Finite Group Databasebeta

← #9477

Subgroups — #9477

Order 256.

367 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 65 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#2482C211yesmaximal, characteristic
2#2273C211yesmaximal, characteristic
3#1826C212yesmaximal, complemented
4#1462C211yesmaximal, characteristic
5#1284C212yesmaximal, complemented
6C2×#117C2211yescharacteristic
7C22×C22⋊C4—21no—
8C22×C22⋊C4C2211yescharacteristic
9#407C2212yes—
10#399C412yescomplemented
11#397C412yescomplemented
12#388—22no—
13#385—21no—
14#381C2211yescharacteristic
15#379—21no—
16C4×C22⋊C4—21no—
17C4×C22⋊C4—24no—
18C4×C22⋊C4C2212yes—
19#339—24no—
20#323—22no—
21C25D411yescharacteristic
22C23×C4—22no—
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C2×M16—22no—
26#117—22no—
27#117—22no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4C2×C411yescharacteristic, complemented
30C2×C4⋊C4C2×C411yescharacteristic, complemented
31C2×C22⋊C4—28no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4C2×C412yes—
41C2×C22⋊C4D412yes—
42C2×C42—21no—
43C2×C42—22no—
44C2×C42D414yes—
45C22⋊C8—22no—
46C22⋊C8—22no—
47#95—22no—
48#95—22no—
49#95—22no—
50#95—21no—
51#95C2×C412yescomplemented
52C24—21no—
53C24—24no—
54C24—21no—
55C24—42no—
56C24—81no—
57C24C22⋊C412yes—
58C24C2×D411yescharacteristic
59C22×C4—21no—
60C22×C4—24no—
61C22×C4—28no—
62C22×C4—21no—
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—42no—
67C22×C4—81no—
68C22×C4—82no—
69C22×C4—82no—
70C22×C4C22⋊C412yes—
71C22×C4C22×C411yescharacteristic
72C22×C4C2×D412yes—
73C22×C4C2×D411yescharacteristic
74C22×C4#4112yes—
75M16—44no—
76C2×C8—42no—
77C4⋊C4—22no—
78C4⋊C4—22no—
79C4⋊C4—82no—
80C22⋊C4—28no—
81C22⋊C4—28no—
82C22⋊C4—24no—
83C22⋊C4—24no—
84C22⋊C4—42no—
85C22⋊C4—44no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C22⋊C4—82no—
89C42—44no—
90C42—42no—
91C42—82no—
92C23—22no—
93C23—22no—
94C23—24no—
95C23—22no—
96C23—22no—
97C23—22no—
98C23—22no—
99C23—41no—
100C23—42no—
101C23—42no—
102C23—42no—
103C23—42no—
104C23—81no—
105C23—81no—
106C23—88no—
107C23#10414yes—
108C23C2×C22⋊C411yescharacteristic
109C23#12011yescharacteristic
110C23#12311yescharacteristic
111C2×C4—24no—
112C2×C4—28no—
113C2×C4—22no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—44no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—81no—
129C2×C4—88no—
130C2×C4C4×D412yes—
131C2×C4#12112yes—
132C8—82no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—21no—
138C22—24no—
139C22—21no—
140C22—21no—
141C22—44no—
142C22—44no—
143C22—44no—
144C22—44no—
145C22—42no—
146C22—42no—
147C22—82no—
148C22—84no—
149C22#38611yescharacteristic
150C22C2×#10412yes—
151C22#45812yes—
152C4—22no—
153C4—22no—
154C4—48no—
155C4—42no—
156C4—82no—
157C4—81no—
158C4—81no—
159C2—22no—
160C2—22no—
161C2—41no—
162C2—44no—
163C2#189011yescharacteristic
164C2#189611yescharacteristic
165C2#191111yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1826C2semidirect
5. #1284C2semidirect
10. #399C4semidirect
11. #397C4semidirect
29. C2×C4⋊C4C2×C4semidirect
30. C2×C4⋊C4C2×C4semidirect
51. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#24822
#22732
#18262
#14622
#12842