Finite Group Databasebeta

← #13604

Subgroups — #13604

Order 256.

161 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 45 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#3142C211yesmaximal, characteristic
2#2132C211yesmaximal, characteristic
3#2126C211yesmaximal, characteristic
4#1414C214yesmaximal, complemented
5#532C2211yescharacteristic
6C2×#125C411yescharacteristic, complemented
7#491—21no—
8#490—21no—
9#488C412yescomplemented
10#488C2212yes—
11#417—21no—
12#417—21no—
13#413—21no—
14#410—21no—
15C4×M16C411yescharacteristic, complemented
16#356—21no—
17#355C2212yes—
18#354C2212yes—
19#353—21no—
20C2×#41D411yescharacteristic
21C2×#41D411yescharacteristic
22C2×M16—21no—
23C2×M16—41no—
24C4⋊Q8D411yescharacteristic
25#127D411yescharacteristic
26#125—21no—
27#125—41no—
28#125C2×C412yes—
29#124C2×C411yescharacteristic
30#124C2311yescharacteristic
31#121—41no—
32C4×D4—41no—
33#117—41no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—41no—
36C2×C42C2×C411yescharacteristic
37#103—22no—
38#103—22no—
39#102—22no—
40#102—22no—
41#101—22no—
42#100—41no—
43#99—22no—
44#99—41no—
45#97—21no—
46C4×C8C2×C412yes—
47#41—41no—
48#41—41no—
49#41—41no—
50#41—41no—
51C2×Q8C22⋊C411yescharacteristic
52C2×Q8C2×D411yescharacteristic
53C2×D4—21no—
54C2×D4—21no—
55C2×D4—41no—
56C2×D4C22⋊C411yescharacteristic
57C2×D4C2×D411yescharacteristic
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—41no—
62C22×C4C22⋊C411yescharacteristic
63M16—41no—
64M16—41no—
65M16—42no—
66M16—81no—
67C2×C8—22no—
68C2×C8—81no—
69C4⋊C4—22no—
70C4⋊C4—41no—
71C4⋊C4—42no—
72C4⋊C4—42no—
73C4⋊C4—41no—
74C4⋊C4—81no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C22⋊C4—42no—
78C42—21no—
79C42—81no—
80C42C22⋊C411yescharacteristic
81C42C22×C411yescharacteristic
82C23—21no—
83C23—21no—
84C23#9911yescharacteristic
85Q8—41no—
86Q8—41no—
87D4—41no—
88D4—41no—
89D4—81no—
90D4—81no—
91D4—82no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—81no—
104C2×C4—82no—
105C2×C4—82no—
106C2×C4#9911yescharacteristic
107C2×C4C2×C22⋊C411yescharacteristic
108C8—42no—
109C8—82no—
110C22—21no—
111C22—41no—
112C22—41no—
113C22—41no—
114C22—81no—
115C22—81no—
116C22C2×#9911yescharacteristic
117C4—21no—
118C4—41no—
119C4—41no—
120C4—41no—
121C4—41no—
122C4—82no—
123C4—162no—
124C4#35411yescharacteristic
125C4#35411yescharacteristic
126C2—21no—
127C2—41no—
128C2—81no—
129C2—81no—
130C2C2×#35411yescharacteristic
subgroupG/H
center, socleC2C2×#354
commutatorC42C22×C4
Frattini, cosocle#124C23
prime radical#532C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G
LECSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1414C2semidirect
6. C2×#125C4semidirect
9. #488C4semidirect
15. C4×M16C4semidirect

Maximal subgroups

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

subgroupindex
#31422
#21322
#21262
#14142