Finite Group Databasebeta

← #11778

Subgroups — #11778

Order 256.

223 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 51 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#2536C211yesmaximal, characteristic
2C4×#101C211yesmaximal, characteristic, complemented
3#1467C211yesmaximal, characteristic, complemented
4#1346C212yesmaximal, complemented
5#1345C212yesmaximal
6C2×C22⋊Q8—21no—
7C2×C4×Q8C2211yescharacteristic
8C2×C4×D4C411yescharacteristic
9#432—21no—
10C2×#101—21no—
11C4×M16—21no—
12#398C411yescharacteristic
13#398C2211yescharacteristic
14#394C411yescharacteristic
15#394C2211yescharacteristic
16#388—21no—
17#388—21no—
18#382C411yescharacteristic
19#380—21no—
20#343C2212yes—
21#324—21no—
22#324C2212yes—
23C22×Q8D411yescharacteristic
24C22×D4C2×C411yescharacteristic
25C23×C4—21no—
26C2×M16—22no—
27C22⋊Q8—44no—
28C4×Q8—41no—
29C4×Q8—41no—
30C4×D4—41no—
31C4×D4—41no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4C2311yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C42C2×C411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42D411yescharacteristic
47C4⋊C8—41no—
48#101—24no—
49C22⋊C8—22no—
50#97—41no—
51#97—41no—
52C4×C8—41no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95C2×C411yescharacteristic
57#95C2×C411yescharacteristic
58C24—21no—
59C2×Q8—22no—
60C2×Q8—22no—
61C2×Q8—41no—
62C2×D4—41no—
63C2×D4—42no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22×C411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77M16—24no—
78M16—42no—
79C2×C8—42no—
80C2×C8—42no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—42no—
94C42—21no—
95C42—21no—
96C42—21no—
97C42—21no—
98C42—21no—
99C42—21no—
100C23—22no—
101C23—41no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23C2×C22⋊C411yescharacteristic
108Q8—44no—
109D4—42no—
110D4—42no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
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—84no—
135C2×C4#9911yescharacteristic
136C2×C4#9911yescharacteristic
137C2×C4#10412yes—
138C2×C4AGL(1,ℤ/8)12yes—
139C8—44no—
140C8—82no—
141C22—21no—
142C22—21no—
143C22—42no—
144C22—42no—
145C22—42no—
146C22—81no—
147C22—81no—
148C22—81no—
149C22—81no—
150C22C2×#9911yescharacteristic
151C22#41811yescharacteristic
152C22C2×#10411yescharacteristic
153C4—21no—
154C4—21no—
155C4—22no—
156C4—22no—
157C4—41no—
158C4—41no—
159C4—41no—
160C4—41no—
161C4—81no—
162C4—81no—
163C4—81no—
164C2—21no—
165C2—21no—
166C2—82no—
167C2#150711yescharacteristic
168C2#212011yescharacteristic
169C2#213311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#101C2semidirect
3. #1467C2semidirect
4. #1346C2semidirect

Maximal subgroups

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

subgroupindex
#25362
C4×#1012
#14672
#13462
#13452