Finite Group Databasebeta

← #12347

Subgroups — #12347

Order 256.

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

normal subgroups only

This group has 151 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#3341C211yesmaximal, characteristic
2#2241C211yesmaximal, characteristic, complemented
3#2208C211yesmaximal, characteristic, complemented
4#1669C211yesmaximal, characteristic, complemented
5#1365C211yesmaximal, characteristic, complemented
6#1331C211yesmaximal, characteristic, complemented
7D4⋊C16C211yesmaximal, characteristic, complemented
8#555—21no—
9C2×SD32—21no—
10#500C2211yescharacteristic
11#494C2211yescharacteristic
12#482—22no—
13#476—21no—
14#474—21no—
15#466—21no—
16C4×D8C2211yescharacteristic
17C8×D4C2211yescharacteristic
18#427—21no—
19C4⋊C16C2211yescharacteristic
20#358—21no—
21#358—22no—
22C4×C16C2211yescharacteristic
23#335C2211yescharacteristic
24#332—21no—
25C2×Q16—21no—
26C2×Q16—21no—
27C2×D8D411yescharacteristic
28C22×C8—21no—
29C4⋊Q8—21no—
30C4⋊Q8—21no—
31#124—42no—
32#121—22no—
33#121—41no—
34C4×D4—21no—
35C4×D4D411yescharacteristic
36#117—22no—
37#117—41no—
38SD32—42no—
39C2×C16—21no—
40C2×C16—21no—
41#107—21no—
42#107—41no—
43#107D411yescharacteristic
44#107D412yes—
45#106—22no—
46C4⋊C8—21no—
47C4⋊C8D411yescharacteristic
48#102—22no—
49#102—21no—
50#102—41no—
51C22⋊C8—21no—
52C4×C8C2311yescharacteristic
53C2×Q8—41no—
54C2×Q8—41no—
55C2×D4—21no—
56C2×D4—42no—
57C2×D4D811yescharacteristic
58C22×C4—21no—
59C22×C4—41no—
60Q16—42no—
61Q16—41no—
62Q16—41no—
63D8—21no—
64D8—21no—
65C2×C8—21no—
66C2×C8—41no—
67C2×C8—41no—
68C2×C8—41no—
69C2×C8C2×D411yescharacteristic
70C2×C8C2×D411yescharacteristic
71C4⋊C4—21no—
72C4⋊C4—22no—
73C4⋊C4—41no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4D811yescharacteristic
77C22⋊C4—21no—
78C22⋊C4—42no—
79C22⋊C4—42no—
80C22⋊C4—41no—
81C42—42no—
82C42C2×D411yescharacteristic
83C16—41no—
84C16—41no—
85C23—21no—
86C23—41no—
87Q8—81no—
88Q8—81no—
89Q8—161no—
90D4—21no—
91D4—21no—
92D4—41no—
93D4—41no—
94D4—162no—
95C2×C4—21no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—42no—
100C2×C4—81no—
101C2×C4—81no—
102C2×C4—81no—
103C2×C4#12011yescharacteristic
104C2×C4C2×D811yescharacteristic
105C2×C4AGL(1,ℤ/8)11yescharacteristic
106C8—21no—
107C8—41no—
108C8—81no—
109C8#13511yescharacteristic
110C8AGL(1,ℤ/8)11yescharacteristic
111C22—41no—
112C22—41no—
113C22—41no—
114C22—81no—
115C22—81no—
116C22—81no—
117C22#44711yescharacteristic
118C4—21no—
119C4—41no—
120C4—81no—
121C4—82no—
122C4—161no—
123C4#44911yescharacteristic
124C4#45411yescharacteristic
125C4#50811yescharacteristic
126C4#51011yescharacteristic
127C2—81no—
128C2—161no—
129C2#162511yescharacteristic
130C2#218911yescharacteristic
131C2#219411yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#3341C2
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
2. #2241C2semidirect
3. #2208C2semidirect
4. #1669C2semidirect
5. #1365C2semidirect
6. #1331C2semidirect
7. D4⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
#33412
#22412
#22082
#16692
#13652
#13312
D4⋊C162