Finite Group Databasebeta

← #9491

Subgroups — #9491

Order 256.

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

normal subgroups only

This group has 272 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#2873C211yesmaximal, characteristic
2#2599C211yesmaximal, characteristic
3#1491C212yesmaximal, complemented
4#1464C211yesmaximal, characteristic
5#1288C212yesmaximal, complemented
6C22×M16—21no—
7C2×C4×Q8C2211yescharacteristic
8C2×#117C2211yescharacteristic
9#432C2212yes—
10#405—24no—
11C4×M16—21no—
12C4×M16C2212yes—
13#398C411yescharacteristic
14#397C411yescharacteristic
15#395C411yescharacteristic
16#393C411yescharacteristic
17#388—21no—
18#387—21no—
19#387—21no—
20#385—21no—
21#383C2211yescharacteristic
22Q8⋊C8—24no—
23#324—21no—
24#323—21no—
25C22×Q8C2×C411yescharacteristic
26C23×C4D411yescharacteristic
27C2×M16—24no—
28C2×M16—24no—
29C2×M16—22no—
30C2×M16—22no—
31C22×C8—22no—
32C4×Q8—22no—
33C4×Q8—22no—
34#117—22no—
35#117—22no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4C2×C411yescharacteristic
45C2×C22⋊C4D411yescharacteristic
46C2×C42—21no—
47C2×C42D411yescharacteristic
48C2×C42C2311yescharacteristic
49C4⋊C8—22no—
50C22⋊C8—22no—
51#97—22no—
52#97—22no—
53#97—22no—
54C4×C8—22no—
55C4×C8D414yes—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95C2×C411yescharacteristic
63#95C2×C411yescharacteristic
64C24C22⋊C411yescharacteristic
65C2×Q8—22no—
66C2×Q8—42no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22×C411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81M16—28no—
82M16—24no—
83M16—24no—
84C2×C8—28no—
85C2×C8—24no—
86C2×C8—42no—
87C4⋊C4—22no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—42no—
91C4⋊C4—42no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C42—21no—
99C42—21no—
100C42—42no—
101C42C2×D412yes—
102C42#4112yes—
103C23—21no—
104C23—21no—
105C23—41no—
106C23—41no—
107C23#10412yes—
108C23C2×C22⋊C411yescharacteristic
109Q8—24no—
110Q8—42no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—24no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4#10412yes—
137C2×C4C4×D412yes—
138C2×C4#12011yescharacteristic
139C2×C4#12112yes—
140C2×C4#12311yescharacteristic
141C8—28no—
142C8—82no—
143C22—21no—
144C22—21no—
145C22—22no—
146C22—42no—
147C22—41no—
148C22—41no—
149C22—81no—
150C22#38611yescharacteristic
151C22C2×#10411yescharacteristic
152C22C2×#10411yescharacteristic
153C4—22no—
154C4—21no—
155C4—22no—
156C4—21no—
157C4—22no—
158C4—41no—
159C4—42no—
160C4—41no—
161C4—81no—
162C4—81no—
163C4—81no—
164C2—21no—
165C2—21no—
166C2—42no—
167C2#189111yescharacteristic
168C2#189611yescharacteristic
169C2#190911yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊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. #1491C2semidirect
5. #1288C2semidirect

Maximal subgroups

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

subgroupindex
#28732
#25992
#14912
#14642
#12882