Finite Group Databasebeta

← #12807

Subgroups — #12807

Order 256.

214 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 63 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#2931C211yesmaximal, characteristic
2#2126C212yesmaximal
3#1812C212yesmaximal, complemented
4#1391C212yesmaximal, complemented
5C2×#131C411yescharacteristic, complemented
6C2×#131C2211yescharacteristic
7#532C2211yescharacteristic
8#436—22no—
9C8×D4—22no—
10#427—22no—
11#423C411yescharacteristic, complemented
12#410C2214yes—
13#408C411yescharacteristic, complemented
14#408C2211yescharacteristic
15C2×C4×C8C411yescharacteristic, complemented
16#354—22no—
17#353—22no—
18H(3,ℤ/4)—22no—
19C2×#41C2×C411yescharacteristic
20C2×#41C2311yescharacteristic
21#131—41no—
22#131—41no—
23#131—41no—
24#131—41no—
25C2×M16—21no—
26C2×M16C2×C411yescharacteristic
27C2×M16C2×C411yescharacteristic
28C2×M16D412yes—
29C22×C8—21no—
30C22×C8C2×C411yescharacteristic
31C22×C8C2×C411yescharacteristic
32C22×C8D412yes—
33C4⋊Q8Q811yescharacteristic
34#127D411yescharacteristic
35#124D411yescharacteristic
36#124Q811yescharacteristic
37#121—22no—
38C4×D4—22no—
39#117—24no—
40C2×C42C2×C411yescharacteristic
41#107—42no—
42#106—42no—
43C4⋊C8—22no—
44#99—24no—
45#99—24no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48C4×C8—21no—
49C4×C8—21no—
50#41—41no—
51#41—41no—
52#41—41no—
53#41—41no—
54C2×Q8C4⋊C411yescharacteristic
55C2×Q8C2×Q811yescharacteristic
56C2×D4—21no—
57C2×D4—22no—
58C2×D4C4⋊C411yescharacteristic
59C2×D4C2×D411yescharacteristic
60C2×D4#4112yes—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4C22×C411yescharacteristic
64C22×C4C2×D412yes—
65M16—41no—
66M16—42no—
67M16—42no—
68M16—41no—
69C2×C8—22no—
70C2×C8—22no—
71C2×C8—21no—
72C2×C8—21no—
73C2×C8—21no—
74C2×C8—21no—
75C2×C8—41no—
76C2×C8—41no—
77C2×C8—41no—
78C2×C8C22⋊C412yes—
79C2×C8C22⋊C411yescharacteristic
80C2×C8C22⋊C411yescharacteristic
81C4⋊C4—22no—
82C4⋊C4—44no—
83C22⋊C4—22no—
84C22⋊C4—22no—
85C22⋊C4—44no—
86C42—21no—
87C42—44no—
88C42C4⋊C411yescharacteristic
89C42C4⋊C411yescharacteristic
90C23—21no—
91C23#11711yescharacteristic
92C23#12112yes—
93Q8—41no—
94Q8—41no—
95D4—41no—
96D4—42no—
97D4—41no—
98D4—44no—
99D4—42no—
100C2×C4—21no—
101C2×C4—22no—
102C2×C4—21no—
103C2×C4—22no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—84no—
111C2×C4C2×C22⋊C411yescharacteristic
112C2×C4C2×C4⋊C411yescharacteristic
113C2×C4C22⋊Q812yes—
114C8—22no—
115C8—21no—
116C8—41no—
117C8—41no—
118C8#9911yescharacteristic
119C8#9911yescharacteristic
120C22—22no—
121C22—21no—
122C22—41no—
123C22—42no—
124C22—42no—
125C22—41no—
126C22#38011yescharacteristic
127C4—22no—
128C4—21no—
129C4—22no—
130C4—41no—
131C4—41no—
132C4—164no—
133C4C2×#9911yescharacteristic
134C4#41011yescharacteristic
135C2—21no—
136C2—42no—
137C2—41no—
138C2—81no—
139C2#179911yescharacteristic
subgroupG/H
centerC4#410
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
socleC2#1799
prime radical#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1812C2semidirect
4. #1391C2semidirect
5. C2×#131C4semidirect
11. #423C4semidirect
13. #408C4semidirect
15. C2×C4×C8C4semidirect

Maximal subgroups

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

subgroupindex
#29312
#21262
#18122
#13912