Finite Group Databasebeta

← #33692

Subgroups — #33692

Order 256.

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

normal subgroups only

This group has 268 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#3176C211yesmaximal, characteristic, complemented
2#2955C211yesmaximal, characteristic, complemented
3#2238C212yesmaximal, complemented
4#2231C212yesmaximal, complemented
5#2213C212yesmaximal, complemented
6#2190C212yesmaximal, complemented
7AGL(1,ℤ/16)C214yesmaximal, complemented
8#2166C211yesmaximal, characteristic, complemented
9#575C2211yescharacteristic
10C2×#131—21no—
11#540—21no—
12#509C2214yes—
13#499C2211yescharacteristic, complemented
14#493C2211yescharacteristic, complemented
15#490—21no—
16#469C2212yes—
17#466—21no—
18#444—21no—
19#444C2212yes—
20#443—21no—
21#443C2212yes—
22#442C2212yescomplemented
23C4×D8—21no—
24#430C2211yescharacteristic
25#421—21no—
26#405C2211yescharacteristic
27#365C2214yescomplemented
28#364C2212yescomplemented
29#361C2214yescomplemented
30#360C2214yes—
31#349C2212yes—
32#347C2212yescomplemented
33C2×#41—21no—
34#137—22no—
35AGL(1,ℤ/8)—22no—
36#135—21no—
37#135C2312yes—
38C2×D8—21no—
39C2×D8C2312yes—
40#131—22no—
41#131—22no—
42#131—41no—
43#131—41no—
44C2×M16—21no—
45C2×M16—41no—
46C2×M16C2311yescharacteristic
47C22×C8—41no—
48C22×C8D411yescharacteristic
49#127—21no—
50#125—21no—
51#123—42no—
52#121—22no—
53#121—42no—
54C4×D4—22no—
55#117D411yescharacteristic
56SD32—24no—
57D16—24no—
58M32C2314yes—
59#108—21no—
60#108C2312yes—
61#107—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—
75C2×Q8—21no—
76C2×D4—21no—
77C2×D4—22no—
78C2×D4—42no—
79C2×D4—41no—
80C2×D4—42no—
81C22×C4—41no—
82C22×C4—42no—
83C22×C4C2×D411yescharacteristic
84Q16—21no—
85Q16#4112yes—
86SD16—22no—
87SD16—22no—
88D8—21no—
89D8—24no—
90D8—41no—
91D8—41no—
92D8#4112yes—
93M16—22no—
94M16—22no—
95M16—42no—
96M16—41no—
97M16—42no—
98C2×C8—21no—
99C2×C8—21no—
100C2×C8—42no—
101C2×C8—41no—
102C2×C8—42no—
103C2×C8C2×D411yescharacteristic
104C2×C8C2×D411yescharacteristic
105C2×C8C2×D411yescharacteristic
106C2×C8C2411yescharacteristic
107C4⋊C4—21no—
108C4⋊C4—22no—
109C4⋊C4—42no—
110C22⋊C4—21no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C16—24no—
116C23—41no—
117C23—42no—
118C23AGL(1,ℤ/8)11yescharacteristic
119Q8—22no—
120Q8—41no—
121D4—22no—
122D4—41no—
123D4—42no—
124D4—42no—
125D4—42no—
126D4—42no—
127D4—162no—
128D4—162no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—162no—
138C2×C4AGL(1,ℤ/8)11yescharacteristic
139C2×C4C22×D411yescharacteristic
140C8—21no—
141C8—21no—
142C8—42no—
143C8AGL(1,ℤ/8)11yescharacteristic
144C8AGL(1,ℤ/8)11yescharacteristic
145C8C2×#4112yes—
146C22—21no—
147C22—41no—
148C22—42no—
149C22—82no—
150C22—82no—
151C22—82no—
152C22—82no—
153C22C2×AGL(1,ℤ/8)11yescharacteristic
154C4—21no—
155C4—41no—
156C4—41no—
157C4—42no—
158C4—82no—
159C4#52511yescharacteristic
160C4C2×AGL(1,ℤ/8)11yescharacteristic
161C2—21no—
162C2—41no—
163C2—82no—
164C2—162no—
165C2#310911yescharacteristic
subgroupG/H
center, socleC2#3109
commutator, Frattini, cosocleC2×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3176C2semidirect
2. #2955C2semidirect
3. #2238C2semidirect
4. #2231C2semidirect
5. #2213C2semidirect
6. #2190C2semidirect
7. AGL(1,ℤ/16)C2semidirect
8. #2166C2semidirect
13. #499C22semidirect
14. #493C22semidirect
22. #442C22semidirect
27. #365C22semidirect
28. #364C22semidirect
29. #361C22semidirect
32. #347C22semidirect

Maximal subgroups

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

subgroupindex
#31762
#29552
#22382
#22312
#22132
#21902
AGL(1,ℤ/16)2
#21662