Finite Group Databasebeta

← #9466

Subgroups — #9466

Order 256.

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

normal subgroups only

This group has 363 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#2436C211yesmaximal, characteristic
2#2310C211yesmaximal, characteristic, complemented
3#1784C211yesmaximal, characteristic, complemented
4#1510C212yesmaximal
5#1284C212yesmaximal, complemented
6C23×Q8C2211yescharacteristic
7C2×#124C412yescomplemented
8C2×#121C2211yescharacteristic
9C2×#117—21no—
10C2×#101—22no—
11#407C2212yes—
12#393—22no—
13#391—24no—
14#391C412yescomplemented
15#386—21no—
16#386—21no—
17#381—21no—
18#381C2212yes—
19C4×C22⋊C4C2211yescharacteristic
20#327—24no—
21C22×Q8—22no—
22C22×Q8—22no—
23C22×Q8—44no—
24C22×Q8C2×C412yes—
25C22×Q8D412yes—
26C22×D4—21no—
27C22×D4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4C2311yescharacteristic
31C2×M16—22no—
32#124—44no—
33#121—22no—
34#121—22no—
35#117—44no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C22⋊C4—22no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C42—22no—
44C2×C42C2×C412yes—
45#101—44no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48#95—24no—
49#95—22no—
50#95—22no—
51#95—21no—
52#95—22no—
53#95D414yes—
54C24—21no—
55C24C2×D411yescharacteristic
56C2×Q8—24no—
57C2×Q8—24no—
58C2×Q8—24no—
59C2×Q8—24no—
60C2×Q8—42no—
61C2×Q8—42no—
62C2×Q8—44no—
63C2×Q8—44no—
64C2×Q8—48no—
65C2×D4—22no—
66C2×D4—42no—
67C2×D4—44no—
68C2×D4—41no—
69C2×D4—41no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—24no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—24no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4C22⋊C412yes—
81C22×C4C22⋊C412yes—
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D412yes—
85C22×C4#4112yes—
86M16—44no—
87C2×C8—42no—
88C4⋊C4—22no—
89C4⋊C4—44no—
90C22⋊C4—44no—
91C22⋊C4—42no—
92C22⋊C4—44no—
93C22⋊C4—44no—
94C22⋊C4—42no—
95C42—42no—
96C42—44no—
97C23—22no—
98C23—22no—
99C23—41no—
100C23—42no—
101C23—42no—
102C23—41no—
103C23—42no—
104C23C2×C22⋊C411yescharacteristic
105C23#12011yescharacteristic
106C23#12311yescharacteristic
107Q8—48no—
108Q8—48no—
109D4—82no—
110D4—82no—
111D4—82no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—24no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—44no—
127C2×C4—42no—
128C2×C4—44no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—44no—
134C2×C4#9914yes—
135C2×C4C4×D412yes—
136C2×C4#12112yes—
137C8—82no—
138C22—21no—
139C22—21no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—21no—
144C22—21no—
145C22—44no—
146C22—84no—
147C22—81no—
148C22—81no—
149C22—81no—
150C22#38611yescharacteristic
151C22C2×#9912yes—
152C22#45412yes—
153C4—22no—
154C4—24no—
155C4—22no—
156C4—42no—
157C4—84no—
158C4—82no—
159C4—81no—
160C2—22no—
161C2—22no—
162C2—41no—
163C2—161no—
164C2#188811yescharacteristic
165C2#189811yescharacteristic
166C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
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
2. #2310C2semidirect
3. #1784C2semidirect
5. #1284C2semidirect
7. C2×#124C4semidirect
14. #391C4semidirect

Maximal subgroups

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

subgroupindex
#24362
#23102
#17842
#15102
#12842