Finite Group Databasebeta

← #11530

Subgroups — #11530

Order 256.

222 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 55 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
1C4×#124C211yesmaximal, characteristic
2#1467C212yesmaximal, complemented
3#1324C212yesmaximal, complemented
4#1318C212yesmaximal
5C2×#124C411yescharacteristic
6C2×C4×Q8C2211yescharacteristic
7C2×C4×D4C411yescharacteristic
8#432—22no—
9#391C411yescharacteristic
10#390C2211yescharacteristic
11#388—22no—
12#382C2211yescharacteristic
13C4×C22⋊C4—22no—
14C43C411yescharacteristic
15#324—22no—
16#324C2214yes—
17C22×Q8C2×C411yescharacteristic
18C22×D4C2×C411yescharacteristic
19C23×C4—21no—
20#124—22no—
21#124—22no—
22C4×Q8—41no—
23C4×Q8—41no—
24C4×D4—41no—
25C4×D4—41no—
26C2×C4⋊C4C2×C411yescharacteristic
27C2×C4⋊C4D412yes—
28C2×C4⋊C4C2311yescharacteristic
29C2×C22⋊C4—22no—
30C2×C22⋊C4—21no—
31C2×C22⋊C4—22no—
32C2×C42—21no—
33C2×C42—21no—
34C2×C42C2×C411yescharacteristic
35C2×C42C2×C411yescharacteristic
36C2×C42C2×C411yescharacteristic
37C2×C42D412yes—
38C4⋊C8—42no—
39C22⋊C8—24no—
40#97—42no—
41#95—22no—
42#95—22no—
43C24—21no—
44C2×Q8—21no—
45C2×Q8—21no—
46C2×Q8—41no—
47C2×Q8—41no—
48C2×D4—21no—
49C2×D4—41no—
50C2×D4—41no—
51C2×D4M1612yes—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—44no—
56C22×C4—41no—
57C22×C4—41no—
58C22×C4C22⋊C412yes—
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22×C411yescharacteristic
62C22×C4C2×D412yes—
63C2×C8—44no—
64C4⋊C4—42no—
65C4⋊C4—41no—
66C4⋊C4—41no—
67C22⋊C4—28no—
68C22⋊C4—44no—
69C22⋊C4—42no—
70C42—22no—
71C42—22no—
72C42—21no—
73C42—21no—
74C42—21no—
75C42—41no—
76C42—42no—
77C42—42no—
78C42M1612yes—
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—21no—
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—21no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—22no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—82no—
108C2×C4—82no—
109C2×C4#9911yescharacteristic
110C2×C4#9911yescharacteristic
111C2×C4C2×M1611yescharacteristic
112C2×C4C2×M1611yescharacteristic
113C8—84no—
114C22—21no—
115C22—21no—
116C22—44no—
117C22—42no—
118C22—82no—
119C22—81no—
120C22—81no—
121C22#40711yescharacteristic
122C22C2×#9911yescharacteristic
123C22#41311yescharacteristic
124C4—21no—
125C4—21no—
126C4—21no—
127C4—42no—
128C4—42no—
129C4—42no—
130C4—41no—
131C4—42no—
132C4—81no—
133C4#35512yes—
134C2—21no—
135C2—21no—
136C2—82no—
137C2#146511yescharacteristic
138C2#212611yescharacteristic
139C2C2×#35511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1467C2semidirect
3. #1324C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1242
#14672
#13242
#13182