Finite Group Databasebeta

← #9451

Subgroups — #9451

Order 256.

259 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 63 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#2973C211yesmaximal, characteristic
2#2547C211yesmaximal, characteristic
3#1558C212yesmaximal, complemented
4#1466C211yesmaximal, characteristic
5#1285C212yesmaximal, complemented
6C2×C4⋊Q8C2211yescharacteristic
7C2×#124C412yes—
8C2×C4×D4C2211yescharacteristic
9#435—24no—
10#432C2212yes—
11#432C2212yes—
12#407—21no—
13C2×C22⋊C8—21no—
14#397—21no—
15#397—21no—
16#391C412yes—
17#380—21no—
18#380—21no—
19C4×C4⋊C4C2211yescharacteristic
20#333—24no—
21#323—21no—
22#323—21no—
23C22×Q8C2×C412yes—
24C22×D4C2×C411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4D411yescharacteristic
27C2×M16—22no—
28C22×C8—22no—
29C4⋊Q8—22no—
30C4⋊Q8—22no—
31#124—44no—
32C4×D4—22no—
33C4×D4—22no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4D411yescharacteristic
41C2×C22⋊C4D411yescharacteristic
42C2×C42C2×C412yes—
43C2×C42C2311yescharacteristic
44C4⋊C8—22no—
45C4⋊C8D414yes—
46C22⋊C8—22no—
47C22⋊C8—22no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50#97—22no—
51#97—22no—
52#95—22no—
53#95—22no—
54#95—21no—
55C24C22⋊C411yescharacteristic
56C24C22⋊C411yescharacteristic
57C2×Q8—24no—
58C2×Q8—42no—
59C2×Q8—42no—
60C2×D4—22no—
61C2×D4—22no—
62C2×D4—41no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22×C411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76M16—82no—
77C2×C8—24no—
78C2×C8—42no—
79C2×C8—82no—
80C4⋊C4—22no—
81C4⋊C4—42no—
82C4⋊C4—41no—
83C4⋊C4—42no—
84C4⋊C4—41no—
85C22⋊C4—22no—
86C22⋊C4—22no—
87C22⋊C4—44no—
88C22⋊C4—44no—
89C42—42no—
90C42C2×D412yes—
91C42#4112yes—
92C23—21no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23—21no—
97C23—41no—
98C23—41no—
99C23—41no—
100C23—41no—
101C23#10012yes—
102C23C2×C22⋊C411yescharacteristic
103Q8—84no—
104D4—44no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—22no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—42no—
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—42no—
121C2×C4—81no—
122C2×C4—81no—
123C2×C4C4×D412yes—
124C2×C4#12011yescharacteristic
125C2×C4#12112yes—
126C2×C4#12311yescharacteristic
127C8—44no—
128C8—82no—
129C22—21no—
130C22—21no—
131C22—22no—
132C22—42no—
133C22—42no—
134C22—41no—
135C22—41no—
136C22—41no—
137C22—41no—
138C22—41no—
139C22—81no—
140C22—81no—
141C22#38611yescharacteristic
142C22C2×#10011yescharacteristic
143C22#41311yescharacteristic
144C4—21no—
145C4—22no—
146C4—22no—
147C4—41no—
148C4—41no—
149C4—82no—
150C4—82no—
151C4#45512yes—
152C2—21no—
153C2—21no—
154C2—42no—
155C2—81no—
156C2#188511yescharacteristic
157C2#190011yescharacteristic
158C2#190911yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1558C2semidirect
5. #1285C2semidirect

Maximal subgroups

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

subgroupindex
#29732
#25472
#15582
#14662
#12852