Finite Group Databasebeta

← #11801

Subgroups — #11801

Order 256.

201 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 49 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#2621C211yesmaximal, characteristic
2#1820C211yesmaximal, characteristic
3#1464C211yesmaximal, characteristic
4#1344C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C22⋊Q8—21no—
7C2×#117C2211yescharacteristic
8#432—21no—
9#398—21no—
10#398C412yescomplemented
11#397—21no—
12#397C2212yes—
13#391—21no—
14#382C411yescharacteristic, complemented
15#381—21no—
16#342—21no—
17#342C2212yes—
18#324—21no—
19#323C2212yes—
20C22×Q8—21no—
21C23×C4D411yescharacteristic
22C4⋊Q8—41no—
23C4⋊Q8—41no—
24C22⋊Q8—44no—
25#117—41no—
26#117—41no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4C2×C412yes—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4D411yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4D411yescharacteristic
36C2×C22⋊C4C2311yescharacteristic
37C2×C42C2×C411yescharacteristic
38C2×C42D411yescharacteristic
39C4⋊C8—41no—
40C22⋊C8—22no—
41#97—41no—
42#95—21no—
43#95—22no—
44#95—22no—
45#95—41no—
46#95—41no—
47#95C2×C412yes—
48C24C2×D411yescharacteristic
49C2×Q8—42no—
50C2×Q8—42no—
51C2×Q8—41no—
52C2×Q8—41no—
53C22×C4—22no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—41no—
59C22×C4—42no—
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22⋊C411yescharacteristic
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22×C411yescharacteristic
65C22×C4C2×D411yescharacteristic
66C2×C8—42no—
67C4⋊C4—41no—
68C4⋊C4—41no—
69C4⋊C4—42no—
70C4⋊C4—42no—
71C4⋊C4—44no—
72C4⋊C4—42no—
73C22⋊C4—41no—
74C22⋊C4—41no—
75C22⋊C4—44no—
76C22⋊C4—42no—
77C22⋊C4—42no—
78C42—21no—
79C42—21no—
80C42—21no—
81C42—21no—
82C23—21no—
83C23—21no—
84C23—41no—
85C23—41no—
86C23C2×C22⋊C411yescharacteristic
87C23AGL(1,ℤ/8)12yes—
88Q8—82no—
89Q8—82no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—82no—
112C2×C4—82no—
113C2×C4—82no—
114C2×C4#9911yescharacteristic
115C2×C4#9911yescharacteristic
116C8—82no—
117C22—21no—
118C22—21no—
119C22—22no—
120C22—42no—
121C22—41no—
122C22—41no—
123C22—81no—
124C22C2×#9911yescharacteristic
125C22#41811yescharacteristic
126C22#42111yescharacteristic
127C4—21no—
128C4—21no—
129C4—41no—
130C4—41no—
131C4—41no—
132C4—41no—
133C4—41no—
134C4—82no—
135C4—82no—
136C4—162no—
137C2—21no—
138C2—21no—
139C2—42no—
140C2#151011yescharacteristic
141C2#212311yescharacteristic
142C2#212911yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. C2×C4⋊Q8C4semidirect
10. #398C4semidirect
14. #382C4semidirect

Maximal subgroups

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

subgroupindex
#26212
#18202
#14642
#13442