Finite Group Databasebeta

← #7635

Subgroups — #7635

Order 256.

260 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×#124C211yesmaximal, characteristic
2#1767C212yesmaximal, complemented
3#1272C212yesmaximal, complemented
4#1270C212yesmaximal
5C2×#124C411yescharacteristic
6C2×C4×Q8C411yescharacteristic
7C2×C4×D4C411yescharacteristic
8C2×C4⋊C8—22no—
9C2×#97—22no—
10#391C2211yescharacteristic
11#390C2211yescharacteristic
12#388—22no—
13#382C411yescharacteristic
14C4×C22⋊C4—22no—
15C43C2211yescharacteristic
16#336C2214yes—
17C22×Q8C2×C411yescharacteristic
18C22×D4C2×C411yescharacteristic
19C23×C4—21no—
20C22×C8—24no—
21#124—22no—
22#124—22no—
23C4×Q8—42no—
24C4×D4—42no—
25C2×C4⋊C4—21no—
26C2×C4⋊C4C2×C411yescharacteristic
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C22⋊C4—22no—
29C2×C22⋊C4—21no—
30C2×C22⋊C4—22no—
31C2×C42—21no—
32C2×C42C2×C411yescharacteristic
33C2×C42C2×C411yescharacteristic
34C2×C42D412yes—
35C2×C42D412yes—
36C2×C42C2311yescharacteristic
37C4⋊C8—28no—
38#97—44no—
39#95—22no—
40#95—22no—
41C24—21no—
42C2×Q8—21no—
43C2×Q8—41no—
44C2×Q8—41no—
45C2×Q8M1612yes—
46C2×D4—21no—
47C2×D4—41no—
48C2×D4—41no—
49C2×D4M1612yes—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—42no—
54C22×C4—44no—
55C22×C4C22⋊C412yes—
56C22×C4C22⋊C411yescharacteristic
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22×C411yescharacteristic
59C22×C4C2×D412yes—
60C2×C8—28no—
61C2×C8—44no—
62C2×C8—44no—
63C4⋊C4—41no—
64C4⋊C4—41no—
65C4⋊C4—42no—
66C22⋊C4—28no—
67C22⋊C4—42no—
68C22⋊C4—42no—
69C22⋊C4—42no—
70C42—22no—
71C42—24no—
72C42—21no—
73C42—21no—
74C42—22no—
75C42—22no—
76C42—41no—
77C42—42no—
78C23—22no—
79C23—41no—
80C23—41no—
81C23—42no—
82C23—42no—
83C23C2×C22⋊C411yescharacteristic
84Q8—42no—
85Q8—42no—
86D4—42no—
87D4—42no—
88C2×C4—22no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—21no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—84no—
105C2×C4#10012yes—
106C2×C4#10414yes—
107C2×C4C2×M1611yescharacteristic
108C2×C4C2×M1611yescharacteristic
109C2×C4AGL(1,ℤ/8)12yes—
110C2×C4#13712yes—
111C8—48no—
112C22—42no—
113C22—44no—
114C22—81no—
115C22—82no—
116C22—81no—
117C22#40711yescharacteristic
118C22#41011yescharacteristic
119C22C2×#10011yescharacteristic
120C22#41712yes—
121C22C2×#10412yes—
122C4—21no—
123C4—21no—
124C4—22no—
125C4—22no—
126C4—41no—
127C4—42no—
128C4—42no—
129C4—81no—
130C4—81no—
131C4#32914yes—
132C2—82no—
133C2#146611yescharacteristic
134C2#148912yes—
135C2#150912yes—
136C2C2×#32911yescharacteristic
137C2#156011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1767C2semidirect
3. #1272C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1242
#17672
#12722
#12702