Finite Group Databasebeta

← #12343

Subgroups — #12343

Order 256.

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

normal subgroups only

This group has 159 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, complemented
2#2240C211yesmaximal, characteristic, complemented
3#2206C211yesmaximal, characteristic, complemented
4#1667C211yesmaximal, characteristic, complemented
5#1365C211yesmaximal, characteristic, complemented
6#1331C211yesmaximal, characteristic, complemented
7D4⋊C16C211yesmaximal, characteristic, complemented
8#555—21no—
9C2×D16—21no—
10#500C2211yescharacteristic
11#493C2211yescharacteristic
12#482—22no—
13#476—21no—
14#466—21no—
15#459—21no—
16C4×D8C2211yescharacteristic
17C8×D4C2211yescharacteristic
18#427—21no—
19C4⋊C16C2211yescharacteristic
20#357—21no—
21#357—22no—
22C4×C16C2211yescharacteristic
23#335C2211yescharacteristic
24#331—21no—
25C2×D8—21no—
26C2×D8—21no—
27C2×D8—41no—
28C2×D8D411yescharacteristic
29C22×C8—21no—
30C4⋊Q8—21no—
31#127—21no—
32#124—42no—
33#121—22no—
34#121—41no—
35C4×D4—21no—
36C4×D4D411yescharacteristic
37#117—22no—
38#117—41no—
39D16—42no—
40C2×C16—21no—
41C2×C16—21no—
42#107—21no—
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×D4—21no—
55C2×D4—42no—
56C2×D4—41no—
57C2×D4—41no—
58C2×D4—41no—
59C2×D4D811yescharacteristic
60C22×C4—21no—
61C22×C4—41no—
62D8—21no—
63D8—21no—
64D8—42no—
65D8—41no—
66D8—41no—
67D8—82no—
68C2×C8—21no—
69C2×C8—41no—
70C2×C8—41no—
71C2×C8—41no—
72C2×C8C2×D411yescharacteristic
73C2×C8C2×D411yescharacteristic
74C4⋊C4—21no—
75C4⋊C4—22no—
76C4⋊C4—41no—
77C4⋊C4D811yescharacteristic
78C22⋊C4—21no—
79C22⋊C4—42no—
80C22⋊C4—42no—
81C22⋊C4—41no—
82C42—42no—
83C42C2×D411yescharacteristic
84C16—41no—
85C16—41no—
86C23—21no—
87C23—41no—
88C23—81no—
89Q8—161no—
90D4—21no—
91D4—21no—
92D4—41no—
93D4—41no—
94D4—81no—
95D4—81no—
96D4—81no—
97D4—81no—
98D4—161no—
99D4—162no—
100C2×C4—21no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—81no—
106C2×C4—81no—
107C2×C4#12011yescharacteristic
108C2×C4C2×D811yescharacteristic
109C2×C4AGL(1,ℤ/8)11yescharacteristic
110C8—21no—
111C8—41no—
112C8—81no—
113C8#13511yescharacteristic
114C8AGL(1,ℤ/8)11yescharacteristic
115C22—41no—
116C22—41no—
117C22—41no—
118C22—81no—
119C22—81no—
120C22—81no—
121C22—161no—
122C22—161no—
123C22—161no—
124C22#44711yescharacteristic
125C4—21no—
126C4—41no—
127C4—81no—
128C4—82no—
129C4#44911yescharacteristic
130C4#45411yescharacteristic
131C4#50811yescharacteristic
132C4#50911yescharacteristic
133C2—81no—
134C2—161no—
135C2—321no—
136C2#162511yescharacteristic
137C2#218611yescharacteristic
138C2#219211yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
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. #3341C2semidirect
2. #2240C2semidirect
3. #2206C2semidirect
4. #1667C2semidirect
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
#22402
#22062
#16672
#13652
#13312
D4⋊C162