Finite Group Databasebeta

← #11483

Subgroups — #11483

Order 256.

191 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 63 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#2929C211yesmaximal, characteristic
2#2115C212yesmaximal, complemented
3#1328C212yesmaximal, complemented
4#1320C212yesmaximal, complemented
5C2×#131C2211yescharacteristic
6C23×C8C411yescharacteristic
7#525C411yescharacteristic
8C2×M32—22no—
9#435—22no—
10C8×D4—22no—
11#424C2211yescharacteristic
12#408C411yescharacteristic
13#407C2211yescharacteristic
14#405C411yescharacteristic
15#349—22no—
16#349C2214yes—
17C2×#41C2×C411yescharacteristic
18C23×C4C2×C411yescharacteristic
19#131—41no—
20#131—41no—
21C2×M16C2×C411yescharacteristic
22C2×M16D412yes—
23C2×M16C2311yescharacteristic
24C22×C8—21no—
25C22×C8—21no—
26C22×C8—41no—
27C22×C8—41no—
28C22×C8C2×C411yescharacteristic
29C22×C8C2×C411yescharacteristic
30C22×C8D412yes—
31#123C812yes—
32C22⋊Q8—21no—
33#121—21no—
34#120C812yes—
35C4×D4—22no—
36#117C2×C411yescharacteristic
37M32—24no—
38M32—42no—
39C2×C16—42no—
40C4⋊C8—22no—
41C22⋊C8—22no—
42C22⋊C8—22no—
43#97—21no—
44C4×C8—21no—
45C24C2×C811yescharacteristic
46#41—41no—
47#41—41no—
48C2×Q8M1611yescharacteristic
49C2×D4—22no—
50C2×D4C2×C812yes—
51C2×D4M1611yescharacteristic
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—41no—
55C22×C4—41no—
56C22×C4C22⋊C412yes—
57C22×C4C2×C811yescharacteristic
58C22×C4C22×C411yescharacteristic
59M16—42no—
60M16—41no—
61M16—41no—
62C2×C8—22no—
63C2×C8—21no—
64C2×C8—22no—
65C2×C8—21no—
66C2×C8—21no—
67C2×C8—21no—
68C2×C8—42no—
69C2×C8—41no—
70C2×C8—41no—
71C2×C8—41no—
72C2×C8C22⋊C411yescharacteristic
73C2×C8C22⋊C411yescharacteristic
74C2×C8C2×D412yes—
75C4⋊C4—22no—
76C4⋊C4—21no—
77C22⋊C4—22no—
78C22⋊C4—22no—
79C22⋊C4C2×C812yes—
80C42—21no—
81C16—44no—
82C23—21no—
83C23—41no—
84C23—42no—
85C23C22⋊C812yes—
86C23C22×C811yescharacteristic
87Q8—41no—
88D4—42no—
89D4—41no—
90D4—44no—
91C2×C4—21no—
92C2×C4—22no—
93C2×C4—21no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—21no—
97C2×C4—42no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4C22⋊C812yes—
102C2×C4C2×C22⋊C411yescharacteristic
103C2×C4C2×M1611yescharacteristic
104C8—22no—
105C8—21no—
106C8—41no—
107C8—41no—
108C8#9911yescharacteristic
109C8#9911yescharacteristic
110C22—22no—
111C22—21no—
112C22—22no—
113C22—42no—
114C22—41no—
115C22—41no—
116C22—42no—
117C22—42no—
118C22—41no—
119C22C2×C22⋊C811yescharacteristic
120C4—21no—
121C4—22no—
122C4—42no—
123C4—41no—
124C4C2×#9911yescharacteristic
125C4#41311yescharacteristic
126C2—21no—
127C2—42no—
128C2—42no—
129C2—41no—
130C2#145911yescharacteristic
subgroupG/H
centerC8#99
commutatorC23C22×C8
Frattini, cosocleC2×M16C23
socleC2#1459
prime radical#525C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2115C2semidirect
3. #1328C2semidirect
4. #1320C2semidirect

Maximal subgroups

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

subgroupindex
#29292
#21152
#13282
#13202