Finite Group Databasebeta

← #9476

Subgroups — #9476

Order 256.

409 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 65 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#2406C211yesmaximal, characteristic
2#2274C211yesmaximal, characteristic, complemented
3#1789C212yesmaximal
4#1463C211yesmaximal, characteristic, complemented
5#1284C212yesmaximal, complemented
6C22×#41C2211yescharacteristic
7C2×#123—21no—
8C2×#117—21no—
9C22×C4⋊C4C2211yescharacteristic
10C2×#104—24no—
11#407C2212yes—
12#395C412yescomplemented
13#394C412yescomplemented
14#386—21no—
15#385—22no—
16#381C2211yescharacteristic
17C4×C4⋊C4—24no—
18C4×C22⋊C4—21no—
19C4×C22⋊C4C2212yes—
20#324—22no—
21C2×#41—22no—
22C2×#41—22no—
23C2×#41—44no—
24C22×Q8C2×C411yescharacteristic
25C22×D4—21no—
26C22×D4C2×C411yescharacteristic
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4C2311yescharacteristic
31C2×M16—22no—
32#123—44no—
33#117—28no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4C2×C412yes—
39C2×C4⋊C4D412yes—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C42—22no—
45C2×C42—22no—
46C2×C42D414yes—
47#104—48no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50#95—22no—
51#95—22no—
52#95—22no—
53#95—21no—
54#95C2×C412yes—
55C24—21no—
56C24C2×D411yescharacteristic
57#41—28no—
58#41—44no—
59#41—44no—
60#41—44no—
61C2×Q8—22no—
62C2×Q8—42no—
63C2×D4—24no—
64C2×D4—22no—
65C2×D4—42no—
66C2×D4—42no—
67C2×D4—41no—
68C2×D4—41no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—24no—
72C22×C4—21no—
73C22×C4—24no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—42no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4C22⋊C412yes—
83C22×C4C22⋊C412yes—
84C22×C4C22×C411yescharacteristic
85C22×C4C2×D412yes—
86C22×C4C2×D411yescharacteristic
87C22×C4#4112yes—
88M16—44no—
89C2×C8—42no—
90C4⋊C4—28no—
91C4⋊C4—24no—
92C4⋊C4—24no—
93C4⋊C4—44no—
94C22⋊C4—28no—
95C22⋊C4—44no—
96C22⋊C4—42no—
97C22⋊C4—44no—
98C42—28no—
99C42—44no—
100C42—82no—
101C23—22no—
102C23—22no—
103C23—22no—
104C23—41no—
105C23—41no—
106C23—42no—
107C23—41no—
108C23—42no—
109C23C2×C22⋊C411yescharacteristic
110C23#12011yescharacteristic
111C23#12311yescharacteristic
112Q8—24no—
113Q8—42no—
114D4—24no—
115D4—42no—
116D4—44no—
117D4—44no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—24no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—24no—
124C2×C4—28no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—48no—
131C2×C4—44no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—44no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—81no—
141C2×C4#10414yescomplemented
142C2×C4C4×D412yes—
143C2×C4#12112yes—
144C8—82no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—44no—
153C22—44no—
154C22—42no—
155C22—82no—
156C22—81no—
157C22—81no—
158C22#38611yescharacteristic
159C22C2×#10412yes—
160C22B(4,𝔽2)12yes—
161C4—22no—
162C4—22no—
163C4—24no—
164C4—48no—
165C4—42no—
166C4—42no—
167C4—82no—
168C2—22no—
169C2—22no—
170C2—41no—
171C2—82no—
172C2#188911yescharacteristic
173C2#189611yescharacteristic
174C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2274C2semidirect
4. #1463C2semidirect
5. #1284C2semidirect
12. #395C4semidirect
13. #394C4semidirect
141. C2×C4#104semidirect

Maximal subgroups

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

subgroupindex
#24062
#22742
#17892
#14632
#12842