Finite Group Databasebeta

← #9514

Subgroups — #9514

Order 256.

239 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 59 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#2921C211yesmaximal, characteristic
2#2484C211yesmaximal, characteristic
3#1822C211yesmaximal, characteristic
4#1463C212yesmaximal
5#1306C212yesmaximal
6C2×#117C2211yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8#433—24no—
9#423—21no—
10#407C2212yes—
11#407C2212yes—
12C4×M16—21no—
13#400—21no—
14#398—21no—
15#387—21no—
16#387—21no—
17#385C411yescharacteristic
18#385C411yescharacteristic
19#381C411yescharacteristic
20#381C411yescharacteristic
21#381C2211yescharacteristic
22#343—21no—
23#343—21no—
24#324—24no—
25C23×C4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2311yescharacteristic
28C2×M16—22no—
29C2×M16—22no—
30#117—22no—
31#117—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4D411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4C2×C411yescharacteristic
41C2×C22⋊C4C2×C411yescharacteristic
42C2×C22⋊C4C2×C411yescharacteristic
43C2×C22⋊C4C2×C411yescharacteristic
44C2×C42D411yescharacteristic
45C2×C42D411yescharacteristic
46C4⋊C8—24no—
47C22⋊C8—22no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50C22⋊C8D414yes—
51#97—22no—
52C4×C8—22no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61C24C22×C411yescharacteristic
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—22no—
68C22×C4—42no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22⋊C411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D412yes—
80C22×C4#4112yes—
81M16—42no—
82M16—42no—
83M16—42no—
84M16—42no—
85C2×C8—24no—
86C2×C8—42no—
87C4⋊C4—22no—
88C4⋊C4—22no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—41no—
92C4⋊C4—42no—
93C4⋊C4—41no—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C42—21no—
99C42—21no—
100C42—21no—
101C42—21no—
102C23—21no—
103C23—21no—
104C23—41no—
105C23—41no—
106C23C2×C22⋊C411yescharacteristic
107C23C4×D412yes—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—22no—
113C2×C4—24no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—41no—
121C2×C4—42no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—82no—
137C2×C4#12011yescharacteristic
138C2×C4#12112yes—
139C2×C4#12311yescharacteristic
140C8—44no—
141C8—82no—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—42no—
148C22—42no—
149C22#38611yescharacteristic
150C22#41011yescharacteristic
151C22#41011yescharacteristic
152C4—21no—
153C4—22no—
154C4—21no—
155C4—42no—
156C4—41no—
157C4—41no—
158C4—41no—
159C4—41no—
160C4—81no—
161C4—81no—
162C4—81no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#189111yescharacteristic
167C2#189911yescharacteristic
168C2#191511yescharacteristic
subgroupG/H
center, socleC22#386
commutator, prime radicalC24C22×C4
Frattini, cosocleC23×C4C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#29212
#24842
#18222
#14632
#13062