Finite Group Databasebeta

← #7632

Subgroups — #7632

Order 256.

295 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 71 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#2296C211yesmaximal, characteristic
2#1728C212yesmaximal
3#1271C214yesmaximal, complemented
4C22×M16—22no—
5C2×#117C411yescharacteristic
6C2×#117C411yescharacteristic
7C22×C42C2211yescharacteristic
8#389C412yes—
9#389C2212yes—
10#385—22no—
11C4×C4⋊C4—22no—
12#336—22no—
13#336C2214yes—
14C23×C4C2×C411yescharacteristic
15C23×C4D412yes—
16C2×M16—28no—
17C2×M16—44no—
18C2×M16—44no—
19C22×C8—24no—
20#117—24no—
21#117—42no—
22C2×C4⋊C4—22no—
23C2×C4⋊C4—22no—
24C2×C4⋊C4C2×C412yes—
25C2×C4⋊C4C2×C412yes—
26C2×C22⋊C4—21no—
27C2×C22⋊C4—21no—
28C2×C42—21no—
29C2×C42—21no—
30C2×C42—42no—
31C2×C42C2×C411yescharacteristic
32C2×C42D412yes—
33C2×C42C2311yescharacteristic
34#95—22no—
35#95—22no—
36C24C22⋊C411yescharacteristic
37C22×C4—22no—
38C22×C4—24no—
39C22×C4—22no—
40C22×C4—24no—
41C22×C4—22no—
42C22×C4—22no—
43C22×C4—22no—
44C22×C4—22no—
45C22×C4—22no—
46C22×C4C22⋊C411yescharacteristic
47C22×C4C22⋊C412yes—
48C22×C4C22×C411yescharacteristic
49C22×C4C2×D412yes—
50M16—416no—
51C2×C8—28no—
52C2×C8—44no—
53C2×C8—44no—
54C4⋊C4—22no—
55C4⋊C4—42no—
56C4⋊C4—44no—
57C4⋊C4—42no—
58C4⋊C4—42no—
59C4⋊C4M1614yescomplemented
60C22⋊C4—24no—
61C22⋊C4—42no—
62C42—24no—
63C42—41no—
64C42—42no—
65C42—41no—
66C42—42no—
67C23—21no—
68C23—21no—
69C23—22no—
70C23—22no—
71C23#9912yes—
72C23C2×C22⋊C411yescharacteristic
73C2×C4—28no—
74C2×C4—28no—
75C2×C4—28no—
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—22no—
80C2×C4—22no—
81C2×C4—21no—
82C2×C4—22no—
83C2×C4—22no—
84C2×C4—24no—
85C2×C4—42no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4#10112yes—
91C2×C4#10414yes—
92C2×C4C2×M1612yes—
93C2×C4AGL(1,ℤ/8)12yes—
94C2×C4#13712yes—
95C8—48no—
96C22—22no—
97C22—24no—
98C22—41no—
99C22—42no—
100C22—41no—
101C22#40711yescharacteristic
102C22C2×#9911yescharacteristic
103C22C2×#10111yescharacteristic
104C22#41712yes—
105C22C2×#10412yes—
106C4—22no—
107C4—22no—
108C4—22no—
109C4—24no—
110C4—22no—
111C4—44no—
112C4—82no—
113C2—42no—
114C2#146311yescharacteristic
115C2#148912yes—
116C2#150412yes—
117C2#152611yescharacteristic
118C2#156111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1271C2semidirect
59. C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
#22962
#17282
#12712