Finite Group Databasebeta

← #17740

Subgroups — #17740

Order 256.

486 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 179 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#3444C211yesmaximal, characteristic
2#2920C212yesmaximal, complemented
3#1559C214yesmaximal, complemented
4#1557C214yesmaximal, complemented
5#1544C212yesmaximal, complemented
6#1543C212yesmaximal, complemented
7#533C2212yes—
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#532C2212yes—
11#529—22no—
12#529—22no—
13#529C414yes—
14#525—22no—
15#525C412yes—
16C4×#41—22no—
17C4×#41C412yes—
18C22×C42C2211yescharacteristic
19#432—22no—
20#432C2214yes—
21#431—22no—
22#431C2214yes—
23#423C2214yes—
24#407—22no—
25C4×M16—22no—
26#333C2214yescomplemented
27#332C2218yescomplemented
28#331C2214yescomplemented
29C2×#41—22no—
30C2×#41C2×C412yes—
31C23×C4C2×C411yescharacteristic
32C23×C4D412yes—
33C2×M16—24no—
34C4⋊Q8C2311yescharacteristic
35C4⋊Q8C2312yes—
36#127C2311yescharacteristic
37#126—24no—
38#125—22no—
39#125C2×C412yes—
40#124—22no—
41#124C2×C412yes—
42#123—24no—
43#123—22no—
44C22⋊Q8—22no—
45C22⋊Q8—22no—
46C22⋊Q8C2×C414yes—
47#121—22no—
48#121—22no—
49#121C2×C414yes—
50#120—22no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×Q8C2×C414yes—
54C4×D4—24no—
55C4×D4—24no—
56C4×D4—24no—
57C4×D4—22no—
58C4×D4—22no—
59C4×D4C2×C414yes—
60#117—22no—
61#117—24no—
62#117—22no—
63#117C2×C412yes—
64C2×C42—22no—
65C2×C42C2×C411yescharacteristic
66C2×C42C2×C412yes—
67C2×C42D412yes—
68C2×C42D412yes—
69C2×C42D412yes—
70C2×C42C2312yes—
71C2×C42C2311yescharacteristic
72C4⋊C8—24no—
73C4⋊C8C2318yes—
74C22⋊C8—24no—
75C22⋊C8—24no—
76#97—24no—
77C4×C8—24no—
78C24C22⋊C411yescharacteristic
79#41—48no—
80#41—44no—
81C2×Q8—22no—
82C2×Q8C22×C412yes—
83C2×D4—22no—
84C2×D4—24no—
85C2×D4—22no—
86C2×D4C22×C412yes—
87C22×C4—24no—
88C22×C4—22no—
89C22×C4—42no—
90C22×C4—44no—
91C22×C4C22⋊C412yes—
92C22×C4C22⋊C412yes—
93C22×C4C22⋊C412yes—
94C22×C4C22⋊C412yes—
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22×C411yescharacteristic
97C22×C4C22×C412yes—
98C22×C4C2×D412yes—
99C22×C4C2×D412yes—
100C22×C4C2×D412yes—
101M16—48no—
102C2×C8—28no—
103C4⋊C4—24no—
104C4⋊C4—24no—
105C4⋊C4—24no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4C22×C414yes—
109C22⋊C4—24no—
110C22⋊C4—22no—
111C22⋊C4—24no—
112C22⋊C4—24no—
113C22⋊C4—24no—
114C42—24no—
115C42—22no—
116C42C22⋊C412yes—
117C42C22⋊C412yes—
118C42C22⋊C411yescharacteristic
119C42C22⋊C411yescharacteristic
120C42C22×C412yes—
121C42C22×C411yescharacteristic
122C42C2×D412yes—
123C42C2×D412yes—
124C42C2×D412yes—
125C42C2411yescharacteristic
126C23—22no—
127C23—41no—
128C23—42no—
129C23C2×C22⋊C411yescharacteristic
130C23C2×C22⋊C412yes—
131Q8—42no—
132Q8—44no—
133D4—42no—
134D4—44no—
135D4—48no—
136D4—44no—
137C2×C4—24no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—24no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—48no—
150C2×C4—44no—
151C2×C4C2×C22⋊C412yes—
152C2×C4C2×C22⋊C412yes—
153C2×C4C2×C22⋊C412yes—
154C2×C4C2×C22⋊C412yes—
155C2×C4C2×C22⋊C411yescharacteristic
156C2×C4C23×C411yescharacteristic
157C2×C4C22×D412yes—
158C8—48no—
159C22—21no—
160C22—22no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—44no—
165C22—42no—
166C22—44no—
167C22C22×C22⋊C411yescharacteristic
168C4—22no—
169C4—21no—
170C4—22no—
171C4—22no—
172C4—22no—
173C4—42no—
174C4—44no—
175C4#41312yes—
176C4#57514yes—
177C2—41no—
178C2—42no—
179C2—82no—
180C2C2×#41311yescharacteristic
181C2#289612yes—
subgroupG/H
centerC2×C4C2×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
socleC22C22×C22⋊C4
prime radical#3444C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2920C2semidirect
3. #1559C2semidirect
4. #1557C2semidirect
5. #1544C2semidirect
6. #1543C2semidirect
26. #333C22semidirect
27. #332C22semidirect
28. #331C22semidirect

Maximal subgroups

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

subgroupindex
#34442
#29202
#15592
#15572
#15442
#15432