Finite Group Databasebeta

← #11485

Subgroups — #11485

Order 256.

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

normal subgroups only

This group has 186 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#2931C211yesmaximal, characteristic
2#2115C212yesmaximal, complemented
3#1327C212yesmaximal, complemented
4#1321C212yesmaximal, complemented
5C2×#131C411yescharacteristic
6C2×#131C2211yescharacteristic
7#532C411yescharacteristic
8C2×M32—22no—
9#436—22no—
10C8×D4—22no—
11#423C2211yescharacteristic
12#408C411yescharacteristic
13#408C2211yescharacteristic
14C2×C4×C8C411yescharacteristic
15#349—22no—
16#349C2214yes—
17C2×#41C2×C411yescharacteristic
18C2×#41C2×C411yescharacteristic
19#131—41no—
20#131—41no—
21#131—41no—
22#131—41no—
23C2×M16—21no—
24C2×M16C2×C411yescharacteristic
25C2×M16D412yes—
26C2×M16C2311yescharacteristic
27C22×C8—21no—
28C22×C8C2×C411yescharacteristic
29C22×C8C2×C411yescharacteristic
30C22×C8D412yes—
31C4⋊Q8C811yescharacteristic
32#127C811yescharacteristic
33#124C811yescharacteristic
34#124C811yescharacteristic
35#121—22no—
36C4×D4—22no—
37C2×C42C2×C411yescharacteristic
38M32—24no—
39M32—42no—
40C2×C16—42no—
41C4⋊C8—22no—
42C22⋊C8—22no—
43C22⋊C8—22no—
44C4×C8—21no—
45C4×C8—21no—
46#41—41no—
47#41—41no—
48#41—41no—
49#41—41no—
50C2×Q8C2×C811yescharacteristic
51C2×Q8C2×C811yescharacteristic
52C2×D4—22no—
53C2×D4—21no—
54C2×D4C2×C811yescharacteristic
55C2×D4C2×C811yescharacteristic
56C2×D4M1612yes—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4C22⋊C412yes—
60C22×C4C22×C411yescharacteristic
61M16—42no—
62M16—41no—
63M16—42no—
64M16—41no—
65C2×C8—22no—
66C2×C8—21no—
67C2×C8—22no—
68C2×C8—21no—
69C2×C8—21no—
70C2×C8—21no—
71C2×C8—41no—
72C2×C8—41no—
73C2×C8—41no—
74C2×C8C22⋊C411yescharacteristic
75C2×C8C22⋊C411yescharacteristic
76C2×C8C2×D412yes—
77C4⋊C4—22no—
78C22⋊C4—22no—
79C22⋊C4—22no—
80C42—21no—
81C42C2×C811yescharacteristic
82C42C2×C811yescharacteristic
83C16—44no—
84C23—21no—
85C23C22⋊C812yes—
86C23C2×M1611yescharacteristic
87Q8—41no—
88Q8—41no—
89D4—41no—
90D4—42no—
91D4—41no—
92D4—44no—
93D4—42no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—22no—
97C2×C4—21no—
98C2×C4—22no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—42no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4C22⋊C812yes—
105C2×C4C2×C22⋊C411yescharacteristic
106C2×C4C22×C811yescharacteristic
107C8—22no—
108C8—21no—
109C8—41no—
110C8—41no—
111C8#9911yescharacteristic
112C8#9911yescharacteristic
113C22—21no—
114C22—22no—
115C22—41no—
116C22—42no—
117C22—42no—
118C22—41no—
119C22C2×C22⋊C811yescharacteristic
120C4—22no—
121C4—22no—
122C4—21no—
123C4—41no—
124C4—41no—
125C4C2×#9911yescharacteristic
126C4#41311yescharacteristic
127C2—21no—
128C2—41no—
129C2—42no—
130C2—81no—
131C2#145911yescharacteristic
subgroupG/H
centerC8#99
commutatorC2×C4C22×C8
Frattini, cosocleC2×M16C23
socleC2#1459
prime radical#532C4
LCSG ▷ C2×C4 ▷ 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. #1327C2semidirect
4. #1321C2semidirect

Maximal subgroups

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

subgroupindex
#29312
#21152
#13272
#13212