Finite Group Databasebeta

← #12295

Subgroups — #12295

Order 256.

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

normal subgroups only

This group has 145 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#3262C211yesmaximal, characteristic
2#2235C211yesmaximal, characteristic, complemented
3#2233C211yesmaximal, characteristic
4#2089C211yesmaximal, characteristic, complemented
5#1350C211yesmaximal, characteristic, complemented
6#1349C211yesmaximal, characteristic
7#1323C211yesmaximal, characteristic, complemented
8#542—21no—
9#477—21no—
10#474—21no—
11#468—21no—
12#467C2211yescharacteristic
13#466C2211yescharacteristic
14#464—21no—
15#463—21no—
16#428—21no—
17C2×#107C2211yescharacteristic
18#424C2211yescharacteristic
19#367—21no—
20#367—21no—
21#358—21no—
22#357—21no—
23C22⋊C16C2211yescharacteristic
24C22⋊C16C2211yescharacteristic
25#343—21no—
26#341C2211yescharacteristic
27C2×Q16D411yescharacteristic
28C2×SD16—21no—
29C2×D8D411yescharacteristic
30C2×M16—21no—
31C2×M16D411yescharacteristic
32C22×C8C2311yescharacteristic
33C4⋊Q8—21no—
34#126—41no—
35#124—21no—
36#124—41no—
37#123—41no—
38C22⋊Q8—21no—
39C22⋊Q8—41no—
40#121—21no—
41C4×Q8—21no—
42C4×D4—21no—
43#117D411yescharacteristic
44C2×C4⋊C4—21no—
45C2×C16—21no—
46C2×C16—21no—
47#107—21no—
48#107—21no—
49#107—21no—
50#107—41no—
51#107D411yescharacteristic
52#107D411yescharacteristic
53#106—41no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56#103—21no—
57#103—21no—
58#102—21no—
59#102—21no—
60C2×Q8—21no—
61C2×Q8—41no—
62C2×D4—21no—
63C2×D4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4C2×D411yescharacteristic
67Q16—41no—
68SD16—81no—
69D8—41no—
70M16—22no—
71M16—41no—
72M16—41no—
73C2×C8—21no—
74C2×C8—41no—
75C2×C8—41no—
76C2×C8C2×D411yescharacteristic
77C2×C8C2×D411yescharacteristic
78C4⋊C4—21no—
79C4⋊C4—21no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4D811yescharacteristic
86C4⋊C4D811yescharacteristic
87C22⋊C4—21no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C42—21no—
94C42—41no—
95C16—41no—
96C16—41no—
97C23—41no—
98C23AGL(1,ℤ/8)11yescharacteristic
99Q8—41no—
100Q8—41no—
101Q8—161no—
102D4—41no—
103D4—41no—
104D4—161no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—81no—
113C2×C4—161no—
114C2×C4—161no—
115C2×C4#12011yescharacteristic
116C2×C4C2×D811yescharacteristic
117C8—21no—
118C8—21no—
119C8—42no—
120C8—41no—
121C22—21no—
122C22—21no—
123C22—21no—
124C22—81no—
125C22—81no—
126C22—81no—
127C22#44711yescharacteristic
128C4—21no—
129C4—41no—
130C4—41no—
131C4—81no—
132C4—81no—
133C4—81no—
134C4—161no—
135C4#45611yescharacteristic
136C4#45811yescharacteristic
137C2—41no—
138C2—161no—
139C2#161011yescharacteristic
140C2#219311yescharacteristic
141C2#219411yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2235C2semidirect
4. #2089C2semidirect
5. #1350C2semidirect
7. #1323C2semidirect

Maximal subgroups

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

subgroupindex
#32622
#22352
#22332
#20892
#13502
#13492
#13232