Finite Group Databasebeta

← #12306

Subgroups — #12306

Order 256.

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

normal subgroups only

This group has 137 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#3263C211yesmaximal, characteristic
2#2239C211yesmaximal, characteristic, complemented
3#2238C211yesmaximal, characteristic
4#2103C211yesmaximal, characteristic, complemented
5#1342C211yesmaximal, characteristic, complemented
6#1341C211yesmaximal, characteristic
7#1328C211yesmaximal, characteristic, complemented
8#542—21no—
9#476—21no—
10#475—21no—
11#470C2211yescharacteristic
12#469C2211yescharacteristic
13#465—21no—
14#464—21no—
15#463—21no—
16#430C2211yescharacteristic
17#424C2211yescharacteristic
18#365—21no—
19#365—21no—
20#362—21no—
21#361—21no—
22#349C2211yescharacteristic
23#349C2211yescharacteristic
24#344—21no—
25#341—21no—
26#338C2211yescharacteristic
27C2×Q16D411yescharacteristic
28C2×SD16—21no—
29C2×D8D411yescharacteristic
30C2×M16—21no—
31C2×M16—21no—
32C2×M16C2311yescharacteristic
33C22×C8D411yescharacteristic
34C4⋊Q8—21no—
35#126—41no—
36#124—21no—
37#124—41no—
38#123—41no—
39C22⋊Q8—21no—
40C22⋊Q8—41no—
41#121—21no—
42C4×Q8—21no—
43C4×D4—21no—
44#117D411yescharacteristic
45M32—21no—
46M32—21no—
47#108—21no—
48#108—21no—
49#106—21no—
50#106D411yescharacteristic
51#106D411yescharacteristic
52C4⋊C8—21no—
53C4⋊C8—21no—
54#103—21no—
55#103—21no—
56#102—21no—
57#102—21no—
58C2×Q8—21no—
59C2×Q8—41no—
60C2×D4—21no—
61C2×D4—41no—
62C22×C4—41no—
63C22×C4C2×D411yescharacteristic
64Q16—41no—
65SD16—81no—
66D8—41no—
67M16—41no—
68M16—41no—
69M16—41no—
70M16—41no—
71M16—41no—
72C2×C8—21no—
73C2×C8—41no—
74C2×C8—41no—
75C2×C8—41no—
76C2×C8C2×D411yescharacteristic
77C2×C8C2×D411yescharacteristic
78C4⋊C4—21no—
79C4⋊C4—21no—
80C4⋊C4—21no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C22⋊C4—41no—
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4SD1611yescharacteristic
90C22⋊C4SD1611yescharacteristic
91C42—21no—
92C42—41no—
93C16—41no—
94C16—41no—
95C23—41no—
96C23C2×SD1611yescharacteristic
97Q8—41no—
98Q8—41no—
99Q8—161no—
100D4—41no—
101D4—41no—
102D4—161no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—161no—
111C2×C4#12011yescharacteristic
112C2×C4AGL(1,ℤ/8)11yescharacteristic
113C8—21no—
114C8—21no—
115C8—41no—
116C8—81no—
117C8—81no—
118C22—21no—
119C22—41no—
120C22—81no—
121C22—81no—
122C22—81no—
123C22#45011yescharacteristic
124C4—21no—
125C4—41no—
126C4—41no—
127C4—81no—
128C4—81no—
129C4—81no—
130C4#45411yescharacteristic
131C4#45811yescharacteristic
132C2—21no—
133C2—41no—
134C2—161no—
135C2#161611yescharacteristic
subgroupG/H
center, socleC2#1616
commutator, Frattini, cosocleC2×M16C23
prime radical#469C22
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. #2103C2semidirect
5. #1342C2semidirect
7. #1328C2semidirect

Maximal subgroups

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

subgroupindex
#32632
#22392
#22382
#21032
#13422
#13412
#13282