Finite Group Databasebeta

← #12808

Subgroups — #12808

Order 256.

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

normal subgroups only

This group has 210 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#2932C211yesmaximal, characteristic
2#2126C212yesmaximal
3#1812C212yesmaximal, complemented
4#1391C212yesmaximal, complemented
5C2×#131C411yescharacteristic, complemented
6C2×#131C2211yescharacteristic
7#532C2211yescharacteristic
8#435—22no—
9#435—22no—
10#427—22no—
11#423C411yescharacteristic, complemented
12#410C2214yes—
13#408C411yescharacteristic, complemented
14#408C2211yescharacteristic
15C2×#97C411yescharacteristic, complemented
16#354—22no—
17#353—22no—
18H(3,ℤ/4)—22no—
19C2×#41C2×C411yescharacteristic
20C2×#41C2311yescharacteristic
21#131—42no—
22#131—42no—
23C2×M16—21no—
24C2×M16C2×C411yescharacteristic
25C2×M16C2×C411yescharacteristic
26C2×M16D412yes—
27C22×C8—21no—
28C22×C8C2×C411yescharacteristic
29C22×C8C2×C411yescharacteristic
30C22×C8D412yes—
31C4⋊Q8Q811yescharacteristic
32#127D411yescharacteristic
33#124D411yescharacteristic
34#124Q811yescharacteristic
35#121—22no—
36C4×D4—22no—
37#117—24no—
38C2×C42C2×C411yescharacteristic
39#107—42no—
40#106—42no—
41C4⋊C8—22no—
42#99—24no—
43#99—24no—
44C22⋊C8—22no—
45C22⋊C8—22no—
46#97—21no—
47#97—21no—
48#41—41no—
49#41—41no—
50#41—41no—
51#41—41no—
52C2×Q8C4⋊C411yescharacteristic
53C2×Q8C2×Q811yescharacteristic
54C2×D4—21no—
55C2×D4—22no—
56C2×D4C4⋊C411yescharacteristic
57C2×D4C2×D411yescharacteristic
58C2×D4#4112yes—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4C22×C411yescharacteristic
62C22×C4C2×D412yes—
63M16—41no—
64M16—42no—
65M16—42no—
66M16—41no—
67C2×C8—22no—
68C2×C8—22no—
69C2×C8—22no—
70C2×C8—21no—
71C2×C8—21no—
72C2×C8—42no—
73C2×C8—41no—
74C2×C8C22⋊C412yes—
75C2×C8C22⋊C411yescharacteristic
76C2×C8C22⋊C411yescharacteristic
77C4⋊C4—22no—
78C4⋊C4—44no—
79C22⋊C4—22no—
80C22⋊C4—22no—
81C22⋊C4—44no—
82C42—21no—
83C42—44no—
84C42C4⋊C411yescharacteristic
85C42C4⋊C411yescharacteristic
86C23—21no—
87C23#11711yescharacteristic
88C23#12112yes—
89Q8—41no—
90Q8—41no—
91D4—41no—
92D4—42no—
93D4—41no—
94D4—42no—
95D4—82no—
96C2×C4—22no—
97C2×C4—21no—
98C2×C4—22no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—81no—
106C2×C4—84no—
107C2×C4C2×C22⋊C411yescharacteristic
108C2×C4C2×C4⋊C411yescharacteristic
109C2×C4C22⋊Q812yes—
110C8—21no—
111C8—22no—
112C8—41no—
113C8—41no—
114C8#9912yes—
115C22—22no—
116C22—21no—
117C22—41no—
118C22—42no—
119C22—42no—
120C22—41no—
121C22#38011yescharacteristic
122C4—21no—
123C4—22no—
124C4—41no—
125C4—41no—
126C4—41no—
127C4—164no—
128C4C2×#9911yescharacteristic
129C4#41011yescharacteristic
130C2—21no—
131C2—42no—
132C2—41no—
133C2—81no—
134C2#179911yescharacteristic
subgroupG/H
centerC4#410
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
socleC2#1799
prime radical#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1812C2semidirect
4. #1391C2semidirect
5. C2×#131C4semidirect
11. #423C4semidirect
13. #408C4semidirect
15. C2×#97C4semidirect

Maximal subgroups

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

subgroupindex
#29322
#21262
#18122
#13912