Finite Group Databasebeta

← #12339

Subgroups — #12339

Order 256.

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

normal subgroups only

This group has 154 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#3402C211yesmaximal, characteristic
2#2251C211yesmaximal, characteristic
3#2209C211yesmaximal, characteristic, complemented
4#1690C211yesmaximal, characteristic, complemented
5#1360C211yesmaximal, characteristic
6#1338C211yesmaximal, characteristic, complemented
7#1334C211yesmaximal, characteristic, complemented
8#556—21no—
9C2×Q32—21no—
10C2×Q32—21no—
11C2×SD32—21no—
12#497—21no—
13#495C2211yescharacteristic
14#494C2211yescharacteristic
15#475—21no—
16#464—21no—
17#463—21no—
18#462—21no—
19#446C2211yescharacteristic
20#440—21no—
21C4×Q16C2211yescharacteristic
22C4⋊C16C2211yescharacteristic
23#358—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#332—21no—
27C2×Q16—21no—
28C2×Q16—21no—
29C2×Q16—21no—
30C2×Q16D411yescharacteristic
31C2×Q16D411yescharacteristic
32C2×SD16—21no—
33C2×SD16—21no—
34C2×D8D411yescharacteristic
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37#126—41no—
38#126—41no—
39#124—21no—
40#124—21no—
41C22⋊Q8—41no—
42C4×Q8—21no—
43C4×Q8—21no—
44C4×Q8D411yescharacteristic
45C4×D4—21no—
46#117—41no—
47Q32—42no—
48Q32—42no—
49SD32—42no—
50C2×C16—21no—
51C2×C16—21no—
52#107D411yescharacteristic
53#106—41no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8D411yescharacteristic
57#103—21no—
58#103—21no—
59#103—41no—
60#102—21no—
61#97—21no—
62C4×C8C2311yescharacteristic
63C2×Q8—21no—
64C2×Q8—21no—
65C2×Q8—41no—
66C2×Q8D811yescharacteristic
67C2×D4—21no—
68C22×C4—41no—
69Q16—21no—
70Q16—21no—
71Q16—41no—
72Q16—41no—
73Q16—41no—
74Q16—41no—
75Q16—41no—
76Q16—81no—
77SD16—41no—
78SD16—41no—
79SD16—81no—
80D8—41no—
81C2×C8—21no—
82C2×C8—41no—
83C2×C8C2×D411yescharacteristic
84C2×C8C2×D411yescharacteristic
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4D811yescharacteristic
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C42—21no—
100C42—41no—
101C42—41no—
102C42C2×D411yescharacteristic
103C16—22no—
104C16—41no—
105C23—41no—
106Q8—41no—
107Q8—41no—
108Q8—41no—
109Q8—41no—
110Q8—41no—
111Q8—81no—
112Q8—81no—
113D4—41no—
114D4—41no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—82no—
121C2×C4—81no—
122C2×C4#12011yescharacteristic
123C2×C4C2×D811yescharacteristic
124C2×C4AGL(1,ℤ/8)11yescharacteristic
125C8—21no—
126C8—41no—
127C8—81no—
128C8#13511yescharacteristic
129C8AGL(1,ℤ/8)11yescharacteristic
130C22—81no—
131C22—81no—
132C22—81no—
133C22#44711yescharacteristic
134C4—21no—
135C4—21no—
136C4—41no—
137C4—41no—
138C4—81no—
139C4—81no—
140C4—81no—
141C4—161no—
142C4#44911yescharacteristic
143C4#45311yescharacteristic
144C2—161no—
145C2#162111yescharacteristic
146C2#219111yescharacteristic
147C2#219511yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radicalC2×D8D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2209C2semidirect
4. #1690C2semidirect
6. #1338C2semidirect
7. #1334C2semidirect

Maximal subgroups

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

subgroupindex
#34022
#22512
#22092
#16902
#13602
#13382
#13342