Finite Group Databasebeta

← #9462

Subgroups — #9462

Order 256.

350 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 65 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#2434C211yesmaximal, characteristic
2#2310C211yesmaximal, characteristic
3#1784C211yesmaximal, characteristic
4#1511C212yesmaximal
5#1284C212yesmaximal
6C23×Q8C2211yescharacteristic
7C2×C4⋊Q8C412yescomplemented
8C2×C22⋊Q8C2211yescharacteristic
9C2×#117—21no—
10C2×#101—22no—
11#407C2212yes—
12#395—22no—
13#391—24no—
14#391C412yescomplemented
15#385—21no—
16#385—21no—
17#381—21no—
18#381C2212yes—
19C4×C22⋊C4C2211yescharacteristic
20#328—24no—
21C22×Q8—22no—
22C22×Q8—22no—
23C22×Q8—44no—
24C22×Q8C2×C411yescharacteristic
25C22×Q8C2×C412yes—
26C22×Q8D412yes—
27C23×C4D411yescharacteristic
28C23×C4D411yescharacteristic
29C23×C4C2311yescharacteristic
30C2×M16—22no—
31C4⋊Q8—44no—
32C22⋊Q8—22no—
33C22⋊Q8—22no—
34#117—44no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C22⋊C4—21no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C42—22no—
44C2×C42C2×C412yes—
45#101—44no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48#95—24no—
49#95—22no—
50#95—22no—
51#95—21no—
52#95—22no—
53#95D414yes—
54C24C2×D411yescharacteristic
55C2×Q8—24no—
56C2×Q8—24no—
57C2×Q8—24no—
58C2×Q8—24no—
59C2×Q8—22no—
60C2×Q8—42no—
61C2×Q8—42no—
62C2×Q8—44no—
63C2×Q8—44no—
64C2×Q8—48no—
65C2×Q8—42no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—24no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—24no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4C22⋊C412yes—
78C22×C4C22⋊C412yes—
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4#4112yes—
83M16—44no—
84C2×C8—42no—
85C4⋊C4—22no—
86C4⋊C4—44no—
87C4⋊C4—44no—
88C4⋊C4—44no—
89C4⋊C4—42no—
90C22⋊C4—24no—
91C22⋊C4—44no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C42—42no—
95C42—44no—
96C23—22no—
97C23—22no—
98C23—41no—
99C23C2×C22⋊C411yescharacteristic
100C23#12011yescharacteristic
101C23#12311yescharacteristic
102Q8—48no—
103Q8—48no—
104Q8—82no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—24no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—24no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—44no—
123C2×C4—44no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—44no—
129C2×C4#9914yes—
130C2×C4C4×D412yes—
131C2×C4#12112yes—
132C8—82no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—44no—
141C22#38611yescharacteristic
142C22C2×#9912yes—
143C22#45612yes—
144C4—22no—
145C4—24no—
146C4—22no—
147C4—44no—
148C4—42no—
149C4—84no—
150C4—81no—
151C4—81no—
152C2—22no—
153C2—22no—
154C2—41no—
155C2#188611yescharacteristic
156C2#189811yescharacteristic
157C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
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
7. C2×C4⋊Q8C4semidirect
14. #391C4semidirect

Maximal subgroups

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

subgroupindex
#24342
#23102
#17842
#15112
#12842