Finite Group Databasebeta

← #11513

Subgroups — #11513

Order 256.

223 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 55 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#2285C211yesmaximal, characteristic
2#1464C212yesmaximal
3#1325C212yesmaximal, complemented
4#1324C212yesmaximal, complemented
5C2×#117C411yescharacteristic
6C2×#117C2211yescharacteristic
7C22×C42C411yescharacteristic
8#432—22no—
9#384C411yescharacteristic
10#384C2211yescharacteristic
11#382C411yescharacteristic
12#382C2211yescharacteristic
13#380—22no—
14C4×C4⋊C4—22no—
15#324C2214yes—
16#323—22no—
17C23×C4—21no—
18C23×C4C2×C411yescharacteristic
19#117—41no—
20#117—41no—
21#117—41no—
22#117—41no—
23C2×C4⋊C4—22no—
24C2×C4⋊C4—22no—
25C2×C4⋊C4C2×C411yescharacteristic
26C2×C4⋊C4C2×C411yescharacteristic
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C4⋊C4C2311yescharacteristic
29C2×C22⋊C4—21no—
30C2×C22⋊C4D412yes—
31C2×C42—21no—
32C2×C42—22no—
33C2×C42—22no—
34C2×C42—21no—
35C2×C42C2×C411yescharacteristic
36C2×C42C2×C411yescharacteristic
37C2×C42D412yes—
38C4⋊C8—42no—
39C22⋊C8—24no—
40#97—42no—
41#95—22no—
42#95—22no—
43C24C22⋊C411yescharacteristic
44C22×C4—22no—
45C22×C4—21no—
46C22×C4—21no—
47C22×C4—21no—
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—24no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—42no—
55C22×C4—42no—
56C22×C4C22⋊C412yes—
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22×C411yescharacteristic
59C22×C4C2×D412yes—
60C2×C8—44no—
61C4⋊C4—41no—
62C4⋊C4—41no—
63C4⋊C4—41no—
64C4⋊C4—41no—
65C4⋊C4—44no—
66C4⋊C4—44no—
67C22⋊C4—42no—
68C22⋊C4—42no—
69C42—24no—
70C42—22no—
71C42—22no—
72C42—21no—
73C42—21no—
74C42—41no—
75C42—41no—
76C42M1612yes—
77C42M1612yes—
78C23—21no—
79C23—21no—
80C23—41no—
81C23—41no—
82C23#10112yes—
83C23C2×C22⋊C411yescharacteristic
84C2×C4—22no—
85C2×C4—22no—
86C2×C4—22no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—21no—
91C2×C4—22no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—48no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4#9911yescharacteristic
108C2×C4#9911yescharacteristic
109C2×C4C2×M1611yescharacteristic
110C2×C4C2×M1611yescharacteristic
111C8—84no—
112C22—21no—
113C22—21no—
114C22—22no—
115C22—42no—
116C22—42no—
117C22—41no—
118C22—41no—
119C22#40711yescharacteristic
120C22C2×#9911yescharacteristic
121C22C2×#10111yescharacteristic
122C4—24no—
123C4—21no—
124C4—21no—
125C4—21no—
126C4—21no—
127C4—42no—
128C4—42no—
129C4—42no—
130C4—81no—
131C4—81no—
132C2—21no—
133C2—21no—
134C2—42no—
135C2#146311yescharacteristic
136C2#212611yescharacteristic
137C2#213211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1325C2semidirect
4. #1324C2semidirect

Maximal subgroups

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

subgroupindex
#22852
#14642
#13252
#13242