Finite Group Databasebeta

← #8456

Subgroups — #8456

Order 256.

513 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 77 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#3496C211yesmaximal, characteristic
2#1955C211yesmaximal, characteristic, complemented
3#1551C212yesmaximal, complemented
4#1530C212yesmaximal, complemented
5#1466C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)C412yes—
7#547—22no—
8#546—24no—
9D42—22no—
10#537C412yes—
11#534—22no—
12#532C2212yes—
13C2×#121—22no—
14C2×C4×D4C2211yescharacteristic
15#432C2212yescomplemented
16C2×C4⋊C8C2211yescharacteristic, complemented
17C2×C22⋊C8—22no—
18C2×#97C2211yescharacteristic, complemented
19#333—22no—
20#331—22no—
21#329—24no—
22#323—22no—
23H(5,𝔽2)—48no—
24C2×#41—24no—
25C2×#41C2×C414yes—
26C22×D4—24no—
27C22×D4—22no—
28C22×D4—24no—
29C22×D4C2×C411yescharacteristic
30C23×C4D412yes—
31C22×C8—21no—
32C22×C8—21no—
33C4⋊Q8D412yes—
34#127D412yes—
35#124D414yes—
36#123—28no—
37C22⋊Q8—24no—
38#121—24no—
39#121—24no—
40#121—24no—
41#121—24no—
42#121—22no—
43#121—22no—
44#120—28no—
45C4×D4—22no—
46C4×D4—22no—
47C4×D4—22no—
48C4×D4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4C2×C411yescharacteristic
51C2×C22⋊C4—24no—
52C2×C22⋊C4D412yes—
53C2×C42C2311yescharacteristic
54C4⋊C8—22no—
55C4⋊C8—22no—
56C22⋊C8—22no—
57C22⋊C8—44no—
58#97—22no—
59#97—22no—
60C24—24no—
61C24C22⋊C412yes—
62#41—48no—
63#41—416no—
64C2×Q8C22⋊C414yes—
65C2×D4—24no—
66C2×D4—28no—
67C2×D4—24no—
68C2×D4—24no—
69C2×D4—22no—
70C2×D4—22no—
71C2×D4—22no—
72C2×D4—416no—
73C2×D4—48no—
74C2×D4—48no—
75C2×D4—82no—
76C2×D4—82no—
77C2×D4C22⋊C414yes—
78C22×C4—24no—
79C22×C4—24no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4C22⋊C412yes—
85C22×C4C22×C411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C2×C8—22no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—42no—
92C2×C8—42no—
93C2×C8—41no—
94C4⋊C4—24no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C22⋊C4—28no—
99C22⋊C4—24no—
100C22⋊C4—24no—
101C22⋊C4—24no—
102C42C2×D412yes—
103C42C2×D412yes—
104C23—22no—
105C23—22no—
106C23—24no—
107C23—42no—
108C23—42no—
109C23—48no—
110C23—84no—
111C23—84no—
112C23#10014yes—
113C23C2×C22⋊C411yescharacteristic
114Q8—44no—
115D4—28no—
116D4—44no—
117D4—416no—
118D4—48no—
119D4—42no—
120D4—42no—
121D4—82no—
122D4—84no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—24no—
128C2×C4—41no—
129C2×C4—48no—
130C2×C4—48no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—82no—
135C2×C4C2×C22⋊C412yes—
136C2×C4#12012yes—
137C2×C4#12011yescharacteristic
138C2×C4#12011yescharacteristic
139C8—42no—
140C8—82no—
141C8—81no—
142C22—21no—
143C22—21no—
144C22—24no—
145C22—44no—
146C22—42no—
147C22—42no—
148C22—44no—
149C22—44no—
150C22—44no—
151C22—82no—
152C22—168no—
153C22#37911yescharacteristic
154C22C2×#10012yes—
155C4—21no—
156C4—22no—
157C4—22no—
158C4—42no—
159C4—44no—
160C4#45311yescharacteristic
161C4#45511yescharacteristic
162C2—22no—
163C2—44no—
164C2—84no—
165C2#178311yescharacteristic
166C2#179211yescharacteristic
167C2#179411yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1955C2semidirect
3. #1551C2semidirect
4. #1530C2semidirect
5. #1466C2semidirect
15. #432C22semidirect
16. C2×C4⋊C8C22semidirect
18. C2×#97C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#19552
#15512
#15302
#14662