Finite Group Databasebeta

← #9145

Subgroups — #9145

Order 256.

347 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 79 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×#376C211yesmaximal, characteristic
3#1817C211yesmaximal, characteristic
4C2×#328C212yesmaximal
5#1294C212yesmaximal
6C2×C22⋊Q8C2211yescharacteristic
7C22×C42—21no—
8C22×C42C2211yescharacteristic
9C2×C22⋊C8C2212yes—
10#391C412yescomplemented
11#389C412yescomplemented
12#387—22no—
13#385—21no—
14#381—21no—
15#380C2211yescharacteristic
16#376—24no—
17C2×#95—21no—
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—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4C2×C412yes—
33C2×C22⋊C4—21no—
34C2×C22⋊C4—21no—
35C2×C42—24no—
36C2×C42—28no—
37C2×C42—22no—
38C2×C42—22no—
39C2×C42C2×C412yes—
40C2×C42D412yes—
41C22⋊C8—22no—
42C22⋊C8—22no—
43#95—24no—
44#95—22no—
45#95—22no—
46#95—22no—
47#95—22no—
48#95—21no—
49#95D414yes—
50C24C2×D411yescharacteristic
51C2×Q8—22no—
52C2×Q8—42no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—24no—
56C22×C4—28no—
57C22×C4—21no—
58C22×C4—24no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—42no—
66C22×C4—44no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4C22⋊C412yes—
70C22×C4C22⋊C412yes—
71C22×C4C22×C411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D412yes—
74C22×C4#4112yes—
75C2×C8—44no—
76C2×C8—42no—
77C4⋊C4—22no—
78C4⋊C4—42no—
79C4⋊C4—44no—
80C4⋊C4—42no—
81C22⋊C4—41no—
82C22⋊C4—41no—
83C22⋊C4—42no—
84C42—216no—
85C42—24no—
86C42—24no—
87C23—22no—
88C23—22no—
89C23C2×C22⋊C411yescharacteristic
90C23#12011yescharacteristic
91C23#12311yescharacteristic
92C23#13512yes—
93C23#13712yes—
94Q8—82no—
95C2×C4—24no—
96C2×C4—24no—
97C2×C4—22no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—28no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—44no—
108C2×C4—48no—
109C2×C4—48no—
110C2×C4—42no—
111C2×C4—44no—
112C2×C4—42no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4#10414yes—
121C2×C4C4×D412yes—
122C2×C4#12112yes—
123C8—82no—
124C22—21no—
125C22—21no—
126C22—21no—
127C22—21no—
128C22—24no—
129C22—24no—
130C22#38611yescharacteristic
131C22#41012yes—
132C22#41611yescharacteristic
133C22#41911yescharacteristic
134C22C2×#10412yes—
135C22#45212yes—
136C22#48512yes—
137C4—24no—
138C4—22no—
139C4—22no—
140C4—48no—
141C4—42no—
142C4—81no—
143C4—81no—
144C4—82no—
145C2—22no—
146C2—22no—
147C2#150012yes—
148C2#150612yes—
149C2#187311yescharacteristic
150C2#189611yescharacteristic
151C2#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
10. #391C4semidirect
11. #389C4semidirect

Maximal subgroups

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

subgroupindex
#28552
C2×#3762
#18172
C2×#3282
#12942