Finite Group Databasebeta

← #10259

Subgroups — #10259

Order 256.

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

normal subgroups only

This group has 237 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#3249C211yesmaximal, characteristic
2#3078C212yesmaximal
3#1984C211yesmaximal, characteristic
4#1493C212yesmaximal
5#1292C211yesmaximal, characteristic
6#533C2212yes—
7C2×C4×Q8C2211yescharacteristic
8#479—24no—
9#477—22no—
10#477—22no—
11#475—22no—
12#467—22no—
13#462—22no—
14#428—22no—
15#423C2212yes—
16C2×C4⋊C8C2211yescharacteristic
17#419—22no—
18C2×C4×C8C2211yescharacteristic
19#336—22no—
20Q8⋊C8—24no—
21C22×Q8D411yescharacteristic
22C2×Q16—22no—
23C2×M16—22no—
24C22×C8—21no—
25C22×C8D412yes—
26C4⋊Q8D412yes—
27C4⋊Q8D412yes—
28#125—22no—
29C22⋊Q8—22no—
30C22⋊Q8—22no—
31C4×Q8—22no—
32C4×Q8—22no—
33C4×Q8—22no—
34C4×Q8—22no—
35#117—22no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4D411yescharacteristic
38C2×C42—21no—
39C2×C42C2311yescharacteristic
40#107—24no—
41#107—22no—
42#106—24no—
43C4⋊C8—22no—
44C4⋊C8—22no—
45C4⋊C8D414yes—
46#103—24no—
47#103—24no—
48#103—24no—
49C4×C8—22no—
50C2×Q8—21no—
51C2×Q8—22no—
52C2×Q8—22no—
53C2×Q8—22no—
54C2×Q8—22no—
55C2×Q8Q1612yescomplemented
56C22×C4—21no—
57C22×C4—21no—
58C22×C4C2×D411yescharacteristic
59C22×C4C2×D411yescharacteristic
60C22×C4#4111yescharacteristic
61Q16—82no—
62M16—42no—
63M16—42no—
64C2×C8—22no—
65C2×C8—22no—
66C2×C8—24no—
67C2×C8—41no—
68C2×C8—41no—
69C2×C8—42no—
70C4⋊C4—22no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—22no—
75C4⋊C4—42no—
76C4⋊C4—42no—
77C4⋊C4—42no—
78C22⋊C4—42no—
79C42—41no—
80C42—42no—
81C42—41no—
82C42C2×D412yes—
83C42C2×D412yes—
84C23#12111yescharacteristic
85Q8—24no—
86Q8—44no—
87Q8—42no—
88Q8—42no—
89Q8—42no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—22no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—42no—
97C2×C4—41no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4#12012yes—
101C2×C4#12011yescharacteristic
102C2×C4#12112yes—
103C2×C4#12711yescharacteristic
104C2×C4C2×Q1611yescharacteristic
105C2×C4#13511yescharacteristic
106C2×C4#13712yes—
107C8—42no—
108C8—44no—
109C8—42no—
110C22—21no—
111C22—21no—
112C22#39211yescharacteristic
113C22#46711yescharacteristic
114C22#47011yescharacteristic
115C4—21no—
116C4—22no—
117C4—42no—
118C4—82no—
119C4—81no—
120C4—82no—
121C4#45112yes—
122C4#45212yes—
123C4#45511yescharacteristic
124C4#45611yescharacteristic
125C2—21no—
126C2—21no—
127C2#200111yescharacteristic
128C2#200711yescharacteristic
129C2#201711yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC23#121
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
55. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#32492
#30782
#19842
#14932
#12922