Finite Group Databasebeta

← #12382

Subgroups — #12382

Order 256.

144 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 43 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#3395C211yesmaximal, characteristic
2#2241C211yesmaximal, characteristic
3#2225C211yesmaximal, characteristic
4#1674C211yesmaximal, characteristic
5#1362C211yesmaximal, characteristic
6Q8⋊C16C211yesmaximal, characteristic
7#1332C211yesmaximal, characteristic
8#557—21no—
9#500C2211yescharacteristic
10#494—21no—
11#494C2211yescharacteristic
12#462—22no—
13#462—21no—
14#462—21no—
15C8×Q8C2211yescharacteristic
16C4×Q16—21no—
17C4×Q16C2211yescharacteristic
18#367—21no—
19C4⋊C16C2211yescharacteristic
20#358—21no—
21#358—22no—
22C4×C16C2211yescharacteristic
23#335C2211yescharacteristic
24#332—21no—
25C2×Q16—22no—
26C2×Q16—21no—
27C2×Q16—41no—
28C2×Q16D411yescharacteristic
29C2×Q16D412yes—
30C4⋊Q8—22no—
31C4⋊Q8—21no—
32C4⋊Q8—21no—
33#125—42no—
34#125—41no—
35C4×Q8—22no—
36C4×Q8—21no—
37C4×Q8—41no—
38C4×Q8D411yescharacteristic
39C2×C16—21no—
40C2×C16—21no—
41#107—21no—
42#107—21no—
43#107D411yescharacteristic
44C4⋊C8—21no—
45C4⋊C8—21no—
46C4⋊C8D411yescharacteristic
47#103—22no—
48#103—21no—
49#103—41no—
50C4×C8—21no—
51C4×C8C2311yescharacteristic
52C2×Q8—21no—
53C2×Q8—22no—
54C2×Q8—41no—
55C2×Q8D811yescharacteristic
56Q16—21no—
57Q16—21no—
58Q16—42no—
59Q16—41no—
60Q16—41no—
61Q16—82no—
62Q16—82no—
63C2×C8—21no—
64C2×C8—41no—
65C2×C8C2×D411yescharacteristic
66C2×C8C2×D411yescharacteristic
67C4⋊C4—21no—
68C4⋊C4—21no—
69C4⋊C4—42no—
70C4⋊C4—42no—
71C4⋊C4—42no—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C4⋊C4D811yescharacteristic
78C42—21no—
79C42—42no—
80C42—41no—
81C42C2×D411yescharacteristic
82C16—41no—
83C16—41no—
84Q8—21no—
85Q8—41no—
86Q8—42no—
87Q8—42no—
88Q8—41no—
89Q8—81no—
90Q8—81no—
91Q8SD3212yes—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—41no—
95C2×C4—42no—
96C2×C4—41no—
97C2×C4—81no—
98C2×C4#12011yescharacteristic
99C2×C4C2×D811yescharacteristic
100C2×C4AGL(1,ℤ/8)11yescharacteristic
101C8—21no—
102C8—41no—
103C8—81no—
104C8#13511yescharacteristic
105C8#13711yescharacteristic
106C22#44711yescharacteristic
107C4—22no—
108C4—21no—
109C4—41no—
110C4—81no—
111C4—82no—
112C4—81no—
113C4—161no—
114C4#45211yescharacteristic
115C4#45511yescharacteristic
116C4C2×SD3211yescharacteristic
117C4#51011yescharacteristic
118C2#164111yescharacteristic
119C2#218511yescharacteristic
120C2#219511yescharacteristic
subgroupG/H
center, socle, prime radicalC22#447
commutator, Frattini, cosocleC4×C8C23
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#33952
#22412
#22252
#16742
#13622
Q8⋊C162
#13322