Finite Group Databasebeta

← #7631

Subgroups — #7631

Order 256.

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

normal subgroups only

This group has 365 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#1728C212yesmaximal, complemented
3#1272C212yesmaximal, complemented
4#1270C212yesmaximal, complemented
5C22×#41C411yescharacteristic
6C22×M16—22no—
7C2×#117C411yescharacteristic
8C22×C42C2211yescharacteristic
9#391C411yescharacteristic
10#391C2211yescharacteristic
11#390C411yescharacteristic
12#390C2211yescharacteristic
13#386—22no—
14C4×C22⋊C4—22no—
15#336—22no—
16#336C2214yes—
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—28no—
28C2×M16—44no—
29C2×M16—44no—
30C22×C8—24no—
31#117—42no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C22⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C42—21no—
38C2×C42—42no—
39C2×C42C2×C411yescharacteristic
40C2×C42D412yes—
41C2×C42C2311yescharacteristic
42#95—22no—
43#95—22no—
44C24—21no—
45C24C22⋊C411yescharacteristic
46#41—48no—
47#41—48no—
48C2×Q8—21no—
49C2×Q8—41no—
50C2×Q8—41no—
51C2×Q8M1612yescomplemented
52C2×D4—24no—
53C2×D4—21no—
54C2×D4—41no—
55C2×D4—41no—
56C2×D4—42no—
57C2×D4—42no—
58C2×D4M1612yescomplemented
59C22×C4—24no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—24no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—24no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C412yes—
75C22×C4C22×C411yescharacteristic
76C22×C4C2×D412yes—
77M16—416no—
78C2×C8—28no—
79C2×C8—44no—
80C2×C8—44no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C22⋊C4—42no—
84C22⋊C4—44no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C42—41no—
88C42—42no—
89C42—41no—
90C42—42no—
91C23—21no—
92C23—21no—
93C23—22no—
94C23—22no—
95C23—22no—
96C23—41no—
97C23—41no—
98C23—42no—
99C23—42no—
100C23#9912yes—
101C23C2×C22⋊C411yescharacteristic
102Q8—42no—
103Q8—42no—
104D4—42no—
105D4—42no—
106D4—48no—
107C2×C4—28no—
108C2×C4—28no—
109C2×C4—24no—
110C2×C4—24no—
111C2×C4—21no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—48no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4#10012yes—
128C2×C4#10414yes—
129C2×C4C2×M1611yescharacteristic
130C2×C4C2×M1611yescharacteristic
131C2×C4AGL(1,ℤ/8)12yes—
132C2×C4#13712yes—
133C8—48no—
134C22—22no—
135C22—24no—
136C22—41no—
137C22—42no—
138C22—41no—
139C22—42no—
140C22—44no—
141C22—81no—
142C22—82no—
143C22—81no—
144C22#40711yescharacteristic
145C22C2×#9911yescharacteristic
146C22C2×#10011yescharacteristic
147C22#41712yes—
148C22C2×#10412yes—
149C4—22no—
150C4—21no—
151C4—21no—
152C4—22no—
153C4—24no—
154C4—22no—
155C4—42no—
156C4—81no—
157C4—81no—
158C2—42no—
159C2—82no—
160C2#146211yescharacteristic
161C2#148912yes—
162C2#150412yes—
163C2#152511yescharacteristic
164C2#156011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1728C2semidirect
3. #1272C2semidirect
4. #1270C2semidirect
51. C2×Q8M16semidirect
58. C2×D4M16semidirect

Maximal subgroups

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

subgroupindex
#22972
#17282
#12722
#12702