Finite Group Databasebeta

← #9024

Subgroups — #9024

Order 256.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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×#399C211yesmaximal, characteristic
2#2312C211yesmaximal, characteristic
3#1824C212yesmaximal, complemented
4C2×#323C211yesmaximal, characteristic
5#1294C212yesmaximal, complemented
6C22×C4⋊C4—21no—
7C22×C4⋊C4C2211yescharacteristic
8C22×C22⋊C4—21no—
9C22×C22⋊C4C2211yescharacteristic
10C2×C22⋊C8C2212yes—
11#399—22no—
12#399C414yescomplemented
13#388—24no—
14#380C2212yes—
15#379—21no—
16C4×C22⋊C4—21no—
17C2×#95C2211yescharacteristic
18#340—24no—
19#323—22no—
20C25D411yescharacteristic
21C23×C4—21no—
22C23×C4—21no—
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—24no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—44no—
31C2×C4⋊C4C2×C412yes—
32C2×C4⋊C4D414yes—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—24no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—44no—
40C2×C22⋊C4C2×C412yes—
41C2×C22⋊C4D412yes—
42C2×C42—22no—
43C22⋊C8—22no—
44C22⋊C8—22no—
45#95—22no—
46#95—21no—
47#95—22no—
48#95C2×C412yes—
49C24—21no—
50C24—22no—
51C24—21no—
52C24—22no—
53C24—42no—
54C24—42no—
55C24C22⋊C412yes—
56C24C2×D411yescharacteristic
57C22×C4—24no—
58C22×C4—22no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—22no—
62C22×C4—24no—
63C22×C4—42no—
64C22×C4—42no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4C22⋊C412yes—
72C22×C4C22×C411yescharacteristic
73C22×C4C2×D412yes—
74C22×C4C2×D411yescharacteristic
75C22×C4#4112yes—
76C2×C8—44no—
77C2×C8—42no—
78C4⋊C4—24no—
79C4⋊C4—44no—
80C4⋊C4—48no—
81C4⋊C4—42no—
82C22⋊C4—24no—
83C22⋊C4—24no—
84C22⋊C4—48no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C22⋊C4—82no—
88C22⋊C4—82no—
89C42—82no—
90C23—22no—
91C23—22no—
92C23—22no—
93C23—22no—
94C23—22no—
95C23—42no—
96C23—42no—
97C23—48no—
98C23—42no—
99C23—42no—
100C23—42no—
101C23—42no—
102C23—81no—
103C23—81no—
104C23—84no—
105C23C2×C22⋊C411yescharacteristic
106C23#12011yescharacteristic
107C23#12311yescharacteristic
108C23#13512yes—
109C23#13712yes—
110C2×C4—24no—
111C2×C4—24no—
112C2×C4—22no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—48no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—44no—
125C2×C4—44no—
126C2×C4—44no—
127C2×C4—81no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4C4×D412yes—
131C2×C4#12112yes—
132C8—82no—
133C22—24no—
134C22—24no—
135C22—21no—
136C22—21no—
137C22—21no—
138C22—21no—
139C22—44no—
140C22—42no—
141C22—42no—
142C22—42no—
143C22—82no—
144C22—84no—
145C22—84no—
146C22#38611yescharacteristic
147C22#41611yescharacteristic
148C22#41911yescharacteristic
149C22#45212yes—
150C22#45312yes—
151C22#45812yes—
152C22#48512yes—
153C4—22no—
154C4—22no—
155C4—44no—
156C4—42no—
157C4—84no—
158C4—82no—
159C2—21no—
160C2—21no—
161C2—22no—
162C2—82no—
163C2#161814yes—
164C2#187311yescharacteristic
165C2#188711yescharacteristic
166C2#189011yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1824C2semidirect
5. #1294C2semidirect
12. #399C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
#23122
#18242
C2×#3232
#12942