Finite Group Databasebeta

← #8913

Subgroups — #8913

Order 256.

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

normal subgroups only

This group has 214 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#1847C212yesmaximal
4C2×#330C211yesmaximal, characteristic
5#1295C212yesmaximal
6C2×#125C2211yescharacteristic
7C2×#107—22no—
8C2×#106—22no—
9C2×#97C2211yescharacteristic
10C2×#97C2212yes—
11#389—22no—
12#387—22no—
13#387—22no—
14#384C2211yescharacteristic
15#382C2212yes—
16#340—22no—
17#330C414yescomplemented
18C22×C8—22no—
19C22×C8—21no—
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#107—44no—
33#106—44no—
34#97—22no—
35#97—22no—
36#97C2×C414yes—
37#95—22no—
38#95—22no—
39#95—22no—
40#95—22no—
41C22×C4—22no—
42C22×C4—22no—
43C22×C4—22no—
44C22×C4—41no—
45C22×C4C2×D411yescharacteristic
46C22×C4C2×D412yes—
47C2×C8—24no—
48C2×C8—24no—
49C2×C8—24no—
50C2×C8—22no—
51C2×C8—41no—
52C2×C8—41no—
53C4⋊C4—24no—
54C4⋊C4—24no—
55C4⋊C4—22no—
56C4⋊C4—42no—
57C4⋊C4—42no—
58C4⋊C4—42no—
59C4⋊C4—42no—
60C4⋊C4—42no—
61C4⋊C4—82no—
62C4⋊C4C4⋊C414yes—
63C42—82no—
64C42C22×C411yescharacteristic
65C42C2×Q811yescharacteristic
66C42#4112yes—
67C23#12011yescharacteristic
68C2×C4—22no—
69C2×C4—21no—
70C2×C4—21no—
71C2×C4—22no—
72C2×C4—24no—
73C2×C4—42no—
74C2×C4—42no—
75C2×C4—42no—
76C2×C4—42no—
77C2×C4—42no—
78C2×C4—42no—
79C2×C4—42no—
80C2×C4—42no—
81C2×C4—82no—
82C2×C4—81no—
83C2×C4C2×C4⋊C411yescharacteristic
84C2×C4C4×D412yes—
85C2×C4C22⋊Q812yes—
86C2×C4#12311yescharacteristic
87C8—28no—
88C8—42no—
89C22#38511yescharacteristic
90C22#45312yes—
91C22#45411yescharacteristic
92C22#45611yescharacteristic
93C22#45612yes—
94C4—21no—
95C4—21no—
96C4—22no—
97C4—22no—
98C4—44no—
99C4—82no—
100C4—82no—
101C4—161no—
102C2#165611yescharacteristic
103C2#166012yes—
104C2#166311yescharacteristic
105C2#185711yescharacteristic
106C2#185912yes—
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
#18472
C2×#3302
#12952