Finite Group Databasebeta

← #12157

Subgroups — #12157

Order 256.

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

normal subgroups only

This group has 186 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#2822C211yesmaximal, characteristic, complemented
2#2040C211yesmaximal, characteristic
3#1610C212yesmaximal
4#1346C212yesmaximal, complemented
5#1324C211yesmaximal, characteristic, complemented
6#537C2211yescharacteristic
7C2×#125—21no—
8#480—22no—
9C2×#104—21no—
10#417—21no—
11#400—22no—
12#398—21no—
13#398—22no—
14#398C2212yes—
15#382C2211yescharacteristic
16#343C2211yescharacteristic
17#327—22no—
18#324C2212yes—
19C2×#41—21no—
20C2×M16—21no—
21#125—42no—
22#125—42no—
23C22⋊Q8—41no—
24#121—22no—
25#121—41no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4D411yescharacteristic
32C2×C4⋊C4D412yes—
33C2×C4⋊C4C2311yescharacteristic
34C2×C42D411yescharacteristic
35#107—42no—
36#104—42no—
37#103—41no—
38#102—41no—
39#102—42no—
40C22⋊C8—22no—
41#95—22no—
42#95—21no—
43#95—22no—
44#95—22no—
45#95D412yes—
46#41—41no—
47#41—41no—
48#41—41no—
49#41—41no—
50C2×Q8—21no—
51C2×D4—21no—
52C2×D4—22no—
53C2×D4—41no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—41no—
59C22×C4C2×D411yescharacteristic
60C22×C4C2×D412yes—
61M16—41no—
62M16—41no—
63C2×C8—42no—
64C2×C8—41no—
65C4⋊C4—22no—
66C4⋊C4—41no—
67C4⋊C4—41no—
68C4⋊C4—42no—
69C4⋊C4—42no—
70C4⋊C4—44no—
71C4⋊C4—42no—
72C4⋊C4—42no—
73C4⋊C4—42no—
74C4⋊C4—81no—
75C22⋊C4—42no—
76C42—21no—
77C42D812yes—
78C23—41no—
79C23#12011yescharacteristic
80Q8—41no—
81Q8—41no—
82D4—41no—
83D4—41no—
84D4—82no—
85D4—81no—
86D4—81no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—42no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—41no—
101C2×C4—42no—
102C2×C4—41no—
103C2×C4—81no—
104C2×C4—81no—
105C2×C4C2×D811yescharacteristic
106C2×C4AGL(1,ℤ/8)11yescharacteristic
107C8—81no—
108C8—82no—
109C22—21no—
110C22—21no—
111C22—81no—
112C22—81no—
113C22—81no—
114C22#44711yescharacteristic
115C22#45411yescharacteristic
116C22B(4,𝔽2)11yescharacteristic
117C4—21no—
118C4—21no—
119C4—42no—
120C4—42no—
121C4—82no—
122C4—82no—
123C4—81no—
124C4—81no—
125C2—21no—
126C2—21no—
127C2—161no—
128C2#159811yescharacteristic
129C2#220111yescharacteristic
130C2#220311yescharacteristic
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. #2822C2semidirect
4. #1346C2semidirect
5. #1324C2semidirect

Maximal subgroups

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

subgroupindex
#28222
#20402
#16102
#13462
#13242