Finite Group Databasebeta

← #12341

Subgroups — #12341

Order 256.

153 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 153 conjugacy classes of subgroups — too many to draw. Only its 41 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#3342C211yesmaximal, characteristic, complemented
2#2240C211yesmaximal, characteristic, complemented
3#2207C211yesmaximal, characteristic, complemented
4#1667C211yesmaximal, characteristic
5#1364C211yesmaximal, characteristic, complemented
6#1332C211yesmaximal, characteristic, complemented
7D4⋊C16C211yesmaximal, characteristic, complemented
8#556—21no—
9C2×SD32—21no—
10#500C2211yescharacteristic
11#493C2211yescharacteristic
12#485—22no—
13#475—21no—
14#467—21no—
15#459—21no—
16C4×Q16C2211yescharacteristic
17C8×D4C2211yescharacteristic
18#427—21no—
19C4⋊C16C2211yescharacteristic
20#357—21no—
21#357—22no—
22C4×C16C2211yescharacteristic
23#335C2211yescharacteristic
24#331—21no—
25C2×Q16D411yescharacteristic
26C2×D8—21no—
27C2×D8—21no—
28C2×D8—41no—
29C22×C8—21no—
30C4⋊Q8—21no—
31#127—21no—
32#126—42no—
33#124—41no—
34C22⋊Q8—22no—
35C4×Q8—21no—
36C4×Q8—41no—
37C4×D4D411yescharacteristic
38#117—22no—
39SD32—42no—
40C2×C16—21no—
41C2×C16—21no—
42#107—21no—
43#107D411yescharacteristic
44#107D412yes—
45#106—22no—
46C4⋊C8—21no—
47C4⋊C8D411yescharacteristic
48#103—22no—
49#103—21no—
50#102—41no—
51C22⋊C8—21no—
52C4×C8C2311yescharacteristic
53C2×Q8—21no—
54C2×Q8—41no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4—41no—
58C2×D4D811yescharacteristic
59C22×C4—21no—
60Q16—21no—
61Q16—21no—
62D8—42no—
63D8—41no—
64D8—41no—
65D8—82no—
66C2×C8—21no—
67C2×C8—41no—
68C2×C8—41no—
69C2×C8—41no—
70C2×C8C2×D411yescharacteristic
71C2×C8C2×D411yescharacteristic
72C4⋊C4—21no—
73C4⋊C4—22no—
74C4⋊C4—42no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C4⋊C4D811yescharacteristic
78C22⋊C4—21no—
79C22⋊C4—42no—
80C22⋊C4—42no—
81C42—42no—
82C42—41no—
83C42C2×D411yescharacteristic
84C16—41no—
85C16—41no—
86C23—21no—
87C23—81no—
88Q8—41no—
89Q8—41no—
90Q8—81no—
91Q8—81no—
92D4—21no—
93D4—21no—
94D4—81no—
95D4—81no—
96D4—81no—
97D4—81no—
98D4—161no—
99C2×C4—21no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—42no—
104C2×C4—41no—
105C2×C4#12011yescharacteristic
106C2×C4C2×D811yescharacteristic
107C2×C4AGL(1,ℤ/8)11yescharacteristic
108C8—21no—
109C8—41no—
110C8—81no—
111C8#13511yescharacteristic
112C8AGL(1,ℤ/8)11yescharacteristic
113C22—41no—
114C22—41no—
115C22—41no—
116C22—161no—
117C22—161no—
118C22—161no—
119C22#44711yescharacteristic
120C4—21no—
121C4—41no—
122C4—81no—
123C4—82no—
124C4—81no—
125C4#44911yescharacteristic
126C4#45411yescharacteristic
127C4#50811yescharacteristic
128C4#50911yescharacteristic
129C2—81no—
130C2—321no—
131C2#162511yescharacteristic
132C2#218611yescharacteristic
133C2#219311yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#1667C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3342C2semidirect
2. #2240C2semidirect
3. #2207C2semidirect
5. #1364C2semidirect
6. #1332C2semidirect
7. D4⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
#33422
#22402
#22072
#16672
#13642
#13322
D4⋊C162