Finite Group Databasebeta

← #8915

Subgroups — #8915

Order 256.

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

normal subgroups only

This group has 198 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#2382C211yesmaximal, characteristic
2#1992C211yesmaximal, characteristic
3#1836C212yesmaximal
4C2×#330C211yesmaximal, characteristic
5#1295C212yesmaximal
6C2×#125C2211yescharacteristic
7C2×#97C2211yescharacteristic
8C2×#97C2212yes—
9#389—22no—
10#387—22no—
11#387—22no—
12#384C2211yescharacteristic
13#382C2212yes—
14#340—22no—
15#340—22no—
16#340—22no—
17#330C414yescomplemented
18C22×C8—21no—
19C22×C8—22no—
20#125—22no—
21#125C2×C412yes—
22#125Q812yes—
23C2×C4⋊C4—22no—
24C2×C4⋊C4—21no—
25C2×C4⋊C4—22no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4D412yes—
28C2×C4⋊C4D412yes—
29C2×C4⋊C4D412yes—
30C2×C42—22no—
31C2×C42C2311yescharacteristic
32#97—22no—
33#97—22no—
34#97C2×C414yes—
35#95—22no—
36#95—22no—
37#95—22no—
38#95—22no—
39C22×C4—22no—
40C22×C4—22no—
41C22×C4—22no—
42C22×C4—41no—
43C22×C4C2×D411yescharacteristic
44C22×C4C2×D412yes—
45C2×C8—22no—
46C2×C8—24no—
47C2×C8—41no—
48C2×C8—42no—
49C2×C8—42no—
50C2×C8—41no—
51C4⋊C4—24no—
52C4⋊C4—24no—
53C4⋊C4—22no—
54C4⋊C4—42no—
55C4⋊C4—42no—
56C4⋊C4—42no—
57C4⋊C4—44no—
58C4⋊C4—82no—
59C4⋊C4C4⋊C414yes—
60C42—82no—
61C42C22×C411yescharacteristic
62C42C2×Q811yescharacteristic
63C42#4112yes—
64C23#12011yescharacteristic
65C2×C4—21no—
66C2×C4—22no—
67C2×C4—22no—
68C2×C4—21no—
69C2×C4—24no—
70C2×C4—42no—
71C2×C4—42no—
72C2×C4—42no—
73C2×C4—42no—
74C2×C4—42no—
75C2×C4—42no—
76C2×C4—42no—
77C2×C4—42no—
78C2×C4—81no—
79C2×C4—82no—
80C2×C4C2×C4⋊C411yescharacteristic
81C2×C4C4×D412yes—
82C2×C4C22⋊Q812yes—
83C2×C4#12311yescharacteristic
84C8—42no—
85C8—44no—
86C22#38511yescharacteristic
87C22#45411yescharacteristic
88C22#45412yes—
89C22#45512yes—
90C22#45611yescharacteristic
91C4—21no—
92C4—21no—
93C4—22no—
94C4—22no—
95C4—44no—
96C4—82no—
97C4—82no—
98C4—161no—
99C2#165811yescharacteristic
100C2#166211yescharacteristic
101C2#166412yes—
102C2#185711yescharacteristic
103C2#186012yes—
subgroupG/H
center, socle, prime radicalC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #330C4semidirect

Maximal subgroups

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

subgroupindex
#23822
#19922
#18362
C2×#3302
#12952