Finite Group Databasebeta

← #17626

Subgroups — #17626

Order 256.

627 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 183 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#3496C211yesmaximal, characteristic
2#2975C211yesmaximal, characteristic, complemented
3#2357C211yesmaximal, characteristic, complemented
4#1517C212yesmaximal, complemented
5#1510C214yesmaximal, complemented
6#1507C214yesmaximal, complemented
7#1502C212yesmaximal, complemented
8C2×H(5,𝔽2)—21no—
9#547C414yescomplemented
10#546—24no—
11D42C414yescomplemented
12#537—21no—
13#534—22no—
14#532C2212yes—
15C2×#121C2214yes—
16C2×C4×D4C2211yescharacteristic
17#436—24no—
18#431C2212yescomplemented
19#407C2214yescomplemented
20#390—21no—
21#388—22no—
22#382C2211yescharacteristic, complemented
23#381C2212yescomplemented
24C4×C4⋊C4—21no—
25C4×C4⋊C4C2211yescharacteristic, complemented
26C4×C22⋊C4C2212yescomplemented
27#327C2214yescomplemented
28#327C2214yescomplemented
29#327C2218yescomplemented
30H(5,𝔽2)—44no—
31H(5,𝔽2)—44no—
32C2×#41—24no—
33C2×#41—22no—
34C22×D4—24no—
35C22×D4—24no—
36C22×D4C2×C414yes—
37C22×D4D411yescharacteristic
38C23×C4C2312yes—
39C2×M16—24no—
40C4⋊Q8C2×C412yes—
41#127C2×C412yes—
42#124—22no—
43#123—28no—
44C22⋊Q8—24no—
45#121—24no—
46#121—24no—
47#121—22no—
48#121—24no—
49#121C2×C418yes—
50#121C2314yes—
51#120—28no—
52C4×D4—22no—
53C4×D4C2×C414yes—
54C4×D4C2×C414yes—
55C4×D4D414yes—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2×C414yes—
59C2×C4⋊C4D411yescharacteristic
60C2×C22⋊C4—24no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4D412yes—
64C2×C42—21no—
65C2×C42—22no—
66C2×C42C2311yescharacteristic
67C4⋊C8—22no—
68C22⋊C8—24no—
69C22⋊C8C2314yes—
70C4×C8—22no—
71#95—22no—
72#95—21no—
73#95C2311yescharacteristic
74#95C2312yes—
75#95C2311yescharacteristic
76C24—24no—
77C24C2×D412yes—
78#41—416no—
79#41—44no—
80#41—44no—
81C2×Q8—22no—
82C2×D4—28no—
83C2×D4—24no—
84C2×D4—22no—
85C2×D4—22no—
86C2×D4—24no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4—416no—
90C2×D4—48no—
91C2×D4—48no—
92C2×D4—84no—
93C2×D4C22⋊C414yes—
94C2×D4C22×C414yes—
95C22×C4—24no—
96C22×C4—21no—
97C22×C4—24no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—82no—
101C22×C4C22×C414yes—
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D414yes—
105C22×C4C2×D411yescharacteristic
106C22×C4C2411yescharacteristic
107M16—84no—
108C2×C8—24no—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—42no—
112C4⋊C4—42no—
113C4⋊C4C22⋊C414yes—
114C4⋊C4C22×C414yes—
115C22⋊C4—28no—
116C22⋊C4—24no—
117C22⋊C4—24no—
118C22⋊C4—44no—
119C22⋊C4—82no—
120C22⋊C4C22⋊C418yes—
121C42—42no—
122C42—82no—
123C42C22×C412yes—
124C42C2×D412yes—
125C23—22no—
126C23—22no—
127C23—24no—
128C23—42no—
129C23—42no—
130C23—48no—
131C23—84no—
132C23—84no—
133C23C2×C22⋊C414yes—
134C23C22×D411yescharacteristic
135Q8—44no—
136D4—28no—
137D4—44no—
138D4—416no—
139D4—44no—
140D4—44no—
141D4—44no—
142D4—82no—
143D4—84no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—44no—
149C2×C4—41no—
150C2×C4—44no—
151C2×C4—41no—
152C2×C4—48no—
153C2×C4—41no—
154C2×C4—48no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4C2×C22⋊C412yes—
162C2×C4C2×C22⋊C414yes—
163C2×C4C2×C22⋊C412yes—
164C2×C4AGL(1,ℤ/8)12yes—
165C2×C4AGL(1,ℤ/8)12yes—
166C2×C4C23×C411yescharacteristic
167C2×C4C22×D411yescharacteristic
168C8—44no—
169C22—21no—
170C22—21no—
171C22—24no—
172C22—24no—
173C22—24no—
174C22—44no—
175C22—44no—
176C22—44no—
177C22—82no—
178C22—82no—
179C22—168no—
180C22C22×C22⋊C411yescharacteristic
181C22C2×AGL(1,ℤ/8)11yescharacteristic
182C22C2×AGL(1,ℤ/8)11yescharacteristic
183C4—21no—
184C4—21no—
185C4—24no—
186C4—22no—
187C4—42no—
188C4—44no—
189C4—81no—
190C4—82no—
191C4#41814yes—
192C2—21no—
193C2—21no—
194C2—44no—
195C2—84no—
196C2#288311yescharacteristic
197C2C2×#41811yescharacteristic
198C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3496C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2975C2semidirect
3. #2357C2semidirect
4. #1517C2semidirect
5. #1510C2semidirect
6. #1507C2semidirect
7. #1502C2semidirect
9. #547C4semidirect
11. D42C4semidirect
18. #431C22semidirect
19. #407C22semidirect
22. #382C22semidirect
23. #381C22semidirect
25. C4×C4⋊C4C22semidirect
26. C4×C22⋊C4C22semidirect
27. #327C22semidirect
28. #327C22semidirect
29. #327C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#29752
#23572
#15172
#15102
#15072
#15022