Finite Group Databasebeta

← #12413

Subgroups — #12413

Order 256.

133 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 133 conjugacy classes of subgroups — too many to draw. Only its 41 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#3390C211yesmaximal, characteristic
2#2255C211yesmaximal, characteristic
3#2225C211yesmaximal, characteristic
4#1696C211yesmaximal, characteristic
5#1363C211yesmaximal, characteristic
6#1338C211yesmaximal, characteristic
7#1334C211yesmaximal, characteristic
8#557—21no—
9#501—21no—
10#500C2211yescharacteristic
11#499C2211yescharacteristic
12#479—21no—
13#479—21no—
14#477—21no—
15#477—21no—
16#446C2211yescharacteristic
17#441—21no—
18C4×Q16C2211yescharacteristic
19#367—21no—
20#367—21no—
21#366—21no—
22C4⋊C16C2211yescharacteristic
23#358—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#333—21no—
27C2×Q16D411yescharacteristic
28C4⋊Q8—21no—
29C4⋊Q8—21no—
30C4⋊Q8—41no—
31#125—21no—
32#125—21no—
33#125—41no—
34#125—41no—
35C4×Q8—21no—
36C4×Q8—21no—
37C4×Q8—21no—
38C4×Q8—41no—
39C4×Q8D411yescharacteristic
40C2×C16—21no—
41C2×C16—21no—
42#107—21no—
43#107—21no—
44#107—21no—
45#107—41no—
46#107D411yescharacteristic
47#107D411yescharacteristic
48#107D411yescharacteristic
49#106—21no—
50#106—21no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8D411yescharacteristic
54#103—21no—
55#103—21no—
56#103—21no—
57#103—41no—
58#97—21no—
59C4×C8C2311yescharacteristic
60C2×Q8—21no—
61C2×Q8—41no—
62C2×Q8—41no—
63C2×Q8D811yescharacteristic
64Q16—21no—
65Q16Q1612yes—
66C2×C8—21no—
67C2×C8—41no—
68C2×C8C2×D411yescharacteristic
69C2×C8C2×D411yescharacteristic
70C4⋊C4—21no—
71C4⋊C4—21no—
72C4⋊C4—21no—
73C4⋊C4—21no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4D811yescharacteristic
85C42—21no—
86C42—41no—
87C42—41no—
88C42—41no—
89C42C2×D411yescharacteristic
90C16—22no—
91C16—41no—
92Q8—22no—
93Q8—41no—
94Q8—41no—
95Q8—82no—
96Q8—161no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—81no—
104C2×C4#12011yescharacteristic
105C2×C4C2×D811yescharacteristic
106C2×C4AGL(1,ℤ/8)11yescharacteristic
107C8—21no—
108C8—41no—
109C8—81no—
110C8C2×Q1611yescharacteristic
111C8#13711yescharacteristic
112C22#44711yescharacteristic
113C4—21no—
114C4—21no—
115C4—42no—
116C4—41no—
117C4—41no—
118C4—81no—
119C4—81no—
120C4—81no—
121C4—161no—
122C4#45111yescharacteristic
123C4#45611yescharacteristic
124C2#164911yescharacteristic
125C2#219111yescharacteristic
126C2#219311yescharacteristic
subgroupG/H
center, socle, prime radicalC22#447
commutator, Frattini, cosocleC4×C8C23
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#33902
#22552
#22252
#16962
#13632
#13382
#13342