Finite Group Databasebeta

← #215667

Subgroups — #215667

Order 512.

902 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 226 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
1C2×#2859C211yesmaximal, characteristic
2#20778C212yesmaximal
3#9894C218yesmaximal, complemented
4#8719C214yesmaximal
5C22×#125C412yes—
6#2859—22no—
7#2859—22no—
8#2859C414yes—
9#2859C2214yes—
10C2×#389—22no—
11C2×#389—22no—
12C2×#384C412yes—
13C2×#382C2212yes—
14C2×C43C2211yescharacteristic
15#1839—22no—
16#1831—22no—
17#1817—22no—
18#1797—22no—
19#1767C2214yes—
20#1749C2214yes—
21#1745C2214yes—
22#1295C22116yes—
23C2×#125—22no—
24C2×#125—22no—
25C2×#125—48no—
26C2×#125C2×C418yes—
27C22×C4⋊C4—24no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4C2×C414yes—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D412yes—
34C22×C42D412yes—
35C22×C42C2311yescharacteristic
36C2×C4⋊C8—24no—
37C2×C22⋊C8—24no—
38C2×#97C2318yes—
39#389—24no—
40#389—24no—
41#389—24no—
42#389—24no—
43#389—24no—
44#389—24no—
45#389—24no—
46#389—24no—
47#384—22no—
48#384—22no—
49#384C2×C414yes—
50#384C2×C414yes—
51#382—24no—
52#382C2×C414yes—
53#382C2314yes—
54C2×#95—24no—
55C43C2×C412yes—
56C43D412yes—
57C43D412yes—
58C43C2312yes—
59#336—24no—
60#336—24no—
61#336—24no—
62#336—24no—
63C23×C4—24no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D412yes—
69C22×C8—28no—
70#125—416no—
71#125—48no—
72#125—48no—
73C2×C4⋊C4—28no—
74C2×C4⋊C4—28no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—48no—
81C2×C4⋊C4—48no—
82C2×C4⋊C4—416no—
83C2×C4⋊C4—48no—
84C2×C4⋊C4C22×C418yes—
85C2×C42—24no—
86C2×C42—24no—
87C2×C42—24no—
88C2×C42—44no—
89C2×C42—44no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42C22⋊C412yes—
92C2×C42C22⋊C412yes—
93C2×C42C22⋊C412yes—
94C2×C42C22⋊C412yes—
95C2×C42C22⋊C412yes—
96C2×C42C22⋊C411yescharacteristic
97C2×C42C22×C412yes—
98C2×C42C22×C411yescharacteristic
99C2×C42C22×C412yes—
100C2×C42C2×D412yes—
101C2×C42C2×D412yes—
102C2×C42C2×D412yes—
103C2×C42C2×D412yes—
104C2×C42C2×D412yes—
105C2×C42C2411yescharacteristic
106C4⋊C8—48no—
107C22⋊C8—48no—
108#97—216no—
109#95—28no—
110#95—28no—
111C24C2×C22⋊C411yescharacteristic
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—28no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—48no—
126C22×C4—48no—
127C22×C4—48no—
128C22×C4#10114yes—
129C22×C4C2×C22⋊C412yes—
130C22×C4C2×C22⋊C412yes—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4C2×C22⋊C412yes—
133C22×C4C2×C22⋊C412yes—
134C22×C4C2×C22⋊C412yes—
135C22×C4C23×C411yescharacteristic
136C22×C4C22×D412yes—
137C2×C8—48no—
138C2×C8—48no—
139C2×C8—48no—
140C4⋊C4—48no—
141C4⋊C4—48no—
142C4⋊C4—48no—
143C4⋊C4—48no—
144C4⋊C4—416no—
145C4⋊C4—416no—
146C4⋊C4—48no—
147C4⋊C4—48no—
148C42—22no—
149C42—24no—
150C42—22no—
151C42—24no—
152C42—24no—
153C42—24no—
154C42—24no—
155C42—24no—
156C42—42no—
157C42—42no—
158C42—42no—
159C42—42no—
160C42—42no—
161C42—42no—
162C42#10014yes—
163C422−1+414yes—
164C23—21no—
165C23—21no—
166C23—22no—
167C23C2×#10112yes—
168C23#42114yes—
169C23C22×C22⋊C411yescharacteristic
170C2×C4—28no—
171C2×C4—28no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—48no—
192C2×C4—48no—
193C2×C4—48no—
194C2×C4—48no—
195C2×C4—48no—
196C2×C4—816no—
197C2×C4C2×#10014yes—
198C2×C4C2×#10012yes—
199C2×C4C2×#10114yes—
200C2×C4#42114yes—
201C2×C4#51912yes—
202C2×C4#53614yes—
203C8—88no—
204C22—22no—
205C22—24no—
206C22—22no—
207C22—24no—
208C22#153414yes—
209C22#231011yescharacteristic
210C22C22×#10011yescharacteristic
211C22C22×#10111yescharacteristic
212C22C2×#42112yes—
213C22C2×#42112yes—
214C4—24no—
215C4—24no—
216C4—22no—
217C4—22no—
218C4—24no—
219C4—24no—
220C4—42no—
221C4—42no—
222C4—88no—
223C2—24no—
224C2C2×#153411yescharacteristic
225C2#1766711yescharacteristic
226C2#1767112yes—
227C2#2182911yescharacteristic
228C2#2184512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9894C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
#207782
#98942
#87192