Finite Group Databasebeta

← #13568

Subgroups — #13568

Order 256.

249 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 45 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#3018C211yesmaximal, characteristic
2#2126C211yesmaximal, characteristic, complemented
3#1525C211yesmaximal, characteristic, complemented
4#1404C214yesmaximal, complemented
5#585C411yescharacteristic, complemented
6#575—21no—
7#532C2211yescharacteristic
8#456C412yescomplemented
9#455—21no—
10#454C2212yes—
11#453—21no—
12C2×#104—21no—
13#413C411yescharacteristic, complemented
14#410—21no—
15C4×M16—21no—
16#354C2212yescomplemented
17#353—21no—
18#329—21no—
19#329C2212yescomplemented
202−1+4—41no—
212−1+4C2×C412yes—
22H(5,𝔽2)—21no—
23H(5,𝔽2)—41no—
24H(5,𝔽2)—41no—
25C2×#41—21no—
26C2×#41—41no—
27C2×#41—42no—
28C2×#41C2×C411yescharacteristic
29C2×#41D411yescharacteristic
30#137—22no—
31#137—41no—
32AGL(1,ℤ/8)—22no—
33AGL(1,ℤ/8)—41no—
34#135—42no—
35C2×M16—21no—
36C2×M16—41no—
37C4⋊Q8C2×C411yescharacteristic
38#127D411yescharacteristic
39#124D411yescharacteristic
40#124C2311yescharacteristic
41#121—41no—
42C4×D4—41no—
43#117—41no—
44C2×C42D411yescharacteristic
45#104—22no—
46#104—22no—
47#101C2×C412yes—
48#100—21no—
49#99—22no—
50#99—41no—
51#97—22no—
52C4×C8—41no—
53#41—22no—
54#41—22no—
55#41—41no—
56#41—42no—
57#41—41no—
58#41—41no—
59#41—41no—
60#41—41no—
61#41—42no—
62#41—42no—
63#41—44no—
64#41—42no—
65#41—41no—
66#41—42no—
67C2×Q8—21no—
68C2×Q8—42no—
69C2×Q8—41no—
70C2×Q8C22⋊C411yescharacteristic
71C2×Q8C22×C411yescharacteristic
72C2×D4—21no—
73C2×D4—21no—
74C2×D4—21no—
75C2×D4—41no—
76C2×D4—42no—
77C2×D4—42no—
78C2×D4—44no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4C22⋊C411yescharacteristic
82C2×D4C2×D411yescharacteristic
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—41no—
87C22×C4—42no—
88C22×C4C22⋊C411yescharacteristic
89Q16—42no—
90SD16—42no—
91SD16—42no—
92D8—42no—
93M16—22no—
94M16—41no—
95M16—44no—
96C2×C8—42no—
97C2×C8—41no—
98C4⋊C4—41no—
99C4⋊C4—81no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—42no—
103C42—21no—
104C42—81no—
105C42C22⋊C411yescharacteristic
106C42C2×D411yescharacteristic
107C23—21no—
108C23—21no—
109C23—41no—
110C23—42no—
111C23#9911yescharacteristic
112Q8—22no—
113Q8—41no—
114Q8—41no—
115Q8—42no—
116Q8—41no—
117D4—22no—
118D4—41no—
119D4—41no—
120D4—42no—
121D4—42no—
122D4—42no—
123D4—44no—
124D4—42no—
125D4—41no—
126D4—82no—
127D4—81no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—44no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—82no—
145C2×C4#9911yescharacteristic
146C2×C4C2×C22⋊C411yescharacteristic
147C8—42no—
148C8—44no—
149C22—21no—
150C22—41no—
151C22—42no—
152C22—41no—
153C22—41no—
154C22—81no—
155C22—81no—
156C22—161no—
157C22—162no—
158C22C2×#9911yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—41no—
163C4—41no—
164C4—42no—
165C4—162no—
166C4AΣL(1,𝔽16)11yescharacteristic
167C4AΣL(1,𝔽16)11yescharacteristic
168C2—21no—
169C2—41no—
170C2—81no—
171C2—81no—
172C2—82no—
173C2C2×AΣL(1,𝔽16)11yescharacteristic
subgroupG/H
centerC4AΣL(1,𝔽16)
commutatorC2×Q8C22×C4
Frattini, cosocle#124C23
socleC2C2×AΣL(1,𝔽16)
prime radical#3018C2
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G
LECSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2126C2semidirect
3. #1525C2semidirect
4. #1404C2semidirect
5. #585C4semidirect
8. #456C4semidirect
13. #413C4semidirect
16. #354C22semidirect
19. #329C22semidirect

Maximal subgroups

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

subgroupindex
#30182
#21262
#15252
#14042