Finite Group Databasebeta

← #9464

Subgroups — #9464

Order 256.

305 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 61 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#2501C211yesmaximal, characteristic
2#2313C211yesmaximal, characteristic, complemented
3#1821C211yesmaximal, characteristic, complemented
4#1510C212yesmaximal
5#1284C212yesmaximal, complemented
6C2×#123—21no—
7C2×#123—21no—
8C2×#121C2211yescharacteristic
9C22×C4⋊C4C2211yescharacteristic
10#407C2212yes—
11#399—22no—
12#390C412yescomplemented
13#387—24no—
14#384C412yescomplemented
15#381C2212yes—
16#380—21no—
17C4×C22⋊C4—21no—
18C4×C22⋊C4C2211yescharacteristic
19#343—22no—
20#327—24no—
21C22×D4—21no—
22C22×D4C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C2×M16—22no—
27#123—44no—
28#123—44no—
29#121—22no—
30#121—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C412yes—
38C2×C4⋊C4D412yes—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—21no—
44C2×C42—22no—
45C2×C42C2×C412yes—
46C22⋊C8—22no—
47C22⋊C8—22no—
48#95—22no—
49#95—22no—
50#95—21no—
51#95—22no—
52#95D414yes—
53C24—21no—
54C24C2×D411yescharacteristic
55C2×D4—22no—
56C2×D4—42no—
57C2×D4—42no—
58C2×D4—42no—
59C2×D4—41no—
60C2×D4—41no—
61C22×C4—22no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—24no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4C22⋊C412yes—
72C22×C4C22⋊C412yes—
73C22×C4C22×C411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D412yes—
76C22×C4#4112yes—
77M16—44no—
78C2×C8—42no—
79C4⋊C4—24no—
80C4⋊C4—24no—
81C4⋊C4—22no—
82C4⋊C4—44no—
83C4⋊C4—44no—
84C4⋊C4—82no—
85C22⋊C4—24no—
86C22⋊C4—44no—
87C22⋊C4—44no—
88C22⋊C4—44no—
89C22⋊C4—42no—
90C42—42no—
91C42—82no—
92C23—22no—
93C23—22no—
94C23—41no—
95C23—41no—
96C23—41no—
97C23—42no—
98C23—41no—
99C23—42no—
100C23C2×C22⋊C411yescharacteristic
101C23#12011yescharacteristic
102C23#12311yescharacteristic
103D4—82no—
104D4—82no—
105D4—82no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—24no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—44no—
123C2×C4—41no—
124C2×C4—44no—
125C2×C4—81no—
126C2×C4—81no—
127C2×C4C4×D412yes—
128C2×C4#12112yes—
129C8—82no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—44no—
138C22—82no—
139C22—82no—
140C22—81no—
141C22—81no—
142C22—81no—
143C22#38611yescharacteristic
144C22#41012yes—
145C22#45312yes—
146C4—22no—
147C4—22no—
148C4—44no—
149C4—42no—
150C4—42no—
151C4—84no—
152C4—81no—
153C2—22no—
154C2—22no—
155C2—41no—
156C2—161no—
157C2#188711yescharacteristic
158C2#189911yescharacteristic
159C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2313C2semidirect
3. #1821C2semidirect
5. #1284C2semidirect
12. #390C4semidirect
14. #384C4semidirect

Maximal subgroups

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

subgroupindex
#25012
#23132
#18212
#15102
#12842