Finite Group Databasebeta

← #99138

Subgroups — #99138

Order 512.

660 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 90 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#13852C211yesmaximal, characteristic
2#8300C212yesmaximal
3#7017C214yesmaximal, complemented
4C22×#117C2211yescharacteristic
5C2×C4×C4⋊C4C412yes—
6C2×C4×C22⋊C4C412yes—
7#1817C2214yes—
8C2×#324—22no—
9C2×#323—22no—
10#1441—22no—
11#1439—22no—
12C4×#95C2212yes—
13C24×C4C2×C411yescharacteristic
14C2×#117—24no—
15C2×#117—44no—
16C22×C4⋊C4—22no—
17C22×C4⋊C4D412yes—
18C22×C22⋊C4—22no—
19C22×C22⋊C4D412yes—
20C22×C42C2×C411yescharacteristic
21C22×C42C2×C414yes—
22C22×C42C2311yescharacteristic
23C2×C22⋊C8—24no—
24C4×C4⋊C4—44no—
25C4×C22⋊C4—44no—
26C2×#95—22no—
27C2×#95—22no—
28C2×#95—24no—
29#336—28no—
30#324—44no—
31#323—44no—
32C25C22⋊C411yescharacteristic
33C23×C4—22no—
34C23×C4—22no—
35C23×C4—22no—
36C23×C4—24no—
37C23×C4—24no—
38C23×C4—42no—
39C23×C4C22⋊C411yescharacteristic
40C23×C4C22⋊C412yes—
41C23×C4C22×C411yescharacteristic
42C23×C4C2×D412yes—
43C22×C8—44no—
44#117—416no—
45C2×C4⋊C4—24no—
46C2×C4⋊C4—24no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—48no—
49C2×C4⋊C4—44no—
50C2×C22⋊C4—24no—
51C2×C22⋊C4—24no—
52C2×C22⋊C4—42no—
53C2×C22⋊C4—48no—
54C2×C22⋊C4—44no—
55C2×C42—24no—
56C2×C42—22no—
57C2×C42—28no—
58C2×C42—42no—
59C2×C42—44no—
60C2×C42—44no—
61C2×C42#4114yes—
62C22⋊C8—44no—
63C22⋊C8—44no—
64#95—48no—
65#95—44no—
66#95—44no—
67C24—21no—
68C24—21no—
69C24—22no—
70C24—22no—
71C24—44no—
72C24#10012yes—
73C24C2×C22⋊C411yescharacteristic
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—28no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—28no—
87C22×C4—48no—
88C22×C4—44no—
89C22×C4—48no—
90C22×C4—44no—
91C22×C4—44no—
92C22×C4—44no—
93C22×C4—44no—
94C22×C4—44no—
95C22×C4—44no—
96C22×C4—44no—
97C22×C4—44no—
98C22×C4—44no—
99C22×C4#9914yes—
100C22×C4#10112yes—
101C22×C4#11712yes—
102C22×C4#12314yes—
103C22×C4#13114yes—
104C2×C8—84no—
105C2×C8—88no—
106C4⋊C4—48no—
107C4⋊C4—416no—
108C22⋊C4—48no—
109C22⋊C4—416no—
110C42—28no—
111C42—44no—
112C42—88no—
113C23—24no—
114C23—22no—
115C23—24no—
116C23—44no—
117C23—44no—
118C23—41no—
119C23—41no—
120C23—44no—
121C23—42no—
122C23—84no—
123C23—82no—
124C23—82no—
125C23#38111yescharacteristic
126C23#40812yes—
127C23C2×#9912yes—
128C23C2×#10011yescharacteristic
129C23C2×#10111yescharacteristic
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—24no—
133C2×C4—24no—
134C2×C4—24no—
135C2×C4—24no—
136C2×C4—22no—
137C2×C4—28no—
138C2×C4—24no—
139C2×C4—416no—
140C2×C4—44no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—44no—
144C2×C4—42no—
145C2×C4—44no—
146C2×C4—48no—
147C2×C4—48no—
148C2×C4—48no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—84no—
152C2×C4—84no—
153C2×C4—84no—
154C2×C4—84no—
155C2×C4#43314yes—
156C8—164no—
157C22—24no—
158C22—22no—
159C22—22no—
160C22—22no—
161C22—22no—
162C22—42no—
163C22—44no—
164C22—84no—
165C22—84no—
166C22—82no—
167C22—82no—
168C22#147112yes—
169C22#147212yes—
170C22#181811yescharacteristic
171C22#181911yescharacteristic
172C22#182111yescharacteristic
173C22#182714yes—
174C4—24no—
175C4—42no—
176C4—44no—
177C4—44no—
178C4—88no—
179C2—22no—
180C2—22no—
181C2—82no—
182C2#870411yescharacteristic
183C2#873212yes—
184C2#873812yes—
185C2#874212yes—
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7017C2semidirect

Maximal subgroups

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

subgroupindex
#138522
#83002
#70172