Finite Group Databasebeta

← #7646

Subgroups — #7646

Order 256.

245 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 67 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#1817C212yesmaximal
3#1271C214yesmaximal, complemented
4C2×#117C411yescharacteristic
5C2×#117C411yescharacteristic
6C22×C42C2211yescharacteristic
7C2×C22⋊C8—22no—
8#389C412yes—
9#389C2212yes—
10#385—22no—
11C4×C4⋊C4—22no—
12#336—22no—
13#336C2214yes—
14C23×C4C2×C411yescharacteristic
15C23×C4D412yes—
16C22×C8—24no—
17#117—24no—
18#117—42no—
19C2×C4⋊C4—22no—
20C2×C4⋊C4—22no—
21C2×C4⋊C4C2×C412yes—
22C2×C4⋊C4C2×C412yes—
23C2×C22⋊C4—21no—
24C2×C22⋊C4—21no—
25C2×C42—21no—
26C2×C42—21no—
27C2×C42—42no—
28C2×C42C2×C411yescharacteristic
29C2×C42D412yes—
30C2×C42C2311yescharacteristic
31C22⋊C8—44no—
32#95—22no—
33#95—22no—
34C24C22⋊C411yescharacteristic
35C22×C4—22no—
36C22×C4—22no—
37C22×C4—22no—
38C22×C4—24no—
39C22×C4—22no—
40C22×C4—24no—
41C22×C4—22no—
42C22×C4—42no—
43C22×C4C22⋊C412yes—
44C22×C4C22⋊C411yescharacteristic
45C22×C4C22×C411yescharacteristic
46C22×C4C2×D412yes—
47C2×C8—44no—
48C2×C8—44no—
49C2×C8—44no—
50C4⋊C4—22no—
51C4⋊C4—42no—
52C4⋊C4—44no—
53C4⋊C4—44no—
54C4⋊C4M1614yes—
55C22⋊C4—24no—
56C22⋊C4—42no—
57C42—24no—
58C42—41no—
59C42—42no—
60C42—41no—
61C42—42no—
62C23—21no—
63C23—21no—
64C23—22no—
65C23—22no—
66C23#9912yes—
67C23C2×C22⋊C411yescharacteristic
68C2×C4—28no—
69C2×C4—21no—
70C2×C4—21no—
71C2×C4—21no—
72C2×C4—22no—
73C2×C4—21no—
74C2×C4—22no—
75C2×C4—24no—
76C2×C4—22no—
77C2×C4—22no—
78C2×C4—22no—
79C2×C4—48no—
80C2×C4—42no—
81C2×C4—42no—
82C2×C4—42no—
83C2×C4—42no—
84C2×C4—42no—
85C2×C4#10112yes—
86C2×C4C2×M1612yes—
87C2×C4#13514yes—
88C8—84no—
89C22—22no—
90C22—24no—
91C22—41no—
92C22—41no—
93C22—42no—
94C22#40711yescharacteristic
95C22C2×#9911yescharacteristic
96C22C2×#10111yescharacteristic
97C22#41612yes—
98C22#42112yes—
99C4—22no—
100C4—22no—
101C4—24no—
102C4—44no—
103C4—42no—
104C4—82no—
105C2—42no—
106C2#146311yescharacteristic
107C2#148812yes—
108C2#151612yes—
109C2#153711yescharacteristic
110C2#154411yescharacteristic
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

Maximal subgroups

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

subgroupindex
#22962
#18172
#12712