Finite Group Databasebeta

← #33690

Subgroups — #33690

Order 256.

263 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 93 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#3183C211yesmaximal, characteristic, complemented
2#2956C211yesmaximal, characteristic, complemented
3#2239C211yesmaximal, characteristic, complemented
4#2238C211yesmaximal, characteristic, complemented
5#2231C212yesmaximal, complemented
6#2212C212yesmaximal, complemented
7#2191C211yesmaximal, characteristic, complemented
8#2190C211yesmaximal, characteristic
9AGL(1,ℤ/16)C212yesmaximal, complemented
10#2180C212yesmaximal, complemented
11#2165C211yesmaximal, characteristic, complemented
12#575C2211yescharacteristic
13#568—21no—
14#529—21no—
15#510C2212yes—
16#509C2212yes—
17#499C2211yescharacteristic, complemented
18#495C2211yescharacteristic, complemented
19#488—21no—
20#470C2211yescharacteristic
21#469C2211yescharacteristic
22#468—21no—
23#444—21no—
24#444C2212yes—
25#443—21no—
26#443C2212yes—
27#442C2211yescharacteristic, complemented
28#441C2211yescharacteristic, complemented
29#440—21no—
30#430C2211yescharacteristic
31C2×#104—21no—
32C4×M16C2211yescharacteristic
33#365C2212yescomplemented
34#365C2212yescomplemented
35#364C2212yescomplemented
36#362C2212yescomplemented
37#361C2212yescomplemented
38#360C2212yes—
39#360C2212yes—
40#349C2211yescharacteristic
41#349C2211yescharacteristic
42#347C2212yescomplemented
43C2×#41—21no—
44#137—22no—
45AGL(1,ℤ/8)—22no—
46#135—21no—
47#135C2312yes—
48C2×Q16C2311yescharacteristic
49C2×SD16—21no—
50C2×D8C2311yescharacteristic
51#131—22no—
52#131—22no—
53#131—42no—
54C2×M16—21no—
55C2×M16—42no—
56C2×M16D411yescharacteristic
57C2×M16C2311yescharacteristic
58#126—41no—
59#125—21no—
60#124—21no—
61#123—41no—
62C22⋊Q8—21no—
63#121—21no—
64C4×Q8—21no—
65C4×D4—21no—
66C4×D4—41no—
67#117—41no—
68C2×C42D411yescharacteristic
69Q32—22no—
70SD32—22no—
71SD32—22no—
72D16—22no—
73M32C2312yes—
74M32C2312yes—
75#108—21no—
76#108C2312yes—
77#107—21no—
78#106C2311yescharacteristic
79#106C2311yescharacteristic
80#104—22no—
81#104—22no—
82#103—21no—
83#103—21no—
84#102—21no—
85#102—21no—
86#97D411yescharacteristic
87#97C2311yescharacteristic
88C4×C8D411yescharacteristic
89C4×C8C2311yescharacteristic
90#41—22no—
91#41—22no—
92#41—41no—
93#41—41no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—41no—
99C2×D4—41no—
100C22×C4—21no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4C2×D411yescharacteristic
104Q16—21no—
105Q16—22no—
106Q16#4112yes—
107SD16—22no—
108SD16—22no—
109SD16—42no—
110D8—21no—
111D8—22no—
112D8#4112yes—
113M16—22no—
114M16—22no—
115M16—22no—
116M16—42no—
117M16—41no—
118M16—42no—
119C2×C8—42no—
120C2×C8—41no—
121C2×C8—42no—
122C2×C8C2×D411yescharacteristic
123C2×C8C2×D411yescharacteristic
124C2×C8C2×D411yescharacteristic
125C2×C8C2411yescharacteristic
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C42—21no—
138C42—41no—
139C42C2×D411yescharacteristic
140C42C2×D411yescharacteristic
141C16—22no—
142C16—22no—
143C23—41no—
144C23—41no—
145C23AGL(1,ℤ/8)11yescharacteristic
146Q8—22no—
147Q8—41no—
148Q8—41no—
149Q8—41no—
150D4—22no—
151D4—41no—
152D4—42no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—82no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—81no—
168C2×C4—82no—
169C2×C4AGL(1,ℤ/8)11yescharacteristic
170C2×C4C22×D411yescharacteristic
171C8—21no—
172C8—21no—
173C8—42no—
174C8AGL(1,ℤ/8)12yes—
175C8C2×#4112yes—
176C22—21no—
177C22—41no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—161no—
183C22C2×AGL(1,ℤ/8)11yescharacteristic
184C4—21no—
185C4—41no—
186C4—41no—
187C4—42no—
188C4—81no—
189C4—81no—
190C4—81no—
191C4#52511yescharacteristic
192C4C2×AGL(1,ℤ/8)11yescharacteristic
193C2—21no—
194C2—41no—
195C2—82no—
196C2—161no—
197C2#310911yescharacteristic
subgroupG/H
center, socleC2#3109
commutator, Frattini, cosocleC2×C8C24
prime radical#2190C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3183C2semidirect
2. #2956C2semidirect
3. #2239C2semidirect
4. #2238C2semidirect
5. #2231C2semidirect
6. #2212C2semidirect
7. #2191C2semidirect
9. AGL(1,ℤ/16)C2semidirect
10. #2180C2semidirect
11. #2165C2semidirect
17. #499C22semidirect
18. #495C22semidirect
27. #442C22semidirect
28. #441C22semidirect
33. #365C22semidirect
34. #365C22semidirect
35. #364C22semidirect
36. #362C22semidirect
37. #361C22semidirect
42. #347C22semidirect

Maximal subgroups

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

subgroupindex
#31832
#29562
#22392
#22382
#22312
#22122
#21912
#21902
AGL(1,ℤ/16)2
#21802
#21652