Finite Group Databasebeta

← #10302

Subgroups — #10302

Order 256.

258 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 53 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#3255C211yesmaximal, characteristic, complemented
2#3254C212yesmaximal
3#1994C212yesmaximal, complemented
4#1962C211yesmaximal, characteristic
5#1309C211yesmaximal, characteristic, complemented
6#533C2212yes—
7#532C2211yescharacteristic
8#477—22no—
9#475—22no—
10#474—22no—
11#470—22no—
12#468—24no—
13#462—22no—
14#461—22no—
15#460—22no—
16#423C2211yescharacteristic
17#423C2212yes—
18C2×#104—22no—
19C43C2211yescharacteristic
20#339—24no—
21C2×#41—21no—
22C2×Q16—22no—
23C2×SD16—24no—
24C2×M16D414yes—
25C2×M16D412yes—
26C4⋊Q8D411yescharacteristic
27C4⋊Q8D412yes—
28C4⋊Q8D412yes—
29#127D411yescharacteristic
30#125—22no—
31#124—21no—
32C22⋊Q8—22no—
33C22⋊Q8—22no—
34#121—22no—
35C4×Q8—22no—
36C4×Q8—22no—
37C4×D4—22no—
38#117—22no—
39C2×C42—22no—
40C2×C42—21no—
41C2×C42C2311yescharacteristic
42#107—24no—
43#106—22no—
44C4⋊C8—22no—
45C4⋊C8—22no—
46C4⋊C8—22no—
47#104—44no—
48#103—24no—
49#103—24no—
50#102—24no—
51#41—42no—
52#41—42no—
53C2×Q8—22no—
54C2×Q8—22no—
55C2×Q8—21no—
56C2×D4—22no—
57C2×D4—21no—
58C2×D4—41no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4C2×D412yes—
62C22×C4C2×D411yescharacteristic
63Q16—82no—
64SD16—84no—
65M16—42no—
66M16—44no—
67C2×C8—24no—
68C2×C8—22no—
69C4⋊C4—22no—
70C4⋊C4—22no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C22⋊C4—42no—
77C22⋊C4—42no—
78C42—22no—
79C42—24no—
80C42—41no—
81C42—42no—
82C42—42no—
83C42C2×D412yes—
84C42C2×D411yescharacteristic
85C42#4111yescharacteristic
86C23—41no—
87C23#12711yescharacteristic
88Q8—42no—
89Q8—42no—
90Q8—42no—
91Q8—42no—
92Q8—42no—
93D4—42no—
94D4—42no—
95D4—42no—
96D4—82no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—42no—
102C2×C4—44no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—41no—
106C2×C4—81no—
107C2×C4—81no—
108C2×C4#12011yescharacteristic
109C2×C4#12012yes—
110C2×C4#12112yes—
111C2×C4#12111yescharacteristic
112C8—42no—
113C8—44no—
114C22—21no—
115C22—22no—
116C22—81no—
117C22—82no—
118C22#39211yescharacteristic
119C4—21no—
120C4—22no—
121C4—44no—
122C4—81no—
123C4—82no—
124C4—82no—
125C4#45311yescharacteristic
126C4#45411yescharacteristic
127C4#45512yes—
128C4#45612yes—
129C2—41no—
130C2—161no—
131C2#200611yescharacteristic
132C2#200712yes—
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×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
1. #3255C2semidirect
3. #1994C2semidirect
5. #1309C2semidirect

Maximal subgroups

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

subgroupindex
#32552
#32542
#19942
#19622
#13092