Finite Group Databasebeta

← #11454

Subgroups — #11454

Order 256.

217 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 71 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#2317C211yesmaximal, characteristic
2#1458C211yesmaximal, characteristic
3#1458C211yesmaximal, characteristic
4#1326C214yesmaximal
5C2×#117C411yescharacteristic
6C2×#117C2211yescharacteristic
7C22×C4⋊C4C411yescharacteristic
8#431—21no—
9#431—21no—
10#389—21no—
11#389—21no—
12#385C412yes—
13#385C2212yes—
14#384—21no—
15#382—21no—
16#324—21no—
17#324—21no—
18#323C2212yes—
19#323C2212yes—
20C23×C4C2×C411yescharacteristic
21C23×C4C2×C412yes—
22#117—22no—
23#117—41no—
24#117—41no—
25#117C814yescomplemented
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—41no—
32C2×C4⋊C4—41no—
33C2×C4⋊C4D411yescharacteristic
34C2×C4⋊C4D411yescharacteristic
35C2×C22⋊C4C2×C412yes—
36C2×C22⋊C4C2×C411yescharacteristic
37C2×C22⋊C4C2311yescharacteristic
38C2×C42—21no—
39C2×C42D411yescharacteristic
40C2×C42D411yescharacteristic
41C4⋊C8—41no—
42C4⋊C8—41no—
43C22⋊C8—22no—
44C22⋊C8—22no—
45C4×C8—41no—
46C4×C8—41no—
47#95—22no—
48#95—22no—
49C24C22×C411yescharacteristic
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—42no—
56C22×C4—82no—
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4C2×C812yes—
62C22×C4M1612yes—
63C22×C4C2×D411yescharacteristic
64C22×C4C2×D411yescharacteristic
65C2×C8—42no—
66C2×C8—42no—
67C4⋊C4—24no—
68C4⋊C4—41no—
69C4⋊C4—41no—
70C4⋊C4—42no—
71C4⋊C4—41no—
72C4⋊C4—41no—
73C4⋊C4—82no—
74C4⋊C4—82no—
75C22⋊C4—22no—
76C22⋊C4—41no—
77C22⋊C4—41no—
78C22⋊C4C2×C814yes—
79C42—24no—
80C42—21no—
81C42—21no—
82C42—21no—
83C42—21no—
84C23—21no—
85C23—21no—
86C23—41no—
87C23—41no—
88C23C2×C22⋊C411yescharacteristic
89C23C22×C811yescharacteristic
90C23C2×M1611yescharacteristic
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—24no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—84no—
112C2×C4C22⋊C812yes—
113C2×C4C22⋊C812yes—
114C2×C4#9911yescharacteristic
115C2×C4#9911yescharacteristic
116C2×C4#10112yes—
117C8—82no—
118C8—82no—
119C22—21no—
120C22—21no—
121C22—21no—
122C22—21no—
123C22—41no—
124C22—41no—
125C22—42no—
126C22—41no—
127C22—41no—
128C22C2×C22⋊C811yescharacteristic
129C22C2×#9911yescharacteristic
130C22C2×#10111yescharacteristic
131C4—22no—
132C4—44no—
133C4—41no—
134C4—41no—
135C4—41no—
136C4—41no—
137C4—82no—
138C4#32412yes—
139C4#32412yes—
140C2—21no—
141C2—21no—
142C2—41no—
143C2—41no—
144C2C2×#32411yescharacteristic
145C2#212211yescharacteristic
146C2#212311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC23C22×C8
Frattini, cosocleC2×C22⋊C4C23
prime radicalC24C22×C4
LCSG ▷ C23 ▷ C22 ▷ 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
25. #117C8semidirect

Maximal subgroups

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

subgroupindex
#23172
#14582
#13262