Finite Group Databasebeta

← #12364

Subgroups — #12364

Order 256.

179 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 39 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#3291C211yesmaximal, characteristic
2#2230C212yesmaximal, complemented
3#1705C211yesmaximal, characteristic, complemented
4#1359C211yesmaximal, characteristic, complemented
5#1335C212yesmaximal, complemented
6#577—22no—
7#498—21no—
8#498C2211yescharacteristic, complemented
9#492—21no—
10#492C2211yescharacteristic
11#471—22no—
12#456—21no—
13#453—21no—
14#443C2212yes—
15#365—22no—
16#364C2212yescomplemented
17#360—22no—
18C82C2211yescharacteristic, complemented
192−1+4—21no—
20H(5,𝔽2)—21no—
21#137—22no—
22#137—42no—
23AGL(1,ℤ/8)—22no—
24AGL(1,ℤ/8)—42no—
25#135—42no—
26#135D412yes—
27C2×SD16—22no—
28C2×SD16—42no—
29C2×SD16—42no—
30#131—22no—
31C4⋊Q8—21no—
32C4⋊Q8D411yescharacteristic
33#127D411yescharacteristic
34M32—22no—
35#108D412yes—
36#106—22no—
37#106—42no—
38#104—22no—
39#101—21no—
40#100—21no—
41C4×C8—21no—
42C4×C8C2311yescharacteristic
43#41—22no—
44#41—42no—
45#41—42no—
46#41—81no—
47C2×Q8—21no—
48C2×Q8—41no—
49C2×Q8—42no—
50C2×D4—21no—
51C2×D4—41no—
52C2×D4—42no—
53C2×D4—81no—
54Q16—42no—
55Q16SD1612yescomplemented
56SD16—22no—
57SD16—42no—
58SD16—42no—
59SD16—42no—
60SD16—42no—
61SD16—84no—
62D8—42no—
63D8SD1612yescomplemented
64M16—22no—
65M16—42no—
66C2×C8—22no—
67C2×C8—42no—
68C2×C8C2×D412yes—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C4⋊C4—41no—
72C42C2×D411yescharacteristic
73C16—42no—
74C23—41no—
75C23—81no—
76Q8—22no—
77Q8—42no—
78Q8—82no—
79Q8—81no—
80D4—22no—
81D4—42no—
82D4—42no—
83D4—82no—
84D4—82no—
85D4—81no—
86C2×C4—21no—
87C2×C4—42no—
88C2×C4—41no—
89C2×C4—81no—
90C2×C4—82no—
91C2×C4—81no—
92C2×C4#12011yescharacteristic
93C8—24no—
94C8—42no—
95C8C2×SD1612yes—
96C8AGL(1,ℤ/8)12yes—
97C22—42no—
98C22—81no—
99C22—161no—
100C22—321no—
101C22#45311yescharacteristic
102C4—22no—
103C4—42no—
104C4—81no—
105C4—161no—
106C4#45012yes—
107C2—21no—
108C2—82no—
109C2—161no—
110C2#162911yescharacteristic
subgroupG/H
center, socleC2#1629
commutator, Frattini, cosocleC4×C8C23
prime radical#3291C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2230C2semidirect
3. #1705C2semidirect
4. #1359C2semidirect
5. #1335C2semidirect
8. #498C22semidirect
16. #364C22semidirect
18. C82C22semidirect
55. Q16SD16semidirect
63. D8SD16semidirect

Maximal subgroups

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

subgroupindex
#32912
#22302
#17052
#13592
#13352