Finite Group Databasebeta

← #10015

Subgroups — #10015

Order 256.

348 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 61 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#3101C212yesmaximal, complemented
2#2397C211yesmaximal, characteristic
3#1995C211yesmaximal, characteristic, complemented
4#1486C212yesmaximal, complemented
5#1274C211yesmaximal, characteristic, complemented
6#533C2211yescharacteristic
7C2×C4×D4—22no—
8C2×C4×D4C2212yes—
9#483—22no—
10#476—22no—
11#461—22no—
12#450—22no—
13C4×SD16—24no—
14#431C2212yes—
15#423C2211yescharacteristic
16#399—22no—
17#392—22no—
18#376C2211yescharacteristic
19#339—22no—
20D4⋊C8—24no—
21C22×D4—22no—
22C22×D4D412yes—
23C23×C4—22no—
24C2×SD16—24no—
25C2×M16—21no—
26C4⋊Q8D412yescomplemented
27#125—21no—
28C22⋊Q8—22no—
29C4×Q8—22no—
30C4×D4—22no—
31C4×D4—22no—
32C4×D4—22no—
33C4×D4—22no—
34C4×D4—44no—
35C4×D4—44no—
36#117—21no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4D412yes—
39C2×C22⋊C4—24no—
40C2×C22⋊C4—22no—
41C2×C42D412yes—
42C2×C42C2311yescharacteristic
43#106—24no—
44C4⋊C8—22no—
45C4⋊C8—22no—
46#103—24no—
47#102—24no—
48C22⋊C8—22no—
49C4×C8D414yescomplemented
50#95—22no—
51C24—22no—
52C2×Q8—22no—
53C2×D4—22no—
54C2×D4—22no—
55C2×D4—44no—
56C2×D4—44no—
57C2×D4—48no—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—42no—
61C22×C4—42no—
62C22×C4—44no—
63C22×C4—44no—
64C22×C4C2×D412yes—
65C22×C4C2×D411yescharacteristic
66SD16—48no—
67M16—42no—
68C2×C8—24no—
69C2×C8—41no—
70C4⋊C4—22no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—44no—
75C4⋊C4—42no—
76C22⋊C4—48no—
77C22⋊C4—41no—
78C22⋊C4—42no—
79C22⋊C4—42no—
80C42—22no—
81C42—41no—
82C42SD1614yescomplemented
83C42C2×D411yescharacteristic
84C42C2×D412yes—
85C42#4111yescharacteristic
86C23—24no—
87C23—44no—
88C23—42no—
89C23—42no—
90C23—44no—
91C23#12011yescharacteristic
92Q8—44no—
93D4—24no—
94D4—24no—
95D4—42no—
96D4—42no—
97D4—416no—
98C2×C4—24no—
99C2×C4—42no—
100C2×C4—44no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—44no—
104C2×C4—84no—
105C2×C4—82no—
106C2×C4—82no—
107C2×C4#12012yes—
108C2×C4#12112yes—
109C2×C4#12111yescharacteristic
110C2×C4#12711yescharacteristic
111C2×C4C2×SD1612yes—
112C2×C4AGL(1,ℤ/8)12yes—
113C8—44no—
114C8—81no—
115C22—22no—
116C22—48no—
117C22—44no—
118C22—84no—
119C22—82no—
120C22—82no—
121C22#39211yescharacteristic
122C22#45012yes—
123C4—22no—
124C4—22no—
125C4—44no—
126C4—82no—
127C4—82no—
128C4#46012yes—
129C4#49211yescharacteristic
130C4#49611yescharacteristic
131C2—22no—
132C2—84no—
133C2#201012yes—
134C2#201411yescharacteristic
subgroupG/H
centerC2×C4#127
commutator, Frattini, cosocleC2×C42C23
socleC22#392
prime radical#2397C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3101C2semidirect
3. #1995C2semidirect
4. #1486C2semidirect
5. #1274C2semidirect
26. C4⋊Q8D4semidirect
49. C4×C8D4semidirect
82. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#31012
#23972
#19952
#14862
#12742