Finite Group Databasebeta

← #8492

Subgroups — #8492

Order 256.

346 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 71 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#3550C211yesmaximal, characteristic
2#1963C211yesmaximal, characteristic
3#1559C212yesmaximal, complemented
4#1536C211yesmaximal, characteristic
5#1506C212yesmaximal, complemented
6#563—22no—
7#559C412yescomplemented
8Q82C412yescomplemented
9#556—22no—
10D4×Q8—22no—
11#542—24no—
12#536—22no—
13#533C2212yes—
14#532C2211yescharacteristic
15#432C2212yes—
16#423C2211yescharacteristic
17C2×#104—22no—
18#376C2211yescharacteristic
19#333—22no—
20#332—22no—
21#329—22no—
22#328—24no—
23C2×#41D412yes—
24C22×Q8—22no—
25C2×M16—21no—
26C4⋊Q8—24no—
27C4⋊Q8—24no—
28C4⋊Q8C2×C411yescharacteristic
29C4⋊Q8D412yes—
30C4⋊Q8D412yes—
31#127C2×C411yescharacteristic
32#126—28no—
33#125—22no—
34#124—24no—
35#124—24no—
36#124D412yes—
37#123—24no—
38#123—24no—
39C22⋊Q8—24no—
40C22⋊Q8—28no—
41C22⋊Q8—22no—
42C22⋊Q8D414yes—
43#121—22no—
44C4×Q8—24no—
45C4×Q8—22no—
46C4×Q8C2×C414yes—
47C4×D4—24no—
48C4×D4—22no—
49#117—22no—
50#117—22no—
51C2×C42—21no—
52C2×C42C2311yescharacteristic
53C4⋊C8—22no—
54C4⋊C8—22no—
55#104—44no—
56C22⋊C8—22no—
57#97—22no—
58#95—22no—
59#41—42no—
60#41—42no—
61C2×Q8—24no—
62C2×Q8—22no—
63C2×Q8—82no—
64C2×Q8—82no—
65C2×Q8C22⋊C414yes—
66C2×Q8C22⋊C412yes—
67C2×D4—22no—
68C2×D4—22no—
69C2×D4C22⋊C412yes—
70C22×C4—24no—
71C22×C4—22no—
72C22×C4—41no—
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D412yes—
75M16—42no—
76C2×C8—42no—
77C2×C8—41no—
78C4⋊C4—28no—
79C4⋊C4—22no—
80C4⋊C4—24no—
81C4⋊C4—22no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—24no—
85C4⋊C4C22⋊C414yes—
86C22⋊C4—24no—
87C22⋊C4—28no—
88C22⋊C4—24no—
89C22⋊C4—24no—
90C42—24no—
91C42—24no—
92C42—22no—
93C42—41no—
94C42C22×C411yescharacteristic
95C42C2×D411yescharacteristic
96C42C2×D412yes—
97C23—22no—
98C23#12011yescharacteristic
99Q8—44no—
100Q8—42no—
101Q8—42no—
102Q8—42no—
103Q8—84no—
104D4—42no—
105D4—44no—
106D4—82no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—24no—
110C2×C4—22no—
111C2×C4—24no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—84no—
117C2×C4—82no—
118C2×C4C2×C22⋊C412yes—
119C2×C4C2×C22⋊C411yescharacteristic
120C2×C4#12012yes—
121C2×C4#12011yescharacteristic
122C8—82no—
123C8—81no—
124C22—21no—
125C22—22no—
126C22—42no—
127C22—44no—
128C22#37911yescharacteristic
129C4—21no—
130C4—22no—
131C4—22no—
132C4—42no—
133C4—44no—
134C4—44no—
135C4—82no—
136C4#45311yescharacteristic
137C4#45511yescharacteristic
138C2—41no—
139C2—82no—
140C2#179211yescharacteristic
141C2#179612yes—
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1559C2semidirect
5. #1506C2semidirect
7. #559C4semidirect
8. Q82C4semidirect

Maximal subgroups

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

subgroupindex
#35502
#19632
#15592
#15362
#15062