Finite Group Databasebeta

← #12302

Subgroups — #12302

Order 256.

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

normal subgroups only

This group has 155 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#3202C211yesmaximal, characteristic
2#2239C212yesmaximal, complemented
3#2098C211yesmaximal, characteristic
4#1340C212yesmaximal, complemented
5#1321C211yesmaximal, characteristic
6#536—21no—
7#485—22no—
8#470C2212yes—
9#467—21no—
10#451—22no—
11#430C2211yescharacteristic
12#424—21no—
13#421—21no—
14#408C2211yescharacteristic
15#365—22no—
16#362—22no—
17#349C2212yes—
18H(3,ℤ/4)C2211yescharacteristic
19C2×#41D411yescharacteristic
20C22×Q8—21no—
21C2×Q16—21no—
22C2×Q16D412yes—
23C2×M16—21no—
24C2×M16C2311yescharacteristic
25C22×C8D411yescharacteristic
26C4⋊Q8—41no—
27#124—41no—
28#123—41no—
29#123—41no—
30C22⋊Q8—22no—
31C22⋊Q8—22no—
32#117—21no—
33#117—21no—
34M32—22no—
35#108—22no—
36#107—21no—
37#106D412yes—
38C4⋊C8—42no—
39#104—42no—
40#103—22no—
41#103—22no—
42C22⋊C8—22no—
43#41—41no—
44#41—41no—
45C2×Q8—22no—
46C2×Q8—41no—
47C2×Q8—81no—
48C2×Q8—81no—
49C2×Q8SD1611yescharacteristic
50C2×D4—21no—
51C2×D4SD1611yescharacteristic
52C22×C4—21no—
53C22×C4—41no—
54C22×C4C2×D411yescharacteristic
55Q16—42no—
56Q16—81no—
57M16—41no—
58M16—41no—
59M16—41no—
60C2×C8—21no—
61C2×C8—41no—
62C2×C8—41no—
63C2×C8C2×D412yes—
64C4⋊C4—22no—
65C4⋊C4—41no—
66C4⋊C4—42no—
67C4⋊C4—42no—
68C22⋊C4—41no—
69C22⋊C4—42no—
70C22⋊C4—41no—
71C22⋊C4—42no—
72C42—22no—
73C42—41no—
74C16—42no—
75C23—21no—
76C23AGL(1,ℤ/8)11yescharacteristic
77Q8—41no—
78Q8—42no—
79Q8—42no—
80Q8—82no—
81D4—41no—
82D4—81no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4—42no—
91C2×C4—82no—
92C2×C4#12011yescharacteristic
93C2×C4C2×SD1611yescharacteristic
94C8—22no—
95C8—41no—
96C8—81no—
97C22—21no—
98C22—41no—
99C22—42no—
100C22—41no—
101C22#45011yescharacteristic
102C4—21no—
103C4—41no—
104C4—82no—
105C4—82no—
106C4—82no—
107C4#45311yescharacteristic
108C4#45811yescharacteristic
109C2—21no—
110C2—41no—
111C2—81no—
112C2#161311yescharacteristic
subgroupG/H
center, socleC2#1613
commutator, Frattini, cosocleC2×M16C23
prime radicalC2×#41D4
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. #1340C2semidirect

Maximal subgroups

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

subgroupindex
#32022
#22392
#20982
#13402
#13212