Finite Group Databasebeta

← #7649

Subgroups — #7649

Order 256.

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

normal subgroups only

This group has 234 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
1C4×#124C211yesmaximal, characteristic
2#1839C212yesmaximal, complemented
3#1272C212yesmaximal, complemented
4#1270C212yesmaximal
5C2×#124C411yescharacteristic
6C2×C4×Q8C411yescharacteristic
7C2×C4×D4C411yescharacteristic
8#391C2211yescharacteristic
9#390C2211yescharacteristic
10#388—22no—
11#382C411yescharacteristic
12C4×C22⋊C4—22no—
13C43C2211yescharacteristic
14#336—22no—
15#336—22no—
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
37#95—22no—
38#95—22no—
39C24—21no—
40C2×Q8—21no—
41C2×Q8—41no—
42C2×Q8—41no—
43C2×Q8M1612yes—
44C2×D4—21no—
45C2×D4—41no—
46C2×D4—41no—
47C2×D4M1612yes—
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—21no—
51C22×C4—44no—
52C22×C4—42no—
53C22×C4C22⋊C412yes—
54C22×C4C22⋊C411yescharacteristic
55C22×C4C22⋊C411yescharacteristic
56C22×C4C22×C411yescharacteristic
57C22×C4C2×D412yes—
58C2×C8—44no—
59C2×C8—44no—
60C2×C8—44no—
61C4⋊C4—41no—
62C4⋊C4—41no—
63C4⋊C4—42no—
64C22⋊C4—28no—
65C22⋊C4—42no—
66C22⋊C4—44no—
67C42—22no—
68C42—21no—
69C42—21no—
70C42—24no—
71C42—42no—
72C42—41no—
73C42—42no—
74C23—22no—
75C23—41no—
76C23—41no—
77C23—42no—
78C23—42no—
79C23C2×C22⋊C411yescharacteristic
80Q8—42no—
81Q8—42no—
82D4—42no—
83D4—42no—
84C2×C4—22no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—22no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—21no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—84no—
102C2×C4#10012yes—
103C2×C4C2×M1611yescharacteristic
104C2×C4C2×M1611yescharacteristic
105C2×C4#13514yes—
106C8—84no—
107C22—42no—
108C22—44no—
109C22—81no—
110C22—82no—
111C22—81no—
112C22#40711yescharacteristic
113C22#41011yescharacteristic
114C22C2×#10011yescharacteristic
115C22#41612yes—
116C22#42112yes—
117C4—21no—
118C4—21no—
119C4—24no—
120C4—22no—
121C4—41no—
122C4—42no—
123C4—42no—
124C4—81no—
125C4—81no—
126C2—82no—
127C2#146611yescharacteristic
128C2#148812yes—
129C2#152112yes—
130C2#153311yescharacteristic
131C2#154311yescharacteristic
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. #1839C2semidirect
3. #1272C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1242
#18392
#12722
#12702