Finite Group Databasebeta

← #12108

Subgroups — #12108

Order 256.

221 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 37 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#3199C211yesmaximal, characteristic, complemented
2#2184C212yesmaximal, complemented
3#2018C211yesmaximal, characteristic, complemented
4#1349C212yesmaximal, complemented
5#1315C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)—21no—
7C22×D8C2211yescharacteristic, complemented
8#535—21no—
9C2×D16—22no—
10#482—22no—
11#469—21no—
12#466C2212yes—
13#447—22no—
14C2×#100—21no—
15#408C2211yescharacteristic
16#357—22no—
17C22⋊C16C2212yescomplemented
18#341C2211yescharacteristic, complemented
19C2×#41D411yescharacteristic
20C22×D4—21no—
21C22×D4—21no—
22AGL(1,ℤ/8)—42no—
23AGL(1,ℤ/8)—42no—
24C2×SD16—41no—
25C2×D8—21no—
26C2×D8—22no—
27C2×D8—42no—
28C2×D8—41no—
29C2×D8D412yes—
30C2×M16—21no—
31C2×M16D411yescharacteristic
32C22×C8C2311yescharacteristic
33#127—41no—
34#124—41no—
35#121—22no—
36#121—22no—
37#120—41no—
38#120—41no—
39#117—21no—
40D16—44no—
41C2×C16—22no—
42#107D412yes—
43#106—21no—
44#102—22no—
45#102—22no—
46#100—42no—
47C22⋊C8—22no—
48C24—41no—
49C24—41no—
50#41—41no—
51#41—41no—
52C2×Q8D811yescharacteristic
53C2×D4—22no—
54C2×D4—21no—
55C2×D4—22no—
56C2×D4—41no—
57C2×D4—41no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4—42no—
61C2×D4—42no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—81no—
65C2×D4—81no—
66C2×D4D811yescharacteristic
67C22×C4—21no—
68C22×C4C2×D411yescharacteristic
69SD16—44no—
70D8—24no—
71D8—42no—
72D8—42no—
73D8—44no—
74D8—81no—
75M16—22no—
76M16—41no—
77M16—41no—
78C2×C8—21no—
79C2×C8—41no—
80C2×C8—41no—
81C2×C8C2×D412yes—
82C4⋊C4—22no—
83C22⋊C4—42no—
84C22⋊C4—41no—
85C22⋊C4—42no—
86C42—41no—
87C16—42no—
88C23—21no—
89C23—42no—
90C23—42no—
91C23—81no—
92C23—82no—
93C23—81no—
94C23—162no—
95C23—162no—
96C23AGL(1,ℤ/8)11yescharacteristic
97Q8—41no—
98D4—41no—
99D4—42no—
100D4—42no—
101D4—42no—
102D4—42no—
103D4—81no—
104D4—81no—
105D4—82no—
106D4—81no—
107D4—162no—
108D4—162no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—42no—
115C2×C4#12011yescharacteristic
116C2×C4C2×D811yescharacteristic
117C8—22no—
118C8—42no—
119C8—41no—
120C22—21no—
121C22—22no—
122C22—41no—
123C22—42no—
124C22—82no—
125C22—82no—
126C22—82no—
127C22—82no—
128C22—162no—
129C22—162no—
130C22—164no—
131C22—162no—
132C22#44711yescharacteristic
133C4—21no—
134C4—41no—
135C4—82no—
136C4#45311yescharacteristic
137C4B(4,𝔽2)11yescharacteristic
138C2—41no—
139C2—81no—
140C2—162no—
141C2—162no—
142C2#159511yescharacteristic
143C2#219212yes—
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3199C2semidirect
2. #2184C2semidirect
3. #2018C2semidirect
4. #1349C2semidirect
5. #1315C2semidirect
7. C22×D8C22semidirect
17. C22⋊C16C22semidirect
18. #341C22semidirect

Maximal subgroups

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

subgroupindex
#31992
#21842
#20182
#13492
#13152