Finite Group Databasebeta

← #12131

Subgroups — #12131

Order 256.

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

normal subgroups only

This group has 199 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#2809C211yesmaximal, characteristic, complemented
2#2007C211yesmaximal, characteristic
3#1616C212yesmaximal
4#1353C212yesmaximal, complemented
5#1319C211yesmaximal, characteristic, complemented
6C2×#137—21no—
7#537C2211yescharacteristic
8#481—22no—
9C2×#104—21no—
10#400C2212yes—
11#398—22no—
12#391—21no—
13#391C2211yescharacteristic
14#389—21no—
15#387—22no—
16#343C2211yescharacteristic
17#327—22no—
18#324C2212yes—
19C2×#41—21no—
20C22×Q8D411yescharacteristic
21#137—42no—
22#137—42no—
23C2×Q16—41no—
24C2×SD16—41no—
25C2×M16—21no—
26C22⋊Q8—41no—
27#121—22no—
28#121—41no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4D412yes—
34C2×C4⋊C4C2311yescharacteristic
35C2×C42—21no—
36C2×C42D411yescharacteristic
37#106—42no—
38#104—42no—
39#102—42no—
40C22⋊C8—22no—
41#95—22no—
42#95—22no—
43#95—22no—
44#95—22no—
45#95D412yes—
46#41—41no—
47#41—41no—
48#41—41no—
49#41—41no—
50C2×Q8—21no—
51C2×Q8—21no—
52C2×Q8—41no—
53C2×Q8—41no—
54C2×Q8SD1612yes—
55C2×D4—21no—
56C2×D4—22no—
57C2×D4—41no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—41no—
63C22×C4C2×D411yescharacteristic
64C22×C4C2×D412yes—
65Q16—44no—
66SD16—44no—
67M16—22no—
68M16—41no—
69C2×C8—41no—
70C2×C8—42no—
71C4⋊C4—22no—
72C4⋊C4—41no—
73C4⋊C4—42no—
74C4⋊C4—44no—
75C4⋊C4—42no—
76C4⋊C4—81no—
77C4⋊C4—81no—
78C22⋊C4—42no—
79C42—41no—
80C42—81no—
81C23—41no—
82C23#12011yescharacteristic
83Q8—42no—
84Q8—41no—
85Q8—41no—
86Q8—81no—
87D4—41no—
88D4—41no—
89D4—82no—
90D4—81no—
91D4—81no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—81no—
110C2×C4—81no—
111C2×C4C2×SD1611yescharacteristic
112C2×C4#13711yescharacteristic
113C8—42no—
114C8—82no—
115C22—21no—
116C22—21no—
117C22—81no—
118C22—81no—
119C22—81no—
120C22#44811yescharacteristic
121C22#45311yescharacteristic
122C22B(4,𝔽2)11yescharacteristic
123C4—21no—
124C4—21no—
125C4—42no—
126C4—42no—
127C4—42no—
128C4—82no—
129C4—81no—
130C4—82no—
131C2—21no—
132C2—21no—
133C2—161no—
134C2#159611yescharacteristic
135C2#220111yescharacteristic
136C2#220411yescharacteristic
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. #2809C2semidirect
4. #1353C2semidirect
5. #1319C2semidirect

Maximal subgroups

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

subgroupindex
#28092
#20072
#16162
#13532
#13192