Finite Group Databasebeta

← #9196

Subgroups — #9196

Order 256.

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

normal subgroups only

This group has 239 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#2921C211yesmaximal, characteristic
2#2621C211yesmaximal, characteristic
3#1546C212yesmaximal, complemented
4#1458C211yesmaximal, characteristic
5#1286C212yesmaximal, complemented
6C2×C4⋊Q8C2211yescharacteristic
7C2×C22⋊Q8—21no—
8C2×#117C2211yescharacteristic
9#433—24no—
10#431C2212yes—
11#423C2212yes—
12#407—21no—
13C4×M16—21no—
14#398—21no—
15#398C412yes—
16#397—21no—
17#397C412yes—
18#391—21no—
19#382C2211yescharacteristic
20#332—24no—
21#324—21no—
22#323—21no—
23C22×Q8—21no—
24C23×C4D411yescharacteristic
25C2×M16—22no—
26C4⋊Q8—22no—
27C4⋊Q8—22no—
28C22⋊Q8—44no—
29#117—22no—
30#117—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4C2×C412yes—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4D411yescharacteristic
37C2×C22⋊C4—22no—
38C2×C22⋊C4C2×C411yescharacteristic
39C2×C22⋊C4D411yescharacteristic
40C2×C42D411yescharacteristic
41C2×C42C2311yescharacteristic
42C4⋊C8—22no—
43C4⋊C8—22no—
44C4⋊C8D414yes—
45C22⋊C8—24no—
46C22⋊C8—22no—
47#97—22no—
48C4×C8—22no—
49C4×C8—22no—
50#95—22no—
51#95—22no—
52#95—21no—
53#95C2×C412yes—
54C24C22⋊C411yescharacteristic
55C2×Q8—41no—
56C2×Q8—41no—
57C2×Q8—42no—
58C2×Q8—42no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D411yescharacteristic
71M16—42no—
72M16—42no—
73C2×C8—24no—
74C2×C8—42no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C4⋊C4—24no—
78C4⋊C4—44no—
79C4⋊C4—42no—
80C4⋊C4—42no—
81C22⋊C4—22no—
82C22⋊C4—22no—
83C22⋊C4—44no—
84C42—21no—
85C42—21no—
86C42C2×D412yes—
87C42#4112yes—
88C23—21no—
89C23—21no—
90C23—21no—
91C23—41no—
92C23—41no—
93C23C2×C22⋊C411yescharacteristic
94Q8—82no—
95Q8—82no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4C4×D412yes—
114C2×C4#12011yescharacteristic
115C2×C4#12112yes—
116C2×C4#12311yescharacteristic
117C2×C4#13512yes—
118C2×C4AGL(1,ℤ/8)12yes—
119C8—44no—
120C8—82no—
121C22—21no—
122C22—21no—
123C22—42no—
124C22—41no—
125C22—41no—
126C22—41no—
127C22—81no—
128C22#38611yescharacteristic
129C22#41611yescharacteristic
130C22#41811yescharacteristic
131C4—22no—
132C4—22no—
133C4—41no—
134C4—41no—
135C4—41no—
136C4—82no—
137C4—82no—
138C4#44912yes—
139C4#48212yes—
140C2—21no—
141C2—21no—
142C2—81no—
143C2#187511yescharacteristic
144C2#190911yescharacteristic
145C2#191511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
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
3. #1546C2semidirect
5. #1286C2semidirect

Maximal subgroups

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

subgroupindex
#29212
#26212
#15462
#14582
#12862