Finite Group Databasebeta

← #18403

Subgroups — #18403

Order 256.

499 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 129 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#3502C211yesmaximal, characteristic, complemented
2#3208C212yesmaximal
3#2548C211yesmaximal, characteristic
4#1599C218yesmaximal, complemented
5#1503C212yesmaximal, complemented
6#1461C211yesmaximal, characteristic, complemented
7Q82—22no—
8#557—22no—
9#557—24no—
10#544—21no—
11#544C2214yes—
12#533—22no—
13#533C2212yes—
14C2×#127C2211yescharacteristic
15C2×#124—21no—
16#519—21no—
17C2×C4×Q8C2211yescharacteristic
18#492—22no—
19#486—22no—
20#460—24no—
21#450C2218yes—
22#431C2212yes—
23#394C2214yes—
24#390—21no—
25#390—22no—
26C4×C4⋊C4C2211yescharacteristic
27#328C2218yes—
28#324C2214yes—
29C22×Q8D411yescharacteristic
30C22×D4—21no—
31C22×D4—21no—
32C22×D4C2312yes—
33C2×SD16—28no—
34C4⋊Q8—22no—
35C4⋊Q8—22no—
36C4⋊Q8—24no—
37C4⋊Q8—28no—
38C4⋊Q8D412yes—
39C4⋊Q8D412yes—
40#127—22no—
41#127—22no—
42#125—24no—
43#125—24no—
44#125—28no—
45#125—22no—
46#124—44no—
47C22⋊Q8—22no—
48C22⋊Q8—24no—
49C22⋊Q8C2314yes—
50C4×Q8—24no—
51C4×Q8—24no—
52C4×Q8—24no—
53C4×Q8—24no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8D414yes—
58#117—24no—
59#117—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C42—21no—
66C2×C42D412yes—
67C2×C42C2311yescharacteristic
68C4⋊C8—22no—
69#102—28no—
70C22⋊C8C2314yes—
71C4×C8—22no—
72#95C2312yes—
73C24—22no—
74C2×Q8—24no—
75C2×Q8—22no—
76C2×Q8—22no—
77C2×Q8—22no—
78C2×Q8SD1614yescomplemented
79C2×Q8C2×D414yes—
80C2×D4—24no—
81C2×D4—44no—
82C2×D4—41no—
83C2×D4—42no—
84C2×D4—41no—
85C2×D4—42no—
86C2×D4—42no—
87C2×D4—82no—
88C22×C4—22no—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D412yes—
91C22×C4C2×D412yes—
92C22×C4C2×D411yescharacteristic
93C22×C4C2411yescharacteristic
94SD16—88no—
95C2×C8—24no—
96C4⋊C4—24no—
97C4⋊C4—24no—
98C4⋊C4—24no—
99C4⋊C4—28no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—24no—
105C4⋊C4—24no—
106C4⋊C4—24no—
107C4⋊C4—42no—
108C4⋊C4C2×D414yes—
109C22⋊C4—24no—
110C22⋊C4—48no—
111C22⋊C4—84no—
112C42—28no—
113C42—22no—
114C42—22no—
115C42—24no—
116C42—42no—
117C42C2×D412yes—
118C42C2×D412yes—
119C23—44no—
120C23—44no—
121C23—42no—
122C23—42no—
123C23—42no—
124C23C22×D411yescharacteristic
125Q8—28no—
126Q8—44no—
127Q8—44no—
128Q8—84no—
129D4—48no—
130D4—82no—
131D4—84no—
132D4—82no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4#12011yescharacteristic
145C2×C4#12012yes—
146C2×C4#12011yescharacteristic
147C2×C4C2×SD1614yes—
148C2×C4C2×SD1612yes—
149C2×C4#13712yes—
150C2×C4C22×D412yes—
151C8—44no—
152C22—21no—
153C22—21no—
154C22—84no—
155C22—84no—
156C22—82no—
157C22—82no—
158C22—82no—
159C22C2×#12011yescharacteristic
160C22C22×SD1611yescharacteristic
161C22C2×#13711yescharacteristic
162C4—24no—
163C4—21no—
164C4—22no—
165C4—44no—
166C4—42no—
167C4—44no—
168C4—82no—
169C4#44814yes—
170C4#45412yes—
171C2—21no—
172C2—21no—
173C2—162no—
174C2C2×#44811yescharacteristic
175C2C2×#45411yescharacteristic
176C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3502C2semidirect
4. #1599C2semidirect
5. #1503C2semidirect
6. #1461C2semidirect
78. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#35022
#32082
#25482
#15992
#15032
#14612