Finite Group Databasebeta

← #13583

Subgroups — #13583

Order 256.

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

normal subgroups only

This group has 208 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#3147C211yesmaximal, characteristic
2#2126C212yesmaximal, complemented
3#1409C212yesmaximal, complemented
4#1408C212yesmaximal, complemented
5C2×#135—22no—
6#532C411yescharacteristic, complemented
7#532C2211yescharacteristic
8#495—22no—
9#494C411yescharacteristic, complemented
10#493C2211yescharacteristic
11#492C411yescharacteristic, complemented
12#492C2211yescharacteristic
13#410—22no—
14C2×C4×C8C411yescharacteristic, complemented
15#354—22no—
16#353C2214yes—
17C2×#41—21no—
18C2×#41D412yes—
19#135—44no—
20#135—44no—
21C2×Q16—22no—
22C2×SD16—22no—
23C2×SD16—22no—
24C2×D8—22no—
25C22×C8—21no—
26C4⋊Q8C2×C411yescharacteristic
27C4⋊Q8C2×C411yescharacteristic
28#127C2×C411yescharacteristic
29#127C2311yescharacteristic
30#124—21no—
31#124D412yes—
32#121—41no—
33#121—41no—
34C4×D4—41no—
35C4×D4—41no—
36#117—42no—
37C2×C42C2×C411yescharacteristic
38#99—24no—
39#99—42no—
40C4×C8—21no—
41C4×C8C2×C411yescharacteristic
42C4×C8C2×C411yescharacteristic
43#41—42no—
44#41—42no—
45#41—42no—
46#41—42no—
47C2×Q8—21no—
48C2×Q8C22⋊C412yes—
49C2×D4—22no—
50C2×D4—21no—
51C2×D4—41no—
52C2×D4—41no—
53C2×D4—41no—
54C2×D4C2×D412yes—
55C22×C4—21no—
56C22×C4—22no—
57C22×C4—41no—
58C22×C4C22⋊C411yescharacteristic
59Q16—44no—
60SD16—44no—
61SD16—44no—
62D8—44no—
63C2×C8—22no—
64C2×C8—22no—
65C2×C8—21no—
66C2×C8—21no—
67C4⋊C4—41no—
68C4⋊C4—41no—
69C4⋊C4—82no—
70C22⋊C4—42no—
71C22⋊C4—42no—
72C22⋊C4—44no—
73C42—21no—
74C42—82no—
75C42C22⋊C411yescharacteristic
76C42C22×C411yescharacteristic
77C23—22no—
78C23—41no—
79C23#9911yescharacteristic
80Q8—42no—
81Q8—42no—
82D4—42no—
83D4—42no—
84D4—82no—
85D4—82no—
86D4—82no—
87D4—82no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—22no—
93C2×C4—42no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—41no—
97C2×C4—81no—
98C2×C4—81no—
99C2×C4—84no—
100C2×C4#9911yescharacteristic
101C2×C4C2×C22⋊C411yescharacteristic
102C8—22no—
103C8—22no—
104C22—21no—
105C22—41no—
106C22—42no—
107C22—82no—
108C22—82no—
109C22—81no—
110C22C2×#9911yescharacteristic
111C4—22no—
112C4—21no—
113C4—41no—
114C4—42no—
115C4—81no—
116C4—164no—
117C4#35311yescharacteristic
118C4#35311yescharacteristic
119C2—21no—
120C2—41no—
121C2—82no—
122C2—161no—
123C2C2×#35311yescharacteristic
subgroupG/H
centerC4#353
commutatorC42C22×C4
Frattini, cosocle#127C23
socleC2C2×#353
prime radical#3147C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G
LECSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2126C2semidirect
3. #1409C2semidirect
4. #1408C2semidirect
6. #532C4semidirect
9. #494C4semidirect
11. #492C4semidirect
14. C2×C4×C8C4semidirect

Maximal subgroups

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

subgroupindex
#31472
#21262
#14092
#14082