Finite Group Databasebeta

← #99154

Subgroups — #99154

Order 512.

962 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 86 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#14341C211yesmaximal, characteristic
2#8298C212yesmaximal
3#7032C214yesmaximal, complemented
4C23×C22⋊C4C2211yescharacteristic
5C2×#379C412yes—
6#1779C2214yes—
7C2×#323—24no—
8#1440—24no—
9C22⋊#95C412yes—
10C22⋊#95C2212yes—
11C26C2×C411yescharacteristic
12C24×C4C2×C411yescharacteristic
13C24×C4C2311yescharacteristic
14C22×C22⋊C4—24no—
15C22×C22⋊C4—22no—
16C22×C22⋊C4—22no—
17C22×C22⋊C4—44no—
18C22×C22⋊C4C2×C414yes—
19C22×C22⋊C4D414yes—
20C2×C22⋊C8—24no—
21C2×C22⋊C8—28no—
22#379—44no—
23C2×#95—24no—
24C2×#95—24no—
25#323—48no—
26C25—24no—
27C25—22no—
28C25—24no—
29C25—42no—
30C25—82no—
31C25—82no—
32C25C22⋊C412yes—
33C25C22×C411yescharacteristic
34C23×C4—22no—
35C23×C4—24no—
36C23×C4—42no—
37C23×C4—42no—
38C23×C4—82no—
39C23×C4—82no—
40C23×C4C22⋊C412yes—
41C23×C4C2×D412yes—
42C23×C4#4114yes—
43C22×C8—44no—
44C2×C22⋊C4—28no—
45C2×C22⋊C4—28no—
46C2×C22⋊C4—28no—
47C2×C22⋊C4—44no—
48C2×C22⋊C4—416no—
49C2×C22⋊C4—44no—
50C2×C22⋊C4—44no—
51C2×C22⋊C4—44no—
52C2×C22⋊C4—44no—
53C2×C22⋊C4—84no—
54C2×C22⋊C4—84no—
55C22⋊C8—48no—
56C22⋊C8—88no—
57#95—84no—
58#95—84no—
59C24—22no—
60C24—22no—
61C24—28no—
62C24—22no—
63C24—22no—
64C24—24no—
65C24—24no—
66C24—42no—
67C24—42no—
68C24—416no—
69C24—44no—
70C24—44no—
71C24—44no—
72C24—44no—
73C24—84no—
74C24—84no—
75C24—88no—
76C24—88no—
77C24—816no—
78C24—816no—
79C24#9914yes—
80C24C2×C22⋊C411yescharacteristic
81C24#11712yes—
82C24#13114yes—
83C22×C4—24no—
84C22×C4—22no—
85C22×C4—24no—
86C22×C4—44no—
87C22×C4—48no—
88C22×C4—44no—
89C22×C4—44no—
90C22×C4—44no—
91C22×C4—48no—
92C22×C4—48no—
93C22×C4—44no—
94C22×C4—44no—
95C22×C4—44no—
96C22×C4—84no—
97C22×C4—88no—
98C22×C4—88no—
99C22×C4#12314yes—
100C2×C8—84no—
101C2×C8—88no—
102C22⋊C4—416no—
103C22⋊C4—88no—
104C22⋊C4—88no—
105C22⋊C4—88no—
106C23—22no—
107C23—22no—
108C23—22no—
109C23—22no—
110C23—24no—
111C23—24no—
112C23—28no—
113C23—24no—
114C23—28no—
115C23—48no—
116C23—44no—
117C23—44no—
118C23—48no—
119C23—44no—
120C23—416no—
121C23—44no—
122C23—44no—
123C23—44no—
124C23—42no—
125C23—42no—
126C23—44no—
127C23—44no—
128C23—44no—
129C23—88no—
130C23—88no—
131C23—816no—
132C23—88no—
133C23—88no—
134C23—864no—
135C23—816no—
136C23#38111yescharacteristic
137C23#40812yes—
138C23C2×#9912yes—
139C23#41312yes—
140C23#43314yes—
141C2×C4—24no—
142C2×C4—28no—
143C2×C4—42no—
144C2×C4—44no—
145C2×C4—84no—
146C2×C4—88no—
147C2×C4—88no—
148C2×C4—84no—
149C2×C4—84no—
150C2×C4—84no—
151C2×C4—88no—
152C2×C4—88no—
153C2×C4—84no—
154C8—164no—
155C22—24no—
156C22—22no—
157C22—22no—
158C22—24no—
159C22—28no—
160C22—24no—
161C22—416no—
162C22—42no—
163C22—42no—
164C22—44no—
165C22—44no—
166C22—44no—
167C22—44no—
168C22—88no—
169C22—88no—
170C22—832no—
171C22—84no—
172C22—84no—
173C22#146814yes—
174C22#181811yescharacteristic
175C22#181911yescharacteristic
176C22#182211yescharacteristic
177C22#182812yes—
178C22#182912yes—
179C4—44no—
180C4—82no—
181C4—164no—
182C2—24no—
183C2—44no—
184C2—84no—
185C2#870211yescharacteristic
186C2#873612yes—
187C2#874611yescharacteristic
188C2#875211yescharacteristic
189C2#875912yes—
subgroupG/H
center, socleC23#381
commutatorC25C22×C4
Frattini, cosocleC24×C4C23
prime radicalC26C2×C4
LCSG ▷ C25 ▷ 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
3. #7032C2semidirect

Maximal subgroups

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

subgroupindex
#143412
#82982
#70322