Finite Group Databasebeta

← #8472

Subgroups — #8472

Order 256.

417 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 75 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#3508C211yesmaximal, characteristic
2#1965C212yesmaximal, complemented
3#1561C211yesmaximal, characteristic, complemented
4#1526C211yesmaximal, characteristic, complemented
5#1489C212yesmaximal, complemented
6#559—22no—
7#550—22no—
8#546—24no—
9#540—22no—
10#535C412yes—
11#533C2211yescharacteristic
12#532—22no—
13C2×#127C2211yescharacteristic
14#518C412yes—
15C4×#41C2211yescharacteristic
16#423C2212yes—
17C2×#100—24no—
18C4×M16C2212yes—
19#332—22no—
20#330—22no—
21Q8⋊C8—22no—
22D4⋊C8—22no—
23C2×#41—24no—
24C2×#41C2×C411yescharacteristic
25C22×D4C2×C412yes—
26C22×D4D414yes—
27C2×M16—22no—
28C2×M16—22no—
29C4⋊Q8D412yes—
30#127—22no—
31#127—21no—
32#127—21no—
33#127—22no—
34#125D412yes—
35#124—24no—
36#124—22no—
37#123—28no—
38C22⋊Q8—22no—
39#121—24no—
40#121—28no—
41#121—24no—
42#120—24no—
43C4×Q8—22no—
44C4×Q8D412yes—
45C4×D4—24no—
46C4×D4—22no—
47C4×D4—24no—
48C4×D4—24no—
49C4×D4D412yes—
50#117—21no—
51#117C2×C411yescharacteristic
52#117C2×C412yes—
53C2×C22⋊C4—24no—
54C2×C42—21no—
55C2×C42C2311yescharacteristic
56C4⋊C8—22no—
57C4⋊C8—22no—
58#100—48no—
59#97—22no—
60C4×C8—22no—
61C24C22⋊C414yes—
62#41—42no—
63#41—84no—
64#41—84no—
65C2×Q8—22no—
66C2×Q8C22⋊C411yescharacteristic
67C2×D4—21no—
68C2×D4—28no—
69C2×D4—22no—
70C2×D4—22no—
71C2×D4—24no—
72C2×D4—24no—
73C2×D4—44no—
74C2×D4—44no—
75C2×D4—44no—
76C2×D4C22⋊C411yescharacteristic
77C22×C4—28no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D412yes—
82M16—24no—
83M16—42no—
84M16—44no—
85C2×C8—42no—
86C2×C8—42no—
87C4⋊C4—22no—
88C4⋊C4—22no—
89C4⋊C4—24no—
90C4⋊C4C22⋊C412yes—
91C4⋊C4C22⋊C414yes—
92C22⋊C4—28no—
93C22⋊C4—28no—
94C22⋊C4—21no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42—21no—
98C42—22no—
99C42—22no—
100C42C2×D412yes—
101C42C2×D411yescharacteristic
102C42C2×D411yescharacteristic
103C23—24no—
104C23—21no—
105C23—48no—
106C23—44no—
107C23C2×C22⋊C411yescharacteristic
108Q8—22no—
109Q8—42no—
110Q8—42no—
111D4—22no—
112D4—44no—
113D4—42no—
114D4—44no—
115D4—44no—
116D4—44no—
117D4—84no—
118D4—88no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—88no—
129C2×C4C2×C22⋊C411yescharacteristic
130C2×C4C2×C22⋊C411yescharacteristic
131C2×C4#12012yes—
132C2×C4#12012yes—
133C8—44no—
134C8—82no—
135C22—22no—
136C22—21no—
137C22—42no—
138C22—48no—
139C22—41no—
140C22—44no—
141C22—164no—
142C22#37911yescharacteristic
143C4—22no—
144C4—21no—
145C4—42no—
146C4—44no—
147C4—41no—
148C4#41312yes—
149C4#45312yes—
150C4#45512yes—
151C2—41no—
152C2—81no—
153C2—84no—
154C2#178511yescharacteristic
155C2#179212yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#3508C2
LCSG ▷ C22×C4 ▷ 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
2. #1965C2semidirect
3. #1561C2semidirect
4. #1526C2semidirect
5. #1489C2semidirect

Maximal subgroups

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

subgroupindex
#35082
#19652
#15612
#15262
#14892