Finite Group Databasebeta

← #12153

Subgroups — #12153

Order 256.

212 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 37 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#2582C211yesmaximal, characteristic, complemented
2#2039C211yesmaximal, characteristic
3#1610C212yesmaximal
4#1346C212yesmaximal, complemented
5#1325C211yesmaximal, characteristic, complemented
6#537C2211yescharacteristic
7C2×C4×Q8—21no—
8#480—22no—
9C2×#104—21no—
10#417—21no—
11#398—21no—
12#398C2212yes—
13#391—22no—
14#389—22no—
15#384C2211yescharacteristic
16#343C2211yescharacteristic
17#327—22no—
18#324C2212yes—
19C2×#41—21no—
20C22×Q8—21no—
21C2×M16—21no—
22C22⋊Q8—41no—
23#121—22no—
24#121—41no—
25C4×Q8—42no—
26C4×Q8—42no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4D411yescharacteristic
32C2×C4⋊C4D412yes—
33C2×C4⋊C4C2311yescharacteristic
34C2×C42—22no—
35C2×C42D411yescharacteristic
36#107—42no—
37#104—42no—
38#103—41no—
39#102—41no—
40#102—42no—
41C22⋊C8—22no—
42#95—22no—
43#95—21no—
44#95—22no—
45#95D412yes—
46#41—41no—
47#41—41no—
48#41—41no—
49#41—41no—
50C2×Q8—24no—
51C2×Q8—21no—
52C2×Q8—42no—
53C2×Q8—42no—
54C2×D4—21no—
55C2×D4—22no—
56C2×D4—41no—
57C22×C4—21no—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—41no—
62C22×C4C2×D411yescharacteristic
63C22×C4C2×D412yes—
64M16—41no—
65M16—41no—
66C2×C8—42no—
67C2×C8—41no—
68C4⋊C4—22no—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C4⋊C4—42no—
72C4⋊C4—44no—
73C4⋊C4—42no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C4⋊C4—81no—
77C22⋊C4—42no—
78C42—21no—
79C42—44no—
80C42D812yes—
81C23—41no—
82C23#12011yescharacteristic
83Q8—48no—
84Q8—41no—
85Q8—41no—
86D4—41no—
87D4—41no—
88D4—82no—
89D4—81no—
90D4—81no—
91C2×C4—22no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—24no—
97C2×C4—22no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—81no—
108C2×C4—81no—
109C2×C4C2×D811yescharacteristic
110C2×C4AGL(1,ℤ/8)11yescharacteristic
111C8—81no—
112C8—82no—
113C22—21no—
114C22—21no—
115C22—81no—
116C22—81no—
117C22—81no—
118C22#44711yescharacteristic
119C22#45411yescharacteristic
120C22B(4,𝔽2)11yescharacteristic
121C4—24no—
122C4—21no—
123C4—21no—
124C4—44no—
125C4—42no—
126C4—82no—
127C4—81no—
128C4—81no—
129C2—21no—
130C2—21no—
131C2—161no—
132C2#159811yescharacteristic
133C2#219811yescharacteristic
134C2#220511yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2582C2semidirect
4. #1346C2semidirect
5. #1325C2semidirect

Maximal subgroups

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

subgroupindex
#25822
#20392
#16102
#13462
#13252