Finite Group Databasebeta

← #33688

Subgroups — #33688

Order 256.

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

normal subgroups only

This group has 282 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#3184C211yesmaximal, characteristic, complemented
2#2956C211yesmaximal, characteristic, complemented
3#2238C212yesmaximal, complemented
4#2230C212yesmaximal, complemented
5#2213C212yesmaximal, complemented
6#2190C212yesmaximal, complemented
7AGL(1,ℤ/16)C214yesmaximal, complemented
8#2165C211yesmaximal, characteristic, complemented
9#575C2211yescharacteristic
10#568—21no—
11#532—21no—
12#509C2214yes—
13#498C2211yescharacteristic, complemented
14#493C2211yescharacteristic, complemented
15#486—21no—
16#469—21no—
17#469C2212yes—
18#444—21no—
19#444C2212yes—
20#443—21no—
21#443C2212yes—
22#442—21no—
23#442C2212yescomplemented
24#430C2211yescharacteristic
25C2×#104—21no—
26C4×M16C2211yescharacteristic
27#365C2214yescomplemented
28#364C2212yescomplemented
29#361C2214yescomplemented
30#360C2214yes—
31#349C2212yes—
32#347C2212yescomplemented
33C2×#41—21no—
34C2×#41—41no—
35#137—22no—
36AGL(1,ℤ/8)—22no—
37#135—21no—
38#135C2312yes—
39C2×D8—21no—
40C2×D8C2312yes—
41#131—22no—
42#131—22no—
43#131—42no—
44C2×M16—21no—
45C2×M16—42no—
46C2×M16D411yescharacteristic
47C2×M16C2311yescharacteristic
48C4⋊Q8—21no—
49#127—21no—
50#124—41no—
51#121—22no—
52#121—41no—
53C4×D4—22no—
54C4×D4—41no—
55C2×C42D411yescharacteristic
56SD32—24no—
57D16—24no—
58M32C2314yes—
59#108—21no—
60#108C2312yes—
61#106—21no—
62#106C2312yes—
63#104—22no—
64#104—22no—
65#102—22no—
66#102—22no—
67#97D411yescharacteristic
68#97C2311yescharacteristic
69C4×C8D411yescharacteristic
70C4×C8C2311yescharacteristic
71#41—22no—
72#41—22no—
73#41—41no—
74#41—41no—
75#41—84no—
76C2×Q8—21no—
77C2×Q8—41no—
78C2×D4—21no—
79C2×D4—22no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—42no—
84C22×C4—21no—
85C22×C4—41no—
86C22×C4—42no—
87C22×C4C2×D411yescharacteristic
88Q16—21no—
89Q16#4112yescomplemented
90SD16—22no—
91SD16—22no—
92D8—21no—
93D8—24no—
94D8—42no—
95D8#4112yescomplemented
96M16—22no—
97M16—22no—
98M16—22no—
99M16—42no—
100M16—41no—
101M16—42no—
102C2×C8—42no—
103C2×C8—41no—
104C2×C8—42no—
105C2×C8C2×D411yescharacteristic
106C2×C8C2×D411yescharacteristic
107C2×C8C2×D411yescharacteristic
108C2×C8C2411yescharacteristic
109C4⋊C4—22no—
110C4⋊C4—41no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C42—21no—
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C16—24no—
117C23—41no—
118C23—42no—
119C23AGL(1,ℤ/8)11yescharacteristic
120Q8—22no—
121Q8—41no—
122Q8—82no—
123D4—22no—
124D4—41no—
125D4—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—82no—
130D4—82no—
131D4—84no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—84no—
142C2×C4AGL(1,ℤ/8)11yescharacteristic
143C2×C4C22×D411yescharacteristic
144C8—21no—
145C8—21no—
146C8—42no—
147C8AGL(1,ℤ/8)12yes—
148C8C2×#4112yes—
149C22—21no—
150C22—41no—
151C22—42no—
152C22—82no—
153C22—82no—
154C22—82no—
155C22—161no—
156C22C2×AGL(1,ℤ/8)11yescharacteristic
157C4—22no—
158C4—21no—
159C4—41no—
160C4—42no—
161C4—82no—
162C4#52511yescharacteristic
163C4C2×AGL(1,ℤ/8)11yescharacteristic
164C2—21no—
165C2—41no—
166C2—82no—
167C2—162no—
168C2#310911yescharacteristic
subgroupG/H
center, socleC2#3109
commutator, Frattini, cosocleC2×C8C24
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. #3184C2semidirect
2. #2956C2semidirect
3. #2238C2semidirect
4. #2230C2semidirect
5. #2213C2semidirect
6. #2190C2semidirect
7. AGL(1,ℤ/16)C2semidirect
8. #2165C2semidirect
13. #498C22semidirect
14. #493C22semidirect
23. #442C22semidirect
27. #365C22semidirect
28. #364C22semidirect
29. #361C22semidirect
32. #347C22semidirect
89. Q16#41semidirect
95. D8#41semidirect

Maximal subgroups

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

subgroupindex
#31842
#29562
#22382
#22302
#22132
#21902
AGL(1,ℤ/16)2
#21652