Finite Group Databasebeta

← #13465

Subgroups — #13465

Order 256.

192 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 49 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#2809C211yesmaximal, characteristic
2#1857C212yesmaximal
3#1857C211yesmaximal, characteristic
4#1394C211yesmaximal, characteristic
5#1394C212yesmaximal
6#519C2211yescharacteristic
7#428—22no—
8#428—21no—
9C2×#101C2212yes—
10C2×#101C2211yescharacteristic
11#400—21no—
12#398—22no—
13#391C2211yescharacteristic
14#391C2212yes—
15#389—21no—
16#387—22no—
17#339—22no—
18#339—21no—
19C22×Q8C2311yescharacteristic
20C2×M16—22no—
21C2×M16—21no—
22C4×Q8—41no—
23C4×Q8—41no—
24#117—22no—
25#117—21no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4D412yes—
31C2×C4⋊C4D411yescharacteristic
32C2×C42D412yes—
33C2×C42D411yescharacteristic
34#107—42no—
35#107—41no—
36#106—42no—
37#106—41no—
38#101—22no—
39#101—21no—
40#101Q814yes—
41#101Q812yes—
42#95—21no—
43#95—22no—
44#95—22no—
45#95—22no—
46#95—22no—
47C2×Q8—41no—
48C2×Q8—41no—
49C2×Q8C2×Q812yes—
50C2×Q8C2×Q811yescharacteristic
51C2×Q8#4111yescharacteristic
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4C2×D412yes—
57C22×C4C2×D411yescharacteristic
58M16—24no—
59M16—22no—
60M16—41no—
61M16—42no—
62C2×C8—42no—
63C2×C8—41no—
64C4⋊C4—22no—
65C4⋊C4—21no—
66C4⋊C4—21no—
67C4⋊C4—22no—
68C4⋊C4—42no—
69C4⋊C4—41no—
70C4⋊C4—44no—
71C4⋊C4—42no—
72C4⋊C4—81no—
73C4⋊C4—81no—
74C22⋊C4—41no—
75C42—41no—
76C42—42no—
77C42—42no—
78C42—41no—
79C23#12011yescharacteristic
80Q8—42no—
81Q8—41no—
82Q8—41no—
83C2×C4—22no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—22no—
87C2×C4—22no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4—42no—
91C2×C4—42no—
92C2×C4—42no—
93C2×C4—42no—
94C2×C4—42no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4C22⋊Q812yes—
101C2×C4C22⋊Q811yescharacteristic
102C2×C4C4⋊Q812yes—
103C2×C4C4⋊Q811yescharacteristic
104C8—44no—
105C8—42no—
106C22—21no—
107C22—21no—
108C22#39511yescharacteristic
109C22B(4,𝔽2)11yescharacteristic
110C22#45811yescharacteristic
111C4—22no—
112C4—21no—
113C4—22no—
114C4—21no—
115C4—42no—
116C4—82no—
117C4—81no—
118C4—81no—
119C4—81no—
120C4—82no—
121C2—21no—
122C2—21no—
123C2#206211yescharacteristic
124C2#220311yescharacteristic
125C2#220411yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23
prime radicalC23#120
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#28092
#18572
#13942