Finite Group Databasebeta

← #12309

Subgroups — #12309

Order 256.

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

normal subgroups only

This group has 140 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#3256C211yesmaximal, characteristic
2#2239C211yesmaximal, characteristic, complemented
3#2238C211yesmaximal, characteristic
4#2102C211yesmaximal, characteristic, complemented
5#1340C211yesmaximal, characteristic, complemented
6#1339C211yesmaximal, characteristic
7#1327C211yesmaximal, characteristic, complemented
8#529—21no—
9#479—21no—
10#478—21no—
11#470C2211yescharacteristic
12#469C2211yescharacteristic
13#468—21no—
14#464—21no—
15#463—21no—
16#430C2211yescharacteristic
17#423C2211yescharacteristic
18#365—21no—
19#365—21no—
20#362—21no—
21#361—21no—
22#349C2211yescharacteristic
23#349C2211yescharacteristic
24#344—21no—
25#339—21no—
26H(3,ℤ/4)C2211yescharacteristic
27C2×Q16D411yescharacteristic
28C2×SD16—21no—
29C2×D8D411yescharacteristic
30C2×M16—21no—
31C2×M16D411yescharacteristic
32C2×M16C2311yescharacteristic
33#126—41no—
34#125—21no—
35#124—21no—
36#123—41no—
37C22⋊Q8—21no—
38#121—21no—
39C4×Q8—21no—
40C4×D4—21no—
41C4×D4—41no—
42#117—21no—
43#117—41no—
44C2×C42D411yescharacteristic
45M32—21no—
46M32—21no—
47#108—21no—
48#108—21no—
49#107—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×D4—21no—
60C2×D4—41no—
61C22×C4—21no—
62C22×C4—41no—
63C22×C4C2×D411yescharacteristic
64Q16—41no—
65SD16—81no—
66D8—41no—
67M16—41no—
68M16—41no—
69M16—41no—
70M16—41no—
71C2×C8—21no—
72C2×C8—41no—
73C2×C8C2×D411yescharacteristic
74C2×C8C2×D411yescharacteristic
75C4⋊C4—21no—
76C4⋊C4—21no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C22⋊C4—41no—
83C22⋊C4—41no—
84C22⋊C4—41no—
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C42—21no—
89C42—41no—
90C42—41no—
91C42SD1611yescharacteristic
92C42SD1611yescharacteristic
93C16—41no—
94C16—41no—
95C23—41no—
96C23AGL(1,ℤ/8)11yescharacteristic
97Q8—41no—
98Q8—41no—
99D4—41no—
100D4—41no—
101D4—82no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—81no—
110C2×C4—82no—
111C2×C4—81no—
112C2×C4#12011yescharacteristic
113C2×C4C2×SD1611yescharacteristic
114C8—21no—
115C8—21no—
116C8—41no—
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—161no—
131C4#45411yescharacteristic
132C4#45811yescharacteristic
133C2—21no—
134C2—41no—
135C2—161no—
136C2#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. #2102C2semidirect
5. #1340C2semidirect
7. #1327C2semidirect

Maximal subgroups

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

subgroupindex
#32562
#22392
#22382
#21022
#13402
#13392
#13272