Finite Group Databasebeta

← #12627

Subgroups — #12627

Order 256.

282 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 81 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#2326C211yesmaximal, characteristic
2C2×#354C212yesmaximal
3C4×#99C212yesmaximal
4#1393C212yesmaximal
5C2×#124C2211yescharacteristic
6C2×C4×Q8C411yescharacteristic, complemented
7C2×C4×D4C2211yescharacteristic
8C2×#99C2214yes—
9#391C411yescharacteristic, complemented
10#390C2211yescharacteristic
11#388—22no—
12#385—22no—
13#384C411yescharacteristic, complemented
14C4×C22⋊C4—22no—
15#376C411yescharacteristic, complemented
16#354C418yescomplemented
17#342—22no—
18C22×Q8C2×C411yescharacteristic
19C22×D4C2311yescharacteristic
20C23×C4D412yes—
21#124—22no—
22#124C2×C414yes—
23C4×Q8—42no—
24C4×D4—42no—
25C2×C4⋊C4—21no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C22⋊C4—22no—
30C2×C22⋊C4—24no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4D412yes—
33C2×C42—21no—
34C2×C42—42no—
35C2×C42C2×C411yescharacteristic
36C2×C42C2×C411yescharacteristic
37C2×C42C2×C411yescharacteristic
38#99—24no—
39#99C2×C418yes—
40#95—21no—
41#95—22no—
42#95—21no—
43#95—42no—
44C24C2×D412yes—
45C2×Q8—21no—
46C2×Q8—41no—
47C2×Q8—41no—
48C2×Q8C4212yescomplemented
49C2×D4—41no—
50C2×D4—41no—
51C2×D4C22×C412yes—
52C2×D4#4112yes—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—44no—
60C22×C4—82no—
61C22×C4C22⋊C411yescharacteristic
62C22×C4C22⋊C412yes—
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22×C411yescharacteristic
65C4⋊C4—41no—
66C4⋊C4—41no—
67C4⋊C4—42no—
68C4⋊C4—82no—
69C22⋊C4—28no—
70C22⋊C4—28no—
71C22⋊C4—42no—
72C22⋊C4—44no—
73C22⋊C4—82no—
74C42—22no—
75C42—21no—
76C42—41no—
77C42—42no—
78C42—84no—
79C42C4212yescomplemented
80C23—22no—
81C23—22no—
82C23—42no—
83C23—42no—
84C23C2×C22⋊C411yescharacteristic
85C23C4×D414yes—
86Q8—42no—
87Q8—42no—
88D4—42no—
89D4—42no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—21no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—44no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—41no—
104C2×C4—48no—
105C2×C4—41no—
106C2×C4—42no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—82no—
111C2×C4—84no—
112C2×C4—84no—
113C2×C4#9912yes—
114C2×C4C2×C4211yescharacteristic
115C2×C4#11711yescharacteristic
116C22—21no—
117C22—21no—
118C22—24no—
119C22—44no—
120C22—42no—
121C22—42no—
122C22—82no—
123C22C4×C22⋊C411yescharacteristic
124C22C2×#9911yescharacteristic
125C22#41011yescharacteristic
126C4—22no—
127C4—21no—
128C4—21no—
129C4—42no—
130C4—42no—
131C4—41no—
132C4—42no—
133C4—81no—
134C4—88no—
135C4—81no—
136C2—21no—
137C2—21no—
138C2—44no—
139C2C4×#9911yescharacteristic
140C2#212711yescharacteristic
141C2#212911yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC2×C4C2×C42
Frattini, prime radical, cosocleC22×D4C23
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×C4×Q8C4semidirect
9. #391C4semidirect
13. #384C4semidirect
15. #376C4semidirect
16. #354C4semidirect
48. C2×Q8C42semidirect
79. C42C42semidirect

Maximal subgroups

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

subgroupindex
#23262
C2×#3542
C4×#992
#13932