Finite Group Databasebeta

← #11525

Subgroups — #11525

Order 256.

369 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 55 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
1C4×#120C211yesmaximal, characteristic
2#1466C211yesmaximal, characteristic, complemented
3#1465C211yesmaximal, characteristic, complemented
4#1326C212yesmaximal, complemented
5#1316C212yesmaximal, complemented
6C24×C4C411yescharacteristic
7C2×#120C411yescharacteristic
8C2×C4×D4—21no—
9C2×C4×D4C2211yescharacteristic
10#432—21no—
11#407—21no—
12#386—21no—
13#386C411yescharacteristic
14#385—21no—
15#385C2211yescharacteristic
16#379C2211yescharacteristic
17C4×C22⋊C4—21no—
18C4×C22⋊C4C411yescharacteristic
19#324—21no—
20#323—21no—
21#323C2212yes—
22#323C2212yes—
23C25C2×C411yescharacteristic
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C23×C4—21no—
27C23×C4—21no—
28C23×C4—44no—
29C23×C4—41no—
30C23×C4—41no—
31C23×C4C2×C411yescharacteristic
32C23×C4C2×C411yescharacteristic
33C23×C4D411yescharacteristic
34C2×M16—41no—
35#120—22no—
36#120—22no—
37C4×D4—41no—
38C4×D4—41no—
39C4×D4—44no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4D411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4C2×C411yescharacteristic
47C2×C22⋊C4C2×C411yescharacteristic
48C2×C22⋊C4D411yescharacteristic
49C2×C22⋊C4C2311yescharacteristic
50C2×C42—21no—
51C2×C42D411yescharacteristic
52C4⋊C8—41no—
53C22⋊C8—22no—
54C22⋊C8—22no—
55C22⋊C8—41no—
56#97—41no—
57#95—21no—
58#95—21no—
59#95—21no—
60C24—21no—
61C24—21no—
62C24—42no—
63C24—41no—
64C24—42no—
65C24—41no—
66C24C22⋊C411yescharacteristic
67C24M1612yescomplemented
68C24C22×C411yescharacteristic
69C2×D4—24no—
70C2×D4—21no—
71C2×D4—21no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—42no—
75C2×D4—42no—
76C22×C4—24no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—44no—
92C22×C4—44no—
93C22×C4—44no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—44no—
98C22×C4C22⋊C411yescharacteristic
99C22×C4C22⋊C411yescharacteristic
100C22×C4C22⋊C411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103M16—82no—
104C2×C8—42no—
105C2×C8—42no—
106C4⋊C4—41no—
107C4⋊C4—42no—
108C22⋊C4—24no—
109C22⋊C4—21no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C22⋊C4—42no—
115C22⋊C4—41no—
116C22⋊C4M1612yescomplemented
117C42—21no—
118C42—21no—
119C42—42no—
120C23—22no—
121C23—22no—
122C23—22no—
123C23—21no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—41no—
129C23—44no—
130C23—42no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—44no—
135C23—42no—
136C23—41no—
137C23—41no—
138C23—42no—
139C23—42no—
140C23—42no—
141C23—48no—
142C23—42no—
143C23—41no—
144C23—41no—
145C23C2×C22⋊C411yescharacteristic
146C23C2×M1611yescharacteristic
147C23C2×M1611yescharacteristic
148D4—48no—
149D4—42no—
150D4—42no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—44no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—48no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—81no—
179C2×C4#9911yescharacteristic
180C2×C4#9911yescharacteristic
181C8—82no—
182C8—82no—
183C22—21no—
184C22—21no—
185C22—24no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—42no—
195C22—44no—
196C22—44no—
197C22—44no—
198C22—44no—
199C22—42no—
200C22—41no—
201C22—42no—
202C22—41no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—81no—
208C22#40711yescharacteristic
209C22C2×#9911yescharacteristic
210C22#41311yescharacteristic
211C4—21no—
212C4—42no—
213C4—41no—
214C4—42no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4AΣL(1,𝔽16)12yes—
220C2—21no—
221C2—21no—
222C2—44no—
223C2—41no—
224C2—41no—
225C2—81no—
226C2#146511yescharacteristic
227C2C2×AΣL(1,𝔽16)11yescharacteristic
228C2#212011yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC2×#120C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1466C2semidirect
3. #1465C2semidirect
4. #1326C2semidirect
5. #1316C2semidirect
67. C24M16semidirect
116. C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C4×#1202
#14662
#14652
#13262
#13162