Finite Group Databasebeta

← #8474

Subgroups — #8474

Order 256.

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

normal subgroups only

This group has 341 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#3509C211yesmaximal, characteristic
2#1997C212yesmaximal, complemented
3#1561C211yesmaximal, characteristic, complemented
4#1526C211yesmaximal, characteristic, complemented
5#1489C212yesmaximal
6#557—22no—
7#556—22no—
8#552—24no—
9#541—22no—
10#536C412yes—
11#533—22no—
12#533C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14#519C412yes—
15C4×#41C2211yescharacteristic
16#423C2212yes—
17C4×M16C2212yes—
18#343—24no—
19#332—22no—
20#330—22no—
21Q8⋊C8—22no—
22D4⋊C8—22no—
23C2×#41C2×C411yescharacteristic
24C22×Q8C2×C412yes—
25C2×M16—22no—
26C2×M16—22no—
27C4⋊Q8—22no—
28C4⋊Q8—21no—
29C4⋊Q8—21no—
30C4⋊Q8—22no—
31C4⋊Q8D412yes—
32#126—28no—
33#125—24no—
34#125—28no—
35#125D412yes—
36#124—22no—
37#123—24no—
38C22⋊Q8—24no—
39C22⋊Q8—22no—
40C22⋊Q8—24no—
41C4×Q8—24no—
42C4×Q8—24no—
43C4×Q8—24no—
44C4×Q8—22no—
45C4×Q8D412yes—
46C4×D4—22no—
47C4×D4D412yes—
48#117—24no—
49#117—24no—
50#117—21no—
51#117C2×C411yescharacteristic
52#117C2×C412yes—
53C2×C4⋊C4D414yes—
54C2×C42—21no—
55C2×C42C2311yescharacteristic
56C4⋊C8—22no—
57C4⋊C8—22no—
58#97—22no—
59C4×C8—22no—
60#41—42no—
61C2×Q8—22no—
62C2×Q8—22no—
63C2×Q8—22no—
64C2×Q8—44no—
65C2×Q8C22⋊C411yescharacteristic
66C2×D4—21no—
67C2×D4C22⋊C411yescharacteristic
68C22×C4—21no—
69C22×C4—22no—
70C22×C4C22⋊C414yes—
71C22×C4C22×C411yescharacteristic
72C22×C4C2×D412yes—
73M16—24no—
74M16—42no—
75M16—44no—
76C2×C8—42no—
77C2×C8—42no—
78C4⋊C4—28no—
79C4⋊C4—28no—
80C4⋊C4—24no—
81C4⋊C4—24no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—24no—
85C4⋊C4C22⋊C412yes—
86C4⋊C4C22⋊C414yes—
87C22⋊C4—28no—
88C22⋊C4—21no—
89C22⋊C4—22no—
90C22⋊C4—22no—
91C42—28no—
92C42—21no—
93C42—22no—
94C42—22no—
95C42C2×D412yes—
96C42C2×D411yescharacteristic
97C42C2×D411yescharacteristic
98C23—21no—
99C23C2×C22⋊C411yescharacteristic
100Q8—22no—
101Q8—44no—
102Q8—44no—
103Q8—42no—
104Q8—42no—
105D4—22no—
106D4—42no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—24no—
112C2×C4—24no—
113C2×C4—21no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—44no—
118C2×C4C2×C22⋊C411yescharacteristic
119C2×C4C2×C22⋊C411yescharacteristic
120C2×C4#12012yes—
121C2×C4#12012yes—
122C8—44no—
123C8—82no—
124C22—22no—
125C22—21no—
126C22—42no—
127C22—41no—
128C22#37911yescharacteristic
129C4—22no—
130C4—21no—
131C4—42no—
132C4—44no—
133C4—41no—
134C4—44no—
135C4#41312yes—
136C4#45412yes—
137C4#45612yes—
138C2—41no—
139C2—81no—
140C2#178511yescharacteristic
141C2#179312yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#41C2×C4
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. #1997C2semidirect
3. #1561C2semidirect
4. #1526C2semidirect

Maximal subgroups

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

subgroupindex
#35092
#19972
#15612
#15262
#14892