Finite Group Databasebeta

← #12292

Subgroups — #12292

Order 256.

155 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 155 conjugacy classes of subgroups — too many to draw. Only its 41 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#3249C211yesmaximal, characteristic
2#2236C212yesmaximal
3#2087C211yesmaximal, characteristic
4#1348C212yesmaximal
5#1322C211yesmaximal, characteristic
6#533—21no—
7#477—22no—
8#467—21no—
9#467C2212yes—
10#462—22no—
11C2×#107C2211yescharacteristic
12C2×C4⋊C8C2211yescharacteristic
13#389—21no—
14#366—22no—
15#358—22no—
16C22⋊C16C2212yes—
17#340—21no—
18#340C2211yescharacteristic
19C2×Q16—21no—
20C2×Q16D412yes—
21C22×C8D411yescharacteristic
22C22×C8C2311yescharacteristic
23C4⋊Q8—21no—
24C4⋊Q8—21no—
25#125—41no—
26C22⋊Q8—22no—
27C22⋊Q8—41no—
28C4×Q8—22no—
29C4×Q8—41no—
30#117—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—41no—
34C2×C42D411yescharacteristic
35C2×C16—22no—
36#107—21no—
37#107—22no—
38#107D412yes—
39C4⋊C8—22no—
40#103—22no—
41#103—22no—
42#95—41no—
43C2×Q8—22no—
44C2×Q8—41no—
45C22×C4—41no—
46C22×C4—41no—
47C22×C4D811yescharacteristic
48C22×C4D811yescharacteristic
49C22×C4C2×D411yescharacteristic
50Q16—42no—
51Q16—81no—
52C2×C8—21no—
53C2×C8—41no—
54C2×C8—41no—
55C2×C8C2×D412yes—
56C4⋊C4—22no—
57C4⋊C4—22no—
58C4⋊C4—41no—
59C4⋊C4—41no—
60C4⋊C4—41no—
61C4⋊C4—42no—
62C4⋊C4—41no—
63C4⋊C4—42no—
64C4⋊C4—41no—
65C4⋊C4—41no—
66C4⋊C4—82no—
67C22⋊C4—42no—
68C42—21no—
69C42—21no—
70C42—42no—
71C16—42no—
72C23C2×D811yescharacteristic
73Q8—42no—
74Q8—42no—
75Q8—82no—
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—41no—
82C2×C4—42no—
83C2×C4—42no—
84C2×C4—42no—
85C2×C4—81no—
86C2×C4—81no—
87C2×C4—81no—
88C2×C4—81no—
89C2×C4—81no—
90C2×C4Q3212yes—
91C2×C4#12011yescharacteristic
92C2×C4AGL(1,ℤ/8)11yescharacteristic
93C8—22no—
94C8—41no—
95C22—21no—
96C22—21no—
97C22#44711yescharacteristic
98C22C2×Q3211yescharacteristic
99C22#51011yescharacteristic
100C4—21no—
101C4—22no—
102C4—41no—
103C4—82no—
104C4—82no—
105C4—82no—
106C4—161no—
107C4#45611yescharacteristic
108C4#45811yescharacteristic
109C2—21no—
110C2—21no—
111C2#161011yescharacteristic
112C2#218811yescharacteristic
113C2#219011yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radicalC23C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#32492
#22362
#20872
#13482
#13222