Finite Group Databasebeta

← #12349

Subgroups — #12349

Order 256.

151 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 151 conjugacy classes of subgroups — too many to draw. Only its 39 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#3352C211yesmaximal, characteristic, complemented
2#2244C211yesmaximal, characteristic, complemented
3#2207C211yesmaximal, characteristic, complemented
4#1687C211yesmaximal, characteristic
5#1363C211yesmaximal, characteristic, complemented
6#1334C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic, complemented
8#552—21no—
9C2×SD32—21no—
10#499C2211yescharacteristic
11#493C2211yescharacteristic
12#485—21no—
13#484—21no—
14#479—21no—
15#470—21no—
16#459—21no—
17C4×Q16C2211yescharacteristic
18#436C2211yescharacteristic
19#428—21no—
20C4⋊C16C2211yescharacteristic
21#357—21no—
22#357—21no—
23#357—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#331—21no—
27C2×Q16D411yescharacteristic
28C2×D8—21no—
29C2×D8—21no—
30C2×D8—41no—
31C2×M16—21no—
32#127—21no—
33#126—41no—
34#125—21no—
35#125—41no—
36#123—41no—
37C22⋊Q8—21no—
38C22⋊Q8—21no—
39C4×Q8—21no—
40C4×D4D411yescharacteristic
41#117—21no—
42#117—41no—
43C2×C4⋊C4—21no—
44SD32—42no—
45C2×C16—21no—
46C2×C16—21no—
47#107—21no—
48#107D411yescharacteristic
49#107D411yescharacteristic
50#107D411yescharacteristic
51#106—21no—
52#106—21no—
53C4⋊C8—21no—
54C4⋊C8D411yescharacteristic
55#103—21no—
56#103—21no—
57#103—21no—
58#102—41no—
59C22⋊C8—21no—
60C4×C8C2311yescharacteristic
61C2×Q8—21no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4D811yescharacteristic
66C22×C4—21no—
67C22×C4—41no—
68Q16—21no—
69Q16—21no—
70D8—42no—
71D8—41no—
72D8—41no—
73D8—82no—
74M16—41no—
75M16—41no—
76C2×C8—21no—
77C2×C8—41no—
78C2×C8C2×D411yescharacteristic
79C2×C8C2×D411yescharacteristic
80C4⋊C4—21no—
81C4⋊C4—21no—
82C4⋊C4—21no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4D811yescharacteristic
90C22⋊C4—21no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C42—41no—
95C42—41no—
96C42C2×D411yescharacteristic
97C16—41no—
98C16—41no—
99C23—21no—
100C23—81no—
101Q8—41no—
102Q8—41no—
103D4—41no—
104D4—81no—
105D4—81no—
106D4—81no—
107D4—81no—
108D4—161no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—82no—
117C2×C4#12011yescharacteristic
118C2×C4C2×D811yescharacteristic
119C2×C4AGL(1,ℤ/8)11yescharacteristic
120C8—21no—
121C8—41no—
122C8—81no—
123C8#13511yescharacteristic
124C8AGL(1,ℤ/8)11yescharacteristic
125C22—41no—
126C22—41no—
127C22—41no—
128C22—161no—
129C22—161no—
130C22—161no—
131C22#44711yescharacteristic
132C4—21no—
133C4—21no—
134C4—41no—
135C4—81no—
136C4—81no—
137C4—81no—
138C4—81no—
139C4#44911yescharacteristic
140C4#45411yescharacteristic
141C2—81no—
142C2—321no—
143C2#162511yescharacteristic
144C2#219011yescharacteristic
145C2#219311yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#1687C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #436 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3352C2semidirect
2. #2244C2semidirect
3. #2207C2semidirect
5. #1363C2semidirect
6. #1334C2semidirect
7. #1330C2semidirect

Maximal subgroups

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

subgroupindex
#33522
#22442
#22072
#16872
#13632
#13342
#13302