Finite Group Databasebeta

← #12804

Subgroups — #12804

Order 256.

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

normal subgroups only

This group has 216 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#2327C211yesmaximal, characteristic
2C2×#356C212yesmaximal
3#1801C212yesmaximal
4#1394C212yesmaximal
5C2×C4⋊Q8C2211yescharacteristic
6C2×C4×Q8C411yescharacteristic
7C2×C4×Q8C2211yescharacteristic
8#423—22no—
9C2×#101C2214yes—
10#391C411yescharacteristic
11#391C2211yescharacteristic
12#387—22no—
13#387—22no—
14#382C411yescharacteristic
15#376C411yescharacteristic
16#356—24no—
17#343—22no—
18C22×Q8C2×C411yescharacteristic
19C22×Q8C2311yescharacteristic
20C2×M16—24no—
21C4⋊Q8—22no—
22C4⋊Q8D412yes—
23C4⋊Q8Q812yes—
24C4×Q8—42no—
25C4×Q8—42no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4D412yes—
31C2×C42—21no—
32C2×C42C2×C411yescharacteristic
33C2×C42C2×C411yescharacteristic
34C2×C42C2×C411yescharacteristic
35C2×C42D412yes—
36C4⋊C8—44no—
37#101—24no—
38#101—24no—
39#95—21no—
40#95—22no—
41#95—22no—
42#95—21no—
43#95—22no—
44C2×Q8—21no—
45C2×Q8—41no—
46C2×Q8—41no—
47C2×Q8—41no—
48C2×Q8—41no—
49C2×Q8C4⋊C412yes—
50C2×Q8C2×D411yescharacteristic
51C2×Q8C2×Q811yescharacteristic
52C2×Q8#4112yes—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4C22⋊C412yes—
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22×C411yescharacteristic
61C22×C4C2×D412yes—
62M16—44no—
63M16—44no—
64C2×C8—44no—
65C4⋊C4—24no—
66C4⋊C4—24no—
67C4⋊C4—41no—
68C4⋊C4—42no—
69C4⋊C4—41no—
70C4⋊C4—42no—
71C42—24no—
72C42—22no—
73C42—21no—
74C42—41no—
75C42—42no—
76C42C4⋊C412yes—
77C23C2×C22⋊C411yescharacteristic
78Q8—41no—
79Q8—41no—
80Q8—42no—
81Q8—42no—
82Q8—42no—
83C2×C4—21no—
84C2×C4—22no—
85C2×C4—21no—
86C2×C4—22no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—42no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—42no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4#9912yes—
104C2×C4C2×C4⋊C411yescharacteristic
105C2×C4#11711yescharacteristic
106C2×C4#12112yes—
107C2×C4C22⋊Q812yes—
108C8—84no—
109C22—21no—
110C22—21no—
111C22#38011yescharacteristic
112C22C2×#9911yescharacteristic
113C22#41011yescharacteristic
114C4—22no—
115C4—21no—
116C4—21no—
117C4—22no—
118C4—22no—
119C4—42no—
120C4—42no—
121C4—41no—
122C4—42no—
123C4—81no—
124C4—81no—
125C2—21no—
126C2—21no—
127C2#179911yescharacteristic
128C2#213311yescharacteristic
129C2#213511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC22×Q8C23
prime radicalC23C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#23272
C2×#3562
#18012
#13942