Finite Group Databasebeta

← #11508

Subgroups — #11508

Order 256.

287 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 55 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#2311C211yesmaximal, characteristic
2#1463C212yesmaximal, complemented
3#1319C212yesmaximal, complemented
4#1318C212yesmaximal, complemented
5C22×#41C411yescharacteristic
6C2×#117C411yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8#407—22no—
9#391C411yescharacteristic
10#391C2211yescharacteristic
11#390C411yescharacteristic
12#390C2211yescharacteristic
13#386—22no—
14#380—22no—
15#324—22no—
16#324C2214yes—
17C2×#41—22no—
18C2×#41—22no—
19C2×#41—42no—
20C2×#41—42no—
21C22×Q8C2×C411yescharacteristic
22C22×D4—21no—
23C22×D4C2×C411yescharacteristic
24C23×C4—21no—
25C23×C4C2×C411yescharacteristic
26C23×C4D412yes—
27C2×M16—42no—
28#117—42no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—42no—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4D412yes—
33C2×C4⋊C4C2311yescharacteristic
34C2×C22⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C42C2×C411yescharacteristic
38C2×C42C2×C411yescharacteristic
39C22⋊C8—24no—
40C22⋊C8—42no—
41#95—22no—
42#95—22no—
43C24—21no—
44C24C22⋊C411yescharacteristic
45#41—48no—
46#41—48no—
47C2×Q8—21no—
48C2×Q8—41no—
49C2×Q8—41no—
50C2×Q8M1612yescomplemented
51C2×D4—24no—
52C2×D4—21no—
53C2×D4—41no—
54C2×D4—41no—
55C2×D4—42no—
56C2×D4—42no—
57C2×D4M1612yescomplemented
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—24no—
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—22no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—82no—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C412yes—
73C22×C4C22×C411yescharacteristic
74C22×C4C2×D412yes—
75M16—84no—
76C2×C8—44no—
77C4⋊C4—41no—
78C4⋊C4—42no—
79C4⋊C4—41no—
80C4⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—82no—
83C22⋊C4—82no—
84C42—41no—
85C42—41no—
86C23—21no—
87C23—21no—
88C23—22no—
89C23—41no—
90C23—41no—
91C23—42no—
92C23—42no—
93C23—41no—
94C23—41no—
95C23#9912yes—
96C23C2×C22⋊C411yescharacteristic
97Q8—42no—
98Q8—42no—
99D4—42no—
100D4—42no—
101D4—48no—
102C2×C4—24no—
103C2×C4—24no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—48no—
112C2×C4—44no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—82no—
124C2×C4#10012yes—
125C2×C4C2×M1611yescharacteristic
126C2×C4C2×M1611yescharacteristic
127C8—84no—
128C22—21no—
129C22—21no—
130C22—22no—
131C22—22no—
132C22—22no—
133C22—42no—
134C22—42no—
135C22—44no—
136C22—42no—
137C22—82no—
138C22—81no—
139C22—81no—
140C22#40711yescharacteristic
141C22C2×#9911yescharacteristic
142C22C2×#10011yescharacteristic
143C4—22no—
144C4—21no—
145C4—21no—
146C4—42no—
147C4—42no—
148C4—42no—
149C4—81no—
150C4—81no—
151C2—21no—
152C2—21no—
153C2—42no—
154C2—82no—
155C2#146211yescharacteristic
156C2#212711yescharacteristic
157C2#213311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1463C2semidirect
3. #1319C2semidirect
4. #1318C2semidirect
50. C2×Q8M16semidirect
57. C2×D4M16semidirect

Maximal subgroups

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

subgroupindex
#23112
#14632
#13192
#13182