Finite Group Databasebeta

← #12632

Subgroups — #12632

Order 256.

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

normal subgroups only

This group has 256 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×#355C212yesmaximal
3C4×#101C212yesmaximal, complemented
4#1394C212yesmaximal, complemented
5C2×#124C2211yescharacteristic
6C2×C4×Q8C2211yescharacteristic
7C2×C4×D4C411yescharacteristic
8C2×#101C2214yes—
9C4×M16—22no—
10#391C2211yescharacteristic
11#390C411yescharacteristic
12#388—22no—
13#385—22no—
14#384C411yescharacteristic
15#376C411yescharacteristic
16#355C418yescomplemented
17#343—22no—
18C22×Q8C2311yescharacteristic
19C22×D4C2×C411yescharacteristic
20C23×C4—21no—
21C2×M16—24no—
22#124—22no—
23#124C2×C414yes—
24C4×Q8—42no—
25C4×D4—42no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4C2×C411yescharacteristic
28C2×C4⋊C4C2×C411yescharacteristic
29C2×C4⋊C4D412yes—
30C2×C22⋊C4—22no—
31C2×C22⋊C4—21no—
32C2×C22⋊C4—22no—
33C2×C42C2×C411yescharacteristic
34C2×C42C2×C411yescharacteristic
35C2×C42C2×C411yescharacteristic
36C2×C42D412yes—
37#101—24no—
38#101C2×C418yescomplemented
39#97—42no—
40C4×C8—42no—
41#95—21no—
42#95—21no—
43#95—22no—
44C24—21no—
45C2×Q8—41no—
46C2×Q8—41no—
47C2×Q8C22×C412yes—
48C2×Q8#4112yes—
49C2×D4—21no—
50C2×D4—41no—
51C2×D4—41no—
52C2×D4C4212yes—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—42no—
57C22×C4—82no—
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C412yes—
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22×C411yescharacteristic
62C22×C4C2×D412yes—
63M16—28no—
64M16—44no—
65C2×C8—44no—
66C4⋊C4—41no—
67C4⋊C4—42no—
68C4⋊C4—41no—
69C4⋊C4—82no—
70C22⋊C4—28no—
71C22⋊C4—42no—
72C22⋊C4—82no—
73C42—22no—
74C42—22no—
75C42—22no—
76C42—21no—
77C42—41no—
78C42C4212yes—
79C23—22no—
80C23—42no—
81C23—42no—
82C23—41no—
83C23—41no—
84C23C2×C22⋊C411yescharacteristic
85Q8—42no—
86Q8—42no—
87D4—42no—
88D4—42no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—21no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—21no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—42no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—84no—
107C2×C4#9912yes—
108C2×C4C2×C4211yescharacteristic
109C2×C4#11711yescharacteristic
110C2×C4C4×D414yes—
111C8—48no—
112C22—21no—
113C22—21no—
114C22—44no—
115C22—42no—
116C22—82no—
117C22—81no—
118C22—81no—
119C22C4×C22⋊C411yescharacteristic
120C22C2×#9911yescharacteristic
121C22#41011yescharacteristic
122C4—22no—
123C4—24no—
124C4—21no—
125C4—21no—
126C4—42no—
127C4—42no—
128C4—41no—
129C4—81no—
130C4—81no—
131C2—21no—
132C2—21no—
133C2—82no—
134C2C4×#9911yescharacteristic
135C2#213311yescharacteristic
136C2#213411yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC2×C4C2×C42
Frattini, cosocleC22×Q8C23
prime radicalC22×D4C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#101C2semidirect
4. #1394C2semidirect
16. #355C4semidirect
38. #101C2×C4semidirect

Maximal subgroups

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

subgroupindex
#23262
C2×#3552
C4×#1012
#13942