Finite Group Databasebeta

← #9515

Subgroups — #9515

Order 256.

314 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 61 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#2973C211yesmaximal, characteristic, complemented
2#2534C211yesmaximal, characteristic
3#1465C211yesmaximal, characteristic, complemented
4#1462C212yesmaximal, complemented
5#1305C212yesmaximal, complemented
6C2×#123C411yescharacteristic
7C2×#120C411yescharacteristic
8C2×C4×D4C2211yescharacteristic
9C22×C22⋊C4C2211yescharacteristic
10#435—24no—
11#432—21no—
12#407C2212yes—
13#407C2212yes—
14C2×C22⋊C8—21no—
15#396—21no—
16#394—21no—
17#388—21no—
18#386—21no—
19#381C411yescharacteristic
20#380C2211yescharacteristic
21#379C411yescharacteristic
22#324—21no—
23#323—21no—
24#323—24no—
25C25C2×C411yescharacteristic
26C22×D4—21no—
27C22×D4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4C2311yescharacteristic
31C2×M16—22no—
32C2×M16—22no—
33C22×C8—22no—
34#123—42no—
35#120—42no—
36C4×D4—22no—
37C4×D4—22no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4D411yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4C2×C411yescharacteristic
50C2×C22⋊C4C2×C411yescharacteristic
51C2×C22⋊C4C2×C411yescharacteristic
52C2×C22⋊C4D411yescharacteristic
53C2×C42D411yescharacteristic
54C4⋊C8—22no—
55C22⋊C8—22no—
56C22⋊C8—22no—
57C22⋊C8—22no—
58C22⋊C8—22no—
59C22⋊C8D414yescomplemented
60#97—22no—
61#95—21no—
62#95—21no—
63#95—21no—
64C24—21no—
65C24—22no—
66C24—42no—
67C24—41no—
68C24—41no—
69C24—81no—
70C24C22⋊C411yescharacteristic
71C24C22×C411yescharacteristic
72C2×D4—22no—
73C2×D4—22no—
74C2×D4—41no—
75C2×D4—42no—
76C2×D4—42no—
77C2×D4—81no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—81no—
87C22×C4—81no—
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C2×D412yes—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4#4112yes—
95M16—82no—
96M16—82no—
97C2×C8—24no—
98C2×C8—42no—
99C2×C8—82no—
100C4⋊C4—22no—
101C4⋊C4—42no—
102C4⋊C4—81no—
103C22⋊C4—22no—
104C22⋊C4—22no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—41no—
109C22⋊C4—42no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—81no—
114C42—21no—
115C42—21no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—22no—
121C23—24no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—42no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—82no—
134C23—82no—
135C23—84no—
136C23—84no—
137C23#10012yes—
138C23C2×C22⋊C411yescharacteristic
139C23C4×D412yes—
140D4—44no—
141D4—84no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—81no—
164C2×C4—82no—
165C2×C4#12011yescharacteristic
166C2×C4#12112yes—
167C2×C4#12311yescharacteristic
168C8—44no—
169C8—82no—
170C22—21no—
171C22—21no—
172C22—22no—
173C22—22no—
174C22—22no—
175C22—22no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22—41no—
183C22—42no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22—84no—
188C22—84no—
189C22—81no—
190C22—81no—
191C22#38611yescharacteristic
192C22C2×#10011yescharacteristic
193C22#41311yescharacteristic
194C4—21no—
195C4—22no—
196C4—21no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—81no—
201C4—81no—
202C4—81no—
203C2—21no—
204C2—21no—
205C2—42no—
206C2—42no—
207C2—82no—
208C2#189111yescharacteristic
209C2#190011yescharacteristic
210C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#120C4
LCSG ▷ C24 ▷ 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
1. #2973C2semidirect
3. #1465C2semidirect
4. #1462C2semidirect
5. #1305C2semidirect
59. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#29732
#25342
#14652
#14622
#13052