Finite Group Databasebeta

← #8494

Subgroups — #8494

Order 256.

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

normal subgroups only

This group has 366 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#3548C211yesmaximal, characteristic
2#1995C211yesmaximal, characteristic, complemented
3#1557C212yesmaximal, complemented
4#1537C211yesmaximal, characteristic, complemented
5#1505C212yesmaximal, complemented
6#562—22no—
7#554—22no—
8#550—22no—
9#547C414yescomplemented
10#540—22no—
11#537—24no—
12#533C2211yescharacteristic
13#532C2212yes—
14#432C2212yes—
15#423C2211yescharacteristic
16#376C2211yescharacteristic
17#339—22no—
18#332—22no—
19#331—22no—
20#330—22no—
21#327—24no—
22C2×#41—24no—
23C2×#41—22no—
24C2×M16—21no—
25C4⋊Q8C2×C412yes—
26C4⋊Q8D412yes—
27#127—22no—
28#127D412yes—
29#125—22no—
30#125D412yes—
31#124—22no—
32#123—28no—
33C22⋊Q8—28no—
34C22⋊Q8—22no—
35C22⋊Q8—24no—
36#121—28no—
37#121—28no—
38#121—24no—
39#121—22no—
40#121D414yes—
41C4×Q8—22no—
42C4×D4—24no—
43C4×D4—22no—
44C4×D4C2×C414yes—
45#117D412yes—
46C2×C4⋊C4—24no—
47C2×C42—21no—
48C2×C42C2311yescharacteristic
49C4⋊C8—22no—
50C4⋊C8—22no—
51C22⋊C8—22no—
52#97—22no—
53#95—22no—
54#41—84no—
55#41—88no—
56C2×Q8—22no—
57C2×Q8—22no—
58C2×D4—28no—
59C2×D4—24no—
60C2×D4—22no—
61C2×D4—22no—
62C2×D4C22⋊C414yes—
63C22×C4—28no—
64C22×C4—24no—
65C22×C4—41no—
66C22×C4C2×D411yescharacteristic
67C22×C4C2×D412yes—
68M16—42no—
69C2×C8—42no—
70C2×C8—41no—
71C4⋊C4—28no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—24no—
75C4⋊C4C22⋊C414yes—
76C4⋊C4C22⋊C414yes—
77C22⋊C4—28no—
78C22⋊C4—22no—
79C22⋊C4—24no—
80C22⋊C4—24no—
81C42—22no—
82C42—22no—
83C42—41no—
84C42C22×C411yescharacteristic
85C42C2×D411yescharacteristic
86C42C2×D412yes—
87C23—24no—
88C23#12011yescharacteristic
89Q8—44no—
90Q8—82no—
91D4—44no—
92D4—42no—
93D4—42no—
94D4—82no—
95D4—84no—
96D4—88no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—24no—
100C2×C4—24no—
101C2×C4—42no—
102C2×C4—48no—
103C2×C4—42no—
104C2×C4—88no—
105C2×C4—82no—
106C2×C4C2×C22⋊C411yescharacteristic
107C2×C4C2×C22⋊C412yes—
108C2×C4#12012yes—
109C2×C4#12011yescharacteristic
110C8—82no—
111C8—81no—
112C22—21no—
113C22—22no—
114C22—44no—
115C22—44no—
116C22—44no—
117C22#37911yescharacteristic
118C4—21no—
119C4—22no—
120C4—22no—
121C4—44no—
122C4—44no—
123C4—82no—
124C4#45411yescharacteristic
125C4#45611yescharacteristic
126C2—41no—
127C2—84no—
128C2#179311yescharacteristic
129C2#179512yes—
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3548C2
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
2. #1995C2semidirect
3. #1557C2semidirect
4. #1537C2semidirect
5. #1505C2semidirect
9. #547C4semidirect

Maximal subgroups

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

subgroupindex
#35482
#19952
#15572
#15372
#15052