Finite Group Databasebeta

← #29653

Subgroups — #29653

Order 256.

478 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#544C211yesmaximal, characteristic
2#3095C212yesmaximal
3#2502C211yesmaximal, characteristic
4#2060C214yesmaximal, complemented
5#2028C214yesmaximal, complemented
6#1826C212yesmaximal
7#1821C211yesmaximal, characteristic
8#544—24no—
9#544—22no—
10#544C2214yes—
11C2×C4⋊Q8—22no—
12C2×#125—22no—
13C2×C22⋊Q8—22no—
14C2×C22⋊Q8C2211yescharacteristic
15C2×#117C2212yes—
16C22×C4⋊C4C2211yescharacteristic
17#485—24no—
18#484—22no—
19#483—22no—
20#428C2214yes—
21#419C2214yes—
22#407C2212yes—
23#398—22no—
24#391C2212yes—
25#385—22no—
26#384C2212yes—
27C4×C22⋊C4C2211yescharacteristic
28#343C2214yes—
29#339C2218yes—
30C22×Q8—22no—
31C22×Q8C2311yescharacteristic
32C23×C4D412yes—
33C23×C4C2311yescharacteristic
34C2×M16C2314yes—
35C4⋊Q8—28no—
36C4⋊Q8—28no—
37#125—28no—
38#125—28no—
39C22⋊Q8—28no—
40C22⋊Q8—28no—
41C22⋊Q8—22no—
42C22⋊Q8—22no—
43#117—22no—
44#117—22no—
45#117D414yes—
46#117C2314yes—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4D412yes—
54C2×C4⋊C4D412yes—
55C2×C4⋊C4D412yes—
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2312yes—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C42—22no—
62C2×C42C2312yes—
63#107—24no—
64#106—24no—
65#103—24no—
66#103—24no—
67C22⋊C8—22no—
68C22⋊C8—22no—
69#95—21no—
70#95—24no—
71C24C2×D411yescharacteristic
72C2×Q8—28no—
73C2×Q8—22no—
74C2×Q8—24no—
75C2×Q8—84no—
76C22×C4—24no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—82no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D412yes—
87C22×C4C2×D412yes—
88C22×C4C2×D414yes—
89C22×C4C2411yescharacteristic
90M16—28no—
91C2×C8—24no—
92C4⋊C4—28no—
93C4⋊C4—216no—
94C4⋊C4—24no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—24no—
98C4⋊C4—22no—
99C4⋊C4—42no—
100C4⋊C4—84no—
101C4⋊C4C2×D414yes—
102C4⋊C4C2×D414yes—
103C22⋊C4—24no—
104C22⋊C4—24no—
105C22⋊C4—24no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C42—24no—
109C42—24no—
110C42—24no—
111C23—22no—
112C23—22no—
113C23—41no—
114C23#12011yescharacteristic
115C23#12011yescharacteristic
116C23C22×D411yescharacteristic
117Q8—44no—
118Q8—88no—
119C2×C4—22no—
120C2×C4—24no—
121C2×C4—24no—
122C2×C4—24no—
123C2×C4—24no—
124C2×C4—24no—
125C2×C4—24no—
126C2×C4—42no—
127C2×C4—44no—
128C2×C4—44no—
129C2×C4—42no—
130C2×C4—44no—
131C2×C4—44no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—82no—
137C2×C4#12012yes—
138C2×C4C22×D412yes—
139C2×C42−1+414yes—
140C8—44no—
141C22—21no—
142C22—21no—
143C22—21no—
144C22—21no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22—44no—
149C22#45612yes—
150C22C2×#12011yescharacteristic
151C22#53612yes—
152C4—22no—
153C4—44no—
154C4—44no—
155C4—44no—
156C4—44no—
157C4—81no—
158C4—81no—
159C4#53614yes—
160C2—22no—
161C2—22no—
162C2—41no—
163C2#261411yescharacteristic
164C2C2×#45611yescharacteristic
165C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2060C2semidirect
5. #2028C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#30952
#25022
#20602
#20282
#18262
#18212