Finite Group Databasebeta

← #11721

Subgroups — #11721

Order 256.

226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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#2396C211yesmaximal, characteristic
2#1742C211yesmaximal, characteristic
3#1458C211yesmaximal, characteristic
4#1354C212yesmaximal, complemented
5#1353C212yesmaximal, complemented
6C22×M16—21no—
7C2×C4×Q8C411yescharacteristic
8C2×#117C2211yescharacteristic
9#431—21no—
10#400C2211yescharacteristic
11#399C411yescharacteristic
12#398—21no—
13#396—21no—
14#395C2211yescharacteristic
15#393C411yescharacteristic
16C4×C4⋊C4—21no—
17C4×C4⋊C4C411yescharacteristic
18C4×C22⋊C4—21no—
19#343—21no—
20#343C2212yes—
21#324C2212yes—
22#323—21no—
23C22×Q8C2×C411yescharacteristic
24C23×C4D411yescharacteristic
25C2×M16—24no—
26C2×M16—22no—
27C2×M16—41no—
28C2×M16—42no—
29C22×C8—41no—
30C4×Q8—41no—
31C4×Q8—41no—
32#117—41no—
33#117—41no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C4⋊C4C2311yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4C2×C411yescharacteristic
44C2×C22⋊C4D411yescharacteristic
45C2×C42—21no—
46C2×C42—21no—
47C2×C42C2×C411yescharacteristic
48C2×C42D411yescharacteristic
49C4⋊C8—41no—
50C22⋊C8—22no—
51C4×C8—41no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95C2×C411yescharacteristic
57#95C2×C411yescharacteristic
58C24C22⋊C411yescharacteristic
59C2×Q8—42no—
60C2×Q8—41no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4SD1612yes—
72C22×C4Q1612yes—
73C22×C4C22×C411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76M16—24no—
77M16—44no—
78M16—42no—
79C2×C8—42no—
80C2×C8—44no—
81C2×C8—42no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—42no—
92C22⋊C4—42no—
93C42—21no—
94C42—21no—
95C42—21no—
96C42—21no—
97C42—41no—
98C42—41no—
99C42—42no—
100C23—21no—
101C23—21no—
102C23—41no—
103C23—41no—
104C23#10312yes—
105C23C2×C22⋊C411yescharacteristic
106Q8—42no—
107Q8—42no—
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—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4#9911yescharacteristic
135C2×C4#9911yescharacteristic
136C2×C4#10312yes—
137C2×C4#10412yes—
138C2×C4C2×SD1611yescharacteristic
139C2×C4C2×Q1611yescharacteristic
140C8—44no—
141C8—82no—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—42no—
146C22—41no—
147C22—41no—
148C22—81no—
149C22C2×#9911yescharacteristic
150C22C2×#10311yescharacteristic
151C22C2×#10411yescharacteristic
152C4—21no—
153C4—21no—
154C4—42no—
155C4—41no—
156C4—42no—
157C4—41no—
158C4—41no—
159C4—41no—
160C4—81no—
161C4—81no—
162C4#32812yes—
163C4#32812yes—
164C2—21no—
165C2—21no—
166C2—42no—
167C2C2×#32811yescharacteristic
168C2#212011yescharacteristic
169C2#213211yescharacteristic
subgroupG/H
centerC2×C4#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
socleC22C2×#99
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
4. #1354C2semidirect
5. #1353C2semidirect

Maximal subgroups

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

subgroupindex
#23962
#17422
#14582
#13542
#13532