Finite Group Databasebeta

← #12800

Subgroups — #12800

Order 256.

262 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 65 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
3#1799C212yesmaximal
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
14#380—22no—
15#376C411yescharacteristic, complemented
16#354—24no—
17#342—22no—
18C22×Q8C2×C411yescharacteristic
19C22×D4C2311yescharacteristic
20C23×C4D412yes—
21#124—22no—
22#124D412yes—
23#124Q812yes—
24C4×Q8—42no—
25C4×D4—42no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—42no—
29C2×C4⋊C4C2×C411yescharacteristic
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C22⋊C4—22no—
32C2×C22⋊C4—24no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4D412yes—
35C2×C42—21no—
36C2×C42C2×C411yescharacteristic
37C2×C42C2×C411yescharacteristic
38C2×C42C2×C411yescharacteristic
39#99—24no—
40#99—24no—
41#95—21no—
42#95—22no—
43#95—21no—
44#95—42no—
45C24C2×D412yes—
46C2×Q8—21no—
47C2×Q8—41no—
48C2×Q8—41no—
49C2×Q8C4⋊C412yescomplemented
50C2×D4—41no—
51C2×D4—41no—
52C2×D4C2×D411yescharacteristic
53C2×D4C2×Q811yescharacteristic
54C2×D4#4112yes—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—44no—
62C22×C4—82no—
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22⋊C412yes—
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22×C411yescharacteristic
67C4⋊C4—41no—
68C4⋊C4—41no—
69C4⋊C4—42no—
70C4⋊C4—82no—
71C4⋊C4—84no—
72C22⋊C4—24no—
73C22⋊C4—24no—
74C22⋊C4—42no—
75C22⋊C4—44no—
76C22⋊C4—44no—
77C22⋊C4—82no—
78C42—22no—
79C42—21no—
80C42—41no—
81C42—42no—
82C42C4⋊C412yescomplemented
83C23—22no—
84C23—22no—
85C23—42no—
86C23—42no—
87C23C2×C22⋊C411yescharacteristic
88C23#12112yes—
89C23C22⋊Q812yes—
90Q8—42no—
91Q8—42no—
92D4—41no—
93D4—41no—
94D4—42no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—22no—
98C2×C4—22no—
99C2×C4—21no—
100C2×C4—22no—
101C2×C4—41no—
102C2×C4—44no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—42no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—82no—
115C2×C4—84no—
116C2×C4—84no—
117C2×C4—84no—
118C2×C4#9912yes—
119C2×C4C2×C4⋊C411yescharacteristic
120C2×C4#11711yescharacteristic
121C22—21no—
122C22—21no—
123C22—22no—
124C22—22no—
125C22—42no—
126C22—42no—
127C22—42no—
128C22—42no—
129C22—82no—
130C22#38011yescharacteristic
131C22C2×#9911yescharacteristic
132C22#41011yescharacteristic
133C4—21no—
134C4—21no—
135C4—22no—
136C4—42no—
137C4—42no—
138C4—42no—
139C4—41no—
140C4—81no—
141C4—81no—
142C4—164no—
143C2—21no—
144C2—21no—
145C2—42no—
146C2—42no—
147C2#179911yescharacteristic
148C2#212711yescharacteristic
149C2#212911yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, prime radical, cosocleC22×D4C23
LCSG ▷ C22×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
49. C2×Q8C4⋊C4semidirect
82. C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#23262
C2×#3542
#17992
#13932