Finite Group Databasebeta

← #11567

Subgroups — #11567

Order 256.

231 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 53 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#2548C211yesmaximal, characteristic
2#1467C211yesmaximal, characteristic, complemented
3#1461C211yesmaximal, characteristic, complemented
4#1318C214yesmaximal, complemented
5C2×#127C411yescharacteristic
6C2×#124C412yes—
7C2×C4×Q8C2211yescharacteristic
8#432—21no—
9#431—21no—
10#394—21no—
11#394—21no—
12#390—21no—
13#390—21no—
14#390C2212yes—
15C4×C4⋊C4C411yescharacteristic
16#324—21no—
17#324—21no—
18#324C2212yes—
19#324C2212yes—
20C22×Q8C2×C411yescharacteristic
21C22×D4—21no—
22C22×D4—21no—
23C22×D4C2×C412yes—
24#127—41no—
25#127—41no—
26#124—44no—
27C4×Q8—41no—
28C4×Q8—41no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4D411yescharacteristic
31C2×C4⋊C4D411yescharacteristic
32C2×C4⋊C4C2311yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C42C2×C411yescharacteristic
38C2×C42C2×C412yes—
39C2×C42D411yescharacteristic
40C2×C42D411yescharacteristic
41C4⋊C8—41no—
42C4⋊C8—41no—
43C22⋊C8—22no—
44C22⋊C8—22no—
45#97—41no—
46C4×C8—41no—
47#95—21no—
48C24—22no—
49C2×Q8—22no—
50C2×Q8—22no—
51C2×Q8—41no—
52C2×D4—24no—
53C2×D4—42no—
54C2×D4—42no—
55C2×D4—42no—
56C2×D4—42no—
57C2×D4—42no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4—82no—
61C22×C4—22no—
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22×C411yescharacteristic
67C22×C4C2×D411yescharacteristic
68C22×C4C2×D411yescharacteristic
69C2×C8—42no—
70C2×C8—42no—
71C4⋊C4—41no—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C4⋊C4—42no—
75C22⋊C4—44no—
76C22⋊C4—44no—
77C22⋊C4—82no—
78C22⋊C4—82no—
79C42—21no—
80C42—21no—
81C42—21no—
82C42—21no—
83C42—21no—
84C42—21no—
85C42—42no—
86C23—42no—
87C23—42no—
88C23—42no—
89C23—42no—
90C23—42no—
91C23—42no—
92C23—42no—
93C23C2×C22⋊C411yescharacteristic
94Q8—44no—
95D4—84no—
96D4—82no—
97D4—84no—
98D4—82no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—22no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4#9911yescharacteristic
117C2×C4#9911yescharacteristic
118C2×C4#10012yes—
119C2×C4#13112yes—
120C8—82no—
121C8—82no—
122C22—21no—
123C22—21no—
124C22—82no—
125C22—82no—
126C22—82no—
127C22—82no—
128C22—82no—
129C22—82no—
130C22—82no—
131C22#40811yescharacteristic
132C22C2×#9911yescharacteristic
133C22C2×#10011yescharacteristic
134C4—21no—
135C4—22no—
136C4—22no—
137C4—41no—
138C4—41no—
139C4—41no—
140C4—41no—
141C4—82no—
142C4#35512yes—
143C2—21no—
144C2—21no—
145C2—162no—
146C2#147111yescharacteristic
147C2#212811yescharacteristic
148C2C2×#35511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC2×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1467C2semidirect
3. #1461C2semidirect
4. #1318C2semidirect

Maximal subgroups

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

subgroupindex
#25482
#14672
#14612
#13182