Finite Group Databasebeta

← #9019

Subgroups — #9019

Order 256.

486 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#394C211yesmaximal, characteristic
2#2316C211yesmaximal, characteristic, complemented
3#1786C212yesmaximal
4C2×#324C211yesmaximal, characteristic, complemented
5#1294C212yesmaximal, complemented
6C23×D4C2211yescharacteristic
7C2×#117—21no—
8C22×C4⋊C4C2211yescharacteristic
9C22×C22⋊C4—21no—
10C22×C22⋊C4—21no—
11C2×#102—24no—
12C2×C22⋊C8C2212yes—
13#394—22no—
14#394C414yescomplemented
15#389—24no—
16#380—21no—
17#380C2212yes—
18C2×#95C2211yescharacteristic
19#324—22no—
20C25—21no—
21C22×D4—22no—
22C22×D4—21no—
23C22×D4—22no—
24C22×D4—44no—
25C22×D4C2×C412yes—
26C23×C4—21no—
27C23×C4D411yescharacteristic
28C23×C4D411yescharacteristic
29C23×C4C2311yescharacteristic
30C22×C8—22no—
31#117—42no—
32#117—42no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4C2×C412yes—
38C2×C4⋊C4D412yes—
39C2×C4⋊C4D414yes—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—24no—
42C2×C22⋊C4—24no—
43C2×C22⋊C4—44no—
44C2×C22⋊C4—42no—
45C2×C22⋊C4—42no—
46C2×C42—22no—
47#102—48no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50#95—22no—
51#95—21no—
52#95—22no—
53#95C2×C412yes—
54C24—24no—
55C24—22no—
56C24—42no—
57C24—42no—
58C24—42no—
59C24—41no—
60C24—41no—
61C24—41no—
62C24—41no—
63C24—41no—
64C24—41no—
65C24C2×D411yescharacteristic
66C2×D4—28no—
67C2×D4—24no—
68C2×D4—44no—
69C2×D4—44no—
70C2×D4—48no—
71C2×D4—44no—
72C2×D4—42no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—24no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4C22⋊C412yes—
88C22×C4C22⋊C412yes—
89C22×C4D812yes—
90C22×C4SD1612yes—
91C22×C4C22×C411yescharacteristic
92C22×C4C2×D412yes—
93C22×C4C2×D411yescharacteristic
94C22×C4#4112yes—
95C2×C8—44no—
96C2×C8—42no—
97C4⋊C4—24no—
98C4⋊C4—24no—
99C4⋊C4—44no—
100C4⋊C4—44no—
101C4⋊C4—82no—
102C22⋊C4—48no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C22⋊C4—48no—
106C42—44no—
107C23—22no—
108C23—22no—
109C23—24no—
110C23—42no—
111C23—44no—
112C23—44no—
113C23—44no—
114C23—44no—
115C23—44no—
116C23—42no—
117C23—42no—
118C23—42no—
119C23—42no—
120C23—42no—
121C23—44no—
122C23—82no—
123C23—82no—
124C23—82no—
125C23—82no—
126C23—82no—
127C23—81no—
128C23—81no—
129C23—82no—
130C23—81no—
131C23—81no—
132C23C2×C22⋊C411yescharacteristic
133C23#12011yescharacteristic
134C23#12311yescharacteristic
135C23C2×D811yescharacteristic
136C23C2×SD1611yescharacteristic
137C23AGL(1,ℤ/8)11yescharacteristic
138C23#13711yescharacteristic
139D4—28no—
140D4—48no—
141D4—44no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—44no—
161C2×C4—81no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4#10214yescomplemented
165C2×C4C4×D412yes—
166C2×C4#12112yes—
167C8—82no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—44no—
177C22—44no—
178C22—44no—
179C22—82no—
180C22—82no—
181C22—84no—
182C22—84no—
183C22—84no—
184C22—84no—
185C22—82no—
186C22—82no—
187C22#38611yescharacteristic
188C22C2×#10211yescharacteristic
189C22#41711yescharacteristic
190C22#44711yescharacteristic
191C22#44811yescharacteristic
192C22#45412yes—
193C22B(4,𝔽2)12yes—
194C22#48011yescharacteristic
195C22#48111yescharacteristic
196C4—24no—
197C4—22no—
198C4—22no—
199C4—42no—
200C4—82no—
201C4—84no—
202C2—21no—
203C2—21no—
204C2—21no—
205C2—21no—
206C2—84no—
207C2#159812yes—
208C2#159912yes—
209C2#187411yescharacteristic
210C2#188811yescharacteristic
211C2#188911yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2316C2semidirect
4. C2×#324C2semidirect
5. #1294C2semidirect
14. #394C4semidirect
164. C2×C4#102semidirect

Maximal subgroups

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

subgroupindex
C2×#3942
#23162
#17862
C2×#3242
#12942