Finite Group Databasebeta

← #11431

Subgroups — #11431

Order 256.

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

normal subgroups only

This group has 301 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#2297C211yesmaximal, characteristic
2#1458C212yesmaximal, complemented
3#1319C212yesmaximal, complemented
4#1318C212yesmaximal, complemented
5C22×#41C411yescharacteristic
6C2×#117C2211yescharacteristic
7C22×C42C411yescharacteristic
8#431—22no—
9#391C411yescharacteristic
10#391C2211yescharacteristic
11#390C411yescharacteristic
12#390C2211yescharacteristic
13#386—22no—
14C4×C22⋊C4—22no—
15#324C2214yes—
16#323—22no—
17C2×#41—22no—
18C2×#41—41no—
19C2×#41—41no—
20C2×#41—42no—
21C2×#41C814yescomplemented
22C22×Q8C2×C411yescharacteristic
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4—21no—
26C23×C4—21no—
27C23×C4C2×C411yescharacteristic
28#117—41no—
29#117—41no—
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C4⋊C4C2311yescharacteristic
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4D412yes—
35C2×C42—21no—
36C2×C42—42no—
37C2×C42C2×C411yescharacteristic
38C2×C42C2×C411yescharacteristic
39C2×C42D412yes—
40C4⋊C8—42no—
41C22⋊C8—24no—
42C4×C8—42no—
43#95—22no—
44#95—22no—
45C24—21no—
46C24C22⋊C411yescharacteristic
47#41—44no—
48#41—44no—
49#41—44no—
50#41—44no—
51C2×Q8—21no—
52C2×Q8—41no—
53C2×Q8—41no—
54C2×Q8C2×C812yes—
55C2×D4—24no—
56C2×D4—21no—
57C2×D4—41no—
58C2×D4—41no—
59C2×D4—42no—
60C2×D4—42no—
61C2×D4C2×C812yes—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—24no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—42no—
75C22×C4C22⋊C412yes—
76C22×C4C22⋊C411yescharacteristic
77C22×C4C2×C812yes—
78C22×C4M1612yes—
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D412yes—
81C2×C8—44no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C22⋊C4—42no—
85C22⋊C4—44no—
86C22⋊C4—44no—
87C42—22no—
88C42—22no—
89C42—41no—
90C42—41no—
91C42—42no—
92C23—21no—
93C23—21no—
94C23—22no—
95C23—41no—
96C23—41no—
97C23—42no—
98C23—42no—
99C23—41no—
100C23—41no—
101C23C22⋊C812yes—
102C23C2×C22⋊C411yescharacteristic
103Q8—42no—
104Q8—42no—
105D4—42no—
106D4—42no—
107D4—48no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—48no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—44no—
127C2×C4—44no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4C22⋊C812yes—
131C2×C4#9911yescharacteristic
132C2×C4#9911yescharacteristic
133C2×C4#10012yes—
134C2×C4C22×C811yescharacteristic
135C2×C4C2×M1611yescharacteristic
136C8—84no—
137C22—21no—
138C22—21no—
139C22—22no—
140C22—42no—
141C22—42no—
142C22—44no—
143C22—41no—
144C22—41no—
145C22—42no—
146C22—82no—
147C22—81no—
148C22—81no—
149C22C2×C22⋊C811yescharacteristic
150C22C2×#9911yescharacteristic
151C22C2×#10011yescharacteristic
152C4—21no—
153C4—21no—
154C4—42no—
155C4—42no—
156C4—42no—
157C4—42no—
158C4—42no—
159C4#32312yes—
160C4#32312yes—
161C2—21no—
162C2—21no—
163C2—42no—
164C2—82no—
165C2C2×#32311yescharacteristic
166C2#212611yescharacteristic
167C2#213211yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC2×C4C22×C8
Frattini, cosocleC2×C4⋊C4C23
socleC22C2×#99
prime radicalC22×#41C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1458C2semidirect
3. #1319C2semidirect
4. #1318C2semidirect
21. C2×#41C8semidirect

Maximal subgroups

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

subgroupindex
#22972
#14582
#13192
#13182