Finite Group Databasebeta

← #241267

Subgroups — #241267

Order 512.

904 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 218 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#37643C211yesmaximal, characteristic
2#20800C212yesmaximal
3#20764C211yesmaximal, characteristic
4#10585C218yesmaximal, complemented
5#8028C212yesmaximal
6#8028C211yesmaximal, characteristic
7C2×#428—24no—
8C2×C4×M16C2211yescharacteristic
9C2×C4×M16C2212yes—
10#2845C2214yes—
11#2629—24no—
12#2629—22no—
13#2552—22no—
14#2552—22no—
15#2552—22no—
16#2285C2211yescharacteristic
17#2285C2212yes—
18C2×#384C2211yescharacteristic
19#1847C2218yes—
20#1845—22no—
21#1844—22no—
22#1836C2214yes—
23#1834—22no—
24#1736—22no—
25#1295C2214yes—
26#1295C2218yes—
27C22×M16—22no—
28C22×M16—21no—
29C2×#125—24no—
30C2×#125—22no—
31C2×#117D414yes—
32C2×#117D412yes—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C42C2311yescharacteristic
36#428—416no—
37C2×#107—28no—
38C2×#106—28no—
39C4×M16—24no—
40C4×M16—22no—
41C4×M16Q818yes—
42C4×M16Q814yes—
43C2×#97C2312yes—
44C2×#97C2314yes—
45C2×C4×C8—21no—
46C2×C4×C8—22no—
47#398—28no—
48#398—28no—
49#398—28no—
50#396—28no—
51#389—24no—
52#389—24no—
53#389—24no—
54#384—24no—
55#384—22no—
56#384—21no—
57#384—22no—
58#384D412yes—
59#384D412yes—
60#384D412yes—
61#384C2312yes—
62#382C2312yes—
63#382C2314yes—
64#380—22no—
65#380—22no—
66#380—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70#340—28no—
71C23×C4—41no—
72C23×C4C2×D412yes—
73C23×C4C2×D411yescharacteristic
74C2×M16—28no—
75C2×M16—28no—
76C2×M16—28no—
77C2×M16—24no—
78C2×M16—42no—
79C2×M16—42no—
80C22×C8—24no—
81C22×C8—22no—
82#125—88no—
83#125—84no—
84#117—44no—
85#117—48no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—48no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4C2×D414yes—
102C2×C4⋊C4C2×D414yes—
103C2×C4⋊C4C2×D414yes—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—22no—
106C2×C42—24no—
107C2×C42—22no—
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D412yes—
110C2×C42C2×Q812yes—
111C2×C42C2×Q814yes—
112C2×C42#4112yes—
113C2×C42C2411yescharacteristic
114#107—416no—
115#106—416no—
116#97—24no—
117#97—22no—
118#97C2×Q818yes—
119#97C2×Q814yes—
120C4×C8—28no—
121C4×C8—24no—
122#95—28no—
123#95—24no—
124#95—24no—
125#95—24no—
126C24#12011yescharacteristic
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—42no—
135C22×C4—84no—
136C22×C4—82no—
137C22×C4#12012yes—
138C22×C4#12011yescharacteristic
139C22×C4C22⋊Q814yes—
140C22×C4C22⋊Q812yes—
141C22×C4C4⋊Q814yes—
142C22×C4C4⋊Q812yes—
143C22×C4C22×D412yes—
144C22×C4C22×D411yescharacteristic
145M16—232no—
146M16—48no—
147C2×C8—28no—
148C2×C8—28no—
149C2×C8—28no—
150C2×C8—24no—
151C2×C8—42no—
152C2×C8—42no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—44no—
156C4⋊C4—48no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—88no—
165C4⋊C4—88no—
166C4⋊C4—88no—
167C4⋊C4—84no—
168C4⋊C4—82no—
169C22⋊C4—84no—
170C22⋊C4—82no—
171C42—48no—
172C42—44no—
173C42C22⋊Q812yes—
174C42C22⋊Q814yes—
175C42C4⋊Q812yes—
176C42C4⋊Q814yes—
177C42C22×Q812yes—
178C42C22×Q811yescharacteristic
179C42C2×#4111yescharacteristic
180C23—22no—
181C23—21no—
182C23—21no—
183C23—22no—
184C23#39512yes—
185C23C2×#12011yescharacteristic
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—28no—
191C2×C4—28no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—84no—
204C2×C4—82no—
205C2×C4—164no—
206C2×C4#39514yes—
207C2×C4#39512yes—
208C2×C4#45314yes—
209C2×C4#45412yes—
210C2×C4#45512yes—
211C2×C4#45614yes—
212C2×C4C2×C22⋊Q812yes—
213C2×C4C2×C22⋊Q811yescharacteristic
214C2×C4C2×C4⋊Q812yes—
215C2×C4C2×C4⋊Q811yescharacteristic
216C8—216no—
217C8—44no—
218C22—24no—
219C22—22no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22C2×#39511yescharacteristic
224C22C2×#45312yes—
225C22C2×#45511yescharacteristic
226C22C2×#45411yescharacteristic
227C22C2×#45612yes—
228C4—22no—
229C4—22no—
230C4—21no—
231C4—21no—
232C4—84no—
233C4—84no—
234C4—84no—
235C4—162no—
236C4#206018yes—
237C4#206114yes—
238C2—42no—
239C2#1907511yescharacteristic
240C2#1907612yes—
241C2#1908411yescharacteristic
242C2C2×#206012yes—
243C2C2×#206111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
4. #10585C2semidirect

Maximal subgroups

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

subgroupindex
#376432
#208002
#207642
#105852
#80282