Finite Group Databasebeta

← #12367

Subgroups — #12367

Order 256.

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

normal subgroups only

This group has 168 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#3293C211yesmaximal, characteristic
2#2230C212yesmaximal, complemented
3#1716C211yesmaximal, characteristic, complemented
4#1358C211yesmaximal, characteristic, complemented
5#1336C212yesmaximal, complemented
6#578—22no—
7#498—21no—
8#498C2211yescharacteristic
9#494C2211yescharacteristic
10#487—21no—
11#473—22no—
12#456—21no—
13#456—21no—
14#443C2212yes—
15#365—22no—
16#364C2212yes—
17#360—22no—
18#322C2211yescharacteristic
192−1+4—21no—
202−1+4—21no—
21#137—22no—
22#137—22no—
23#137—42no—
24#137—42no—
25#135—42no—
26#135D412yes—
27C2×Q16—22no—
28C2×Q16—42no—
29#131—22no—
30C4⋊Q8—21no—
31C4⋊Q8D411yescharacteristic
32C4⋊Q8D411yescharacteristic
33M32—22no—
34#108D412yes—
35#106—22no—
36#106—42no—
37#104—22no—
38#103—42no—
39#101—21no—
40#101—21no—
41C4×C8—21no—
42C4×C8C2311yescharacteristic
43#41—22no—
44#41—42no—
45#41—42no—
46#41—81no—
47#41—81no—
48C2×Q8—21no—
49C2×Q8—21no—
50C2×Q8—41no—
51C2×Q8—42no—
52C2×Q8—42no—
53Q16—42no—
54Q16—42no—
55Q16—42no—
56Q16—42no—
57Q16SD1612yes—
58SD16—22no—
59SD16—42no—
60SD16—42no—
61D8SD1612yes—
62M16—22no—
63M16—42no—
64C2×C8—22no—
65C2×C8—42no—
66C2×C8C2×D412yes—
67C4⋊C4—41no—
68C4⋊C4—41no—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C42C2×D411yescharacteristic
72C16—42no—
73Q8—22no—
74Q8—42no—
75Q8—42no—
76Q8—82no—
77Q8—81no—
78Q8—81no—
79D4—22no—
80D4—42no—
81D4—81no—
82D4—81no—
83C2×C4—21no—
84C2×C4—42no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—82no—
88C2×C4—82no—
89C2×C4—81no—
90C2×C4#12011yescharacteristic
91C8—42no—
92C8—42no—
93C8C2×SD1612yes—
94C8AGL(1,ℤ/8)12yes—
95C22—42no—
96C22#45311yescharacteristic
97C4—22no—
98C4—42no—
99C4—81no—
100C4—81no—
101C4—161no—
102C4#45012yes—
103C2—21no—
104C2—82no—
105C2#162911yescharacteristic
subgroupG/H
center, socleC2#1629
commutator, Frattini, cosocleC4×C8C23
prime radical#3293C2
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. #1716C2semidirect
4. #1358C2semidirect
5. #1336C2semidirect

Maximal subgroups

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

subgroupindex
#32932
#22302
#17162
#13582
#13362