Finite Group Databasebeta

← #12203

Subgroups — #12203

Order 256.

266 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 37 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#2483C211yesmaximal, characteristic
2#2029C211yesmaximal, characteristic
3#1608C212yesmaximal, complemented
4#1344C212yesmaximal, complemented
5#1316C211yesmaximal, characteristic
6#543C2211yescharacteristic
7C22×C22⋊C4—21no—
8#483—22no—
9#397—21no—
10#397C2212yes—
11#385—22no—
12#381—22no—
13#379—21no—
14#379C2211yescharacteristic
15#342—21no—
16#342C2211yescharacteristic
17#328—22no—
18#323C2212yes—
19C25D411yescharacteristic
20C23×C4—21no—
21C23×C4—21no—
22C22⋊Q8—22no—
23C22⋊Q8—42no—
24C2×C4⋊C4—21no—
25C2×C4⋊C4D412yes—
26C2×C22⋊C4—22no—
27C2×C22⋊C4—22no—
28C2×C22⋊C4—21no—
29C2×C22⋊C4—22no—
30C2×C22⋊C4—21no—
31C2×C22⋊C4—42no—
32C2×C22⋊C4—42no—
33C2×C22⋊C4—42no—
34C2×C22⋊C4D411yescharacteristic
35C2×C22⋊C4C2311yescharacteristic
36#106—42no—
37#103—42no—
38C22⋊C8—22no—
39#95—21no—
40#95—22no—
41#95—22no—
42#95D412yes—
43C24—21no—
44C24—21no—
45C24—41no—
46C24—44no—
47C24—41no—
48C24SD1612yes—
49C24C2×D411yescharacteristic
50C2×Q8—22no—
51C22×C4—22no—
52C22×C4—21no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—42no—
56C22×C4—42no—
57C22×C4—44no—
58C22×C4—41no—
59C22×C4—81no—
60C22×C4—81no—
61C22×C4C2×D412yes—
62C2×C8—42no—
63C4⋊C4—22no—
64C4⋊C4—42no—
65C4⋊C4—42no—
66C4⋊C4—81no—
67C22⋊C4—41no—
68C22⋊C4—44no—
69C22⋊C4—44no—
70C22⋊C4—41no—
71C22⋊C4—41no—
72C22⋊C4—42no—
73C22⋊C4—42no—
74C22⋊C4—41no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C22⋊C4—84no—
78C23—24no—
79C23—22no—
80C23—21no—
81C23—21no—
82C23—42no—
83C23—44no—
84C23—42no—
85C23—42no—
86C23—44no—
87C23—41no—
88C23—41no—
89C23—44no—
90C23—81no—
91C23—84no—
92C23—82no—
93C23#12011yescharacteristic
94C23C2×SD1611yescharacteristic
95C23AGL(1,ℤ/8)11yescharacteristic
96Q8—82no—
97C2×C4—22no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—44no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—84no—
108C2×C4—84no—
109C2×C4—82no—
110C2×C4—81no—
111C2×C4—82no—
112C8—82no—
113C22—22no—
114C22—24no—
115C22—21no—
116C22—21no—
117C22—41no—
118C22—41no—
119C22—42no—
120C22—44no—
121C22—44no—
122C22—42no—
123C22—42no—
124C22—44no—
125C22—81no—
126C22—84no—
127C22—82no—
128C22#45011yescharacteristic
129C22#45511yescharacteristic
130C22B(4,𝔽2)11yescharacteristic
131C4—42no—
132C4—81no—
133C4—82no—
134C4—82no—
135C4—82no—
136C4—161no—
137C2—21no—
138C2—21no—
139C2—41no—
140C2—44no—
141C2—41no—
142C2#160111yescharacteristic
143C2#219712yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radicalC25D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1608C2semidirect
4. #1344C2semidirect

Maximal subgroups

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

subgroupindex
#24832
#20292
#16082
#13442
#13162