Finite Group Databasebeta

← #193071

Subgroups — #193071

Order 512.

1144 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 274 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#34368C211yesmaximal, characteristic
2#21200C211yesmaximal, characteristic
3#20764C212yesmaximal, complemented
4C2×#1525C211yesmaximal, characteristic
5#8914C218yesmaximal, complemented
6#8028C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#417—22no—
9C2×C4×M16C2211yescharacteristic
10C2×C4×M16C2212yes—
11#2379C2214yes—
12#2369—21no—
13#2366—21no—
14#2354—22no—
15D4×C4⋊C4—22no—
16#2324—24no—
17#2311—22no—
18#2285C2212yes—
19C2×#384C2211yescharacteristic
20#1959C2214yes—
21#1957—21no—
22#1956—21no—
23#1836C2218yes—
24#1834—22no—
25#1834—22no—
26#1736—24no—
27#1525C418yescomplemented
28C2×#329C2214yes—
29#1295C2218yes—
30C22×#41D412yes—
31C22×M16—21no—
32C22×M16—22no—
33#532—24no—
34#532C2×C414yes—
35#532Q814yes—
36C2×C4⋊Q8D411yescharacteristic
37C2×#127D411yescharacteristic
38C2×#124C2312yes—
39C2×#121—22no—
40C2×C4×D4—24no—
41C2×C4×D4—22no—
42C2×#117—22no—
43C2×#117D414yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C42C2311yescharacteristic
48#417—48no—
49C2×#103—24no—
50C2×#102—24no—
51C4×M16—24no—
52C4×M16—24no—
53C4×M16C2×C418yescomplemented
54C2×#97C2312yes—
55C2×#97C2314yes—
56C2×C4×C8—22no—
57C2×C4×C8—21no—
58#391—24no—
59#390—24no—
60#390—24no—
61#389—24no—
62#388—28no—
63#386—24no—
64#386—24no—
65#385—28no—
66#384—22no—
67#384—21no—
68#384D412yes—
69#384D412yes—
70#384C2312yes—
71#382C2314yes—
72#380—24no—
73#380—22no—
74#380—22no—
75C4×C4⋊C4—24no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78#340—28no—
79#340—28no—
80#329C2×C4116yescomplemented
81C2×#41—24no—
82C2×#41—44no—
83C2×#41—44no—
84C2×#41C4⋊C418yes—
85C22×Q8C2×D412yes—
86C22×D4—22no—
87C22×D4—22no—
88C22×D4C2×D412yes—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4—41no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C2×M16—24no—
95C2×M16—28no—
96C2×M16—44no—
97C2×M16—44no—
98C2×M16—42no—
99C2×M16—42no—
100C22×C8—24no—
101C22×C8—22no—
102C4⋊Q8—22no—
103C4⋊Q8C4⋊C414yes—
104#127—22no—
105#127C4⋊C414yes—
106#124—24no—
107#124C22×C414yes—
108#124C2×Q814yes—
109#121—28no—
110#121—28no—
111C4×D4—28no—
112C4×D4—28no—
113C4×D4—88no—
114#117—48no—
115#117—84no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—48no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—84no—
129C2×C4⋊C4C2×D414yes—
130C2×C4⋊C4C2×D414yes—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—24no—
135C2×C42—24no—
136C2×C42—24no—
137C2×C42C22×C412yes—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D412yes—
140C2×C42C2×Q812yes—
141C2×C42#4114yes—
142C2×C42C2411yescharacteristic
143#103—48no—
144#102—48no—
145#97—24no—
146#97—24no—
147#97C22×C418yes—
148C4×C8—28no—
149C4×C8—42no—
150#95—28no—
151#95—24no—
152#95—24no—
153C24—22no—
154C24#12011yescharacteristic
155#41—416no—
156#41—88no—
157C2×Q8—22no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8C2×C4⋊C414yes—
161C2×D4—28no—
162C2×D4—22no—
163C2×D4—28no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—82no—
169C2×D4—82no—
170C2×D4C2×C4⋊C414yes—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—28no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—84no—
183C22×C4—88no—
184C22×C4—82no—
185C22×C4—84no—
186C22×C4C2×C4⋊C412yes—
187C22×C4C4×D414yes—
188C22×C4#12011yescharacteristic
189C22×C4#12012yes—
190C22×C4C22⋊Q814yes—
191C22×C4#12312yes—
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D412yes—
194M16—48no—
195M16—416no—
196C2×C8—24no—
197C2×C8—28no—
198C2×C8—44no—
199C2×C8—42no—
200C2×C8—44no—
201C2×C8—42no—
202C4⋊C4—28no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—84no—
212C4⋊C4—82no—
213C4⋊C4—84no—
214C22⋊C4—28no—
215C22⋊C4—28no—
216C22⋊C4—84no—
217C22⋊C4—84no—
218C22⋊C4—88no—
219C42—48no—
220C42—84no—
221C42C2×C4⋊C412yes—
222C42C4×D414yes—
223C42C22⋊Q814yes—
224C42#12312yes—
225C42C23×C411yescharacteristic
226C42C22×Q811yescharacteristic
227C42C2×#4112yes—
228C23—21no—
229C23—22no—
230C23—21no—
231C23—22no—
232C23—24no—
233C23—42no—
234C23—42no—
235C23—44no—
236C23—44no—
237C23#38512yes—
238C23C2×#12011yescharacteristic
239Q8—44no—
240Q8—82no—
241D4—44no—
242D4—416no—
243D4—82no—
244D4—88no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—28no—
252C2×C4—28no—
253C2×C4—24no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—416no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—82no—
264C2×C4—84no—
265C2×C4—168no—
266C2×C4—164no—
267C2×C4#38512yes—
268C2×C4#38514yes—
269C2×C4#45312yes—
270C2×C4#45414yes—
271C2×C4#45512yes—
272C2×C4#45514yes—
273C2×C4C22×C4⋊C411yescharacteristic
274C2×C4C2×C4×D412yes—
275C2×C4C2×C22⋊Q812yes—
276C2×C4C2×#12311yescharacteristic
277C8—44no—
278C8—48no—
279C22—22no—
280C22—24no—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—44no—
285C22—48no—
286C22—82no—
287C22—82no—
288C22—84no—
289C22C2×#38511yescharacteristic
290C22C2×#45311yescharacteristic
291C22C2×#45511yescharacteristic
292C22C2×#45512yes—
293C22C2×#45412yes—
294C4—21no—
295C4—21no—
296C4—22no—
297C4—22no—
298C4—44no—
299C4—84no—
300C4—84no—
301C4—162no—
302C4#185614yes—
303C4#186018yes—
304C2—42no—
305C2—84no—
306C2#1907411yescharacteristic
307C2#1907711yescharacteristic
308C2#1908012yes—
309C2C2×#185611yescharacteristic
310C2C2×#186012yes—
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20764C2semidirect
5. #8914C2semidirect
6. #8028C2semidirect
27. #1525C4semidirect
53. C4×M16C2×C4semidirect
80. #329C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343682
#212002
#207642
C2×#15252
#89142
#80282