Finite Group Databasebeta

← #12299

Subgroups — #12299

Order 256.

162 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 35 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#3195C211yesmaximal, characteristic
2#2239C212yesmaximal, complemented
3#2091C211yesmaximal, characteristic
4#1342C212yesmaximal, complemented
5#1320C211yesmaximal, characteristic
6C2×C22⋊Q8—21no—
7#483—22no—
8#470—21no—
9#470C2212yes—
10#451—22no—
11#430—21no—
12#430C2211yescharacteristic
13#407—21no—
14#407C2211yescharacteristic
15#365—22no—
16#362—22no—
17#349C2212yes—
18#338C2211yescharacteristic
19C22×Q8—21no—
20C23×C4D411yescharacteristic
21C2×Q16—21no—
22C2×Q16D412yes—
23C2×M16—21no—
24C2×M16—21no—
25C2×M16D411yescharacteristic
26C2×M16C2311yescharacteristic
27C22⋊Q8—22no—
28C22⋊Q8—22no—
29C22⋊Q8—41no—
30C22⋊Q8—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—41no—
33C2×C22⋊C4—41no—
34M32—22no—
35#108—22no—
36#108—42no—
37#106—21no—
38#106D412yes—
39#103—22no—
40#103—22no—
41C22⋊C8—22no—
42C22⋊C8—42no—
43C24SD1611yescharacteristic
44C2×Q8—22no—
45C2×Q8—41no—
46C2×Q8—81no—
47C2×Q8—81no—
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—41no—
51C22×C4—41no—
52C22×C4—41no—
53C22×C4—41no—
54C22×C4SD1611yescharacteristic
55C22×C4C2×D411yescharacteristic
56Q16—42no—
57Q16—81no—
58M16—41no—
59M16—41no—
60M16—41no—
61M16—41no—
62M16—41no—
63M16—41no—
64C2×C8—22no—
65C2×C8—21no—
66C2×C8—41no—
67C2×C8C2×D412yes—
68C4⋊C4—22no—
69C4⋊C4—41no—
70C4⋊C4—42no—
71C4⋊C4—42no—
72C22⋊C4—42no—
73C22⋊C4—42no—
74C16—42no—
75C23—21no—
76C23—41no—
77C23—42no—
78C23C2×SD1611yescharacteristic
79Q8—42no—
80Q8—42no—
81Q8—82no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4—81no—
91C2×C4—82no—
92C2×C4—82no—
93C2×C4#12011yescharacteristic
94C2×C4AGL(1,ℤ/8)11yescharacteristic
95C8—22no—
96C8—42no—
97C8—41no—
98C8—81no—
99C22—22no—
100C22—21no—
101C22—41no—
102C22—41no—
103C22—42no—
104C22—41no—
105C22—81no—
106C22#45011yescharacteristic
107C4—21no—
108C4—41no—
109C4—82no—
110C4—82no—
111C4#45311yescharacteristic
112C4#45811yescharacteristic
113C2—21no—
114C2—42no—
115C2—41no—
116C2#161311yescharacteristic
subgroupG/H
center, socleC2#1613
commutator, Frattini, cosocleC2×M16C23
prime radicalC24SD16
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2239C2semidirect
4. #1342C2semidirect

Maximal subgroups

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

subgroupindex
#31952
#22392
#20912
#13422
#13202