Finite Group Databasebeta

← #19080

Subgroups — #19080

Order 256.

468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 115 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#3548C211yesmaximal, characteristic
2#3141C212yesmaximal, complemented
3#3139C211yesmaximal, characteristic, complemented
4#1662C218yesmaximal, complemented
5#1526C211yesmaximal, characteristic, complemented
6#1525C212yesmaximal, complemented
7#562C2214yes—
8#554—22no—
9#550—22no—
10#547—22no—
11#540—22no—
12#537—24no—
13#533C2211yescharacteristic
14#532C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16#491C2214yes—
17#489C2218yes—
18#487—22no—
19#487—22no—
20#486—22no—
21#419—22no—
22#417—24no—
23C4×M16C2211yescharacteristic
24C4×M16C2212yes—
25#330C2214yes—
26#329C2218yes—
27C2×#41—24no—
28C2×#41D414yes—
29C22×Q8—21no—
30C2×M16—22no—
31C2×M16—21no—
32C4⋊Q8—22no—
33C4⋊Q8—21no—
34C4⋊Q8D412yes—
35C4⋊Q8D412yes—
36C4⋊Q8C2312yes—
37#127—22no—
38#127D412yes—
39#125—22no—
40#125C2312yes—
41#124C2314yes—
42#123—28no—
43C22⋊Q8—24no—
44C22⋊Q8—28no—
45C22⋊Q8—22no—
46#121—28no—
47#121—24no—
48#121—28no—
49#121—22no—
50#121—22no—
51C4×Q8—22no—
52C4×D4—24no—
53C4×D4—22no—
54C4×D4—22no—
55#117D412yes—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—22no—
58C2×C42C2311yescharacteristic
59#103—28no—
60#103—28no—
61#102—28no—
62#97C2312yes—
63#97C2314yes—
64C4×C8—22no—
65C4×C8—21no—
66#41—48no—
67#41—88no—
68C2×Q8—22no—
69C2×Q8—22no—
70C2×Q8—44no—
71C2×Q8—41no—
72C2×Q8C2×D414yes—
73C2×D4—28no—
74C2×D4—24no—
75C2×D4—22no—
76C2×D4—22no—
77C2×D4C2×D414yes—
78C22×C4—28no—
79C22×C4—24no—
80C22×C4—41no—
81C22×C4C2×D412yes—
82C22×C4C2×D411yescharacteristic
83M16—42no—
84M16—44no—
85C2×C8—24no—
86C2×C8—22no—
87C4⋊C4—28no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—24no—
92C4⋊C4—24no—
93C4⋊C4—42no—
94C4⋊C4C2×D414yes—
95C22⋊C4—28no—
96C22⋊C4—22no—
97C22⋊C4—24no—
98C22⋊C4—24no—
99C42—22no—
100C42C2×D411yescharacteristic
101C42C2×D412yes—
102C42C2411yescharacteristic
103C23—24no—
104C23#12011yescharacteristic
105Q8—44no—
106Q8—44no—
107Q8—44no—
108Q8—82no—
109D4—44no—
110D4—44no—
111D4—44no—
112D4—84no—
113D4—88no—
114C2×C4—22no—
115C2×C4—24no—
116C2×C4—24no—
117C2×C4—24no—
118C2×C4—48no—
119C2×C4—42no—
120C2×C4—88no—
121C2×C4—82no—
122C2×C4#12011yescharacteristic
123C2×C4#12012yes—
124C2×C4C22×D411yescharacteristic
125C2×C4C22×D412yes—
126C8—44no—
127C8—42no—
128C22—22no—
129C22—21no—
130C22—44no—
131C22—44no—
132C22—44no—
133C22C2×#12011yescharacteristic
134C4—22no—
135C4—21no—
136C4—44no—
137C4—44no—
138C4—82no—
139C4#45412yes—
140C4#45514yes—
141C2—41no—
142C2—84no—
143C2C2×#45512yes—
144C2C2×#45411yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3141C2semidirect
3. #3139C2semidirect
4. #1662C2semidirect
5. #1526C2semidirect
6. #1525C2semidirect

Maximal subgroups

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

subgroupindex
#35482
#31412
#31392
#16622
#15262
#15252