Finite Group Databasebeta

← #9144

Subgroups — #9144

Order 256.

369 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 87 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#2855C211yesmaximal, characteristic
2C2×C4×C4⋊C4C211yesmaximal, characteristic
3#1817C211yesmaximal, characteristic
4C2×#328C212yesmaximal
5#1294C212yesmaximal
6C2×C22⋊Q8C2211yescharacteristic
7C22×C4⋊C4—21no—
8C22×C42—21no—
9C22×C42C2211yescharacteristic
10C2×C22⋊C8C2212yes—
11#391C412yescomplemented
12#389C412yescomplemented
13#387—22no—
14#385—21no—
15#381—21no—
16#380C2211yescharacteristic
17C4×C4⋊C4—24no—
18C2×#95C2212yes—
19#336—22no—
20#328—24no—
21C22×Q8C2×C411yescharacteristic
22C23×C4—22no—
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27C22⋊Q8—22no—
28C22⋊Q8—22no—
29C2×C4⋊C4—28no—
30C2×C4⋊C4—24no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C412yes—
35C2×C22⋊C4—21no—
36C2×C22⋊C4—21no—
37C2×C42—22no—
38C2×C42—22no—
39C2×C42—24no—
40C2×C42—42no—
41C2×C42—42no—
42C2×C42C2×C412yes—
43C2×C42D412yes—
44C22⋊C8—22no—
45C22⋊C8—22no—
46#95—22no—
47#95—22no—
48#95—22no—
49#95—22no—
50#95—21no—
51#95D414yes—
52C24C2×D411yescharacteristic
53C2×Q8—22no—
54C2×Q8—42no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—24no—
62C22×C4—28no—
63C22×C4—24no—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—42no—
68C22×C4—44no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4C22⋊C412yes—
72C22×C4C22⋊C412yes—
73C22×C4SD1612yes—
74C22×C4Q1612yes—
75C22×C4C22×C411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D412yes—
78C22×C4#4112yes—
79C2×C8—44no—
80C2×C8—42no—
81C4⋊C4—216no—
82C4⋊C4—22no—
83C4⋊C4—42no—
84C4⋊C4—44no—
85C4⋊C4—42no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C42—24no—
91C42—24no—
92C42—48no—
93C23—22no—
94C23—22no—
95C23C2×C22⋊C411yescharacteristic
96C23#12011yescharacteristic
97C23#12311yescharacteristic
98C23C2×SD1611yescharacteristic
99C23C2×Q1611yescharacteristic
100C23AGL(1,ℤ/8)11yescharacteristic
101C23#13711yescharacteristic
102Q8—82no—
103C2×C4—24no—
104C2×C4—24no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—28no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—44no—
119C2×C4—48no—
120C2×C4—48no—
121C2×C4—44no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4#10314yes—
130C2×C4#10414yes—
131C2×C4C4×D412yes—
132C2×C4#12112yes—
133C8—82no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—21no—
138C22—22no—
139C22—22no—
140C22—22no—
141C22—22no—
142C22#38611yescharacteristic
143C22#41012yes—
144C22C2×#10311yescharacteristic
145C22#41711yescharacteristic
146C22C2×#10412yes—
147C22#45011yescharacteristic
148C22#45111yescharacteristic
149C22#48311yescharacteristic
150C22#48411yescharacteristic
151C4—24no—
152C4—24no—
153C4—22no—
154C4—22no—
155C4—48no—
156C4—81no—
157C4—81no—
158C4—82no—
159C2—21no—
160C2—21no—
161C2—21no—
162C2—21no—
163C2#150312yes—
164C2#150912yes—
165C2#187211yescharacteristic
166C2#189611yescharacteristic
167C2#189911yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #391C4semidirect
12. #389C4semidirect

Maximal subgroups

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

subgroupindex
#28552
C2×C4×C4⋊C42
#18172
C2×#3282
#12942