22×C4semidirect
184. C4×C8C22×C4semidirect

Maximal subgroups

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

subgroupindex
#607982
C2×#23802
#177412
#175932
#94282
#94252
#79752