Finite Group Databasebeta

← #11769

Subgroups — #11769

Order 256.

152 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 47 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#3173C211yesmaximal, characteristic
2#2166C211yesmaximal, characteristic, complemented
3#1747C211yesmaximal, characteristic, complemented
4#1342C211yesmaximal, characteristic, complemented
5#1341C211yesmaximal, characteristic
6#1340C211yesmaximal, characteristic, complemented
7#1339C211yesmaximal, characteristic
8#542—21no—
9#498C411yescharacteristic, complemented
10#495C411yescharacteristic, complemented
11#489—21no—
12#470C2211yescharacteristic
13#469C2211yescharacteristic
14#465—21no—
15#442C411yescharacteristic, complemented
16#441C411yescharacteristic, complemented
17C4×SD16—21no—
18#405—21no—
19#405C2211yescharacteristic
20#364—21no—
21#349C2211yescharacteristic
22#349C2211yescharacteristic
23#347—21no—
24#338C2211yescharacteristic
25H(3,ℤ/4)—21no—
26H(3,ℤ/4)C2211yescharacteristic
27C2×Q16C2×C411yescharacteristic
28C2×SD16—21no—
29C2×D8C2×C411yescharacteristic
30C2×M16—21no—
31C2×M16C2311yescharacteristic
32C22×C8D411yescharacteristic
33C4⋊Q8—21no—
34#126—41no—
35#124—21no—
36#124—41no—
37#123—41no—
38C22⋊Q8—21no—
39C22⋊Q8—41no—
40#121—21no—
41C4×Q8—21no—
42C4×D4—21no—
43#117—21no—
44#117D411yescharacteristic
45M32—21no—
46M32—21no—
47#106—21no—
48#106C2×C411yescharacteristic
49#106C2×C411yescharacteristic
50#103—21no—
51#103—21no—
52#102—21no—
53#102—21no—
54#97—22no—
55#97C2×C411yescharacteristic
56#97D411yescharacteristic
57C4×C8—22no—
58C4×C8C2×C411yescharacteristic
59C4×C8D411yescharacteristic
60C2×Q8—21no—
61C2×Q8—41no—
62C2×D4—21no—
63C2×D4—41no—
64C22×C4—41no—
65C22×C4C2×D411yescharacteristic
66Q16—41no—
67SD16—41no—
68SD16—41no—
69D8—41no—
70M16—41no—
71M16—41no—
72M16—41no—
73C2×C8—21no—
74C2×C8—21no—
75C2×C8—22no—
76C2×C8C22⋊C411yescharacteristic
77C2×C8C22⋊C411yescharacteristic
78C2×C8C22×C411yescharacteristic
79C2×C8C2×D411yescharacteristic
80C4⋊C4—21no—
81C4⋊C4—21no—
82C4⋊C4—21no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C22⋊C4—21no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C42—22no—
95C42—41no—
96C42C22⋊C411yescharacteristic
97C42C22⋊C411yescharacteristic
98C16—41no—
99C16—41no—
100C23—41no—
101C23AGL(1,ℤ/8)11yescharacteristic
102Q8—41no—
103Q8—41no—
104Q8—161no—
105D4—41no—
106D4—41no—
107D4—161no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—42no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—161no—
117C2×C4C2×C22⋊C411yescharacteristic
118C2×C4AGL(1,ℤ/8)11yescharacteristic
119C8—21no—
120C8—21no—
121C8—21no—
122C8—42no—
123C8#10411yescharacteristic
124C8#10411yescharacteristic
125C22—21no—
126C22—41no—
127C22—81no—
128C22—81no—
129C22—81no—
130C22#41811yescharacteristic
131C4—21no—
132C4—41no—
133C4—41no—
134C4—81no—
135C4—81no—
136C4—82no—
137C4—81no—
138C4#41011yescharacteristic
139C4C2×#10411yescharacteristic
140C2—21no—
141C2—41no—
142C2—161no—
143C2#150511yescharacteristic
subgroupG/H
center, socleC2#1505
commutatorC2×C8C22×C4
Frattini, cosocleC2×M16C23
prime radical#469C22
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2166C2semidirect
3. #1747C2semidirect
4. #1342C2semidirect
6. #1340C2semidirect
9. #498C4semidirect
10. #495C4semidirect
15. #442C4semidirect
16. #441C4semidirect

Maximal subgroups

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

subgroupindex
#31732
#21662
#17472
#13422
#13412
#13402
#13392