Finite Group Databasebeta

← #13560

Subgroups — #13560

Order 256.

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

normal subgroups only

This group has 243 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#1543C211yesmaximal, characteristic, complemented
4#1402C214yesmaximal, complemented
5#585C411yescharacteristic, complemented
6#575—21no—
7#532C2211yescharacteristic
8#456—21no—
9#455C412yescomplemented
10#454—21no—
11#453C2212yes—
12#431—21no—
13C2×#104—21no—
14#413C411yescharacteristic, complemented
15#410—21no—
16#354—21no—
17#353C2212yescomplemented
18#333—21no—
19#331C2212yescomplemented
202−1+4—21no—
212−1+4—41no—
22H(5,𝔽2)—41no—
23H(5,𝔽2)—41no—
24H(5,𝔽2)C2×C412yes—
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—
36C4⋊Q8D411yescharacteristic
37#127C2311yescharacteristic
38#124C2×C411yescharacteristic
39#124D411yescharacteristic
40#121—41no—
41C4×D4—41no—
42#117—41no—
43C2×C42D411yescharacteristic
44C4⋊C8—22no—
45#104—22no—
46#104—22no—
47#101—21no—
48#100C2×C412yes—
49#99—22no—
50#99—41no—
51C22⋊C8—41no—
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—44no—
78C2×D4—41no—
79C2×D4—42no—
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—
95C2×C8—41no—
96C2×C8—42no—
97C4⋊C4—41no—
98C4⋊C4—81no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—42no—
102C42—21no—
103C42—81no—
104C42C22⋊C411yescharacteristic
105C42C2×D411yescharacteristic
106C23—21no—
107C23—21no—
108C23—41no—
109C23—42no—
110C23#9911yescharacteristic
111Q8—22no—
112Q8—41no—
113Q8—41no—
114Q8—42no—
115Q8—41no—
116D4—22no—
117D4—41no—
118D4—41no—
119D4—42no—
120D4—42no—
121D4—42no—
122D4—44no—
123D4—42no—
124D4—41no—
125D4—82no—
126D4—81no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—44no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—82no—
144C2×C4#9911yescharacteristic
145C2×C4C2×C22⋊C411yescharacteristic
146C8—42no—
147C8—82no—
148C22—21no—
149C22—41no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22—81no—
155C22—161no—
156C22—162no—
157C22C2×#9911yescharacteristic
158C4—22no—
159C4—21no—
160C4—41no—
161C4—41no—
162C4—41no—
163C4—42no—
164C4—162no—
165C4AΣL(1,𝔽16)11yescharacteristic
166C4AΣL(1,𝔽16)11yescharacteristic
167C2—21no—
168C2—41no—
169C2—81no—
170C2—81no—
171C2—82no—
172C2C2×AΣL(1,𝔽16)11yescharacteristic
subgroupG/H
centerC4AΣL(1,𝔽16)
commutatorC2×D4C22×C4
Frattini, cosocle#127C23
socleC2C2×AΣL(1,𝔽16)
prime radical#3018C2
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G
LECSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2126C2semidirect
3. #1543C2semidirect
4. #1402C2semidirect
5. #585C4semidirect
9. #455C4semidirect
14. #413C4semidirect
17. #353C22semidirect
19. #331C22semidirect

Maximal subgroups

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

subgroupindex
#30182
#21262
#15432
#14022