Finite Group Databasebeta

← #33693

Subgroups — #33693

Order 256.

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

normal subgroups only

This group has 256 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#3177C211yesmaximal, characteristic, complemented
2#2955C211yesmaximal, characteristic
3#2239C212yesmaximal, complemented
4#2231C212yesmaximal, complemented
5#2214C212yesmaximal, complemented
6#2191C212yesmaximal
7#2180C214yesmaximal, complemented
8#2166C211yesmaximal, characteristic, complemented
9#575C2211yescharacteristic
10C2×#131—21no—
11#541—21no—
12#510C2214yes—
13#499C2211yescharacteristic, complemented
14#494C2211yescharacteristic, complemented
15#491—21no—
16#470C2212yes—
17#467—21no—
18#444—21no—
19#444C2212yes—
20#443—21no—
21#443C2212yes—
22#441C2212yescomplemented
23C4×Q16—21no—
24#430C2211yescharacteristic
25#421—21no—
26#405C2211yescharacteristic
27#365C2214yescomplemented
28#364C2212yescomplemented
29#362C2214yescomplemented
30#360C2214yes—
31#349C2212yes—
32#347C2212yescomplemented
33C2×#41—21no—
34#137—22no—
35AGL(1,ℤ/8)—22no—
36#135—21no—
37#135C2312yes—
38C2×Q16—21no—
39C2×Q16C2312yes—
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
49C4⋊Q8—21no—
50#126—42no—
51#125—21no—
52#125—42no—
53C22⋊Q8—22no—
54C4×Q8—22no—
55#117D411yescharacteristic
56Q32—24no—
57SD32—24no—
58M32C2314yes—
59#108—21no—
60#108C2312yes—
61#107—21no—
62#106C2312yes—
63#104—22no—
64#104—22no—
65#103—22no—
66#103—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×Q8—22no—
77C2×D4—21no—
78C2×D4—41no—
79C22×C4—41no—
80C22×C4C2×D411yescharacteristic
81Q16—21no—
82Q16—24no—
83Q16—41no—
84Q16—41no—
85Q16#4112yes—
86SD16—22no—
87SD16—22no—
88D8—21no—
89D8#4112yes—
90M16—22no—
91M16—22no—
92M16—42no—
93M16—41no—
94M16—42no—
95C2×C8—21no—
96C2×C8—21no—
97C2×C8—42no—
98C2×C8—41no—
99C2×C8—42no—
100C2×C8C2×D411yescharacteristic
101C2×C8C2×D411yescharacteristic
102C2×C8C2×D411yescharacteristic
103C2×C8C2411yescharacteristic
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C22⋊C4—21no—
111C22⋊C4—42no—
112C42—42no—
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C16—24no—
116C23—41no—
117C23AGL(1,ℤ/8)11yescharacteristic
118Q8—22no—
119Q8—41no—
120Q8—42no—
121Q8—42no—
122D4—22no—
123D4—41no—
124D4—42no—
125D4—42no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4AGL(1,ℤ/8)11yescharacteristic
136C2×C4C22×D411yescharacteristic
137C8—21no—
138C8—21no—
139C8—42no—
140C8AGL(1,ℤ/8)11yescharacteristic
141C8AGL(1,ℤ/8)11yescharacteristic
142C8C2×#4112yes—
143C22—21no—
144C22—41no—
145C22—42no—
146C22—82no—
147C22C2×AGL(1,ℤ/8)11yescharacteristic
148C4—21no—
149C4—41no—
150C4—41no—
151C4—42no—
152C4—82no—
153C4—82no—
154C4#52511yescharacteristic
155C4C2×AGL(1,ℤ/8)11yescharacteristic
156C2—21no—
157C2—41no—
158C2—82no—
159C2#310911yescharacteristic
subgroupG/H
center, socleC2#3109
commutator, Frattini, cosocleC2×C8C24
prime radical#575C22
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. #3177C2semidirect
3. #2239C2semidirect
4. #2231C2semidirect
5. #2214C2semidirect
7. #2180C2semidirect
8. #2166C2semidirect
13. #499C22semidirect
14. #494C22semidirect
22. #441C22semidirect
27. #365C22semidirect
28. #364C22semidirect
29. #362C22semidirect
32. #347C22semidirect

Maximal subgroups

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

subgroupindex
#31772
#29552
#22392
#22312
#22142
#21912
#21802
#21662