Finite Group Databasebeta

← #9487

Subgroups — #9487

Order 256.

259 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 67 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#2975C211yesmaximal, characteristic
2#2580C211yesmaximal, characteristic
3#1800C211yesmaximal, characteristic
4#1487C212yesmaximal, complemented
5#1286C212yesmaximal, complemented
6C2×C4×Q8C2211yescharacteristic
7C2×C4×D4C2211yescharacteristic
8#436—24no—
9#431C2212yes—
10#423C2212yes—
11C2×#100—21no—
12#407—22no—
13#399C412yes—
14#393C412yes—
15#390—21no—
16#389—21no—
17#385—22no—
18#382C2211yescharacteristic
19#343—21no—
20Q8⋊C8—24no—
21C22×Q8C2×C411yescharacteristic
22C22×D4D411yescharacteristic
23C23×C4D412yes—
24C2×M16—24no—
25C2×M16—22no—
26C4×Q8—22no—
27C4×Q8—22no—
28C4×D4—22no—
29C4×D4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4D411yescharacteristic
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4C2×C412yes—
38C2×C42—21no—
39C2×C42C2311yescharacteristic
40C4⋊C8—22no—
41C4⋊C8—22no—
42C4⋊C8—22no—
43#100—42no—
44C22⋊C8—24no—
45C22⋊C8—22no—
46C4×C8D414yes—
47#95—22no—
48#95—21no—
49#95C2×C412yes—
50C24C22⋊C412yes—
51C2×Q8—22no—
52C2×Q8—42no—
53C2×D4—22no—
54C2×D4—21no—
55C2×D4—22no—
56C2×D4—21no—
57C22×C4—21no—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—21no—
62C22×C4—82no—
63C22×C4C22⋊C412yes—
64C22×C4C22×C411yescharacteristic
65C22×C4C2×D411yescharacteristic
66C22×C4C2×D411yescharacteristic
67M16—42no—
68M16—42no—
69M16—84no—
70C2×C8—24no—
71C2×C8—42no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C4⋊C4—41no—
78C4⋊C4—82no—
79C22⋊C4—24no—
80C22⋊C4—44no—
81C22⋊C4—82no—
82C42—42no—
83C42C2×D412yes—
84C42#4112yes—
85C23—22no—
86C23—22no—
87C23—22no—
88C23—42no—
89C23—42no—
90C23C2×C22⋊C411yescharacteristic
91Q8—24no—
92Q8—42no—
93D4—42no—
94D4—42no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—44no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—82no—
110C2×C4C4×D412yes—
111C2×C4#12011yescharacteristic
112C2×C4#12112yes—
113C2×C4#12311yescharacteristic
114C2×C4AGL(1,ℤ/8)11yescharacteristic
115C2×C4#13712yes—
116C2×C4#13711yescharacteristic
117C8—44no—
118C8—82no—
119C22—21no—
120C22—21no—
121C22—44no—
122C22—42no—
123C22—42no—
124C22—42no—
125C22—82no—
126C22#38611yescharacteristic
127C22#41711yescharacteristic
128C22#41911yescharacteristic
129C4—22no—
130C4—21no—
131C4—21no—
132C4—42no—
133C4—42no—
134C4—82no—
135C4—81no—
136C4#46812yes—
137C4#47012yes—
138C2—21no—
139C2—21no—
140C2—82no—
141C2#189111yescharacteristic
142C2#189411yescharacteristic
143C2#191111yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ 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
4. #1487C2semidirect
5. #1286C2semidirect

Maximal subgroups

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

subgroupindex
#29752
#25802
#18002
#14872
#12862