Finite Group Databasebeta

← #99149

Subgroups — #99149

Order 512.

732 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 106 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#13776C211yesmaximal, characteristic
2#8299C212yesmaximal
3#7032C214yesmaximal
4C23×C4⋊C4C2211yescharacteristic
5C2×#381C411yescharacteristic
6C2×#381C411yescharacteristic
7C2×C4×C22⋊C4C412yes—
8#1779C2214yes—
9C2×#324—24no—
10#1436—24no—
11C22⋊#95C2212yes—
12C24×C4C2×C411yescharacteristic
13C24×C4C2×C411yescharacteristic
14C24×C4C2311yescharacteristic
15C22×C4⋊C4—24no—
16C22×C4⋊C4—44no—
17C22×C4⋊C4D414yes—
18C22×C22⋊C4C2×C412yes—
19C22×C22⋊C4C2×C412yes—
20C22×C42—22no—
21C2×C22⋊C8—24no—
22C2×C22⋊C8—28no—
23#381—22no—
24#381—24no—
25#381C814yescomplemented
26C4×C22⋊C4—44no—
27C2×#95—22no—
28C2×#95—22no—
29C2×#95—22no—
30C2×#95—24no—
31#324—48no—
32C25C22×C411yescharacteristic
33C23×C4—22no—
34C23×C4—22no—
35C23×C4—22no—
36C23×C4—24no—
37C23×C4—22no—
38C23×C4—22no—
39C23×C4—42no—
40C23×C4—42no—
41C23×C4—42no—
42C23×C4—82no—
43C23×C4—82no—
44C23×C4C22⋊C412yes—
45C23×C4C22⋊C412yes—
46C23×C4C2×C812yes—
47C23×C4M1612yes—
48C23×C4C2×D412yes—
49C23×C4#4114yes—
50C22×C8—44no—
51C2×C4⋊C4—28no—
52C2×C4⋊C4—28no—
53C2×C4⋊C4—44no—
54C2×C4⋊C4—416no—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—42no—
58C2×C22⋊C4—42no—
59C2×C22⋊C4—42no—
60C2×C22⋊C4—42no—
61C2×C22⋊C4C2×C814yes—
62C2×C22⋊C4M1614yes—
63C2×C42—44no—
64C2×C42—84no—
65C22⋊C8—48no—
66C22⋊C8—88no—
67#95—28no—
68#95—28no—
69#95—44no—
70#95—84no—
71C24—22no—
72C24—22no—
73C24—42no—
74C24—42no—
75C24C2×C22⋊C411yescharacteristic
76C24#11712yes—
77C24C22×C811yescharacteristic
78C24C2×M1611yescharacteristic
79C24C2×M1612yes—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—24no—
85C22×C4—24no—
86C22×C4—24no—
87C22×C4—24no—
88C22×C4—28no—
89C22×C4—24no—
90C22×C4—24no—
91C22×C4—24no—
92C22×C4—44no—
93C22×C4—44no—
94C22×C4—44no—
95C22×C4—48no—
96C22×C4—44no—
97C22×C4—44no—
98C22×C4—44no—
99C22×C4—44no—
100C22×C4—44no—
101C22×C4—44no—
102C22×C4—44no—
103C22×C4—44no—
104C22×C4—44no—
105C22×C4—44no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—84no—
111C22×C4—88no—
112C22×C4—88no—
113C22×C4C22⋊C814yes—
114C22×C4#9914yes—
115C22×C4#12314yes—
116C2×C8—84no—
117C2×C8—88no—
118C4⋊C4—416no—
119C22⋊C4—44no—
120C22⋊C4—44no—
121C22⋊C4—44no—
122C22⋊C4—44no—
123C42—88no—
124C23—22no—
125C23—22no—
126C23—22no—
127C23—22no—
128C23—24no—
129C23—24no—
130C23—22no—
131C23—22no—
132C23—22no—
133C23—22no—
134C23—42no—
135C23—42no—
136C23—44no—
137C23—44no—
138C23—44no—
139C23—48no—
140C23#38111yescharacteristic
141C23C2×C22⋊C811yescharacteristic
142C23#40711yescharacteristic
143C23C2×#9912yes—
144C23#41312yes—
145C23#43112yes—
146C23#43212yes—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—28no—
151C2×C4—28no—
152C2×C4—48no—
153C2×C4—48no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—48no—
161C2×C4—48no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—84no—
165C2×C4—88no—
166C2×C4—88no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4—84no—
172C2×C4—84no—
173C8—164no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—24no—
179C22—24no—
180C22—24no—
181C22—22no—
182C22—22no—
183C22—44no—
184C22—44no—
185C22—44no—
186C22—44no—
187C22—42no—
188C22—42no—
189C22—44no—
190C22#145912yes—
191C22#146512yes—
192C22#181711yescharacteristic
193C22#181911yescharacteristic
194C22#182211yescharacteristic
195C22#182812yes—
196C22#183012yes—
197C4—44no—
198C4—44no—
199C4—44no—
200C4—84no—
201C4—84no—
202C2—22no—
203C2—22no—
204C2—42no—
205C2—42no—
206C2#870011yescharacteristic
207C2#873512yes—
208C2#874611yescharacteristic
209C2#875311yescharacteristic
210C2#875812yes—
subgroupG/H
center, socleC23#381
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #381C8semidirect

Maximal subgroups

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

subgroupindex
#137762
#82992
#70322