Finite Group Databasebeta

← #200877

Subgroups — #200877

Order 512.

1129 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 1129 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#60800C211yesmaximal, characteristic
2C2×#2381C211yesmaximal, characteristic, complemented
3#17746C211yesmaximal, characteristic, complemented
4#17593C212yesmaximal
5#9429C214yesmaximal, complemented
6#9426C214yesmaximal, complemented
7#7975C212yesmaximal, complemented
8#3444C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3218—24no—
11#3209C414yescomplemented
12#3206C414yescomplemented
13#3152C2212yes—
14#3139C2212yes—
15#3039—22no—
16#3004—22no—
17#2920C2212yes—
18#2920C2211yescharacteristic
19#2381—24no—
20#2381—22no—
21#2381C2214yes—
22C2×#391—22no—
23C2×#387—24no—
24C2×#383C2212yes—
25#1558C2214yes—
26#1548—24no—
27#1521—24no—
28#1515C2214yes—
29#1514C2214yes—
30#1504—24no—
31#1288C2218yes—
32C23×Q8D412yes—
33C2×#137—24no—
34#533C2×C412yes—
35#533C2311yescharacteristic
36#532C2×C412yes—
37#532C2311yescharacteristic
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8C2×C414yes—
42C2×C4⋊Q8D412yes—
43C2×C4⋊Q8D414yes—
44C2×C4⋊Q8C2312yes—
45#529—22no—
46#529—22no—
47#529—22no—
48#525—21no—
49#525—22no—
50C4×#41—21no—
51C4×#41—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C42—22no—
55C22×C42C2311yescharacteristic
56#497—24no—
57#494C2×C414yescomplemented
58#492C2×C414yescomplemented
59#491—24no—
60#487C2×C418yescomplemented
61#462—28no—
62#461—28no—
63#451—28no—
64#448—28no—
65#432—22no—
66#432—22no—
67#432C2314yes—
68#431—22no—
69#431—22no—
70#431C2×C414yes—
71#423—21no—
72#423—22no—
73#419—24no—
74#407—21no—
75#407—22no—
76C4×M16—22no—
77C4×M16C2312yes—
78#391—28no—
79#391—28no—
80#387—216no—
81#387—216no—
82#383—22no—
83#383—22no—
84#383D414yes—
85#383C2314yes—
86C2×#95—24no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D414yes—
90#333—28no—
91#328—28no—
92#327—28no—
93C2×#41—22no—
94C2×#41—21no—
95C22×Q8—22no—
96C22×Q8—24no—
97C22×Q8—82no—
98C22×Q8—82no—
99C22×Q8C22⋊C418yes—
100C22×Q8C2×D414yes—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105#137—84no—
106#137—84no—
107C2×Q16—28no—
108C2×SD16—28no—
109C2×M16—22no—
110C2×M16—42no—
111C4⋊Q8—24no—
112C4⋊Q8—24no—
113C4⋊Q8—24no—
114C4⋊Q8—24no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—24no—
118C4⋊Q8C22⋊C418yes—
119C4⋊Q8C22×C412yes—
120C4⋊Q8C22×C411yescharacteristic
121C4⋊Q8C22×C414yes—
122C4⋊Q8C2×D414yes—
123#127C22×C411yescharacteristic
124#126—42no—
125#125—22no—
126#125—21no—
127#124—22no—
128#124—21no—
129#123—41no—
130#123—42no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134#121—22no—
135#121—22no—
136#121—22no—
137#120—41no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—42no—
147#117—22no—
148#117—21no—
149#117—42no—
150#117—41no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—84no—
159C2×C42—28no—
160C2×C42—41no—
161C2×C42—84no—
162C2×C42C22×C412yes—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42#4112yes—
167C2×C42#4112yes—
168C2×C42C2411yescharacteristic
169C4⋊C8—24no—
170C4⋊C8—24no—
171C4⋊C8—42no—
172#103—216no—
173C22⋊C8—24no—
174C22⋊C8—24no—
175C22⋊C8—42no—
176#97—22no—
177#97—24no—
178C4×C8—42no—
179C4×C8C22×C414yes—
180#95—216no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95C2×D418yes—
187C24#12011yescharacteristic
188#41—44no—
189#41—84no—
190C2×Q8—24no—
191C2×Q8—28no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8—88no—
199C2×Q8—88no—
200C2×Q8C2×C22⋊C418yes—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—41no—
204C2×D4—42no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—48no—
213C22×C4—416no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4C2×C22⋊C412yes—
217C22×C4C4×D412yes—
218C22×C4C4×D412yes—
219C22×C4#12011yescharacteristic
220C22×C4#12012yes—
221C22×C4#12112yes—
222C22×C4#12112yes—
223C22×C4#12312yes—
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D412yes—
226Q16—88no—
227SD16—88no—
228M16—44no—
229M16—84no—
230C2×C8—24no—
231C2×C8—44no—
232C4⋊C4—28no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—88no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42—41no—
249C42—42no—
250C42—88no—
251C42C2×C22⋊C412yes—
252C42C4×D412yes—
253C42C4×D412yes—
254C42#12112yes—
255C42#12112yes—
256C42#12311yescharacteristic
257C42#12311yescharacteristic
258C42C23×C411yescharacteristic
259C42C22×D411yescharacteristic
260C42C2×#4112yes—
261C23—24no—
262C23—22no—
263C23—41no—
264C23#38612yes—
265C23C2×#12011yescharacteristic
266Q8—48no—
267Q8—42no—
268Q8—84no—
269Q8—88no—
270Q8—82no—
271D4—42no—
272D4—82no—
273D4—84no—
274D4—82no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—28no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—416no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—416no—
290C2×C4—48no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—82no—
294C2×C4—84no—
295C2×C4—816no—
296C2×C4#38612yes—
297C2×C4#38612yes—
298C2×C4#38612yes—
299C2×C4C22×C22⋊C411yescharacteristic
300C2×C4C2×C4×D412yes—
301C2×C4C2×#12112yes—
302C2×C4C2×#12311yescharacteristic
303C8—44no—
304C8—84no—
305C22—28no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—82no—
310C22—81no—
311C22#190514yes—
312C22C2×#38611yescharacteristic
313C22#301912yes—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—21no—
319C4—21no—
320C4—22no—
321C4—48no—
322C4—88no—
323C4—82no—
324C4—81no—
325C2—22no—
326C2—24no—
327C2—161no—
328C2#2095311yescharacteristic
329C2C2×#190512yes—
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
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×#2381C2semidirect
3. #17746C2semidirect
5. #9429C2semidirect
6. #9426C2semidirect
7. #7975C2semidirect
11. #3209C4semidirect
12. #3206C4semidirect
57. #494C2×C4semidirect
58. #492C2×C4semidirect
60. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
#608002
C2×#23812
#177462
#175932
#94292
#94262
#79752