Finite Group Databasebeta

← #11908

Subgroups — #11908

Order 256.

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

normal subgroups only

This group has 194 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#2567C211yesmaximal, characteristic
2#1801C211yesmaximal, characteristic
3#1461C211yesmaximal, characteristic
4#1354C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic
6C2×C4×Q8C2211yescharacteristic
7#431—21no—
8#423—21no—
9C2×#101—21no—
10#395C412yes—
11#395C2212yes—
12#391—21no—
13#389—21no—
14#389—21no—
15#384—21no—
16C4×C4⋊C4C411yescharacteristic
17#343C2212yes—
18#324—21no—
19#324C2212yes—
20C22×Q8C2×C412yes—
21C22×Q8D411yescharacteristic
22C2×M16—22no—
23C4⋊Q8—41no—
24C4⋊Q8—41no—
25C4×Q8—41no—
26C4×Q8—41no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4D411yescharacteristic
34C2×C4⋊C4C2311yescharacteristic
35C2×C42—21no—
36C2×C42C2×C411yescharacteristic
37C2×C42D411yescharacteristic
38C2×C42D411yescharacteristic
39C4⋊C8—41no—
40C4⋊C8—41no—
41C4⋊C8—41no—
42#101—42no—
43C22⋊C8—22no—
44C4×C8—41no—
45#95—22no—
46#95—22no—
47#95C2×C412yes—
48C2×Q8—41no—
49C2×Q8—41no—
50C2×Q8—42no—
51C2×Q8—82no—
52C2×Q8SD1612yes—
53C2×Q8Q1612yes—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4C22⋊C411yescharacteristic
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22×C411yescharacteristic
61C22×C4C2×D411yescharacteristic
62C22×C4C2×D411yescharacteristic
63M16—42no—
64M16—42no—
65C2×C8—42no—
66C2×C8—42no—
67C4⋊C4—24no—
68C4⋊C4—41no—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C4⋊C4—42no—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C4⋊C4—82no—
75C4⋊C4—82no—
76C42—21no—
77C42—21no—
78C42—21no—
79C42—21no—
80C42—21no—
81C42—21no—
82C42—42no—
83C23C2×C22⋊C411yescharacteristic
84Q8—42no—
85Q8—42no—
86Q8—84no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—22no—
94C2×C4—24no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4#9911yescharacteristic
107C2×C4#9911yescharacteristic
108C2×C4#10312yes—
109C2×C4#10312yes—
110C2×C4C2×SD1611yescharacteristic
111C2×C4C2×Q1611yescharacteristic
112C8—82no—
113C8—82no—
114C22—21no—
115C22—21no—
116C22C2×#9911yescharacteristic
117C22C2×#10311yescharacteristic
118C22#42111yescharacteristic
119C4—21no—
120C4—21no—
121C4—22no—
122C4—22no—
123C4—21no—
124C4—44no—
125C4—41no—
126C4—41no—
127C4—41no—
128C4—82no—
129C4#35612yes—
130C2—21no—
131C2—21no—
132C2#152011yescharacteristic
133C2#212211yescharacteristic
134C2C2×#35611yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC23C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#25672
#18012
#14612
#13542