Finite Group Databasebeta

← #10030

Subgroups — #10030

Order 256.

260 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 51 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, complemented
2#3110C212yesmaximal
3#1955C211yesmaximal, characteristic
4#1531C212yesmaximal, complemented
5#1290C211yesmaximal, characteristic, complemented
6#533C2212yes—
7C2×C4×D4C2211yescharacteristic
8#484—22no—
9#477—22no—
10#474—22no—
11#467—22no—
12#462—22no—
13#461—22no—
14#450—22no—
15#440—24no—
16#432C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×C22⋊C8—22no—
19C2×#97C2211yescharacteristic
20#330—24no—
21C22×D4D411yescharacteristic
22C23×C4—21no—
23C2×Q16—22no—
24C2×SD16—24no—
25C22×C8—21no—
26C22×C8D412yes—
27C4⋊Q8D412yes—
28C4⋊Q8D412yes—
29#125—22no—
30C22⋊Q8—22no—
31C22⋊Q8—22no—
32C4×Q8—22no—
33C4×Q8—22no—
34C4×D4—22no—
35C4×D4—22no—
36#117—22no—
37C2×C4⋊C4D411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C42C2311yescharacteristic
40#107—24no—
41#107—22no—
42C4⋊C8—22no—
43C4⋊C8—22no—
44#103—24no—
45#103—24no—
46#102—24no—
47C22⋊C8—22no—
48C22⋊C8—44no—
49#97—22no—
50#97D414yes—
51C24—21no—
52C2×Q8—22no—
53C2×Q8—22no—
54C2×D4—22no—
55C2×D4—42no—
56C22×C4—21no—
57C22×C4—42no—
58C22×C4—42no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4C2×D411yescharacteristic
62C22×C4C2×D411yescharacteristic
63C22×C4#4111yescharacteristic
64Q16—82no—
65SD16—48no—
66C2×C8—22no—
67C2×C8—24no—
68C2×C8—22no—
69C2×C8—41no—
70C2×C8—41no—
71C2×C8—42no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—22no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C4⋊C4—42no—
78C4⋊C4—42no—
79C22⋊C4—42no—
80C22⋊C4—42no—
81C42—42no—
82C42C2×D412yes—
83C42C2×D412yes—
84C23—22no—
85C23—42no—
86C23—41no—
87C23—41no—
88C23—42no—
89C23#12111yescharacteristic
90Q8—44no—
91Q8—42no—
92Q8—42no—
93D4—24no—
94D4—42no—
95C2×C4—22no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—41no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—42no—
102C2×C4—41no—
103C2×C4—82no—
104C2×C4—81no—
105C2×C4—81no—
106C2×C4#12012yes—
107C2×C4#12011yescharacteristic
108C2×C4#12112yes—
109C2×C4#12711yescharacteristic
110C8—42no—
111C8—44no—
112C8—42no—
113C22—21no—
114C22—21no—
115C22—44no—
116C22—42no—
117C22—82no—
118C22—81no—
119C22—81no—
120C22#39211yescharacteristic
121C22#47312yes—
122C4—21no—
123C4—22no—
124C4—22no—
125C4—81no—
126C4—82no—
127C4—82no—
128C4#45311yescharacteristic
129C4#45411yescharacteristic
130C2—22no—
131C2—82no—
132C2#200611yescharacteristic
133C2#201011yescharacteristic
134C2#202011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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. #3249C2semidirect
4. #1531C2semidirect
5. #1290C2semidirect

Maximal subgroups

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

subgroupindex
#32492
#31102
#19552
#15312
#12902