Finite Group Databasebeta

← #13201

Subgroups — #13201

Order 256.

210 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 69 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#2930C211yesmaximal, characteristic
2#2153C211yesmaximal, characteristic, complemented
3#2117C212yesmaximal, complemented
4C2×#349C212yesmaximal, complemented
5#2110C211yesmaximal, characteristic, complemented
6C2×#131C2211yescharacteristic
7C22×M16C2211yescharacteristic
8#525C411yescharacteristic
9C2×M32C2212yes—
10C2×M32C2212yes—
11#436—22no—
12#435—22no—
13#424C411yescharacteristic
14#408C411yescharacteristic
15#407C411yescharacteristic
16#405C2211yescharacteristic
17#364—22no—
18#350—24no—
19#349—24no—
20#347—22no—
21C2×#41C2×C411yescharacteristic
22C23×C4C2×C411yescharacteristic
23#131—22no—
24#131—22no—
25C2×M16—22no—
26C2×M16—21no—
27C2×M16—21no—
28C2×M16—21no—
29C2×M16—22no—
30C2×M16C2×C411yescharacteristic
31C2×M16C2×C411yescharacteristic
32C22×C8—21no—
33C22×C8C2×C411yescharacteristic
34C22×C8C2311yescharacteristic
35#123—21no—
36C22⋊Q8—21no—
37#121—21no—
38#120—21no—
39C4×D4—22no—
40#117C2×C411yescharacteristic
41M32—22no—
42M32—22no—
43M32—22no—
44M32D414yescomplemented
45C2×C16—22no—
46C2×C16—22no—
47C4⋊C8—22no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50#97D412yes—
51C4×C8D412yes—
52C24M1611yescharacteristic
53#41—22no—
54#41—22no—
55C2×Q8M1611yescharacteristic
56C2×D4—21no—
57C2×D4—22no—
58C2×D4M1611yescharacteristic
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—22no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4M1611yescharacteristic
65C22×C4C22×C411yescharacteristic
66M16—22no—
67M16—22no—
68M16—42no—
69M16—41no—
70M16—41no—
71M16—41no—
72M16—41no—
73C2×C8—21no—
74C2×C8—21no—
75C2×C8—22no—
76C2×C8—42no—
77C2×C8C22⋊C412yes—
78C2×C8C2×D411yescharacteristic
79C2×C8C2×D412yes—
80C2×C8C2×D411yescharacteristic
81C2×C8#4112yes—
82C4⋊C4—22no—
83C4⋊C4—21no—
84C22⋊C4—22no—
85C22⋊C4—22no—
86C22⋊C4—21no—
87C42C22⋊C412yes—
88C16—24no—
89C16—42no—
90C23—21no—
91C23—21no—
92C23—22no—
93C23—81no—
94C23C2×M1611yescharacteristic
95Q8—22no—
96D4—22no—
97D4—42no—
98D4—82no—
99C2×C4—22no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4C2×C22⋊C411yescharacteristic
109C2×C4C4×D412yes—
110C2×C4C2×M1611yescharacteristic
111C2×C4#13112yes—
112C8—22no—
113C8—41no—
114C8—41no—
115C8#12112yes—
116C8#12411yescharacteristic
117C8#12711yescharacteristic
118C22—41no—
119C22—41no—
120C22—42no—
121C22—41no—
122C22—41no—
123C22—41no—
124C22—82no—
125C22#40711yescharacteristic
126C22#43512yes—
127C4—42no—
128C4—41no—
129C4—41no—
130C4#39011yescharacteristic
131C4#40811yescharacteristic
132C4#43512yes—
133C2—21no—
134C2—22no—
135C2—81no—
136C2—81no—
137C2#195511yescharacteristic
subgroupG/H
centerC4#390
commutatorC22×C4C22×C4
Frattini, cosocleC22×C8C23
socleC2#1955
prime radical#525C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2153C2semidirect
3. #2117C2semidirect
4. C2×#349C2semidirect
5. #2110C2semidirect
44. M32D4semidirect

Maximal subgroups

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

subgroupindex
#29302
#21532
#21172
C2×#3492
#21102