Finite Group Databasebeta

← #13584

Subgroups — #13584

Order 256.

237 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 45 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#3151C211yesmaximal, characteristic
2C2×#353C212yesmaximal, complemented
3#1409C212yesmaximal, complemented
4#1408C212yesmaximal, complemented
5C2×AGL(1,ℤ/8)—22no—
6#532C411yescharacteristic, complemented
7C2×#127C2211yescharacteristic
8#496—22no—
9#493C411yescharacteristic, complemented
10#493C2211yescharacteristic
11#492C411yescharacteristic, complemented
12#492C2211yescharacteristic
13C2×#99—22no—
14C4×M16C411yescharacteristic, complemented
15#353—22no—
16#353C2214yes—
17C2×#41—21no—
18C22×D4—41no—
19C22×D4D412yes—
20AGL(1,ℤ/8)—44no—
21AGL(1,ℤ/8)—44no—
22C2×SD16—22no—
23C2×SD16—22no—
24C2×D8—22no—
25C2×D8—22no—
26C2×M16—21no—
27C4⋊Q8C2×C411yescharacteristic, complemented
28#127—41no—
29#127C2×C411yescharacteristic
30#127C2×C411yescharacteristic, complemented
31#127D412yes—
32#127C2311yescharacteristic
33#124—21no—
34#121—41no—
35C4×D4—41no—
36C2×C22⋊C4—42no—
37C2×C42C2×C411yescharacteristic
38#99—24no—
39#99—42no—
40#97—21no—
41C4×C8C2×C411yescharacteristic, complemented
42C4×C8C2×C411yescharacteristic, complemented
43C24—22no—
44#41—42no—
45#41—42no—
46C2×Q8—21no—
47C2×D4—22no—
48C2×D4—21no—
49C2×D4—42no—
50C2×D4—42no—
51C2×D4—41no—
52C2×D4—42no—
53C2×D4—41no—
54C2×D4—41no—
55C2×D4—41no—
56C2×D4—84no—
57C2×D4C22⋊C412yes—
58C2×D4C2×D412yes—
59C22×C4—21no—
60C22×C4—41no—
61C22×C4—82no—
62C22×C4C22⋊C411yescharacteristic
63SD16—44no—
64SD16—44no—
65D8—44no—
66D8—44no—
67M16—24no—
68C2×C8—21no—
69C2×C8—21no—
70C4⋊C4—41no—
71C22⋊C4—42no—
72C22⋊C4—44no—
73C22⋊C4—82no—
74C42—21no—
75C42C22⋊C411yescharacteristic
76C42C22×C411yescharacteristic
77C23—22no—
78C23—22no—
79C23—42no—
80C23—42no—
81C23—41no—
82C23—84no—
83C23#9911yescharacteristic
84Q8—42no—
85D4—42no—
86D4—42no—
87D4—42no—
88D4—82no—
89D4—82no—
90D4—82no—
91D4—82no—
92D4—84no—
93D4—82no—
94C2×C4—22no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—42no—
98C2×C4—41no—
99C2×C4—81no—
100C2×C4—81no—
101C2×C4—84no—
102C2×C4—164no—
103C2×C4#9911yescharacteristic
104C2×C4C2×C22⋊C411yescharacteristic
105C8—22no—
106C8—22no—
107C22—41no—
108C22—42no—
109C22—42no—
110C22—82no—
111C22—82no—
112C22—84no—
113C22—82no—
114C22—81no—
115C22—162no—
116C22#35312yes—
117C22C2×#9911yescharacteristic
118C4—22no—
119C4—21no—
120C4—21no—
121C4—41no—
122C4—81no—
123C4—164no—
124C2—21no—
125C2—22no—
126C2—82no—
127C2—82no—
128C2—161no—
129C2C2×#35311yescharacteristic
subgroupG/H
center, socleC2C2×#353
commutatorC42C22×C4
Frattini, cosocle#127C23
prime radical#3151C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G
LECSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#353C2semidirect
3. #1409C2semidirect
4. #1408C2semidirect
6. #532C4semidirect
9. #493C4semidirect
11. #492C4semidirect
14. C4×M16C4semidirect
27. C4⋊Q8C2×C4semidirect
30. #127C2×C4semidirect
41. C4×C8C2×C4semidirect
42. C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#31512
C2×#3532
#14092
#14082